msne
Mixed Strategy Nash Equilibrium
May 22, 2012
()
Mixed Strategy Nash Equilibrium
May 22, 2012
1 / 12
Drawback of Pure Strategy Nash Equilibrium
One drawback of the pure-strategy Nash equilibrium concept is that it many games do not have a pure-strategy Nash equilibrium.
(Rock paper scissors, matching pennies to name a couple)
Does this mean that we have no solution concept for these types of games?
Not by a long shot!
()
Mixed Strategy Nash Equilibrium
May 22, 2012
2 / 12
Re-Defining the Best Response
Suppose player 2 randomizes according to the following mixed strategy: σ2 (R) = σ2 (P) = σ2 (S) = 1
3
Given σ2 , what would be the payoff to player 1 of playing any pure strategy? Given σ2 , what would be the payoff to player 1 of playing any mixed strategy?
()
Mixed Strategy Nash Equilibrium
May 22, 2012
3 / 12
Re-Defining the Best Response
The previous discussion seems to warrant an extension of the definition of what it means to be a best response.
We say that σi is a best response to σ−i if and only if ui (σi , σ−i ) ≥ ui (si , σ−i ) for all si ∈ Si .
()
Mixed Strategy Nash Equilibrium
May 22, 2012
4 / 12
Mixed Strategy Nash Equilibrium
Definition
Let σ = (σ1 , σ2 , · · · , σn ) be a strategy profile where σi ∈ ∆Si for each i.
Then σ is a mixed strategy Nash equilibrium if ui (σi , σ−i ) ≥ ui (si , σ−i ) for each si ∈ Si and each i ∈ N
()
Mixed Strategy Nash Equilibrium
May 22, 2012
5 / 12
Results on MSNE
Theorem
Every finite game has at least one Nash equilibrium (in pure or mixed strategies) ()
Mixed Strategy Nash Equilibrium
May 22, 2012
6 / 12
Calculating MSNE
There are a few results that make the calculation of MSNE much easier in 2-player games:
()
Mixed Strategy Nash Equilibrium
May 22, 2012
7 / 12
Calculating MSNE: Result 1
Each players’ mixture will only place positive probability on
rationalizable
May 22, 2012
()
Mixed Strategy Nash Equilibrium
May 22, 2012
1 / 12
Drawback of Pure Strategy Nash Equilibrium
One drawback of the pure-strategy Nash equilibrium concept is that it many games do not have a pure-strategy Nash equilibrium.
(Rock paper scissors, matching pennies to name a couple)
Does this mean that we have no solution concept for these types of games?
Not by a long shot!
()
Mixed Strategy Nash Equilibrium
May 22, 2012
2 / 12
Re-Defining the Best Response
Suppose player 2 randomizes according to the following mixed strategy: σ2 (R) = σ2 (P) = σ2 (S) = 1
3
Given σ2 , what would be the payoff to player 1 of playing any pure strategy? Given σ2 , what would be the payoff to player 1 of playing any mixed strategy?
()
Mixed Strategy Nash Equilibrium
May 22, 2012
3 / 12
Re-Defining the Best Response
The previous discussion seems to warrant an extension of the definition of what it means to be a best response.
We say that σi is a best response to σ−i if and only if ui (σi , σ−i ) ≥ ui (si , σ−i ) for all si ∈ Si .
()
Mixed Strategy Nash Equilibrium
May 22, 2012
4 / 12
Mixed Strategy Nash Equilibrium
Definition
Let σ = (σ1 , σ2 , · · · , σn ) be a strategy profile where σi ∈ ∆Si for each i.
Then σ is a mixed strategy Nash equilibrium if ui (σi , σ−i ) ≥ ui (si , σ−i ) for each si ∈ Si and each i ∈ N
()
Mixed Strategy Nash Equilibrium
May 22, 2012
5 / 12
Results on MSNE
Theorem
Every finite game has at least one Nash equilibrium (in pure or mixed strategies) ()
Mixed Strategy Nash Equilibrium
May 22, 2012
6 / 12
Calculating MSNE
There are a few results that make the calculation of MSNE much easier in 2-player games:
()
Mixed Strategy Nash Equilibrium
May 22, 2012
7 / 12
Calculating MSNE: Result 1
Each players’ mixture will only place positive probability on
rationalizable