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### Course: AP®︎/College Microeconomics>Unit 4

Lesson 5: Oligopoly and game theory

# Game Theory

## Problem

Juan and Elsa two of ten players who are participating in a reality TV show that makes players engage in a series of challenges. If a player loses that challenge they are sent home and lose the opportunity for the grand prize. In previous episodes, they have each won a medallion that gives the holder immunity from a challenge that sends them home. However, a player must decide to use it before the challenge starts.
Juan and Elsa have both won immunity medalions and are now facing each other in a challenge that may send one of them home, or someone else could lose the challenge and be sent home. They can use their immunity now or save it for later.
The payoffs to their strategies, in units of utility, are given in the payoff matrix shown here. The first entry represents Elsa's payoff and the second entry represents Juan's payoff.
How many Nash equilibria exist for this game?