Part 3 Extensive Form and Bayesian Games - 25 Sep 2014 © Tom Lenaerts, 2013 © Tom Lenaerts, 2013 Example Example

INFO-F-409 • What? Why? • Rational choice • Strategic games Learning dynamics • Nash Equilibrium Extensive form games, subgame-perfect Nash equilibria • Best response and game theoretical extensions • Dominance • Mixed strategies • Mixed-strategy Nash Equilibria T. Lenaerts and A. Nowé • Support ﬁnding MLG, Université Libre de Bruxelles and • Lemke-Howson algorithm DINF, Vrije Universiteit Brussel

© Tom Lenaerts, 2013 © Tom Lenaerts, 2013 Extensive form games Extensive form games Strategic games assume that each decision maker chooses Strategic games assume that each decision maker chooses her actions once and for all her actions once and for all

Fragment Fragment fromThe Big- fromThe Big- Bang Theory Bang Theory (2008) (2008)

Hence it does not take into account the sequential Hence it does not take into account the sequential structure of decision making structure of decision making

part 3 Extensive form and Bayesian games - 25 Sep 2014 © Tom Lenaerts, 2013 © Tom Lenaerts, 2013 Example Example

In an Entry game there are two players: A (the incumbent) and B (the challenger)

So B may decide to challenge (or to stay out) and if B challenges A See also VUB Bachelor course on Artiﬁcial may either allow entry or ﬁght against entry Intelligence

In an Entry game there are two players: A history is the sequence of actions taken by the players A (the incumbent) and B (the challenger) up to some decision point A terminal history is a history that contains the action choices of all the players up until the point where the payoff is distributed

The Entry game has 3 terminal histories 1.(Challenge, Allow entry), 2.(Challenge, Fight entry ) and 3.(Stay out)

So B may decide to challenge (or to stay out) and if B challenges A A sub-history is a history that contains part of a terminal may either allow entry or ﬁght against entry history

part 3 Extensive form and Bayesian games - 25 Sep 2014 © Tom Lenaerts, 2013 © Tom Lenaerts, 2013 Some terminology Tree representation

One can assign to every sub-history which is not the complete history (= proper sub-history) a player using a player function P(proper sub-history) :

(Challenge) and ∅ are proper sub-histories of (Challenge, Allow entry) and (Challenge,Fight entry)

Thus P(Challenge) indicates that player A (the incumbent) acts after that point Thus P(∅) indicates that player B (the challenger) acts after that point (which is the start of the game)

© Tom Lenaerts, 2013 © Tom Lenaerts, 2013 Tree representation Tree representation B prefers (Challenge, Allow B prefers (Challenge, Allow entry) over (Stay out) over entry) over (Stay out) over (Challenge, Fight entry) (Challenge, Fight entry)

A prefers (Stay out) over (Challenge, Allow entry) over (Challenge, Fight entry)

1 1,2

2 0 2,1 0,0

part 3 Extensive form and Bayesian games - 25 Sep 2014 © Tom Lenaerts, 2013 © Tom Lenaerts, 2013 Extensive game Perfect Information

Deﬁnition : • What: An extensive game with perfect information consists of : • Players know the node they are in • a set of players • a set of terminal histories with the property that none of • They know all the prior choices, including these histories is a proper sub-history of another those of other agents • A player function that assigns a player to every proper • What happens when agents have only incomplete sub-history that can be derived from the terminal knowledge of the actions taken by others or no histories longer remember their past actions? • For each player, preferences over the set of terminal histories • Games with Imperfect Information (see later)

If all terminal histories are finite, then the game has a A(∅) ∈ {Challenge, Stay out} finite horizon If a game has a finite horizon and finitely many terminal histories then the game is called finite A(Challenge) Definition : ∈ {Allow, Fight} A(Stay out)= A strategy of a player i in an extensive game with ∅ 1,2 perfect information is a function that assigns to each Strategies for B are history h after which it is player i’s turn to move {Challenge, Stay out} (P(h)=i, where P is the player function) an action in A(h), i.e. the set of available actions after h 0,0 Strategies for A are 2,1 {Allow Entry, Fight Entry}

part 3 Extensive form and Bayesian games - 25 Sep 2014 © Tom Lenaerts, 2013 © Tom Lenaerts, 2013 Strategies Equilibrium

A(∅) ∈ {C, D} One way to determine the steady states is to transform the extensive game into a strategic game and determine the Nash equilibria in that way

Allow Fight A(C) ∈ {E, F} A(D) ∈ {E, F} 1 0 Challenge 2 0 2 2 Stay out 1 1 Strategies for 1 are {C, D} Strategies for 2 are {CE, CF, DE, DF}

One way to determine the steady states is to transform the extensive One way to determine the steady states is to transform the extensive game into a strategic game and determine the Nash equilibria in that way game into a strategic game and determine the Nash equilibria in that way

Allow Fight Allow Fight 1 0 11 0 Challenge Challenge 22 0 22 0 2 2 22 22 Stay out Stay out 1 11 1 11

part 3 Extensive form and Bayesian games - 25 Sep 2014 © Tom Lenaerts, 2013 © Tom Lenaerts, 2013 Equilibrium Equilibrium

One way to determine the steady states is to transform the extensive game into a strategic game and determine the Nash equilibria in that way This NE ignores the sequential structure of the game since it treats strategies as choices that are made once and for all a different notion of equilibrium is required to describe the Allow Fight This NE seems to assume steady state of an extensive game 11 0 that player B knew that Challenge player A was going to fight and therefore selected No The idea is that this equilibrium requires each player’s 22 0 challenge strategy to be optimal, given the other players’ strategies, not ! only at the start but at every possible history 22 22 This is at odd with the Stay out extensive model where the ? action Stay out will never 1 11 be osberved when player B To reach this new definition, we first need to define the choses not to challenge A notion of a sub-game

© Tom Lenaerts, 2013 © Tom Lenaerts, 2013 Sub-game Sub-game Deﬁnition : The Entry game has 2 sub-games Let Γ be an extensive game with perfect information, with player function P. For any nonterminal history h of Γ, the subgame Γ(h) following the history h is the following extensive game Players The players in Γ 1,2 Terminal histories The set of all sequences h’ of actions such that (h,h’) is a terminal history of Γ Player function The player function P(h,h’) is assigned to each proper sub-history h’of a terminal history 2,1 0,0 Preferences each player prefers h’ to h’’ if and only if she prefers (h,h’) to (h,h’’) in Γ

part 3 Extensive form and Bayesian games - 25 Sep 2014 © Tom Lenaerts, 2013 © Tom Lenaerts, 2013 Sub-game Sub-game

The Entry game has 2 sub-games The Entry game has 2 sub-games S S

1,2 1,2

2,1 0,0 2,1 0,0

The Entry game has 2 sub-games The game below has 3 sub-games S

1,2

2,1 0,0

S2 is a proper sub-game

part 3 Extensive form and Bayesian games - 25 Sep 2014 © Tom Lenaerts, 2013 © Tom Lenaerts, 2013 Sub-game Sub-game

The game below has 3 sub-games The game below has 3 sub-games S1 S1

S2 S3 S2 S3

S2 and S3 are proper sub-games

part 3 Extensive form and Bayesian games - 25 Sep 2014 © Tom Lenaerts, 2013 Sub-game perfect © Tom Lenaerts, 2013 Sub-game perfect equilibrium equilibrium Definition : Definition : the strategy profile s* in an extensive game with perfect Equivalently, for every player i and every history h after which it is information is a sub-game perfect equilibrium if, for every player player i’s turn to move, i, every history h after which it is player i’s turn to move (P(h)=i), and every strategy ri of player i, the terminal history Oh(s*) ui(Oh(s*)) ≥ ui(Oh(ri,s-i*)) for every strategy ri of player i generated by s* after the history h is at least as good according to ! player i’s preferences as the terminal Oh(ri, s-i*) generated by the where ui is a payoff function that represents the player i’s strategy profile (ri,s-i*) in which player i chooses ri while every preferences and Oh(s) is the terminal history consisting of h other player j chooses sj* followed by the sequence of actions generated by s after h

Every sub-game perfect equilibrium is a Nash equilibrium

11 0 11 0 Challenge Best response analysis of the strategic game Challenge Best response analysis of the strategic game 22 0 claims that there are two NE 22 0 claims that there are two NE ! ! 22 22 22 22 Are they also subgame perfect ? Are they also subgame perfect ? Stay out Stay out 1 11 1 11

Take the NE s*=(challenge,allow): Sub-game S1 Take the NE s*=(challenge,allow): Sub-game S2

Player B (i=B) moves at h=∅→Oh(s*) =(challenge, allow) Player A (i=A) moves at h=challenge→Oh(s*) =(challenge, allow)

So now we check the payoffs for every action rB of B, given Oh(s*) So now we check the payoffs for every action rA of A, given Oh(s*)

rB*= Challenge → uB(Oh(s*)) = 2 rA*=allow → uA(Oh(s*)) = 1

rB= Stay out → uB(Oh(rB,s-B*)) = 1 rA=ﬁght → uA(Oh(rA, s-A*)) = 0

part 3 Extensive form and Bayesian games - 25 Sep 2014 © Tom Lenaerts, 2013 © Tom Lenaerts, 2013 Example Example Allow Fight Allow Fight

Take the NE s*=(challenge,allow): Sub-game S2 Take the NE s*=(Stay out,ﬁght): Sub-game S2

Player A (i=A) moves at h=challenge→Oh(s*) =(challenge, allow) Player B (i=B) moves at h=∅→Oh(s*) =(Stay out, ﬁght)

So now we check the payoffs for every action rA of A, given Oh(s*) So now we check the payoffs for every action rB of B, given Oh(s*)

rA*=allow → uA(Oh(s*)) = 1 So (challenge,allow) is a subgame rB*=Stay out → uB(Oh(s*)) = 1

rA=ﬁght → uA(Oh(rA, s-A*)) = 0 perfect equilibirum rB= challenge → uB(Oh(rB, s-B*)) = 0

Take the NE s*=(stay out,ﬁght): Sub-game S2 Take the NE s*=(stay out,ﬁght): Sub-game S2

Player A (i=A) moves at h=challenge→Oh(s*) =(challenge, ﬁght) Player A (i=A) moves at h=challenge→Oh(s*) =(challenge, ﬁght)

So now we check the payoffs for every action rA of A, given Oh(s*) So now we check the payoffs for every action rA of A, given Oh(s*)

rA*=fight → uA(Oh(s*)) = 0 rA*=fight → uA(Oh(s*)) = 0 So (no challenge, fight) is a NOT

rA=allow → ui(Oh(ra, s-A*)) = 1 rA=allow → ui(Oh(ra, s-A*)) = 1 subgame perfect equilibirum

part 3 Extensive form and Bayesian games - 25 Sep 2014 © Tom Lenaerts, 2013 © Tom Lenaerts, 2013 Finding the SPE Finding the SPE In a game with a ﬁnite horizon, the set of subgame perfect In a game with a ﬁnite horizon, the set of subgame perfect equilibria can be found more easily by using a procedure that equilibria can be found more easily by using a procedure that extends the backward induction process : extends the backward induction process :

1,2 1,2

2,1 0,0 2,1 0,0

© Tom Lenaerts, 2013 © Tom Lenaerts, 2013 Finding the SPE Finding the SPE In a game with a ﬁnite horizon, the set of subgame perfect In a game with a ﬁnite horizon, the set of subgame perfect equilibria can be found more easily by using a procedure that equilibria can be found more easily by using a procedure that extends the backward induction process : extends the backward induction process :

1,2 1,2

2,1 0,0 2,1 0,0

1,2 1,2

2,1 0,0 2,1 0,0 unique SPE

part 3 Extensive form and Bayesian games - 25 Sep 2014 © Tom Lenaerts, 2013 © Tom Lenaerts, 2013 Finding the SPE Finding the SPE

unique SPE

Originally invented by Rosenthal in 1982 (with 100 moves)

unique SPE An eight-moves centipede game Note that passing (P) the money always means that you may Backward induction cannot be applied to every extensive receive less than currently possible game with perfect information : e.g. games with inﬁnitely long histories ,... What is the Sub-game perfect Nash equilibrium here?

part 3 Extensive form and Bayesian games - 25 Sep 2014 © Tom Lenaerts, 2013 © Tom Lenaerts, 2013 The Centipede game The Centipede game

Originally invented by Rosenthal in 1982 (with 100 moves) Originally invented by Rosenthal in 1982 (with 100 moves)

part 3 Extensive form and Bayesian games - 25 Sep 2014 © Tom Lenaerts, 2013 © Tom Lenaerts, 2013 The Centipede game The Centipede game

Originally invented by Rosenthal in 1982 (with 100 moves) Originally invented by Rosenthal in 1982 (with 100 moves)

part 3 Extensive form and Bayesian games - 25 Sep 2014 © Tom Lenaerts, 2013 © Tom Lenaerts, 2013 The Centipede game The Centipede game

Originally invented by Rosenthal in 1982 (with 100 moves) Originally invented by Rosenthal in 1982 (with 100 moves)

So the rational choice is to take immediately the money

Yet experiments show different results (see McKelvey and Palfrey, 1992 and Nagel and Tang 1998)

from McKelvey and Palfrey (1992) An Experimental Study of the Centipede Game. Econometrica 60(4):803-836

Opening scene of The Dark Knight

part 3 Extensive form and Bayesian games - 25 Sep 2014 © Tom Lenaerts, 2013 © Tom Lenaerts, 2013 The pirate puzzle The pirate puzzle

Assume 5 pirates (A, B, C, D and E) who have found a treasure of 100 gold coins. They split up the money according to an ancient pirate code that depends on the ferocity of the pirates.

Assume that pirate E is the ﬁercest, then D, then C ... So the strict ferocity ranking is E>D>C>B>A

The code works as follows; 1.The fiercest pirate proposes a split of the gold 2.All the pirates including the fiercest vote whether to accept the split 3.If they agree, the split happens 4.Otherwise the pirate who proposed the split gets thrown Opening scene of The Dark Knight overboard and is eaten by the sharks 5.The process is repeated with the next fiercest pirate, until a proposal is accepted

What proposal should the ﬁercest pirate make ? This problem can be solved by reasoning backwards:

Should he (or she) give away most of the loot?

The code works as follows; 1.The fiercest pirate proposes a split of the gold 2.All the pirates including the fiercest vote whether to accept the split 3.If they agree, the split happens 4.Otherwise the pirate who proposed the split gets thrown E D C B A overboard and is eaten by the sharks 5.The process is repeated with the next fiercest pirate, until a proposal is accepted

part 3 Extensive form and Bayesian games - 25 Sep 2014 © Tom Lenaerts, 2013 © Tom Lenaerts, 2013 The pirate puzzle The pirate puzzle

This problem can be solved by reasoning backwards: This problem can be solved by reasoning backwards:

Let’s start with game where only 2 pirates remain : A and B

E D C B A E D C B A B: 100 - 0 C: 99 - 0 - 1

This problem can be solved by reasoning backwards: This problem can be solved by reasoning backwards:

E D C B A E D C B A

D: 99 - 0 - 1- 0 E: 98 - 0 - 1- 0 -1

part 3 Extensive form and Bayesian games - 25 Sep 2014 © Tom Lenaerts, 2013 © Tom Lenaerts, 2013 The pirate puzzle Extensions Simultaneous moves : So what happened in the opening scene ? In some situations, after some sequence of The joker bribed every weaker robber with actions of the players, the players may need to the promise of a bigger share choose the next action simultaneously Chance moves : reaching in the end the situation of the two pirates , allowing him to steal everything Sometimes, random events may occur that alter the sequence of actions Bayesian games Sometimes you lack information about the opponent

Take for instance the following variant of the battle of the This deﬁnes the following extensive form game: sexes: First one player decides to stay home and • a set of players: {1,2} watch television or to attend a concert. • a set of terminal histories: {Book, (Concert, (B,B)), (Concert, ! (B,S)), (Concert, (S,B)), (Concert, (S,S))} When he or she decided to stay home, the • A player function: P(∅) = {1}, P(Concert)={1,2} game ends • Preferences: ! • Player 1: (Concert,(B,B)) > Book > (Concert,(S,S)) > If he or she decides to attend the concert, (Concert,(S,B)) = (Concert,(B,S)) then both players have to choose • Player 2: (Concert,(S,S)) > Book > (Concert,(B,B)) > simultaneously which concert, Bach or (Concert,(S,B)) = (Concert,(B,S)) Stravinsky

part 3 Extensive form and Bayesian games - 25 Sep 2014 © Tom Lenaerts, 2013 © Tom Lenaerts, 2013 Simultaneous moves Simultaneous moves

How to ﬁnd the equilibrium? How to ﬁnd the equilibrium?

Actions of player 1: A1(∅)={Concert, Book}, A1(Concert)={B, S}, B S Actions of player 2: A2(Concert)={B, S} (concert, B) 3,1 0,0 Strategies player 1: (Concert, B), (Concert, (concert, S) 0,0 1,3 S), (Book, B) and (Book, S) Strategies player 2: B and S (book, B) 2,2 2,2 (book, S) 2,2 2,2

How to ﬁnd the equilibrium? How to ﬁnd the equilibrium?

B S B S (concert, B) 33,1 0,0 (concert, B) 33,11 0,0 (concert, S) 0,0 1,3 (concert, S) 0,0 1,33 (book, B) 2,2 22,2 (book, B) 2,22 22,22 (book, S) 2,2 22,2 (book, S) 2,22 22,22

part 3 Extensive form and Bayesian games - 25 Sep 2014 © Tom Lenaerts, 2013 © Tom Lenaerts, 2013

Simultaneous moves S Simultaneous moves How to ﬁnd the equilibrium? How to ﬁnd the equilibrium?

B S S (concert, B) 33,11 0,0 (concert, S) 0,0 1,33 (book, B) 2,22 22,22 (book, S) 2,22 22,22

S Simultaneous moves S Simultaneous moves How to ﬁnd the equilibrium? How to ﬁnd the equilibrium?

S S

part 3 Extensive form and Bayesian games - 25 Sep 2014 © Tom Lenaerts, 2013 © Tom Lenaerts, 2013

S Simultaneous moves S Simultaneous moves How to ﬁnd the equilibrium? How to ﬁnd the equilibrium?

1. What are the pure strategy 1. What are the pure strategy equilibria for the sub-game S2 equilibria for the sub-game S2 S S

S Simultaneous moves S Simultaneous moves How to ﬁnd the equilibrium? How to ﬁnd the equilibrium?

1. What are the pure strategy 1. What are the pure strategy equilibria for the sub-game S2 equilibria for the sub-game S2 S (B,B) and (S,S) S (B,B) and (S,S)

2. What is the optimal choice in S1 of the the outcome of S2 is (B,B)

3 1 3 1

3. What is the optimal choice in S1 1 3 1 3 of the the outcome of S2 is (S,S)

part 3 Extensive form and Bayesian games - 25 Sep 2014 © Tom Lenaerts, 2013 © Tom Lenaerts, 2013

S Simultaneous moves S Simultaneous moves How to ﬁnd the equilibrium? How to ﬁnd the equilibrium?

1. What are the pure strategy 1. What are the pure strategy equilibria for the sub-game S2 equilibria for the sub-game S2 S (B,B) and (S,S) S (B,B) and (S,S)

2. What is the optimal choice in S1 2. What is the optimal choice in S1 of the the outcome of S2 is (B,B) of the the outcome of S2 is (B,B)

Concert Concert 3 1 3 1

3. What is the optimal choice in S1 3. What is the optimal choice in S1 1 3 1 3 of the the outcome of S2 is (S,S) of the the outcome of S2 is (S,S)

book

S Simultaneous moves Chance moves Thus the sub-game perfect equilibria are : Remember that in the deﬁnition of extensive-form games, there is a function P that assigns a player to each history. S ((concert,B),B) and ((book, S),S) Here, one can also assign chance as opposed to a player

As a consequence, the preferences of the players become deﬁned over the set of lotteries (probability distribution) 3 1 over terminal histories Note : BoS also has a mixed 1 3 strategy Nash equilibrium, which we did not consider here

part 3 Extensive form and Bayesian games - 25 Sep 2014 © Tom Lenaerts, 2013 © Tom Lenaerts, 2013 Chance moves Chance moves

after the move of player 1, chance may direct you to a terminal history or to a decision of player 2

Backward induction: Backward induction:

In sub-game S2, player 2 chooses C

S Chance moves S Chance moves Backward induction: Backward induction:

In sub-game S2, player 2 chooses C In sub-game S2, player 2 chooses C

In sub-game S1, we consider the In sub-game S1, we consider the consequences of the actions of player 1 consequences of the actions of player 1

S S If A is selected then 1 obtains a payoff of 1

S Chance moves S Chance moves Backward induction: Backward induction:

In sub-game S2, player 2 chooses C In sub-game S2, player 2 chooses C

In sub-game S1, we consider the In sub-game S1, we consider the consequences of the actions of player 1 consequences of the actions of player 1

S If A is selected then 1 obtains a payoff of 1 S If A is selected then 1 obtains a payoff of 1 If B is selected then 1 obtains a payoff of 3 If B is selected then 1 obtains a payoff of 3 with probability 1/2 and a payoff of 0 with with probability 1/2 and a payoff of 0 with probability 1/2 probability 1/2 ! ! Hence an expected payoff of 3/2 Hence an expected payoff of 3/2 Thus the SPE is here is (B, C)

part 3 Extensive form and Bayesian games - 25 Sep 2014 © Tom Lenaerts, 2013 © Tom Lenaerts, 2013 Chance moves Chance moves

The duel from the good, the bad and the ugly (1966) The duel from the good, the bad and the ugly (1966)

See the web-site for the ﬁrst assignment Often players do not know the preferences One of them will be this one .... • of their opponents Sequential truel Each of persons A, B, and C has a gun containing a single bullet. Each person, as long as she is alive, may shoot at any surviving person. First A can shoot, then B (if still alive), then C (if still or they may not know how well the alive). • ! opponent knows their preferences Denote by pi the probability that player i hits her intended target; assume that 0 < pi < 1. Assume that each player wish to maximize her probability of survival; among outcomes in which her survival probability is the same, she wants the danger posed by any other survivors Bayesian games allows one to analyze any to be as small as possible. • ! Model this situation as an extensive game with perfect information and chance moves. (Draw situation in which a player is not completely the diagram. Note that the sub-games following histories in which A misses her intended target are the same). informed about an environmental aspect Find the subgame perfect equilibria of the game. (Consider only cases in which pA, pB, and pC are all different.) Explain the logic behind A’s equilibrium action. Show that “weakness is that may be relevant for her choice of action strength” for C: she is better off if pC < pB than if pC > pB. !

part 3 Extensive form and Bayesian games - 25 Sep 2014 © Tom Lenaerts, 2013 © Tom Lenaerts, 2013 Bayesian games Bayesian games These are the two potential games: Consider another variant of the battle of the sexes:

player 1 is unsure whether player 2 prefers to go out with her or prefers to avoid her. ! Let’s assume that there is equal chance for both state M (which can be based on player 1’s personal state V assessment) ! So player 1 beliefs that with probability 1/2 she plays two different games

Player 2 knows which of the two games is being played. What are the equilibria of this kind of game?

What are the equilibria of this kind of What are the equilibria of this kind of game? game?

Player 1 needs to form a belief about which kind of actions player 2 can take We can now calculate the expected when he is either one of the types payoff for all possible combinations of player 2’s types Belief of player 1; When being the left type player 2 will play B action for (B,B) (B,S) (S,B) (S,S) When being the right type player 2 will both types play S B 2 1 1 0

Then πB = 0.5 * 2 + 0.5 * 0 = 1 S 0 0,5 0,5 1 πS = 0.5 * 2 + 0.5 * 0 = 0.5

part 3 Extensive form and Bayesian games - 25 Sep 2014 © Tom Lenaerts, 2013 © Tom Lenaerts, 2013 Bayesian games Bayesian games

Deﬁnition a pure strategy Nash equilibrium Deﬁnition a pure strategy Nash equilibrium is a triple of actions, with the property that is a triple of actions, with the property that The action of player 1 is optimal, given both The action of player 1 is optimal, given both actions of the two player 2 types (and player actions of the two player 2 types (and player 1’s belief about the state) 1’s belief about the state) The action of each player 2 type is optimal, The action of each player 2 type is optimal, given the action of player 1 given the action of player 1 (B,B) (B,S) (S,B) (S,S) (B,B) (B,S) (S,B) (S,S) First, player 1 has First, player 1 has B 2 1 1 0 to determine his B 22 1 11 0 to determine his or her best or her best response given response given S 0 0,5 0,5 1 the actions of S 0 0,5 0,5 1 the actions of both types both types

B 22 1 11 0 B 22 1 11 0

S 0 0,5 0,5 1 S 0 0,5 0,5 1

best response of player 1 best response of player 1

Type “meet” Type “avoid” Type “meet” Type “avoid” Second, Second, determine the 1 0 0 2 determine the 11 0 0 2 best responses best responses of player 2 of player 2 against player 1 0 2 1 0 against player 1 0 2 1 0 in both types in both types

part 3 Extensive form and Bayesian games - 25 Sep 2014 © Tom Lenaerts, 2013 © Tom Lenaerts, 2013 Bayesian games Bayesian games (B,B) (B,S) (S,B) (S,S) (B,B) (B,S) (S,B) (S,S)

B 22 1 11 0 B 22 1 11 0

S 0 0,5 0,5 1 S 0 0,5 0,5 1

best response of player 1 best response of player 1

Type “meet” Type “avoid” Type “meet” Type “avoid” Second, Second, determine the 11 0 0 2 determine the 11 0 0 22 best responses best responses of player 2 of player 2 against player 1 0 22 1 0 against player 1 0 22 1 0 in both types in both types

B 22 1 11 0 B 22 1 11 0

S 0 0,5 0,5 1 S 0 0,5 0,5 1

best response of player 1 best response of player 1 The pure Type “meet” Type “avoid” strategy Nash Type “meet” Type “avoid” Second, equilibria can now be determine the 11 0 0 22 11 0 0 22 best responses determined by ! of player 2 matching these against player 1 0 22 11 0 results 0 22 11 0 in both types

part 3 Extensive form and Bayesian games - 25 Sep 2014 © Tom Lenaerts, 2013 © Tom Lenaerts, 2013 Bayesian games Bayesian games (B,B) (B,S) (S,B) (S,S) (B,B) (B,S) (S,B) (S,S)

B 22 1 11 0 B 22 1 11 0

S 0 0,5 0,5 1 S 0 0,5 0,5 1

best response of player 1 best response of player 1 The pure The pure strategy Nash Type “meet” Type “avoid” strategy Nash Type “meet” Type “avoid” equilibria can equilibria can now be {B,(B,S)} is a now be 11 0 0 22 11 0 0 22 determined by NE determined by ! matching these matching these results 0 22 11 0 results 0 22 11 0

(B,B) (B,S) (S,B) (S,S) We can make the game even more interesting when both players don’t know whether the other one wants to meet or avoid the other one B 22 1 11 0

S 0 0,5 0,5 1 state yy state yn every player wants only player 1 wants best response of player 1 to go out with the to go out other The pure strategy Nash Type “meet” Type “avoid” equilibria can now be 1 2 state nn determined by 1 0 0 2 matching these state ny neither of the players only player 2 wants to go out with the results 0 22 11 0 other ! wants to go out

every player wants

part 3 Extensive form and Bayesian games - 25 Sep 2014 © Tom Lenaerts, 2013 © Tom Lenaerts, 2013 Bayesian games Bayesian games

Note that in this game, player 1 cannot distinguish between states yy and Note that in this game, player 1 cannot distinguish between states yy and yn, and between ny and nn (vice versa for player 2) yn, and between ny and nn (vice versa for player 2)

state yy state yn state yy state yn

How can player 1 distinguish between yy (ny) and yn (nn)? state nn state nn state ny state ny For this the player 1 needs information (same for player 2)

© Tom Lenaerts, 2013 © Tom Lenaerts, 2013 Bayesian games Bayesian games Depending on the signal, each player has to determine what he can expect We can make the game even more interesting when both players don’t in terms of payoff, given the probabilities of each state. know whether the other one wants to meet or avoid the other one signal y1 n1 state yy state yn (B,B) (B,S) (S,B) (S,S) (B,B) (B,S) (S,B) (S,S) 2 1 B 2 1 1 0 B 0 1 1 2 The signals will provide P1 information about the state S 0 1/2 1/2 1 S 1 1/2 1/2 0 each player ﬁnds herself in signal signal signal {B,(B,S)} {B,(B,S)}

state nn B S B S B S B S state ny Each player needs to 1 0 0 2 11 0 0 22 determine the actions it will P2 2 1 1 take for each signal it 0 2 1 0 0 22 1 0 receives. state yy state yn state ny state nn

part 3 Extensive form and Bayesian games - 25 Sep 2014 © Tom Lenaerts, 2013 © Tom Lenaerts, 2013 Bayesian games Bayesian games Depending on the signal, each player has to determine what he can expect in terms of payoff, given the probabilities of each state. Putting things together, one can see that there 2 pure strategy Nash state yy state ny state yn state nn equilibria for this bayesian game y1 n1 B 2 0 B 0 2 B 2 0 B 0 22

P1 S 0 1 S 11 0 S 0 1 S 1 0 {(B,B),(B,S)} P1 {B,(B,S)} {B,(B,S)} {(B,S),B} and {(S,B),S} {(S,B),S}

(B,B) (B,S) (S,B) (S,S) (B,B) (B,S) (S,B) (S,S) {(B,S),B} and {(S,B),S} {(S,B),(S,S)} B 1 2/32/3 1/3 0 B 0 1/3 2/32/3 1 P2 {(S,B),S}

P2 S 0 2/32/3 4/3 2 S 22 4/34/3 2/32/3 0 y2 n2 y2 n2

© Tom Lenaerts, 2013 © Tom Lenaerts, 2013 Bayesian games Bayesian games Players: 1 and 2 States: meet and avoid Definition: A Bayesian game consists of Actions: for each player {B,S} A set of players Signals: For an assumed signal z, the signal function of 1 is A set of states τ1(meet)=τ1(avoid)=z And for each player Player 2 receives one of two signals, m or v. The signal function is τ (meet)= m and τ (avoid)=v A set of actions 2 2 A set of signals that she may receive Beliefs: A signal function that associates a signal with each Player 1 assigns a probability of 1/2 to each state when receiving the signal z state Player 2 assigns a probability 1 to state meet when receiving signal 1 and a probability for each signal of 1 to state avoid when receiving signal v a belief about the states consistent with the signal A payoff function over pairs (a, ω) where a is an payoffs: the payoff ui(a, meet) for each player are given by the first matrix in the figure and the action profile and is a ω state payoff ui(a, avoid) for each player are given by the second matrix in the same figure

part 3 Extensive form and Bayesian games - 25 Sep 2014 © Tom Lenaerts, 2013 Bayesian games Bayesian games Players: 1 and 2 States: {yy, yn, ny, nn} Actions: for each player {B,S} Signals: • These utility function can now be used to Player 1 receives two signals, y1 or n1; the signal function is formally deﬁne the notion of a pure τ1(yy)=τ1(yn)=y1 and τ1(ny)=τ1(nn)=n1 Player 2 receives two signals, y2 or n2; The signal function is strategy Nash equilibrium τ2(yy)=τ2(ny)=y2 and τ2(yn)=τ2(nn)=n2 This can be further extended towards Beliefs: • Player 1 assigns a probability of 1/2 to the states yy and yn when receiving the signal extensive form games, signaling games, ... y1and the probability 1/2 to the states ny and nn when receiving n1 Player 2 assigns a probability 2/3 to state yy and 1/3 to state ny when receiving signal y2 and the probability of 2/3 to state yn and 1/3 to state nn when receiving signal n2 • maybe next year :) payoffs: the payoff ui(a, ω) for each player i for all possible action pairs and states as provided by the ﬁgure above.

part 3 Extensive form and Bayesian games - 25 Sep 2014