Taking Incomplete Information Seriously: the Misunderstanding of John Harsanyi

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Taking Incomplete Information Seriously: The Misunderstanding of John Harsanyi

Stephen Morris

University of Texas at Austin November 2018 I Even if there is not perfect information (e.g., there is uncertainty and maybe even asymmetric information), there is common knowledge about the structure of the environment

I No common knowledge assumptions at all

I Everything is "common knowledge" among economic agents 2. Complete but Imperfect Information

3. Incomplete Information

A Tripartite Distinction of Informational Assumptions in Economic Analysis

1. Perfect Information I Even if there is not perfect information (e.g., there is uncertainty and maybe even asymmetric information), there is common knowledge about the structure of the environment

I No common knowledge assumptions at all

2. Complete but Imperfect Information

3. Incomplete Information

A Tripartite Distinction of Informational Assumptions in Economic Analysis

1. Perfect Information

I Everything is "common knowledge" among economic agents I No common knowledge assumptions at all

I Even if there is not perfect information (e.g., there is uncertainty and maybe even asymmetric information), there is common knowledge about the structure of the environment 3. Incomplete Information

A Tripartite Distinction of Informational Assumptions in Economic Analysis

1. Perfect Information

I Everything is "common knowledge" among economic agents 2. Complete but Imperfect Information I No common knowledge assumptions at all

3. Incomplete Information

A Tripartite Distinction of Informational Assumptions in Economic Analysis

1. Perfect Information

I Everything is "common knowledge" among economic agents 2. Complete but Imperfect Information

I Even if there is not perfect information (e.g., there is uncertainty and maybe even asymmetric information), there is common knowledge about the structure of the environment I No common knowledge assumptions at all

A Tripartite Distinction of Informational Assumptions in Economic Analysis

1. Perfect Information

I Everything is "common knowledge" among economic agents 2. Complete but Imperfect Information

1. Perfect Information

I Everything is "common knowledge" among economic agents 2. Complete but Imperfect Information

I No common knowledge assumptions at all I There is common knowledge of the structure of the game being played: players, rules of the game, feasible strategies, payo¤s, etc....; but may not know past or current actions of other players or exogenous uncertainty I LEADING EXAMPLE: Poker

I There is not common knowledge of the structure of the game being played I LEADING EXAMPLE: Almost all economic environments of interest?

3. Incomplete Information

The Tripartite Distinction in Game Theory von Neumann and Morgenstern "Theory of Games and Economic Behavior" 1944 1. Perfect Information Games I There is common knowledge of the structure of the game being played: players, rules of the game, feasible strategies, payo¤s, etc....; but may not know past or current actions of other players or exogenous uncertainty I LEADING EXAMPLE: Poker

I There is not common knowledge of the structure of the game being played I LEADING EXAMPLE: Almost all economic environments of interest?

I LEADING EXAMPLE: Chess 2. Complete but Imperfect Information

3. Incomplete Information

The Tripartite Distinction in Game Theory von Neumann and Morgenstern "Theory of Games and Economic Behavior" 1944 1. Perfect Information Games

I There is common knowledge of the structure of a game being played: players, the order in which they move, previous moves, payo¤s, etc... I There is common knowledge of the structure of the game being played: players, rules of the game, feasible strategies, payo¤s, etc....; but may not know past or current actions of other players or exogenous uncertainty I LEADING EXAMPLE: Poker

I There is not common knowledge of the structure of the game being played I LEADING EXAMPLE: Almost all economic environments of interest?

2. Complete but Imperfect Information

3. Incomplete Information

The Tripartite Distinction in Game Theory von Neumann and Morgenstern "Theory of Games and Economic Behavior" 1944 1. Perfect Information Games

I There is common knowledge of the structure of a game being played: players, the order in which they move, previous moves, payo¤s, etc... I LEADING EXAMPLE: Chess I There is not common knowledge of the structure of the game being played I LEADING EXAMPLE: Almost all economic environments of interest?

I There is common knowledge of the structure of the game being played: players, rules of the game, feasible strategies, payo¤s, etc....; but may not know past or current actions of other players or exogenous uncertainty I LEADING EXAMPLE: Poker 3. Incomplete Information

The Tripartite Distinction in Game Theory von Neumann and Morgenstern "Theory of Games and Economic Behavior" 1944 1. Perfect Information Games

I There is common knowledge of the structure of a game being played: players, the order in which they move, previous moves, payo¤s, etc... I LEADING EXAMPLE: Chess 2. Complete but Imperfect Information I There is not common knowledge of the structure of the game being played I LEADING EXAMPLE: Almost all economic environments of interest?

I LEADING EXAMPLE: Poker 3. Incomplete Information

The Tripartite Distinction in Game Theory von Neumann and Morgenstern "Theory of Games and Economic Behavior" 1944 1. Perfect Information Games

3. Incomplete Information

The Tripartite Distinction in Game Theory von Neumann and Morgenstern "Theory of Games and Economic Behavior" 1944 1. Perfect Information Games

I There is not common knowledge of the structure of the game being played The Tripartite Distinction in Game Theory von Neumann and Morgenstern "Theory of Games and Economic Behavior" 1944 1. Perfect Information Games

I There is not common knowledge of the structure of the game being played I LEADING EXAMPLE: Almost all economic environments of interest? A Pessimistic Assessment

von Neumann and Morgenstern "Theory of Games and Economic Behavior" 1944 ...we cannot avoid the assumption that all subjects under consideration are completely informed about the physical characteristics of the situation in which they operate

I Aumann (1987) wrote "The common knowledge assumption underlies all of game theory and much of economic theory. Whatever be the model under discussion ... the model itself must be assumed common knowledge; otherwise the model is insu¢ ciently speci…ed, and the analysis incoherent." I we can incorporate any incomplete information without loss of generality!

John Harsanyi 1967/68

I incomplete Information is not a problem John Harsanyi 1967/68

I incomplete Information is not a problem I we can incorporate any incomplete information without loss of generality! I types are "like" your hand in poker

I in game theory, it can encompass payo¤s but also the rules of the game.... I in economic model, it can encompass preferences, technology, etc...

I two players, Ann and Bob (generalize straightforwardly to many players) I each player has a space of possible "types": TA, TB

I write πA (tB , θ tA) for the probability that type tA of Ann j assigns to both Bob being type tB and the state being θ; so we have πA : TA ∆ (TB Θ) ! and analogously

πB : TB ∆ (TA Θ) ! I The state space Θ can embed a lot of stu¤...

John Harsanyi Part 1: Type Spaces

I there is a set of states Θ that we care about I types are "like" your hand in poker

I in game theory, it can encompass payo¤s but also the rules of the game.... I in economic model, it can encompass preferences, technology, etc...

I each player has a space of possible "types": TA, TB

I write πA (tB , θ tA) for the probability that type tA of Ann j assigns to both Bob being type tB and the state being θ; so we have πA : TA ∆ (TB Θ) ! and analogously

πB : TB ∆ (TA Θ) ! I The state space Θ can embed a lot of stu¤...

John Harsanyi Part 1: Type Spaces

I there is a set of states Θ that we care about I two players, Ann and Bob (generalize straightforwardly to many players) I in game theory, it can encompass payo¤s but also the rules of the game.... I in economic model, it can encompass preferences, technology, etc...

I types are "like" your hand in poker

I write πA (tB , θ tA) for the probability that type tA of Ann j assigns to both Bob being type tB and the state being θ; so we have πA : TA ∆ (TB Θ) ! and analogously

πB : TB ∆ (TA Θ) ! I The state space Θ can embed a lot of stu¤...

John Harsanyi Part 1: Type Spaces

I there is a set of states Θ that we care about I two players, Ann and Bob (generalize straightforwardly to many players) I each player has a space of possible "types": TA, TB I in game theory, it can encompass payo¤s but also the rules of the game.... I in economic model, it can encompass preferences, technology, etc...

I write πA (tB , θ tA) for the probability that type tA of Ann j assigns to both Bob being type tB and the state being θ; so we have πA : TA ∆ (TB Θ) ! and analogously

πB : TB ∆ (TA Θ) ! I The state space Θ can embed a lot of stu¤...

John Harsanyi Part 1: Type Spaces

I there is a set of states Θ that we care about I two players, Ann and Bob (generalize straightforwardly to many players) I each player has a space of possible "types": TA, TB I types are "like" your hand in poker I in game theory, it can encompass payo¤s but also the rules of the game.... I in economic model, it can encompass preferences, technology, etc...

I The state space Θ can embed a lot of stu¤...

John Harsanyi Part 1: Type Spaces

I write πA (tB , θ tA) for the probability that type tA of Ann j assigns to both Bob being type tB and the state being θ; so we have πA : TA ∆ (TB Θ) ! and analogously

πB : TB ∆ (TA Θ) ! I in game theory, it can encompass payo¤s but also the rules of the game.... I in economic model, it can encompass preferences, technology, etc...

John Harsanyi Part 1: Type Spaces

I write πA (tB , θ tA) for the probability that type tA of Ann j assigns to both Bob being type tB and the state being θ; so we have πA : TA ∆ (TB Θ) ! and analogously

πB : TB ∆ (TA Θ) ! I The state space Θ can embed a lot of stu¤... I in economic model, it can encompass preferences, technology, etc...

John Harsanyi Part 1: Type Spaces

I write πA (tB , θ tA) for the probability that type tA of Ann j assigns to both Bob being type tB and the state being θ; so we have πA : TA ∆ (TB Θ) ! and analogously

πB : TB ∆ (TA Θ) ! I The state space Θ can embed a lot of stu¤...

I in game theory, it can encompass payo¤s but also the rules of the game.... John Harsanyi Part 1: Type Spaces

I write πA (tB , θ tA) for the probability that type tA of Ann j assigns to both Bob being type tB and the state being θ; so we have πA : TA ∆ (TB Θ) ! and analogously

πB : TB ∆ (TA Θ) ! I The state space Θ can embed a lot of stu¤...

I in game theory, it can encompass payo¤s but also the rules of the game.... I in economic model, it can encompass preferences, technology, etc... 1. her belief about the state 2. her belief about the state and the Bob’sbelief about the state 3. her belief about the state and [Bob’sbelief about the state and Ann’sbelief about the state] 4. and so on....

John Harsanyi Part 2: Universal Type Spaces

I Ann is characterized by... 2. her belief about the state and the Bob’sbelief about the state 3. her belief about the state and [Bob’sbelief about the state and Ann’sbelief about the state] 4. and so on....

John Harsanyi Part 2: Universal Type Spaces

I Ann is characterized by... 1. her belief about the state 3. her belief about the state and [Bob’sbelief about the state and Ann’sbelief about the state] 4. and so on....

John Harsanyi Part 2: Universal Type Spaces

I Ann is characterized by... 1. her belief about the state 2. her belief about the state and the Bob’sbelief about the state 4. and so on....

John Harsanyi Part 2: Universal Type Spaces

I Ann is characterized by... 1. her belief about the state 2. her belief about the state and the Bob’sbelief about the state 3. her belief about the state and [Bob’sbelief about the state and Ann’sbelief about the state] I So Ann is characterized by this in…nite sequence of such higher order beliefs, or universal types I "universal type space" T satis…es T ∆ (T Θ) I We can assume that this structure is common knowledge I Incomplete information is not a problem after all!

John Harsanyi Part 2: Universal Type Spaces

I Ann is characterized by... 1. her belief about the state 2. her belief about the state and the Bob’sbelief about the state 3. her belief about the state and [Bob’sbelief about the state and Ann’sbelief about the state] 4. and so on.... I "universal type space" T satis…es T ∆ (T Θ) I We can assume that this structure is common knowledge I Incomplete information is not a problem after all!

John Harsanyi Part 2: Universal Type Spaces

I So Ann is characterized by this in…nite sequence of such higher order beliefs, or universal types I We can assume that this structure is common knowledge I Incomplete information is not a problem after all!

John Harsanyi Part 2: Universal Type Spaces

I So Ann is characterized by this in…nite sequence of such higher order beliefs, or universal types I "universal type space" T satis…es T ∆ (T Θ) I Incomplete information is not a problem after all!

John Harsanyi Part 2: Universal Type Spaces

I So Ann is characterized by this in…nite sequence of such higher order beliefs, or universal types I "universal type space" T satis…es T ∆ (T Θ) I We can assume that this structure is common knowledge I Incomplete information is not a problem after all! I the economics profession went straight back to make unrealistic complete information assumptions, by working with "small" and otherwise simple type spaces (e.g., independent types = common knowledge of …rst order beliefs) I a little "bait and switch"?

I how incomplete information can be re-visited recognizing that implicit common knowledge assumptions are a real issue I make those implicit common knowledge assumptions explicit and relax them I taking higher-order beliefs seriously

I by working with the universal type space, we can dispense with common knowledge assumptions

I the bad news:

I an interesting research agenda?

The Misunderstanding of John Harsanyi

I the good news: I the economics profession went straight back to make unrealistic complete information assumptions, by working with "small" and otherwise simple type spaces (e.g., independent types = common knowledge of …rst order beliefs) I a little "bait and switch"?

I the bad news:

I an interesting research agenda?

The Misunderstanding of John Harsanyi

I the good news:

I by working with the universal type space, we can dispense with common knowledge assumptions I how incomplete information can be re-visited recognizing that implicit common knowledge assumptions are a real issue I make those implicit common knowledge assumptions explicit and relax them I taking higher-order beliefs seriously

I an interesting research agenda?

The Misunderstanding of John Harsanyi

I the good news:

I by working with the universal type space, we can dispense with common knowledge assumptions

I the bad news: I how incomplete information can be re-visited recognizing that implicit common knowledge assumptions are a real issue I make those implicit common knowledge assumptions explicit and relax them I taking higher-order beliefs seriously

I a little "bait and switch"?

I an interesting research agenda?

The Misunderstanding of John Harsanyi

I the good news:

I by working with the universal type space, we can dispense with common knowledge assumptions

I the bad news:

I the economics profession went straight back to make unrealistic complete information assumptions, by working with "small" and otherwise simple type spaces (e.g., independent types = common knowledge of …rst order beliefs) I how incomplete information can be re-visited recognizing that implicit common knowledge assumptions are a real issue I make those implicit common knowledge assumptions explicit and relax them I taking higher-order beliefs seriously

I an interesting research agenda?

The Misunderstanding of John Harsanyi

I the good news:

I by working with the universal type space, we can dispense with common knowledge assumptions

I the bad news:

I the economics profession went straight back to make unrealistic complete information assumptions, by working with "small" and otherwise simple type spaces (e.g., independent types = common knowledge of …rst order beliefs) I a little "bait and switch"? I how incomplete information can be re-visited recognizing that implicit common knowledge assumptions are a real issue I make those implicit common knowledge assumptions explicit and relax them I taking higher-order beliefs seriously

The Misunderstanding of John Harsanyi

I the good news:

I by working with the universal type space, we can dispense with common knowledge assumptions

I the bad news:

I an interesting research agenda? I make those implicit common knowledge assumptions explicit and relax them I taking higher-order beliefs seriously

The Misunderstanding of John Harsanyi

I the good news:

I by working with the universal type space, we can dispense with common knowledge assumptions

I the bad news:

I an interesting research agenda?

I how incomplete information can be re-visited recognizing that implicit common knowledge assumptions are a real issue I taking higher-order beliefs seriously

The Misunderstanding of John Harsanyi

I the good news:

I by working with the universal type space, we can dispense with common knowledge assumptions

I the bad news:

I an interesting research agenda?

I the good news:

I by working with the universal type space, we can dispense with common knowledge assumptions

I the bad news:

I an interesting research agenda?

Wilson (1987): Game theory....is de…cient to the extent that it assumes other features to be common knowledge, such as one agent’sprobability assessment about another’s preferences or information. I foresee the progress of game theory as depending on successive reductions in the base of common knowledge required to conduct useful analyses of practical problems. Only by repeated weakening of common knowledge assumptions will the theory approximate reality.” 1. modelling coordination ("global games") 2. informationally robust analysis of games (maybe) 3. informationally robust mechanism design (probably not)

I key but subtle observation: relaxing common knowledge is equivalent to allowing richer type spaces

The Misunderstanding of John Harsanyi

I this lecture: review some baby steps in this agenda 2. informationally robust analysis of games (maybe) 3. informationally robust mechanism design (probably not)

I key but subtle observation: relaxing common knowledge is equivalent to allowing richer type spaces

The Misunderstanding of John Harsanyi

I this lecture: review some baby steps in this agenda 1. modelling coordination ("global games") 3. informationally robust mechanism design (probably not)

I key but subtle observation: relaxing common knowledge is equivalent to allowing richer type spaces

The Misunderstanding of John Harsanyi

I this lecture: review some baby steps in this agenda 1. modelling coordination ("global games") 2. informationally robust analysis of games (maybe) I key but subtle observation: relaxing common knowledge is equivalent to allowing richer type spaces

The Misunderstanding of John Harsanyi

I this lecture: review some baby steps in this agenda 1. modelling coordination ("global games") 2. informationally robust analysis of games (maybe) 3. informationally robust mechanism design (probably not) The Misunderstanding of John Harsanyi

I key but subtle observation: relaxing common knowledge is equivalent to allowing richer type spaces I E.g., currency crises, bank runs, …nancial crises, demand externalities.....

I Strategic complementarities are important but what are the implications of multiple equilibria for empirical work, policy analysis or comparative statics more generally?

I CLAIM: It is important for lots of applied economic analysis to think about the implications of relaxing common knowledge assumptions in coordination games

Application 1: Strategic Complementarities

I Many economic problems have "strategic complementarities" and thus, when modelled as a perfect information game, multiple equilibria I Strategic complementarities are important but what are the implications of multiple equilibria for empirical work, policy analysis or comparative statics more generally?

I CLAIM: It is important for lots of applied economic analysis to think about the implications of relaxing common knowledge assumptions in coordination games

Application 1: Strategic Complementarities

I Many economic problems have "strategic complementarities" and thus, when modelled as a perfect information game, multiple equilibria

I E.g., currency crises, bank runs, …nancial crises, demand externalities..... I CLAIM: It is important for lots of applied economic analysis to think about the implications of relaxing common knowledge assumptions in coordination games

Application 1: Strategic Complementarities

I Many economic problems have "strategic complementarities" and thus, when modelled as a perfect information game, multiple equilibria

I E.g., currency crises, bank runs, …nancial crises, demand externalities.....

I Strategic complementarities are important but what are the implications of multiple equilibria for empirical work, policy analysis or comparative statics more generally? Application 1: Strategic Complementarities

I Many economic problems have "strategic complementarities" and thus, when modelled as a perfect information game, multiple equilibria

I E.g., currency crises, bank runs, …nancial crises, demand externalities.....

I Strategic complementarities are important but what are the implications of multiple equilibria for empirical work, policy analysis or comparative statics more generally?

I CLAIM: It is important for lots of applied economic analysis to think about the implications of relaxing common knowledge assumptions in coordination games I Ann chooses row, Bob chooses column

Invest Not Invest Invest θA, θB θA 1, 0 Not Invest 0, θB 1 0

I If 0 < θA < 1 and 0 < θB < 1, then this game has multiple Nash equilibria

Canonical Coordination Game

Today’sTalk:

I Simplest Example of a Coordination Game... I If 0 < θA < 1 and 0 < θB < 1, then this game has multiple Nash equilibria

Canonical Coordination Game

Today’sTalk:

I Simplest Example of a Coordination Game... I Ann chooses row, Bob chooses column

Invest Not Invest Invest θA, θB θA 1, 0 Not Invest 0, θB 1 0 Canonical Coordination Game

Today’sTalk:

I Simplest Example of a Coordination Game... I Ann chooses row, Bob chooses column

Invest Not Invest Invest θA, θB θA 1, 0 Not Invest 0, θB 1 0

I If 0 < θA < 1 and 0 < θB < 1, then this game has multiple Nash equilibria I θI = ω + εI , where the εI are i.i.d. noise distributed on a small interval

I Ann knows θA but forms conjecture about θB by Bayes updating...

I Bob knows θB but forms conjecture about θA by Bayes updating...

I Minor variant of Carlsson and van Damme (1993)

Global Games

Invest Not Invest Invest θA, θB θA 1, 0 Not Invest 0, θB 1 0

I Suppose that state ω is uniformly distributed on an interval containing [0, 1] I Ann knows θA but forms conjecture about θB by Bayes updating...

I Bob knows θB but forms conjecture about θA by Bayes updating...

I Minor variant of Carlsson and van Damme (1993)

Global Games

Invest Not Invest Invest θA, θB θA 1, 0 Not Invest 0, θB 1 0

I Suppose that state ω is uniformly distributed on an interval containing [0, 1]

I θI = ω + εI , where the εI are i.i.d. noise distributed on a small interval I Bob knows θB but forms conjecture about θA by Bayes updating...

I Minor variant of Carlsson and van Damme (1993)

Global Games

Invest Not Invest Invest θA, θB θA 1, 0 Not Invest 0, θB 1 0

I Suppose that state ω is uniformly distributed on an interval containing [0, 1]

I θI = ω + εI , where the εI are i.i.d. noise distributed on a small interval

I Ann knows θA but forms conjecture about θB by Bayes updating... I Minor variant of Carlsson and van Damme (1993)

Global Games

Invest Not Invest Invest θA, θB θA 1, 0 Not Invest 0, θB 1 0

I Suppose that state ω is uniformly distributed on an interval containing [0, 1]

I θI = ω + εI , where the εI are i.i.d. noise distributed on a small interval

I Ann knows θA but forms conjecture about θB by Bayes updating...

I Bob knows θB but forms conjecture about θA by Bayes updating... Global Games

Invest Not Invest Invest θA, θB θA 1, 0 Not Invest 0, θB 1 0

I Suppose that state ω is uniformly distributed on an interval containing [0, 1]

I θI = ω + εI , where the εI are i.i.d. noise distributed on a small interval

I Ann knows θA but forms conjecture about θB by Bayes updating...

I Bob knows θB but forms conjecture about θA by Bayes updating...

I Minor variant of Carlsson and van Damme (1993) I for small noise, the risk dominant Nash equilibrium of the perfect information game is almost always played

I Suppose θA = θ 1 I Ann attaches probability 2 to θB θ 1 I For this to be an equilibrium, we must have θ = 2 I The "risk dominant" action is always played in this equilibrium

Risk Dominance

I Suppose Ann and Bob follow strategies of the form: invest if θI θ I for small noise, the risk dominant Nash equilibrium of the perfect information game is almost always played

1 I Ann attaches probability 2 to θB θ 1 I For this to be an equilibrium, we must have θ = 2 I The "risk dominant" action is always played in this equilibrium

Risk Dominance

I Suppose Ann and Bob follow strategies of the form: invest if θI θ I Suppose θA = θ I for small noise, the risk dominant Nash equilibrium of the perfect information game is almost always played

1 I For this to be an equilibrium, we must have θ = 2 I The "risk dominant" action is always played in this equilibrium

Risk Dominance

I Suppose Ann and Bob follow strategies of the form: invest if θI θ I Suppose θA = θ 1 I Ann attaches probability to θB θ 2 I for small noise, the risk dominant Nash equilibrium of the perfect information game is almost always played

I The "risk dominant" action is always played in this equilibrium

Risk Dominance

I Suppose Ann and Bob follow strategies of the form: invest if θI θ I Suppose θA = θ 1 I Ann attaches probability 2 to θB θ 1 I For this to be an equilibrium, we must have θ = 2 I for small noise, the risk dominant Nash equilibrium of the perfect information game is almost always played

Risk Dominance

I Suppose Ann and Bob follow strategies of the form: invest if θI θ I Suppose θA = θ 1 I Ann attaches probability 2 to θB θ 1 I For this to be an equilibrium, we must have θ = 2 I The "risk dominant" action is always played in this equilibrium Risk Dominance

I for small noise, the risk dominant Nash equilibrium of the perfect information game is almost always played I Let θ be the largest value of θI at which it is rationalizable for either player to not invest 1 I Suppose θ > 2 I In particular, suppose that not invest is rationalizable for Ann when θA = θ and invest is uniquely rationalizable for Bob whenever θB > θ 1 I When θA = θ, she assigns probability 2 to θB > θ 1 I Her expected to payo¤ to investing is at least θ > 0 2 I ....a contradiction

I PROOF:

Unique Rationalizable Play

I In fact, the unique "rationalizable" in this game is to invest if 1 and only if θI 2 I Let θ be the largest value of θI at which it is rationalizable for either player to not invest 1 I Suppose θ > 2 I In particular, suppose that not invest is rationalizable for Ann when θA = θ and invest is uniquely rationalizable for Bob whenever θB > θ 1 I When θA = θ, she assigns probability 2 to θB > θ 1 I Her expected to payo¤ to investing is at least θ > 0 2 I ....a contradiction

Unique Rationalizable Play

I In fact, the unique "rationalizable" in this game is to invest if 1 and only if θI 2 I PROOF: 1 I Suppose θ > 2 I In particular, suppose that not invest is rationalizable for Ann when θA = θ and invest is uniquely rationalizable for Bob whenever θB > θ 1 I When θA = θ, she assigns probability 2 to θB > θ 1 I Her expected to payo¤ to investing is at least θ > 0 2 I ....a contradiction

Unique Rationalizable Play

I In fact, the unique "rationalizable" in this game is to invest if 1 and only if θI 2 I PROOF:

I Let θ be the largest value of θI at which it is rationalizable for either player to not invest I In particular, suppose that not invest is rationalizable for Ann when θA = θ and invest is uniquely rationalizable for Bob whenever θB > θ 1 I When θA = θ, she assigns probability 2 to θB > θ 1 I Her expected to payo¤ to investing is at least θ > 0 2 I ....a contradiction

Unique Rationalizable Play

I In fact, the unique "rationalizable" in this game is to invest if 1 and only if θI 2 I PROOF:

I Let θ be the largest value of θI at which it is rationalizable for either player to not invest 1 I Suppose θ > 2 1 I When θA = θ, she assigns probability 2 to θB > θ 1 I Her expected to payo¤ to investing is at least θ > 0 2 I ....a contradiction

Unique Rationalizable Play

I In fact, the unique "rationalizable" in this game is to invest if 1 and only if θI 2 I PROOF:

Unique Rationalizable Play

I In fact, the unique "rationalizable" in this game is to invest if 1 and only if θI 2 I PROOF:

Unique Rationalizable Play

I In fact, the unique "rationalizable" in this game is to invest if 1 and only if θI 2 I PROOF:

I Let θ be the largest value of θI at which it is rationalizable for either player to not invest 1 I Suppose θ > 2 I In particular, suppose that not invest is rationalizable for Ann when θA = θ and invest is uniquely rationalizable for Bob whenever θB > θ 1 I When θA = θ, she assigns probability 2 to θB > θ 1 I Her expected to payo¤ to investing is at least θ > 0 2 I ....a contradiction I θI = ω + σ.εI , and εI are i.i.d. noise, where σ > 0 is small and f ( ) is a smooth density 1 I If σ 0, then Ann always attaches probability to 2 θB θA I As σ 0, unique rationalizable outcome has each player ! 1 invest if and only if θI 2

Global Games

Invest Not Invest Invest θA, θB θA 1, 0 Not Invest 0, θB 1 0

I Suppose that state ω is distributed according to smooth density g ( ) 1 I If σ 0, then Ann always attaches probability to 2 θB θA I As σ 0, unique rationalizable outcome has each player ! 1 invest if and only if θI 2

Global Games

Invest Not Invest Invest θA, θB θA 1, 0 Not Invest 0, θB 1 0

I Suppose that state ω is distributed according to smooth density g ( ) I θI = ω + σ.εI , and εI are i.i.d. noise, where σ > 0 is small and f ( ) is a smooth density I As σ 0, unique rationalizable outcome has each player ! 1 invest if and only if θI 2

Global Games

Invest Not Invest Invest θA, θB θA 1, 0 Not Invest 0, θB 1 0

I Suppose that state ω is distributed according to smooth density g ( ) I θI = ω + σ.εI , and εI are i.i.d. noise, where σ > 0 is small and f ( ) is a smooth density 1 I If σ 0, then Ann always attaches probability to 2 θB θA I As σ 0, unique rationalizable outcome has each player ! 1 invest if and only if θI 2 I Selected "risk dominant" action becomes best response to uniform distribution over proportion of others investing I Can do lots of interesting comparative statics / policy analysis (and people have done....)

I Analysis extends to many player binary action symmetric payo¤s very cleanly:

I Further extends to general supermodular games

Global Games Extensions

I INTERPRETATION: Relaxing strong and unjusti…ed assumption of common knowledge of payo¤s generates intuitive prediction I Selected "risk dominant" action becomes best response to uniform distribution over proportion of others investing I Can do lots of interesting comparative statics / policy analysis (and people have done....)

I Further extends to general supermodular games

Global Games Extensions

I INTERPRETATION: Relaxing strong and unjusti…ed assumption of common knowledge of payo¤s generates intuitive prediction

I Analysis extends to many player binary action symmetric payo¤s very cleanly: I Can do lots of interesting comparative statics / policy analysis (and people have done....)

I Further extends to general supermodular games

Global Games Extensions

I INTERPRETATION: Relaxing strong and unjusti…ed assumption of common knowledge of payo¤s generates intuitive prediction

I Analysis extends to many player binary action symmetric payo¤s very cleanly:

I Selected "risk dominant" action becomes best response to uniform distribution over proportion of others investing I Further extends to general supermodular games

Global Games Extensions

I INTERPRETATION: Relaxing strong and unjusti…ed assumption of common knowledge of payo¤s generates intuitive prediction

I Analysis extends to many player binary action symmetric payo¤s very cleanly:

I Selected "risk dominant" action becomes best response to uniform distribution over proportion of others investing I Can do lots of interesting comparative statics / policy analysis (and people have done....) Global Games Extensions

I INTERPRETATION: Relaxing strong and unjusti…ed assumption of common knowledge of payo¤s generates intuitive prediction

I Analysis extends to many player binary action symmetric payo¤s very cleanly:

I Further extends to general supermodular games I In particular, we now have the unreasonable implicit and importnat common knowledge assumption about the distribution of signals conditional on the state....

I Let’sgo back to basics and examine our coordination game without making common knowledge assumptions.... or at least making fewer common knowledge assumptions....

Global Games Critique

I BUT isn’tthis a rather ad hoc way of relaxing common knowledge assumptions? I Let’sgo back to basics and examine our coordination game without making common knowledge assumptions.... or at least making fewer common knowledge assumptions....

Global Games Critique

I BUT isn’tthis a rather ad hoc way of relaxing common knowledge assumptions?

I In particular, we now have the unreasonable implicit and importnat common knowledge assumption about the distribution of signals conditional on the state.... Global Games Critique

I BUT isn’tthis a rather ad hoc way of relaxing common knowledge assumptions?

I In particular, we now have the unreasonable implicit and importnat common knowledge assumption about the distribution of signals conditional on the state....

I Let’sgo back to basics and examine our coordination game without making common knowledge assumptions.... or at least making fewer common knowledge assumptions.... I Ann has a set of types TA, where a type is characterized by a payo¤ parameter θ (t ) Θ and a belief π (t ) ∆ (T ) A A 2 A A A 2 B I similarly for Bob b b

I natural to consider slightly di¤erent type spaces:

An Important Restriction on Type Space: Private Values

I suppose that Ann’spreferences are summarized by a parameter θA ΘA (known to Ann), and similarly for Bob ("private values")2 I Ann has a set of types TA, where a type is characterized by a payo¤ parameter θ (t ) Θ and a belief π (t ) ∆ (T ) A A 2 A A A 2 B I similarly for Bob b b

An Important Restriction on Type Space: Private Values

I suppose that Ann’spreferences are summarized by a parameter θA ΘA (known to Ann), and similarly for Bob ("private values")2

I natural to consider slightly di¤erent type spaces: I similarly for Bob

An Important Restriction on Type Space: Private Values

I suppose that Ann’spreferences are summarized by a parameter θA ΘA (known to Ann), and similarly for Bob ("private values")2

I natural to consider slightly di¤erent type spaces:

I Ann has a set of types TA, where a type is characterized by a payo¤ parameter θ (t ) Θ and a belief π (t ) ∆ (T ) A A 2 A A A 2 B b b An Important Restriction on Type Space: Private Values

I suppose that Ann’spreferences are summarized by a parameter θA ΘA (known to Ann), and similarly for Bob ("private values")2

I natural to consider slightly di¤erent type spaces:

I Ann has a set of types TA, where a type is characterized by a payo¤ parameter θ (t ) Θ and a belief π (t ) ∆ (T ) A A 2 A A A 2 B I similarly for Bob b b 1. her belief about Bob’spayo¤ parameter 2. her belief about Bob’sbelief and his payo¤ parameter 3. and so on....

I Ann’suniversal type space is T ΘA ∆ (T ) A B I Is a subset of our …rst universal type space

Universal Type Space with Private Values

I Ann’suniversal type is her payo¤ parameter and 2. her belief about Bob’sbelief and his payo¤ parameter 3. and so on....

I Ann’suniversal type space is T ΘA ∆ (T ) A B I Is a subset of our …rst universal type space

Universal Type Space with Private Values

I Ann’suniversal type is her payo¤ parameter and 1. her belief about Bob’spayo¤ parameter 3. and so on....

I Ann’suniversal type space is T ΘA ∆ (T ) A B I Is a subset of our …rst universal type space

Universal Type Space with Private Values

I Ann’suniversal type is her payo¤ parameter and 1. her belief about Bob’spayo¤ parameter 2. her belief about Bob’sbelief and his payo¤ parameter I Ann’suniversal type space is T ΘA ∆ (T ) A B I Is a subset of our …rst universal type space

Universal Type Space with Private Values

I Ann’suniversal type is her payo¤ parameter and 1. her belief about Bob’spayo¤ parameter 2. her belief about Bob’sbelief and his payo¤ parameter 3. and so on.... I Is a subset of our …rst universal type space

Universal Type Space with Private Values

I Ann’suniversal type is her payo¤ parameter and 1. her belief about Bob’spayo¤ parameter 2. her belief about Bob’sbelief and his payo¤ parameter 3. and so on....

I Ann’suniversal type space is T ΘA ∆ (T ) A B Universal Type Space with Private Values

I Ann’suniversal type is her payo¤ parameter and 1. her belief about Bob’spayo¤ parameter 2. her belief about Bob’sbelief and his payo¤ parameter 3. and so on....

I Ann’suniversal type space is T ΘA ∆ (T ) A B I Is a subset of our …rst universal type space I Suppose Ann is almost sure that that Bob is almost sure that 3 θA 4 I Suppose Ann is almost sure that that Bob is almost sure that 3 Ann is almost sure that θB 4 I and so on up to a arbitrarily large number of …nite levels I technically, Ann’stype is in the universal type space is close in the product topology to the type with common knowledge of 3 3 4 , 4 I what can we say about strategic behavior?

Relaxing Common Knowledge Assumptions in Coordination Game

3 I Suppose Ann is almost sure that θB 4 I Suppose Ann is almost sure that that Bob is almost sure that 3 Ann is almost sure that θB 4 I and so on up to a arbitrarily large number of …nite levels I technically, Ann’stype is in the universal type space is close in the product topology to the type with common knowledge of 3 3 4 , 4 I what can we say about strategic behavior?

Relaxing Common Knowledge Assumptions in Coordination Game

3 I Suppose Ann is almost sure that θB 4 I Suppose Ann is almost sure that that Bob is almost sure that 3 θA 4 I and so on up to a arbitrarily large number of …nite levels I technically, Ann’stype is in the universal type space is close in the product topology to the type with common knowledge of 3 3 4 , 4 I what can we say about strategic behavior?

Relaxing Common Knowledge Assumptions in Coordination Game

3 I Suppose Ann is almost sure that θB 4 I Suppose Ann is almost sure that that Bob is almost sure that 3 θA 4 I Suppose Ann is almost sure that that Bob is almost sure that 3 Ann is almost sure that θB 4 I technically, Ann’stype is in the universal type space is close in the product topology to the type with common knowledge of 3 3 4 , 4 I what can we say about strategic behavior?

Relaxing Common Knowledge Assumptions in Coordination Game

3 I Suppose Ann is almost sure that θB 4 I Suppose Ann is almost sure that that Bob is almost sure that 3 θA 4 I Suppose Ann is almost sure that that Bob is almost sure that 3 Ann is almost sure that θB 4 I and so on up to a arbitrarily large number of …nite levels I technically, Ann’stype is in the universal type space is close in the product topology to the type with common knowledge of 3 3 4 , 4 Relaxing Common Knowledge Assumptions in Coordination Game

3 I Suppose Ann is almost sure that θB 4 I Suppose Ann is almost sure that that Bob is almost sure that 3 θA 4 I Suppose Ann is almost sure that that Bob is almost sure that 3 Ann is almost sure that θB 4 I and so on up to a arbitrarily large number of …nite levels I technically, Ann’stype is in the universal type space is close in the product topology to the type with common knowledge of 3 3 4 , 4 I what can we say about strategic behavior? Electronic Mail Game on Steroids

Rubinstein 89, Weinstein and Yildiz 07 3 3 Suppose that the state may be "good" with (θA, θB ) = 4 , 4 :

Invest Not Invest 3 3 1 Invest 4 , 4 4 , 0 Not Invest 0, 1 0, 0 4 but Bob may have a dominant strategy to not invest, so the state 3 is "bad", with (θA, θB ) = , 1 : 4 Invest Not Invest 3 1 Invest 4 , 1 4 , 0 Not Invest 0, 2 0, 0 I If Ann does not receive a …rst message, she thinks that the state is bad with probability 1 ε I If a player does not receive a con…rmation of his/her message, he/she thinks that the other player did not receive his/her message with probability 1 ε

I If Ann receives the message, she sends a con…rmation to Bob telling her that he received the message

I and so on.... I Suppose that players are pessimistic:

Electronic Mail Game on Steroids

I If the state is good, Bob sends a message to Ann, reporting that the state is good I If Ann does not receive a …rst message, she thinks that the state is bad with probability 1 ε I If a player does not receive a con…rmation of his/her message, he/she thinks that the other player did not receive his/her message with probability 1 ε

I and so on.... I Suppose that players are pessimistic:

Electronic Mail Game on Steroids

I If the state is good, Bob sends a message to Ann, reporting that the state is good

I If Ann receives the message, she sends a con…rmation to Bob telling her that he received the message I If Ann does not receive a …rst message, she thinks that the state is bad with probability 1 ε I If a player does not receive a con…rmation of his/her message, he/she thinks that the other player did not receive his/her message with probability 1 ε

I Suppose that players are pessimistic:

Electronic Mail Game on Steroids

I If the state is good, Bob sends a message to Ann, reporting that the state is good

I If Ann receives the message, she sends a con…rmation to Bob telling her that he received the message

I and so on.... I If Ann does not receive a …rst message, she thinks that the state is bad with probability 1 ε I If a player does not receive a con…rmation of his/her message, he/she thinks that the other player did not receive his/her message with probability 1 ε

Electronic Mail Game on Steroids

I If the state is good, Bob sends a message to Ann, reporting that the state is good

I If Ann receives the message, she sends a con…rmation to Bob telling her that he received the message

I and so on.... I Suppose that players are pessimistic: I If a player does not receive a con…rmation of his/her message, he/she thinks that the other player did not receive his/her message with probability 1 ε

Electronic Mail Game on Steroids

I If the state is good, Bob sends a message to Ann, reporting that the state is good

I If Ann receives the message, she sends a con…rmation to Bob telling her that he received the message

I and so on.... I Suppose that players are pessimistic:

I If Ann does not receive a …rst message, she thinks that the state is bad with probability 1 ε Electronic Mail Game on Steroids

I If the state is good, Bob sends a message to Ann, reporting that the state is good

I If Ann receives the message, she sends a con…rmation to Bob telling her that he received the message

I and so on.... I Suppose that players are pessimistic:

I If Ann does not receive a …rst message, she thinks that the state is bad with probability 1 ε I If a player does not receive a con…rmation of his/her message, he/she thinks that the other player did not receive his/her message with probability 1 ε I In the bad state, Bob does not invest I If Ann does not receive a message, she does not invest I If Bob receives only one message, she does not invest I and so on....

I However, "not invest" is the unique rationalizable (and thus equilibrium) action for this type of Ann

I Proof:

I "On steroids" relative to Rubinstein 89 become we didn’t impose the common prior assumption

I Weinstein Yildiz 07 show that this logic is completely general: (roughly) any action that is rationalizable in a perfect information game is uniquely rationalizable for a nearby type in the product topology

Infection Argument

I If Ann receives many con…rmations, she is "close" (formally, in the product topology on the universal type space) to 3 3 common knowledge that game is (θA, θB ) = 4 , 4 I In the bad state, Bob does not invest I If Ann does not receive a message, she does not invest I If Bob receives only one message, she does not invest I and so on....

I Proof:

I "On steroids" relative to Rubinstein 89 become we didn’t impose the common prior assumption

Infection Argument

I If Ann receives many con…rmations, she is "close" (formally, in the product topology on the universal type space) to 3 3 common knowledge that game is (θA, θB ) = 4 , 4 I However, "not invest" is the unique rationalizable (and thus equilibrium) action for this type of Ann I In the bad state, Bob does not invest I If Ann does not receive a message, she does not invest I If Bob receives only one message, she does not invest I and so on....

I "On steroids" relative to Rubinstein 89 become we didn’t impose the common prior assumption

Infection Argument

I Proof: I If Ann does not receive a message, she does not invest I If Bob receives only one message, she does not invest I and so on....

I "On steroids" relative to Rubinstein 89 become we didn’t impose the common prior assumption

Infection Argument

I Proof:

I In the bad state, Bob does not invest I If Bob receives only one message, she does not invest I and so on....

I "On steroids" relative to Rubinstein 89 become we didn’t impose the common prior assumption

Infection Argument

I Proof:

I In the bad state, Bob does not invest I If Ann does not receive a message, she does not invest I and so on....

I "On steroids" relative to Rubinstein 89 become we didn’t impose the common prior assumption

Infection Argument

I Proof:

I In the bad state, Bob does not invest I If Ann does not receive a message, she does not invest I If Bob receives only one message, she does not invest I "On steroids" relative to Rubinstein 89 become we didn’t impose the common prior assumption

Infection Argument

I Proof:

I In the bad state, Bob does not invest I If Ann does not receive a message, she does not invest I If Bob receives only one message, she does not invest I and so on.... I Weinstein Yildiz 07 show that this logic is completely general: (roughly) any action that is rationalizable in a perfect information game is uniquely rationalizable for a nearby type in the product topology

Infection Argument

I Proof:

I In the bad state, Bob does not invest I If Ann does not receive a message, she does not invest I If Bob receives only one message, she does not invest I and so on....

I "On steroids" relative to Rubinstein 89 become we didn’t impose the common prior assumption Infection Argument

I Proof:

I In the bad state, Bob does not invest I If Ann does not receive a message, she does not invest I If Bob receives only one message, she does not invest I and so on....

I "On steroids" relative to Rubinstein 89 become we didn’t impose the common prior assumption

I we don’thave assume perfect information or nothing

Bad News?

I von Neumann was right about one thing: cannot do much without making common knowledge assumptions.... I we don’thave assume perfect information or nothing

Bad News?

I von Neumann was right about one thing: cannot do much without making common knowledge assumptions....

I but let us be sophisticated about what common knowledge assumptions we make Bad News?

I von Neumann was right about one thing: cannot do much without making common knowledge assumptions....

I but let us be sophisticated about what common knowledge assumptions we make

I we don’thave assume perfect information or nothing I Thus r (tA) = Pr (xB (tB ) < xA (tA) tA) j I for today’stalk, I will be vague about inequalities versus equalities; for simplicity, suppose

r (tA) = Pr (xB (tB ) = xA (tA) tA) = 0 j I Assume common knowledge that both players’rank beliefs are 1 2 I This is a major but explicit common knowledge assumption di¤erent from perfect information, or independent types, or usual assumptions we make.... I Claim: if there is common knowledge that rank beliefs are 1 uniform (i.e., 2 ), then players have unique rationalizable actions. They always play the risk dominant action, i.e., 1 1 invest if xA > 2 and not invest if xA < 2 . I Morris Shin Yildiz 15 have this result, and generalizations I This observation makes explicit the common knowledge assumption implicit in the global games literature

Good News?

I In the (private value) universal type space, a player’s"rank belief" is the probability that she assigns to her return to investment being higher than another player’s I for today’stalk, I will be vague about inequalities versus equalities; for simplicity, suppose

Good News?

I In the (private value) universal type space, a player’s"rank belief" is the probability that she assigns to her return to investment being higher than another player’s I Thus r (tA) = Pr (xB (tB ) < xA (tA) tA) j I Assume common knowledge that both players’rank beliefs are 1 2 I This is a major but explicit common knowledge assumption di¤erent from perfect information, or independent types, or usual assumptions we make.... I Claim: if there is common knowledge that rank beliefs are 1 uniform (i.e., 2 ), then players have unique rationalizable actions. They always play the risk dominant action, i.e., 1 1 invest if xA > 2 and not invest if xA < 2 . I Morris Shin Yildiz 15 have this result, and generalizations I This observation makes explicit the common knowledge assumption implicit in the global games literature

Good News?

r (tA) = Pr (xB (tB ) = xA (tA) tA) = 0 j I This is a major but explicit common knowledge assumption di¤erent from perfect information, or independent types, or usual assumptions we make.... I Claim: if there is common knowledge that rank beliefs are 1 uniform (i.e., 2 ), then players have unique rationalizable actions. They always play the risk dominant action, i.e., 1 1 invest if xA > 2 and not invest if xA < 2 . I Morris Shin Yildiz 15 have this result, and generalizations I This observation makes explicit the common knowledge assumption implicit in the global games literature

Good News?

r (tA) = Pr (xB (tB ) = xA (tA) tA) = 0 j I Assume common knowledge that both players’rank beliefs are 1 2 I Claim: if there is common knowledge that rank beliefs are 1 uniform (i.e., 2 ), then players have unique rationalizable actions. They always play the risk dominant action, i.e., 1 1 invest if xA > 2 and not invest if xA < 2 . I Morris Shin Yildiz 15 have this result, and generalizations I This observation makes explicit the common knowledge assumption implicit in the global games literature

Good News?

r (tA) = Pr (xB (tB ) = xA (tA) tA) = 0 j I Assume common knowledge that both players’rank beliefs are 1 2 I This is a major but explicit common knowledge assumption di¤erent from perfect information, or independent types, or usual assumptions we make.... I Morris Shin Yildiz 15 have this result, and generalizations I This observation makes explicit the common knowledge assumption implicit in the global games literature

Good News?

r (tA) = Pr (xB (tB ) = xA (tA) tA) = 0 j I Assume common knowledge that both players’rank beliefs are 1 2 I This is a major but explicit common knowledge assumption di¤erent from perfect information, or independent types, or usual assumptions we make.... I Claim: if there is common knowledge that rank beliefs are 1 uniform (i.e., 2 ), then players have unique rationalizable actions. They always play the risk dominant action, i.e., 1 1 invest if xA > 2 and not invest if xA < 2 . I This observation makes explicit the common knowledge assumption implicit in the global games literature

Good News?

Informational Robustness

I in economic theory, we often distinguish.... I endogenous variables...

Informational Robustness

I in economic theory, we often distinguish....

I exogenous variables... Informational Robustness

I in economic theory, we often distinguish....

I exogenous variables... I endogenous variables... I description of the game, players, actions, states, payo¤ functions, information structure

I strategies

I exogenous variables...

I endogenous variables...

I …nd equilibrium of …xed game

Informational Robustness

I in economic theory, we often distinguish.... I strategies

I description of the game, players, actions, states, payo¤ functions, information structure

I endogenous variables...

I …nd equilibrium of …xed game

Informational Robustness

I in economic theory, we often distinguish....

I exogenous variables... I strategies

I endogenous variables...

I …nd equilibrium of …xed game

Informational Robustness

I in economic theory, we often distinguish....

I exogenous variables...

I description of the game, players, actions, states, payo¤ functions, information structure I strategies

I …nd equilibrium of …xed game

Informational Robustness

I in economic theory, we often distinguish....

I exogenous variables...

I description of the game, players, actions, states, payo¤ functions, information structure

I endogenous variables... I …nd equilibrium of …xed game

Informational Robustness

I in economic theory, we often distinguish....

I exogenous variables...

I description of the game, players, actions, states, payo¤ functions, information structure

I endogenous variables...

I strategies Informational Robustness

I in economic theory, we often distinguish....

I exogenous variables...

I description of the game, players, actions, states, payo¤ functions, information structure

I endogenous variables...

I strategies

I …nd equilibrium of …xed game I characterize the set of equilibria that might result for all information structures

I relax all common knowledge assumptions (maintaining the common prior assumption)

I why is this useful (and feasible)?

Informationally Robust Analysis

I examine all (or many) information structures at once... I relax all common knowledge assumptions (maintaining the common prior assumption)

I why is this useful (and feasible)?

Informationally Robust Analysis

I examine all (or many) information structures at once... I characterize the set of equilibria that might result for all information structures I why is this useful (and feasible)?

Informationally Robust Analysis

I examine all (or many) information structures at once... I characterize the set of equilibria that might result for all information structures

I relax all common knowledge assumptions (maintaining the common prior assumption) Informationally Robust Analysis

I examine all (or many) information structures at once... I characterize the set of equilibria that might result for all information structures

I relax all common knowledge assumptions (maintaining the common prior assumption)

I why is this useful (and feasible)? 2. Robust Identi…cation 3. Information Design

Informationally Robust Analysis

1. Robust Predictions 3. Information Design

Informationally Robust Analysis

1. Robust Predictions 2. Robust Identi…cation Informationally Robust Analysis

1. Robust Predictions 2. Robust Identi…cation 3. Information Design I producer charges uniform monopoly price, giving consumer surplus u and producer surplus π

I consumer gets zero surplus and producer extracts e¢ cient surplus w > π + u

I no information = uniform price monopoly

I full information = perfect price discrimination

I Demand curve for a good represents single unit demand of a continuum of consumers

I What can happen to consumer surplus, producer surplus and thus total surplus for any information that the seller has about consumers?

I Two special cases:

I Robust Prediction: What can we say about all (consumer surplus, producer surplus) pairs that can arise?

Example: Third Degree Price Discrimination

I Bergemann, Brooks and Morris (2015) I producer charges uniform monopoly price, giving consumer surplus u and producer surplus π

I consumer gets zero surplus and producer extracts e¢ cient surplus w > π + u

I no information = uniform price monopoly

I full information = perfect price discrimination

I What can happen to consumer surplus, producer surplus and thus total surplus for any information that the seller has about consumers?

I Two special cases:

I Robust Prediction: What can we say about all (consumer surplus, producer surplus) pairs that can arise?

Example: Third Degree Price Discrimination

I Bergemann, Brooks and Morris (2015) I Demand curve for a good represents single unit demand of a continuum of consumers I producer charges uniform monopoly price, giving consumer surplus u and producer surplus π

I consumer gets zero surplus and producer extracts e¢ cient surplus w > π + u

I no information = uniform price monopoly

I full information = perfect price discrimination

I Two special cases:

I Robust Prediction: What can we say about all (consumer surplus, producer surplus) pairs that can arise?

Example: Third Degree Price Discrimination

I Bergemann, Brooks and Morris (2015) I Demand curve for a good represents single unit demand of a continuum of consumers

I What can happen to consumer surplus, producer surplus and thus total surplus for any information that the seller has about consumers? I producer charges uniform monopoly price, giving consumer surplus u and producer surplus π

I consumer gets zero surplus and producer extracts e¢ cient surplus w > π + u

I no information = uniform price monopoly

I full information = perfect price discrimination

I Robust Prediction: What can we say about all (consumer surplus, producer surplus) pairs that can arise?

Example: Third Degree Price Discrimination

I Bergemann, Brooks and Morris (2015) I Demand curve for a good represents single unit demand of a continuum of consumers

I What can happen to consumer surplus, producer surplus and thus total surplus for any information that the seller has about consumers?

I Two special cases: I consumer gets zero surplus and producer extracts e¢ cient surplus w > π + u

I producer charges uniform monopoly price, giving consumer surplus u and producer surplus π

I full information = perfect price discrimination

I Robust Prediction: What can we say about all (consumer surplus, producer surplus) pairs that can arise?

Example: Third Degree Price Discrimination

I Bergemann, Brooks and Morris (2015) I Demand curve for a good represents single unit demand of a continuum of consumers

I What can happen to consumer surplus, producer surplus and thus total surplus for any information that the seller has about consumers?

I Two special cases:

I no information = uniform price monopoly I consumer gets zero surplus and producer extracts e¢ cient surplus w > π + u

I full information = perfect price discrimination

I Robust Prediction: What can we say about all (consumer surplus, producer surplus) pairs that can arise?

Example: Third Degree Price Discrimination

I Bergemann, Brooks and Morris (2015) I Demand curve for a good represents single unit demand of a continuum of consumers

I What can happen to consumer surplus, producer surplus and thus total surplus for any information that the seller has about consumers?

I Two special cases:

I no information = uniform price monopoly

I producer charges uniform monopoly price, giving consumer surplus u and producer surplus π I consumer gets zero surplus and producer extracts e¢ cient surplus w > π + u

I Robust Prediction: What can we say about all (consumer surplus, producer surplus) pairs that can arise?

Example: Third Degree Price Discrimination

I Bergemann, Brooks and Morris (2015) I Demand curve for a good represents single unit demand of a continuum of consumers

I What can happen to consumer surplus, producer surplus and thus total surplus for any information that the seller has about consumers?

I Two special cases:

I no information = uniform price monopoly

I producer charges uniform monopoly price, giving consumer surplus u and producer surplus π

I full information = perfect price discrimination I Robust Prediction: What can we say about all (consumer surplus, producer surplus) pairs that can arise?

Example: Third Degree Price Discrimination

I Bergemann, Brooks and Morris (2015) I Demand curve for a good represents single unit demand of a continuum of consumers

I What can happen to consumer surplus, producer surplus and thus total surplus for any information that the seller has about consumers?

I Two special cases:

I no information = uniform price monopoly

I producer charges uniform monopoly price, giving consumer surplus u and producer surplus π

I full information = perfect price discrimination

I consumer gets zero surplus and producer extracts e¢ cient surplus w > π + u Example: Third Degree Price Discrimination

I Bergemann, Brooks and Morris (2015) I Demand curve for a good represents single unit demand of a continuum of consumers

I What can happen to consumer surplus, producer surplus and thus total surplus for any information that the seller has about consumers?

I Two special cases:

I no information = uniform price monopoly

I producer charges uniform monopoly price, giving consumer surplus u and producer surplus π

I full information = perfect price discrimination

I consumer gets zero surplus and producer extracts e¢ cient surplus w > π + u

I Robust Prediction: What can we say about all (consumer surplus, producer surplus) pairs that can arise?

A Pictorial Characterization Producer surplus

0 Consumer surplus

The Uniform Price Monopoly Producersurplus

No information

0 Consumer surplus

I producer charges (uniform) monopoly price I consumers get positive consumer surplus, socially ine¢ cient allocation First Degree Price Discrimination: Perfect Discrimination

Complete information Producersurplus

0 Consumer surplus

I producer extracts full surplus I consumers get zero surplus, but socially e¢ cient allocation Welfare Bounds: Voluntary Participation

Consumer surplus is at least zero Producer surplus

0 Consumer surplus Welfare Bounds: Nonnegative Value of Information

Producer gets at least uniform price profit Producer surplus

0 Consumer surplus Welfare Bounds: Social Surplus

Total surplus is bounded by efficient outcome Producer surplus

0 Consumer surplus Welfare Bounds and Third Degree Price Discrimination

What is the feasible surplus set? Producer surplus

0 Consumer surplus

I set must be convex Main Result: No More Robust Predictions!

Main result Producer surplus

0 Consumer surplus Example 1 1 1 I 3 of consumers have valuation 1, 3 have valuation 2 and 3 have valuation 3 I optimal prices:

Optimal prices

x{2}

p*=2

p*=3 p*=1

x{3} x{1} Splitting

I A segmentation of the three value uniform aggregate market:

v = 1 v = 2 v = 3 weight 1 1 1 2 market 1 2 6 3 3 0 1 2 1 market 2 3 3 6 0 1 0 1 market 3 6 1 1 1 total 3 3 3 "Extremal Segmentation"

v = 1 v = 2 v = 3 weight 1 1 1 2 1, 2, 3 2 6 3 3 f g 0 1 2 1 2, 3 3 3 6 f g 0 1 0 1 2 6 f g 1 1 1 total 3 3 3

I price 2 is optimal in all markets I in fact, seller is always indi¤erent between all prices in the support of the market

I this is always possible to do (this is the meat of our general argument) Geometry of Extremal Markets S I extremal segment x : seller is indi¤erent between all prices in the support of S

Extreme markets

x{2}

x{1,2}

x* x{2,3}

x{1,2,3}

x{3} x{1,3} x{1} Consumer Surplus Maximizing Segmentation

I an optimal policy: always charge lowest price in the support of every segment:

v = 1 v = 2 v = 3 price weight 1 1 1 1 2 1, 2, 3 2 6 3 3 f g 0 1 2 2 1 2, 3 3 3 6 f g 0 1 0 2 1 2 6 f g 1 1 1 1 total 3 3 3 Social Surplus Minimizing Segmentation

I all incentive constraints in the support are binding I another optimal policy: always charge highest price in each segment:

v = 1 v = 2 v = 3 price weight 1 1 1 3 2 1, 2, 3 2 6 3 3 f g 0 1 2 3 1 2, 3 3 3 6 f g 0 1 0 2 1 2 6 f g 1 1 1 1 total 3 3 3 I In other settings, there are.... e.g., …rst price auction

Robust Predictions

I In this example, surprisingly weak Robust Predictions

I In this example, surprisingly weak I In other settings, there are.... e.g., …rst price auction I Very little.....

Robust Identi…cation

I What can be inferred from prices about valuations? Robust Identi…cation

I What can be inferred from prices about valuations? I Very little..... I If the designer had the joint interest of consumers in mind he would pick the bottom right hand corner

I Compare Kamenica and Gentzkow (2011)

Information Design

I Consider the problem of an "information designer" who could pick (and commit to) an information structure to give to the monopolist I Compare Kamenica and Gentzkow (2011)

Information Design

I Consider the problem of an "information designer" who could pick (and commit to) an information structure to give to the monopolist

I If the designer had the joint interest of consumers in mind he would pick the bottom right hand corner Information Design

I Consider the problem of an "information designer" who could pick (and commit to) an information structure to give to the monopolist

I If the designer had the joint interest of consumers in mind he would pick the bottom right hand corner

I Compare Kamenica and Gentzkow (2011) The Misunderstanding of John Harsanyi and Mechanism Design

Wilson (1987): (more complete quote) Game theory has a great advantage in explicitly analyzing the consequences of trading rules that presumably are really common knowledge; it is de…cient to the extent that it assumes other features to be common knowledge, such as one agent’sprobability assessment about another’spreferences or information. I foresee the progress of game theory as depending on successive reductions in the base of common knowledge required to conduct useful analyses of practical problems. Only by repeated weakening of common knowledge assumptions will the theory approximate reality.” I really important to relax common knowledge of the mechanism (John Sutton and IO) I common knowledge of the environment is maybe (at least a bit) less of a problem

I take relaxing common knowledge assumptions seriously and allowing real incomplete information in mechanism design I this may rationalize simple / detail-free mechanisms (and suggest new questions)

I we would really like to assume that there is complete information about the game/mechanism I it is particularly desirable to relax common knowledge assumptions about the environment, because optimal mechanisms are otherwise too …nely tuned I Contrast this with economic theory / game theory

I One response to misunderstanding: do not address "incomplete information", focus on simple mechanisms, computational constraints, worst case analysis, etc... I Another response:

The Misunderstanding of John Harsanyi and Mechanism Design

I Mechanism design I really important to relax common knowledge of the mechanism (John Sutton and IO) I common knowledge of the environment is maybe (at least a bit) less of a problem

I take relaxing common knowledge assumptions seriously and allowing real incomplete information in mechanism design I this may rationalize simple / detail-free mechanisms (and suggest new questions)

I it is particularly desirable to relax common knowledge assumptions about the environment, because optimal mechanisms are otherwise too …nely tuned I Contrast this with economic theory / game theory

I One response to misunderstanding: do not address "incomplete information", focus on simple mechanisms, computational constraints, worst case analysis, etc... I Another response:

The Misunderstanding of John Harsanyi and Mechanism Design

I Mechanism design I we would really like to assume that there is complete information about the game/mechanism I really important to relax common knowledge of the mechanism (John Sutton and IO) I common knowledge of the environment is maybe (at least a bit) less of a problem

I take relaxing common knowledge assumptions seriously and allowing real incomplete information in mechanism design I this may rationalize simple / detail-free mechanisms (and suggest new questions)

I Contrast this with economic theory / game theory

I One response to misunderstanding: do not address "incomplete information", focus on simple mechanisms, computational constraints, worst case analysis, etc... I Another response:

The Misunderstanding of John Harsanyi and Mechanism Design

I really important to relax common knowledge of the mechanism (John Sutton and IO) I common knowledge of the environment is maybe (at least a bit) less of a problem I One response to misunderstanding: do not address "incomplete information", focus on simple mechanisms, computational constraints, worst case analysis, etc... I Another response:

The Misunderstanding of John Harsanyi and Mechanism Design

I common knowledge of the environment is maybe (at least a bit) less of a problem I One response to misunderstanding: do not address "incomplete information", focus on simple mechanisms, computational constraints, worst case analysis, etc... I Another response:

The Misunderstanding of John Harsanyi and Mechanism Design

I One response to misunderstanding: do not address "incomplete information", focus on simple mechanisms, computational constraints, worst case analysis, etc... I Another response:

The Misunderstanding of John Harsanyi and Mechanism Design

I Another response:

The Misunderstanding of John Harsanyi and Mechanism Design

I Mechanism design I we would really like to assume that there is complete information about the game/mechanism I it is particularly desirable to relax common knowledge assumptions about the environment, because optimal mechanisms are otherwise too …nely tuned I Contrast this with economic theory / game theory I really important to relax common knowledge of the mechanism (John Sutton and IO) I common knowledge of the environment is maybe (at least a bit) less of a problem I One response to misunderstanding: do not address "incomplete information", focus on simple mechanisms, computational constraints, worst case analysis, etc... I take relaxing common knowledge assumptions seriously and allowing real incomplete information in mechanism design I this may rationalize simple / detail-free mechanisms (and suggest new questions)

The Misunderstanding of John Harsanyi and Mechanism Design

I Mechanism design I we would really like to assume that there is complete information about the game/mechanism I it is particularly desirable to relax common knowledge assumptions about the environment, because optimal mechanisms are otherwise too …nely tuned I Contrast this with economic theory / game theory I really important to relax common knowledge of the mechanism (John Sutton and IO) I common knowledge of the environment is maybe (at least a bit) less of a problem I One response to misunderstanding: do not address "incomplete information", focus on simple mechanisms, computational constraints, worst case analysis, etc... I Another response: I take relaxing common knowledge assumptions seriously and allowing real incomplete information in mechanism design The Misunderstanding of John Harsanyi and Mechanism Design

I Mechanism design I we would really like to assume that there is complete information about the game/mechanism I it is particularly desirable to relax common knowledge assumptions about the environment, because optimal mechanisms are otherwise too …nely tuned I Contrast this with economic theory / game theory I really important to relax common knowledge of the mechanism (John Sutton and IO) I common knowledge of the environment is maybe (at least a bit) less of a problem I One response to misunderstanding: do not address "incomplete information", focus on simple mechanisms, computational constraints, worst case analysis, etc... I Another response: I take relaxing common knowledge assumptions seriously and allowing real incomplete information in mechanism design I this may rationalize simple / detail-free mechanisms (and suggest new questions) 1. naive type space (identify types with payo¤ parameters) 2. common prior (beliefs could have been derived from common prior and Bayes updating) 3. and either independence or beliefs determine payo¤ parameters (BDP: Neeman 2004) implied by generic beliefs on naive type space

I Sometimes implicitly or explicitly trying to implement on all types spaces in some class (e.g., all naive common prior independent type spaces)

I Implementing on the universal type space is the same (modulo technicalities) as implementing on all types spaces

Type Space Restrictions = Implicit Common Knowledge Assumptions

I Common to assume: 2. common prior (beliefs could have been derived from common prior and Bayes updating) 3. and either independence or beliefs determine payo¤ parameters (BDP: Neeman 2004) implied by generic beliefs on naive type space

I Sometimes implicitly or explicitly trying to implement on all types spaces in some class (e.g., all naive common prior independent type spaces)

I Implementing on the universal type space is the same (modulo technicalities) as implementing on all types spaces

Type Space Restrictions = Implicit Common Knowledge Assumptions

I Common to assume: 1. naive type space (identify types with payo¤ parameters) 3. and either independence or beliefs determine payo¤ parameters (BDP: Neeman 2004) implied by generic beliefs on naive type space

I Sometimes implicitly or explicitly trying to implement on all types spaces in some class (e.g., all naive common prior independent type spaces)

I Implementing on the universal type space is the same (modulo technicalities) as implementing on all types spaces

Type Space Restrictions = Implicit Common Knowledge Assumptions

I Common to assume: 1. naive type space (identify types with payo¤ parameters) 2. common prior (beliefs could have been derived from common prior and Bayes updating) I Sometimes implicitly or explicitly trying to implement on all types spaces in some class (e.g., all naive common prior independent type spaces)

I Implementing on the universal type space is the same (modulo technicalities) as implementing on all types spaces

Type Space Restrictions = Implicit Common Knowledge Assumptions

I Common to assume: 1. naive type space (identify types with payo¤ parameters) 2. common prior (beliefs could have been derived from common prior and Bayes updating) 3. and either independence or beliefs determine payo¤ parameters (BDP: Neeman 2004) implied by generic beliefs on naive type space I Implementing on the universal type space is the same (modulo technicalities) as implementing on all types spaces

Type Space Restrictions = Implicit Common Knowledge Assumptions

I Sometimes implicitly or explicitly trying to implement on all types spaces in some class (e.g., all naive common prior independent type spaces) Type Space Restrictions = Implicit Common Knowledge Assumptions

I Sometimes implicitly or explicitly trying to implement on all types spaces in some class (e.g., all naive common prior independent type spaces)

I Implementing on the universal type space is the same (modulo technicalities) as implementing on all types spaces I players can be given a strictly positive incentive to truthfully announce their types via bets at no expected cost

I BDP does (or does not) hold generically on the universal type space

I There is not full surplus extraction on the universal type space I Take a position on which types in the universal type space are relevant

I with independent naive common prior type space, buyers earn information rent I with BDP naive common prior type space, e¢ cient allocation and full surplus extraction

I One response:

I Nuanced response:

Funny Result 1: Full Surplus Extraction

I Consider the private good allocation problem with private values and transfers. Easy to implement the e¢ cient allocation. But two key results about revenue: I players can be given a strictly positive incentive to truthfully announce their types via bets at no expected cost

I BDP does (or does not) hold generically on the universal type space

I There is not full surplus extraction on the universal type space I Take a position on which types in the universal type space are relevant

I with BDP naive common prior type space, e¢ cient allocation and full surplus extraction

I One response:

I Nuanced response:

Funny Result 1: Full Surplus Extraction

I Consider the private good allocation problem with private values and transfers. Easy to implement the e¢ cient allocation. But two key results about revenue:

I with independent naive common prior type space, buyers earn information rent I BDP does (or does not) hold generically on the universal type space

I There is not full surplus extraction on the universal type space I Take a position on which types in the universal type space are relevant

I players can be given a strictly positive incentive to truthfully announce their types via bets at no expected cost

I One response:

I Nuanced response:

Funny Result 1: Full Surplus Extraction

I Consider the private good allocation problem with private values and transfers. Easy to implement the e¢ cient allocation. But two key results about revenue:

I with independent naive common prior type space, buyers earn information rent I with BDP naive common prior type space, e¢ cient allocation and full surplus extraction I BDP does (or does not) hold generically on the universal type space

I There is not full surplus extraction on the universal type space I Take a position on which types in the universal type space are relevant

I One response:

I Nuanced response:

Funny Result 1: Full Surplus Extraction

I Consider the private good allocation problem with private values and transfers. Easy to implement the e¢ cient allocation. But two key results about revenue:

I with independent naive common prior type space, buyers earn information rent I with BDP naive common prior type space, e¢ cient allocation and full surplus extraction

I players can be given a strictly positive incentive to truthfully announce their types via bets at no expected cost I There is not full surplus extraction on the universal type space I Take a position on which types in the universal type space are relevant

I BDP does (or does not) hold generically on the universal type space

I Nuanced response:

Funny Result 1: Full Surplus Extraction

I Consider the private good allocation problem with private values and transfers. Easy to implement the e¢ cient allocation. But two key results about revenue:

I with independent naive common prior type space, buyers earn information rent I with BDP naive common prior type space, e¢ cient allocation and full surplus extraction

I players can be given a strictly positive incentive to truthfully announce their types via bets at no expected cost

I One response: I There is not full surplus extraction on the universal type space I Take a position on which types in the universal type space are relevant

I Nuanced response:

Funny Result 1: Full Surplus Extraction

I Consider the private good allocation problem with private values and transfers. Easy to implement the e¢ cient allocation. But two key results about revenue:

I with independent naive common prior type space, buyers earn information rent I with BDP naive common prior type space, e¢ cient allocation and full surplus extraction

I players can be given a strictly positive incentive to truthfully announce their types via bets at no expected cost

I One response:

I BDP does (or does not) hold generically on the universal type space I There is not full surplus extraction on the universal type space I Take a position on which types in the universal type space are relevant

Funny Result 1: Full Surplus Extraction

I Consider the private good allocation problem with private values and transfers. Easy to implement the e¢ cient allocation. But two key results about revenue:

I with independent naive common prior type space, buyers earn information rent I with BDP naive common prior type space, e¢ cient allocation and full surplus extraction

I players can be given a strictly positive incentive to truthfully announce their types via bets at no expected cost

I One response:

I BDP does (or does not) hold generically on the universal type space

I Nuanced response: I Take a position on which types in the universal type space are relevant

Funny Result 1: Full Surplus Extraction

I Consider the private good allocation problem with private values and transfers. Easy to implement the e¢ cient allocation. But two key results about revenue:

I with independent naive common prior type space, buyers earn information rent I with BDP naive common prior type space, e¢ cient allocation and full surplus extraction

I players can be given a strictly positive incentive to truthfully announce their types via bets at no expected cost

I One response:

I BDP does (or does not) hold generically on the universal type space

I Nuanced response:

I There is not full surplus extraction on the universal type space Funny Result 1: Full Surplus Extraction

I Consider the private good allocation problem with private values and transfers. Easy to implement the e¢ cient allocation. But two key results about revenue:

I with independent naive common prior type space, buyers earn information rent I with BDP naive common prior type space, e¢ cient allocation and full surplus extraction

I players can be given a strictly positive incentive to truthfully announce their types via bets at no expected cost

I One response:

I BDP does (or does not) hold generically on the universal type space

I Nuanced response:

I There is not full surplus extraction on the universal type space I Take a position on which types in the universal type space are relevant I with independent types, AGV (see also Arrow)

I back to dominant strategies and negative results I prior extraction: ask players to report their common prior and shoot them if they report something di¤erent

I Implemention of the e¢ cient outcome in Bayes Nash equilibrium on universal type space may or may not be equivalent to dominant strategies implementation

I Not possible to implement e¢ cient choice in dominant strategies I Possible to implement e¢ cient choice in (Bayes) Nash equilibrium

I But what if the prior is not known? Two responses:

I Alternative nuanced response: relax union of common prior naive type spaces assumption to universal type space. Nuanced conclusion:

Funny Result 2: Prior Extraction

I Consider a public goods problem with private values and budget balanced transfers. Two key public good results: I with independent types, AGV (see also Arrow)

I back to dominant strategies and negative results I prior extraction: ask players to report their common prior and shoot them if they report something di¤erent

I Implemention of the e¢ cient outcome in Bayes Nash equilibrium on universal type space may or may not be equivalent to dominant strategies implementation

I Possible to implement e¢ cient choice in (Bayes) Nash equilibrium

I But what if the prior is not known? Two responses:

I Alternative nuanced response: relax union of common prior naive type spaces assumption to universal type space. Nuanced conclusion:

Funny Result 2: Prior Extraction

I Consider a public goods problem with private values and budget balanced transfers. Two key public good results:

I Not possible to implement e¢ cient choice in dominant strategies I back to dominant strategies and negative results I prior extraction: ask players to report their common prior and shoot them if they report something di¤erent

I Implemention of the e¢ cient outcome in Bayes Nash equilibrium on universal type space may or may not be equivalent to dominant strategies implementation

I with independent types, AGV (see also Arrow)

I But what if the prior is not known? Two responses:

I Alternative nuanced response: relax union of common prior naive type spaces assumption to universal type space. Nuanced conclusion:

Funny Result 2: Prior Extraction

I Consider a public goods problem with private values and budget balanced transfers. Two key public good results:

I Not possible to implement e¢ cient choice in dominant strategies I Possible to implement e¢ cient choice in (Bayes) Nash equilibrium I back to dominant strategies and negative results I prior extraction: ask players to report their common prior and shoot them if they report something di¤erent

I Implemention of the e¢ cient outcome in Bayes Nash equilibrium on universal type space may or may not be equivalent to dominant strategies implementation

I But what if the prior is not known? Two responses:

I Alternative nuanced response: relax union of common prior naive type spaces assumption to universal type space. Nuanced conclusion:

Funny Result 2: Prior Extraction

I Consider a public goods problem with private values and budget balanced transfers. Two key public good results:

I Not possible to implement e¢ cient choice in dominant strategies I Possible to implement e¢ cient choice in (Bayes) Nash equilibrium

I with independent types, AGV (see also Arrow) I Implemention of the e¢ cient outcome in Bayes Nash equilibrium on universal type space may or may not be equivalent to dominant strategies implementation

I back to dominant strategies and negative results I prior extraction: ask players to report their common prior and shoot them if they report something di¤erent

I Alternative nuanced response: relax union of common prior naive type spaces assumption to universal type space. Nuanced conclusion:

Funny Result 2: Prior Extraction

I Consider a public goods problem with private values and budget balanced transfers. Two key public good results:

I Not possible to implement e¢ cient choice in dominant strategies I Possible to implement e¢ cient choice in (Bayes) Nash equilibrium

I with independent types, AGV (see also Arrow)

I But what if the prior is not known? Two responses: I Implemention of the e¢ cient outcome in Bayes Nash equilibrium on universal type space may or may not be equivalent to dominant strategies implementation

I prior extraction: ask players to report their common prior and shoot them if they report something di¤erent

I Alternative nuanced response: relax union of common prior naive type spaces assumption to universal type space. Nuanced conclusion:

Funny Result 2: Prior Extraction

I Consider a public goods problem with private values and budget balanced transfers. Two key public good results:

I Not possible to implement e¢ cient choice in dominant strategies I Possible to implement e¢ cient choice in (Bayes) Nash equilibrium

I with independent types, AGV (see also Arrow)

I But what if the prior is not known? Two responses:

I back to dominant strategies and negative results I Implemention of the e¢ cient outcome in Bayes Nash equilibrium on universal type space may or may not be equivalent to dominant strategies implementation

I Alternative nuanced response: relax union of common prior naive type spaces assumption to universal type space. Nuanced conclusion:

Funny Result 2: Prior Extraction

I Consider a public goods problem with private values and budget balanced transfers. Two key public good results:

I Not possible to implement e¢ cient choice in dominant strategies I Possible to implement e¢ cient choice in (Bayes) Nash equilibrium

I with independent types, AGV (see also Arrow)

I But what if the prior is not known? Two responses:

I back to dominant strategies and negative results I prior extraction: ask players to report their common prior and shoot them if they report something di¤erent I Implemention of the e¢ cient outcome in Bayes Nash equilibrium on universal type space may or may not be equivalent to dominant strategies implementation

Funny Result 2: Prior Extraction

I Consider a public goods problem with private values and budget balanced transfers. Two key public good results:

I Not possible to implement e¢ cient choice in dominant strategies I Possible to implement e¢ cient choice in (Bayes) Nash equilibrium

I with independent types, AGV (see also Arrow)

I But what if the prior is not known? Two responses:

I back to dominant strategies and negative results I prior extraction: ask players to report their common prior and shoot them if they report something di¤erent

I Alternative nuanced response: relax union of common prior naive type spaces assumption to universal type space. Nuanced conclusion: Funny Result 2: Prior Extraction

I Consider a public goods problem with private values and budget balanced transfers. Two key public good results:

I Not possible to implement e¢ cient choice in dominant strategies I Possible to implement e¢ cient choice in (Bayes) Nash equilibrium

I with independent types, AGV (see also Arrow)

I But what if the prior is not known? Two responses:

I back to dominant strategies and negative results I prior extraction: ask players to report their common prior and shoot them if they report something di¤erent

I Alternative nuanced response: relax union of common prior naive type spaces assumption to universal type space. Nuanced conclusion:

I Implemention of the e¢ cient outcome in Bayes Nash equilibrium on universal type space may or may not be equivalent to dominant strategies implementation I Let’srelax this assumption I Suppose that values are interdependent

I Ann’svalue of an object is vA = θA + γθB for some 0 < γ < 1

I Analogously, Bob’svalue is vB = θB + γθA

Relaxing Private Values Assumption

I Maintained common knowledge assumption in discussion so far: private values I Suppose that values are interdependent

I Ann’svalue of an object is vA = θA + γθB for some 0 < γ < 1

I Analogously, Bob’svalue is vB = θB + γθA

Relaxing Private Values Assumption

I Maintained common knowledge assumption in discussion so far: private values

I Let’srelax this assumption I Ann’svalue of an object is vA = θA + γθB for some 0 < γ < 1

I Analogously, Bob’svalue is vB = θB + γθA

Relaxing Private Values Assumption

I Maintained common knowledge assumption in discussion so far: private values

I Let’srelax this assumption I Suppose that values are interdependent I Analogously, Bob’svalue is vB = θB + γθA

Relaxing Private Values Assumption

I Maintained common knowledge assumption in discussion so far: private values

I Let’srelax this assumption I Suppose that values are interdependent

I Ann’svalue of an object is vA = θA + γθB for some 0 < γ < 1 Relaxing Private Values Assumption

I Maintained common knowledge assumption in discussion so far: private values

I Let’srelax this assumption I Suppose that values are interdependent

I Ann’svalue of an object is vA = θA + γθB for some 0 < γ < 1

I Analogously, Bob’svalue is vB = θB + γθA 2. Ann and Bob each have a signal that confounds a common value and private value component (cannot be distinguished)

Three Interpretations

1. θA is Ann’sconsumption value but it is possible that Ann will have to re-sell to Bob, extracting proportion γ of Bob’svalue Three Interpretations

1. θA is Ann’sconsumption value but it is possible that Ann will have to re-sell to Bob, extracting proportion γ of Bob’svalue 2. Ann and Bob each have a signal that confounds a common value and private value component (cannot be distinguished) I By linear algebra, we have 1 1 θA = (vA γvB ) and θB = (vB γvA) 1 γ2 1 γ2 I So if we considered the player speci…c payo¤ parameter universal type space for (θA, θB ), we were implicitly assuming that there was common knowledge that Ann knows vA γvB and Bob knows vB γvA I Whether this makes sense depends on the interpretation

Implicit Common Knowledge Assumptions and Interdependent Values

I In example, we have single good interdependent values example, we had

vA = θA + γθB and vB = θB + γθA I So if we considered the player speci…c payo¤ parameter universal type space for (θA, θB ), we were implicitly assuming that there was common knowledge that Ann knows vA γvB and Bob knows vB γvA I Whether this makes sense depends on the interpretation

Implicit Common Knowledge Assumptions and Interdependent Values

I In example, we have single good interdependent values example, we had

vA = θA + γθB and vB = θB + γθA

I By linear algebra, we have 1 1 θA = (vA γvB ) and θB = (vB γvA) 1 γ2 1 γ2 I Whether this makes sense depends on the interpretation

Implicit Common Knowledge Assumptions and Interdependent Values

I In example, we have single good interdependent values example, we had

vA = θA + γθB and vB = θB + γθA

I By linear algebra, we have 1 1 θA = (vA γvB ) and θB = (vB γvA) 1 γ2 1 γ2 I So if we considered the player speci…c payo¤ parameter universal type space for (θA, θB ), we were implicitly assuming that there was common knowledge that Ann knows vA γvB and Bob knows vB γvA Implicit Common Knowledge Assumptions and Interdependent Values

I In example, we have single good interdependent values example, we had

vA = θA + γθB and vB = θB + γθA

I By linear algebra, we have 1 1 θA = (vA γvB ) and θB = (vB γvA) 1 γ2 1 γ2 I So if we considered the player speci…c payo¤ parameter universal type space for (θA, θB ), we were implicitly assuming that there was common knowledge that Ann knows vA γvB and Bob knows vB γvA I Whether this makes sense depends on the interpretation I Ann’sbelief about Bob’s…rst order valuation I Ann’svaluation conditional on Bob’s…rst order valuations

I Ann’sbelief about Bob’ssecond order type I Ann’svaluation conditional on Bob’ssecond order type

2. second order belief and valuation:

3. third order belief and valuation:

4. and so on

Canonical Preference Higher-Order Preference Types

Should actually distinguish "higher order preference types", e.g., 1. …rst order valuation: Ann’sunconditional value of an object,

Thus we are looking higher order preferences over acts I Ann’sbelief about Bob’ssecond order type I Ann’svaluation conditional on Bob’ssecond order type

I Ann’sbelief about Bob’s…rst order valuation I Ann’svaluation conditional on Bob’s…rst order valuations 3. third order belief and valuation:

4. and so on

Canonical Preference Higher-Order Preference Types

Should actually distinguish "higher order preference types", e.g., 1. …rst order valuation: Ann’sunconditional value of an object, 2. second order belief and valuation:

Thus we are looking higher order preferences over acts I Ann’sbelief about Bob’ssecond order type I Ann’svaluation conditional on Bob’ssecond order type

I Ann’svaluation conditional on Bob’s…rst order valuations 3. third order belief and valuation:

4. and so on

Canonical Preference Higher-Order Preference Types

Should actually distinguish "higher order preference types", e.g., 1. …rst order valuation: Ann’sunconditional value of an object, 2. second order belief and valuation:

I Ann’sbelief about Bob’s…rst order valuation

Thus we are looking higher order preferences over acts I Ann’sbelief about Bob’ssecond order type I Ann’svaluation conditional on Bob’ssecond order type

3. third order belief and valuation:

4. and so on

Canonical Preference Higher-Order Preference Types

Should actually distinguish "higher order preference types", e.g., 1. …rst order valuation: Ann’sunconditional value of an object, 2. second order belief and valuation:

I Ann’sbelief about Bob’s…rst order valuation I Ann’svaluation conditional on Bob’s…rst order valuations

Thus we are looking higher order preferences over acts I Ann’sbelief about Bob’ssecond order type I Ann’svaluation conditional on Bob’ssecond order type 4. and so on

Canonical Preference Higher-Order Preference Types

Should actually distinguish "higher order preference types", e.g., 1. …rst order valuation: Ann’sunconditional value of an object, 2. second order belief and valuation:

I Ann’sbelief about Bob’s…rst order valuation I Ann’svaluation conditional on Bob’s…rst order valuations 3. third order belief and valuation:

Thus we are looking higher order preferences over acts I Ann’svaluation conditional on Bob’ssecond order type 4. and so on

Canonical Preference Higher-Order Preference Types

Should actually distinguish "higher order preference types", e.g., 1. …rst order valuation: Ann’sunconditional value of an object, 2. second order belief and valuation:

I Ann’sbelief about Bob’s…rst order valuation I Ann’svaluation conditional on Bob’s…rst order valuations 3. third order belief and valuation:

I Ann’sbelief about Bob’ssecond order type

Thus we are looking higher order preferences over acts 4. and so on

Canonical Preference Higher-Order Preference Types

Should actually distinguish "higher order preference types", e.g., 1. …rst order valuation: Ann’sunconditional value of an object, 2. second order belief and valuation:

I Ann’sbelief about Bob’s…rst order valuation I Ann’svaluation conditional on Bob’s…rst order valuations 3. third order belief and valuation:

I Ann’sbelief about Bob’ssecond order type I Ann’svaluation conditional on Bob’ssecond order type

Thus we are looking higher order preferences over acts Canonical Preference Higher-Order Preference Types

Should actually distinguish "higher order preference types", e.g., 1. …rst order valuation: Ann’sunconditional value of an object, 2. second order belief and valuation:

I Ann’sbelief about Bob’s…rst order valuation I Ann’svaluation conditional on Bob’s…rst order valuations 3. third order belief and valuation:

I Ann’sbelief about Bob’ssecond order type I Ann’svaluation conditional on Bob’ssecond order type 4. and so on Thus we are looking higher order preferences over acts I In this representation, Ann’sprobability of state 1 is her unconditional valuation of the object

I Higher order preference types correspond exactly to what would be learnt about players

I selling on the higher-order preference type space is complicated

Universal Higher-Order Preference Type Space

I Can represent all higher-order preference types by universal space T = ∆ (T 0, 1 ) f g I Higher order preference types correspond exactly to what would be learnt about players

I selling on the higher-order preference type space is complicated

Universal Higher-Order Preference Type Space

I Can represent all higher-order preference types by universal space T = ∆ (T 0, 1 ) f g I In this representation, Ann’sprobability of state 1 is her unconditional valuation of the object I selling on the higher-order preference type space is complicated

Universal Higher-Order Preference Type Space

I Can represent all higher-order preference types by universal space T = ∆ (T 0, 1 ) f g I In this representation, Ann’sprobability of state 1 is her unconditional valuation of the object

I Higher order preference types correspond exactly to what would be learnt about players Universal Higher-Order Preference Type Space

I Can represent all higher-order preference types by universal space T = ∆ (T 0, 1 ) f g I In this representation, Ann’sprobability of state 1 is her unconditional valuation of the object

I Higher order preference types correspond exactly to what would be learnt about players

I selling on the higher-order preference type space is complicated Conclusion

I Incomplete information has not been fully incorporated into economic analysis

I Results are driven by implicit common knowledge whose role is sometimes not well understood

I But relaxing all common knowledge assumptions may be possible but unhelpful

I Focus on which are reasonable common knowledge assumptions and make them explicit