How can game theory be applied to the economic settings?
Our first example concerns public goods. In this game, each player can either contribute or not. For example, two roommates can either clean their apartment or not. If they both clean, the apartment is nice. If one cleans, then that roommate does all of the work and the other gets half of the benefits. Finally, if neither cleans, neither is very happy. This suggests the following payoffs as shown in Figure 16.21 "Cleaning the apartment".
Figure 16.21 Cleaning the apartment
You can verify that this game is similar to the prisoner’s dilemma in that the only Nash equilibrium is the pure strategy in which neither player cleans. This is a game-theoretic version of the tragedy of the commons—even though both roommates would be better off if both cleaned, neither do. As a practical matter, roommates do solve this problem, using strategies that we will investigate when we consider dynamic games.
Figure 16.22 Driving on the right
Figure 16.23 Bank location game
Figure 16.24 Political mudslinging
You have probably had the experience of trying to avoid encountering someone, whom we will call Rocky. In this instance, Rocky is actually trying to find you. Here it is Saturday night and you are choosing which party, of two possible parties, to attend. You like Party 1 better and, if Rocky goes to the other party, you get 20. If Rocky attends Party 1, you are going to be uncomfortable and get 5. Similarly, Party 2 is worth 15, unless Rocky attends, in which case it is worth 0. Rocky likes Party 2 better (these different preferences may be part of the reason why you are avoiding him), but he is trying to see you. So he values Party 2 at 10, Party 1 at 5, and your presence at the party he attends is worth 10. These values are reflected in Figure 16.25 "Avoiding Rocky".
Figure 16.25 Avoiding Rocky
Figure 16.26 Price cutting game
The free-rider problem of public goods with two players can be formulated as a game.
Whether to drive on the right or the left is a game similar to battle of the sexes.
Many everyday situations are reasonably formulated as games.
Verify that the bank location game has no pure strategy equilibria and that there is a mixed strategy equilibrium where each city offers a rebate with probability ½.
Show that the only Nash equilibrium of the political mudslinging game is a mixed strategy with equal probabilities of throwing mud and not throwing mud.
Suppose that voters partially forgive a candidate for throwing mud in the political mudslinging game when the rival throws mud, so that the (Mud, Mud) outcome has payoff (2.5, 0.5). How does the equilibrium change?
Show that there are no pure strategy Nash equilibria in the avoiding Rocky game.
Find the mixed strategy Nash equilibria.
Show that the probability that you encounter Rocky is 7 12 .
Show that the firms in the price-cutting game have a dominant strategy to price low, so that the only Nash equilibrium is (Low, Low).