GAME ESSENCE

Games of imperfect information

The simplest two-person zero-sum games of imperfect information are those that have saddle points. (All two-person zero-sum games of perfect information have saddle points.) Such games have a predetermined outcome (assuming rational play), and each player can, by choosing the right, obtain an amount at least equal to this outcome no matter what the other player does. This predetermined outcome is called the value of the game. An example of such a game is described in normal form below.Two campaigning political parties, A and B, must each decide how to handle a controversial issue in a certain town. They can either support the issue, oppose it, or evade it. Each party must make its decision without knowing what its rival will do. Every pair of decisions determines the percentage of the vote that each party receives in the town, and each party wants to maximize its own percentage of the vote. The entries in the matrix represent party A's percentage (the remaining percentage goes to party B); if, for example, A supports the issue and B evades it, A gets 80 percent (and B, 20 percent) of the vote. A's decision seems difficult at first because it depends upon B's strategy. A does best to oppose if B supports, evade if B opposes, and support if B evades. A must therefore consider B's decision before making its own. No matter what A does, B gains the largest percentage of votes by opposing the issue. Once A recognizes this, its strategy should clearly be to evade and settle for 30 percent of the vote. This 30 percent/70 percent division of the vote is the game's saddle point. A more systematic way of finding the saddle point is to determine the maximin and minimax values. Using this method, A first determines the percentage of votes it can obtain for each of its strategies and then finds the maximum of these three minimum values. The minimum percentages A will get if it supports, opposes, or evades are, respectively, 20, 25, and 30; the largest of these, 30, is the maximin value. Similarly, for each strategy it chooses, B determines the maximum percentage of votes A can win (and thus the minimum that B can win). In this case if B supports, opposes, or evades, the maximum A gets is 80, 30, or 80, respectively. B obtains its highest percentage by minimizing A's maximum percentage of the vote. The smallest of A's maximum values is 30, and 30 is therefore B's minimax value. Because both the minimax and the maximin values are 30, 30 is a saddle point. The two parties might as well announce their strategies in advance; neither gains from the knowledge. .