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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 $X$ dollars.a. Draw a 2 \(\times\) 2 payoff matrix describing the decisions the athletes face.b. For what $X$ is taking the drug the Nash equilibrium?c. Does making the drug safer (that is, lowering $X$) make the athletes better or worse off? Explain.
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This example, we are using game theory to create a payoff matrix and determine certain costs of taking a drug for these athletes were given is that if these athletes win whatever competition there in they win a total prize of $10,000. However, if one of these players takes this performance enhancing drug, that player will win and the other player will lose. If they both take the drug they tie and split the prize. If neither takes the drug they tie and split the prize. And with this drug we have an associated cost with it for future health risks equal to some dollar value amount of X. We'd like to first create this payoff matrix for these two players. So Let's start by putting in the information that we know to be true. So let's start with player one taking the drug and player to not taking the drug. Now we no one takes the drug he's going to win and the other is going to lose. So player one takes that drug, he's going to get that $10,000 which is the prize, but he's going to pay the price to take it. Which was that cost of X. Right? So we have $10,000 minus whatever X. Value that is. And the other one is going to lose. They're not gonna get anything. So we can do the same thing and this diagonal box down below. So if player two takes the drug, player one does not, we're going to have the same values just opposite. Player two is going to win that $10,000 minus that cost of X. Taking the drug. And player one is not going to make anything. Now let's start with if they both take the drug we know they're going to split the prize. So they're each going to make that half the prize money which is 5000. But because they're each taking the drug, they also have to pay the cost of it in health risks, which is X. Dollars. And now if neither of them takes the drug, we know that they're just gonna split the prize, which is going to be $5,000. And they don't have to pay any X. Costs because they did not take the drug. So this would be our payoff matrix here for these two players. Now we'd like to know for part B. Is at this drug nash equilibrium, what is this cost of X. That would make it worthwhile for these players? So really what we're looking at is if they are both taking the drug, right? So we are looking at this first box if they're both taking it and we know that the cost is five or the payoff is 5000 minus X. So let's take this 5000 minus X. And in order for this to be worth it to them, they need to make something right? So it should be greater than zero. They don't want to walk away not Making anything at all. So, solving for X, we see that it is 5000 greater than our dollar value X. So therefore for taking these drugs to be the Nash equilibrium, the value of X should be less than $5,000. If it's any more than $5,000, these players aren't going to make anything. Alright now part C is asking us if this drug were to be made safer, would the players be better or worse off? Well, if we're assuming that the drug is made safer, we can probably also assume that the health risk of taking it would be reduced, meaning that this value of X, the cost of taking it is probably going to be lower, so therefore we could see an increase in their payoffs, and as a result, the players are probably going to be better off.
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