# Mixed Strategy Nash Equilibrium: Fight vs. Opera

There are two pure-strategy Nash Equilibria: both go to the opera, both go to the fight.

There's also a mixed strategy Nash equilibrium. In order to achieve it, both players need to be indifferent between going to the opera and going to the fight, and therefore willing to play a mixed strategy. Can you figure out what that equilibrium is?

## Payoff Matrix

The size of the circles represents the probability of the payoff occurring, given the other player's strategy.

## Row Player Payoff

E[\pi_R(\text{Opera})] = c \times {{ params.TLpayoffR }} + (1-c) \times 0 || = {{ params.c | number: 1}} \times {{ params.TLpayoffR }} + {{ 1 - params.c | number: 1 }} \times 0 || = {{ params.c*params.TLpayoffR | number:1 }}
E[\pi_R(\text{Fight})] = c \times 0 + (1-c) \times {{ params.BRpayoffR }} || = {{ params.c | number: 1 }} \times 0 + {{ 1 - params.c | number: 1 }} \times {{ params.BRpayoffR }} || = {{ (1-params.c)*params.BRpayoffR | number:1 }}

## Column Player Payoff

E[\pi_C(\text{Opera})] = r \times {{ params.TLpayoffC }} + (1-r) \times 0 || = {{ params.r | number: 1}} \times {{ params.TLpayoffC }} + {{ 1 - params.r | number: 1 }} \times 0 || = {{ params.r*params.TLpayoffC | number:1 }}
E[\pi_C(\text{Fight})] = r \times 0 + (1-c) \times {{ params.BRpayoffC }} || = {{ params.r | number: 1 }} \times 0 + {{ 1 - params.r | number: 1 }} \times {{ params.BRpayoffC }} || = {{ (1-params.r)*params.BRpayoffC | number:1 }}

## Best Response Functions

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