Copyright (c) Christopher Makler / econgraphs.org

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 }}

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