Game Theory Practice Questions

Question 1

 The payoff matrix above represents the game of chicken. You and I are driving towards each other at high speed, trying to make each other veer out of the way. If we both veer, then nothing in particular happens (say we each get 0 utility from this outcome). If only one of us veers, though, he/she suffers a loss of utility of 2 (humiliation), whereas the other gains 5 units of utility. If neither of us veers, we have a nasty accident and each lose 200.

Find any Nash equilibria of the above game.

Procedure:

Finding the Nash equilibria of simple games like this is straightforward.

1. Pretend you are one player (for example "ME").
2. Suppose that you believe your opponent is playing Veer. Find your best response to "Veer". In this example, your best response is to play "Drive", because 5 > 0.
3. Do the same for each of your opponent's actions
4. Now pretend you are the other player. Repeat steps 2 and 3.
5. For a Nash equilibrium, you need each player to be "best-responding" to what the other player is doing. My action is my best response to what you're doing, and your action is your best response to what I am doing.

Question 2

 The above game repsesents the interaction between a businessman and a politician. The businessman maximizes profits, and the politician maximizes the probability that he will be re-elected to office.

The businessman wants to persuade the politician to give him a monopoly in a certain industry. If he gets the monopoly, the businessman makes $10 million in profits. He is considering contributing $1 million to the re-election campaign of the politician, to try to persuade her to give him the monopoly. However, this kind of transaction is illegal, so they cannot write a contract on it. Assume that his opportunity cost is zero.

As for the politician, the probability that she is reelected if she does nothing is 0.5. If she gives the monopoly to the businessman, this reduces her probability of reelection by 0.5, as she is perceived by the public as being corrupt. If she gets a campaign contribution of $1,000,000, her probability of being re-elected increases by 0.5.

Fill in the payoff matrix, and find any Nash equilibria.

Question 3

 The payoff matrix above represents the Cold War. If the USA and the USSR bomb each other with nuclear weapons, the world comes to an abrupt and fiery end, represented by -200 on the payoff matrix. If only one country bombs the other one, the disaster is localized and the lack of retaliation means that the victim only loses 100, whereas the bomber gets p. If nobody nukes anybody, they both get zero.

1. Suppose p is zero. Find the Nash equilibria of this game.
2. Suppose p is any negative number. Are the Nash equilibria the same as in (a)?
3. Suppose p is any positive number. Are the Nash equilibria the same as in (a)?

Question 4 (Very challenging)

1. Suppose that there is only one firm in the market (monopoly). Calculate the level of output that maximizes profit for this firm, and compute the maximal monopoly profit.
2. Suppose instead that there are two identical firms (duopoly), Airbus and Boeing. Let q^A be the output of Airbus, and q^B be the output Boeing. If Boeing is producing q^B, how much should Airbus produce to maximize profits? If Airbus is producing q^A, how much should Boeing produce to maximize profits?
3. What is the unique Nash equilibrium of this oligopoly game? What are the equilibrium profits of each firm?
4. Airbus and Boeing are considering a collusive agreement to form a cartel and act like a monopoly. Under the agreement, each firm will produce half the monopoly output, and hence each will get half of monopoly profits. However, each firm is free to "cheat" and break the agreement. If a firm cheats, it maximizes profits given what the other firm is doing. Thus if only one firm cheats, it maximizes profits given that the other firm is producing half the monopoly output. If both cheat, then it is as if there were no agreement and each firm makes the duopoly profit it got in (c). Fill out the payoff matrix below.

(e) What is the unique Nash equilibrium of this game? If Nash equilibrium is a reasonable prediction of strategic behavior, will a collusive agreement be signed?