NEW USES OF THE ABSTRACT—BOEHM ay a) 
“rationally” (though mathematicians are not sure just how to define 
“rational” behavior). Obviously, game theory represents a high de- 
gree of abstraction; people are never so purposeful and well informed, 
even in as circumscribed a competition as a game of chess. Yet the 
abstraction of man is valid to the extent that game theory is proving 
useful in analyzing business and military situations. 
When it was first developed in the twenties, chiefly by Emile Borel 
in France and John von Neumann in Germany, game theory was 
limited to the simplest forms of competition. As late as 1944 the 
definitive book on the subject (“Theory of Games and Economic Be- 
havior,” by Von Neumann and Princeton economist Oskar Morgen- 
stern) drew many of its illustrative examples from a form of one- 
card poker with limited betting between two people. Now, however, 
the strategies of two-person, zero-sum games (in which one player 
gains what his opponent loses) have been quite thoroughly analyzed. 
And game theorists have pushed on to more complex types of com- 
petition, which are generally more true to life. 
Early game theory left much to be desired when it assumed that 
every plan should be designed for play against an allwise opponent 
who would find out the strategy and adopt his own most effective 
counterstrategy. In military terms, this amounted to the assumption 
that the enemy’s intelligence service was infallible. The game-theory 
solution was a randomly mixed strategy—one in which each move 
would be dictated by chance, say the roll of dice, so that the enemy 
could not possibly anticipate it. (For much the same reason the 
United States Armed Forces teach intelligence officers to estimate the 
enemy’s capabilities rather than his intentions.) Many mathemati- 
cians have felt that this approach is unrealistically cautious. Re- 
cently game theorists have worked out strategies that will take ad- 
vantage of a careless or inexpert opponent without risking anything 
if he happens to play shrewdly. (For a relatively simple example, 
see diagram, fig. 3.) 
The most difficult games to analyze mathematically are those in 
which the players are not strictly competing with one another. An 
example is a labor-management negotiation; both sides lose unless 
they reach an agreement. Another complicating factor is collusion 
among players—e.g., an agreement between two buyers not to bid 
against each other. Still another is payment of money outside the 
framework of the “game,” as when a large company holds a distrib- 
utor in line by subsidizing him. 
WHO GETS HOW MUCH? 
The biggest problem in analyzing such complex situations has been 
to find a mathematical procedure for distributing profits in such a 
way that “rational” players will be satisfied. One formula has been 
