msne

Mixed Strategy Nash Equilibrium

May 22, 2012

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Mixed Strategy Nash Equilibrium

May 22, 2012

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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!

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Mixed Strategy Nash Equilibrium

May 22, 2012

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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?

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Mixed Strategy Nash Equilibrium

May 22, 2012

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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 .

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Mixed Strategy Nash Equilibrium

May 22, 2012

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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

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Mixed Strategy Nash Equilibrium

May 22, 2012

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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

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Calculating MSNE

There are a few results that make the calculation of MSNE much easier in 2-player games:

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Mixed Strategy Nash Equilibrium

May 22, 2012

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Calculating MSNE: Result 1

Each players’ mixture will only place positive probability on
rationalizable