Carlos Hurtado - University Of Illinoishrtdmrt2/Teaching/GT_2016_19/L3.pdf · Carlos Hurtado...

Rationalizable Strategies and Nash Equilibrium

Carlos Hurtado

Department of EconomicsUniversity of Illinois at Urbana-Champaign

[email protected]

Junel 9th, 2016

C. Hurtado (UIUC - Economics) Game Theory

mailto:[email protected]

On the Agenda

1 Formalizing the Game

2 Rationalizability

3 Exercises

4 Nash Equilibrium

5 Exercises


Formalizing the Game

On the Agenda


2 Rationalizability

3 Exercises

4 Nash Equilibrium

5 Exercises




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 planfor playing the game, which specifies a feasible action of a player’sinformation sets in the game.

- profile of pure strategies: s = (s1, s2, · · · , sN) ∈∏N

i=1 Si = S.Note: let s−i = (s1, s2, · · · , si−1, si+1, · · · , sN) ∈ S−i , we will denotes = (si , s−i ) ∈ (Si , S−i ) = S.

- Payoff function: ui :∏N

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∑si∈Si

σi (si ) = 1.

C. Hurtado (UIUC - Economics) Game Theory 1 / 16



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 overterminal nodes of the game.

I But we can easily extend payoffs again to define payoffs over a profile of mixedstrategies as follows:

ui (σ1, · · · , σN) =∑s∈S

[σ1(s1) · · ·σN(sN)] ui (s1, · · · , sN)

ui (σi , σ−i ) =∑si∈Si

∑s−i∈S−i

[∏j 6=i

σj (sj )

]σi (si )ui (si , s−i )

I For the above formula to make sense, it is critical that each player is randomizingindependently. That is, each player is independently tossing her own die to decideon which pure strategy to play.




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,

ui (si , σ−i ) =∑

s−i∈S−i

[∏j 6=i

σj (sj )

]ui (si , s−i )

and

ui (σi , σ−i ) =∑s̃i∈Si

∑s−i∈S−i

[∏j 6=i

σj (sj )

]σi (s̃i )ui (s̃i , s−i )

Then, ui (si , σ−i )− ui (σi , σ−i ) is

∑s−i∈S−i

(∏j 6=i

σj (sj )

)[ui (si , s−i )−

∑s̃i∈Si

σi (s̃i )ui (s̃i , s−i )

]




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,

ui (si , σ−i ) =∑

s−i∈S−i

[∏j 6=i

σj (sj )

]ui (si , s−i )

and

ui (σi , σ−i ) =∑s̃i∈Si

∑s−i∈S−i

[∏j 6=i

σj (sj )

]σi (s̃i )ui (s̃i , s−i )

Then, ui (si , σ−i )− ui (σi , σ−i ) is

∑s−i∈S−i

(∏j 6=i

σj (sj )

)[ui (si , s−i )−

∑s̃i∈Si


]




I ui (si , σ−i )− ui (σi , σ−i ) is

∑s−i∈S−i

(∏j 6=i

σj (sj )

)[ui (si , s−i )−

∑s̃i∈Si


]

I Since si is strictly dominant, ui (si , s−i ) > ui (s̃i , s−i ) for all s̃i 6= si and all s−i .

I Hence, ui (si , s−i ) >∑s̃i∈Si

σi (s̃i )ui (s̃i , s−i ) for any σi ∈ ∆(Si ) such that σi 6= si

(why?).

I This implies the desired inequality: ui (si , σ−i )− ui (σi , σ−i ) > 0




I We learned that: If si is a strictly dominant strategy for player i , then for allσi ∈ ∆(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 ∈ Siand 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 inrestricting attention to pure strategies for all players when looking for strictlydominant strategies.


Rationalizability

On the Agenda


2 Rationalizability

3 Exercises

4 Nash Equilibrium

5 Exercises


Rationalizability

Rationalizability

I

l rL 4,-4 9,-9M 6,-6 6,-6R 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.


Rationalizability

Rationalizability

I

l rL 4,-4 9,-9M 6,-6 6,-6R 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.


Rationalizability

Rationalizability

I

I Do not shoot to the middleI Do not use a strategy that is never a best response


Rationalizability

Rationalizability

DefinitionA strategy σi ∈ ∆(Si ) is a best response to the strategy profile σ−i ∈ ∆(S−i ) ifu(σi , σ−i ) ≥ u(σ̃i , σ−i ) for all σ̃i ∈ ∆(Si ). A strategy σi ∈ ∆(Si ) is never a bestresponse 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 ofthe 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 theopponents, 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 responseif and only if it is strictly dominated.


Rationalizability

Rationalizability

I In games with more than 2 players, there may be strategies that are not strictlydominated that are nonetheless never best responses.

I As before, it is a consequence of ”rationality” that a player should not play astrategy that is never a best response. That is, we can delete strategies that arenever best responses.

I By iterating on the knowledge of rationality, we iteratively delete strategies thatare never best responses.

I The set of strategies for a player that survives this iterated deletion of never bestresponses 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.


Rationalizability

Rationalizability

I In games with more than 2 players, there may be strategies that are not strictlydominated that are nonetheless never best responses.

I As before, it is a consequence of ”rationality” that a player should not play astrategy that is never a best response. That is, we can delete strategies that arenever best responses.

I By iterating on the knowledge of rationality, we iteratively delete strategies thatare never best responses.

I The set of strategies for a player that survives this iterated deletion of never bestresponses 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.


Rationalizability

Rationalizability

DefinitionI 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)-rationalizablefor player j 6= i .

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


Rationalizability

Rationalizability

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

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

I In this sense, rationalizability is (weakly) more restrictive than iterated deletion ofstrictly 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 bestresponses: those that a rational player can justify, or rationalize, with somereasonable conjecture concerning the behavior of his rivals (reasonable in the sensethat his opponents are not presumed to play strategies that are never bestresponses, etc.).

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


Exercises

On the Agenda


2 Rationalizability

3 Exercises

4 Nash Equilibrium

5 Exercises


Exercises

Exercises

I Exercise 1. Show that there can be no strategy σi ∈ ∆(Si ) such that for all si ∈ Siand 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 responseif and only if it is strictly dominated.

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

I

1/2 b1 b2 b3 b4

a1 0, 7 2, 5 7, 0 0, 1a2 5, 2 3, 3 5, 2 0, 1a3 7, 0 2, 5 0, 7 0, 1a4 0, 0 0,-2 0, 0 10,-1


Nash Equilibrium

On the Agenda


2 Rationalizability

3 Exercises

4 Nash Equilibrium

5 Exercises


Nash Equilibrium

Nash Equilibrium

I Now we turn to the most well-known solution concept in game theory. We’ll firstdiscuss pure strategy Nash equilibrium (PSNE), and then later extend to mixedstrategies.

DefinitionA strategy profile s = (s1, ..., sN) ∈ S is a Pure Strategy Nash Equilibrium (PSNE) if forall i and s̃i ∈ Si , u(si , s−i ) ≥ u(s̃i , s−i ).

I In a Nash equilibrium, each player’s strategy must be a best response to thosestrategies of his opponents that are components of the equilibrium.

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


Nash Equilibrium

Nash Equilibrium

I Unlike with our earlier solution concepts (dominance and rationalizability), Nashequilibrium applies to a profile of strategies rather than any individual’s strategy.When people say ”Nash equilibrium strategy”, what they mean is ”a strategy thatis part of a Nash equilibrium profile”.

I The term equilibrium is used because it connotes that if a player knew that hisopponents were playing the prescribed strategies, then she is playing optimally byfollowing 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 hisopponents 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 thatthe conjectured play of the rivals can also be justified in this way. On the otherhand, Nash requires that a player play optimally with respect to what hisopponents are actually playing. That is to say, the conjecture she holds about heropponents’ play is correct.


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, eachplayer’s strategy in a Nash equilibrium profile is rationalizable, but lots ofrationalizable profiles are not Nash equilibria.


Exercises

On the Agenda


2 Rationalizability

3 Exercises

4 Nash Equilibrium

5 Exercises


Exercises

Exercises

I Find the Nash Equilibria of the following games:

I

I What about Rock, Paper, Scissors?


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