Roger McCain's Game Theory: A Nontechnical Introduction to the Analysis of Strategy
(Revised Edition) is available in Asia now and will be available in
USA, UK, Europe and the rest of the World in October. Should you
require the title for fall term, special arrangements can be made so
please do contact Ms HooiYean Lee (mkt@wspc.com).
If you are considering to use this title after fall and would like to
receive an inspection copy, please send your request via World Scientific Publishing Company's website. You may also visit http://www.worldscibooks.com/economics/7517.html for sample chapters. Meanwhile, at a more advanced level, give some thought to McCain's Game Theory and Public Policy, Elgar, 2010. Part 1, "A Historical and Critical Survey," assesses the literature of game theory particularly from the point of view of the pragmatic purposes of public policy, while Part 2 presents some proposals and research toward a gametheoretic foundation for political economy. 
The maximin strategy is a "rational" solution to all twoperson 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:
Table 42 (Repeated)
Perrier 

Price=$1 
Price=$2 

Apollinaris 
Price=$1 
0,0 
5000, 5000 
Price=$2 
5000, 5000 
0,0 
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:
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 51. 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 highprice competitor. (This example follows one by Warren Nutter).
Table 51
Acme Widgets 

p=1 
p=2 
p=3 

Widgeon Widgets 
p=1 
0,0 
50, 10 
40,20 
p=2 
10,50 
20,20 
90,10 

p=3 
20, 40 
10,90 
50,50 
We can see that this is not a zerosum 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.
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 widgetselling game. First, for example, we can see that the strategy pair p=3 for each player (bottom right) is not a Nashequilibrium. 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 widgetselling game is a lowprice, zeroprofit 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 dominantstrategy equilibrium in Table 42. We will see that it, too, is a NashEquilibrium. (Check it out). Also, look again at the dominantstrategy equilibrium in the Prisoners' Dilemma. It, too, is a NashEquilibrium. 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 zerosum 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 NashEquilibrium in the same game? And what if there were more than one?
Roger A. McCain