Introduction to Game Theory: a Discovery Approach

In the last section we saw that non-zero-sum games can differ on all of the above!

Even with non-zero-sum games, it is helpful to start by finding any equilibrium points.

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.

We saw in Section 4.1, that our methods for analyzing zero-sum games do not work very well on non-zero-sum games. Let’s look a little closer at this.

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, \(E_2(C)\text{,}\) is 20/3; and the expected payoff to Player 2 for playing pure strategy D, \(E_2(D)\text{,}\) 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.

OK, so now, how do we analyze these games?

Can you see that some of the analysis might be better understood with psychology than with mathematics?

In order to better understand non-zero-sum games we look at two classic games.

Since this game, as presented, is probably only playerd once, we can begin by looking for dominated strategies and equilibrium points.

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.

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.

We now turn to another classic example. We can ask similar qustions about Chicken that we ask about Prisoner’s Dilemma.

Again, since this game as presented is probably only played once, we can begin by looking for dominated strategies and equilibrium points.

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.

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.