Two athletes, Adam and Benjamin, compete for a prize. They are equally good. This means that if neither of them or both of them use steroids, then each of them wins the competition with probability 1/2. However, if only one of them uses steroids, the one who uses them wins for sure. Suppose that these athletes do not care about the long run consequences of steroids. Suppose also that the payoff of winning is 1 and the payoff of losing is 0.

A1. What are the strategies available to each athlete? Write down the normal form of the game (the bimatrix of strategies and payoffs).

A2. Does any of the athletes have a dominant strategy? Find the Nash Equilibrium of the game. Interpret.

The anti-drug committee can decide to test one of the athletes right before the competition. The testing method is infallible. If steroids are detected the athlete is disqualified, loses the competition and pays an extra fine c. The other athlete becomes automatically the winner.

A3. Suppose that they choose to test Adam (and both Adam and Benjamin know it beforehand). What is the Nash equilibrium of the game? Is any athlete detected using steroids in the Nash Equilibrium? Is any athlete using steroids in the Nash equilibrium? Suppose that they choose to test Benjamin (and both Adam and Benjamin know it beforehand). What is the Nash equilibrium of the game? Is any athlete detected using steroids in the Nash Equilibrium? Is any athlete using steroids in the Nash equilibrium? Should anti-drug be happy with the outcome in any of these two cases?

A4. Suppose now that anti-drug flips a coin: if it is head, then Adam is tested and if it is tail then Benjamin is tested. Both Adam and Benjamin know that anti-drug uses this method but they have to decide whether to take steroids before the competition (so before anti-drug flips the coin). What is the Nash equilibrium of the game? Is any athlete detected using steroids in the Nash Equilibrium? Is any athlete using steroids in the Nash equilibrium? Should anti-drug be happy with the outcome in this case?