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Rationalizable Strategies and Nash Equilibrium

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Rationalizable Strategies and Nash Equilibrium Rationalizable Strategies and Nash Equilibrium Carlos Hurtado Department of Economics University of Illinois at Urbana-Champaign [email protected] Junel 9th, 2016 C. Hurtado (UIUC - Economics) Game Theory On the Agenda 1 Formalizing the Game 2 Rationalizability 3 Exercises 4 Nash Equilibrium 5 Exercises C. Hurtado (UIUC - Economics) Game Theory Formalizing the Game On the Agenda 1 Formalizing the Game 2 Rationalizability 3 Exercises 4 Nash Equilibrium 5 Exercises C. Hurtado (UIUC - Economics) Game Theory Formalizing the Game Formalizing the Game I Let me fix some Notation: - set of players: I = {1, 2, ··· , N} - set of actions: ∀i ∈ I, ai ∈ Ai , where each player i has a set of actions Ai . - strategies for each player: ∀i ∈ I, si ∈ Si , where each player i has a set of pure strategies Si available to him. A strategy is a complete contingent plan for playing the game, which specifies a feasible action of a player’s information sets in the game. QN - profile of pure strategies: s = (s1, s2, ··· , sN ) ∈ i=1 Si = S. Note: let s−i = (s1, s2, ··· , si−1, si+1, ··· , sN ) ∈ S−i , we will denote s = (si , s−i ) ∈ (Si , S−i ) = S. QN - Payoff function: ui : i=1 Si → R, denoted by ui (si , s−i ) - A mixed strategy for player i is a function σi : Si → [0, 1], which assigns a probability σi (si ) ≥ 0 to each pure strategy si ∈ Si , satisfying P σi (si ) = 1. si ∈Si C. Hurtado (UIUC - Economics) Game Theory 1 / 16 Formalizing the Game Formalizing the Game I Notice now that even if there is no role for nature in a game, when players use (nondegenerate) mixed strategies, this induces a probability distribution over terminal nodes of the game. I But we can easily extend payoffs again to define payoffs over a profile of mixed strategies as follows: X ui (σ1, ··· , σN ) = [σ1(s1) ··· σN (sN )] ui (s1, ··· , sN ) s∈S " # X X Y ui (σi , σ−i ) = σj (sj ) σi (si )ui (si , s−i ) si ∈Si s−i ∈S−i j6=i I For the above formula to make sense, it is critical that each player is randomizing independently. That is, each player is independently tossing her own die to decide on which pure strategy to play. C. Hurtado (UIUC - Economics) Game Theory 2 / 16 Formalizing the Game Formalizing the Game I If si is a strictly dominant strategy for player i, then for all σi ∈ ∆(Si ), σi 6= si , and all σ−i ∈ ∆(S−i ), ui (si , σ−i ) > ui (σi , σ−i ). I Let σi ∈ ∆(Si ), with σi 6= si , and let σ−i ∈ ∆(S−i ). Then, " # X Y ui (si , σ−i ) = σj (sj ) ui (si , s−i ) s−i ∈S−i j6=i and " # X X Y ui (σi , σ−i ) = σj (sj ) σi (˜si )ui (˜si , s−i ) ˜si ∈Si s−i ∈S−i j6=i Then, ui (si , σ−i ) − ui (σi , σ−i ) is !" # X Y X σj (sj ) ui (si , s−i ) − σi (˜si )ui (˜si , s−i ) s−i ∈S−i j6=i ˜si ∈Si C. Hurtado (UIUC - Economics) Game Theory 3 / 16 Formalizing the Game Formalizing the Game I If si is a strictly dominant strategy for player i, then for all σi ∈ ∆(Si ), σi 6= si , and all σ−i ∈ ∆(S−i ), ui (si , σ−i ) > ui (σi , σ−i ). I Let σi ∈ ∆(Si ), with σi 6= si , and let σ−i ∈ ∆(S−i ). Then, " # X Y ui (si , σ−i ) = σj (sj ) ui (si , s−i ) s−i ∈S−i j6=i and " # X X Y ui (σi , σ−i ) = σj (sj ) σi (˜si )ui (˜si , s−i ) ˜si ∈Si s−i ∈S−i j6=i Then, ui (si , σ−i ) − ui (σi , σ−i ) is !" # X Y X σj (sj ) ui (si , s−i ) − σi (˜si )ui (˜si , s−i ) s−i ∈S−i j6=i ˜si ∈Si C. Hurtado (UIUC - Economics) Game Theory 3 / 16 Formalizing the Game Formalizing the Game I ui (si , σ−i ) − ui (σi , σ−i ) is !" # X Y X σj (sj ) ui (si , s−i ) − σi (˜si )ui (˜si , s−i ) s−i ∈S−i j6=i ˜si ∈Si I Since si is strictly dominant, ui (si , s−i ) > ui (˜si , s−i ) for all ˜si 6= si and all s−i . X I Hence, ui (si , s−i ) > σi (˜si )ui (˜si , s−i ) for any σi ∈ ∆(Si ) such that σi 6= si ˜si ∈Si (why?). I This implies the desired inequality: ui (si , σ−i ) − ui (σi , σ−i ) > 0 C. Hurtado (UIUC - Economics) Game Theory 4 / 16 Formalizing the Game Formalizing the Game I We learned that: If si is a strictly dominant strategy for player i, then for all σi ∈ ∆(Si ), σi 6= si , and all σ−i ∈ ∆(S−i ), ui (si , σ−i ) > ui (σi , σ−i ). I Exercise 1. Show that there can be no strategy σi ∈ ∆(Si ) such that for all si ∈ Si and s−i ∈ S−i , ui (σi , s−i ) > ui (si , s−i ). I The preceding Theorem and Exercise show that there is absolutely no loss in restricting attention to pure strategies for all players when looking for strictly dominant strategies. C. Hurtado (UIUC - Economics) Game Theory 5 / 16 Rationalizability On the Agenda 1 Formalizing the Game 2 Rationalizability 3 Exercises 4 Nash Equilibrium 5 Exercises C. Hurtado (UIUC - Economics) Game Theory Rationalizability Rationalizability l r L 4,-4 9,-9 I M 6,-6 6,-6 R 9,-9 4,-4 I Penalty Kick Game is one of the most important games in the world. I This game has no dominant strategies I We need refinements to solve more games. C. Hurtado (UIUC - Economics) Game Theory 6 / 16 Rationalizability Rationalizability l r L 4,-4 9,-9 I M 6,-6 6,-6 R 9,-9 4,-4 I Penalty Kick Game is one of the most important games in the world. I This game has no dominant strategies I We need refinements to solve more games. C. Hurtado (UIUC - Economics) Game Theory 6 / 16 Rationalizability Rationalizability I I Do not shoot to the middle I Do not use a strategy that is never a best response C. Hurtado (UIUC - Economics) Game Theory 7 / 16 Rationalizability Rationalizability Definition A strategy σi ∈ ∆(Si ) is a best response to the strategy profile σ−i ∈ ∆(S−i ) if u(σi , σ−i ) ≥ u(˜σi , σ−i ) for all σ˜i ∈ ∆(Si ). A strategy σi ∈ ∆(Si ) is never a best response if there is no σ−i ∈ ∆(S−i ) for which σi is a best response. I The idea is that a strategy, σi , is a best response if there is some strategy profile of the opponents for which σi does at least as well as any other strategy. I Conversely, σi is never a best response if for every strategy profile of the opponents, there is some strategy that does strictly better than σi . I Clearly, in any game, a strategy that is strictly dominated is never a best response. I Exercise 2. Prove that in 2-player games, a pure strategy is never a best response if and only if it is strictly dominated. C. Hurtado (UIUC - Economics) Game Theory 8 / 16 Rationalizability Rationalizability I In games with more than 2 players, there may be strategies that are not strictly dominated that are nonetheless never best responses. I As before, it is a consequence of ”rationality” that a player should not play a strategy that is never a best response. That is, we can delete strategies that are never best responses. I By iterating on the knowledge of rationality, we iteratively delete strategies that are never best responses. I The set of strategies for a player that survives this iterated deletion of never best responses is called her set of rationalizable strategies. I The rationalizable actions can be computed as follows: 1 Start with the full action set for each player. 2 Remove actions which are never a best responses to any belief about the opponents’ actions. 3 Repeat process with the opponents’ remaining actions until no further actions are eliminated. 4 In this process leaves a non-empty set of actions for each player those are the rationalizable actions. C. Hurtado (UIUC - Economics) Game Theory 9 / 16 Rationalizability Rationalizability I In games with more than 2 players, there may be strategies that are not strictly dominated that are nonetheless never best responses. I As before, it is a consequence of ”rationality” that a player should not play a strategy that is never a best response. That is, we can delete strategies that are never best responses. I By iterating on the knowledge of rationality, we iteratively delete strategies that are never best responses. I The set of strategies for a player that survives this iterated deletion of never best responses is called her set of rationalizable strategies. I The rationalizable actions can be computed as follows: 1 Start with the full action set for each player. 2 Remove actions which are never a best responses to any belief about the opponents’ actions. 3 Repeat process with the opponents’ remaining actions until no further actions are eliminated. 4 In this process leaves a non-empty set of actions for each player those are the rationalizable actions. C. Hurtado (UIUC - Economics) Game Theory 9 / 16 Rationalizability Rationalizability Definition I σi ∈ ∆(Si ) is a 1-rationalizable strategy for player i if it is a best response to some strategy profile σ−i ∈ ∆(S−i ). I σi ∈ ∆(Si ) is a k-rationalizable strategy (k ≥ 2) for player i if it is a best response to some strategy profile σ−i ∈ ∆(S−i ) such that each σj is (k − 1)-rationalizable for player j 6= i.
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