”The tragedy of the commons” in the dynamic context

Agnieszka Wiszniewska- ”The tragedy of the commons” in the Matyszkiel

dynamic context Games Nash eq. Stackelberg eq. Pareto opt. Example-static Agnieszka Wiszniewska-Matyszkiel The tragedy Dynamic games

A simple model Objectives [email protected] Bellman Revised sufficient condition

Social optimum

Nash equilibrium Nash equilibria for n players Finite horizon truncation

Enforcing optimality

Kolokwium Wydziałowe MIM UW Carrying capacity 11 March 2021 Numerics

Continuous time

Conclusions Extraction of a common fishery with possibility of extinction, division into Excusive Economic Zones and inherent constraints with possibility to model many fishermen by the simplest possible model.

”The tragedy of the The motivating example commons” in the dynamic context

Agnieszka Wiszniewska- Matyszkiel

Games Nash eq. Stackelberg eq. Pareto opt. Example-static The tragedy

Dynamic games

A simple model Objectives

Bellman Revised sufficient condition

Social optimum

Nash equilibrium Nash equilibria for n players Finite horizon truncation

Enforcing optimality

Carrying capacity

Numerics

Continuous time

Conclusions with possibility of extinction, division into Excusive Economic Zones and inherent constraints with possibility to model many fishermen by the simplest possible model.

”The tragedy of the The motivating example commons” in the dynamic context

Agnieszka Wiszniewska- Matyszkiel

Games Nash eq. Stackelberg eq. Pareto opt. Example-static The tragedy

Dynamic games

A simple model Objectives

Bellman Revised sufficient condition Extraction of a common fishery Social optimum Nash equilibrium Nash equilibria for n players Finite horizon truncation

Enforcing optimality

Carrying capacity

Numerics

Continuous time

Conclusions and inherent constraints with possibility to model many fishermen by the simplest possible model.

”The tragedy of the The motivating example commons” in the dynamic context

Agnieszka Wiszniewska- Matyszkiel

Games Nash eq. Stackelberg eq. Pareto opt. Example-static The tragedy

Dynamic games

A simple model Objectives

Bellman Revised sufficient condition Extraction of a common fishery with possibility of extinction, Social optimum Nash equilibrium division into Excusive Economic Zones Nash equilibria for n players Finite horizon truncation

Enforcing optimality

Carrying capacity

Numerics

Continuous time

Conclusions with possibility to model many fishermen by the simplest possible model.

”The tragedy of the The motivating example commons” in the dynamic context

Agnieszka Wiszniewska- Matyszkiel

Games Nash eq. Stackelberg eq. Pareto opt. Example-static The tragedy

Dynamic games

A simple model Objectives

Bellman Revised sufficient condition Extraction of a common fishery with possibility of extinction, Social optimum Nash equilibrium division into Excusive Economic Zones and inherent Nash equilibria for n players constraints Finite horizon truncation Enforcing optimality

Carrying capacity

Numerics

Continuous time

Conclusions by the simplest possible model.

”The tragedy of the The motivating example commons” in the dynamic context

Agnieszka Wiszniewska- Matyszkiel

Games Nash eq. Stackelberg eq. Pareto opt. Example-static The tragedy

Dynamic games

A simple model Objectives

Bellman Revised sufficient condition Extraction of a common fishery with possibility of extinction, Social optimum Nash equilibrium division into Excusive Economic Zones and inherent Nash equilibria for n players constraints with possibility to model many fishermen Finite horizon truncation Enforcing optimality

Carrying capacity

Numerics

Continuous time

Conclusions ”The tragedy of the The motivating example commons” in the dynamic context

Agnieszka Wiszniewska- Matyszkiel

Games Nash eq. Stackelberg eq. Pareto opt. Example-static The tragedy

Dynamic games

A simple model Objectives

Bellman Revised sufficient condition Extraction of a common fishery with possibility of extinction, Social optimum Nash equilibrium division into Excusive Economic Zones and inherent Nash equilibria for n players constraints with possibility to model many fishermen by the Finite horizon truncation simplest possible model. Enforcing optimality Carrying capacity

Numerics

Continuous time

Conclusions (called players), each of them having his/her own aim (represented by maximization of his/her payoff), with the realization of that aim influenced by the other’s choices (the payoff is a function of the whole strategy profile– choice of strategies of all players). § At least 2 agents + sets to choose from + aim + interaction.

”The tragedy of the What is the game? commons” in the dynamic context

Agnieszka Wiszniewska- Matyszkiel

Games Nash eq. § Stackelberg eq. Any situation of decision making by at least two agents Pareto opt. Example-static The tragedy

Dynamic games

A simple model Objectives

Bellman Revised sufficient condition

Social optimum

Nash equilibrium Nash equilibria for n players Finite horizon truncation

Enforcing optimality

Carrying capacity

Numerics

Continuous time

Conclusions each of them having his/her own aim (represented by maximization of his/her payoff), with the realization of that aim influenced by the other’s choices (the payoff is a function of the whole strategy profile– choice of strategies of all players). § At least 2 agents + sets to choose from + aim + interaction.

”The tragedy of the What is the game? commons” in the dynamic context

Agnieszka Wiszniewska- Matyszkiel

Games Nash eq. § Stackelberg eq. Any situation of decision making by at least two agents Pareto opt. (called players), Example-static The tragedy

Dynamic games

A simple model Objectives

Bellman Revised sufficient condition

Social optimum

Nash equilibrium Nash equilibria for n players Finite horizon truncation

Enforcing optimality

Carrying capacity

Numerics

Continuous time

Conclusions (represented by maximization of his/her payoff), with the realization of that aim influenced by the other’s choices (the payoff is a function of the whole strategy profile– choice of strategies of all players). § At least 2 agents + sets to choose from + aim + interaction.

”The tragedy of the What is the game? commons” in the dynamic context

Agnieszka Wiszniewska- Matyszkiel

Games Nash eq. § Stackelberg eq. Any situation of decision making by at least two agents Pareto opt. (called players), each of them having his/her own aim Example-static The tragedy

Dynamic games

A simple model Objectives

Bellman Revised sufficient condition

Social optimum

Nash equilibrium Nash equilibria for n players Finite horizon truncation

Enforcing optimality

Carrying capacity

Numerics

Continuous time

Conclusions with the realization of that aim influenced by the other’s choices (the payoff is a function of the whole strategy profile– choice of strategies of all players). § At least 2 agents + sets to choose from + aim + interaction.

”The tragedy of the What is the game? commons” in the dynamic context

Agnieszka Wiszniewska- Matyszkiel

Games Nash eq. § Stackelberg eq. Any situation of decision making by at least two agents Pareto opt. (called players), each of them having his/her own aim Example-static The tragedy

(represented by maximization of his/her payoff), Dynamic games

A simple model Objectives

Bellman Revised sufficient condition

Social optimum

Nash equilibrium Nash equilibria for n players Finite horizon truncation

Enforcing optimality

Carrying capacity

Numerics

Continuous time

Conclusions choice of strategies of all players). § At least 2 agents + sets to choose from + aim + interaction.

”The tragedy of the What is the game? commons” in the dynamic context

Agnieszka Wiszniewska- Matyszkiel

Games Nash eq. § Stackelberg eq. Any situation of decision making by at least two agents Pareto opt. (called players), each of them having his/her own aim Example-static The tragedy

(represented by maximization of his/her payoff), with the Dynamic games

realization of that aim influenced by the other’s choices A simple model (the payoff is a function of the whole strategy profile– Objectives Bellman Revised sufficient condition

Social optimum

Nash equilibrium Nash equilibria for n players Finite horizon truncation

Enforcing optimality

Carrying capacity

Numerics

Continuous time

Conclusions § At least 2 agents + sets to choose from + aim + interaction.

”The tragedy of the What is the game? commons” in the dynamic context

Agnieszka Wiszniewska- Matyszkiel

Games Nash eq. § Stackelberg eq. Any situation of decision making by at least two agents Pareto opt. (called players), each of them having his/her own aim Example-static The tragedy

(represented by maximization of his/her payoff), with the Dynamic games

realization of that aim influenced by the other’s choices A simple model (the payoff is a function of the whole strategy profile– Objectives Bellman choice of strategies of all players). Revised sufficient condition

Social optimum

Nash equilibrium Nash equilibria for n players Finite horizon truncation

Enforcing optimality

Carrying capacity

Numerics

Continuous time

Conclusions ”The tragedy of the What is the game? commons” in the dynamic context

Agnieszka Wiszniewska- Matyszkiel

Games Nash eq. § Stackelberg eq. Any situation of decision making by at least two agents Pareto opt. (called players), each of them having his/her own aim Example-static The tragedy

(represented by maximization of his/her payoff), with the Dynamic games

realization of that aim influenced by the other’s choices A simple model (the payoff is a function of the whole strategy profile– Objectives Bellman choice of strategies of all players). Revised sufficient condition § At least 2 agents + sets to choose from + aim + Social optimum Nash equilibrium interaction. Nash equilibria for n players Finite horizon truncation

Enforcing optimality

Carrying capacity

Numerics

Continuous time

Conclusions § The payoff functions are defined on the set of strategy

profiles, i.e. S “ iPI Si. ¯ § With auxiliary notationŚ rSi, S„is to denote the profile of strategies S¯ with strategy of player i replaced by Si we define Nash equilibrium. § A strategy profile S¯ is a Nash equilibrium iff for all i P I (a.e. for the continuum of players case) ¯ ¯ JipSiq ě Jiprsi, S„isq for every si P Si § i.e. every player maximizes payoff given the strategies of the other players § or best responds to the strategies of the others.

”The tragedy of the A short introduction to game theory commons” in the dynamic context

Agnieszka Wiszniewska- § A game in strategic form is defined by a triple: the set of Matyszkiel players I (usually finite or a continuum), players’ sets of Games available strategies Si and player’s payoff functions Ji. Nash eq. Stackelberg eq. Pareto opt. Example-static The tragedy

Dynamic games

A simple model Objectives

Bellman Revised sufficient condition

Social optimum

Nash equilibrium Nash equilibria for n players Finite horizon truncation

Enforcing optimality

Carrying capacity

Numerics

Continuous time

Conclusions ¯ § With auxiliary notation rSi, S„is to denote the profile of strategies S¯ with strategy of player i replaced by Si we define Nash equilibrium. § A strategy profile S¯ is a Nash equilibrium iff for all i P I (a.e. for the continuum of players case) ¯ ¯ JipSiq ě Jiprsi, S„isq for every si P Si § i.e. every player maximizes payoff given the strategies of the other players § or best responds to the strategies of the others.

”The tragedy of the A short introduction to game theory commons” in the dynamic context

Agnieszka Wiszniewska- § A game in strategic form is defined by a triple: the set of Matyszkiel players I (usually finite or a continuum), players’ sets of Games available strategies Si and player’s payoff functions Ji. Nash eq. Stackelberg eq. § The payoff functions are defined on the set of strategy Pareto opt. Example-static The tragedy profiles, i.e. S “ iPI Si. Dynamic games

Ś A simple model Objectives

Bellman Revised sufficient condition

Social optimum

Nash equilibrium Nash equilibria for n players Finite horizon truncation

Enforcing optimality

Carrying capacity

Numerics

Continuous time

Conclusions § A strategy profile S¯ is a Nash equilibrium iff for all i P I (a.e. for the continuum of players case) ¯ ¯ JipSiq ě Jiprsi, S„isq for every si P Si § i.e. every player maximizes payoff given the strategies of the other players § or best responds to the strategies of the others.

”The tragedy of the A short introduction to game theory commons” in the dynamic context

Agnieszka Wiszniewska- § A game in strategic form is defined by a triple: the set of Matyszkiel players I (usually finite or a continuum), players’ sets of Games available strategies Si and player’s payoff functions Ji. Nash eq. Stackelberg eq. § The payoff functions are defined on the set of strategy Pareto opt. Example-static The tragedy profiles, i.e. S “ iPI Si. ¯ Dynamic games § With auxiliary notation rSi, S„is to denote the profile of Ś A simple model ¯ strategies S with strategy of player i replaced by Si we Objectives define Nash equilibrium. Bellman Revised sufficient condition

Social optimum

Nash equilibrium Nash equilibria for n players Finite horizon truncation

Enforcing optimality

Carrying capacity

Numerics

Continuous time

Conclusions (a.e. for the continuum of players case) ¯ ¯ JipSiq ě Jiprsi, S„isq for every si P Si § i.e. every player maximizes payoff given the strategies of the other players § or best responds to the strategies of the others.

”The tragedy of the A short introduction to game theory commons” in the dynamic context

Agnieszka Wiszniewska- § A game in strategic form is defined by a triple: the set of Matyszkiel players I (usually finite or a continuum), players’ sets of Games available strategies Si and player’s payoff functions Ji. Nash eq. Stackelberg eq. § The payoff functions are defined on the set of strategy Pareto opt. Example-static The tragedy profiles, i.e. S “ iPI Si. ¯ Dynamic games § With auxiliary notation rSi, S„is to denote the profile of Ś A simple model ¯ strategies S with strategy of player i replaced by Si we Objectives define Nash equilibrium. Bellman Revised sufficient condition ¯ § A strategy profile S is a Nash equilibrium iff for all i P I Social optimum

Nash equilibrium Nash equilibria for n players Finite horizon truncation

Enforcing optimality

Carrying capacity

Numerics

Continuous time

Conclusions § i.e. every player maximizes payoff given the strategies of the other players § or best responds to the strategies of the others.

”The tragedy of the A short introduction to game theory commons” in the dynamic context

Agnieszka Wiszniewska- § A game in strategic form is defined by a triple: the set of Matyszkiel players I (usually finite or a continuum), players’ sets of Games available strategies Si and player’s payoff functions Ji. Nash eq. Stackelberg eq. § The payoff functions are defined on the set of strategy Pareto opt. Example-static The tragedy profiles, i.e. S “ iPI Si. ¯ Dynamic games § With auxiliary notation rSi, S„is to denote the profile of Ś A simple model ¯ strategies S with strategy of player i replaced by Si we Objectives define Nash equilibrium. Bellman Revised sufficient condition ¯ § A strategy profile S is a Nash equilibrium iff for all i P I Social optimum (a.e. for the continuum of players case) Nash equilibrium ¯ ¯ Nash equilibria for n players JipSiq ě Jiprsi, S„isq for every si P Si Finite horizon truncation Enforcing optimality

Carrying capacity

Numerics

Continuous time

Conclusions § or best responds to the strategies of the others.

”The tragedy of the A short introduction to game theory commons” in the dynamic context

Agnieszka Wiszniewska- § A game in strategic form is defined by a triple: the set of Matyszkiel players I (usually finite or a continuum), players’ sets of Games available strategies Si and player’s payoff functions Ji. Nash eq. Stackelberg eq. § The payoff functions are defined on the set of strategy Pareto opt. Example-static The tragedy profiles, i.e. S “ iPI Si. ¯ Dynamic games § With auxiliary notation rSi, S„is to denote the profile of Ś A simple model ¯ strategies S with strategy of player i replaced by Si we Objectives define Nash equilibrium. Bellman Revised sufficient condition ¯ § A strategy profile S is a Nash equilibrium iff for all i P I Social optimum (a.e. for the continuum of players case) Nash equilibrium ¯ ¯ Nash equilibria for n players JipSiq ě Jiprsi, S„isq for every si P Si Finite horizon truncation § i.e. every player maximizes payoff given the strategies Enforcing optimality of the other players Carrying capacity Numerics

Continuous time

Conclusions ”The tragedy of the A short introduction to game theory commons” in the dynamic context

Agnieszka Wiszniewska- § A game in strategic form is defined by a triple: the set of Matyszkiel players I (usually finite or a continuum), players’ sets of Games available strategies Si and player’s payoff functions Ji. Nash eq. Stackelberg eq. § The payoff functions are defined on the set of strategy Pareto opt. Example-static The tragedy profiles, i.e. S “ iPI Si. ¯ Dynamic games § With auxiliary notation rSi, S„is to denote the profile of Ś A simple model ¯ strategies S with strategy of player i replaced by Si we Objectives define Nash equilibrium. Bellman Revised sufficient condition ¯ § A strategy profile S is a Nash equilibrium iff for all i P I Social optimum (a.e. for the continuum of players case) Nash equilibrium ¯ ¯ Nash equilibria for n players JipSiq ě Jiprsi, S„isq for every si P Si Finite horizon truncation § i.e. every player maximizes payoff given the strategies Enforcing optimality of the other players Carrying capacity Numerics § or best responds to the strategies of the others. Continuous time

Conclusions (In fact, Bi depends

nontrivially only on S„i, so we are going to abuse notation sometimes and write BipS„iq if needed.) § A profile S¯ is a Nash equilibrium iff S¯ P BpS¯q. § So, calculation of a Nash equilibrium requires solving a set of optimization problems in players’ strategy spaces coupled by finding a fixed point of the resulting best response correspondence in the space of strategy profiles.

”The tragedy of the A short introduction to game theory cont. commons” in the dynamic context

Agnieszka Wiszniewska- Matyszkiel § We can write it as a fixed point of the following Games multivalued correspondence B : S ( S, called the best Nash eq. Stackelberg eq. response correspondence defined by Pareto opt. Example-static BipSq “ Argmax Jiprsi, S„isq. The tragedy si PSi Dynamic games

A simple model Objectives

Bellman Revised sufficient condition

Social optimum

Nash equilibrium Nash equilibria for n players Finite horizon truncation

Enforcing optimality

Carrying capacity

Numerics

Continuous time

Conclusions § So, calculation of a Nash equilibrium requires solving a set of optimization problems in players’ strategy spaces coupled by finding a fixed point of the resulting best response correspondence in the space of strategy profiles.

”The tragedy of the A short introduction to game theory cont. commons” in the dynamic context

Agnieszka Wiszniewska- Matyszkiel § We can write it as a fixed point of the following Games multivalued correspondence B : S ( S, called the best Nash eq. Stackelberg eq. response correspondence defined by Pareto opt. Example-static BipSq “ Argmax Jiprsi, S„isq. (In fact, Bi depends The tragedy si PSi Dynamic games nontrivially only on S„i, so we are going to abuse A simple model notation sometimes and write BipS„iq if needed.) Objectives § A profile S¯ is a Nash equilibrium iff S¯ P BpS¯q. Bellman Revised sufficient condition

Social optimum

Nash equilibrium Nash equilibria for n players Finite horizon truncation

Enforcing optimality

Carrying capacity

Numerics

Continuous time

Conclusions coupled by finding a fixed point of the resulting best response correspondence in the space of strategy profiles.

”The tragedy of the A short introduction to game theory cont. commons” in the dynamic context

Agnieszka Wiszniewska- Matyszkiel § We can write it as a fixed point of the following Games multivalued correspondence B : S ( S, called the best Nash eq. Stackelberg eq. response correspondence defined by Pareto opt. Example-static BipSq “ Argmax Jiprsi, S„isq. (In fact, Bi depends The tragedy si PSi Dynamic games nontrivially only on S„i, so we are going to abuse A simple model notation sometimes and write BipS„iq if needed.) Objectives § A profile S¯ is a Nash equilibrium iff S¯ P BpS¯q. Bellman Revised sufficient condition § So, calculation of a Nash equilibrium requires solving a Social optimum set of optimization problems in players’ strategy spaces Nash equilibrium Nash equilibria for n players Finite horizon truncation

Enforcing optimality

Carrying capacity

Numerics

Continuous time

Conclusions ”The tragedy of the A short introduction to game theory cont. commons” in the dynamic context

Agnieszka Wiszniewska- Matyszkiel § We can write it as a fixed point of the following Games multivalued correspondence B : S ( S, called the best Nash eq. Stackelberg eq. response correspondence defined by Pareto opt. Example-static BipSq “ Argmax Jiprsi, S„isq. (In fact, Bi depends The tragedy si PSi Dynamic games nontrivially only on S„i, so we are going to abuse A simple model notation sometimes and write BipS„iq if needed.) Objectives § A profile S¯ is a Nash equilibrium iff S¯ P BpS¯q. Bellman Revised sufficient condition § So, calculation of a Nash equilibrium requires solving a Social optimum set of optimization problems in players’ strategy spaces Nash equilibrium Nash equilibria for n players coupled by finding a fixed point of the resulting best Finite horizon truncation response correspondence in the space of strategy Enforcing optimality profiles. Carrying capacity Numerics

Continuous time

Conclusions § For two players: the first mover/better informed/higher in hierarchy player 1 – the leader, the other, player 2, behaving as at a Nash equilibrium – the follower. § A profile S¯ is a Stackelberg equilibrium iff there exists a selection b2 P B2 such that S¯2 P b2pS¯1q and S¯1 P Argmax J1pS1, b2pS1qq. S1PS1 § A nested optimization! § For more than two players there may be different level of hierarchy or some players at the same level: e.g. a leader and many followers playing Nash between them given the leader’s strategy (and the leader knows it and takes into its calculation). So, an optimization nested with a set of optimizations coupled by a fixed point...

”The tragedy of the A short introduction to game theory cont. 2 commons” in the dynamic context § What if players either do choose their strategies Agnieszka Wiszniewska- sequentially, there is a hierarchy or one of them has Matyszkiel informational advantage (i.e. s/he can calculate the best Games response function of the other player or players). Then, Nash eq. Stackelberg eq. instead of Nash, we consider a Stackelberg equilibrium. Pareto opt. Example-static The tragedy

Dynamic games

A simple model Objectives

Bellman Revised sufficient condition

Social optimum

Nash equilibrium Nash equilibria for n players Finite horizon truncation

Enforcing optimality

Carrying capacity

Numerics

Continuous time

Conclusions § A profile S¯ is a Stackelberg equilibrium iff there exists a selection b2 P B2 such that S¯2 P b2pS¯1q and S¯1 P Argmax J1pS1, b2pS1qq. S1PS1 § A nested optimization! § For more than two players there may be different level of hierarchy or some players at the same level: e.g. a leader and many followers playing Nash between them given the leader’s strategy (and the leader knows it and takes into its calculation). So, an optimization nested with a set of optimizations coupled by a fixed point...

”The tragedy of the A short introduction to game theory cont. 2 commons” in the dynamic context § What if players either do choose their strategies Agnieszka Wiszniewska- sequentially, there is a hierarchy or one of them has Matyszkiel informational advantage (i.e. s/he can calculate the best Games response function of the other player or players). Then, Nash eq. Stackelberg eq. instead of Nash, we consider a Stackelberg equilibrium. Pareto opt. § Example-static For two players: the first mover/better informed/higher in The tragedy

hierarchy player 1 – the leader, the other, player 2, Dynamic games behaving as at a Nash equilibrium – the follower. A simple model Objectives

Bellman Revised sufficient condition

Social optimum

Nash equilibrium Nash equilibria for n players Finite horizon truncation

Enforcing optimality

Carrying capacity

Numerics

Continuous time

Conclusions and S¯1 P Argmax J1pS1, b2pS1qq. S1PS1 § A nested optimization! § For more than two players there may be different level of hierarchy or some players at the same level: e.g. a leader and many followers playing Nash between them given the leader’s strategy (and the leader knows it and takes into its calculation). So, an optimization nested with a set of optimizations coupled by a fixed point...

”The tragedy of the A short introduction to game theory cont. 2 commons” in the dynamic context § What if players either do choose their strategies Agnieszka Wiszniewska- sequentially, there is a hierarchy or one of them has Matyszkiel informational advantage (i.e. s/he can calculate the best Games response function of the other player or players). Then, Nash eq. Stackelberg eq. instead of Nash, we consider a Stackelberg equilibrium. Pareto opt. § Example-static For two players: the first mover/better informed/higher in The tragedy

hierarchy player 1 – the leader, the other, player 2, Dynamic games behaving as at a Nash equilibrium – the follower. A simple model Objectives § A profile S¯ is a Stackelberg equilibrium iff there exists a ¯ ¯ Bellman selection b2 P B2 such that S2 P b2pS1q Revised sufficient condition Social optimum

Nash equilibrium Nash equilibria for n players Finite horizon truncation

Enforcing optimality

Carrying capacity

Numerics

Continuous time

Conclusions § A nested optimization! § For more than two players there may be different level of hierarchy or some players at the same level: e.g. a leader and many followers playing Nash between them given the leader’s strategy (and the leader knows it and takes into its calculation). So, an optimization nested with a set of optimizations coupled by a fixed point...

”The tragedy of the A short introduction to game theory cont. 2 commons” in the dynamic context § What if players either do choose their strategies Agnieszka Wiszniewska- sequentially, there is a hierarchy or one of them has Matyszkiel informational advantage (i.e. s/he can calculate the best Games response function of the other player or players). Then, Nash eq. Stackelberg eq. instead of Nash, we consider a Stackelberg equilibrium. Pareto opt. § Example-static For two players: the first mover/better informed/higher in The tragedy

hierarchy player 1 – the leader, the other, player 2, Dynamic games behaving as at a Nash equilibrium – the follower. A simple model Objectives § A profile S¯ is a Stackelberg equilibrium iff there exists a ¯ ¯ Bellman selection b2 P B2 such that S2 P b2pS1q and Revised sufficient condition S¯1 P Argmax J1pS1, b2pS1qq. Social optimum S1PS1 Nash equilibrium Nash equilibria for n players Finite horizon truncation

Enforcing optimality

Carrying capacity

Numerics

Continuous time

Conclusions § For more than two players there may be different level of hierarchy or some players at the same level: e.g. a leader and many followers playing Nash between them given the leader’s strategy (and the leader knows it and takes into its calculation). So, an optimization nested with a set of optimizations coupled by a fixed point...

”The tragedy of the A short introduction to game theory cont. 2 commons” in the dynamic context § What if players either do choose their strategies Agnieszka Wiszniewska- sequentially, there is a hierarchy or one of them has Matyszkiel informational advantage (i.e. s/he can calculate the best Games response function of the other player or players). Then, Nash eq. Stackelberg eq. instead of Nash, we consider a Stackelberg equilibrium. Pareto opt. § Example-static For two players: the first mover/better informed/higher in The tragedy

hierarchy player 1 – the leader, the other, player 2, Dynamic games behaving as at a Nash equilibrium – the follower. A simple model Objectives § A profile S¯ is a Stackelberg equilibrium iff there exists a ¯ ¯ Bellman selection b2 P B2 such that S2 P b2pS1q and Revised sufficient condition S¯1 P Argmax J1pS1, b2pS1qq. Social optimum S1PS1 Nash equilibrium § A nested optimization! Nash equilibria for n players Finite horizon truncation

Enforcing optimality

Carrying capacity

Numerics

Continuous time

Conclusions e.g. a leader and many followers playing Nash between them given the leader’s strategy (and the leader knows it and takes into its calculation). So, an optimization nested with a set of optimizations coupled by a fixed point...

”The tragedy of the A short introduction to game theory cont. 2 commons” in the dynamic context § What if players either do choose their strategies Agnieszka Wiszniewska- sequentially, there is a hierarchy or one of them has Matyszkiel informational advantage (i.e. s/he can calculate the best Games response function of the other player or players). Then, Nash eq. Stackelberg eq. instead of Nash, we consider a Stackelberg equilibrium. Pareto opt. § Example-static For two players: the first mover/better informed/higher in The tragedy

hierarchy player 1 – the leader, the other, player 2, Dynamic games behaving as at a Nash equilibrium – the follower. A simple model Objectives § A profile S¯ is a Stackelberg equilibrium iff there exists a ¯ ¯ Bellman selection b2 P B2 such that S2 P b2pS1q and Revised sufficient condition S¯1 P Argmax J1pS1, b2pS1qq. Social optimum S1PS1 Nash equilibrium § A nested optimization! Nash equilibria for n players Finite horizon truncation § For more than two players there may be different level of Enforcing optimality

hierarchy or some players at the same level: Carrying capacity

Numerics

Continuous time

Conclusions So, an optimization nested with a set of optimizations coupled by a fixed point...

”The tragedy of the A short introduction to game theory cont. 2 commons” in the dynamic context § What if players either do choose their strategies Agnieszka Wiszniewska- sequentially, there is a hierarchy or one of them has Matyszkiel informational advantage (i.e. s/he can calculate the best Games response function of the other player or players). Then, Nash eq. Stackelberg eq. instead of Nash, we consider a Stackelberg equilibrium. Pareto opt. § Example-static For two players: the first mover/better informed/higher in The tragedy

hierarchy player 1 – the leader, the other, player 2, Dynamic games behaving as at a Nash equilibrium – the follower. A simple model Objectives § A profile S¯ is a Stackelberg equilibrium iff there exists a ¯ ¯ Bellman selection b2 P B2 such that S2 P b2pS1q and Revised sufficient condition S¯1 P Argmax J1pS1, b2pS1qq. Social optimum S1PS1 Nash equilibrium § A nested optimization! Nash equilibria for n players Finite horizon truncation § For more than two players there may be different level of Enforcing optimality

hierarchy or some players at the same level: e.g. a leader Carrying capacity

and many followers playing Nash between them given the Numerics

leader’s strategy (and the leader knows it and takes into its Continuous time calculation). Conclusions ”The tragedy of the A short introduction to game theory cont. 2 commons” in the dynamic context § What if players either do choose their strategies Agnieszka Wiszniewska- sequentially, there is a hierarchy or one of them has Matyszkiel informational advantage (i.e. s/he can calculate the best Games response function of the other player or players). Then, Nash eq. Stackelberg eq. instead of Nash, we consider a Stackelberg equilibrium. Pareto opt. § Example-static For two players: the first mover/better informed/higher in The tragedy

hierarchy player 1 – the leader, the other, player 2, Dynamic games behaving as at a Nash equilibrium – the follower. A simple model Objectives § A profile S¯ is a Stackelberg equilibrium iff there exists a ¯ ¯ Bellman selection b2 P B2 such that S2 P b2pS1q and Revised sufficient condition S¯1 P Argmax J1pS1, b2pS1qq. Social optimum S1PS1 Nash equilibrium § A nested optimization! Nash equilibria for n players Finite horizon truncation § For more than two players there may be different level of Enforcing optimality

hierarchy or some players at the same level: e.g. a leader Carrying capacity

and many followers playing Nash between them given the Numerics

leader’s strategy (and the leader knows it and takes into its Continuous time calculation). So, an optimization nested with a set of Conclusions optimizations coupled by a fixed point... (a.e. for the continuum of players) § and JipSq ą JipS¯q for some i (in a set of positive measure for the continuum of players). § If the payoffs are monetary (and side payments are possible) then the most obvious Pareto optimal profile is Ji pSq the profile which maximizes iPI #I (or its continuous equivalent for the continuum of players). We call such a profile the social optimum. ř

”The tragedy of the A short introduction to game theory cont. 3 commons” in the dynamic context

Agnieszka Wiszniewska- Matyszkiel

Games § We are also interested in Pareto optimal profiles, i.e. Nash eq. ¯ Stackelberg eq. profiles S such that there exists no profile S with Pareto opt. § ¯ Example-static JipSq ě JipSq for all i The tragedy

Dynamic games

A simple model Objectives

Bellman Revised sufficient condition

Social optimum

Nash equilibrium Nash equilibria for n players Finite horizon truncation

Enforcing optimality

Carrying capacity

Numerics

Continuous time

Conclusions § and JipSq ą JipS¯q for some i (in a set of positive measure for the continuum of players). § If the payoffs are monetary (and side payments are possible) then the most obvious Pareto optimal profile is Ji pSq the profile which maximizes iPI #I (or its continuous equivalent for the continuum of players). We call such a profile the social optimum. ř

”The tragedy of the A short introduction to game theory cont. 3 commons” in the dynamic context

Agnieszka Wiszniewska- Matyszkiel

Games § We are also interested in Pareto optimal profiles, i.e. Nash eq. ¯ Stackelberg eq. profiles S such that there exists no profile S with Pareto opt. § ¯ Example-static JipSq ě JipSq for all i (a.e. for the continuum of players) The tragedy

Dynamic games

A simple model Objectives

Bellman Revised sufficient condition

Social optimum

Nash equilibrium Nash equilibria for n players Finite horizon truncation

Enforcing optimality

Carrying capacity

Numerics

Continuous time

Conclusions (in a set of positive measure for the continuum of players). § If the payoffs are monetary (and side payments are possible) then the most obvious Pareto optimal profile is Ji pSq the profile which maximizes iPI #I (or its continuous equivalent for the continuum of players). We call such a profile the social optimum. ř

”The tragedy of the A short introduction to game theory cont. 3 commons” in the dynamic context

Agnieszka Wiszniewska- Matyszkiel

Games § We are also interested in Pareto optimal profiles, i.e. Nash eq. ¯ Stackelberg eq. profiles S such that there exists no profile S with Pareto opt. § ¯ Example-static JipSq ě JipSq for all i (a.e. for the continuum of players) The tragedy § ¯ and JipSq ą JipSq for some i Dynamic games

A simple model Objectives

Bellman Revised sufficient condition

Social optimum

Nash equilibrium Nash equilibria for n players Finite horizon truncation

Enforcing optimality

Carrying capacity

Numerics

Continuous time

Conclusions § If the payoffs are monetary (and side payments are possible) then the most obvious Pareto optimal profile is Ji pSq the profile which maximizes iPI #I (or its continuous equivalent for the continuum of players). We call such a profile the social optimum. ř

”The tragedy of the A short introduction to game theory cont. 3 commons” in the dynamic context

Agnieszka Wiszniewska- Matyszkiel

Games § We are also interested in Pareto optimal profiles, i.e. Nash eq. ¯ Stackelberg eq. profiles S such that there exists no profile S with Pareto opt. § ¯ Example-static JipSq ě JipSq for all i (a.e. for the continuum of players) The tragedy § ¯ and JipSq ą JipSq for some i (in a set of positive Dynamic games

measure for the continuum of players). A simple model Objectives

Bellman Revised sufficient condition

Social optimum

Nash equilibrium Nash equilibria for n players Finite horizon truncation

Enforcing optimality

Carrying capacity

Numerics

Continuous time

Conclusions (or its continuous equivalent for the continuum of players). We call such a profile the social optimum.

”The tragedy of the A short introduction to game theory cont. 3 commons” in the dynamic context

Agnieszka Wiszniewska- Matyszkiel

Games § We are also interested in Pareto optimal profiles, i.e. Nash eq. ¯ Stackelberg eq. profiles S such that there exists no profile S with Pareto opt. § ¯ Example-static JipSq ě JipSq for all i (a.e. for the continuum of players) The tragedy § ¯ and JipSq ą JipSq for some i (in a set of positive Dynamic games

measure for the continuum of players). A simple model § If the payoffs are monetary (and side payments are Objectives Bellman possible) then the most obvious Pareto optimal profile is Revised sufficient condition Ji pSq Social optimum the profile which maximizes iPI #I Nash equilibrium Nash equilibria for n players ř Finite horizon truncation

Enforcing optimality

Carrying capacity

Numerics

Continuous time

Conclusions We call such a profile the social optimum.

”The tragedy of the A short introduction to game theory cont. 3 commons” in the dynamic context

Agnieszka Wiszniewska- Matyszkiel

Games § We are also interested in Pareto optimal profiles, i.e. Nash eq. ¯ Stackelberg eq. profiles S such that there exists no profile S with Pareto opt. § ¯ Example-static JipSq ě JipSq for all i (a.e. for the continuum of players) The tragedy § ¯ and JipSq ą JipSq for some i (in a set of positive Dynamic games

measure for the continuum of players). A simple model § If the payoffs are monetary (and side payments are Objectives Bellman possible) then the most obvious Pareto optimal profile is Revised sufficient condition Ji pSq Social optimum the profile which maximizes iPI #I (or its continuous equivalent for the continuum of players). Nash equilibrium Nash equilibria for n players ř Finite horizon truncation

Enforcing optimality

Carrying capacity

Numerics

Continuous time

Conclusions ”The tragedy of the A short introduction to game theory cont. 3 commons” in the dynamic context

Agnieszka Wiszniewska- Matyszkiel

Games § We are also interested in Pareto optimal profiles, i.e. Nash eq. ¯ Stackelberg eq. profiles S such that there exists no profile S with Pareto opt. § ¯ Example-static JipSq ě JipSq for all i (a.e. for the continuum of players) The tragedy § ¯ and JipSq ą JipSq for some i (in a set of positive Dynamic games

measure for the continuum of players). A simple model § If the payoffs are monetary (and side payments are Objectives Bellman possible) then the most obvious Pareto optimal profile is Revised sufficient condition Ji pSq Social optimum the profile which maximizes iPI #I (or its continuous equivalent for the continuum of players). We call such a Nash equilibrium ř Nash equilibria for n players profile the social optimum. Finite horizon truncation Enforcing optimality

Carrying capacity

Numerics

Continuous time

Conclusions So, J1pS1, S2q “ p100 ´ pS1 ` S2qqS1 ´ 10S1, J2 analogously. § The best response correspondences B1pS2q “ 90´S2 Argmaxp100 ´ pS1 ` S2qqS1 ´ 10S1 “ 2 , S1ě0 ! ) B2pS1q analogously. 90´S2 S1 “ , § The Nash equilibrium given by 2 So, S “ 90´S1 . " 2 2 S1 “ S2 “ 30 with price 40. § The social optimum Argmaxp100 ´ pS1 ` S2qqS1 ´ 10S1 ` p100 ´ pS1 ` S1,S2ě0 S2qqS2 ´ 10S2 “ tpS1, S2q : S1 ` S2 “ 45u, price 55. So, if additionally, S1 “ S2, then higher profits for both.

”The tragedy of the Example: a static fishery in an unconstrained commons” in the dynamic context word Agnieszka Wiszniewska- § A fishery with two identical fishing firms, Si “ R`, linear Matyszkiel

costs of fishing 10Si, price dependent on the amount of Games fish on the market 100 S S . Nash eq. ´ p 1 ` 2q Stackelberg eq. Pareto opt. Example-static The tragedy

Dynamic games

A simple model Objectives

Bellman Revised sufficient condition

Social optimum

Nash equilibrium Nash equilibria for n players Finite horizon truncation

Enforcing optimality

Carrying capacity

Numerics

Continuous time

Conclusions § The best response correspondences B1pS2q “ 90´S2 Argmaxp100 ´ pS1 ` S2qqS1 ´ 10S1 “ 2 , S1ě0 ! ) B2pS1q analogously. 90´S2 S1 “ , § The Nash equilibrium given by 2 So, S “ 90´S1 . " 2 2 S1 “ S2 “ 30 with price 40. § The social optimum Argmaxp100 ´ pS1 ` S2qqS1 ´ 10S1 ` p100 ´ pS1 ` S1,S2ě0 S2qqS2 ´ 10S2 “ tpS1, S2q : S1 ` S2 “ 45u, price 55. So, if additionally, S1 “ S2, then higher profits for both.

”The tragedy of the Example: a static fishery in an unconstrained commons” in the dynamic context word Agnieszka Wiszniewska- § A fishery with two identical fishing firms, Si “ R`, linear Matyszkiel

costs of fishing 10Si, price dependent on the amount of Games fish on the market 100 S S . So, Nash eq. ´ p 1 ` 2q Stackelberg eq. J S , S 100 S S S 10S , J Pareto opt. 1p 1 2q “ p ´ p 1 ` 2qq 1 ´ 1 2 Example-static analogously. The tragedy Dynamic games

A simple model Objectives

Bellman Revised sufficient condition

Social optimum

Nash equilibrium Nash equilibria for n players Finite horizon truncation

Enforcing optimality

Carrying capacity

Numerics

Continuous time

Conclusions 90´S2 S1 “ , § The Nash equilibrium given by 2 So, S “ 90´S1 . " 2 2 S1 “ S2 “ 30 with price 40. § The social optimum Argmaxp100 ´ pS1 ` S2qqS1 ´ 10S1 ` p100 ´ pS1 ` S1,S2ě0 S2qqS2 ´ 10S2 “ tpS1, S2q : S1 ` S2 “ 45u, price 55. So, if additionally, S1 “ S2, then higher profits for both.

”The tragedy of the Example: a static fishery in an unconstrained commons” in the dynamic context word Agnieszka Wiszniewska- § A fishery with two identical fishing firms, Si “ R`, linear Matyszkiel

costs of fishing 10Si, price dependent on the amount of Games fish on the market 100 S S . So, Nash eq. ´ p 1 ` 2q Stackelberg eq. J S , S 100 S S S 10S , J Pareto opt. 1p 1 2q “ p ´ p 1 ` 2qq 1 ´ 1 2 Example-static analogously. The tragedy Dynamic games § The best response correspondences B1pS2q “ A simple model 90´S2 Argmaxp100 ´ pS1 ` S2qqS1 ´ 10S1 “ 2 , Objectives S1ě0 Bellman ! ) Revised sufficient condition B2pS1q analogously. Social optimum

Nash equilibrium Nash equilibria for n players Finite horizon truncation

Enforcing optimality

Carrying capacity

Numerics

Continuous time

Conclusions So,

S1 “ S2 “ 30 with price 40. § The social optimum Argmaxp100 ´ pS1 ` S2qqS1 ´ 10S1 ` p100 ´ pS1 ` S1,S2ě0 S2qqS2 ´ 10S2 “ tpS1, S2q : S1 ` S2 “ 45u, price 55. So, if additionally, S1 “ S2, then higher profits for both.

”The tragedy of the Example: a static fishery in an unconstrained commons” in the dynamic context word Agnieszka Wiszniewska- § A fishery with two identical fishing firms, Si “ R`, linear Matyszkiel

costs of fishing 10Si, price dependent on the amount of Games fish on the market 100 S S . So, Nash eq. ´ p 1 ` 2q Stackelberg eq. J S , S 100 S S S 10S , J Pareto opt. 1p 1 2q “ p ´ p 1 ` 2qq 1 ´ 1 2 Example-static analogously. The tragedy Dynamic games § The best response correspondences B1pS2q “ A simple model 90´S2 Argmaxp100 ´ pS1 ` S2qqS1 ´ 10S1 “ 2 , Objectives S1ě0 Bellman ! ) Revised sufficient condition B2pS1q analogously. 90´S2 Social optimum S1 “ , § 2 The Nash equilibrium given by 90´S1 Nash equilibrium S2 “ 2 . Nash equilibria for n players " Finite horizon truncation

Enforcing optimality

Carrying capacity

Numerics

Continuous time

Conclusions § The social optimum Argmaxp100 ´ pS1 ` S2qqS1 ´ 10S1 ` p100 ´ pS1 ` S1,S2ě0 S2qqS2 ´ 10S2 “ tpS1, S2q : S1 ` S2 “ 45u, price 55. So, if additionally, S1 “ S2, then higher profits for both.

”The tragedy of the Example: a static fishery in an unconstrained commons” in the dynamic context word Agnieszka Wiszniewska- § A fishery with two identical fishing firms, Si “ R`, linear Matyszkiel

costs of fishing 10Si, price dependent on the amount of Games fish on the market 100 S S . So, Nash eq. ´ p 1 ` 2q Stackelberg eq. J S , S 100 S S S 10S , J Pareto opt. 1p 1 2q “ p ´ p 1 ` 2qq 1 ´ 1 2 Example-static analogously. The tragedy Dynamic games § The best response correspondences B1pS2q “ A simple model 90´S2 Argmaxp100 ´ pS1 ` S2qqS1 ´ 10S1 “ 2 , Objectives S1ě0 Bellman ! ) Revised sufficient condition B2pS1q analogously. 90´S2 Social optimum S1 “ , § 2 The Nash equilibrium given by 90´S1 So, Nash equilibrium S2 “ 2 . Nash equilibria for n players " Finite horizon truncation S1 “ S2 “ 30 with price 40. Enforcing optimality

Carrying capacity

Numerics

Continuous time

Conclusions So, if additionally, S1 “ S2, then higher profits for both.

”The tragedy of the Example: a static fishery in an unconstrained commons” in the dynamic context word Agnieszka Wiszniewska- § A fishery with two identical fishing firms, Si “ R`, linear Matyszkiel

costs of fishing 10Si, price dependent on the amount of Games fish on the market 100 S S . So, Nash eq. ´ p 1 ` 2q Stackelberg eq. J S , S 100 S S S 10S , J Pareto opt. 1p 1 2q “ p ´ p 1 ` 2qq 1 ´ 1 2 Example-static analogously. The tragedy Dynamic games § The best response correspondences B1pS2q “ A simple model 90´S2 Argmaxp100 ´ pS1 ` S2qqS1 ´ 10S1 “ 2 , Objectives S1ě0 Bellman ! ) Revised sufficient condition B2pS1q analogously. 90´S2 Social optimum S1 “ , § 2 The Nash equilibrium given by 90´S1 So, Nash equilibrium S2 “ 2 . Nash equilibria for n players " Finite horizon truncation S1 “ S2 “ 30 with price 40. Enforcing optimality § The social optimum Carrying capacity Argmaxp100 ´ pS1 ` S2qqS1 ´ 10S1 ` p100 ´ pS1 ` Numerics S1,S2ě0 Continuous time S2qqS2 ´ 10S2 “ tpS1, S2q : S1 ` S2 “ 45u, price 55. Conclusions ”The tragedy of the Example: a static fishery in an unconstrained commons” in the dynamic context word Agnieszka Wiszniewska- § A fishery with two identical fishing firms, Si “ R`, linear Matyszkiel

costs of fishing 10Si, price dependent on the amount of Games fish on the market 100 S S . So, Nash eq. ´ p 1 ` 2q Stackelberg eq. J S , S 100 S S S 10S , J Pareto opt. 1p 1 2q “ p ´ p 1 ` 2qq 1 ´ 1 2 Example-static analogously. The tragedy Dynamic games § The best response correspondences B1pS2q “ A simple model 90´S2 Argmaxp100 ´ pS1 ` S2qqS1 ´ 10S1 “ 2 , Objectives S1ě0 Bellman ! ) Revised sufficient condition B2pS1q analogously. 90´S2 Social optimum S1 “ , § 2 The Nash equilibrium given by 90´S1 So, Nash equilibrium S2 “ 2 . Nash equilibria for n players " Finite horizon truncation S1 “ S2 “ 30 with price 40. Enforcing optimality § The social optimum Carrying capacity Argmaxp100 ´ pS1 ` S2qqS1 ´ 10S1 ` p100 ´ pS1 ` Numerics S1,S2ě0 Continuous time S2qqS2 ´ 10S2 “ tpS1, S2q : S1 ` S2 “ 45u, price 55. Conclusions So, if additionally, S1 “ S2, then higher profits for both. 90´S1 S2 “ 2 “ 22.5. Price 32.5. § The leader extracts more than at a Nash equilibrium and gets more payoff that at the symmetric cooperative solution and it makes the follower extract as in the symmetric cooperative solution and get less payoff than at Nash. § That may be only the matter of informational advantage and kindly informing the follower about the resulting choice!

”The tragedy of the Example: a static fishery cont. commons” in the dynamic context

Agnieszka Wiszniewska- Matyszkiel § The Stackelberg equilibrium: After calculating B2pS1q, Games the leader optimizes Nash eq. S Argmax 100 S 90´S1 S 10S 45. Stackelberg eq. 1 P p ´ p 1 ` 2 qq 1 ´ 1 “ Pareto opt. S1ě0 Example-static The tragedy

Dynamic games

A simple model Objectives

Bellman Revised sufficient condition

Social optimum

Nash equilibrium Nash equilibria for n players Finite horizon truncation

Enforcing optimality

Carrying capacity

Numerics

Continuous time

Conclusions Price 32.5. § The leader extracts more than at a Nash equilibrium and gets more payoff that at the symmetric cooperative solution and it makes the follower extract as in the symmetric cooperative solution and get less payoff than at Nash. § That may be only the matter of informational advantage and kindly informing the follower about the resulting choice!

”The tragedy of the Example: a static fishery cont. commons” in the dynamic context

Agnieszka Wiszniewska- Matyszkiel § The Stackelberg equilibrium: After calculating B2pS1q, Games the leader optimizes Nash eq. S Argmax 100 S 90´S1 S 10S 45. Stackelberg eq. 1 P p ´ p 1 ` 2 qq 1 ´ 1 “ Pareto opt. S1ě0 Example-static 90´S1 The tragedy S2 “ “ 22.5. 2 Dynamic games

A simple model Objectives

Bellman Revised sufficient condition

Social optimum

Nash equilibrium Nash equilibria for n players Finite horizon truncation

Enforcing optimality

Carrying capacity

Numerics

Continuous time

Conclusions § The leader extracts more than at a Nash equilibrium and gets more payoff that at the symmetric cooperative solution and it makes the follower extract as in the symmetric cooperative solution and get less payoff than at Nash. § That may be only the matter of informational advantage and kindly informing the follower about the resulting choice!

”The tragedy of the Example: a static fishery cont. commons” in the dynamic context

Agnieszka Wiszniewska- Matyszkiel § The Stackelberg equilibrium: After calculating B2pS1q, Games the leader optimizes Nash eq. S Argmax 100 S 90´S1 S 10S 45. Stackelberg eq. 1 P p ´ p 1 ` 2 qq 1 ´ 1 “ Pareto opt. S1ě0 Example-static 90´S1 The tragedy S2 “ “ 22.5. Price 32.5. 2 Dynamic games

A simple model Objectives

Bellman Revised sufficient condition

Social optimum

Nash equilibrium Nash equilibria for n players Finite horizon truncation

Enforcing optimality

Carrying capacity

Numerics

Continuous time

Conclusions ”The tragedy of the Example: a static fishery cont. commons” in the dynamic context

Agnieszka Wiszniewska- Matyszkiel § The Stackelberg equilibrium: After calculating B2pS1q, Games the leader optimizes Nash eq. S Argmax 100 S 90´S1 S 10S 45. Stackelberg eq. 1 P p ´ p 1 ` 2 qq 1 ´ 1 “ Pareto opt. S1ě0 Example-static 90´S1 The tragedy S2 “ “ 22.5. Price 32.5. 2 Dynamic games

§ The leader extracts more than at a Nash equilibrium A simple model and gets more payoff that at the symmetric cooperative Objectives Bellman solution and it makes the follower extract as in the Revised sufficient condition symmetric cooperative solution and get less payoff than Social optimum at Nash. Nash equilibrium Nash equilibria for n players § That may be only the matter of informational advantage Finite horizon truncation and kindly informing the follower about the resulting Enforcing optimality choice! Carrying capacity Numerics

Continuous time

Conclusions (Hardin 1968) the logic of pursuing individual benefit in commons without constraints results in overexploitation (and sometimes extinction of the harvested species), and it is worse for everybody compared to the result of ”mutual coercion, mutually agreed upon” but even then there is a temptation to cheat... and who is going to control the wardens... § In games related to extraction of common (or interrelated) resources: the fact that the social optimum is not a Nash equilibrium and a/the Nash equilibrium (often unique) is not Pareto optimal and it yields payoffs smaller for all players than the social optimum. § Usually solved by enforcement: changing a game by adding a benevolent social planner – a Stackelberg leader modifying payoffs of the rest of the players by e.g. a tax in order that the previous social optimum is a Nash equilibrium given his strategy.

”The tragedy of the ”The tragedy of the commons” commons” in the dynamic context § Philosophically: Agnieszka Wiszniewska- Matyszkiel

Games Nash eq. Stackelberg eq. Pareto opt. Example-static The tragedy

Dynamic games

A simple model Objectives

Bellman Revised sufficient condition

Social optimum

Nash equilibrium Nash equilibria for n players Finite horizon truncation

Enforcing optimality

Carrying capacity

Numerics

Continuous time

Conclusions and it is worse for everybody compared to the result of ”mutual coercion, mutually agreed upon” but even then there is a temptation to cheat... and who is going to control the wardens... § In games related to extraction of common (or interrelated) resources: the fact that the social optimum is not a Nash equilibrium and a/the Nash equilibrium (often unique) is not Pareto optimal and it yields payoffs smaller for all players than the social optimum. § Usually solved by enforcement: changing a game by adding a benevolent social planner – a Stackelberg leader modifying payoffs of the rest of the players by e.g. a tax in order that the previous social optimum is a Nash equilibrium given his strategy.

”The tragedy of the ”The tragedy of the commons” commons” in the dynamic context § Philosophically: (Hardin 1968) the logic of pursuing Agnieszka Wiszniewska- individual benefit in commons without constraints Matyszkiel results in overexploitation (and sometimes extinction of Games the harvested species), Nash eq. Stackelberg eq. Pareto opt. Example-static The tragedy

Dynamic games

A simple model Objectives

Bellman Revised sufficient condition

Social optimum

Nash equilibrium Nash equilibria for n players Finite horizon truncation

Enforcing optimality

Carrying capacity

Numerics

Continuous time

Conclusions but even then there is a temptation to cheat... and who is going to control the wardens... § In games related to extraction of common (or interrelated) resources: the fact that the social optimum is not a Nash equilibrium and a/the Nash equilibrium (often unique) is not Pareto optimal and it yields payoffs smaller for all players than the social optimum. § Usually solved by enforcement: changing a game by adding a benevolent social planner – a Stackelberg leader modifying payoffs of the rest of the players by e.g. a tax in order that the previous social optimum is a Nash equilibrium given his strategy.

”The tragedy of the ”The tragedy of the commons” commons” in the dynamic context § Philosophically: (Hardin 1968) the logic of pursuing Agnieszka Wiszniewska- individual benefit in commons without constraints Matyszkiel results in overexploitation (and sometimes extinction of Games the harvested species), and it is worse for everybody Nash eq. Stackelberg eq. compared to the result of ”mutual coercion, mutually Pareto opt. Example-static agreed upon” The tragedy Dynamic games

A simple model Objectives

Bellman Revised sufficient condition

Social optimum

Nash equilibrium Nash equilibria for n players Finite horizon truncation

Enforcing optimality

Carrying capacity

Numerics

Continuous time

Conclusions and who is going to control the wardens... § In games related to extraction of common (or interrelated) resources: the fact that the social optimum is not a Nash equilibrium and a/the Nash equilibrium (often unique) is not Pareto optimal and it yields payoffs smaller for all players than the social optimum. § Usually solved by enforcement: changing a game by adding a benevolent social planner – a Stackelberg leader modifying payoffs of the rest of the players by e.g. a tax in order that the previous social optimum is a Nash equilibrium given his strategy.

”The tragedy of the ”The tragedy of the commons” commons” in the dynamic context § Philosophically: (Hardin 1968) the logic of pursuing Agnieszka Wiszniewska- individual benefit in commons without constraints Matyszkiel results in overexploitation (and sometimes extinction of Games the harvested species), and it is worse for everybody Nash eq. Stackelberg eq. compared to the result of ”mutual coercion, mutually Pareto opt. Example-static agreed upon” but even then there is a temptation to The tragedy cheat... Dynamic games A simple model Objectives

Bellman Revised sufficient condition

Social optimum

Nash equilibrium Nash equilibria for n players Finite horizon truncation

Enforcing optimality

Carrying capacity

Numerics

Continuous time

Conclusions § In games related to extraction of common (or interrelated) resources: the fact that the social optimum is not a Nash equilibrium and a/the Nash equilibrium (often unique) is not Pareto optimal and it yields payoffs smaller for all players than the social optimum. § Usually solved by enforcement: changing a game by adding a benevolent social planner – a Stackelberg leader modifying payoffs of the rest of the players by e.g. a tax in order that the previous social optimum is a Nash equilibrium given his strategy.

”The tragedy of the ”The tragedy of the commons” commons” in the dynamic context § Philosophically: (Hardin 1968) the logic of pursuing Agnieszka Wiszniewska- individual benefit in commons without constraints Matyszkiel results in overexploitation (and sometimes extinction of Games the harvested species), and it is worse for everybody Nash eq. Stackelberg eq. compared to the result of ”mutual coercion, mutually Pareto opt. Example-static agreed upon” but even then there is a temptation to The tragedy cheat... and who is going to control the wardens... Dynamic games A simple model Objectives

Bellman Revised sufficient condition

Social optimum

Nash equilibrium Nash equilibria for n players Finite horizon truncation

Enforcing optimality

Carrying capacity

Numerics

Continuous time

Conclusions and a/the Nash equilibrium (often unique) is not Pareto optimal and it yields payoffs smaller for all players than the social optimum. § Usually solved by enforcement: changing a game by adding a benevolent social planner – a Stackelberg leader modifying payoffs of the rest of the players by e.g. a tax in order that the previous social optimum is a Nash equilibrium given his strategy.

”The tragedy of the ”The tragedy of the commons” commons” in the dynamic context § Philosophically: (Hardin 1968) the logic of pursuing Agnieszka Wiszniewska- individual benefit in commons without constraints Matyszkiel results in overexploitation (and sometimes extinction of Games the harvested species), and it is worse for everybody Nash eq. Stackelberg eq. compared to the result of ”mutual coercion, mutually Pareto opt. Example-static agreed upon” but even then there is a temptation to The tragedy cheat... and who is going to control the wardens... Dynamic games § A simple model In games related to extraction of common (or Objectives

interrelated) resources: the fact that the social optimum Bellman is not a Nash equilibrium Revised sufficient condition Social optimum

Nash equilibrium Nash equilibria for n players Finite horizon truncation

Enforcing optimality

Carrying capacity

Numerics

Continuous time

Conclusions it yields payoffs smaller for all players than the social optimum. § Usually solved by enforcement: changing a game by adding a benevolent social planner – a Stackelberg leader modifying payoffs of the rest of the players by e.g. a tax in order that the previous social optimum is a Nash equilibrium given his strategy.

”The tragedy of the ”The tragedy of the commons” commons” in the dynamic context § Philosophically: (Hardin 1968) the logic of pursuing Agnieszka Wiszniewska- individual benefit in commons without constraints Matyszkiel results in overexploitation (and sometimes extinction of Games the harvested species), and it is worse for everybody Nash eq. Stackelberg eq. compared to the result of ”mutual coercion, mutually Pareto opt. Example-static agreed upon” but even then there is a temptation to The tragedy cheat... and who is going to control the wardens... Dynamic games § A simple model In games related to extraction of common (or Objectives

interrelated) resources: the fact that the social optimum Bellman is not a Nash equilibrium and a/the Nash equilibrium Revised sufficient condition Social optimum (often unique) is not Pareto optimal and Nash equilibrium Nash equilibria for n players Finite horizon truncation

Enforcing optimality

Carrying capacity

Numerics

Continuous time

Conclusions § Usually solved by enforcement: changing a game by adding a benevolent social planner – a Stackelberg leader modifying payoffs of the rest of the players by e.g. a tax in order that the previous social optimum is a Nash equilibrium given his strategy.

”The tragedy of the ”The tragedy of the commons” commons” in the dynamic context § Philosophically: (Hardin 1968) the logic of pursuing Agnieszka Wiszniewska- individual benefit in commons without constraints Matyszkiel results in overexploitation (and sometimes extinction of Games the harvested species), and it is worse for everybody Nash eq. Stackelberg eq. compared to the result of ”mutual coercion, mutually Pareto opt. Example-static agreed upon” but even then there is a temptation to The tragedy cheat... and who is going to control the wardens... Dynamic games § A simple model In games related to extraction of common (or Objectives

interrelated) resources: the fact that the social optimum Bellman is not a Nash equilibrium and a/the Nash equilibrium Revised sufficient condition Social optimum (often unique) is not Pareto optimal and it yields payoffs Nash equilibrium smaller for all players than the social optimum. Nash equilibria for n players Finite horizon truncation

Enforcing optimality

Carrying capacity

Numerics

Continuous time

Conclusions ”The tragedy of the ”The tragedy of the commons” commons” in the dynamic context § Philosophically: (Hardin 1968) the logic of pursuing Agnieszka Wiszniewska- individual benefit in commons without constraints Matyszkiel results in overexploitation (and sometimes extinction of Games the harvested species), and it is worse for everybody Nash eq. Stackelberg eq. compared to the result of ”mutual coercion, mutually Pareto opt. Example-static agreed upon” but even then there is a temptation to The tragedy cheat... and who is going to control the wardens... Dynamic games § A simple model In games related to extraction of common (or Objectives

interrelated) resources: the fact that the social optimum Bellman is not a Nash equilibrium and a/the Nash equilibrium Revised sufficient condition Social optimum (often unique) is not Pareto optimal and it yields payoffs Nash equilibrium smaller for all players than the social optimum. Nash equilibria for n players Finite horizon truncation § Usually solved by enforcement: changing a game by Enforcing optimality

adding a benevolent social planner – a Stackelberg Carrying capacity

leader modifying payoffs of the rest of the players by Numerics e.g. a tax in order that the previous social optimum is a Continuous time Nash equilibrium given his strategy. Conclusions § the greenhouse gasses emission, § air and water pollution, § space debris, § congestion of frequencies, § congestion of earth, § antimicrobial resistance problem, § human behaviour concerning e.g. wearing masks during current epidemics.

§ But they all are dynamic problems!

”The tragedy of the ”The tragedy of the commons” – where? commons” in the dynamic context

Agnieszka Wiszniewska- Matyszkiel § common or interrelated fisheries, Games Nash eq. Stackelberg eq. Pareto opt. Example-static The tragedy

Dynamic games

A simple model Objectives

Bellman Revised sufficient condition

Social optimum

Nash equilibrium Nash equilibria for n players Finite horizon truncation

Enforcing optimality

Carrying capacity

Numerics

Continuous time

Conclusions § air and water pollution, § space debris, § congestion of frequencies, § congestion of earth, § antimicrobial resistance problem, § human behaviour concerning e.g. wearing masks during current epidemics.

§ But they all are dynamic problems!

”The tragedy of the ”The tragedy of the commons” – where? commons” in the dynamic context

Agnieszka Wiszniewska- Matyszkiel § common or interrelated fisheries, Games § Nash eq. the greenhouse gasses emission, Stackelberg eq. Pareto opt. Example-static The tragedy

Dynamic games

A simple model Objectives

Bellman Revised sufficient condition

Social optimum

Nash equilibrium Nash equilibria for n players Finite horizon truncation

Enforcing optimality

Carrying capacity

Numerics

Continuous time

Conclusions § space debris, § congestion of frequencies, § congestion of earth, § antimicrobial resistance problem, § human behaviour concerning e.g. wearing masks during current epidemics.

§ But they all are dynamic problems!

”The tragedy of the ”The tragedy of the commons” – where? commons” in the dynamic context

Agnieszka Wiszniewska- Matyszkiel § common or interrelated fisheries, Games § Nash eq. the greenhouse gasses emission, Stackelberg eq. Pareto opt. § air and water pollution, Example-static The tragedy

Dynamic games

A simple model Objectives

Bellman Revised sufficient condition

Social optimum

Nash equilibrium Nash equilibria for n players Finite horizon truncation

Enforcing optimality

Carrying capacity

Numerics

Continuous time

Conclusions § congestion of frequencies, § congestion of earth, § antimicrobial resistance problem, § human behaviour concerning e.g. wearing masks during current epidemics.

§ But they all are dynamic problems!

”The tragedy of the ”The tragedy of the commons” – where? commons” in the dynamic context

Agnieszka Wiszniewska- Matyszkiel § common or interrelated fisheries, Games § Nash eq. the greenhouse gasses emission, Stackelberg eq. Pareto opt. § air and water pollution, Example-static The tragedy

§ space debris, Dynamic games

A simple model Objectives

Bellman Revised sufficient condition

Social optimum

Nash equilibrium Nash equilibria for n players Finite horizon truncation

Enforcing optimality

Carrying capacity

Numerics

Continuous time

Conclusions § congestion of earth, § antimicrobial resistance problem, § human behaviour concerning e.g. wearing masks during current epidemics.

§ But they all are dynamic problems!

”The tragedy of the ”The tragedy of the commons” – where? commons” in the dynamic context

Agnieszka Wiszniewska- Matyszkiel § common or interrelated fisheries, Games § Nash eq. the greenhouse gasses emission, Stackelberg eq. Pareto opt. § air and water pollution, Example-static The tragedy

§ space debris, Dynamic games § congestion of frequencies, A simple model Objectives

Bellman Revised sufficient condition

Social optimum

Nash equilibrium Nash equilibria for n players Finite horizon truncation

Enforcing optimality

Carrying capacity

Numerics

Continuous time

Conclusions § antimicrobial resistance problem, § human behaviour concerning e.g. wearing masks during current epidemics.

§ But they all are dynamic problems!

”The tragedy of the ”The tragedy of the commons” – where? commons” in the dynamic context

Agnieszka Wiszniewska- Matyszkiel § common or interrelated fisheries, Games § Nash eq. the greenhouse gasses emission, Stackelberg eq. Pareto opt. § air and water pollution, Example-static The tragedy

§ space debris, Dynamic games § congestion of frequencies, A simple model Objectives

§ congestion of earth, Bellman Revised sufficient condition

Social optimum

Nash equilibrium Nash equilibria for n players Finite horizon truncation

Enforcing optimality

Carrying capacity

Numerics

Continuous time

Conclusions § human behaviour concerning e.g. wearing masks during current epidemics.

§ But they all are dynamic problems!

”The tragedy of the ”The tragedy of the commons” – where? commons” in the dynamic context

Agnieszka Wiszniewska- Matyszkiel § common or interrelated fisheries, Games § Nash eq. the greenhouse gasses emission, Stackelberg eq. Pareto opt. § air and water pollution, Example-static The tragedy

§ space debris, Dynamic games § congestion of frequencies, A simple model Objectives

§ congestion of earth, Bellman Revised sufficient condition § antimicrobial resistance problem, Social optimum

Nash equilibrium Nash equilibria for n players Finite horizon truncation

Enforcing optimality

Carrying capacity

Numerics

Continuous time

Conclusions § But they all are dynamic problems!

”The tragedy of the ”The tragedy of the commons” – where? commons” in the dynamic context

Agnieszka Wiszniewska- Matyszkiel § common or interrelated fisheries, Games § Nash eq. the greenhouse gasses emission, Stackelberg eq. Pareto opt. § air and water pollution, Example-static The tragedy

§ space debris, Dynamic games § congestion of frequencies, A simple model Objectives

§ congestion of earth, Bellman Revised sufficient condition § antimicrobial resistance problem, Social optimum § human behaviour concerning e.g. wearing masks Nash equilibrium Nash equilibria for n players during current epidemics. Finite horizon truncation

Enforcing optimality

Carrying capacity

Numerics

Continuous time

Conclusions ”The tragedy of the ”The tragedy of the commons” – where? commons” in the dynamic context

Agnieszka Wiszniewska- Matyszkiel § common or interrelated fisheries, Games § Nash eq. the greenhouse gasses emission, Stackelberg eq. Pareto opt. § air and water pollution, Example-static The tragedy

§ space debris, Dynamic games § congestion of frequencies, A simple model Objectives

§ congestion of earth, Bellman Revised sufficient condition § antimicrobial resistance problem, Social optimum § human behaviour concerning e.g. wearing masks Nash equilibrium Nash equilibria for n players during current epidemics. Finite horizon truncation

Enforcing optimality § But they all are dynamic problems! Carrying capacity

Numerics

Continuous time

Conclusions § Strategies are functions which describe what to do, i.e. which decision from a decision set Di to choose at each time instant in T. At least measurability needed in continuous time. § The trajectory of the state variable for discrete time is defined by the strategy profile using a difference/differential equation. § while the payoff is the sum/integral of discounted current payoffs plus a terminal payoff. § Like in optimal control, strategies can be open loop (functions of time, initial condition fixed), feedback (function of state or state and time, initial condition arbitrary), history-dependent... depending on information structure considered.

”The tragedy of the Dynamic games commons” in the dynamic context

§ Dynamic games are games played over time set T, Agnieszka Wiszniewska- continuous or discrete, finite or infinite, with additional Matyszkiel state variable x P X. Games Nash eq. Stackelberg eq. Pareto opt. Example-static The tragedy

Dynamic games

A simple model Objectives

Bellman Revised sufficient condition

Social optimum

Nash equilibrium Nash equilibria for n players Finite horizon truncation

Enforcing optimality

Carrying capacity

Numerics

Continuous time

Conclusions § The trajectory of the state variable for discrete time is defined by the strategy profile using a difference/differential equation. § while the payoff is the sum/integral of discounted current payoffs plus a terminal payoff. § Like in optimal control, strategies can be open loop (functions of time, initial condition fixed), feedback (function of state or state and time, initial condition arbitrary), history-dependent... depending on information structure considered.

”The tragedy of the Dynamic games commons” in the dynamic context

§ Dynamic games are games played over time set T, Agnieszka Wiszniewska- continuous or discrete, finite or infinite, with additional Matyszkiel state variable x P X. Games § Strategies are functions which describe what to do, i.e. Nash eq. Stackelberg eq. which decision from a decision set Di to choose at each Pareto opt. Example-static time instant in T. At least measurability needed in The tragedy continuous time. Dynamic games A simple model Objectives

Bellman Revised sufficient condition

Social optimum

Nash equilibrium Nash equilibria for n players Finite horizon truncation

Enforcing optimality

Carrying capacity

Numerics

Continuous time

Conclusions § while the payoff is the sum/integral of discounted current payoffs plus a terminal payoff. § Like in optimal control, strategies can be open loop (functions of time, initial condition fixed), feedback (function of state or state and time, initial condition arbitrary), history-dependent... depending on information structure considered.

”The tragedy of the Dynamic games commons” in the dynamic context

§ Dynamic games are games played over time set T, Agnieszka Wiszniewska- continuous or discrete, finite or infinite, with additional Matyszkiel state variable x P X. Games § Strategies are functions which describe what to do, i.e. Nash eq. Stackelberg eq. which decision from a decision set Di to choose at each Pareto opt. Example-static time instant in T. At least measurability needed in The tragedy continuous time. Dynamic games A simple model § The trajectory of the state variable for discrete time is Objectives defined by the strategy profile using a Bellman difference/differential equation. Revised sufficient condition Social optimum

Nash equilibrium Nash equilibria for n players Finite horizon truncation

Enforcing optimality

Carrying capacity

Numerics

Continuous time

Conclusions § Like in optimal control, strategies can be open loop (functions of time, initial condition fixed), feedback (function of state or state and time, initial condition arbitrary), history-dependent... depending on information structure considered.

”The tragedy of the Dynamic games commons” in the dynamic context

§ Dynamic games are games played over time set T, Agnieszka Wiszniewska- continuous or discrete, finite or infinite, with additional Matyszkiel state variable x P X. Games § Strategies are functions which describe what to do, i.e. Nash eq. Stackelberg eq. which decision from a decision set Di to choose at each Pareto opt. Example-static time instant in T. At least measurability needed in The tragedy continuous time. Dynamic games A simple model § The trajectory of the state variable for discrete time is Objectives defined by the strategy profile using a Bellman difference/differential equation. Revised sufficient condition Social optimum § while the payoff is the sum/integral of discounted Nash equilibrium current payoffs plus a terminal payoff. Nash equilibria for n players Finite horizon truncation

Enforcing optimality

Carrying capacity

Numerics

Continuous time

Conclusions , feedback (function of state or state and time, initial condition arbitrary), history-dependent... depending on information structure considered.

”The tragedy of the Dynamic games commons” in the dynamic context

§ Dynamic games are games played over time set T, Agnieszka Wiszniewska- continuous or discrete, finite or infinite, with additional Matyszkiel state variable x P X. Games § Strategies are functions which describe what to do, i.e. Nash eq. Stackelberg eq. which decision from a decision set Di to choose at each Pareto opt. Example-static time instant in T. At least measurability needed in The tragedy continuous time. Dynamic games A simple model § The trajectory of the state variable for discrete time is Objectives defined by the strategy profile using a Bellman difference/differential equation. Revised sufficient condition Social optimum § while the payoff is the sum/integral of discounted Nash equilibrium current payoffs plus a terminal payoff. Nash equilibria for n players Finite horizon truncation § Like in optimal control, strategies can be open loop Enforcing optimality (functions of time, initial condition fixed) Carrying capacity Numerics

Continuous time

Conclusions or state and time, initial condition arbitrary), history-dependent... depending on information structure considered.

”The tragedy of the Dynamic games commons” in the dynamic context

§ Dynamic games are games played over time set T, Agnieszka Wiszniewska- continuous or discrete, finite or infinite, with additional Matyszkiel state variable x P X. Games § Strategies are functions which describe what to do, i.e. Nash eq. Stackelberg eq. which decision from a decision set Di to choose at each Pareto opt. Example-static time instant in T. At least measurability needed in The tragedy continuous time. Dynamic games A simple model § The trajectory of the state variable for discrete time is Objectives defined by the strategy profile using a Bellman difference/differential equation. Revised sufficient condition Social optimum § while the payoff is the sum/integral of discounted Nash equilibrium current payoffs plus a terminal payoff. Nash equilibria for n players Finite horizon truncation § Like in optimal control, strategies can be open loop Enforcing optimality (functions of time, initial condition fixed), feedback Carrying capacity (function of state Numerics Continuous time

Conclusions history-dependent... depending on information structure considered.

”The tragedy of the Dynamic games commons” in the dynamic context

§ Dynamic games are games played over time set T, Agnieszka Wiszniewska- continuous or discrete, finite or infinite, with additional Matyszkiel state variable x P X. Games § Strategies are functions which describe what to do, i.e. Nash eq. Stackelberg eq. which decision from a decision set Di to choose at each Pareto opt. Example-static time instant in T. At least measurability needed in The tragedy continuous time. Dynamic games A simple model § The trajectory of the state variable for discrete time is Objectives defined by the strategy profile using a Bellman difference/differential equation. Revised sufficient condition Social optimum § while the payoff is the sum/integral of discounted Nash equilibrium current payoffs plus a terminal payoff. Nash equilibria for n players Finite horizon truncation § Like in optimal control, strategies can be open loop Enforcing optimality (functions of time, initial condition fixed), feedback Carrying capacity (function of state or state and time, initial condition Numerics arbitrary), Continuous time Conclusions depending on information structure considered.

”The tragedy of the Dynamic games commons” in the dynamic context

§ Dynamic games are games played over time set T, Agnieszka Wiszniewska- continuous or discrete, finite or infinite, with additional Matyszkiel state variable x P X. Games § Strategies are functions which describe what to do, i.e. Nash eq. Stackelberg eq. which decision from a decision set Di to choose at each Pareto opt. Example-static time instant in T. At least measurability needed in The tragedy continuous time. Dynamic games A simple model § The trajectory of the state variable for discrete time is Objectives defined by the strategy profile using a Bellman difference/differential equation. Revised sufficient condition Social optimum § while the payoff is the sum/integral of discounted Nash equilibrium current payoffs plus a terminal payoff. Nash equilibria for n players Finite horizon truncation § Like in optimal control, strategies can be open loop Enforcing optimality (functions of time, initial condition fixed), feedback Carrying capacity (function of state or state and time, initial condition Numerics arbitrary), history-dependent... Continuous time Conclusions ”The tragedy of the Dynamic games commons” in the dynamic context

§ Dynamic games are games played over time set T, Agnieszka Wiszniewska- continuous or discrete, finite or infinite, with additional Matyszkiel state variable x P X. Games § Strategies are functions which describe what to do, i.e. Nash eq. Stackelberg eq. which decision from a decision set Di to choose at each Pareto opt. Example-static time instant in T. At least measurability needed in The tragedy continuous time. Dynamic games A simple model § The trajectory of the state variable for discrete time is Objectives defined by the strategy profile using a Bellman difference/differential equation. Revised sufficient condition Social optimum § while the payoff is the sum/integral of discounted Nash equilibrium current payoffs plus a terminal payoff. Nash equilibria for n players Finite horizon truncation § Like in optimal control, strategies can be open loop Enforcing optimality (functions of time, initial condition fixed), feedback Carrying capacity (function of state or state and time, initial condition Numerics arbitrary), history-dependent... depending on Continuous time information structure considered. Conclusions Usually open loop and feedback Nash equilibria do not lead to the same state trajectories!

For the hare slightly slower than the wolf no open loop Nash equilibrium while quite obvious feedback Nash equilibrium. § Open loop Nash equilibria do not have good properties: if one player makes an error at one time instant, the suboptimal solution path is further chosen or the new solution has to be recalculated. § Feedback Nash equilibria are resilient to such errors - this is called subgame perfection.

”The tragedy of the Dynamic games – surprise 1 commons” in the dynamic context

§ Unlike in optimal control, information structure matters! Agnieszka Wiszniewska- Matyszkiel

Games Nash eq. Stackelberg eq. Pareto opt. Example-static The tragedy

Dynamic games

A simple model Objectives

Bellman Revised sufficient condition

Social optimum

Nash equilibrium Nash equilibria for n players Finite horizon truncation

Enforcing optimality

Carrying capacity

Numerics

Continuous time

Conclusions For the hare slightly slower than the wolf no open loop Nash equilibrium while quite obvious feedback Nash equilibrium. § Open loop Nash equilibria do not have good properties: if one player makes an error at one time instant, the suboptimal solution path is further chosen or the new solution has to be recalculated. § Feedback Nash equilibria are resilient to such errors - this is called subgame perfection.

”The tragedy of the Dynamic games – surprise 1 commons” in the dynamic context

§ Unlike in optimal control, information structure matters! Agnieszka Wiszniewska- Usually open loop and feedback Nash equilibria do not Matyszkiel lead to the same state trajectories! Games Nash eq. Stackelberg eq. Pareto opt. Example-static The tragedy

Dynamic games

A simple model Objectives

Bellman Revised sufficient condition

Social optimum

Nash equilibrium Nash equilibria for n players Finite horizon truncation

Enforcing optimality

Carrying capacity

Numerics

Continuous time

Conclusions : if one player makes an error at one time instant, the suboptimal solution path is further chosen or the new solution has to be recalculated. § Feedback Nash equilibria are resilient to such errors - this is called subgame perfection.

”The tragedy of the Dynamic games – surprise 1 commons” in the dynamic context

§ Unlike in optimal control, information structure matters! Agnieszka Wiszniewska- Usually open loop and feedback Nash equilibria do not Matyszkiel lead to the same state trajectories! Games Nash eq. Stackelberg eq. Pareto opt. Example-static The tragedy

Dynamic games

A simple model Objectives

Bellman Revised sufficient condition

For the hare slightly slower than the wolf no open loop Nash equilibrium Social optimum

while quite obvious feedback Nash equilibrium. Nash equilibrium § Nash equilibria for n players Open loop Nash equilibria do not have good properties Finite horizon truncation

Enforcing optimality

Carrying capacity

Numerics

Continuous time

Conclusions § Feedback Nash equilibria are resilient to such errors - this is called subgame perfection.

”The tragedy of the Dynamic games – surprise 1 commons” in the dynamic context

§ Unlike in optimal control, information structure matters! Agnieszka Wiszniewska- Usually open loop and feedback Nash equilibria do not Matyszkiel lead to the same state trajectories! Games Nash eq. Stackelberg eq. Pareto opt. Example-static The tragedy

Dynamic games

A simple model Objectives

Bellman Revised sufficient condition

For the hare slightly slower than the wolf no open loop Nash equilibrium Social optimum

while quite obvious feedback Nash equilibrium. Nash equilibrium § Nash equilibria for n players Open loop Nash equilibria do not have good properties: Finite horizon truncation

if one player makes an error at one time instant, the Enforcing optimality suboptimal solution path is further chosen or the new Carrying capacity solution has to be recalculated. Numerics Continuous time

Conclusions ”The tragedy of the Dynamic games – surprise 1 commons” in the dynamic context

§ Unlike in optimal control, information structure matters! Agnieszka Wiszniewska- Usually open loop and feedback Nash equilibria do not Matyszkiel lead to the same state trajectories! Games Nash eq. Stackelberg eq. Pareto opt. Example-static The tragedy

Dynamic games

A simple model Objectives

Bellman Revised sufficient condition

For the hare slightly slower than the wolf no open loop Nash equilibrium Social optimum

while quite obvious feedback Nash equilibrium. Nash equilibrium § Nash equilibria for n players Open loop Nash equilibria do not have good properties: Finite horizon truncation

if one player makes an error at one time instant, the Enforcing optimality suboptimal solution path is further chosen or the new Carrying capacity solution has to be recalculated. Numerics § Feedback Nash equilibria are resilient to such errors - Continuous time this is called subgame perfection. Conclusions ”The tragedy of the Dynamic games – surprise 1 commons” in the dynamic context

§ Unlike in optimal control, information structure matters! Agnieszka Wiszniewska- Usually open loop and feedback Nash equilibria do not Matyszkiel lead to the same state trajectories! Games Nash eq. Stackelberg eq. Pareto opt. Example-static The tragedy

Dynamic games

A simple model Objectives

Bellman Revised sufficient condition

For the hare slightly slower than the wolf no open loop Nash equilibrium Social optimum

while quite obvious feedback Nash equilibrium. Nash equilibrium § Nash equilibria for n players Open loop Nash equilibria do not have good properties: Finite horizon truncation

if one player makes an error at one time instant, the Enforcing optimality suboptimal solution path is further chosen or the new Carrying capacity solution has to be recalculated. Numerics § Feedback Nash equilibria are resilient to such errors - Continuous time this is called subgame perfection. Conclusions with parameters in the feedback strategy spaces coupled by finding a fixed point of the resulting best response correspondence (in the space of strategy profiles). § The choice of information structure determines the choice of the solution method: a coupled set of Bellman/ Hamilton-Jacobi-Bellman equations (for feedback) vs a coupled set of necessary conditions given by KKT multipliers or Pontryagin Maximum Principle (for open loop). Generally they yield different results! § For some problems with a continuum of players, also a decomposition method (introduced and developed in A. Wiszniewska-Matyszkiel: Positivity 2002, C& C 2003, IGTR 2002, 2003, JOTA 2014) can be used and the results for open loop and feedback are equivalent in a wider class of problems (JOTA 2014).

”The tragedy of the Dynamic games cont. commons” in the dynamic context § Finding a Nash equilibrium requires solving a set of Agnieszka Wiszniewska- parametrized optimal control problems Matyszkiel

Games Nash eq. Stackelberg eq. Pareto opt. Example-static The tragedy

Dynamic games

A simple model Objectives

Bellman Revised sufficient condition

Social optimum

Nash equilibrium Nash equilibria for n players Finite horizon truncation

Enforcing optimality

Carrying capacity

Numerics

Continuous time

Conclusions coupled by finding a fixed point of the resulting best response correspondence (in the space of strategy profiles). § The choice of information structure determines the choice of the solution method: a coupled set of Bellman/ Hamilton-Jacobi-Bellman equations (for feedback) vs a coupled set of necessary conditions given by KKT multipliers or Pontryagin Maximum Principle (for open loop). Generally they yield different results! § For some problems with a continuum of players, also a decomposition method (introduced and developed in A. Wiszniewska-Matyszkiel: Positivity 2002, C& C 2003, IGTR 2002, 2003, JOTA 2014) can be used and the results for open loop and feedback are equivalent in a wider class of problems (JOTA 2014).

”The tragedy of the Dynamic games cont. commons” in the dynamic context § Finding a Nash equilibrium requires solving a set of Agnieszka Wiszniewska- parametrized optimal control problems with parameters Matyszkiel in the feedback strategy spaces Games Nash eq. Stackelberg eq. Pareto opt. Example-static The tragedy

Dynamic games

A simple model Objectives

Bellman Revised sufficient condition

Social optimum

Nash equilibrium Nash equilibria for n players Finite horizon truncation

Enforcing optimality

Carrying capacity

Numerics

Continuous time

Conclusions § The choice of information structure determines the choice of the solution method: a coupled set of Bellman/ Hamilton-Jacobi-Bellman equations (for feedback) vs a coupled set of necessary conditions given by KKT multipliers or Pontryagin Maximum Principle (for open loop). Generally they yield different results! § For some problems with a continuum of players, also a decomposition method (introduced and developed in A. Wiszniewska-Matyszkiel: Positivity 2002, C& C 2003, IGTR 2002, 2003, JOTA 2014) can be used and the results for open loop and feedback are equivalent in a wider class of problems (JOTA 2014).

”The tragedy of the Dynamic games cont. commons” in the dynamic context § Finding a Nash equilibrium requires solving a set of Agnieszka Wiszniewska- parametrized optimal control problems with parameters Matyszkiel in the feedback strategy spaces coupled by finding a Games fixed point of the resulting best response Nash eq. Stackelberg eq. correspondence (in the space of strategy profiles). Pareto opt. Example-static The tragedy

Dynamic games

A simple model Objectives

Bellman Revised sufficient condition

Social optimum

Nash equilibrium Nash equilibria for n players Finite horizon truncation

Enforcing optimality

Carrying capacity

Numerics

Continuous time

Conclusions : a coupled set of Bellman/ Hamilton-Jacobi-Bellman equations (for feedback) vs a coupled set of necessary conditions given by KKT multipliers or Pontryagin Maximum Principle (for open loop). Generally they yield different results! § For some problems with a continuum of players, also a decomposition method (introduced and developed in A. Wiszniewska-Matyszkiel: Positivity 2002, C& C 2003, IGTR 2002, 2003, JOTA 2014) can be used and the results for open loop and feedback are equivalent in a wider class of problems (JOTA 2014).

”The tragedy of the Dynamic games cont. commons” in the dynamic context § Finding a Nash equilibrium requires solving a set of Agnieszka Wiszniewska- parametrized optimal control problems with parameters Matyszkiel in the feedback strategy spaces coupled by finding a Games fixed point of the resulting best response Nash eq. Stackelberg eq. correspondence (in the space of strategy profiles). Pareto opt. Example-static § The choice of information structure determines the The tragedy choice of the solution method Dynamic games A simple model Objectives

Bellman Revised sufficient condition

Social optimum

Nash equilibrium Nash equilibria for n players Finite horizon truncation

Enforcing optimality

Carrying capacity

Numerics

Continuous time

Conclusions Generally they yield different results! § For some problems with a continuum of players, also a decomposition method (introduced and developed in A. Wiszniewska-Matyszkiel: Positivity 2002, C& C 2003, IGTR 2002, 2003, JOTA 2014) can be used and the results for open loop and feedback are equivalent in a wider class of problems (JOTA 2014).

”The tragedy of the Dynamic games cont. commons” in the dynamic context § Finding a Nash equilibrium requires solving a set of Agnieszka Wiszniewska- parametrized optimal control problems with parameters Matyszkiel in the feedback strategy spaces coupled by finding a Games fixed point of the resulting best response Nash eq. Stackelberg eq. correspondence (in the space of strategy profiles). Pareto opt. Example-static § The choice of information structure determines the The tragedy choice of the solution method: a coupled set of Bellman/ Dynamic games A simple model Hamilton-Jacobi-Bellman equations (for feedback) vs a Objectives

coupled set of necessary conditions given by KKT Bellman multipliers or Pontryagin Maximum Principle (for open Revised sufficient condition Social optimum loop). Nash equilibrium Nash equilibria for n players Finite horizon truncation

Enforcing optimality

Carrying capacity

Numerics

Continuous time

Conclusions § For some problems with a continuum of players, also a decomposition method (introduced and developed in A. Wiszniewska-Matyszkiel: Positivity 2002, C& C 2003, IGTR 2002, 2003, JOTA 2014) can be used and the results for open loop and feedback are equivalent in a wider class of problems (JOTA 2014).

”The tragedy of the Dynamic games cont. commons” in the dynamic context § Finding a Nash equilibrium requires solving a set of Agnieszka Wiszniewska- parametrized optimal control problems with parameters Matyszkiel in the feedback strategy spaces coupled by finding a Games fixed point of the resulting best response Nash eq. Stackelberg eq. correspondence (in the space of strategy profiles). Pareto opt. Example-static § The choice of information structure determines the The tragedy choice of the solution method: a coupled set of Bellman/ Dynamic games A simple model Hamilton-Jacobi-Bellman equations (for feedback) vs a Objectives

coupled set of necessary conditions given by KKT Bellman multipliers or Pontryagin Maximum Principle (for open Revised sufficient condition Social optimum loop). Generally they yield different results! Nash equilibrium Nash equilibria for n players Finite horizon truncation

Enforcing optimality

Carrying capacity

Numerics

Continuous time

Conclusions ”The tragedy of the Dynamic games cont. commons” in the dynamic context § Finding a Nash equilibrium requires solving a set of Agnieszka Wiszniewska- parametrized optimal control problems with parameters Matyszkiel in the feedback strategy spaces coupled by finding a Games fixed point of the resulting best response Nash eq. Stackelberg eq. correspondence (in the space of strategy profiles). Pareto opt. Example-static § The choice of information structure determines the The tragedy choice of the solution method: a coupled set of Bellman/ Dynamic games A simple model Hamilton-Jacobi-Bellman equations (for feedback) vs a Objectives

coupled set of necessary conditions given by KKT Bellman multipliers or Pontryagin Maximum Principle (for open Revised sufficient condition Social optimum loop). Generally they yield different results! Nash equilibrium § For some problems with a continuum of players, also a Nash equilibria for n players Finite horizon truncation

decomposition method (introduced and developed in A. Enforcing optimality

Wiszniewska-Matyszkiel: Positivity 2002, C& C 2003, Carrying capacity

IGTR 2002, 2003, JOTA 2014) can be used and the Numerics results for open loop and feedback are equivalent in a Continuous time wider class of problems (JOTA 2014). Conclusions § There are various generalizations.Some of them are not subgame perfect, some of them may result in a need to recalculate the leaders’s strategy during the game (and, consequently the follower’s).

”The tragedy of the Dynamic games – surprise 2 commons” in the dynamic context

Agnieszka Wiszniewska- Matyszkiel

Games Nash eq. Stackelberg eq. § Definition of feedback Stackelberg equilibrium is not Pareto opt. Example-static straightforward – ”the best response to every strategy of The tragedy the leader” – is not a well posed problem! Dynamic games A simple model Objectives

Bellman Revised sufficient condition

Social optimum

Nash equilibrium Nash equilibria for n players Finite horizon truncation

Enforcing optimality

Carrying capacity

Numerics

Continuous time

Conclusions Some of them are not subgame perfect, some of them may result in a need to recalculate the leaders’s strategy during the game (and, consequently the follower’s).

”The tragedy of the Dynamic games – surprise 2 commons” in the dynamic context

Agnieszka Wiszniewska- Matyszkiel

Games Nash eq. Stackelberg eq. § Definition of feedback Stackelberg equilibrium is not Pareto opt. Example-static straightforward – ”the best response to every strategy of The tragedy the leader” – is not a well posed problem! Dynamic games A simple model § There are various generalizations. Objectives

Bellman Revised sufficient condition

Social optimum

Nash equilibrium Nash equilibria for n players Finite horizon truncation

Enforcing optimality

Carrying capacity

Numerics

Continuous time

Conclusions ”The tragedy of the Dynamic games – surprise 2 commons” in the dynamic context

Agnieszka Wiszniewska- Matyszkiel

Games Nash eq. Stackelberg eq. § Definition of feedback Stackelberg equilibrium is not Pareto opt. Example-static straightforward – ”the best response to every strategy of The tragedy the leader” – is not a well posed problem! Dynamic games A simple model § There are various generalizations.Some of them are not Objectives subgame perfect, some of them may result in a need to Bellman recalculate the leaders’s strategy during the game (and, Revised sufficient condition Social optimum consequently the follower’s). Nash equilibrium Nash equilibria for n players Finite horizon truncation

Enforcing optimality

Carrying capacity

Numerics

Continuous time

Conclusions § Linear quadratic dynamic games(LQDG) with linear state equation and quadratic current and terminal payoffs are most extensively studied (besides fully linear games) and have good economic interpretation. § So, let’s add the inherent constraints to LQDG and we will have a nice model, with quite standard and nice results.

”The tragedy of the Dynamic games cont.2 commons” in the dynamic context

Agnieszka Wiszniewska- Matyszkiel

Games § The complexity of the problem results in the fact that we Nash eq. Stackelberg eq. still do not know much about equilibria of dynamic Pareto opt. Example-static games in feedback form. The tragedy

Dynamic games

A simple model Objectives

Bellman Revised sufficient condition

Social optimum

Nash equilibrium Nash equilibria for n players Finite horizon truncation

Enforcing optimality

Carrying capacity

Numerics

Continuous time

Conclusions § So, let’s add the inherent constraints to LQDG and we will have a nice model, with quite standard and nice results.

”The tragedy of the Dynamic games cont.2 commons” in the dynamic context

Agnieszka Wiszniewska- Matyszkiel

Games § The complexity of the problem results in the fact that we Nash eq. Stackelberg eq. still do not know much about equilibria of dynamic Pareto opt. Example-static games in feedback form. The tragedy § Linear quadratic dynamic games(LQDG) with linear Dynamic games state equation and quadratic current and terminal A simple model Objectives

payoffs are most extensively studied (besides fully Bellman linear games) and have good economic interpretation. Revised sufficient condition Social optimum

Nash equilibrium Nash equilibria for n players Finite horizon truncation

Enforcing optimality

Carrying capacity

Numerics

Continuous time

Conclusions ”The tragedy of the Dynamic games cont.2 commons” in the dynamic context

Agnieszka Wiszniewska- Matyszkiel

Games § The complexity of the problem results in the fact that we Nash eq. Stackelberg eq. still do not know much about equilibria of dynamic Pareto opt. Example-static games in feedback form. The tragedy § Linear quadratic dynamic games(LQDG) with linear Dynamic games state equation and quadratic current and terminal A simple model Objectives

payoffs are most extensively studied (besides fully Bellman linear games) and have good economic interpretation. Revised sufficient condition Social optimum § So, let’s add the inherent constraints to LQDG and we Nash equilibrium will have a nice model, with quite standard and nice Nash equilibria for n players Finite horizon truncation

results. Enforcing optimality

Carrying capacity

Numerics

Continuous time

Conclusions § The set of states of the resource is R`. § Discrete time, infinite horizon (first). § At each time moment, player i extracts amount si ě 0, these si, in common, constitute a static profile s. § Constraint: given state x, the decisions have to fulfil si P r0, cxs. § Each of the players has cost function 1 2 costpsiq “ fsi ` 2 si . § The catch is sold at a common market at a price pricepsq “ A ´ u, where u is the aggregate extraction of s. § Aggregate extraction influences also the state of the resource.

”The tragedy of the ”The tragedy” in a simple model – the commons” in the dynamic context motivating example revisited Agnieszka Wiszniewska- § We consider a exploitation of a common renewable Matyszkiel

resource with set of players I from [1] R. Singh, A Games Nash eq. Wiszniewska-Matyszkiel, 2018, Linear Quadratic Game of Exploitation of Common Renewable Stackelberg eq. Resources with Inherent Constraints, Topological Methods in Nonlinear Analysis 51, 23–54. Pareto opt. Example-static The tragedy

Dynamic games

A simple model Objectives

Bellman Revised sufficient condition

Social optimum

Nash equilibrium Nash equilibria for n players Finite horizon truncation

Enforcing optimality

Carrying capacity

Numerics

Continuous time

Conclusions § Discrete time, infinite horizon (first). § At each time moment, player i extracts amount si ě 0, these si, in common, constitute a static profile s. § Constraint: given state x, the decisions have to fulfil si P r0, cxs. § Each of the players has cost function 1 2 costpsiq “ fsi ` 2 si . § The catch is sold at a common market at a price pricepsq “ A ´ u, where u is the aggregate extraction of s. § Aggregate extraction influences also the state of the resource.

”The tragedy of the ”The tragedy” in a simple model – the commons” in the dynamic context motivating example revisited Agnieszka Wiszniewska- § We consider a exploitation of a common renewable Matyszkiel

resource with set of players I from [1] R. Singh, A Games Nash eq. Wiszniewska-Matyszkiel, 2018, Linear Quadratic Game of Exploitation of Common Renewable Stackelberg eq. Resources with Inherent Constraints, Topological Methods in Nonlinear Analysis 51, 23–54. Pareto opt. Example-static § The set of states of the resource is R`. The tragedy Dynamic games

A simple model Objectives

Bellman Revised sufficient condition

Social optimum

Nash equilibrium Nash equilibria for n players Finite horizon truncation

Enforcing optimality

Carrying capacity

Numerics

Continuous time

Conclusions § At each time moment, player i extracts amount si ě 0, these si, in common, constitute a static profile s. § Constraint: given state x, the decisions have to fulfil si P r0, cxs. § Each of the players has cost function 1 2 costpsiq “ fsi ` 2 si . § The catch is sold at a common market at a price pricepsq “ A ´ u, where u is the aggregate extraction of s. § Aggregate extraction influences also the state of the resource.

”The tragedy of the ”The tragedy” in a simple model – the commons” in the dynamic context motivating example revisited Agnieszka Wiszniewska- § We consider a exploitation of a common renewable Matyszkiel

resource with set of players I from [1] R. Singh, A Games Nash eq. Wiszniewska-Matyszkiel, 2018, Linear Quadratic Game of Exploitation of Common Renewable Stackelberg eq. Resources with Inherent Constraints, Topological Methods in Nonlinear Analysis 51, 23–54. Pareto opt. Example-static § The set of states of the resource is R`. The tragedy § Discrete time, infinite horizon (first). Dynamic games A simple model Objectives

Bellman Revised sufficient condition

Social optimum

Nash equilibrium Nash equilibria for n players Finite horizon truncation

Enforcing optimality

Carrying capacity

Numerics

Continuous time

Conclusions § Constraint: given state x, the decisions have to fulfil si P r0, cxs. § Each of the players has cost function 1 2 costpsiq “ fsi ` 2 si . § The catch is sold at a common market at a price pricepsq “ A ´ u, where u is the aggregate extraction of s. § Aggregate extraction influences also the state of the resource.

”The tragedy of the ”The tragedy” in a simple model – the commons” in the dynamic context motivating example revisited Agnieszka Wiszniewska- § We consider a exploitation of a common renewable Matyszkiel

resource with set of players I from [1] R. Singh, A Games Nash eq. Wiszniewska-Matyszkiel, 2018, Linear Quadratic Game of Exploitation of Common Renewable Stackelberg eq. Resources with Inherent Constraints, Topological Methods in Nonlinear Analysis 51, 23–54. Pareto opt. Example-static § The set of states of the resource is R`. The tragedy § Discrete time, infinite horizon (first). Dynamic games § A simple model At each time moment, player i extracts amount si ě 0, Objectives these si, in common, constitute a static profile s. Bellman Revised sufficient condition

Social optimum

Nash equilibrium Nash equilibria for n players Finite horizon truncation

Enforcing optimality

Carrying capacity

Numerics

Continuous time

Conclusions § Each of the players has cost function 1 2 costpsiq “ fsi ` 2 si . § The catch is sold at a common market at a price pricepsq “ A ´ u, where u is the aggregate extraction of s. § Aggregate extraction influences also the state of the resource.

”The tragedy of the ”The tragedy” in a simple model – the commons” in the dynamic context motivating example revisited Agnieszka Wiszniewska- § We consider a exploitation of a common renewable Matyszkiel

resource with set of players I from [1] R. Singh, A Games Nash eq. Wiszniewska-Matyszkiel, 2018, Linear Quadratic Game of Exploitation of Common Renewable Stackelberg eq. Resources with Inherent Constraints, Topological Methods in Nonlinear Analysis 51, 23–54. Pareto opt. Example-static § The set of states of the resource is R`. The tragedy § Discrete time, infinite horizon (first). Dynamic games § A simple model At each time moment, player i extracts amount si ě 0, Objectives these si, in common, constitute a static profile s. Bellman § Constraint: given state x, the decisions have to fulfil Revised sufficient condition Social optimum si P r0, cxs. Nash equilibrium Nash equilibria for n players Finite horizon truncation

Enforcing optimality

Carrying capacity

Numerics

Continuous time

Conclusions § The catch is sold at a common market at a price pricepsq “ A ´ u, where u is the aggregate extraction of s. § Aggregate extraction influences also the state of the resource.

”The tragedy of the ”The tragedy” in a simple model – the commons” in the dynamic context motivating example revisited Agnieszka Wiszniewska- § We consider a exploitation of a common renewable Matyszkiel

resource with set of players I from [1] R. Singh, A Games Nash eq. Wiszniewska-Matyszkiel, 2018, Linear Quadratic Game of Exploitation of Common Renewable Stackelberg eq. Resources with Inherent Constraints, Topological Methods in Nonlinear Analysis 51, 23–54. Pareto opt. Example-static § The set of states of the resource is R`. The tragedy § Discrete time, infinite horizon (first). Dynamic games § A simple model At each time moment, player i extracts amount si ě 0, Objectives these si, in common, constitute a static profile s. Bellman § Constraint: given state x, the decisions have to fulfil Revised sufficient condition Social optimum si P r0, cxs. Nash equilibrium § Each of the players has cost function Nash equilibria for n players Finite horizon truncation costps q “ fs ` 1 s2. i i 2 i Enforcing optimality

Carrying capacity

Numerics

Continuous time

Conclusions at a price pricepsq “ A ´ u, where u is the aggregate extraction of s. § Aggregate extraction influences also the state of the resource.

”The tragedy of the ”The tragedy” in a simple model – the commons” in the dynamic context motivating example revisited Agnieszka Wiszniewska- § We consider a exploitation of a common renewable Matyszkiel

resource with set of players I from [1] R. Singh, A Games Nash eq. Wiszniewska-Matyszkiel, 2018, Linear Quadratic Game of Exploitation of Common Renewable Stackelberg eq. Resources with Inherent Constraints, Topological Methods in Nonlinear Analysis 51, 23–54. Pareto opt. Example-static § The set of states of the resource is R`. The tragedy § Discrete time, infinite horizon (first). Dynamic games § A simple model At each time moment, player i extracts amount si ě 0, Objectives these si, in common, constitute a static profile s. Bellman § Constraint: given state x, the decisions have to fulfil Revised sufficient condition Social optimum si P r0, cxs. Nash equilibrium § Each of the players has cost function Nash equilibria for n players Finite horizon truncation costps q “ fs ` 1 s2. i i 2 i Enforcing optimality § The catch is sold at a common market Carrying capacity

Numerics

Continuous time

Conclusions where u is the aggregate extraction of s. § Aggregate extraction influences also the state of the resource.

”The tragedy of the ”The tragedy” in a simple model – the commons” in the dynamic context motivating example revisited Agnieszka Wiszniewska- § We consider a exploitation of a common renewable Matyszkiel

resource with set of players I from [1] R. Singh, A Games Nash eq. Wiszniewska-Matyszkiel, 2018, Linear Quadratic Game of Exploitation of Common Renewable Stackelberg eq. Resources with Inherent Constraints, Topological Methods in Nonlinear Analysis 51, 23–54. Pareto opt. Example-static § The set of states of the resource is R`. The tragedy § Discrete time, infinite horizon (first). Dynamic games § A simple model At each time moment, player i extracts amount si ě 0, Objectives these si, in common, constitute a static profile s. Bellman § Constraint: given state x, the decisions have to fulfil Revised sufficient condition Social optimum si P r0, cxs. Nash equilibrium § Each of the players has cost function Nash equilibria for n players Finite horizon truncation costps q “ fs ` 1 s2. i i 2 i Enforcing optimality § The catch is sold at a common market at a price Carrying capacity

pricepsq “ A ´ u, Numerics

Continuous time

Conclusions § Aggregate extraction influences also the state of the resource.

”The tragedy of the ”The tragedy” in a simple model – the commons” in the dynamic context motivating example revisited Agnieszka Wiszniewska- § We consider a exploitation of a common renewable Matyszkiel

resource with set of players I from [1] R. Singh, A Games Nash eq. Wiszniewska-Matyszkiel, 2018, Linear Quadratic Game of Exploitation of Common Renewable Stackelberg eq. Resources with Inherent Constraints, Topological Methods in Nonlinear Analysis 51, 23–54. Pareto opt. Example-static § The set of states of the resource is R`. The tragedy § Discrete time, infinite horizon (first). Dynamic games § A simple model At each time moment, player i extracts amount si ě 0, Objectives these si, in common, constitute a static profile s. Bellman § Constraint: given state x, the decisions have to fulfil Revised sufficient condition Social optimum si P r0, cxs. Nash equilibrium § Each of the players has cost function Nash equilibria for n players Finite horizon truncation costps q “ fs ` 1 s2. i i 2 i Enforcing optimality § The catch is sold at a common market at a price Carrying capacity

pricepsq “ A ´ u, where u is the aggregate extraction of Numerics

s. Continuous time

Conclusions ”The tragedy of the ”The tragedy” in a simple model – the commons” in the dynamic context motivating example revisited Agnieszka Wiszniewska- § We consider a exploitation of a common renewable Matyszkiel

resource with set of players I from [1] R. Singh, A Games Nash eq. Wiszniewska-Matyszkiel, 2018, Linear Quadratic Game of Exploitation of Common Renewable Stackelberg eq. Resources with Inherent Constraints, Topological Methods in Nonlinear Analysis 51, 23–54. Pareto opt. Example-static § The set of states of the resource is R`. The tragedy § Discrete time, infinite horizon (first). Dynamic games § A simple model At each time moment, player i extracts amount si ě 0, Objectives these si, in common, constitute a static profile s. Bellman § Constraint: given state x, the decisions have to fulfil Revised sufficient condition Social optimum si P r0, cxs. Nash equilibrium § Each of the players has cost function Nash equilibria for n players Finite horizon truncation costps q “ fs ` 1 s2. i i 2 i Enforcing optimality § The catch is sold at a common market at a price Carrying capacity

pricepsq “ A ´ u, where u is the aggregate extraction of Numerics

s. Continuous time

§ Aggregate extraction influences also the state of the Conclusions resource. § but decomposing the decision making structure (the whole world – ... – continents – countries – ... – firms).

”The tragedy of the The model – cont. commons” in the dynamic context

Agnieszka Wiszniewska- Matyszkiel § Increasing number of players does not mean Games introducing additional users, Nash eq. Stackelberg eq. Pareto opt. Example-static The tragedy

Dynamic games

A simple model Objectives

Bellman Revised sufficient condition

Social optimum

Nash equilibrium Nash equilibria for n players Finite horizon truncation

Enforcing optimality

Carrying capacity

Numerics

Continuous time

Conclusions (the whole world – ... – continents – countries – ... – firms).

”The tragedy of the The model – cont. commons” in the dynamic context

Agnieszka Wiszniewska- Matyszkiel § Increasing number of players does not mean Games introducing additional users, Nash eq. Stackelberg eq. § Pareto opt. but decomposing the decision making structure Example-static The tragedy

Dynamic games

A simple model Objectives

Bellman Revised sufficient condition

Social optimum

Nash equilibrium Nash equilibria for n players Finite horizon truncation

Enforcing optimality

Carrying capacity

Numerics

Continuous time

Conclusions ”The tragedy of the The model – cont. commons” in the dynamic context

Agnieszka Wiszniewska- Matyszkiel § Increasing number of players does not mean Games introducing additional users, Nash eq. Stackelberg eq. § Pareto opt. but decomposing the decision making structure (the Example-static whole world – ... – continents – countries – ... – firms). The tragedy Dynamic games

A simple model Objectives

Bellman Revised sufficient condition

Social optimum

Nash equilibrium Nash equilibria for n players Finite horizon truncation

Enforcing optimality

Carrying capacity

Numerics

Continuous time

Conclusions ”The tragedy of the The model – cont. commons” in the dynamic context

Agnieszka Wiszniewska- Matyszkiel § Increasing number of players does not mean Games introducing additional users, Nash eq. Stackelberg eq. § Pareto opt. but decomposing the decision making structure (the Example-static whole world – ... – continents – countries – ... – firms). The tragedy Dynamic games

A simple model Objectives

Bellman Revised sufficient condition

Social optimum

Nash equilibrium Nash equilibria for n players Finite horizon truncation

Enforcing optimality

Carrying capacity

Numerics

Continuous time

Conclusions ”The tragedy of the The model – cont. commons” in the dynamic context

Agnieszka Wiszniewska- Matyszkiel § Increasing number of players does not mean Games introducing additional users, Nash eq. Stackelberg eq. § Pareto opt. but decomposing the decision making structure (the Example-static whole world – ... – continents – countries – ... – firms). The tragedy Dynamic games

A simple model Objectives

Bellman Revised sufficient condition

Social optimum

Nash equilibrium Nash equilibria for n players Finite horizon truncation

Enforcing optimality

Carrying capacity

Numerics

Continuous time

Conclusions ”The tragedy of the The model – cont. commons” in the dynamic context

Agnieszka Wiszniewska- Matyszkiel § Increasing number of players does not mean Games introducing additional users, Nash eq. Stackelberg eq. § Pareto opt. but decomposing the decision making structure (the Example-static whole world – ... – continents – countries – ... – firms). The tragedy Dynamic games

A simple model Objectives

Bellman Revised sufficient condition

Social optimum

Nash equilibrium Nash equilibria for n players Finite horizon truncation

Enforcing optimality

Carrying capacity

Numerics

Continuous time

Conclusions ”The tragedy of the The model – cont. commons” in the dynamic context

Agnieszka Wiszniewska- Matyszkiel § Increasing number of players does not mean Games introducing additional users, Nash eq. Stackelberg eq. § Pareto opt. but decomposing the decision making structure (the Example-static whole world – ... – continents – countries – ... – firms). The tragedy Dynamic games

A simple model Objectives

Bellman Revised sufficient condition

Social optimum

Nash equilibrium Nash equilibria for n players Finite horizon truncation

Enforcing optimality

Carrying capacity

Numerics

Continuous time

Conclusions § and players are measured by the uniform normalized measure on each I. § Consequently, the aggregate extraction n si u “ i“1 n in the case of n players and u “ r0,1s sidλpiq (λ means the Lebesgue measure) in theř case of continuum of players. ş § Rule of growth with fishing Xipt ` 1q “ p1 ` ξqXptq ´ Uptq (generalized later). § To make depletion possible, we set c “ p1 ` ξq. § 1 Discounting by a discount factor β “ 1`ξ (the golden rule).

§ Current payoff of player i: Pipsq “ pricepsqsi ´ costpsiq (auxiliary notation Ppsi, uq or Ppsi, s„iq). § So we have a linear-quadratic dynamic game with linear state-dependent constraints on controls.

”The tragedy of the The model – cont.2 commons” in the dynamic context § To model this, the set of players I is t1,..., nu or r0, 1s Agnieszka Wiszniewska- Matyszkiel

Games Nash eq. Stackelberg eq. Pareto opt. Example-static The tragedy

Dynamic games

A simple model Objectives

Bellman Revised sufficient condition

Social optimum

Nash equilibrium Nash equilibria for n players Finite horizon truncation

Enforcing optimality

Carrying capacity

Numerics

Continuous time

Conclusions § Consequently, the aggregate extraction n si u “ i“1 n in the case of n players and u “ r0,1s sidλpiq (λ means the Lebesgue measure) in theř case of continuum of players. ş § Rule of growth with fishing Xipt ` 1q “ p1 ` ξqXptq ´ Uptq (generalized later). § To make depletion possible, we set c “ p1 ` ξq. § 1 Discounting by a discount factor β “ 1`ξ (the golden rule).

§ Current payoff of player i: Pipsq “ pricepsqsi ´ costpsiq (auxiliary notation Ppsi, uq or Ppsi, s„iq). § So we have a linear-quadratic dynamic game with linear state-dependent constraints on controls.

”The tragedy of the The model – cont.2 commons” in the dynamic context § To model this, the set of players I is t1,..., nu or r0, 1s Agnieszka Wiszniewska- § and players are measured by the uniform normalized Matyszkiel

measure on each I. Games Nash eq. Stackelberg eq. Pareto opt. Example-static The tragedy

Dynamic games

A simple model Objectives

Bellman Revised sufficient condition

Social optimum

Nash equilibrium Nash equilibria for n players Finite horizon truncation

Enforcing optimality

Carrying capacity

Numerics

Continuous time

Conclusions and u “ r0,1s sidλpiq (λ means the Lebesgue measure) in the case of continuum of players. ş § Rule of growth with fishing Xipt ` 1q “ p1 ` ξqXptq ´ Uptq (generalized later). § To make depletion possible, we set c “ p1 ` ξq. § 1 Discounting by a discount factor β “ 1`ξ (the golden rule).

§ Current payoff of player i: Pipsq “ pricepsqsi ´ costpsiq (auxiliary notation Ppsi, uq or Ppsi, s„iq). § So we have a linear-quadratic dynamic game with linear state-dependent constraints on controls.

”The tragedy of the The model – cont.2 commons” in the dynamic context § To model this, the set of players I is t1,..., nu or r0, 1s Agnieszka Wiszniewska- § and players are measured by the uniform normalized Matyszkiel

measure on each I. Games Nash eq. § Consequently, the aggregate extraction Stackelberg eq. n si Pareto opt. u “ i“1 n in the case of n players Example-static The tragedy ř Dynamic games A simple model Objectives

Bellman Revised sufficient condition

Social optimum

Nash equilibrium Nash equilibria for n players Finite horizon truncation

Enforcing optimality

Carrying capacity

Numerics

Continuous time

Conclusions § Rule of growth with fishing Xipt ` 1q “ p1 ` ξqXptq ´ Uptq (generalized later). § To make depletion possible, we set c “ p1 ` ξq. § 1 Discounting by a discount factor β “ 1`ξ (the golden rule).

§ Current payoff of player i: Pipsq “ pricepsqsi ´ costpsiq (auxiliary notation Ppsi, uq or Ppsi, s„iq). § So we have a linear-quadratic dynamic game with linear state-dependent constraints on controls.

”The tragedy of the The model – cont.2 commons” in the dynamic context § To model this, the set of players I is t1,..., nu or r0, 1s Agnieszka Wiszniewska- § and players are measured by the uniform normalized Matyszkiel

measure on each I. Games Nash eq. § Consequently, the aggregate extraction Stackelberg eq. n si Pareto opt. u “ i“1 n in the case of n players Example-static The tragedy and u “ sidλpiq (λ means the Lebesgue measure) r0,1s Dynamic games in theř case of continuum of players. ş A simple model Objectives

Bellman Revised sufficient condition

Social optimum

Nash equilibrium Nash equilibria for n players Finite horizon truncation

Enforcing optimality

Carrying capacity

Numerics

Continuous time

Conclusions § To make depletion possible, we set c “ p1 ` ξq. § 1 Discounting by a discount factor β “ 1`ξ (the golden rule).

§ Current payoff of player i: Pipsq “ pricepsqsi ´ costpsiq (auxiliary notation Ppsi, uq or Ppsi, s„iq). § So we have a linear-quadratic dynamic game with linear state-dependent constraints on controls.

”The tragedy of the The model – cont.2 commons” in the dynamic context § To model this, the set of players I is t1,..., nu or r0, 1s Agnieszka Wiszniewska- § and players are measured by the uniform normalized Matyszkiel

measure on each I. Games Nash eq. § Consequently, the aggregate extraction Stackelberg eq. n si Pareto opt. u “ i“1 n in the case of n players Example-static The tragedy and u “ sidλpiq (λ means the Lebesgue measure) r0,1s Dynamic games in theř case of continuum of players. ş A simple model § Rule of growth with fishing Objectives Bellman Xipt ` 1q “ p1 ` ξqXptq ´ Uptq (generalized later). Revised sufficient condition

Social optimum

Nash equilibrium Nash equilibria for n players Finite horizon truncation

Enforcing optimality

Carrying capacity

Numerics

Continuous time

Conclusions § 1 Discounting by a discount factor β “ 1`ξ (the golden rule).

§ Current payoff of player i: Pipsq “ pricepsqsi ´ costpsiq (auxiliary notation Ppsi, uq or Ppsi, s„iq). § So we have a linear-quadratic dynamic game with linear state-dependent constraints on controls.

”The tragedy of the The model – cont.2 commons” in the dynamic context § To model this, the set of players I is t1,..., nu or r0, 1s Agnieszka Wiszniewska- § and players are measured by the uniform normalized Matyszkiel

measure on each I. Games Nash eq. § Consequently, the aggregate extraction Stackelberg eq. n si Pareto opt. u “ i“1 n in the case of n players Example-static The tragedy and u “ sidλpiq (λ means the Lebesgue measure) r0,1s Dynamic games in theř case of continuum of players. ş A simple model § Rule of growth with fishing Objectives Bellman Xipt ` 1q “ p1 ` ξqXptq ´ Uptq (generalized later). Revised sufficient condition § To make depletion possible, we set c “ p1 ` ξq. Social optimum Nash equilibrium Nash equilibria for n players Finite horizon truncation

Enforcing optimality

Carrying capacity

Numerics

Continuous time

Conclusions § Current payoff of player i: Pipsq “ pricepsqsi ´ costpsiq (auxiliary notation Ppsi, uq or Ppsi, s„iq). § So we have a linear-quadratic dynamic game with linear state-dependent constraints on controls.

”The tragedy of the The model – cont.2 commons” in the dynamic context § To model this, the set of players I is t1,..., nu or r0, 1s Agnieszka Wiszniewska- § and players are measured by the uniform normalized Matyszkiel

measure on each I. Games Nash eq. § Consequently, the aggregate extraction Stackelberg eq. n si Pareto opt. u “ i“1 n in the case of n players Example-static The tragedy and u “ sidλpiq (λ means the Lebesgue measure) r0,1s Dynamic games in theř case of continuum of players. ş A simple model § Rule of growth with fishing Objectives Bellman Xipt ` 1q “ p1 ` ξqXptq ´ Uptq (generalized later). Revised sufficient condition § To make depletion possible, we set c “ p1 ` ξq. Social optimum § Discounting by a discount factor β 1 (the golden Nash equilibrium “ 1`ξ Nash equilibria for n players rule). Finite horizon truncation Enforcing optimality

Carrying capacity

Numerics

Continuous time

Conclusions (auxiliary notation Ppsi, uq or Ppsi, s„iq). § So we have a linear-quadratic dynamic game with linear state-dependent constraints on controls.

”The tragedy of the The model – cont.2 commons” in the dynamic context § To model this, the set of players I is t1,..., nu or r0, 1s Agnieszka Wiszniewska- § and players are measured by the uniform normalized Matyszkiel

measure on each I. Games Nash eq. § Consequently, the aggregate extraction Stackelberg eq. n si Pareto opt. u “ i“1 n in the case of n players Example-static The tragedy and u “ sidλpiq (λ means the Lebesgue measure) r0,1s Dynamic games in theř case of continuum of players. ş A simple model § Rule of growth with fishing Objectives Bellman Xipt ` 1q “ p1 ` ξqXptq ´ Uptq (generalized later). Revised sufficient condition § To make depletion possible, we set c “ p1 ` ξq. Social optimum § Discounting by a discount factor β 1 (the golden Nash equilibrium “ 1`ξ Nash equilibria for n players rule). Finite horizon truncation Enforcing optimality § Current payoff of player i: P s price s s cost s ip q “ p q i ´ p iq Carrying capacity

Numerics

Continuous time

Conclusions § So we have a linear-quadratic dynamic game with linear state-dependent constraints on controls.

”The tragedy of the The model – cont.2 commons” in the dynamic context § To model this, the set of players I is t1,..., nu or r0, 1s Agnieszka Wiszniewska- § and players are measured by the uniform normalized Matyszkiel

measure on each I. Games Nash eq. § Consequently, the aggregate extraction Stackelberg eq. n si Pareto opt. u “ i“1 n in the case of n players Example-static The tragedy and u “ sidλpiq (λ means the Lebesgue measure) r0,1s Dynamic games in theř case of continuum of players. ş A simple model § Rule of growth with fishing Objectives Bellman Xipt ` 1q “ p1 ` ξqXptq ´ Uptq (generalized later). Revised sufficient condition § To make depletion possible, we set c “ p1 ` ξq. Social optimum § Discounting by a discount factor β 1 (the golden Nash equilibrium “ 1`ξ Nash equilibria for n players rule). Finite horizon truncation Enforcing optimality § Current payoff of player i: P s price s s cost s ip q “ p q i ´ p iq Carrying capacity

(auxiliary notation Ppsi, uq or Ppsi, s„iq). Numerics

Continuous time

Conclusions ”The tragedy of the The model – cont.2 commons” in the dynamic context § To model this, the set of players I is t1,..., nu or r0, 1s Agnieszka Wiszniewska- § and players are measured by the uniform normalized Matyszkiel

measure on each I. Games Nash eq. § Consequently, the aggregate extraction Stackelberg eq. n si Pareto opt. u “ i“1 n in the case of n players Example-static The tragedy and u “ sidλpiq (λ means the Lebesgue measure) r0,1s Dynamic games in theř case of continuum of players. ş A simple model § Rule of growth with fishing Objectives Bellman Xipt ` 1q “ p1 ` ξqXptq ´ Uptq (generalized later). Revised sufficient condition § To make depletion possible, we set c “ p1 ` ξq. Social optimum § Discounting by a discount factor β 1 (the golden Nash equilibrium “ 1`ξ Nash equilibria for n players rule). Finite horizon truncation Enforcing optimality § Current payoff of player i: P s price s s cost s ip q “ p q i ´ p iq Carrying capacity

(auxiliary notation Ppsi, uq or Ppsi, s„iq). Numerics § So we have a linear-quadratic dynamic game with linear Continuous time state-dependent constraints on controls. Conclusions § The objective—payoff function of player i is 8 t JipSq “ PipSpXptqqqβ (for feedback controls). t“0 § We wantř to calculate the social optima,

§ i.e. profiles which maximize aggregate payoff; § and Nash equilibria, § i.e. profiles at which each player maximizes their payoff given strategies of remaining players. § Calculation of both require solving dynamic optimization problems. § In the case of Nash equilibrium, a set of dynamic optimization problems coupled by finding a fixed point in the space of feedback strategy profiles.

”The tragedy of the Objectives commons” in the dynamic context § We consider feedback strategies – choices of decisions Agnieszka Wiszniewska- as functions of state, Sipxq. Matyszkiel

Games Nash eq. Stackelberg eq. Pareto opt. Example-static The tragedy

Dynamic games

A simple model Objectives

Bellman Revised sufficient condition

Social optimum

Nash equilibrium Nash equilibria for n players Finite horizon truncation

Enforcing optimality

Carrying capacity

Numerics

Continuous time

Conclusions § We want to calculate the social optima,

§ i.e. profiles which maximize aggregate payoff; § and Nash equilibria, § i.e. profiles at which each player maximizes their payoff given strategies of remaining players. § Calculation of both require solving dynamic optimization problems. § In the case of Nash equilibrium, a set of dynamic optimization problems coupled by finding a fixed point in the space of feedback strategy profiles.

”The tragedy of the Objectives commons” in the dynamic context § We consider feedback strategies – choices of decisions Agnieszka Wiszniewska- as functions of state, Sipxq. Matyszkiel

§ The objective—payoff function of player i is Games 8 Nash eq. t Stackelberg eq. JipSq “ PipSpXptqqqβ (for feedback controls). Pareto opt. t“0 Example-static ř The tragedy Dynamic games

A simple model Objectives

Bellman Revised sufficient condition

Social optimum

Nash equilibrium Nash equilibria for n players Finite horizon truncation

Enforcing optimality

Carrying capacity

Numerics

Continuous time

Conclusions § i.e. profiles which maximize aggregate payoff; § and Nash equilibria, § i.e. profiles at which each player maximizes their payoff given strategies of remaining players. § Calculation of both require solving dynamic optimization problems. § In the case of Nash equilibrium, a set of dynamic optimization problems coupled by finding a fixed point in the space of feedback strategy profiles.

”The tragedy of the Objectives commons” in the dynamic context § We consider feedback strategies – choices of decisions Agnieszka Wiszniewska- as functions of state, Sipxq. Matyszkiel

§ The objective—payoff function of player i is Games 8 Nash eq. t Stackelberg eq. JipSq “ PipSpXptqqqβ (for feedback controls). Pareto opt. t“0 Example-static § We wantř to calculate the social optima, The tragedy Dynamic games

A simple model Objectives

Bellman Revised sufficient condition

Social optimum

Nash equilibrium Nash equilibria for n players Finite horizon truncation

Enforcing optimality

Carrying capacity

Numerics

Continuous time

Conclusions § and Nash equilibria, § i.e. profiles at which each player maximizes their payoff given strategies of remaining players. § Calculation of both require solving dynamic optimization problems. § In the case of Nash equilibrium, a set of dynamic optimization problems coupled by finding a fixed point in the space of feedback strategy profiles.

”The tragedy of the Objectives commons” in the dynamic context § We consider feedback strategies – choices of decisions Agnieszka Wiszniewska- as functions of state, Sipxq. Matyszkiel

§ The objective—payoff function of player i is Games 8 Nash eq. t Stackelberg eq. JipSq “ PipSpXptqqqβ (for feedback controls). Pareto opt. t“0 Example-static § We wantř to calculate the social optima, The tragedy Dynamic games § i.e. profiles which maximize aggregate payoff; A simple model Objectives

Bellman Revised sufficient condition

Social optimum

Nash equilibrium Nash equilibria for n players Finite horizon truncation

Enforcing optimality

Carrying capacity

Numerics

Continuous time

Conclusions § Calculation of both require solving dynamic optimization problems. § In the case of Nash equilibrium, a set of dynamic optimization problems coupled by finding a fixed point in the space of feedback strategy profiles.

”The tragedy of the Objectives commons” in the dynamic context § We consider feedback strategies – choices of decisions Agnieszka Wiszniewska- as functions of state, Sipxq. Matyszkiel

§ The objective—payoff function of player i is Games 8 Nash eq. t Stackelberg eq. JipSq “ PipSpXptqqqβ (for feedback controls). Pareto opt. t“0 Example-static § We wantř to calculate the social optima, The tragedy Dynamic games § i.e. profiles which maximize aggregate payoff; A simple model Objectives § and Nash equilibria, Bellman § i.e. profiles at which each player maximizes their payoff Revised sufficient condition given strategies of remaining players. Social optimum Nash equilibrium Nash equilibria for n players Finite horizon truncation

Enforcing optimality

Carrying capacity

Numerics

Continuous time

Conclusions § In the case of Nash equilibrium, a set of dynamic optimization problems coupled by finding a fixed point in the space of feedback strategy profiles.

”The tragedy of the Objectives commons” in the dynamic context § We consider feedback strategies – choices of decisions Agnieszka Wiszniewska- as functions of state, Sipxq. Matyszkiel

§ The objective—payoff function of player i is Games 8 Nash eq. t Stackelberg eq. JipSq “ PipSpXptqqqβ (for feedback controls). Pareto opt. t“0 Example-static § We wantř to calculate the social optima, The tragedy Dynamic games § i.e. profiles which maximize aggregate payoff; A simple model Objectives § and Nash equilibria, Bellman § i.e. profiles at which each player maximizes their payoff Revised sufficient condition given strategies of remaining players. Social optimum Nash equilibrium § Calculation of both require solving dynamic optimization Nash equilibria for n players problems. Finite horizon truncation Enforcing optimality

Carrying capacity

Numerics

Continuous time

Conclusions coupled by finding a fixed point in the space of feedback strategy profiles.

”The tragedy of the Objectives commons” in the dynamic context § We consider feedback strategies – choices of decisions Agnieszka Wiszniewska- as functions of state, Sipxq. Matyszkiel

§ The objective—payoff function of player i is Games 8 Nash eq. t Stackelberg eq. JipSq “ PipSpXptqqqβ (for feedback controls). Pareto opt. t“0 Example-static § We wantř to calculate the social optima, The tragedy Dynamic games § i.e. profiles which maximize aggregate payoff; A simple model Objectives § and Nash equilibria, Bellman § i.e. profiles at which each player maximizes their payoff Revised sufficient condition given strategies of remaining players. Social optimum Nash equilibrium § Calculation of both require solving dynamic optimization Nash equilibria for n players problems. Finite horizon truncation Enforcing optimality § In the case of Nash equilibrium, a set of dynamic Carrying capacity

optimization problems Numerics

Continuous time

Conclusions ”The tragedy of the Objectives commons” in the dynamic context § We consider feedback strategies – choices of decisions Agnieszka Wiszniewska- as functions of state, Sipxq. Matyszkiel

§ The objective—payoff function of player i is Games 8 Nash eq. t Stackelberg eq. JipSq “ PipSpXptqqqβ (for feedback controls). Pareto opt. t“0 Example-static § We wantř to calculate the social optima, The tragedy Dynamic games § i.e. profiles which maximize aggregate payoff; A simple model Objectives § and Nash equilibria, Bellman § i.e. profiles at which each player maximizes their payoff Revised sufficient condition given strategies of remaining players. Social optimum Nash equilibrium § Calculation of both require solving dynamic optimization Nash equilibria for n players problems. Finite horizon truncation Enforcing optimality § In the case of Nash equilibrium, a set of dynamic Carrying capacity

optimization problems Numerics

coupled by finding a fixed point in the space of feedback Continuous time

strategy profiles. Conclusions § for X defined by Xpt ` 1q “ fpXptq, UpXptq, tq, tq with initial condition Xp¯tq “ x¯. § We assume Jp¯t, x¯, Uq is always well defined, although it can be ´8.

§ If a function V : X ˆ N Ñ R fulfils the Bellman equation:

§ (BE) Vpx, tq “ sup gpx, u, tq ` δ Vpfpx, u, tq, t ` 1q with uPU the terminal condition: § (TC) for every trajectory X, lim sup VpXptq, tq δt “ 0 tÑ8 § then V is the value function of the dynamic optimization problem,while any selection from the Argmax of the rhs. of the (BE) is an optimal control.

”The tragedy of the Standard infinite horizon Bellman sufficient commons” in the dynamic context condition Agnieszka Wiszniewska- § For a dynamic optimization problem Matyszkiel 8 ¯ Games § ¯ t´t maximize Jpt, x¯, Uq “ gpXptq, UpXptq, tq, tqδ , Nash eq. t“¯t Stackelberg eq. Pareto opt. ř Example-static The tragedy

Dynamic games

A simple model Objectives

Bellman Revised sufficient condition

Social optimum

Nash equilibrium Nash equilibria for n players Finite horizon truncation

Enforcing optimality

Carrying capacity

Numerics

Continuous time

Conclusions with initial condition Xp¯tq “ x¯. § We assume Jp¯t, x¯, Uq is always well defined, although it can be ´8.

§ If a function V : X ˆ N Ñ R fulfils the Bellman equation:

§ (BE) Vpx, tq “ sup gpx, u, tq ` δ Vpfpx, u, tq, t ` 1q with uPU the terminal condition: § (TC) for every trajectory X, lim sup VpXptq, tq δt “ 0 tÑ8 § then V is the value function of the dynamic optimization problem,while any selection from the Argmax of the rhs. of the (BE) is an optimal control.

”The tragedy of the Standard infinite horizon Bellman sufficient commons” in the dynamic context condition Agnieszka Wiszniewska- § For a dynamic optimization problem Matyszkiel 8 ¯ Games § ¯ t´t maximize Jpt, x¯, Uq “ gpXptq, UpXptq, tq, tqδ , Nash eq. t“¯t Stackelberg eq. § Pareto opt. for X defined by Xpt `ř1q “ fpXptq, UpXptq, tq, tq Example-static The tragedy

Dynamic games

A simple model Objectives

Bellman Revised sufficient condition

Social optimum

Nash equilibrium Nash equilibria for n players Finite horizon truncation

Enforcing optimality

Carrying capacity

Numerics

Continuous time

Conclusions § We assume Jp¯t, x¯, Uq is always well defined, although it can be ´8.

§ If a function V : X ˆ N Ñ R fulfils the Bellman equation:

§ (BE) Vpx, tq “ sup gpx, u, tq ` δ Vpfpx, u, tq, t ` 1q with uPU the terminal condition: § (TC) for every trajectory X, lim sup VpXptq, tq δt “ 0 tÑ8 § then V is the value function of the dynamic optimization problem,while any selection from the Argmax of the rhs. of the (BE) is an optimal control.

”The tragedy of the Standard infinite horizon Bellman sufficient commons” in the dynamic context condition Agnieszka Wiszniewska- § For a dynamic optimization problem Matyszkiel 8 ¯ Games § ¯ t´t maximize Jpt, x¯, Uq “ gpXptq, UpXptq, tq, tqδ , Nash eq. t“¯t Stackelberg eq. § Pareto opt. for X defined by Xpt `ř1q “ fpXptq, UpXptq, tq, tq with Example-static initial condition Xp¯tq “ x¯. The tragedy Dynamic games

A simple model Objectives

Bellman Revised sufficient condition

Social optimum

Nash equilibrium Nash equilibria for n players Finite horizon truncation

Enforcing optimality

Carrying capacity

Numerics

Continuous time

Conclusions § If a function V : X ˆ N Ñ R fulfils the Bellman equation:

§ (BE) Vpx, tq “ sup gpx, u, tq ` δ Vpfpx, u, tq, t ` 1q with uPU the terminal condition: § (TC) for every trajectory X, lim sup VpXptq, tq δt “ 0 tÑ8 § then V is the value function of the dynamic optimization problem,while any selection from the Argmax of the rhs. of the (BE) is an optimal control.

”The tragedy of the Standard infinite horizon Bellman sufficient commons” in the dynamic context condition Agnieszka Wiszniewska- § For a dynamic optimization problem Matyszkiel 8 ¯ Games § ¯ t´t maximize Jpt, x¯, Uq “ gpXptq, UpXptq, tq, tqδ , Nash eq. t“¯t Stackelberg eq. § Pareto opt. for X defined by Xpt `ř1q “ fpXptq, UpXptq, tq, tq with Example-static initial condition Xp¯tq “ x¯. The tragedy § We assume Jp¯t, x¯, Uq is always well defined, although it Dynamic games can be ´8. A simple model Objectives

Bellman Revised sufficient condition

Social optimum

Nash equilibrium Nash equilibria for n players Finite horizon truncation

Enforcing optimality

Carrying capacity

Numerics

Continuous time

Conclusions with the terminal condition: § (TC) for every trajectory X, lim sup VpXptq, tq δt “ 0 tÑ8 § then V is the value function of the dynamic optimization problem,while any selection from the Argmax of the rhs. of the (BE) is an optimal control.

”The tragedy of the Standard infinite horizon Bellman sufficient commons” in the dynamic context condition Agnieszka Wiszniewska- § For a dynamic optimization problem Matyszkiel 8 ¯ Games § ¯ t´t maximize Jpt, x¯, Uq “ gpXptq, UpXptq, tq, tqδ , Nash eq. t“¯t Stackelberg eq. § Pareto opt. for X defined by Xpt `ř1q “ fpXptq, UpXptq, tq, tq with Example-static initial condition Xp¯tq “ x¯. The tragedy § We assume Jp¯t, x¯, Uq is always well defined, although it Dynamic games can be ´8. A simple model Objectives

§ Bellman If a function V : X ˆ N Ñ R fulfils the Bellman equation: Revised sufficient condition

Social optimum

§ (BE) Vpx, tq “ sup gpx, u, tq ` δ Vpfpx, u, tq, t ` 1q Nash equilibrium uPU Nash equilibria for n players Finite horizon truncation

Enforcing optimality

Carrying capacity

Numerics

Continuous time

Conclusions § then V is the value function of the dynamic optimization problem,while any selection from the Argmax of the rhs. of the (BE) is an optimal control.

”The tragedy of the Standard infinite horizon Bellman sufficient commons” in the dynamic context condition Agnieszka Wiszniewska- § For a dynamic optimization problem Matyszkiel 8 ¯ Games § ¯ t´t maximize Jpt, x¯, Uq “ gpXptq, UpXptq, tq, tqδ , Nash eq. t“¯t Stackelberg eq. § Pareto opt. for X defined by Xpt `ř1q “ fpXptq, UpXptq, tq, tq with Example-static initial condition Xp¯tq “ x¯. The tragedy § We assume Jp¯t, x¯, Uq is always well defined, although it Dynamic games can be ´8. A simple model Objectives

§ Bellman If a function V : X ˆ N Ñ R fulfils the Bellman equation: Revised sufficient condition

Social optimum

§ (BE) Vpx, tq “ sup gpx, u, tq ` δ Vpfpx, u, tq, t ` 1q with Nash equilibrium uPU Nash equilibria for n players the terminal condition: Finite horizon truncation § (TC) for every trajectory X, lim sup VpXptq, tq δt “ 0 Enforcing optimality tÑ8 Carrying capacity

Numerics

Continuous time

Conclusions while any selection from the Argmax of the rhs. of the (BE) is an optimal control.

”The tragedy of the Standard infinite horizon Bellman sufficient commons” in the dynamic context condition Agnieszka Wiszniewska- § For a dynamic optimization problem Matyszkiel 8 ¯ Games § ¯ t´t maximize Jpt, x¯, Uq “ gpXptq, UpXptq, tq, tqδ , Nash eq. t“¯t Stackelberg eq. § Pareto opt. for X defined by Xpt `ř1q “ fpXptq, UpXptq, tq, tq with Example-static initial condition Xp¯tq “ x¯. The tragedy § We assume Jp¯t, x¯, Uq is always well defined, although it Dynamic games can be ´8. A simple model Objectives

§ Bellman If a function V : X ˆ N Ñ R fulfils the Bellman equation: Revised sufficient condition

Social optimum

§ (BE) Vpx, tq “ sup gpx, u, tq ` δ Vpfpx, u, tq, t ` 1q with Nash equilibrium uPU Nash equilibria for n players the terminal condition: Finite horizon truncation § (TC) for every trajectory X, lim sup VpXptq, tq δt “ 0 Enforcing optimality tÑ8 Carrying capacity § then V is the value function of the dynamic optimization Numerics

problem, Continuous time

Conclusions ”The tragedy of the Standard infinite horizon Bellman sufficient commons” in the dynamic context condition Agnieszka Wiszniewska- § For a dynamic optimization problem Matyszkiel 8 ¯ Games § ¯ t´t maximize Jpt, x¯, Uq “ gpXptq, UpXptq, tq, tqδ , Nash eq. t“¯t Stackelberg eq. § Pareto opt. for X defined by Xpt `ř1q “ fpXptq, UpXptq, tq, tq with Example-static initial condition Xp¯tq “ x¯. The tragedy § We assume Jp¯t, x¯, Uq is always well defined, although it Dynamic games can be ´8. A simple model Objectives

§ Bellman If a function V : X ˆ N Ñ R fulfils the Bellman equation: Revised sufficient condition

Social optimum

§ (BE) Vpx, tq “ sup gpx, u, tq ` δ Vpfpx, u, tq, t ` 1q with Nash equilibrium uPU Nash equilibria for n players the terminal condition: Finite horizon truncation § (TC) for every trajectory X, lim sup VpXptq, tq δt “ 0 Enforcing optimality tÑ8 Carrying capacity § then V is the value function of the dynamic optimization Numerics

problem,while any selection from the Argmax of the rhs. Continuous time

of the (BE) is an optimal control. Conclusions § (b) and if lim sup VpXptq, tq δt ă 0, then Jpt, x, Uq “ ´8 tÑ8 for every U such that trajectory X is corresponding to it.

[2] A. Wiszniewska-Matyszkiel, 2011, On the terminal condition for the Bellman equation for dynamic

optimization with an infinite horizon, Applied Mathematics Letters 24, 943–949.

§ (b) is necessary! and (a) is also necessary under very week condition

[3] A Wiszniewska-Matyszkiel, R. Singh, 2020, Necessity of the Terminal Condition in the Infinite

Horizon Dynamic Optimization Problems with Unbounded Payoff, Automatica,

https://doi.org/10.1016/j.automatica.2020.109332.

”The tragedy of the Revised sufficient condition commons” in the dynamic context

Agnieszka Wiszniewska- § (TC) can be replaced by Matyszkiel

§ (TC’) Games Nash eq. § (a) for every admissible trajectory X Stackelberg eq. lim sup VpXptq, tq δt ď 0 Pareto opt. Example-static t Ñ8 The tragedy

Dynamic games

A simple model Objectives

Bellman Revised sufficient condition

Social optimum

Nash equilibrium Nash equilibria for n players Finite horizon truncation

Enforcing optimality

Carrying capacity

Numerics

Continuous time

Conclusions then Jpt, x, Uq “ ´8 for every U such that trajectory X is corresponding to it.

[2] A. Wiszniewska-Matyszkiel, 2011, On the terminal condition for the Bellman equation for dynamic

optimization with an infinite horizon, Applied Mathematics Letters 24, 943–949.

§ (b) is necessary! and (a) is also necessary under very week condition

[3] A Wiszniewska-Matyszkiel, R. Singh, 2020, Necessity of the Terminal Condition in the Infinite

Horizon Dynamic Optimization Problems with Unbounded Payoff, Automatica,

https://doi.org/10.1016/j.automatica.2020.109332.

”The tragedy of the Revised sufficient condition commons” in the dynamic context

Agnieszka Wiszniewska- § (TC) can be replaced by Matyszkiel

§ (TC’) Games Nash eq. § (a) for every admissible trajectory X Stackelberg eq. lim sup VpXptq, tq δt ď 0 Pareto opt. Example-static t Ñ8 The tragedy § (b) and if lim sup VpXptq, tq δt ă 0, Dynamic games tÑ8 A simple model Objectives

Bellman Revised sufficient condition

Social optimum

Nash equilibrium Nash equilibria for n players Finite horizon truncation

Enforcing optimality

Carrying capacity

Numerics

Continuous time

Conclusions [2] A. Wiszniewska-Matyszkiel, 2011, On the terminal condition for the Bellman equation for dynamic

optimization with an infinite horizon, Applied Mathematics Letters 24, 943–949.

§ (b) is necessary! and (a) is also necessary under very week condition

[3] A Wiszniewska-Matyszkiel, R. Singh, 2020, Necessity of the Terminal Condition in the Infinite

Horizon Dynamic Optimization Problems with Unbounded Payoff, Automatica,

https://doi.org/10.1016/j.automatica.2020.109332.

”The tragedy of the Revised sufficient condition commons” in the dynamic context

Agnieszka Wiszniewska- § (TC) can be replaced by Matyszkiel

§ (TC’) Games Nash eq. § (a) for every admissible trajectory X Stackelberg eq. lim sup VpXptq, tq δt ď 0 Pareto opt. Example-static t Ñ8 The tragedy § (b) and if lim sup VpXptq, tq δt ă 0, then Jpt, x, Uq “ ´8 Dynamic games tÑ8 for every U such that trajectory X is corresponding to it. A simple model Objectives

Bellman Revised sufficient condition

Social optimum

Nash equilibrium Nash equilibria for n players Finite horizon truncation

Enforcing optimality

Carrying capacity

Numerics

Continuous time

Conclusions § (b) is necessary! and (a) is also necessary under very week condition

[3] A Wiszniewska-Matyszkiel, R. Singh, 2020, Necessity of the Terminal Condition in the Infinite

Horizon Dynamic Optimization Problems with Unbounded Payoff, Automatica,

https://doi.org/10.1016/j.automatica.2020.109332.

”The tragedy of the Revised sufficient condition commons” in the dynamic context

Agnieszka Wiszniewska- § (TC) can be replaced by Matyszkiel

§ (TC’) Games Nash eq. § (a) for every admissible trajectory X Stackelberg eq. lim sup VpXptq, tq δt ď 0 Pareto opt. Example-static t Ñ8 The tragedy § (b) and if lim sup VpXptq, tq δt ă 0, then Jpt, x, Uq “ ´8 Dynamic games tÑ8 for every U such that trajectory X is corresponding to it. A simple model Objectives

[2] A. Wiszniewska-Matyszkiel, 2011, On the terminal condition for the Bellman equation for dynamic Bellman Revised sufficient condition optimization with an infinite horizon, Applied Mathematics Letters 24, 943–949. Social optimum

Nash equilibrium Nash equilibria for n players Finite horizon truncation

Enforcing optimality

Carrying capacity

Numerics

Continuous time

Conclusions and (a) is also necessary under very week condition

[3] A Wiszniewska-Matyszkiel, R. Singh, 2020, Necessity of the Terminal Condition in the Infinite

Horizon Dynamic Optimization Problems with Unbounded Payoff, Automatica,

https://doi.org/10.1016/j.automatica.2020.109332.

”The tragedy of the Revised sufficient condition commons” in the dynamic context

Agnieszka Wiszniewska- § (TC) can be replaced by Matyszkiel

§ (TC’) Games Nash eq. § (a) for every admissible trajectory X Stackelberg eq. lim sup VpXptq, tq δt ď 0 Pareto opt. Example-static t Ñ8 The tragedy § (b) and if lim sup VpXptq, tq δt ă 0, then Jpt, x, Uq “ ´8 Dynamic games tÑ8 for every U such that trajectory X is corresponding to it. A simple model Objectives

[2] A. Wiszniewska-Matyszkiel, 2011, On the terminal condition for the Bellman equation for dynamic Bellman Revised sufficient condition optimization with an infinite horizon, Applied Mathematics Letters 24, 943–949. Social optimum § (b) is necessary! Nash equilibrium Nash equilibria for n players Finite horizon truncation

Enforcing optimality

Carrying capacity

Numerics

Continuous time

Conclusions [3] A Wiszniewska-Matyszkiel, R. Singh, 2020, Necessity of the Terminal Condition in the Infinite

Horizon Dynamic Optimization Problems with Unbounded Payoff, Automatica,

https://doi.org/10.1016/j.automatica.2020.109332.

”The tragedy of the Revised sufficient condition commons” in the dynamic context

Agnieszka Wiszniewska- § (TC) can be replaced by Matyszkiel

§ (TC’) Games Nash eq. § (a) for every admissible trajectory X Stackelberg eq. lim sup VpXptq, tq δt ď 0 Pareto opt. Example-static t Ñ8 The tragedy § (b) and if lim sup VpXptq, tq δt ă 0, then Jpt, x, Uq “ ´8 Dynamic games tÑ8 for every U such that trajectory X is corresponding to it. A simple model Objectives

[2] A. Wiszniewska-Matyszkiel, 2011, On the terminal condition for the Bellman equation for dynamic Bellman Revised sufficient condition optimization with an infinite horizon, Applied Mathematics Letters 24, 943–949. Social optimum § (b) is necessary! and (a) is also necessary under very Nash equilibrium Nash equilibria for n players week condition Finite horizon truncation Enforcing optimality

Carrying capacity

Numerics

Continuous time

Conclusions ”The tragedy of the Revised sufficient condition commons” in the dynamic context

Agnieszka Wiszniewska- § (TC) can be replaced by Matyszkiel

§ (TC’) Games Nash eq. § (a) for every admissible trajectory X Stackelberg eq. lim sup VpXptq, tq δt ď 0 Pareto opt. Example-static t Ñ8 The tragedy § (b) and if lim sup VpXptq, tq δt ă 0, then Jpt, x, Uq “ ´8 Dynamic games tÑ8 for every U such that trajectory X is corresponding to it. A simple model Objectives

[2] A. Wiszniewska-Matyszkiel, 2011, On the terminal condition for the Bellman equation for dynamic Bellman Revised sufficient condition optimization with an infinite horizon, Applied Mathematics Letters 24, 943–949. Social optimum § (b) is necessary! and (a) is also necessary under very Nash equilibrium Nash equilibria for n players week condition Finite horizon truncation

[3] A Wiszniewska-Matyszkiel, R. Singh, 2020, Necessity of the Terminal Condition in the Infinite Enforcing optimality Carrying capacity Horizon Dynamic Optimization Problems with Unbounded Payoff, Automatica, Numerics https://doi.org/10.1016/j.automatica.2020.109332. Continuous time

Conclusions § We solve the problem assuming quadratic value function Vpxq “ hx2 ` gx ` k (by undetermined coefficient method). § By considering the point of 0 derivative in rhs. of (BE), we obtain two possible h, negative or 0, then g (unique only for nonzero h) and, consequently, unique k. § Of all those solutions, only Vpxq “ hx2 ` gx with negative h solves (BE) on the whole domain! § For this V, the optimum of the rhs. of (BE) is ξx, § which results in constant state trajectory. § But Vpxq “ hx2 ` gx with negative h which solves (BE) (and it is the only quadratic solution of it) is not the value function.

”The tragedy of the Social optimum commons” in the dynamic context

Agnieszka Wiszniewska- § The solution is symmetric. Matyszkiel

Games Nash eq. Stackelberg eq. Pareto opt. Example-static The tragedy

Dynamic games

A simple model Objectives

Bellman Revised sufficient condition

Social optimum

Nash equilibrium Nash equilibria for n players Finite horizon truncation

Enforcing optimality

Carrying capacity

Numerics

Continuous time

Conclusions § By considering the point of 0 derivative in rhs. of (BE), we obtain two possible h, negative or 0, then g (unique only for nonzero h) and, consequently, unique k. § Of all those solutions, only Vpxq “ hx2 ` gx with negative h solves (BE) on the whole domain! § For this V, the optimum of the rhs. of (BE) is ξx, § which results in constant state trajectory. § But Vpxq “ hx2 ` gx with negative h which solves (BE) (and it is the only quadratic solution of it) is not the value function.

”The tragedy of the Social optimum commons” in the dynamic context

Agnieszka Wiszniewska- § The solution is symmetric. Matyszkiel § We solve the problem assuming quadratic value Games 2 function Vpxq “ hx ` gx ` k (by undetermined Nash eq. Stackelberg eq. coefficient method). Pareto opt. Example-static The tragedy

Dynamic games

A simple model Objectives

Bellman Revised sufficient condition

Social optimum

Nash equilibrium Nash equilibria for n players Finite horizon truncation

Enforcing optimality

Carrying capacity

Numerics

Continuous time

Conclusions § Of all those solutions, only Vpxq “ hx2 ` gx with negative h solves (BE) on the whole domain! § For this V, the optimum of the rhs. of (BE) is ξx, § which results in constant state trajectory. § But Vpxq “ hx2 ` gx with negative h which solves (BE) (and it is the only quadratic solution of it) is not the value function.

”The tragedy of the Social optimum commons” in the dynamic context

Agnieszka Wiszniewska- § The solution is symmetric. Matyszkiel § We solve the problem assuming quadratic value Games 2 function Vpxq “ hx ` gx ` k (by undetermined Nash eq. Stackelberg eq. coefficient method). Pareto opt. Example-static § By considering the point of 0 derivative in rhs. of (BE), The tragedy we obtain two possible h, negative or 0, then g (unique Dynamic games A simple model only for nonzero h) and, consequently, unique k. Objectives Bellman Revised sufficient condition

Social optimum

Nash equilibrium Nash equilibria for n players Finite horizon truncation

Enforcing optimality

Carrying capacity

Numerics

Continuous time

Conclusions § For this V, the optimum of the rhs. of (BE) is ξx, § which results in constant state trajectory. § But Vpxq “ hx2 ` gx with negative h which solves (BE) (and it is the only quadratic solution of it) is not the value function.

”The tragedy of the Social optimum commons” in the dynamic context

Agnieszka Wiszniewska- § The solution is symmetric. Matyszkiel § We solve the problem assuming quadratic value Games 2 function Vpxq “ hx ` gx ` k (by undetermined Nash eq. Stackelberg eq. coefficient method). Pareto opt. Example-static § By considering the point of 0 derivative in rhs. of (BE), The tragedy we obtain two possible h, negative or 0, then g (unique Dynamic games A simple model only for nonzero h) and, consequently, unique k. Objectives § 2 Bellman Of all those solutions, only Vpxq “ hx ` gx with Revised sufficient condition

negative h solves (BE) on the whole domain! Social optimum

Nash equilibrium Nash equilibria for n players Finite horizon truncation

Enforcing optimality

Carrying capacity

Numerics

Continuous time

Conclusions § which results in constant state trajectory. § But Vpxq “ hx2 ` gx with negative h which solves (BE) (and it is the only quadratic solution of it) is not the value function.

”The tragedy of the Social optimum commons” in the dynamic context

Agnieszka Wiszniewska- § The solution is symmetric. Matyszkiel § We solve the problem assuming quadratic value Games 2 function Vpxq “ hx ` gx ` k (by undetermined Nash eq. Stackelberg eq. coefficient method). Pareto opt. Example-static § By considering the point of 0 derivative in rhs. of (BE), The tragedy we obtain two possible h, negative or 0, then g (unique Dynamic games A simple model only for nonzero h) and, consequently, unique k. Objectives § 2 Bellman Of all those solutions, only Vpxq “ hx ` gx with Revised sufficient condition

negative h solves (BE) on the whole domain! Social optimum § For this V, the optimum of the rhs. of (BE) is ξx, Nash equilibrium Nash equilibria for n players Finite horizon truncation

Enforcing optimality

Carrying capacity

Numerics

Continuous time

Conclusions § But Vpxq “ hx2 ` gx with negative h which solves (BE) (and it is the only quadratic solution of it) is not the value function.

”The tragedy of the Social optimum commons” in the dynamic context

Agnieszka Wiszniewska- § The solution is symmetric. Matyszkiel § We solve the problem assuming quadratic value Games 2 function Vpxq “ hx ` gx ` k (by undetermined Nash eq. Stackelberg eq. coefficient method). Pareto opt. Example-static § By considering the point of 0 derivative in rhs. of (BE), The tragedy we obtain two possible h, negative or 0, then g (unique Dynamic games A simple model only for nonzero h) and, consequently, unique k. Objectives § 2 Bellman Of all those solutions, only Vpxq “ hx ` gx with Revised sufficient condition

negative h solves (BE) on the whole domain! Social optimum § For this V, the optimum of the rhs. of (BE) is ξx, Nash equilibrium Nash equilibria for n players § which results in constant state trajectory. Finite horizon truncation Enforcing optimality

Carrying capacity

Numerics

Continuous time

Conclusions ”The tragedy of the Social optimum commons” in the dynamic context

Agnieszka Wiszniewska- § The solution is symmetric. Matyszkiel § We solve the problem assuming quadratic value Games 2 function Vpxq “ hx ` gx ` k (by undetermined Nash eq. Stackelberg eq. coefficient method). Pareto opt. Example-static § By considering the point of 0 derivative in rhs. of (BE), The tragedy we obtain two possible h, negative or 0, then g (unique Dynamic games A simple model only for nonzero h) and, consequently, unique k. Objectives § 2 Bellman Of all those solutions, only Vpxq “ hx ` gx with Revised sufficient condition

negative h solves (BE) on the whole domain! Social optimum § For this V, the optimum of the rhs. of (BE) is ξx, Nash equilibrium Nash equilibria for n players § which results in constant state trajectory. Finite horizon truncation Enforcing optimality § 2 But Vpxq “ hx ` gx with negative h which solves (BE) Carrying capacity

(and it is the only quadratic solution of it) is not the Numerics

value function. Continuous time

Conclusions § It holds also for n “ 1, i.e. simple dynamic optimization problem. § Of course, (TC’) is not fulfilled. § The Bellman equation, if we neglect constraints, has also continuum of linear solutions, gx ` kˆ for arbitrary g. § The solution corresponding to the quadratic V is ξx. It guarantees sustainability – so it is not enough to check (TC) along the trajectory corresponding to maximizer of rhs of (BE), as it is sometimes done. § The solutions with nonzero g also violate (TC’). § g “ 0 does not solve (BE) for small x. § There is also a solution with the only piecewise quadratic V that fulfils both (BE) and (TC).

”The tragedy of the Counterexample!!! commons” in the dynamic context

Agnieszka § The only quadratic solution of (BE) is not the value Wiszniewska- function! Matyszkiel

Games Nash eq. Stackelberg eq. Pareto opt. Example-static The tragedy

Dynamic games

A simple model Objectives

Bellman Revised sufficient condition

Social optimum

Nash equilibrium Nash equilibria for n players Finite horizon truncation

Enforcing optimality

Carrying capacity

Numerics

Continuous time

Conclusions § Of course, (TC’) is not fulfilled. § The Bellman equation, if we neglect constraints, has also continuum of linear solutions, gx ` kˆ for arbitrary g. § The solution corresponding to the quadratic V is ξx. It guarantees sustainability – so it is not enough to check (TC) along the trajectory corresponding to maximizer of rhs of (BE), as it is sometimes done. § The solutions with nonzero g also violate (TC’). § g “ 0 does not solve (BE) for small x. § There is also a solution with the only piecewise quadratic V that fulfils both (BE) and (TC).

”The tragedy of the Counterexample!!! commons” in the dynamic context

Agnieszka § The only quadratic solution of (BE) is not the value Wiszniewska- function! Matyszkiel § It holds also for n “ 1, i.e. simple dynamic optimization Games Nash eq. problem. Stackelberg eq. Pareto opt. Example-static The tragedy

Dynamic games

A simple model Objectives

Bellman Revised sufficient condition

Social optimum

Nash equilibrium Nash equilibria for n players Finite horizon truncation

Enforcing optimality

Carrying capacity

Numerics

Continuous time

Conclusions § The Bellman equation, if we neglect constraints, has also continuum of linear solutions, gx ` kˆ for arbitrary g. § The solution corresponding to the quadratic V is ξx. It guarantees sustainability – so it is not enough to check (TC) along the trajectory corresponding to maximizer of rhs of (BE), as it is sometimes done. § The solutions with nonzero g also violate (TC’). § g “ 0 does not solve (BE) for small x. § There is also a solution with the only piecewise quadratic V that fulfils both (BE) and (TC).

”The tragedy of the Counterexample!!! commons” in the dynamic context

Agnieszka § The only quadratic solution of (BE) is not the value Wiszniewska- function! Matyszkiel § It holds also for n “ 1, i.e. simple dynamic optimization Games Nash eq. problem. Stackelberg eq. Pareto opt. § Example-static Of course, (TC’) is not fulfilled. The tragedy

Dynamic games

A simple model Objectives

Bellman Revised sufficient condition

Social optimum

Nash equilibrium Nash equilibria for n players Finite horizon truncation

Enforcing optimality

Carrying capacity

Numerics

Continuous time

Conclusions § The solution corresponding to the quadratic V is ξx. It guarantees sustainability – so it is not enough to check (TC) along the trajectory corresponding to maximizer of rhs of (BE), as it is sometimes done. § The solutions with nonzero g also violate (TC’). § g “ 0 does not solve (BE) for small x. § There is also a solution with the only piecewise quadratic V that fulfils both (BE) and (TC).

”The tragedy of the Counterexample!!! commons” in the dynamic context

Agnieszka § The only quadratic solution of (BE) is not the value Wiszniewska- function! Matyszkiel § It holds also for n “ 1, i.e. simple dynamic optimization Games Nash eq. problem. Stackelberg eq. Pareto opt. § Example-static Of course, (TC’) is not fulfilled. The tragedy § The Bellman equation, if we neglect constraints, has Dynamic games also continuum of linear solutions, gx ` kˆ for arbitrary g. A simple model Objectives

Bellman Revised sufficient condition

Social optimum

Nash equilibrium Nash equilibria for n players Finite horizon truncation

Enforcing optimality

Carrying capacity

Numerics

Continuous time

Conclusions § The solutions with nonzero g also violate (TC’). § g “ 0 does not solve (BE) for small x. § There is also a solution with the only piecewise quadratic V that fulfils both (BE) and (TC).

”The tragedy of the Counterexample!!! commons” in the dynamic context

Agnieszka § The only quadratic solution of (BE) is not the value Wiszniewska- function! Matyszkiel § It holds also for n “ 1, i.e. simple dynamic optimization Games Nash eq. problem. Stackelberg eq. Pareto opt. § Example-static Of course, (TC’) is not fulfilled. The tragedy § The Bellman equation, if we neglect constraints, has Dynamic games also continuum of linear solutions, gx ` kˆ for arbitrary g. A simple model Objectives § The solution corresponding to the quadratic V is ξx. It Bellman guarantees sustainability – so it is not enough to check Revised sufficient condition Social optimum (TC) along the trajectory corresponding to maximizer of Nash equilibrium rhs of (BE), as it is sometimes done. Nash equilibria for n players Finite horizon truncation

Enforcing optimality

Carrying capacity

Numerics

Continuous time

Conclusions § g “ 0 does not solve (BE) for small x. § There is also a solution with the only piecewise quadratic V that fulfils both (BE) and (TC).

”The tragedy of the Counterexample!!! commons” in the dynamic context

Agnieszka § The only quadratic solution of (BE) is not the value Wiszniewska- function! Matyszkiel § It holds also for n “ 1, i.e. simple dynamic optimization Games Nash eq. problem. Stackelberg eq. Pareto opt. § Example-static Of course, (TC’) is not fulfilled. The tragedy § The Bellman equation, if we neglect constraints, has Dynamic games also continuum of linear solutions, gx ` kˆ for arbitrary g. A simple model Objectives § The solution corresponding to the quadratic V is ξx. It Bellman guarantees sustainability – so it is not enough to check Revised sufficient condition Social optimum (TC) along the trajectory corresponding to maximizer of Nash equilibrium rhs of (BE), as it is sometimes done. Nash equilibria for n players Finite horizon truncation

§ The solutions with nonzero g also violate (TC’). Enforcing optimality

Carrying capacity

Numerics

Continuous time

Conclusions § There is also a solution with the only piecewise quadratic V that fulfils both (BE) and (TC).

”The tragedy of the Counterexample!!! commons” in the dynamic context

Agnieszka § The only quadratic solution of (BE) is not the value Wiszniewska- function! Matyszkiel § It holds also for n “ 1, i.e. simple dynamic optimization Games Nash eq. problem. Stackelberg eq. Pareto opt. § Example-static Of course, (TC’) is not fulfilled. The tragedy § The Bellman equation, if we neglect constraints, has Dynamic games also continuum of linear solutions, gx ` kˆ for arbitrary g. A simple model Objectives § The solution corresponding to the quadratic V is ξx. It Bellman guarantees sustainability – so it is not enough to check Revised sufficient condition Social optimum (TC) along the trajectory corresponding to maximizer of Nash equilibrium rhs of (BE), as it is sometimes done. Nash equilibria for n players Finite horizon truncation

§ The solutions with nonzero g also violate (TC’). Enforcing optimality § g “ 0 does not solve (BE) for small x. Carrying capacity Numerics

Continuous time

Conclusions ”The tragedy of the Counterexample!!! commons” in the dynamic context

Agnieszka § The only quadratic solution of (BE) is not the value Wiszniewska- function! Matyszkiel § It holds also for n “ 1, i.e. simple dynamic optimization Games Nash eq. problem. Stackelberg eq. Pareto opt. § Example-static Of course, (TC’) is not fulfilled. The tragedy § The Bellman equation, if we neglect constraints, has Dynamic games also continuum of linear solutions, gx ` kˆ for arbitrary g. A simple model Objectives § The solution corresponding to the quadratic V is ξx. It Bellman guarantees sustainability – so it is not enough to check Revised sufficient condition Social optimum (TC) along the trajectory corresponding to maximizer of Nash equilibrium rhs of (BE), as it is sometimes done. Nash equilibria for n players Finite horizon truncation

§ The solutions with nonzero g also violate (TC’). Enforcing optimality § g “ 0 does not solve (BE) for small x. Carrying capacity Numerics § There is also a solution with the only piecewise Continuous time quadratic V that fulfils both (BE) and (TC). Conclusions ˆ A´f ˆ ˆ for s “ 3 , h “ ´3 ξ p1 ` ξq , g “ pA ´ fqp1 ` ξq, and ˜ pA´fq2p1`ξq k “ 6ξ . and it is independent of the number of players (both n ě 1 and continuum).

×109 9

8

7

6

5

4

3

2

Value function per user for social optima 1

0 0 0.5 1 1.5 2 2.5 3 State (x) ×104 Figure: Value function per player for social optimum

”The tragedy of the Social optimum cont. commons” in the dynamic context Theorem 1 Agnieszka (a) The value function per player is Wiszniewska- Matyszkiel ˆ gˆ ¨ x ` h ¨ x2 if x P p0, ˆs q, V¯pxq “ 2 ξ Games ˜ Nash eq. #k otherwise, Stackelberg eq. Pareto opt. Example-static The tragedy

Dynamic games

A simple model Objectives

Bellman Revised sufficient condition

Social optimum

Nash equilibrium Nash equilibria for n players Finite horizon truncation

Enforcing optimality

Carrying capacity

Numerics

Continuous time

Conclusions and it is independent of the number of players (both n ě 1 and continuum).

×109 9

8

7

6

5

4

3

2

Value function per user for social optima 1

0 0 0.5 1 1.5 2 2.5 3 State (x) ×104 Figure: Value function per player for social optimum

”The tragedy of the Social optimum cont. commons” in the dynamic context Theorem 1 Agnieszka (a) The value function per player is Wiszniewska- Matyszkiel ˆ gˆ ¨ x ` h ¨ x2 if x P p0, ˆs q, V¯pxq “ 2 ξ Games ˜ Nash eq. #k otherwise, Stackelberg eq. Pareto opt. ˆ A´f ˆ ˆ Example-static for s “ 3 , h “ ´3 ξ p1 ` ξq , g “ pA ´ fqp1 ` ξq, and The tragedy 2 ˜ pA´fq p1`ξq Dynamic games k “ 6ξ . A simple model Objectives

Bellman Revised sufficient condition

Social optimum

Nash equilibrium Nash equilibria for n players Finite horizon truncation

Enforcing optimality

Carrying capacity

Numerics

Continuous time

Conclusions ×109 9

8

7

6

5

4

3

2

Value function per user for social optima 1

0 0 0.5 1 1.5 2 2.5 3 State (x) ×104 Figure: Value function per player for social optimum

”The tragedy of the Social optimum cont. commons” in the dynamic context Theorem 1 Agnieszka (a) The value function per player is Wiszniewska- Matyszkiel ˆ gˆ ¨ x ` h ¨ x2 if x P p0, ˆs q, V¯pxq “ 2 ξ Games ˜ Nash eq. #k otherwise, Stackelberg eq. Pareto opt. ˆ A´f ˆ ˆ Example-static for s “ 3 , h “ ´3 ξ p1 ` ξq , g “ pA ´ fqp1 ` ξq, and The tragedy 2 ˜ pA´fq p1`ξq Dynamic games k “ 6ξ . A simple model and it is independent of the number of players (both n ě 1 Objectives and continuum). Bellman Revised sufficient condition

Social optimum

Nash equilibrium Nash equilibria for n players Finite horizon truncation

Enforcing optimality

Carrying capacity

Numerics

Continuous time

Conclusions ”The tragedy of the Social optimum cont. commons” in the dynamic context Theorem 1 Agnieszka (a) The value function per player is Wiszniewska- Matyszkiel ˆ gˆ ¨ x ` h ¨ x2 if x P p0, ˆs q, V¯pxq “ 2 ξ Games ˜ Nash eq. #k otherwise, Stackelberg eq. Pareto opt. ˆ A´f ˆ ˆ Example-static for s “ 3 , h “ ´3 ξ p1 ` ξq , g “ pA ´ fqp1 ` ξq, and The tragedy 2 ˜ pA´fq p1`ξq Dynamic games k “ 6ξ . A simple model and it is independent of the number of players (both n ě 1 Objectives and continuum). Bellman Revised sufficient condition

×109 9 Social optimum 8

7 Nash equilibrium Nash equilibria for n players 6 Finite horizon truncation 5

4 Enforcing optimality

3 Carrying capacity 2

Value function per user for social optima 1 Numerics

0 0 0.5 1 1.5 2 2.5 3 Continuous time State (x) ×104 Conclusions Figure: Value function per player for social optimum For piecewise defined V¯ and s, the Bellman equation has to be checked again!

”The tragedy of the Social optimum cont. 2 commons” in the dynamic context Theorem 1 cont. (b) A profile defined by Agnieszka ˆs Wiszniewska- ξx, x 0, , Matyszkiel ˆ SO P p ξ q Si pxq “ #sˆ otherwise, Games Nash eq. is the unique social optimum both for n players and the Stackelberg eq. Pareto opt. continuum of players. Example-static The tragedy

350 Dynamic games

300 A simple model Objectives 250

200 Bellman Revised sufficient condition 150 Social optimum 100 Strategy for social optima

50 Nash equilibrium Nash equilibria for n players 0 0 0.5 1 1.5 2 2.5 3 Finite horizon truncation State (x) ×104 Enforcing optimality

Figure: Strategy of each player at social optimum Carrying capacity

Numerics

Continuous time

Conclusions ”The tragedy of the Social optimum cont. 2 commons” in the dynamic context Theorem 1 cont. (b) A profile defined by Agnieszka ˆs Wiszniewska- ξx, x 0, , Matyszkiel ˆ SO P p ξ q Si pxq “ #sˆ otherwise, Games Nash eq. is the unique social optimum both for n players and the Stackelberg eq. Pareto opt. continuum of players. Example-static The tragedy

350 Dynamic games

300 A simple model Objectives 250

200 Bellman Revised sufficient condition 150 Social optimum 100 Strategy for social optima

50 Nash equilibrium Nash equilibria for n players 0 0 0.5 1 1.5 2 2.5 3 Finite horizon truncation State (x) ×104 Enforcing optimality

Figure: Strategy of each player at social optimum Carrying capacity

Numerics

Continuous time

For piecewise defined V¯ and s, the Bellman equation has to Conclusions be checked again! – a decomposition method (a dynamic game decomposed into a sequence of static games). § by A. Wiszniewska-Matyszkiel (Positivity 2002, C& C 2003, IGTR 2002, 2003, JOTA 2014).

600

500

400

300

200 Strategy for Nash equilibrium 100

0 0 100 200 300 400 500 600 700 800 900 1000 State (x) Figure: Strategy of each player at the Nash equilibria

§ Exploitation many times larger than at the social optimum.

”The tragedy of the Nash equilibrium for continuum of players commons” in the dynamic context § Different method of calculation Agnieszka Wiszniewska- Matyszkiel

Games Nash eq. Stackelberg eq. Pareto opt. Example-static The tragedy

Dynamic games

A simple model Objectives

Bellman Revised sufficient condition

Social optimum

Nash equilibrium Nash equilibria for n players Finite horizon truncation

Enforcing optimality

Carrying capacity

Numerics

Continuous time

Conclusions § by A. Wiszniewska-Matyszkiel (Positivity 2002, C& C 2003, IGTR 2002, 2003, JOTA 2014).

600

500

400

300

200 Strategy for Nash equilibrium 100

0 0 100 200 300 400 500 600 700 800 900 1000 State (x) Figure: Strategy of each player at the Nash equilibria

§ Exploitation many times larger than at the social optimum.

”The tragedy of the Nash equilibrium for continuum of players commons” in the dynamic context § Different method of calculation – a decomposition Agnieszka Wiszniewska- method (a dynamic game decomposed into a sequence Matyszkiel of static games). Games Nash eq. Stackelberg eq. Pareto opt. Example-static The tragedy

Dynamic games

A simple model Objectives

Bellman Revised sufficient condition

Social optimum

Nash equilibrium Nash equilibria for n players Finite horizon truncation

Enforcing optimality

Carrying capacity

Numerics

Continuous time

Conclusions 600

500

400

300

200 Strategy for Nash equilibrium 100

0 0 100 200 300 400 500 600 700 800 900 1000 State (x) Figure: Strategy of each player at the Nash equilibria

§ Exploitation many times larger than at the social optimum.

”The tragedy of the Nash equilibrium for continuum of players commons” in the dynamic context § Different method of calculation – a decomposition Agnieszka Wiszniewska- method (a dynamic game decomposed into a sequence Matyszkiel of static games). Games § by A. Wiszniewska-Matyszkiel (Positivity 2002, C& C Nash eq. Stackelberg eq. 2003, IGTR 2002, 2003, JOTA 2014). Pareto opt. Example-static The tragedy

Dynamic games

A simple model Objectives

Bellman Revised sufficient condition

Social optimum

Nash equilibrium Nash equilibria for n players Finite horizon truncation

Enforcing optimality

Carrying capacity

Numerics

Continuous time

Conclusions § Exploitation many times larger than at the social optimum.

”The tragedy of the Nash equilibrium for continuum of players commons” in the dynamic context § Different method of calculation – a decomposition Agnieszka Wiszniewska- method (a dynamic game decomposed into a sequence Matyszkiel of static games). Games § by A. Wiszniewska-Matyszkiel (Positivity 2002, C& C Nash eq. Stackelberg eq. 2003, IGTR 2002, 2003, JOTA 2014). Pareto opt. Example-static The tragedy 600 Dynamic games 500 A simple model 400 Objectives

300 Bellman Revised sufficient condition 200

Strategy for Nash equilibrium Social optimum 100 Nash equilibrium 0 0 100 200 300 400 500 600 700 800 900 1000 Nash equilibria for n players State (x) Finite horizon truncation

Figure: Strategy of each player at the Nash equilibria Enforcing optimality

Carrying capacity

Numerics

Continuous time

Conclusions ”The tragedy of the Nash equilibrium for continuum of players commons” in the dynamic context § Different method of calculation – a decomposition Agnieszka Wiszniewska- method (a dynamic game decomposed into a sequence Matyszkiel of static games). Games § by A. Wiszniewska-Matyszkiel (Positivity 2002, C& C Nash eq. Stackelberg eq. 2003, IGTR 2002, 2003, JOTA 2014). Pareto opt. Example-static The tragedy 600 Dynamic games 500 A simple model 400 Objectives

300 Bellman Revised sufficient condition 200

Strategy for Nash equilibrium Social optimum 100 Nash equilibrium 0 0 100 200 300 400 500 600 700 800 900 1000 Nash equilibria for n players State (x) Finite horizon truncation

Figure: Strategy of each player at the Nash equilibria Enforcing optimality

Carrying capacity

Numerics § Exploitation many times larger than at the social Continuous time optimum. Conclusions ”The tragedy of the Nash equilibrium for continuum of players commons” in the dynamic context

Agnieszka Wiszniewska- Matyszkiel

Games Nash eq. Stackelberg eq. Pareto opt. Theorem 2 (a) The profile defined by Example-static The tragedy 1 ξ x for x xˆ , ˆ NE p ` q ď 1 Dynamic games Si pxq “ A´f otherwise, A simple model # 2 Objectives ˆ A´f for x1 “ 2p1`ξq , is the only feedback Nash equilibrium profile Bellman (up to measure equivalence). Revised sufficient condition Social optimum

Nash equilibrium Nash equilibria for n players Finite horizon truncation

Enforcing optimality

Carrying capacity

Numerics

Continuous time

Conclusions ”The tragedy of the Nash equilibrium for continuum of players commons” in the dynamic context Theorem 2 cont. (b) The function defined by Agnieszka NE V¯ pxq “ Wiszniewska- i Matyszkiel Pdeplpxq, for x ď xˆ1 N 2 k´1 N k´1 Games k“1pA´fq β N N pA´fq k“1 β ` β Pdepl p1 ` ξq x ´ for x P pxˆN, xˆNNash`1 eq.s, $ 8 2 Stackelberg eq. ř 2 ř ’ pA´fq p1`ξq ´ ¯ Pareto opt. & 8 ¨ ξ otherwise, Example-static The tragedy for Pdeplpxq “ Ppp1 ` ξqx, p1 ` ξqxq (payoff resulting from ’ Dynamic games % ˆ A´f N k immediate depletion of the resource) and xN “ 2 k“1 β A simple model for N ě 1, is the value function for optimization problem for Objectives ř Bellman the continuum of players game. Revised sufficient condition

×106 Social optimum 7

6 Nash equilibrium

5 Nash equilibria for n players Finite horizon truncation 4

3 Enforcing optimality

2 Carrying capacity Value function for Nash equilibrium 1 Numerics 0 0 0.5 1 1.5 2 2.5 3 State (x) × 4 10 Continuous time Figure: Value function at Nash equilibrium for continuum of players Conclusions ”The tragedy of the Value function through a magnifying glass commons” in the dynamic context

Agnieszka Wiszniewska- ×106 3 Matyszkiel

Games 2.5 Nash eq. Stackelberg eq. Pareto opt. 2 Example-static The tragedy

1.5 Dynamic games

A simple model 1 Objectives

Bellman

Value function for Nash equilibrium 0.5 Revised sufficient condition Social optimum

0 Nash equilibrium 0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000 State (x) Nash equilibria for n players Finite horizon truncation Figure: Value function at Nash equilibrium for continuum of players Enforcing optimality – zoomed view Carrying capacity Numerics

Continuous time

Conclusions ”The tragedy of the Nash equilibrium for continuum of players cont2. commons” in the dynamic context Theorem 2 cont. (c) For x P pxˆN, xˆN 1s with xˆ0 “ 0, the ` Agnieszka resource will be depleted/extracted in N ` 1 stages, while for Wiszniewska- Matyszkiel x ě xˆ8 “ limNÑ8 xˆN, the resource will never be depleted. Games Nash eq. 30 Stackelberg eq. Pareto opt. Example-static 25 The tragedy

Dynamic games 20 A simple model Objectives 15 Bellman Revised sufficient condition

10 Social optimum

Nash equilibrium 5 Nash equilibria for n players Number of time instants to exhaustion Finite horizon truncation

0 Enforcing optimality 0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000 Initial state Carrying capacity

Numerics Figure: Number of time moments to resource exhaustion at Nash equilibrium for continuum of players Continuous time Conclusions § However, it is not possible for analogous form of equilibrium strategies, piecewise linear with two intervals. § The only thing we were able to prove (with reasonable length of proof) is that the number of pieces in both equilibrium and value function is greater than two. § Any attempt to determine the symmetric solution (with possibly infinitely many ”switches”) assuming continuity (with respect to state) of: the value functions or the equilibrium strategies or the rhs of the Bellman equation along the optimal equilibrium strategy or another function related to depletion of resources was unsuccessful. So...

”The tragedy of the Nash equilibria for n players commons” in the dynamic context § For n players, a similar value function to that for Agnieszka Wiszniewska- continuum of players, with number of stages to Matyszkiel depletion nonstrictly increasing as x increases, can be Games expected. Nash eq. Stackelberg eq. Pareto opt. Example-static The tragedy

Dynamic games

A simple model Objectives

Bellman Revised sufficient condition

Social optimum

Nash equilibrium Nash equilibria for n players Finite horizon truncation

Enforcing optimality

Carrying capacity

Numerics

Continuous time

Conclusions § The only thing we were able to prove (with reasonable length of proof) is that the number of pieces in both equilibrium and value function is greater than two. § Any attempt to determine the symmetric solution (with possibly infinitely many ”switches”) assuming continuity (with respect to state) of: the value functions or the equilibrium strategies or the rhs of the Bellman equation along the optimal equilibrium strategy or another function related to depletion of resources was unsuccessful. So...

”The tragedy of the Nash equilibria for n players commons” in the dynamic context § For n players, a similar value function to that for Agnieszka Wiszniewska- continuum of players, with number of stages to Matyszkiel depletion nonstrictly increasing as x increases, can be Games expected. Nash eq. Stackelberg eq. § However, it is not possible for analogous form of Pareto opt. Example-static equilibrium strategies, piecewise linear with two The tragedy intervals. Dynamic games A simple model Objectives

Bellman Revised sufficient condition

Social optimum

Nash equilibrium Nash equilibria for n players Finite horizon truncation

Enforcing optimality

Carrying capacity

Numerics

Continuous time

Conclusions § Any attempt to determine the symmetric solution (with possibly infinitely many ”switches”) assuming continuity (with respect to state) of: the value functions or the equilibrium strategies or the rhs of the Bellman equation along the optimal equilibrium strategy or another function related to depletion of resources was unsuccessful. So...

”The tragedy of the Nash equilibria for n players commons” in the dynamic context § For n players, a similar value function to that for Agnieszka Wiszniewska- continuum of players, with number of stages to Matyszkiel depletion nonstrictly increasing as x increases, can be Games expected. Nash eq. Stackelberg eq. § However, it is not possible for analogous form of Pareto opt. Example-static equilibrium strategies, piecewise linear with two The tragedy intervals. Dynamic games A simple model § The only thing we were able to prove (with reasonable Objectives length of proof) is that the number of pieces in both Bellman Revised sufficient condition equilibrium and value function is greater than two. Social optimum

Nash equilibrium Nash equilibria for n players Finite horizon truncation

Enforcing optimality

Carrying capacity

Numerics

Continuous time

Conclusions (with possibly infinitely many ”switches”) assuming continuity (with respect to state) of: the value functions or the equilibrium strategies or the rhs of the Bellman equation along the optimal equilibrium strategy or another function related to depletion of resources was unsuccessful. So...

”The tragedy of the Nash equilibria for n players commons” in the dynamic context § For n players, a similar value function to that for Agnieszka Wiszniewska- continuum of players, with number of stages to Matyszkiel depletion nonstrictly increasing as x increases, can be Games expected. Nash eq. Stackelberg eq. § However, it is not possible for analogous form of Pareto opt. Example-static equilibrium strategies, piecewise linear with two The tragedy intervals. Dynamic games A simple model § The only thing we were able to prove (with reasonable Objectives length of proof) is that the number of pieces in both Bellman Revised sufficient condition equilibrium and value function is greater than two. Social optimum

§ Any attempt to determine the symmetric solution Nash equilibrium Nash equilibria for n players Finite horizon truncation

Enforcing optimality

Carrying capacity

Numerics

Continuous time

Conclusions or the equilibrium strategies or the rhs of the Bellman equation along the optimal equilibrium strategy or another function related to depletion of resources was unsuccessful. So...

”The tragedy of the Nash equilibria for n players commons” in the dynamic context § For n players, a similar value function to that for Agnieszka Wiszniewska- continuum of players, with number of stages to Matyszkiel depletion nonstrictly increasing as x increases, can be Games expected. Nash eq. Stackelberg eq. § However, it is not possible for analogous form of Pareto opt. Example-static equilibrium strategies, piecewise linear with two The tragedy intervals. Dynamic games A simple model § The only thing we were able to prove (with reasonable Objectives length of proof) is that the number of pieces in both Bellman Revised sufficient condition equilibrium and value function is greater than two. Social optimum

§ Any attempt to determine the symmetric solution (with Nash equilibrium Nash equilibria for n players possibly infinitely many ”switches”) assuming continuity Finite horizon truncation

(with respect to state) of: the value functions Enforcing optimality

Carrying capacity

Numerics

Continuous time

Conclusions or the rhs of the Bellman equation along the optimal equilibrium strategy or another function related to depletion of resources was unsuccessful. So...

”The tragedy of the Nash equilibria for n players commons” in the dynamic context § For n players, a similar value function to that for Agnieszka Wiszniewska- continuum of players, with number of stages to Matyszkiel depletion nonstrictly increasing as x increases, can be Games expected. Nash eq. Stackelberg eq. § However, it is not possible for analogous form of Pareto opt. Example-static equilibrium strategies, piecewise linear with two The tragedy intervals. Dynamic games A simple model § The only thing we were able to prove (with reasonable Objectives length of proof) is that the number of pieces in both Bellman Revised sufficient condition equilibrium and value function is greater than two. Social optimum

§ Any attempt to determine the symmetric solution (with Nash equilibrium Nash equilibria for n players possibly infinitely many ”switches”) assuming continuity Finite horizon truncation

(with respect to state) of: the value functions or the Enforcing optimality equilibrium strategies Carrying capacity Numerics

Continuous time

Conclusions or another function related to depletion of resources was unsuccessful. So...

”The tragedy of the Nash equilibria for n players commons” in the dynamic context § For n players, a similar value function to that for Agnieszka Wiszniewska- continuum of players, with number of stages to Matyszkiel depletion nonstrictly increasing as x increases, can be Games expected. Nash eq. Stackelberg eq. § However, it is not possible for analogous form of Pareto opt. Example-static equilibrium strategies, piecewise linear with two The tragedy intervals. Dynamic games A simple model § The only thing we were able to prove (with reasonable Objectives length of proof) is that the number of pieces in both Bellman Revised sufficient condition equilibrium and value function is greater than two. Social optimum

§ Any attempt to determine the symmetric solution (with Nash equilibrium Nash equilibria for n players possibly infinitely many ”switches”) assuming continuity Finite horizon truncation

(with respect to state) of: the value functions or the Enforcing optimality equilibrium strategies or the rhs of the Bellman equation Carrying capacity along the optimal equilibrium strategy Numerics Continuous time

Conclusions was unsuccessful. So...

”The tragedy of the Nash equilibria for n players commons” in the dynamic context § For n players, a similar value function to that for Agnieszka Wiszniewska- continuum of players, with number of stages to Matyszkiel depletion nonstrictly increasing as x increases, can be Games expected. Nash eq. Stackelberg eq. § However, it is not possible for analogous form of Pareto opt. Example-static equilibrium strategies, piecewise linear with two The tragedy intervals. Dynamic games A simple model § The only thing we were able to prove (with reasonable Objectives length of proof) is that the number of pieces in both Bellman Revised sufficient condition equilibrium and value function is greater than two. Social optimum

§ Any attempt to determine the symmetric solution (with Nash equilibrium Nash equilibria for n players possibly infinitely many ”switches”) assuming continuity Finite horizon truncation

(with respect to state) of: the value functions or the Enforcing optimality equilibrium strategies or the rhs of the Bellman equation Carrying capacity along the optimal equilibrium strategy or another Numerics function related to depletion of resources Continuous time Conclusions So...

”The tragedy of the Nash equilibria for n players commons” in the dynamic context § For n players, a similar value function to that for Agnieszka Wiszniewska- continuum of players, with number of stages to Matyszkiel depletion nonstrictly increasing as x increases, can be Games expected. Nash eq. Stackelberg eq. § However, it is not possible for analogous form of Pareto opt. Example-static equilibrium strategies, piecewise linear with two The tragedy intervals. Dynamic games A simple model § The only thing we were able to prove (with reasonable Objectives length of proof) is that the number of pieces in both Bellman Revised sufficient condition equilibrium and value function is greater than two. Social optimum

§ Any attempt to determine the symmetric solution (with Nash equilibrium Nash equilibria for n players possibly infinitely many ”switches”) assuming continuity Finite horizon truncation

(with respect to state) of: the value functions or the Enforcing optimality equilibrium strategies or the rhs of the Bellman equation Carrying capacity along the optimal equilibrium strategy or another Numerics function related to depletion of resources was Continuous time unsuccessful. Conclusions ”The tragedy of the Nash equilibria for n players commons” in the dynamic context § For n players, a similar value function to that for Agnieszka Wiszniewska- continuum of players, with number of stages to Matyszkiel depletion nonstrictly increasing as x increases, can be Games expected. Nash eq. Stackelberg eq. § However, it is not possible for analogous form of Pareto opt. Example-static equilibrium strategies, piecewise linear with two The tragedy intervals. Dynamic games A simple model § The only thing we were able to prove (with reasonable Objectives length of proof) is that the number of pieces in both Bellman Revised sufficient condition equilibrium and value function is greater than two. Social optimum

§ Any attempt to determine the symmetric solution (with Nash equilibrium Nash equilibria for n players possibly infinitely many ”switches”) assuming continuity Finite horizon truncation

(with respect to state) of: the value functions or the Enforcing optimality equilibrium strategies or the rhs of the Bellman equation Carrying capacity along the optimal equilibrium strategy or another Numerics function related to depletion of resources was Continuous time unsuccessful. So... Conclusions and allow the Nash equilibrium strategies to be

§ discontinuous at the points at which the number of time moments to depletion changes; § constant strategies and value function for x above some level; § proving that requires more compound tools than the continuum of players Nash equilibrium and any social optimum; § a symmetric piecewice linear Nash equilibrium, if it exists, is discontinuous (and we can state its general form up to location the points of discontinuity and checking the Bellman inclusion for the discontinuous, non-quasi concave function at the rhs.) and § it is a limit of Nash equilibria for finite horizon truncations of the game § and the irregularity is inherited from finite horizon truncations of the game.

”The tragedy of the Nash equilibria for n players continued commons” in the dynamic context

Agnieszka § Let us skip the continuity assumption Wiszniewska- Matyszkiel

Games Nash eq. Stackelberg eq. Pareto opt. Example-static The tragedy

Dynamic games

A simple model Objectives

Bellman Revised sufficient condition

Social optimum

Nash equilibrium Nash equilibria for n players Finite horizon truncation

Enforcing optimality

Carrying capacity

Numerics

Continuous time

Conclusions § discontinuous at the points at which the number of time moments to depletion changes; § constant strategies and value function for x above some level; § proving that requires more compound tools than the continuum of players Nash equilibrium and any social optimum; § a symmetric piecewice linear Nash equilibrium, if it exists, is discontinuous (and we can state its general form up to location the points of discontinuity and checking the Bellman inclusion for the discontinuous, non-quasi concave function at the rhs.) and § it is a limit of Nash equilibria for finite horizon truncations of the game § and the irregularity is inherited from finite horizon truncations of the game.

”The tragedy of the Nash equilibria for n players continued commons” in the dynamic context

Agnieszka § Let us skip the continuity assumption and allow the Wiszniewska- Nash equilibrium strategies to be Matyszkiel

Games Nash eq. Stackelberg eq. Pareto opt. Example-static The tragedy

Dynamic games

A simple model Objectives

Bellman Revised sufficient condition

Social optimum

Nash equilibrium Nash equilibria for n players Finite horizon truncation

Enforcing optimality

Carrying capacity

Numerics

Continuous time

Conclusions § constant strategies and value function for x above some level; § proving that requires more compound tools than the continuum of players Nash equilibrium and any social optimum; § a symmetric piecewice linear Nash equilibrium, if it exists, is discontinuous (and we can state its general form up to location the points of discontinuity and checking the Bellman inclusion for the discontinuous, non-quasi concave function at the rhs.) and § it is a limit of Nash equilibria for finite horizon truncations of the game § and the irregularity is inherited from finite horizon truncations of the game.

”The tragedy of the Nash equilibria for n players continued commons” in the dynamic context

Agnieszka § Let us skip the continuity assumption and allow the Wiszniewska- Nash equilibrium strategies to be Matyszkiel

§ discontinuous at the points at which the number of time Games Nash eq. moments to depletion changes; Stackelberg eq. Pareto opt. Example-static The tragedy

Dynamic games

A simple model Objectives

Bellman Revised sufficient condition

Social optimum

Nash equilibrium Nash equilibria for n players Finite horizon truncation

Enforcing optimality

Carrying capacity

Numerics

Continuous time

Conclusions § proving that requires more compound tools than the continuum of players Nash equilibrium and any social optimum; § a symmetric piecewice linear Nash equilibrium, if it exists, is discontinuous (and we can state its general form up to location the points of discontinuity and checking the Bellman inclusion for the discontinuous, non-quasi concave function at the rhs.) and § it is a limit of Nash equilibria for finite horizon truncations of the game § and the irregularity is inherited from finite horizon truncations of the game.

”The tragedy of the Nash equilibria for n players continued commons” in the dynamic context

Agnieszka § Let us skip the continuity assumption and allow the Wiszniewska- Nash equilibrium strategies to be Matyszkiel

§ discontinuous at the points at which the number of time Games Nash eq. moments to depletion changes; Stackelberg eq. Pareto opt. § constant strategies and value function for x above some Example-static level; The tragedy Dynamic games

A simple model Objectives

Bellman Revised sufficient condition

Social optimum

Nash equilibrium Nash equilibria for n players Finite horizon truncation

Enforcing optimality

Carrying capacity

Numerics

Continuous time

Conclusions § a symmetric piecewice linear Nash equilibrium, if it exists, is discontinuous (and we can state its general form up to location the points of discontinuity and checking the Bellman inclusion for the discontinuous, non-quasi concave function at the rhs.) and § it is a limit of Nash equilibria for finite horizon truncations of the game § and the irregularity is inherited from finite horizon truncations of the game.

”The tragedy of the Nash equilibria for n players continued commons” in the dynamic context

Agnieszka § Let us skip the continuity assumption and allow the Wiszniewska- Nash equilibrium strategies to be Matyszkiel

§ discontinuous at the points at which the number of time Games Nash eq. moments to depletion changes; Stackelberg eq. Pareto opt. § constant strategies and value function for x above some Example-static level; The tragedy § proving that requires more compound tools than the Dynamic games continuum of players Nash equilibrium and any social A simple model Objectives optimum; Bellman Revised sufficient condition

Social optimum

Nash equilibrium Nash equilibria for n players Finite horizon truncation

Enforcing optimality

Carrying capacity

Numerics

Continuous time

Conclusions § it is a limit of Nash equilibria for finite horizon truncations of the game § and the irregularity is inherited from finite horizon truncations of the game.

”The tragedy of the Nash equilibria for n players continued commons” in the dynamic context

Agnieszka § Let us skip the continuity assumption and allow the Wiszniewska- Nash equilibrium strategies to be Matyszkiel

§ discontinuous at the points at which the number of time Games Nash eq. moments to depletion changes; Stackelberg eq. Pareto opt. § constant strategies and value function for x above some Example-static level; The tragedy § proving that requires more compound tools than the Dynamic games continuum of players Nash equilibrium and any social A simple model Objectives optimum; Bellman § a symmetric piecewice linear Nash equilibrium, if it Revised sufficient condition exists, is discontinuous (and we can state its general Social optimum form up to location the points of discontinuity and Nash equilibrium Nash equilibria for n players checking the Bellman inclusion for the discontinuous, Finite horizon truncation

non-quasi concave function at the rhs.) and Enforcing optimality

Carrying capacity

Numerics

Continuous time

Conclusions § and the irregularity is inherited from finite horizon truncations of the game.

”The tragedy of the Nash equilibria for n players continued commons” in the dynamic context

Agnieszka § Let us skip the continuity assumption and allow the Wiszniewska- Nash equilibrium strategies to be Matyszkiel

§ discontinuous at the points at which the number of time Games Nash eq. moments to depletion changes; Stackelberg eq. Pareto opt. § constant strategies and value function for x above some Example-static level; The tragedy § proving that requires more compound tools than the Dynamic games continuum of players Nash equilibrium and any social A simple model Objectives optimum; Bellman § a symmetric piecewice linear Nash equilibrium, if it Revised sufficient condition exists, is discontinuous (and we can state its general Social optimum form up to location the points of discontinuity and Nash equilibrium Nash equilibria for n players checking the Bellman inclusion for the discontinuous, Finite horizon truncation

non-quasi concave function at the rhs.) and Enforcing optimality § it is a limit of Nash equilibria for finite horizon Carrying capacity truncations of the game Numerics

Continuous time

Conclusions ”The tragedy of the Nash equilibria for n players continued commons” in the dynamic context

Agnieszka § Let us skip the continuity assumption and allow the Wiszniewska- Nash equilibrium strategies to be Matyszkiel

§ discontinuous at the points at which the number of time Games Nash eq. moments to depletion changes; Stackelberg eq. Pareto opt. § constant strategies and value function for x above some Example-static level; The tragedy § proving that requires more compound tools than the Dynamic games continuum of players Nash equilibrium and any social A simple model Objectives optimum; Bellman § a symmetric piecewice linear Nash equilibrium, if it Revised sufficient condition exists, is discontinuous (and we can state its general Social optimum form up to location the points of discontinuity and Nash equilibrium Nash equilibria for n players checking the Bellman inclusion for the discontinuous, Finite horizon truncation

non-quasi concave function at the rhs.) and Enforcing optimality § it is a limit of Nash equilibria for finite horizon Carrying capacity truncations of the game Numerics § and the irregularity is inherited from finite horizon Continuous time truncations of the game. Conclusions starting from two stages

from [4] R. Singh, A. Wiszniewska-Matyszkiel, 2019, Discontinuous Nash equilibria in a two-stage

linear-quadratic dynamic game with linear constraints, IEEE Trans. on Aut. Control, 64, 3074–3079 § In the two stage truncation of the game: a continuum of discountinous symmetric equilibria and no continuous symmetric equilibrium!

Figure: Two stage truncation of the game

(a) two symmetric Nash (b) two symmetric Nash equilibria equilibria—zoomed view

”The tragedy of the Nash equilibria in the n “ 2 players truncated commons” in the dynamic context game Agnieszka Wiszniewska- § So, we analyse truncations Matyszkiel

Games Nash eq. Stackelberg eq. Pareto opt. Example-static The tragedy

Dynamic games

A simple model Objectives

Bellman Revised sufficient condition

Social optimum

Nash equilibrium Nash equilibria for n players Finite horizon truncation

Enforcing optimality

Carrying capacity

Numerics

Continuous time

Conclusions § In the two stage truncation of the game: a continuum of discountinous symmetric equilibria and no continuous symmetric equilibrium!

Figure: Two stage truncation of the game

(a) two symmetric Nash (b) two symmetric Nash equilibria equilibria—zoomed view

”The tragedy of the Nash equilibria in the n “ 2 players truncated commons” in the dynamic context game Agnieszka Wiszniewska- § So, we analyse truncations starting from two stages Matyszkiel

from [4] R. Singh, A. Wiszniewska-Matyszkiel, 2019, Discontinuous Nash equilibria in a two-stage Games Nash eq. linear-quadratic dynamic game with linear constraints, IEEE Trans. on Aut. Control, 64, 3074–3079 Stackelberg eq. Pareto opt. Example-static The tragedy

Dynamic games

A simple model Objectives

Bellman Revised sufficient condition

Social optimum

Nash equilibrium Nash equilibria for n players Finite horizon truncation

Enforcing optimality

Carrying capacity

Numerics

Continuous time

Conclusions : a continuum of discountinous symmetric equilibria and no continuous symmetric equilibrium!

Figure: Two stage truncation of the game

(a) two symmetric Nash (b) two symmetric Nash equilibria equilibria—zoomed view

”The tragedy of the Nash equilibria in the n “ 2 players truncated commons” in the dynamic context game Agnieszka Wiszniewska- § So, we analyse truncations starting from two stages Matyszkiel

from [4] R. Singh, A. Wiszniewska-Matyszkiel, 2019, Discontinuous Nash equilibria in a two-stage Games Nash eq. linear-quadratic dynamic game with linear constraints, IEEE Trans. on Aut. Control, 64, 3074–3079 Stackelberg eq. § In the two stage truncation of the game Pareto opt. Example-static The tragedy

Dynamic games

A simple model Objectives

Bellman Revised sufficient condition

Social optimum

Nash equilibrium Nash equilibria for n players Finite horizon truncation

Enforcing optimality

Carrying capacity

Numerics

Continuous time

Conclusions discountinous symmetric equilibria and no continuous symmetric equilibrium!

Figure: Two stage truncation of the game

(a) two symmetric Nash (b) two symmetric Nash equilibria equilibria—zoomed view

”The tragedy of the Nash equilibria in the n “ 2 players truncated commons” in the dynamic context game Agnieszka Wiszniewska- § So, we analyse truncations starting from two stages Matyszkiel

from [4] R. Singh, A. Wiszniewska-Matyszkiel, 2019, Discontinuous Nash equilibria in a two-stage Games Nash eq. linear-quadratic dynamic game with linear constraints, IEEE Trans. on Aut. Control, 64, 3074–3079 Stackelberg eq. § In the two stage truncation of the game: a continuum of Pareto opt. Example-static The tragedy

Dynamic games

A simple model Objectives

Bellman Revised sufficient condition

Social optimum

Nash equilibrium Nash equilibria for n players Finite horizon truncation

Enforcing optimality

Carrying capacity

Numerics

Continuous time

Conclusions and no continuous symmetric equilibrium!

Figure: Two stage truncation of the game

(a) two symmetric Nash (b) two symmetric Nash equilibria equilibria—zoomed view

”The tragedy of the Nash equilibria in the n “ 2 players truncated commons” in the dynamic context game Agnieszka Wiszniewska- § So, we analyse truncations starting from two stages Matyszkiel

from [4] R. Singh, A. Wiszniewska-Matyszkiel, 2019, Discontinuous Nash equilibria in a two-stage Games Nash eq. linear-quadratic dynamic game with linear constraints, IEEE Trans. on Aut. Control, 64, 3074–3079 Stackelberg eq. § In the two stage truncation of the game: a continuum of Pareto opt. Example-static discountinous symmetric equilibria The tragedy Dynamic games

A simple model Objectives

Bellman Revised sufficient condition

Social optimum

Nash equilibrium Nash equilibria for n players Finite horizon truncation

Enforcing optimality

Carrying capacity

Numerics

Continuous time

Conclusions Figure: Two stage truncation of the game

(a) two symmetric Nash (b) two symmetric Nash equilibria equilibria—zoomed view

”The tragedy of the Nash equilibria in the n “ 2 players truncated commons” in the dynamic context game Agnieszka Wiszniewska- § So, we analyse truncations starting from two stages Matyszkiel

from [4] R. Singh, A. Wiszniewska-Matyszkiel, 2019, Discontinuous Nash equilibria in a two-stage Games Nash eq. linear-quadratic dynamic game with linear constraints, IEEE Trans. on Aut. Control, 64, 3074–3079 Stackelberg eq. § In the two stage truncation of the game: a continuum of Pareto opt. Example-static discountinous symmetric equilibria and no continuous The tragedy symmetric equilibrium! Dynamic games A simple model Objectives

Bellman Revised sufficient condition

Social optimum

Nash equilibrium Nash equilibria for n players Finite horizon truncation

Enforcing optimality

Carrying capacity

Numerics

Continuous time

Conclusions (b) two symmetric Nash equilibria—zoomed view

”The tragedy of the Nash equilibria in the n “ 2 players truncated commons” in the dynamic context game Agnieszka Wiszniewska- § So, we analyse truncations starting from two stages Matyszkiel

from [4] R. Singh, A. Wiszniewska-Matyszkiel, 2019, Discontinuous Nash equilibria in a two-stage Games Nash eq. linear-quadratic dynamic game with linear constraints, IEEE Trans. on Aut. Control, 64, 3074–3079 Stackelberg eq. § In the two stage truncation of the game: a continuum of Pareto opt. Example-static discountinous symmetric equilibria and no continuous The tragedy symmetric equilibrium! Dynamic games A simple model Objectives

Figure: Two stage truncation of the game Bellman Revised sufficient condition

Social optimum

Nash equilibrium Nash equilibria for n players Finite horizon truncation

Enforcing optimality

Carrying capacity

Numerics (a) two symmetric Nash Continuous time equilibria Conclusions ”The tragedy of the Nash equilibria in the n “ 2 players truncated commons” in the dynamic context game Agnieszka Wiszniewska- § So, we analyse truncations starting from two stages Matyszkiel

from [4] R. Singh, A. Wiszniewska-Matyszkiel, 2019, Discontinuous Nash equilibria in a two-stage Games Nash eq. linear-quadratic dynamic game with linear constraints, IEEE Trans. on Aut. Control, 64, 3074–3079 Stackelberg eq. § In the two stage truncation of the game: a continuum of Pareto opt. Example-static discountinous symmetric equilibria and no continuous The tragedy symmetric equilibrium! Dynamic games A simple model Objectives

Figure: Two stage truncation of the game Bellman Revised sufficient condition

Social optimum

Nash equilibrium Nash equilibria for n players Finite horizon truncation

Enforcing optimality

Carrying capacity

Numerics (a) two symmetric Nash (b) two symmetric Nash Continuous time equilibria equilibria—zoomed view Conclusions ”The tragedy of the Nash equilibria in the n “ 2 players truncated commons” in the dynamic context game Agnieszka Wiszniewska- Matyszkiel

Games Nash eq. Stackelberg eq. Pareto opt. Example-static The tragedy

Dynamic games

A simple model Objectives

Bellman Revised sufficient condition

Social optimum

Nash equilibrium Nash equilibria for n players Figure: Two stage truncation of the game—the value functions at Finite horizon truncation two symmetric Nash equilibria Enforcing optimality Carrying capacity

Numerics

Continuous time

Conclusions (Ppsi, s„iq { Ppsi, s„iq ´ Tpsi, xq) in order to obtain socially optimal profile as a Nash equilibrium in the modified game. § The rate of linear tax (Tpsi, xq “ τpxqsi) enforcing social optimality in the continuum of players game is given by A ´ f ´ 2ξx if x ď A´f , τ x 3ξ p q “ A´f # 3 otherwise.

1000

900

800

700

600

500 Tax rate enforcing social optimality 400

300 0 0.5 1 1.5 2 2.5 3 State (x) ×104 Figure: Rate of tax enforcing social optimality for continuum of players

”The tragedy of the Enforcing social optimality by a tax or commons” in the dynamic context tax-subsidy system Agnieszka Wiszniewska- § Introduction of a regulatory tax Matyszkiel

Games Nash eq. Stackelberg eq. Pareto opt. Example-static The tragedy

Dynamic games

A simple model Objectives

Bellman Revised sufficient condition

Social optimum

Nash equilibrium Nash equilibria for n players Finite horizon truncation

Enforcing optimality

Carrying capacity

Numerics

Continuous time

Conclusions in order to obtain socially optimal profile as a Nash equilibrium in the modified game. § The rate of linear tax (Tpsi, xq “ τpxqsi) enforcing social optimality in the continuum of players game is given by A ´ f ´ 2ξx if x ď A´f , τ x 3ξ p q “ A´f # 3 otherwise.

1000

900

800

700

600

500 Tax rate enforcing social optimality 400

300 0 0.5 1 1.5 2 2.5 3 State (x) ×104 Figure: Rate of tax enforcing social optimality for continuum of players

”The tragedy of the Enforcing social optimality by a tax or commons” in the dynamic context tax-subsidy system Agnieszka Wiszniewska- § Introduction of a regulatory tax Matyszkiel

(Ppsi, s„iq { Ppsi, s„iq ´ Tpsi, xq) Games Nash eq. Stackelberg eq. Pareto opt. Example-static The tragedy

Dynamic games

A simple model Objectives

Bellman Revised sufficient condition

Social optimum

Nash equilibrium Nash equilibria for n players Finite horizon truncation

Enforcing optimality

Carrying capacity

Numerics

Continuous time

Conclusions § The rate of linear tax (Tpsi, xq “ τpxqsi) enforcing social optimality in the continuum of players game is given by A ´ f ´ 2ξx if x ď A´f , τ x 3ξ p q “ A´f # 3 otherwise.

1000

900

800

700

600

500 Tax rate enforcing social optimality 400

300 0 0.5 1 1.5 2 2.5 3 State (x) ×104 Figure: Rate of tax enforcing social optimality for continuum of players

”The tragedy of the Enforcing social optimality by a tax or commons” in the dynamic context tax-subsidy system Agnieszka Wiszniewska- § Introduction of a regulatory tax Matyszkiel

(Ppsi, s„iq { Ppsi, s„iq ´ Tpsi, xq) in order to obtain Games socially optimal profile as a Nash equilibrium in the Nash eq. Stackelberg eq. modified game. Pareto opt. Example-static The tragedy

Dynamic games

A simple model Objectives

Bellman Revised sufficient condition

Social optimum

Nash equilibrium Nash equilibria for n players Finite horizon truncation

Enforcing optimality

Carrying capacity

Numerics

Continuous time

Conclusions (Tpsi, xq “ τpxqsi) enforcing social optimality in the continuum of players game is given by A ´ f ´ 2ξx if x ď A´f , τ x 3ξ p q “ A´f # 3 otherwise.

1000

900

800

700

600

500 Tax rate enforcing social optimality 400

300 0 0.5 1 1.5 2 2.5 3 State (x) ×104 Figure: Rate of tax enforcing social optimality for continuum of players

”The tragedy of the Enforcing social optimality by a tax or commons” in the dynamic context tax-subsidy system Agnieszka Wiszniewska- § Introduction of a regulatory tax Matyszkiel

(Ppsi, s„iq { Ppsi, s„iq ´ Tpsi, xq) in order to obtain Games socially optimal profile as a Nash equilibrium in the Nash eq. Stackelberg eq. modified game. Pareto opt. Example-static § The rate of linear tax The tragedy Dynamic games

A simple model Objectives

Bellman Revised sufficient condition

Social optimum

Nash equilibrium Nash equilibria for n players Finite horizon truncation

Enforcing optimality

Carrying capacity

Numerics

Continuous time

Conclusions enforcing social optimality in the continuum of players game is given by A ´ f ´ 2ξx if x ď A´f , τ x 3ξ p q “ A´f # 3 otherwise.

1000

900

800

700

600

500 Tax rate enforcing social optimality 400

300 0 0.5 1 1.5 2 2.5 3 State (x) ×104 Figure: Rate of tax enforcing social optimality for continuum of players

”The tragedy of the Enforcing social optimality by a tax or commons” in the dynamic context tax-subsidy system Agnieszka Wiszniewska- § Introduction of a regulatory tax Matyszkiel

(Ppsi, s„iq { Ppsi, s„iq ´ Tpsi, xq) in order to obtain Games socially optimal profile as a Nash equilibrium in the Nash eq. Stackelberg eq. modified game. Pareto opt. Example-static § The rate of linear tax (Tpsi, xq “ τpxqsi) The tragedy Dynamic games

A simple model Objectives

Bellman Revised sufficient condition

Social optimum

Nash equilibrium Nash equilibria for n players Finite horizon truncation

Enforcing optimality

Carrying capacity

Numerics

Continuous time

Conclusions in the continuum of players game is given by A ´ f ´ 2ξx if x ď A´f , τ x 3ξ p q “ A´f # 3 otherwise.

1000

900

800

700

600

500 Tax rate enforcing social optimality 400

300 0 0.5 1 1.5 2 2.5 3 State (x) ×104 Figure: Rate of tax enforcing social optimality for continuum of players

”The tragedy of the Enforcing social optimality by a tax or commons” in the dynamic context tax-subsidy system Agnieszka Wiszniewska- § Introduction of a regulatory tax Matyszkiel

(Ppsi, s„iq { Ppsi, s„iq ´ Tpsi, xq) in order to obtain Games socially optimal profile as a Nash equilibrium in the Nash eq. Stackelberg eq. modified game. Pareto opt. Example-static § The rate of linear tax (Tpsi, xq “ τpxqsi) enforcing social The tragedy optimality Dynamic games A simple model Objectives

Bellman Revised sufficient condition

Social optimum

Nash equilibrium Nash equilibria for n players Finite horizon truncation

Enforcing optimality

Carrying capacity

Numerics

Continuous time

Conclusions is given by A ´ f ´ 2ξx if x ď A´f , τ x 3ξ p q “ A´f # 3 otherwise.

1000

900

800

700

600

500 Tax rate enforcing social optimality 400

300 0 0.5 1 1.5 2 2.5 3 State (x) ×104 Figure: Rate of tax enforcing social optimality for continuum of players

”The tragedy of the Enforcing social optimality by a tax or commons” in the dynamic context tax-subsidy system Agnieszka Wiszniewska- § Introduction of a regulatory tax Matyszkiel

(Ppsi, s„iq { Ppsi, s„iq ´ Tpsi, xq) in order to obtain Games socially optimal profile as a Nash equilibrium in the Nash eq. Stackelberg eq. modified game. Pareto opt. Example-static § The rate of linear tax (Tpsi, xq “ τpxqsi) enforcing social The tragedy optimality in the continuum of players game Dynamic games A simple model Objectives

Bellman Revised sufficient condition

Social optimum

Nash equilibrium Nash equilibria for n players Finite horizon truncation

Enforcing optimality

Carrying capacity

Numerics

Continuous time

Conclusions 1000

900

800

700

600

500 Tax rate enforcing social optimality 400

300 0 0.5 1 1.5 2 2.5 3 State (x) ×104 Figure: Rate of tax enforcing social optimality for continuum of players

”The tragedy of the Enforcing social optimality by a tax or commons” in the dynamic context tax-subsidy system Agnieszka Wiszniewska- § Introduction of a regulatory tax Matyszkiel

(Ppsi, s„iq { Ppsi, s„iq ´ Tpsi, xq) in order to obtain Games socially optimal profile as a Nash equilibrium in the Nash eq. Stackelberg eq. modified game. Pareto opt. Example-static § The rate of linear tax (Tpsi, xq “ τpxqsi) enforcing social The tragedy optimality in the continuum of players game is given by Dynamic games A´f A simple model A ´ f ´ 2ξx if x ď , Objectives τpxq “ 3ξ A´f otherwise. Bellman # 3 Revised sufficient condition

Social optimum

Nash equilibrium Nash equilibria for n players Finite horizon truncation

Enforcing optimality

Carrying capacity

Numerics

Continuous time

Conclusions ”The tragedy of the Enforcing social optimality by a tax or commons” in the dynamic context tax-subsidy system Agnieszka Wiszniewska- § Introduction of a regulatory tax Matyszkiel

(Ppsi, s„iq { Ppsi, s„iq ´ Tpsi, xq) in order to obtain Games socially optimal profile as a Nash equilibrium in the Nash eq. Stackelberg eq. modified game. Pareto opt. Example-static § The rate of linear tax (Tpsi, xq “ τpxqsi) enforcing social The tragedy optimality in the continuum of players game is given by Dynamic games A´f A simple model A ´ f ´ 2ξx if x ď , Objectives τpxq “ 3ξ A´f otherwise. Bellman # 3 Revised sufficient condition

Social optimum

1000 Nash equilibrium 900 Nash equilibria for n players

800 Finite horizon truncation

700 Enforcing optimality

600 Carrying capacity

500 Numerics Tax rate enforcing social optimality 400 Continuous time 300 0 0.5 1 1.5 2 2.5 3 State (x) ×104 Conclusions Figure: Rate of tax enforcing social optimality for continuum of players It is not a problem, since: § if from time 0 on the regulator chooses the tax rate τpx0q, then the state is constantly x0 and the resulting Nash equilibrium is equal to social optimum in the initial problem; § generally, if instead of ”tax” we use the term ”environmental levy”, then increasing the levy as the state of the environment deteriorates seems justified. § The resulting tax paid is

×104 14

12

10

8

6

4 Tax enforcing social optimality

2

0 0 0.5 1 1.5 2 2.5 3 State (x) ×104 Figure: Tax enforcing social optimality for continuum of players

”The tragedy of the Enforcing social optimality by a tax – cont. commons” in the dynamic context § Variable tax rate? Agnieszka Wiszniewska- Matyszkiel

Games Nash eq. Stackelberg eq. Pareto opt. Example-static The tragedy

Dynamic games

A simple model Objectives

Bellman Revised sufficient condition

Social optimum

Nash equilibrium Nash equilibria for n players Finite horizon truncation

Enforcing optimality

Carrying capacity

Numerics

Continuous time

Conclusions , since: § if from time 0 on the regulator chooses the tax rate τpx0q, then the state is constantly x0 and the resulting Nash equilibrium is equal to social optimum in the initial problem; § generally, if instead of ”tax” we use the term ”environmental levy”, then increasing the levy as the state of the environment deteriorates seems justified. § The resulting tax paid is

×104 14

12

10

8

6

4 Tax enforcing social optimality

2

0 0 0.5 1 1.5 2 2.5 3 State (x) ×104 Figure: Tax enforcing social optimality for continuum of players

”The tragedy of the Enforcing social optimality by a tax – cont. commons” in the dynamic context § Variable tax rate? It is not a problem Agnieszka Wiszniewska- Matyszkiel

Games Nash eq. Stackelberg eq. Pareto opt. Example-static The tragedy

Dynamic games

A simple model Objectives

Bellman Revised sufficient condition

Social optimum

Nash equilibrium Nash equilibria for n players Finite horizon truncation

Enforcing optimality

Carrying capacity

Numerics

Continuous time

Conclusions § generally, if instead of ”tax” we use the term ”environmental levy”, then increasing the levy as the state of the environment deteriorates seems justified. § The resulting tax paid is

×104 14

12

10

8

6

4 Tax enforcing social optimality

2

0 0 0.5 1 1.5 2 2.5 3 State (x) ×104 Figure: Tax enforcing social optimality for continuum of players

”The tragedy of the Enforcing social optimality by a tax – cont. commons” in the dynamic context § Variable tax rate? It is not a problem, since: Agnieszka § if from time 0 on the regulator chooses the tax rate Wiszniewska- Matyszkiel τpx0q, then the state is constantly x0 and the resulting Nash equilibrium is equal to social optimum in the initial Games Nash eq. problem; Stackelberg eq. Pareto opt. Example-static The tragedy

Dynamic games

A simple model Objectives

Bellman Revised sufficient condition

Social optimum

Nash equilibrium Nash equilibria for n players Finite horizon truncation

Enforcing optimality

Carrying capacity

Numerics

Continuous time

Conclusions § The resulting tax paid is

×104 14

12

10

8

6

4 Tax enforcing social optimality

2

0 0 0.5 1 1.5 2 2.5 3 State (x) ×104 Figure: Tax enforcing social optimality for continuum of players

”The tragedy of the Enforcing social optimality by a tax – cont. commons” in the dynamic context § Variable tax rate? It is not a problem, since: Agnieszka § if from time 0 on the regulator chooses the tax rate Wiszniewska- Matyszkiel τpx0q, then the state is constantly x0 and the resulting Nash equilibrium is equal to social optimum in the initial Games Nash eq. problem; Stackelberg eq. § generally, if instead of ”tax” we use the term Pareto opt. Example-static ”environmental levy”, then increasing the levy as the The tragedy state of the environment deteriorates seems justified. Dynamic games

A simple model Objectives

Bellman Revised sufficient condition

Social optimum

Nash equilibrium Nash equilibria for n players Finite horizon truncation

Enforcing optimality

Carrying capacity

Numerics

Continuous time

Conclusions ”The tragedy of the Enforcing social optimality by a tax – cont. commons” in the dynamic context § Variable tax rate? It is not a problem, since: Agnieszka § if from time 0 on the regulator chooses the tax rate Wiszniewska- Matyszkiel τpx0q, then the state is constantly x0 and the resulting Nash equilibrium is equal to social optimum in the initial Games Nash eq. problem; Stackelberg eq. § generally, if instead of ”tax” we use the term Pareto opt. Example-static ”environmental levy”, then increasing the levy as the The tragedy state of the environment deteriorates seems justified. Dynamic games § The resulting tax paid is A simple model Objectives ×104 14 Bellman 12 Revised sufficient condition

10 Social optimum

8 Nash equilibrium 6 Nash equilibria for n players Finite horizon truncation 4 Tax enforcing social optimality

2 Enforcing optimality

0 0 0.5 1 1.5 2 2.5 3 Carrying capacity State (x) ×104 Numerics

Figure: Tax enforcing social optimality for continuum of Continuous time

players Conclusions – then the results are ¯ SO equivalent (i.e. the same τ enforces Si ), but no tax is paid. § If we consider the tax rate τ calculated for the continuum of players but consider taxing overexploitation only ¯ SO ` i.e. Tpsi, xq “ τpxqpsi ´ Si q then the tax rate τpxq enforces social optimality for every number of players n. § So, the continuum of players model helped us to solve the problem of enforcement for n players although we are not able to calculate the Nash equilibrium for n players.

”The tragedy of the Enforcing social optimality by a tax – cont. commons” in the dynamic context

Agnieszka Wiszniewska- § If we consider a tax-subsidy system with Matyszkiel ¯ SO Tpsi, xq “ τpxqpsi ´ Si q Games Nash eq. Stackelberg eq. Pareto opt. Example-static The tragedy

Dynamic games

A simple model Objectives

Bellman Revised sufficient condition

Social optimum

Nash equilibrium Nash equilibria for n players Finite horizon truncation

Enforcing optimality

Carrying capacity

Numerics

Continuous time

Conclusions ¯ SO (i.e. the same τ enforces Si ), but no tax is paid. § If we consider the tax rate τ calculated for the continuum of players but consider taxing overexploitation only ¯ SO ` i.e. Tpsi, xq “ τpxqpsi ´ Si q then the tax rate τpxq enforces social optimality for every number of players n. § So, the continuum of players model helped us to solve the problem of enforcement for n players although we are not able to calculate the Nash equilibrium for n players.

”The tragedy of the Enforcing social optimality by a tax – cont. commons” in the dynamic context

Agnieszka Wiszniewska- § If we consider a tax-subsidy system with Matyszkiel ¯ SO Tpsi, xq “ τpxqpsi ´ Si q – then the results are Games equivalent Nash eq. Stackelberg eq. Pareto opt. Example-static The tragedy

Dynamic games

A simple model Objectives

Bellman Revised sufficient condition

Social optimum

Nash equilibrium Nash equilibria for n players Finite horizon truncation

Enforcing optimality

Carrying capacity

Numerics

Continuous time

Conclusions but no tax is paid. § If we consider the tax rate τ calculated for the continuum of players but consider taxing overexploitation only ¯ SO ` i.e. Tpsi, xq “ τpxqpsi ´ Si q then the tax rate τpxq enforces social optimality for every number of players n. § So, the continuum of players model helped us to solve the problem of enforcement for n players although we are not able to calculate the Nash equilibrium for n players.

”The tragedy of the Enforcing social optimality by a tax – cont. commons” in the dynamic context

Agnieszka Wiszniewska- § If we consider a tax-subsidy system with Matyszkiel ¯ SO Tpsi, xq “ τpxqpsi ´ Si q – then the results are Games equivalent (i.e. the same τ enforces S¯ SO), Nash eq. i Stackelberg eq. Pareto opt. Example-static The tragedy

Dynamic games

A simple model Objectives

Bellman Revised sufficient condition

Social optimum

Nash equilibrium Nash equilibria for n players Finite horizon truncation

Enforcing optimality

Carrying capacity

Numerics

Continuous time

Conclusions § If we consider the tax rate τ calculated for the continuum of players but consider taxing overexploitation only ¯ SO ` i.e. Tpsi, xq “ τpxqpsi ´ Si q then the tax rate τpxq enforces social optimality for every number of players n. § So, the continuum of players model helped us to solve the problem of enforcement for n players although we are not able to calculate the Nash equilibrium for n players.

”The tragedy of the Enforcing social optimality by a tax – cont. commons” in the dynamic context

Agnieszka Wiszniewska- § If we consider a tax-subsidy system with Matyszkiel ¯ SO Tpsi, xq “ τpxqpsi ´ Si q – then the results are Games equivalent (i.e. the same τ enforces S¯ SO), but no tax is Nash eq. i Stackelberg eq. paid. Pareto opt. Example-static The tragedy

Dynamic games

A simple model Objectives

Bellman Revised sufficient condition

Social optimum

Nash equilibrium Nash equilibria for n players Finite horizon truncation

Enforcing optimality

Carrying capacity

Numerics

Continuous time

Conclusions but consider taxing overexploitation only ¯ SO ` i.e. Tpsi, xq “ τpxqpsi ´ Si q then the tax rate τpxq enforces social optimality for every number of players n. § So, the continuum of players model helped us to solve the problem of enforcement for n players although we are not able to calculate the Nash equilibrium for n players.

”The tragedy of the Enforcing social optimality by a tax – cont. commons” in the dynamic context

Agnieszka Wiszniewska- § If we consider a tax-subsidy system with Matyszkiel ¯ SO Tpsi, xq “ τpxqpsi ´ Si q – then the results are Games equivalent (i.e. the same τ enforces S¯ SO), but no tax is Nash eq. i Stackelberg eq. paid. Pareto opt. Example-static § If we consider the tax rate τ calculated for the The tragedy Dynamic games continuum of players A simple model Objectives

Bellman Revised sufficient condition

Social optimum

Nash equilibrium Nash equilibria for n players Finite horizon truncation

Enforcing optimality

Carrying capacity

Numerics

Continuous time

Conclusions ¯ SO ` i.e. Tpsi, xq “ τpxqpsi ´ Si q then the tax rate τpxq enforces social optimality for every number of players n. § So, the continuum of players model helped us to solve the problem of enforcement for n players although we are not able to calculate the Nash equilibrium for n players.

”The tragedy of the Enforcing social optimality by a tax – cont. commons” in the dynamic context

Agnieszka Wiszniewska- § If we consider a tax-subsidy system with Matyszkiel ¯ SO Tpsi, xq “ τpxqpsi ´ Si q – then the results are Games equivalent (i.e. the same τ enforces S¯ SO), but no tax is Nash eq. i Stackelberg eq. paid. Pareto opt. Example-static § If we consider the tax rate τ calculated for the The tragedy Dynamic games continuum of players but consider taxing A simple model overexploitation only Objectives Bellman Revised sufficient condition

Social optimum

Nash equilibrium Nash equilibria for n players Finite horizon truncation

Enforcing optimality

Carrying capacity

Numerics

Continuous time

Conclusions then the tax rate τpxq enforces social optimality for every number of players n. § So, the continuum of players model helped us to solve the problem of enforcement for n players although we are not able to calculate the Nash equilibrium for n players.

”The tragedy of the Enforcing social optimality by a tax – cont. commons” in the dynamic context

Agnieszka Wiszniewska- § If we consider a tax-subsidy system with Matyszkiel ¯ SO Tpsi, xq “ τpxqpsi ´ Si q – then the results are Games equivalent (i.e. the same τ enforces S¯ SO), but no tax is Nash eq. i Stackelberg eq. paid. Pareto opt. Example-static § If we consider the tax rate τ calculated for the The tragedy Dynamic games continuum of players but consider taxing A simple model overexploitation only Objectives SO ` i.e. Tpsi, xq “ τpxqpsi ´ S¯ q Bellman i Revised sufficient condition

Social optimum

Nash equilibrium Nash equilibria for n players Finite horizon truncation

Enforcing optimality

Carrying capacity

Numerics

Continuous time

Conclusions § So, the continuum of players model helped us to solve the problem of enforcement for n players although we are not able to calculate the Nash equilibrium for n players.

”The tragedy of the Enforcing social optimality by a tax – cont. commons” in the dynamic context

Agnieszka Wiszniewska- § If we consider a tax-subsidy system with Matyszkiel ¯ SO Tpsi, xq “ τpxqpsi ´ Si q – then the results are Games equivalent (i.e. the same τ enforces S¯ SO), but no tax is Nash eq. i Stackelberg eq. paid. Pareto opt. Example-static § If we consider the tax rate τ calculated for the The tragedy Dynamic games continuum of players but consider taxing A simple model overexploitation only Objectives SO ` i.e. Tpsi, xq “ τpxqpsi ´ S¯ q Bellman i Revised sufficient condition

then the tax rate τpxq enforces social optimality for Social optimum

every number of players n. Nash equilibrium Nash equilibria for n players Finite horizon truncation

Enforcing optimality

Carrying capacity

Numerics

Continuous time

Conclusions although we are not able to calculate the Nash equilibrium for n players.

”The tragedy of the Enforcing social optimality by a tax – cont. commons” in the dynamic context

Agnieszka Wiszniewska- § If we consider a tax-subsidy system with Matyszkiel ¯ SO Tpsi, xq “ τpxqpsi ´ Si q – then the results are Games equivalent (i.e. the same τ enforces S¯ SO), but no tax is Nash eq. i Stackelberg eq. paid. Pareto opt. Example-static § If we consider the tax rate τ calculated for the The tragedy Dynamic games continuum of players but consider taxing A simple model overexploitation only Objectives SO ` i.e. Tpsi, xq “ τpxqpsi ´ S¯ q Bellman i Revised sufficient condition

then the tax rate τpxq enforces social optimality for Social optimum

every number of players n. Nash equilibrium Nash equilibria for n players § So, the continuum of players model helped us to solve Finite horizon truncation the problem of enforcement for n players Enforcing optimality Carrying capacity

Numerics

Continuous time

Conclusions ”The tragedy of the Enforcing social optimality by a tax – cont. commons” in the dynamic context

Agnieszka Wiszniewska- § If we consider a tax-subsidy system with Matyszkiel ¯ SO Tpsi, xq “ τpxqpsi ´ Si q – then the results are Games equivalent (i.e. the same τ enforces S¯ SO), but no tax is Nash eq. i Stackelberg eq. paid. Pareto opt. Example-static § If we consider the tax rate τ calculated for the The tragedy Dynamic games continuum of players but consider taxing A simple model overexploitation only Objectives SO ` i.e. Tpsi, xq “ τpxqpsi ´ S¯ q Bellman i Revised sufficient condition

then the tax rate τpxq enforces social optimality for Social optimum

every number of players n. Nash equilibrium Nash equilibria for n players § So, the continuum of players model helped us to solve Finite horizon truncation the problem of enforcement for n players although we Enforcing optimality are not able to calculate the Nash equilibrium for n Carrying capacity players. Numerics Continuous time

Conclusions if we appropriately ˆs modify the dynamics of the state above ξ in order to take into account the carrying capacity of the environment.

”The tragedy of the Extensions of the model and introducing commons” in the dynamic context carrying capacity Agnieszka Wiszniewska- Matyszkiel

Games Nash eq. Stackelberg eq. Pareto opt. Example-static The tragedy § All the above results remain valid Dynamic games A simple model Objectives

Bellman Revised sufficient condition

Social optimum

Nash equilibrium Nash equilibria for n players Finite horizon truncation

Enforcing optimality

Carrying capacity

Numerics

Continuous time

Conclusions ”The tragedy of the Extensions of the model and introducing commons” in the dynamic context carrying capacity Agnieszka Wiszniewska- Matyszkiel

Games Nash eq. Stackelberg eq. Pareto opt. Example-static The tragedy § All the above results remain valid if we appropriately Dynamic games ˆs A simple model modify the dynamics of the state above ξ in order to Objectives take into account the carrying capacity of the Bellman environment. Revised sufficient condition Social optimum

Nash equilibrium Nash equilibria for n players Finite horizon truncation

Enforcing optimality

Carrying capacity

Numerics

Continuous time

Conclusions § A class of optimal control problems (i.e. n “ 1), analogous to our social optimality but for a whole interval of possible discount factors (a slightly more impatient decision makers): for a candidate V f for the value function calculated analogously as in Theorem 1 and a control Sf from the rhs of the (BE) with V f , for every  ą 0, there is a discount factor close to the golden rule such that the Bellman equation is fulfilled everywhere besides an -neighbourhood of 0, while Sf is far from the optimal control while V f from the value function on the set of all reasonable states (i.e. below ˆs ξ ). [5] R. Singh, A. Wiszniewska-Matyszkiel, 2020, A class of linear quadratic dynamic optimization problems with state dependent constraints, MMOR, 91, 325–355. § Nested induction (backward and forward) plus concave analysis needed to derive the optimal control analytically – piecewise linear with infinitely many pieces.

”The tragedy of the Be careful with numerics! commons” in the dynamic context § Solving (BE) numerically is costly. Agnieszka Wiszniewska- Matyszkiel

Games Nash eq. Stackelberg eq. Pareto opt. Example-static The tragedy

Dynamic games

A simple model Objectives

Bellman Revised sufficient condition

Social optimum

Nash equilibrium Nash equilibria for n players Finite horizon truncation

Enforcing optimality

Carrying capacity

Numerics

Continuous time

Conclusions ”The tragedy of the Be careful with numerics! commons” in the dynamic context § Solving (BE) numerically is costly. Agnieszka § Wiszniewska- A class of optimal control problems (i.e. n “ 1), Matyszkiel analogous to our social optimality but for a whole Games interval of possible discount factors (a slightly more Nash eq. f Stackelberg eq. impatient decision makers): for a candidate V for the Pareto opt. Example-static value function calculated analogously as in Theorem 1 The tragedy f f and a control S from the rhs of the (BE) with V , for Dynamic games every  ą 0, there is a discount factor close to the A simple model golden rule such that the Bellman equation is fulfilled Objectives f Bellman everywhere besides an -neighbourhood of 0, while S Revised sufficient condition is far from the optimal control while V f from the value Social optimum function on the set of all reasonable states (i.e. below Nash equilibrium Nash equilibria for n players ˆs Finite horizon truncation ξ ). [5] R. Singh, A. Wiszniewska-Matyszkiel, 2020, A class of linear quadratic dynamic Enforcing optimality optimization problems with state dependent constraints, MMOR, 91, 325–355. § Nested induction (backward and forward) plus concave Carrying capacity analysis needed to derive the optimal control Numerics Continuous time analytically – piecewise linear with infinitely many Conclusions pieces. § General theory for such problems still not developed (viscosity solutions for infinite horizon, sufficiency and necessity, etc.)

”The tragedy of the Continuous time commons” in the dynamic context

Agnieszka Wiszniewska- Matyszkiel

Games Nash eq. Stackelberg eq. Pareto opt. Example-static § We considered a model from similar class in continuous The tragedy time to model a cryptocurrency mining game. Dynamic games

A simple model Objectives

Bellman Revised sufficient condition

Social optimum

Nash equilibrium Nash equilibria for n players Finite horizon truncation

Enforcing optimality

Carrying capacity

Numerics

Continuous time

Conclusions ”The tragedy of the Continuous time commons” in the dynamic context

Agnieszka Wiszniewska- Matyszkiel

Games Nash eq. Stackelberg eq. Pareto opt. Example-static § We considered a model from similar class in continuous The tragedy time to model a cryptocurrency mining game. Dynamic games § General theory for such problems still not developed A simple model Objectives

(viscosity solutions for infinite horizon, sufficiency and Bellman necessity, etc.) Revised sufficient condition Social optimum

Nash equilibrium Nash equilibria for n players Finite horizon truncation

Enforcing optimality

Carrying capacity

Numerics

Continuous time

Conclusions § Problems of feedback Nash equilibria require solving a set of coupled parametrized dynamic optimization problems, with strategies of the others as parameters. § Problems of feedback Stackelberg equilibria are even more complicated. § Only few classes of such games have been solved and some proofs are still incomplete. § Models lack realistic constraints. § Adding even very inherent constraints can change the solutions drastically, with several surpsises.

”The tragedy of the Conclusions ”The tragedy of the commons” in commons” in the dynamic context dynamic context Agnieszka Wiszniewska- Matyszkiel § To model most of the tragedy of the commons Games problems, tools of dynamic games are required, Nash eq. Stackelberg eq. especially feedback Nash and Stackelberg equilibria. Pareto opt. Example-static The tragedy

Dynamic games

A simple model Objectives

Bellman Revised sufficient condition

Social optimum

Nash equilibrium Nash equilibria for n players Finite horizon truncation

Enforcing optimality

Carrying capacity

Numerics

Continuous time

Conclusions § Problems of feedback Stackelberg equilibria are even more complicated. § Only few classes of such games have been solved and some proofs are still incomplete. § Models lack realistic constraints. § Adding even very inherent constraints can change the solutions drastically, with several surpsises.

”The tragedy of the Conclusions ”The tragedy of the commons” in commons” in the dynamic context dynamic context Agnieszka Wiszniewska- Matyszkiel § To model most of the tragedy of the commons Games problems, tools of dynamic games are required, Nash eq. Stackelberg eq. especially feedback Nash and Stackelberg equilibria. Pareto opt. Example-static § Problems of feedback Nash equilibria require solving a The tragedy Dynamic games set of coupled parametrized dynamic optimization A simple model problems, with strategies of the others as parameters. Objectives Bellman Revised sufficient condition

Social optimum

Nash equilibrium Nash equilibria for n players Finite horizon truncation

Enforcing optimality

Carrying capacity

Numerics

Continuous time

Conclusions § Only few classes of such games have been solved and some proofs are still incomplete. § Models lack realistic constraints. § Adding even very inherent constraints can change the solutions drastically, with several surpsises.

”The tragedy of the Conclusions ”The tragedy of the commons” in commons” in the dynamic context dynamic context Agnieszka Wiszniewska- Matyszkiel § To model most of the tragedy of the commons Games problems, tools of dynamic games are required, Nash eq. Stackelberg eq. especially feedback Nash and Stackelberg equilibria. Pareto opt. Example-static § Problems of feedback Nash equilibria require solving a The tragedy Dynamic games set of coupled parametrized dynamic optimization A simple model problems, with strategies of the others as parameters. Objectives § Bellman Problems of feedback Stackelberg equilibria are even Revised sufficient condition

more complicated. Social optimum

Nash equilibrium Nash equilibria for n players Finite horizon truncation

Enforcing optimality

Carrying capacity

Numerics

Continuous time

Conclusions § Models lack realistic constraints. § Adding even very inherent constraints can change the solutions drastically, with several surpsises.

”The tragedy of the Conclusions ”The tragedy of the commons” in commons” in the dynamic context dynamic context Agnieszka Wiszniewska- Matyszkiel § To model most of the tragedy of the commons Games problems, tools of dynamic games are required, Nash eq. Stackelberg eq. especially feedback Nash and Stackelberg equilibria. Pareto opt. Example-static § Problems of feedback Nash equilibria require solving a The tragedy Dynamic games set of coupled parametrized dynamic optimization A simple model problems, with strategies of the others as parameters. Objectives § Bellman Problems of feedback Stackelberg equilibria are even Revised sufficient condition

more complicated. Social optimum § Only few classes of such games have been solved and Nash equilibrium Nash equilibria for n players some proofs are still incomplete. Finite horizon truncation Enforcing optimality

Carrying capacity

Numerics

Continuous time

Conclusions § Adding even very inherent constraints can change the solutions drastically, with several surpsises.

”The tragedy of the Conclusions ”The tragedy of the commons” in commons” in the dynamic context dynamic context Agnieszka Wiszniewska- Matyszkiel § To model most of the tragedy of the commons Games problems, tools of dynamic games are required, Nash eq. Stackelberg eq. especially feedback Nash and Stackelberg equilibria. Pareto opt. Example-static § Problems of feedback Nash equilibria require solving a The tragedy Dynamic games set of coupled parametrized dynamic optimization A simple model problems, with strategies of the others as parameters. Objectives § Bellman Problems of feedback Stackelberg equilibria are even Revised sufficient condition

more complicated. Social optimum § Only few classes of such games have been solved and Nash equilibrium Nash equilibria for n players some proofs are still incomplete. Finite horizon truncation Enforcing optimality § Models lack realistic constraints. Carrying capacity

Numerics

Continuous time

Conclusions ”The tragedy of the Conclusions ”The tragedy of the commons” in commons” in the dynamic context dynamic context Agnieszka Wiszniewska- Matyszkiel § To model most of the tragedy of the commons Games problems, tools of dynamic games are required, Nash eq. Stackelberg eq. especially feedback Nash and Stackelberg equilibria. Pareto opt. Example-static § Problems of feedback Nash equilibria require solving a The tragedy Dynamic games set of coupled parametrized dynamic optimization A simple model problems, with strategies of the others as parameters. Objectives § Bellman Problems of feedback Stackelberg equilibria are even Revised sufficient condition

more complicated. Social optimum § Only few classes of such games have been solved and Nash equilibrium Nash equilibria for n players some proofs are still incomplete. Finite horizon truncation Enforcing optimality § Models lack realistic constraints. Carrying capacity

§ Adding even very inherent constraints can change the Numerics

solutions drastically, with several surpsises. Continuous time

Conclusions and making exhaustion possible, a linear quadratic game of resource extraction yields results which are contrary to standard results in LQ dynamic games. § The calculated unique socially optimal profile, independent on the number of players, guarantees sustainability for every initial state. § This calculation indicates that we have to be very careful about terminal condition for Bellman equation. § Social optimum for this problem is a simple counterexample to the correctness of commonly used skipping checking terminal condition —the only ”nice” solution of (BE) is not the value function), which it started a research on necessity of the terminal condition.

”The tragedy of the Conclusions – LQ games with constraints commons” in the dynamic context

Agnieszka § After imposing natural constraints (by the amount of Wiszniewska- resource available) Matyszkiel

Games Nash eq. Stackelberg eq. Pareto opt. Example-static The tragedy

Dynamic games

A simple model Objectives

Bellman Revised sufficient condition

Social optimum

Nash equilibrium Nash equilibria for n players Finite horizon truncation

Enforcing optimality

Carrying capacity

Numerics

Continuous time

Conclusions a linear quadratic game of resource extraction yields results which are contrary to standard results in LQ dynamic games. § The calculated unique socially optimal profile, independent on the number of players, guarantees sustainability for every initial state. § This calculation indicates that we have to be very careful about terminal condition for Bellman equation. § Social optimum for this problem is a simple counterexample to the correctness of commonly used skipping checking terminal condition —the only ”nice” solution of (BE) is not the value function), which it started a research on necessity of the terminal condition.

”The tragedy of the Conclusions – LQ games with constraints commons” in the dynamic context

Agnieszka § After imposing natural constraints (by the amount of Wiszniewska- resource available) and making exhaustion possible, Matyszkiel

Games Nash eq. Stackelberg eq. Pareto opt. Example-static The tragedy

Dynamic games

A simple model Objectives

Bellman Revised sufficient condition

Social optimum

Nash equilibrium Nash equilibria for n players Finite horizon truncation

Enforcing optimality

Carrying capacity

Numerics

Continuous time

Conclusions § The calculated unique socially optimal profile, independent on the number of players, guarantees sustainability for every initial state. § This calculation indicates that we have to be very careful about terminal condition for Bellman equation. § Social optimum for this problem is a simple counterexample to the correctness of commonly used skipping checking terminal condition —the only ”nice” solution of (BE) is not the value function), which it started a research on necessity of the terminal condition.

”The tragedy of the Conclusions – LQ games with constraints commons” in the dynamic context

Agnieszka § After imposing natural constraints (by the amount of Wiszniewska- resource available) and making exhaustion possible, a Matyszkiel

linear quadratic game of resource extraction yields Games Nash eq. results which are contrary to standard results in LQ Stackelberg eq. Pareto opt. dynamic games. Example-static The tragedy

Dynamic games

A simple model Objectives

Bellman Revised sufficient condition

Social optimum

Nash equilibrium Nash equilibria for n players Finite horizon truncation

Enforcing optimality

Carrying capacity

Numerics

Continuous time

Conclusions independent on the number of players, guarantees sustainability for every initial state. § This calculation indicates that we have to be very careful about terminal condition for Bellman equation. § Social optimum for this problem is a simple counterexample to the correctness of commonly used skipping checking terminal condition —the only ”nice” solution of (BE) is not the value function), which it started a research on necessity of the terminal condition.

”The tragedy of the Conclusions – LQ games with constraints commons” in the dynamic context

Agnieszka § After imposing natural constraints (by the amount of Wiszniewska- resource available) and making exhaustion possible, a Matyszkiel

linear quadratic game of resource extraction yields Games Nash eq. results which are contrary to standard results in LQ Stackelberg eq. Pareto opt. dynamic games. Example-static The tragedy § The calculated unique socially optimal profile, Dynamic games

A simple model Objectives

Bellman Revised sufficient condition

Social optimum

Nash equilibrium Nash equilibria for n players Finite horizon truncation

Enforcing optimality

Carrying capacity

Numerics

Continuous time

Conclusions guarantees sustainability for every initial state. § This calculation indicates that we have to be very careful about terminal condition for Bellman equation. § Social optimum for this problem is a simple counterexample to the correctness of commonly used skipping checking terminal condition —the only ”nice” solution of (BE) is not the value function), which it started a research on necessity of the terminal condition.

”The tragedy of the Conclusions – LQ games with constraints commons” in the dynamic context

Agnieszka § After imposing natural constraints (by the amount of Wiszniewska- resource available) and making exhaustion possible, a Matyszkiel

linear quadratic game of resource extraction yields Games Nash eq. results which are contrary to standard results in LQ Stackelberg eq. Pareto opt. dynamic games. Example-static The tragedy § The calculated unique socially optimal profile, Dynamic games

independent on the number of players, A simple model Objectives

Bellman Revised sufficient condition

Social optimum

Nash equilibrium Nash equilibria for n players Finite horizon truncation

Enforcing optimality

Carrying capacity

Numerics

Continuous time

Conclusions § This calculation indicates that we have to be very careful about terminal condition for Bellman equation. § Social optimum for this problem is a simple counterexample to the correctness of commonly used skipping checking terminal condition —the only ”nice” solution of (BE) is not the value function), which it started a research on necessity of the terminal condition.

”The tragedy of the Conclusions – LQ games with constraints commons” in the dynamic context

Agnieszka § After imposing natural constraints (by the amount of Wiszniewska- resource available) and making exhaustion possible, a Matyszkiel

linear quadratic game of resource extraction yields Games Nash eq. results which are contrary to standard results in LQ Stackelberg eq. Pareto opt. dynamic games. Example-static The tragedy § The calculated unique socially optimal profile, Dynamic games

independent on the number of players, guarantees A simple model sustainability for every initial state. Objectives Bellman Revised sufficient condition

Social optimum

Nash equilibrium Nash equilibria for n players Finite horizon truncation

Enforcing optimality

Carrying capacity

Numerics

Continuous time

Conclusions § Social optimum for this problem is a simple counterexample to the correctness of commonly used skipping checking terminal condition —the only ”nice” solution of (BE) is not the value function), which it started a research on necessity of the terminal condition.

”The tragedy of the Conclusions – LQ games with constraints commons” in the dynamic context

Agnieszka § After imposing natural constraints (by the amount of Wiszniewska- resource available) and making exhaustion possible, a Matyszkiel

linear quadratic game of resource extraction yields Games Nash eq. results which are contrary to standard results in LQ Stackelberg eq. Pareto opt. dynamic games. Example-static The tragedy § The calculated unique socially optimal profile, Dynamic games

independent on the number of players, guarantees A simple model sustainability for every initial state. Objectives Bellman § This calculation indicates that we have to be very Revised sufficient condition careful about terminal condition for Bellman equation. Social optimum Nash equilibrium Nash equilibria for n players Finite horizon truncation

Enforcing optimality

Carrying capacity

Numerics

Continuous time

Conclusions —the only ”nice” solution of (BE) is not the value function), which it started a research on necessity of the terminal condition.

”The tragedy of the Conclusions – LQ games with constraints commons” in the dynamic context

Agnieszka § After imposing natural constraints (by the amount of Wiszniewska- resource available) and making exhaustion possible, a Matyszkiel

linear quadratic game of resource extraction yields Games Nash eq. results which are contrary to standard results in LQ Stackelberg eq. Pareto opt. dynamic games. Example-static The tragedy § The calculated unique socially optimal profile, Dynamic games

independent on the number of players, guarantees A simple model sustainability for every initial state. Objectives Bellman § This calculation indicates that we have to be very Revised sufficient condition careful about terminal condition for Bellman equation. Social optimum Nash equilibrium § Social optimum for this problem is a simple Nash equilibria for n players counterexample to the correctness of commonly used Finite horizon truncation Enforcing optimality skipping checking terminal condition Carrying capacity

Numerics

Continuous time

Conclusions which it started a research on necessity of the terminal condition.

”The tragedy of the Conclusions – LQ games with constraints commons” in the dynamic context

Agnieszka § After imposing natural constraints (by the amount of Wiszniewska- resource available) and making exhaustion possible, a Matyszkiel

linear quadratic game of resource extraction yields Games Nash eq. results which are contrary to standard results in LQ Stackelberg eq. Pareto opt. dynamic games. Example-static The tragedy § The calculated unique socially optimal profile, Dynamic games

independent on the number of players, guarantees A simple model sustainability for every initial state. Objectives Bellman § This calculation indicates that we have to be very Revised sufficient condition careful about terminal condition for Bellman equation. Social optimum Nash equilibrium § Social optimum for this problem is a simple Nash equilibria for n players counterexample to the correctness of commonly used Finite horizon truncation Enforcing optimality skipping checking terminal condition —the only ”nice” Carrying capacity solution of (BE) is not the value function), Numerics

Continuous time

Conclusions ”The tragedy of the Conclusions – LQ games with constraints commons” in the dynamic context

Agnieszka § After imposing natural constraints (by the amount of Wiszniewska- resource available) and making exhaustion possible, a Matyszkiel

linear quadratic game of resource extraction yields Games Nash eq. results which are contrary to standard results in LQ Stackelberg eq. Pareto opt. dynamic games. Example-static The tragedy § The calculated unique socially optimal profile, Dynamic games

independent on the number of players, guarantees A simple model sustainability for every initial state. Objectives Bellman § This calculation indicates that we have to be very Revised sufficient condition careful about terminal condition for Bellman equation. Social optimum Nash equilibrium § Social optimum for this problem is a simple Nash equilibria for n players counterexample to the correctness of commonly used Finite horizon truncation Enforcing optimality skipping checking terminal condition —the only ”nice” Carrying capacity solution of (BE) is not the value function), which it Numerics

started a research on necessity of the terminal Continuous time

condition. Conclusions § For Nash equilibrium for n players, if it exists, it is piecewise linear with infinitely many pieces and infinitely many points of discontinuity – negative results proved for regular solutions. We can calculate them up to discontinuity points. § Discountinuity appears already in the two stage truncation of the game. § The results are unchanged if the linear dynamic is modified above some level to capture carrying capacity of the environment. § We also found tax rate of linear tax enforcing social optimality. § We can calculate such a tax although we cannot calculate Nash equilibria for the original problem. § The continuum of players game helps to find solutions for n players games! § Be careful with numerics!

§ ”The tragedy of the Nash equilibrium for the continuum of players case is commons” in the piecewise linear with value function piecewise quadratic dynamic context Agnieszka with infinitely many pieces, non-monotone, Wiszniewska- non-differentiable. Matyszkiel

Games Nash eq. Stackelberg eq. Pareto opt. Example-static The tragedy

Dynamic games

A simple model Objectives

Bellman Revised sufficient condition

Social optimum

Nash equilibrium Nash equilibria for n players Finite horizon truncation

Enforcing optimality

Carrying capacity

Numerics

Continuous time

Conclusions – negative results proved for regular solutions. We can calculate them up to discontinuity points. § Discountinuity appears already in the two stage truncation of the game. § The results are unchanged if the linear dynamic is modified above some level to capture carrying capacity of the environment. § We also found tax rate of linear tax enforcing social optimality. § We can calculate such a tax although we cannot calculate Nash equilibria for the original problem. § The continuum of players game helps to find solutions for n players games! § Be careful with numerics!

§ ”The tragedy of the Nash equilibrium for the continuum of players case is commons” in the piecewise linear with value function piecewise quadratic dynamic context Agnieszka with infinitely many pieces, non-monotone, Wiszniewska- non-differentiable. Matyszkiel

§ For Nash equilibrium for n players, if it exists, it is Games Nash eq. piecewise linear with infinitely many pieces and infinitely Stackelberg eq. Pareto opt. many points of discontinuity Example-static The tragedy

Dynamic games

A simple model Objectives

Bellman Revised sufficient condition

Social optimum

Nash equilibrium Nash equilibria for n players Finite horizon truncation

Enforcing optimality

Carrying capacity

Numerics

Continuous time

Conclusions § Discountinuity appears already in the two stage truncation of the game. § The results are unchanged if the linear dynamic is modified above some level to capture carrying capacity of the environment. § We also found tax rate of linear tax enforcing social optimality. § We can calculate such a tax although we cannot calculate Nash equilibria for the original problem. § The continuum of players game helps to find solutions for n players games! § Be careful with numerics!

§ ”The tragedy of the Nash equilibrium for the continuum of players case is commons” in the piecewise linear with value function piecewise quadratic dynamic context Agnieszka with infinitely many pieces, non-monotone, Wiszniewska- non-differentiable. Matyszkiel

§ For Nash equilibrium for n players, if it exists, it is Games Nash eq. piecewise linear with infinitely many pieces and infinitely Stackelberg eq. Pareto opt. many points of discontinuity – negative results proved Example-static for regular solutions. We can calculate them up to The tragedy discontinuity points. Dynamic games A simple model Objectives

Bellman Revised sufficient condition

Social optimum

Nash equilibrium Nash equilibria for n players Finite horizon truncation

Enforcing optimality

Carrying capacity

Numerics

Continuous time

Conclusions § The results are unchanged if the linear dynamic is modified above some level to capture carrying capacity of the environment. § We also found tax rate of linear tax enforcing social optimality. § We can calculate such a tax although we cannot calculate Nash equilibria for the original problem. § The continuum of players game helps to find solutions for n players games! § Be careful with numerics!

§ ”The tragedy of the Nash equilibrium for the continuum of players case is commons” in the piecewise linear with value function piecewise quadratic dynamic context Agnieszka with infinitely many pieces, non-monotone, Wiszniewska- non-differentiable. Matyszkiel

§ For Nash equilibrium for n players, if it exists, it is Games Nash eq. piecewise linear with infinitely many pieces and infinitely Stackelberg eq. Pareto opt. many points of discontinuity – negative results proved Example-static for regular solutions. We can calculate them up to The tragedy discontinuity points. Dynamic games A simple model § Discountinuity appears already in the two stage Objectives truncation of the game. Bellman Revised sufficient condition

Social optimum

Nash equilibrium Nash equilibria for n players Finite horizon truncation

Enforcing optimality

Carrying capacity

Numerics

Continuous time

Conclusions § We also found tax rate of linear tax enforcing social optimality. § We can calculate such a tax although we cannot calculate Nash equilibria for the original problem. § The continuum of players game helps to find solutions for n players games! § Be careful with numerics!

§ ”The tragedy of the Nash equilibrium for the continuum of players case is commons” in the piecewise linear with value function piecewise quadratic dynamic context Agnieszka with infinitely many pieces, non-monotone, Wiszniewska- non-differentiable. Matyszkiel

§ For Nash equilibrium for n players, if it exists, it is Games Nash eq. piecewise linear with infinitely many pieces and infinitely Stackelberg eq. Pareto opt. many points of discontinuity – negative results proved Example-static for regular solutions. We can calculate them up to The tragedy discontinuity points. Dynamic games A simple model § Discountinuity appears already in the two stage Objectives truncation of the game. Bellman Revised sufficient condition § The results are unchanged if the linear dynamic is Social optimum modified above some level to capture carrying capacity Nash equilibrium of the environment. Nash equilibria for n players Finite horizon truncation

Enforcing optimality

Carrying capacity

Numerics

Continuous time

Conclusions § We can calculate such a tax although we cannot calculate Nash equilibria for the original problem. § The continuum of players game helps to find solutions for n players games! § Be careful with numerics!

§ ”The tragedy of the Nash equilibrium for the continuum of players case is commons” in the piecewise linear with value function piecewise quadratic dynamic context Agnieszka with infinitely many pieces, non-monotone, Wiszniewska- non-differentiable. Matyszkiel

§ For Nash equilibrium for n players, if it exists, it is Games Nash eq. piecewise linear with infinitely many pieces and infinitely Stackelberg eq. Pareto opt. many points of discontinuity – negative results proved Example-static for regular solutions. We can calculate them up to The tragedy discontinuity points. Dynamic games A simple model § Discountinuity appears already in the two stage Objectives truncation of the game. Bellman Revised sufficient condition § The results are unchanged if the linear dynamic is Social optimum modified above some level to capture carrying capacity Nash equilibrium of the environment. Nash equilibria for n players Finite horizon truncation § We also found tax rate of linear tax enforcing social Enforcing optimality

optimality. Carrying capacity

Numerics

Continuous time

Conclusions § The continuum of players game helps to find solutions for n players games! § Be careful with numerics!

§ ”The tragedy of the Nash equilibrium for the continuum of players case is commons” in the piecewise linear with value function piecewise quadratic dynamic context Agnieszka with infinitely many pieces, non-monotone, Wiszniewska- non-differentiable. Matyszkiel

§ For Nash equilibrium for n players, if it exists, it is Games Nash eq. piecewise linear with infinitely many pieces and infinitely Stackelberg eq. Pareto opt. many points of discontinuity – negative results proved Example-static for regular solutions. We can calculate them up to The tragedy discontinuity points. Dynamic games A simple model § Discountinuity appears already in the two stage Objectives truncation of the game. Bellman Revised sufficient condition § The results are unchanged if the linear dynamic is Social optimum modified above some level to capture carrying capacity Nash equilibrium of the environment. Nash equilibria for n players Finite horizon truncation § We also found tax rate of linear tax enforcing social Enforcing optimality

optimality. Carrying capacity

§ We can calculate such a tax although we cannot Numerics

calculate Nash equilibria for the original problem. Continuous time

Conclusions § ”The tragedy of the Nash equilibrium for the continuum of players case is commons” in the piecewise linear with value function piecewise quadratic dynamic context Agnieszka with infinitely many pieces, non-monotone, Wiszniewska- non-differentiable. Matyszkiel

§ For Nash equilibrium for n players, if it exists, it is Games Nash eq. piecewise linear with infinitely many pieces and infinitely Stackelberg eq. Pareto opt. many points of discontinuity – negative results proved Example-static for regular solutions. We can calculate them up to The tragedy discontinuity points. Dynamic games A simple model § Discountinuity appears already in the two stage Objectives truncation of the game. Bellman Revised sufficient condition § The results are unchanged if the linear dynamic is Social optimum modified above some level to capture carrying capacity Nash equilibrium of the environment. Nash equilibria for n players Finite horizon truncation § We also found tax rate of linear tax enforcing social Enforcing optimality

optimality. Carrying capacity

§ We can calculate such a tax although we cannot Numerics

calculate Nash equilibria for the original problem. Continuous time § The continuum of players game helps to find solutions Conclusions for n players games! § Be careful with numerics! ”The tragedy of the commons” in the dynamic context

Agnieszka Wiszniewska- Matyszkiel

Games Nash eq. Stackelberg eq. Pareto opt. Example-static The tragedy Thank you for your attention! Dynamic games A simple model Objectives

Bellman Revised sufficient condition

Social optimum

Nash equilibrium Nash equilibria for n players Finite horizon truncation

Enforcing optimality

Carrying capacity

Numerics

Continuous time

Conclusions