Rationalizable Strategies

Transcription

1 Rationalizable Strategies Carlos Hurtado Department of Economics University of Illinois at Urbana-Champaign Jun 1st, 2015 C. Hurtado (UIUC - Economics) Game Theory

2 On the Agenda 1 Formalizing the Game 2 Rationalizability 3 Exercises 4 Nash Equilibrium 5 Exercises C. Hurtado (UIUC - Economics) Game Theory

3 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 1 / 19

4 Formalizing the Game Formalizing the Game Let me fix some Notation: - set of players: I = {1, 2,, N} - set of actions: i I, a i A i, where each player i has a set of actions A i. - strategies for each player: i I, s i S i, where each player i has a set of pure strategies S i 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. - profile of pure strategies: s = (s 1, s 2,, s N ) N Si = S. i=1 Note: let s i = (s 1, s 2,, s i 1, s i+1,, s N ) S i, we will denote s = (s i, s i) (S i, S i) = S. - Payoff function: u i : N Si R, denoted by ui(si, s i) i=1 - A mixed strategy for player i is a function σ i : S i [0, 1], which assigns a probability σ i(s i) 0 to each pure strategy s i S i, satisfying s i S i σ i(s i) = 1. C. Hurtado (UIUC - Economics) Game Theory 2 / 19

5 Formalizing the Game Formalizing the Game 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. But we can easily extend payoffs again to define payoffs over a profile of mixed strategies as follows: u i (σ 1,, σ N ) = u i (σ i, σ i ) = [σ 1 (s 1 ) σ N (s N )] u i (s 1,, s N ) s S s i S i s i S i j i [ ] σ j (s j ) σ i (s i )u i (s i, s 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 3 / 19

6 Formalizing the Game Formalizing the Game If s i is a strictly dominant strategy for player i, then for all σ i (S i), σ i s i, and all σ i (S i), u i(s i, σ i) > u i(σ i, σ i). Let σ i (S i), with σ i s i, and let σ i (S i). Then, [ ] u i(s i, σ i) = σ j(s j) u i(s i, s i) and u i(σ i, σ i) = s i S i s i S i s i S i j i [ ] σ j(s j) σ i( s i)u i( s i, s i) Then, u i(s i, σ i) u i(σ i, σ i) is ( ) [ ] σ j(s j) u i(s i, s i) σ i( s i)u i( s i, s i) j i s i S i s i S i j i C. Hurtado (UIUC - Economics) Game Theory 4 / 19

7 Formalizing the Game Formalizing the Game u i(s i, σ i) u i(σ i, σ i) is ( ) [ ] σ j(s j) u i(s i, s i) σ i( s i)u i( s i, s i) j i s i S i s i S i Since s i is strictly dominant, u i(s i, s i) > u i( s i, s i) for all s i s i and all s i. Hence, u i(s i, s i) > σ i( s i)u i( s i, s i) for any σ i (S i) such that σ i s i s i S i (why?). This implies the desired inequality: u i(s i, σ i) u i(σ i, σ i) > 0 C. Hurtado (UIUC - Economics) Game Theory 5 / 19

8 Formalizing the Game Formalizing the Game We learned that: If s i is a strictly dominant strategy for player i, then for all σ i (S i), σ i s i, and all σ i (S i), u i(s i, σ i) > u i(σ i, σ i). Exercise 1. Show that there can be no strategy σ i (S i) such that for all s i S i and s i S i, u i(σ i, s i) > u i(s i, s 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 6 / 19

9 Rationalizability On the Agenda 1 Formalizing the Game 2 Rationalizability 3 Exercises 4 Nash Equilibrium 5 Exercises C. Hurtado (UIUC - Economics) Game Theory 7 / 19

10 Rationalizability Rationalizability Definition A strategy σ i (S i) is a best response to the strategy profile σ i (S i) if u(σ i, σ i) u( σ i, σ i) for all σ i (S i). A strategy σ i (S i) is never a best response if there is no σ i (S i) for which σ i is a best response. The idea is that a strategy, σ i, is a best response if there is some strategy profile of the opponents for which σ i does at least as well as any other strategy. Conversely, σ i is never a best response if for every strategy profile of the opponents, there is some strategy that does strictly better than σ i. Clearly, in any game, a strategy that is strictly dominated is never a best response. 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 / 19

11 Rationalizability Rationalizability In games with more than 2 players, there may be strategies that are not strictly dominated that are nonetheless never best responses. 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. By iterating on the knowledge of rationality, we iteratively delete strategies that are never best responses. The set of strategies for a player that survives this iterated deletion of never best responses is called her set of rationalizable strategies. C. Hurtado (UIUC - Economics) Game Theory 9 / 19

12 Rationalizability Rationalizability Definition 1 σ i (S i) is a 1-rationalizable strategy for player i if it is a best response to some strategy profile σ i (S i). 2 σ i (S i) 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 σ j is (k âĺš 1)-rationalizable for player j i. 3 σ i (S i) is a rationalizable for player i if it is k-rationalizable for all k 1. C. Hurtado (UIUC - Economics) Game Theory 10 / 19

13 Rationalizability Rationalizability Note that the set of rationalizable strategies can no be larger that the set of strategies surviving iterative removal of strictly dominated strategies. This follows from the earlier comment that a strictly dominated strategy is never a best response. In this sense, rationalizability is (weakly) more restrictive than iterated deletion of strictly dominated strategies. It turns out that in 2-player games, the two concepts coincide. In n-player games (n > 2), they don t have to. Strategies that remain after iterative elimination of strategies that are never best responses: those that a rational player can justify, or rationalize, with some reasonable conjecture concerning the behavior of his rivals (reasonable in the sense that his opponents are not presumed to play strategies that are never best responses, etc.). Rationalizable intuitively means that there is a plausible explanation that would justify the use of the strategy. C. Hurtado (UIUC - Economics) Game Theory 11 / 19

14 Exercises On the Agenda 1 Formalizing the Game 2 Rationalizability 3 Exercises 4 Nash Equilibrium 5 Exercises C. Hurtado (UIUC - Economics) Game Theory 12 / 19

15 Exercises Exercises Exercise 1. Show that there can be no strategy σ i (S i) such that for all s i S i and s i S i, u i(σ i, s i) > u i(s i, s 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. Determine the set of rationalizable pure strategies for the following game: 1/2 b 1 b 2 b 3 b 4 a 1 0, 7 2, 5 7, 0 0, 1 a 2 5, 2 3, 3 5, 2 0, 1 a 3 7, 0 2, 5 0, 7 0, 1 a 4 0, 0 0,2 0, 0 10,1 C. Hurtado (UIUC - Economics) Game Theory 13 / 19

16 Nash Equilibrium On the Agenda 1 Formalizing the Game 2 Rationalizability 3 Exercises 4 Nash Equilibrium 5 Exercises C. Hurtado (UIUC - Economics) Game Theory 14 / 19

17 Nash Equilibrium Nash Equilibrium Now we turn to the most well-known solution concept in game theory. We ll first discuss pure strategy Nash equilibrium (PSNE), and then later extend to mixed strategies. Definition A strategy profile s = (s 1,..., s N ) S is a Pure Strategy Nash Equilibrium (PSNE) if for all i and s i S i, u(s i, s i) u( s i, s i). In a Nash equilibrium, each player s strategy must be a best response to those strategies of his opponents that are components of the equilibrium. Remark: Every finite game of perfect information has a pure strategy Nash equilibrium. C. Hurtado (UIUC - Economics) Game Theory 15 / 19

18 Nash Equilibrium Nash Equilibrium Unlike with our earlier solution concepts (dominance and rationalizability), Nash equilibrium applies to a profile of strategies rather than any individual s strategy. When people say Nash equilibrium strategy, what they mean is a strategy that is part of a Nash equilibrium profile. The term equilibrium is used because it connotes that if a player knew that his opponents were playing the prescribed strategies, then she is playing optimally by following her prescribed strategy. In a sense, this is like a rational expectations equilibrium, in that in a Nash equilibrium, a player s beliefs about what his opponents will do get confirmed (where the beliefs are precisely the opponents prescribed strategies). Rationalizability only requires a player play optimally with respect to some reasonable conjecture about the opponents play, where reasonable means that the conjectured play of the rivals can also be justified in this way. On the other hand, Nash requires that a player play optimally with respect to what his opponents are actually playing. That is to say, the conjecture she holds about her opponents play is correct. C. Hurtado (UIUC - Economics) Game Theory 16 / 19

19 Nash Equilibrium Nash Equilibrium The above point makes clear that Nash equilibrium is not simply a consequence of (common knowledge of) rationality and the structure of the game. Clearly, each player s strategy in a Nash equilibrium profile is rationalizable, but lots of rationalizable profiles are not Nash equilibria. C. Hurtado (UIUC - Economics) Game Theory 17 / 19

20 Exercises On the Agenda 1 Formalizing the Game 2 Rationalizability 3 Exercises 4 Nash Equilibrium 5 Exercises C. Hurtado (UIUC - Economics) Game Theory 18 / 19

21 Exercises Exercises Find the Nash Equilibria of the following games: What about Rock, Paper, Scissors? C. Hurtado (UIUC - Economics) Game Theory 19 / 19