Question

Two athletes of equal ability are competing for a prize of $10,000. Each is deciding whether to take a dangerous performance-enhancing drug. If one athlete takes the drug and the other does not, the one who takes the drug wins the prize. If both or neither take the drug, they tie and split the prize. Taking the drug imposes health risks that are equivalent to a loss of XX dollars.

Complete the following payoff matrix describing the decisions the athletes face. Enter Player One's payoff on the left in each situation, Player Two's on the right.

		Player Two's Decision
		Take Drug	Don't Take Drug
Player One's Decision	Take Drug	,	,
	Don't Take Drug	,	,      True or False: The Nash equilibrium is taking the drug if X is less than $10,000. True False Suppose there was a way to make the drug safer (that is, have lower XX). Which of the following statements are true about the effects of making the drug safer? Check all that apply. It increases the payoff of taking the drug. It has no effect on the athletes' decision to take the drug if X remains greater than $5,000. It lowers the likelihood of taking the drug.

Answer 1

For the given pay off matrix and given that XX is less than 10,000 and further assume its more than 5000.

If player 1 takes the drug, player 2 has two options - take the drug and recieve a negative pay off or not take the drug an get 0 pay off - In this case he will not take the drug

If player 1 does not take the drug, player 2 can take the drug an recieve a payoff ranging from 4999 to 1 or not take the drug and recieve 5000 pay off. obviously he does not take the drug is the better option.

If player 2 takes the drug, player 1 can take the drug and recieve negative pay off or not take the drug and get 0 pay off. It is better not to take the drug.

If player 2 does not take the drug, player 1 can take the drug and reciev pay off ranging from 4999 to 1 or not take the drug and get 5000 pay off. So not taking the drug is better choice.

Hence for both not taking the drug is dominant strategy and hence the Nash Equilibrium.

But now also consider that XX can be less than 5000. then on player 1 choice of taking drug, player 2 can take drug and recieve pay off from 1 to 4999 or not take the drug and recieve 0. So he would take the drug. If player 1 does not take drug, player 2 can take and recieve pay off from 5001 to 9999, while he can not take the drug and recieve, 5000. Hence he would take the drug. The same argument would apply the other way for player 1 and here nash equilibrium would be to take the drug.

Based on the above discussion, if X is less than 10,000 it is not necessary the nash equilibrium will be to take the drug. Taking the drug is nash equilibrium only if x was with certainity is less than 5000. Hence the statement is false.

If taking the drug is safer, it means X has become further less, and can even be less than 5000 also. Hence - It increases the pay off of taking the drug, Also it has no effect on the athletes' decision to take the drug if X remains greater than $5,000 as then Nash Equilibrium is not taking the drug. Only last option is incorrect as lowering of X makes taking the drug more likely.