As we saw in Section 4.1, the equilibrium points in non-zero-sum games need not have the same values. Does Player 1 prefer one of the equilibria from Activity 4.2.2 over the other?
Since it is now possible for both players to benefit at the same time, it might be a good idea for players to communicate with each other. For example, if Player 1 says that she will choose A no matter what, then it is in Player 2’s best interest to choose D. If communication is allowed in the game, then we say the non-zero-sum game is cooperative. If no communication is allowed, we say it is non-cooperative.
If we apply the graphical method for Player 1 to the game in Table 4.2.2, we get that Player 1 should play a (1/3, 2/3) mixed strategy for an expected payoff of 10/3. Similarly we can determine that Player 2 should play a (2/3, 1/3) mixed strategy for an expected payoff of 10/3. Recall we developed this strategy as a “super defensive” strategy. But are our players motivated to play as defensively in a non-zero-sum game? Not necessarily! It is no longer true that Player 2 needs to keep Player 1 from gaining.
Now suppose, Player 1 plays the (1/3, 2/3) strategy. Then the expected payoff to Player 2 for playing pure strategy C, , is 20/3; and the expected payoff to Player 2 for playing pure strategy D, , is 5/3. Thus Player 2 prefers C over D. But if Player 2 plays only C, then Player 1 should abandon her (1/3, 2/3) strategy and just play B. This results in the payoff vector (5, 10). Notice, that now the expected value for Player 1 is 5, which is better than 10/3! Again, since Player 2 is not trying to keep Player 1 from gaining, there is no reason to apply the maximin strategy to non-zero-sum games. Similarly, we don’t want to apply the expected value solution since Player 1 does not care if Player 2’s expected values are equal. Each player only cares about his or her own payoff, not the payoff of the other player. It is also useful to note that the mixed strategy is not an equilibrium strategy since at least one player wants to change strategy.
What are some possible strategies for each player in Table 4.2.2? Might some strategies depend on communicating with the other player? Might some strategies depend on what a player knows about her opponent, especially if communication is not allowed?
Two partners in crime are arrested for burglary and sent to separate rooms. They are each offered a deal: if they confess and rat on their partner, they will receive a reduced sentence. So if one confesses and the other doesn’t, the confessor only gets 3 months in prison, while the partner serves 10 years. If both confess, then they each get 8 years. However, if neither confess, there isn’t enough evidence, and each gets just one year. We can represent the situation with the following matrix.
Table4.2.4.The Prisoner’s Dilemma (years in prison).
Does the matrix in Table 4.2.4 have any dominated strategies for Player 1? Does it have any dominated strategies for Player 2? Keep in mind that a prisoner prefers smaller numbers since prison time is bad.
If you were to be one of the prisoners, what would you do? Do you think everyone would do that, too? What would our perfectly rational player do? Would your strategy change if you are allowed to communicate? We examine some of these questions in the next few activities.
Suppose you are Prisoner 1. What should you do? Why? Suppose you are Prisoner 2. What should you do? Why? Does your choice of strategies result in an equilibrium pair?
Now suppose both prisoners are perfectly rational, so that any decision Prisoner 1 makes would also be the decision Prisoner 2 makes. Further, suppose both prisoners know that their opponent is perfectly rational. What should each prisoner do?
Suppose Prisoner 2 is unpredictable and is likely to confess with 50/50 chance. What should Prisoner 1 do? Does it change if Prisoner 2 confesses with a 75% chance? What if he confesses with a 25% chance.
Explain why Prisoner’s Dilemma is a “dilemma” for the prisoners. Is it likely they will choose a strategy which leads to the best outcome for both? You might want to consider whether there are dominated strategies.
You should now have some idea about why we call this game a dilemma, since the players may in fact have difficulty deciding on whether to confess or not. Even two perfectly rational players may not be able to get the best payoff.
Two drivers drive towards each other. If one driver swerves, he is considered a “chicken.” If a driver doesn’t swerve (drives straight), he is considered the winner. Of course if neither swerves, they crash and neither wins. A possible payoff matrix for this game is given in the following matrix.
What strategy results in the best outcome for Driver 1? What strategy results in the best outcome for Driver 2? What happens if they both choose that strategy?
Consider the strategy pair with outcome . Does Driver 1 have any regrets about his choice? What about Driver 2? Is this an equilibrium point? Are there any other points in which neither driver would regret his or her choice?
If you were to be one of the drivers, what would you do? Do you think everyone would do that, too? What would our perfectly rational player do? Would your strategy change if you are allowed to communicate? We examine some of these questions in the next few activities.
Now suppose both drivers in the game of Chicken are perfectly rational, so that any decision Driver 1 makes would also be the decision Driver 2 makes. Further, suppose both drivers know that their opponent is perfectly rational. What should each driver do?
Suppose Driver 2 is poorly programmed self-driving car that will swerve or drive straight with a 50/50 chance. What should Driver 1 do? Does it change if the self-driving car swerves with 75% chance?
Compare Prisoner’s Dilemma and Chicken. Are there dominated strategies in both games? Are there equilibrium pairs? Are players likely to reach the optimal payoff for one player, both players, or neither player? Does a player’s choice depend on what he knows about his opponent (perfectly rational or perfectly random)?
Both Prisoner’s Dilemma and Chicken are models of games where we describe the choice of strategy as Cooperate and Defect. In Prisoner’s Dilemma, we think of cooperating as cooperating with the other player, and defecting as turning against the other player. So if both players cooperate (with each other, not the law), they will get the higher payoff of only one year in prison. They defect by ratting on each other. In Chicken, players cooperate by swerving and defect by driving straight. Using the examples of Prisoner’s Dilemma and Chicken, think about how these games can model other everyday interactions where you could describe your choices as cooperating and defecting. The remainder of the chapter looks more closely at situations where players can cooperate or defect.