Game Theory: An Introductory Sketch

"Solutions" to Nonconstant Sum Games

Roger McCain's Game Theory: A Nontechnical Introduction to the Analysis of Strategy is out of print, as of the spring of 2008. Dr. McCain retains the copyright and can authorize copying for those who wish to use the book. Contact him. The instructors' guide, without correction of known problems, is here.

The maximin strategy is a "rational" solution to all two-person zero come games. However, it is not a solution for nonconstant sum games. The difficulty is that there are a number of different solution concepts for nonconstant sum games, and no one is clearly the "right" answer in every case. The different solution concepts may overlap, though. We have already seen one possible solution concept for nonconstant sum games: the dominant strategy equilibrium. Let's take another look at the example of the two mineral water companies. Their payoff table was:

We saw that the maximin solution was for each company to cut price to $1. That is also a dominant strategy equilibrium. It's easy to check that: Apollinaris can reason that either Perrier cuts to $1 or not. If they do, Apollinaris is better off cutting to 1 to avoid a loss of $5000. But if Perrier doesn't cut, Apollinaris can earn a profit of 5000 by cutting. And Perrier can reason in the same way, so cutting is a dominant strategy for each competitor.

But this is, of course, a very simplified -- even unrealistic -- conception of price competition. Let's look at a more complicated, perhaps more realistic pricing example:

Another Price Competition Example

Following a long tradition in economics, we will think of two companies selling "widgets" at a price of one, two, or three dollars per widget. the payoffs are profits -- after allowing for costs of all kinds -- and are shown in Table 5-1. The general idea behind the example is that the company that charges a lower price will get more customers and thus, within limits, more profits than the high-price competitor. (This example follows one by Warren Nutter).

We can see that this is not a zero-sum game. Profits may add up to 100, 20, 40, or zero, depending on the strategies that the two competitors choose. Thus, the maximin solution does not apply. We can also see fairly easily that there is no dominant strategy equilibrium. Widgeon company can reason as follows: if Acme were to choose a price of 3, then Widgeon's best price is 2, but otherwise Widgeon's best price is 1 -- neither is dominant.

Nash Equilibrium

We will need another, broader concept of equilibrium if we are to do anything with this game. The concept we need is called the Nash Equilibrium, after Nobel Laureate (in economics) and mathematician John Nash. Nash, a student of Tucker's, contributed several key concepts to game theory around 1950. The Nash Equilibrium conception was one of these, and is probably the most widely used "solution concept" in game theory.

DEFINITION: Nash Equilibrium If there is a set of strategies with the property that no player can benefit by changing her strategy while the other players keep their strategies unchanged, then that set of strategies and the corresponding payoffs constitute the Nash Equilibrium.

Let's apply that definition to the widget-selling game. First, for example, we can see that the strategy pair p=3 for each player (bottom right) is not a Nash-equilibrium. From that pair, each competitor can benefit by cutting price, if the other player keeps her strategy unchanged. Or consider the bottom middle -- Widgeon charges $3 but Acme charges $2. From that pair, Widgeon benefits by cutting to $1. In this way, we can eliminate any strategy pair except the upper left, at which both competitors charge $1.

We see that the Nash equilibrium in the widget-selling game is a low-price, zero-profit equilibrium. Many economists believe that result is descriptive of real, highly competitive markets -- although there is, of course, a great deal about this example that is still "unrealistic."

Let's go back and take a look at that dominant-strategy equilibrium in Table 4-2. We will see that it, too, is a Nash-Equilibrium. (Check it out). Also, look again at the dominant-strategy equilibrium in the Prisoners' Dilemma. It, too, is a Nash-Equilibrium. In fact, any dominant strategy equilibrium is also a Nash Equilibrium. The Nash equilibrium is an extension of the concepts of dominant strategy equilibrium and of the maximin solution for zero-sum games.

It would be nice to say that that answers all our questions. But, of course, it does not. Here is just the first of the questions it does not answer: could there be more than one Nash-Equilibrium in the same game? And what if there were more than one?

Roger A. McCain