316 ANNUAL REPORT SMITHSONIAN INSTITUTION, 1959 
developed by Lloyd Shapley of the Rand Corp. An outside arbitra- 
tor must decide the payments. The formula tells him how to give 
the players payments appropriate to the strength of their bargaining 
powers, and it also maximizes the total payment. There are obvious 
practical difficulties in applying Shapley’s “arbitration value.” In 
the first place, the payment, or value, each player receives can seldom 
be measured simply in dollars. Thus the arbitrator would have a 
hard time deciding on the proper distribution if the players were to 
y 
Calleither 
Hea oss” 
4 eet: 
+= Cal Tails” Ill cat “Heads 
Follow formula 
Proportion of “Heads” to date 
Proportion of correct calls te date 
Figure 3.—How to play smarter than safe. Early workers in game theory designed strat- 
egies that were safe to use against infallible opponents, but mathematicians now know 
ways to take advantage of a careless opponent without risking anything. The diagram 
dictates the best strategy for guessing whether your opponent has placed a concealed coin 
heads up or tails up. If he were wise, he would mix heads and tails randomly, simply by 
flipping the coin each time. In that case, you could do no better than break even in the 
long run. But if he tries to anticipate your guesses, the strategy in the diagram enables 
you to win whenever he follows any regular pattern; and in any event you will do no 
worse than break even in the long run. 
As the game progresses, you keep track of the proportion of times your opponent has 
placed the coin heads up and the proportion of times you have won. ‘This determines the 
point Q. When Q is in the black or gray triangles, you follow the pure strategies shown 
in the diagram. But when @Q is in the white triangle, you must adopt a mixed strategy, 
which you calculate as follows: Draw a line connecting the center of the diagram with Q 
and extending to the base line. The length x determines your strategy. Since x is in 
this case 44, you should adopt some random way of calling heads or tails that makes it 
three times as likely that you will call tails. (You might put four slips of paper in a hat— 
three of them marked tails—and draw one.) This method takes advantage of your op- 
ponent’s apparent tendency to place the coin tails up, yet it keeps him from guessing your 
strategy. If you follow this plan, the point Q should ultimately end up in the black 
triangle, which represents a profit for you. 
