9.4.1 Two-Player Games

To help make the connection to Section 9.3 smoother, two-player games will be considered first. This case is also easier to understand because the notation is simpler. The ideas are then extended without difficulty from two players to many players. The game is formulated as follows.

- The same components as in Formulation 9.7, except the cost function.
- A function,
,
called the
*cost function for*. - A function,
,
called the
*cost function for*.

Paralleling Section 9.3, first consider applying deterministic strategies to solve the game. As before, one possibility is that a player can apply its security strategy. To accomplish this, it does not even need to look at the cost function of the other player. It seems somewhat inappropriate, however, to neglect the consideration of both cost functions when making a decision. In most cases, the security strategy results in regret, which makes it inappropriate for nonzero-sum games.

A strategy that avoids regret will now be given. A pair
of actions is defined to be a *Nash equilibrium*
if

and

These expressions imply that neither nor has regret. Equation (9.66) indicates that is satisfied with its action, , given the action, , chosen by . cannot reduce its cost any further by changing its action. Likewise, (9.67) indicates that is satisfied with its action .

The game in Formulation 9.8 can be completely represented using two cost matrices. Let and denote the cost matrices for and , respectively. Recall that Figure 9.2 showed a pattern for detecting a saddle point. A Nash equilibrium can be detected as shown in Figure 9.5. Think about the relationship between the two. If , then can be negated and superimposed on top of . This will yield the pattern in Figure 9.2 (each becomes because of negation). The values and coincide in this case. This observation implies that if , then the Nash equilibrium is actually the same concept as a saddle point. It applies, however, to much more general games.

By applying (9.66) and (9.67), or by using the patterns in Figure 9.5, it can be seen that and is a Nash equilibrium. The resulting costs are and . Another Nash equilibrium appears at and . This yields costs and , which is better for both players.

For zero-sum games, the existence of multiple saddle points did not
cause any problem; however, for nonzero-sum games, there are great
troubles. In the example shown here, one Nash equilibrium is clearly
better than the other for both players. Therefore, it may seem
reasonable that a rational DM would choose the better one. The issue
of multiple Nash equilibria will be discussed next.