Rationalizable Strategies and Nash Equilibrium

Home , Best response, Rationalizability, Strategy (game theory)

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 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

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.

I σi ∈ ∆(Si ) is a rationalizable for player i if it is k-rationalizable for all k ≥ 1.

C. Hurtado (UIUC - Economics) Game Theory 10 / 16 Rationalizability Rationalizability

I Note that the set of rationalizable strategies can no be larger that the set of strategies surviving iterative removal of strictly dominated strategies.

I This follows from the earlier comment that a strictly dominated strategy is never a best response.

I In this sense, rationalizability is (weakly) more restrictive than iterated deletion of strictly dominated strategies.

I It turns out that in 2-player games, the two concepts coincide. In n-player games (n > 2), they don’t have to.

I 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.).

I ”Rationalizable” intuitively means that there is a plausible explanation that would justify the use of the strategy.

C. Hurtado (UIUC - Economics) Game Theory 11 / 16 Exercises On the Agenda

1 Formalizing the Game

2 Rationalizability

3 Exercises

4 Nash Equilibrium

5 Exercises

C. Hurtado (UIUC - Economics) Game Theory Exercises Exercises

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 Exercise 2. Prove that in 2-player games, a pure strategy is never a best response if and only if it is strictly dominated.

I Determine the set of rationalizable pure strategies for the following game:

1/2 b1 b2 b3 b4

a1 0, 7 2, 5 7, 0 0, 1 I a2 5, 2 3, 3 5, 2 0, 1 a3 7, 0 2, 5 0, 7 0, 1

a4 0, 0 0,-2 0, 0 10,-1

C. Hurtado (UIUC - Economics) Game Theory 12 / 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 Nash Equilibrium Nash Equilibrium

I 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 = (s1, ..., sN ) ∈ S is a Pure Strategy Nash Equilibrium (PSNE) if for all i and ˜si ∈ Si , u(si , s−i ) ≥ u(˜si , s−i ).

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.

I Remark: Every ﬁnite game of perfect information has a pure strategy Nash equilibrium.

C. Hurtado (UIUC - Economics) Game Theory 13 / 16 Nash Equilibrium Nash Equilibrium

I 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”.

I 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).

I 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 14 / 16 Nash Equilibrium Nash Equilibrium

I 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 15 / 16 Exercises On the Agenda

1 Formalizing the Game

2 Rationalizability

3 Exercises

4 Nash Equilibrium

5 Exercises

C. Hurtado (UIUC - Economics) Game Theory Exercises

Exercises

I Find the Nash Equilibria of the following games:

I What about Rock, Paper, Scissors?

C. Hurtado (UIUC - Economics) Game Theory 16 / 16