Strategic Games

BIBLIOGRAPHY

The concept of a strategic game forms part of the central idea behind the theory of games. A strategy is a plan of action that a player can use to guide the player in selecting moves throughout the game. It may be trivially simple, such as “whenever you are called on to move, select randomly among the moves,” or it may contain detailed complex instructions to cover all contingencies.

With few exceptions, virtually all games of any interest have a strategic component. Even tic-tac-toe, which is often utilized to illustrate an inessential game, or one that is not worth playing, has a strategic component. It is regarded as inessential because with only a little thought both players can enforce a draw; nevertheless, it has a set of strategies from which each player can select his actions. Even in a game as simple as this it is possible to select a losing strategy.

In the formulation and mathematical analysis of games of strategy, John von Neumann and Oskar Morgenstern (1944) made the great contribution of providing a complete language for the study of individual and multiperson conscious strategic choice. Concepts that had been extremely vague were defined precisely and operationally in a manner so that they could be analyzed mathematically. This includes the precise meanings of the terms choice, move, information, strategy, outcome, and payoff. But in many of the social sciences, particularly political science, sociology, and anthropology, some research questions are rather abstract, taking into account context and much of the richness and intangibles of areas of investigation. For many of the critical problems in the disciplines these subtleties cannot be ignored, and unfortunately the formal language of strategic games does not pick them up. With its mass markets and anonymous rational conscious behavior, economics has many more questions that are congenial to the strategic game analogy than do the other social sciences.

When the analogy is made between formal games of strategy and strategic behavior in politics or society, the gap between them is manifested in the assumptions that are made concerning knowledge of the rules of the game and common knowledge. When we consider the game chess, two implicit assumptions are made: It is assumed that all players know the rules of the game, and that each player knows that the other player knows the rules of the game. In society, even the best politicians know that much is unknown.

In the formal definition, strategy is defined as a complete book of instructions that a player could give to a delegate to play for him that describes what he wants the delegate to do under every contingency that might arise. A game as simple as chess has a hyper-astronomical number of strategies, so large that there is no way that they can all be enumerated in practice. It is clear that chess could be considered an inessential game in the sense that if a player could calculate all strategies it would be simple to select an optimal strategy, and if each player did that he would not bother playing—he would merely submit his optimal strategies and the game would be declared a draw or a win for the first mover. Yet years of experience and calculations on the size of the calculations tell us that the way chess is actually played by the best of human chess players is not by the enumeration of all strategies.

The difference between the abstraction of certain concepts and their manifestation in everyday life is nicely illustrated in the formal concept of strategy and how it is manifested in both military and corporate arenas. A strategy in the military or at the top of a corporation is an overall plan that, in its scope, bears some resemblance to the ideal strategy utilized in strategic games, but it has at least two critical modifications: It recognizes the critical role of delegation and aggregation. The general knows that neither he nor his opponent knows all the rules of the game, and he knows that he has to delegate decision-making to those who have greater special information than he does.

In spite of these caveats, there are many questions in economics, political science, social psychology, and even law, biology, and anthropology that can use the formal game structure profitably. Two basic applications to political science serve as examples. The first is the Condorcet Paradox and the second the Shapley-Shubik Power Index.

The Condorcet Paradox was established in 1785 by Marquis de Condorcet (b. 1743). His Essay on the Application of Analysis to the Probability of Majority Decisions describes describes the intransitivity of majority preference. According to the Condorcet Paradox majority wishes can be in conflict with each other. Consider three individuals named A, B, and C and the issues called I, II, and III to be put to the vote. The preferences of A are given by I pr II pr III (where pr means “preferred to”), the preferences of B are III pr I pr II, and the preferences of C are II pr III pr I. Consider a simple majority vote between any two social choices. By a 2:1 majority, a vote between I and II selects I, a vote between I and III selects III, and a vote between II and III selects II.

The Shapley-Shubik Power Index (Shapley and Shubik 1954) gives an intrinsic measure for how power varies with the accumulation of votes. Consider a committee with five votes. If there are three individuals with one person having three votes and the other two people having one each, and the rule was simple majority vote, the individual with three votes would have all the power. If there were five individuals, each with one vote, the power would be spread evenly. The index gives the nonlinear formula to measure power as a function of distribution of the votes. For example, in the five-vote simple-majority voting game, if there were four individuals with one having two of the five votes and the others having one each, the first person’s power would be one-half, and the others’ would be one-sixteenth each. This measure provides a benchmark, assuming that all individuals have equal chances of forming any coalition. In application, a correction for coalition possibilities must be made. A natural application of this is in predicting outcomes in Supreme Court cases.

SEE ALSO Strategic Behavior; Strategy and Voting Games; Subgame Perfection

BIBLIOGRAPHY

Condorcet, Jean-Antoine-Nicolas de Caritat, Marquis de. 1785. Essai sur l’application de l’analyse à la probabilité des décisions rendues à la pluralité des voix (Essay on the Application of Analysis to the Probability of Majority Decisions ). Paris: Impr. Royale. New York: Chelsea, 1972.

O’Neill, Barry. 1999. Honor, Symbols, and War. Ann Arbor:University of Michigan Press.

Schelling, Thomas C. 1960. The Strategy of Conflict. Cambridge, MA: Harvard University Press.

Shapley, Lloyd S., and Martin Shubik. 1954. A Method for Evaluating the Distribution of Power in a Committee System. American Political Science Review 48, no. 3 (September): 787–792.

von Neumann, John, and Oskar Morgenstern. 1944. Theory of Games and Economic Behavior. Princeton, NJ: Princeton University Press.

Martin Shubik