principleMinor
Mixed-strategy Nash equilibria
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Problem
Is the following statement always true:
if there is a mixed-strategy Nash equilibria then it is unique.
I know that there can be several pure strategy Nash equilibrias.
if there is a mixed-strategy Nash equilibria then it is unique.
I know that there can be several pure strategy Nash equilibrias.
Solution
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No. The easiest way to prove this is by counterexample. Expanding matching pennies, we can create a game with two mixed strategies (players either mix between A and B or between C and D):
What might be true about that statement is for a mix of particular pure strategies for two players, there is a unique mixed strategy equilibrium. This presumes that there is a threshold after which you switch from using strategy $0$ to strategy $1$. At this threshold, your opponent is ambivalent between her strategies. As depicted below, the intersection (and hence mix) will be unique.
on Area 51 that you should totally follow!
No. The easiest way to prove this is by counterexample. Expanding matching pennies, we can create a game with two mixed strategies (players either mix between A and B or between C and D):
What might be true about that statement is for a mix of particular pure strategies for two players, there is a unique mixed strategy equilibrium. This presumes that there is a threshold after which you switch from using strategy $0$ to strategy $1$. At this threshold, your opponent is ambivalent between her strategies. As depicted below, the intersection (and hence mix) will be unique.
Context
StackExchange Computer Science Q#2344, answer score: 4
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