# 2 Game Theory of Performance Enhancing Drugs Suppose Barry Bonds an...

## Question

2 Game Theory of Performance Enhancing Drugs Suppose Barry Bonds and Roger Clemens, who compete against each other regularly, are deciding whether or not to take performance enhancing drugs (PEDs). They decide simultaneously before they begin their career. If they both take PEDs, they will each earn a \$100 million for their career. If they both choose to not take PEDs, they will each earn a \$80 million. If Bonds takes PEDs and Clemens does not, then Bonds hits a lot of home runs and earns \$125 million and Clemens earns \$50 million. If Clemens takes PHDs and Bonds does not, then Clemens earns \$140 million and Bonds earns \$50 million. There is a chance that a player who takes PEDs gets caught, which is given by the probability p. If they are caught, the player's entire salary is forfeited (i.e., they earn 0 for their career).

1. What is the expected salary for each player as a function of p if they both choose to take PEDs?
2. Draw the normal form of this game (i.e., the game matrix). Hint: payoffs will be in terms of the players' expected salary.
3. Assume p = 0.25. Find all the pure strategy Nash Equilibria of this game.
4. Suppose the MLB wants to get rid of PEDs. To do so, they enhance their screening for PEDs, which increases p. What is the lowest level of p where both players choose not to take PEDs in the equilibrium?

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