Question

Consider two fighters training for an upcoming fight. Each can choose to use performance enhancing drugs, or not, which affect your chance of winning.

• All else equal, each fighter has an equal (50%) chance of winning.
• If one fighter uses performance enhancing drugs while the other does not, their chance of winning increases to 80%.
• Each fighter values winning as a payoff of 30. (A 50% chance of winning is worth 15, an 80% chance of winning is worth 24.)
• Using performance enhancing drugs causes negative long-term health effects of 6, subtracted from the payoff for winning.

a) There is only one fight, and the two fighters choose their strategies simultaneously. What is the Nash equilibrium of this game? Explain.

Answer 1

The game can be represented by the table below:

The optimal strategy for both the players is as follows from the payoff matrix. If the other player chooses to not drug, then the best response is to use drugs. If the other player chooses drugs, then the best response is to not use drugs.

Thus, there are two Nash equilibrium:

(1) fighter 1 uses drugs and Fighter 2 does not use drugs

(2) fighter 2  uses drugs and Fighter 1 does not use drugs