An Introduction to Decision Theory

In the final sections of Chapter 11 we showed that all (two-person) zero-sum games can be solved by determining a set of equilibrium strategies. From a purely theoretical perspective, there is little more to say about zero-sum games. However, nonzero-sum games are more interesting, and require further attention by game theorists. This chapter gives a brief overview of what we currently know about nonzero-sum games. First, we will give a proper introduction to the equilibrium concept tacitly taken for granted in the previous chapter. We will then go on to analyze a couple of well-known nonzero-sum games. The chapter ends with a discussion of whether game theory has any implications for ethics, biology and other subjects, which some scholars believe is the case.

The Nash Equilibrium

The prisoner's dilemma is an example of a nonzero-sum game. This is because the loss or gain made by each player is not the exact opposite of that made by the other player. If we sum up the utilities in each box they do not always equal zero. In the prisoner's dilemma each prisoner will have to spend some time in prison no matter which strategy he chooses, so the total sum for both of them is always negative. To see this point more clearly, it is helpful to take a second look at the game matrix of the prisoner's dilemma (Table 12.1).

Obviously, R2 is dominated by R1, whereas C2 is dominated by C1. Hence, rational players will select the pair of strategies (R1, C1). It follows that (R1, C1) is in equilibrium, because if Row knew that Col was going to confess (play C1), then he would not be better off by switching to another strategy; and if Col knew that Row was going to confess (play R1), then he would also be best off by not switching his strategy. The notion of equilibrium tacitly employed here was originally proposed by John Nash, a Princeton mathematician who was awarded the Nobel Prize in Economics for his contributions to game theory. Nash's basic idea was very simple: Rational players will do whatever they can to insure that they do not feel unnecessarily unhappy about their decision.