Strategy and Voting Games

BIBLIOGRAPHY

In game theory, a player’s strategy refers to a complete contingent plan for all possible scenarios that might arise. When voting is modeled in a game theoretical framework, voters are treated as strategic players, and the framework describes the strategic interactions among the voters. A voting game is characterized by three key elements: the set of voters, the set of strategies available to each voter, and each voter’s preferences on the set of voting outcomes. The set of feasible strategies in a particular voting game depends on the voting system in use. For example, under the plurality voting system, each voter can vote for only one candidate. With preferential voting, each voter ranks a list of candidates in order of preference. Therefore, a voter’s strategy in a plurality voting game specifies which candidate to vote for each combination of other players’ votes, whereas in a preferential voting game, a strategy specifies a preference list for each combination of other players’ rankings. A voting outcome is determined by all voters’ strategies (called a strategy profile) under a particular voting system. For example, in a single-winner plurality voting game, the candidate with the most votes wins the election. In proportional representation systems, the percentage of votes received by a party determines the percentage of seats allocated to the party. A voter’s preference on the set of voting outcomes is usually represented by a payoff function that assigns a numeric value to each strategy profile.

The concept of strategy, together with strategic game (or normal form game), was first formally introduced in 1944 in The Theory of Games and Economic Behavior, by John von Neumann and Oskar Morgenstern. Later, in 1950, John F. Nash Jr. developed the solution concept of the Nash equilibrium, which describes any stable state in which each player’s strategy is optimal (in the sense that it maximizes a player’s payoffs among his or her other strategies) given other players’ equilibrium strategies. Since then, game theory has become an important tool for analyzing problems in various fields, including economics, biology, and political science. An early application of game theory to voting situations was offered by Robin Farquharson in his influential 1969 book, Theory of Voting. Farquharson’s approach departed from earlier studies, which usually assumed sincere voting by disregarding the strategic aspects of voting. A voter’s strategy is sincere if the person votes according to his or her true preferences regardless of other voters’ strategies. In general, a sincere voting profile does not constitute an equilibrium for a voting game. Allan Gibbard (1973) and Mark Allen Satterthwaite (1975) proved a theorem showing that strategic voting is universal in common democratic systems. Consequently, modeling voting as a strategic game has been widely accepted in economics and political science.

Although strategic voting modeling has its theoretical attractiveness, the degree of empirical support varies for different voting systems. In addition, two lines of criticism of game theoretical voting models have arisen. From a descriptive view, it is not clear whether real-life voters have full knowledge of game structures and are able to perform complex strategic calculus. Normatively, strategic voting rationalizes manipulations of systems by voters who misrepresent their preferences, and accordingly by candidates who use media influence to shape voters’ perceptions. Such manipulations are generally considered undesirable for a democratic system.

SEE ALSO Strategic Behavior; Strategic Games; Strategy

BIBLIOGRAPHY

Farquharson, Robin. 1969. Theory of Voting. New Haven, CT:Yale University Press.

Gibbard, Allan. 1973. Manipulation of Voting Schemes: A General Result. Econometrica 41 (4): 587–601.

Nash, John F., Jr. 1950. Equilibrium Points in n -Person Games. Proceedings of the National Academy of Sciences of the United States of America 36 (1): 48–49.

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

Satterthwaite, Mark Allen. 1975. Strategy-Proofness and Arrow’s Conditions: Existence and Correspondence Theorems for Voting Procedures and Social Welfare Functions. Journal of Economic Theory 10 (2): 187–217.

Yuelan Chen