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The term formal political theory is a catch phrase for myriad modeling traditions in political science. The various traditions include game theory, social choice theory, spatial theory of elections, decision theory, and behavioral game theory. At the heart of formal theory is an axiomatic system or mathematical approach to model political behavior. The underlying premise that drives this approach is the assumption made about human behavior (or players) relating to rationality. This assumption specifies that players who populate the various models take actions that maximize their (expected) utility or payoffs. While there are many assumptions that define a utility function or how players have preferences over outcomes, the basic premise is that given the choice of two or more alternatives a player is able to rank the alternatives from best to worse or specify indifference between alternatives, and when given a choice between two or more alternatives players will always behave rationally and select the alternative that has the highest value to themselves regardless of how this affects other players. Using this assumption, formal political theory allows a mathematical method to model human behavior within political institutions.

Game Theory

John Von Neumann and Oscar Morgenstern pioneered game theory in 1944 by building on earlier work by Ernest Zermelo (1913), Emile Borel (1921), and John Von Neumann (1928). Game theory studies the interdependence of actions of players or their strategic interaction. This approach is concerned with how the adoption of particular strategies by players results in optimal (or nonoptimal) outcomes. Since incentives are held fixed, the resulting outcome usually is determined by the payoffs assigned to particular strategies, the information that players have about the actions of other players or assigned payoffs, and the institutional rules that have been imposed on the players’ environment.

Game theory can be divided into two broad categories: cooperative (or coalition) games and noncooperative (or strategic) games. (Cooperative game theory is also called “social choice theory” and will be discussed later.) In noncooperative games players cannot enter into binding agreements, whereas cooperative games players are allowed to enter into binding agreements. The classic distinction used to revolve around whether players can communicate with each other, but noncooperative games that allow for communication have been developed, such as cheap talk games. Within noncooperative games there is a further distinction between normal form games (games represented by a payoff matrix) and extensive form games (games represented by a game tree).The extensive form is useful to model games where players have more than one move and also games where players might have imperfect information such as the case wherein a player may not know what action another player has taken. Normal form games consider a situation in which the players must make choices simultaneously so that they make choices without knowing how the other player is going to choose. There are also repeated games in which players play the same game for either a finite or an infinite number of times. Games can also have incomplete information wherein players do not know all of the payoffs (or probabilities of payoffs) of the other players.

In the early 1950s, John Nash proposed a solution for noncooperative games referred to as a Nash equilibrium, in which all players’ play strategies are optimal or a best response given what other players have played. A Nash equilibrium is established when all players have selected strategies and no player has a positive incentive to change her or his strategy given what strategies other players have adopted. Games can have two types of equilibrium, either a pure-strategy equilibrium (where players select a single strategy) or a mixed-strategy equilibrium (where a probability distribution is calculated over the strategy set). A mixed strategy assigns a particular probability to each strategy in the set, for example, it might specify that a player will select strategy A 45 percent of the time and strategy B 55 percent of the time. Nash proved that all games with a finite strategy set have at least one Nash equilibrium either in pure or mixed strategies, although games can and do have more than one Nash equilibrium.

Subsequent equilibrium concepts have been refinements of the Nash equilibrium. In terms of solving games, one concept that is often invoked is Pareto efficiency, named after Vilfredo Pareto (1935). When a game has more than one equilibrium, then a Pareto-optimal equilibrium is an equilibrium wherein all players jointly are better off than another equilibrium wherein there is a lower payoff for all the players. A weaker version of the selection of a Pareto-optimal equilibrium is the case in which some players are indifferent to payoffs in another equilibrium but other players receive a positive gain from its selection.

Other noncooperative equilibrium concepts or refinements have included the notion of iterated elimination of dominated strategies. This concept simply eliminates a player’s strategy if all payoff components of a player’s particular strategy are lower than another viable strategy. Subgame perfect equilibrium for extensive form games eliminates outcomes that are not reasonable given the game’s structure. This procedure eliminates outcomes that would not be chosen given that the players are rational. For Bayesian games that incorporate incomplete information, a Bayesian Nash equilibrium was developed to predict outcomes when uncertainty over a state of nature or when another player’s move is involved. In this refinement, Bayes rules are used to establish that players’ beliefs are consistent with the history of game play.

Classic Noncooperative Games

The first classic noncooperative game is the chicken game. In this game two players (in the original game they are portrayed as rebellious teens) are in their cars, and they drive their cars at high rates of speed directly at each other; the first to swerve away is considered to be the chicken and loses the game. In this game if each player chooses not to swerve they end up in the worse payoff (possible death), but if one player swerves and the other does not, then the one who does not swerve wins since the one who swerves is considered a chicken. There are two equilibria to this game, in which one player swerves and the other player does not and vice versa. However, the best (or Pareto outcome) is for both players to swerve. One aspect of this game is the existence of a credible threat (i.e., reputation building). For example, if one of the drivers were to throw his or her steering wheel out of the car, then that action would be a credible threat and the other player should swerve.

The second game examines social dilemmas and is referred to as the Prisoner’s Dilemma. The classic story behind this game is that two suspected criminals have been arrested for a crime, and the district attorney separates them, giving each the same deal. The deal is that if one criminal reveals the details of the crime before the other, the one who confesses first will get a lesser sentence in court than the other suspected criminal. Hence, each suspected criminal achieves a better outcome if he or she confesses or reveals the details of the crime while the other criminal remains silent. The best outcome is a cooperative outcome (or Pareto outcome) if both criminal suspects remain silent and do not confess the details of the crime. The worse outcome for both criminal suspects is for both of them to confess. The dilemma in this game is that both players have a dominated strategy to confess or defect, which yields an outcome that is worse than the outcome if they both cooperated. This game illustrates the problem of cooperation because even if the two players could communicate and agree not to confess, they both still have incentives to renege and to confess.

The last is the battle of the sexes, which is a coordination game. In this game a husband and wife have to decide whether to go to a ballet or to a boxing match. The man most prefers to go to the boxing match, whereas the wife most prefers to go to the ballet. Each prefers to be together rather than to be alone, but the husband gets more utility if they go to the boxing match and the wife gets more utility if they go to the ballet. Each receives his or her lowest utility if he or she goes alone. In this game there are two equilibria in which they both go to the boxing match or they both go to the ballet. Hence, both players have an incentive to cooperate, but the problem is coordination since they do not agree on which outcome is better. In this game bargaining may lead to a solution, or the wife could restrict alternatives and say that she faints at the sight of blood, forcing the husband to go to the ballet.

Social Choice Theory

Social choice theory, or cooperative game theory, examines how the aggregation of individual preferences translates into a group choice or collective preference (sometimes referred to as “collect choice”). The problem explored is generating a social choice function (a voting mechanism that determines voting outcomes) that is fair and equitable. The roots of social choice theory date back to 1299 when Ramon Llull proposed that candidates in multicandidate elections compete against each other two at a time.

Cooperative game theory, unlike noncooperative game theory, assumes players compete as teams as opposed to individuals, where coalitions are a team of individuals who have agreed to coordinate on a common set of strategies to play in the game. It is assumed that cooperation can be enforced by an outside party.

Cooperative games are derived and solved either axiomatically or spatially. Using the axiomatic method, conditions (or axioms) are specified, and then these conditions are checked for consistency. Spatially (or geometrically), the political environment is composed of dimensions (vectors in Euclidean space), where the dimensions can be thought of as policy spaces. Players within the space are assumed to have utility functions represented as points on the dimension for one dimension and within the dimension for two or more dimensions. Utility functions defined over dimensions measure the distance an alternative is from a player’s ideal point (or a player’s personal location in the dimensional space) and assume that the closer the alternative is to the ideal point, the more that player prefers that alternative.

The central problems investigated within this area can be illustrated by the voter’s paradox. Assume that three equal-sized groups of voters have to decide among three alternatives: x, y, and z, where group 1 most prefers x to y to z, group 2 prefers y to z to x, and group 3 prefers z to x to y. Now assume that the voting rule requires pairwise comparisons of alternatives. If x is paired against z, then z wins since groups 2 and 3 prefer z to x. Now pair z against y where groups 1 and 2 prefer y to z. Pairing y against x yields x as the eventual winner. This configuration of preferences yields what is called an intransitive social ordering, in which the winner of the election depends on the matchup that voters are presented with. In other words there is no Condorcet winner (named after Marquis de Condorcet 1785), which is a winner that beats all other alternatives in a pairwise comparison.

This example is generalized by the general impossibility theorem or the Arrow’s impossibility theorem (named after Kenneth Arrow 1951), which proves that when the social choice function assumes four reasonable assumptions about preferences and there are more than two alternatives, then it is impossible to develop a perfect system of voting. The following are Arrow’s four assumptions: assumption 1 is “universal domain,” which assumes social preferences should be transitive and complete in the sense that given a choice between alternatives y and x, x is preferred to y, y is preferred to x, or the two are indifferent. Assumption 2 is Pareto efficiency, positing that if each individual prefers one alternative to another, then that alternative should be the winner. Assumption 3 is “nondictatorship,” in which one individual should not dictate the outcome. Finally, assumption 4 is “independence from irrelevant alternative,” wherein the social preference of y compared with x should be independent of preferences for other alternatives. Arrow proved that these four assumptions are inconsistent and result in a voting system that is a dictatorship. In other words, there is no voting rule, such as majority voting, for establishing social preferences from arbitrary individual preferences.

Furthermore, Alan Gibbard’s (1973) and Mark Satterthwaite’s (1975) impossibility theorems show that any reasonable voting scheme such as majority-rule voting is susceptible to manipulation even when considering strategic or sophisticated voting. Strategic voting is voting for a player’s lesser-preferred alternative to gain a more desirable outcome.

Formally, there have been two ways to bypass these results. First, it is argued that interpersonal comparison of utility is not scientifically sound. That is, what one person derives from a utility, such as food for a homeless person, is different from the utility a person derives if he or she is wealthy. Hence, it is impossible to assign the same utility for food to the two individuals. Second, the impossibility theorem can be solved if players are assumed to have single-peaked preferences. Single peaked preferences assume a single dimension where x comes before y and y comes before z, each individual has a preference over the dimension where there is a unique ideal point over their most preferred preference (the highest peak), and the second and third preferences descend below the first. In the voter’s paradox example, if the dimension is arranged such that x appears first, y second, and z third, then the voter’s third preference ordered of z, x, y would be disallowed since it is not single peaked (x is second in order instead of y).

When social choice functions are examined spatially, one of the primary efforts is trying to define a core. Like the Nash equilibrium, a core is a point or strategies in which no coalition of players has a positive incentive to defect (i.e., it is the set of maximal choices). When a majority-rule core exists, then it is also known as a Condorcet winner (see Peter Ordeshook’s 1986 Game Theory and Political Theory for variants on the properties of the core.)

Assuming two dimensions and using Euclidean preferences (which allow players to have preferences over two dimensions), a core is guaranteed only by the very strict Plott-McKelvey symmetry condition. Since this condition requires precise placement of ideal points in the dimensional space, which is unlikely or rare in the real world, it is conjectured that equilibrium in two-dimensional space is unlikely unless institutional rules are imposed. Lacking institutional structure within two dimensions, Richard McKelvey (“Intransitivities in Multidimensional Voting Bodies” 1976) proposed what is referred to as the “chaos theorem,” which means there is instability within the dimensions since there is no Condorcet winner. Assuming all players vote sincerely and that one player (an agenda setter) can determine the location of the alternatives that are voted on within the dimensions, then this player can achieve any desired outcome (including his or her ideal point). Again this result implies that majority-rule institutions can be dictatorial.

Instability within the dimensions can be overcome by imposing institutional structure such as issue-by-issue voting in which players vote for one alternative on one dimension first and then they vote for another alternative on the second dimension. This effectively reduces the two-dimensional model to two one-dimensional models, wherein the equilibrium is the intersection of the median of the two one-dimensional models. Other institutional features create a structurally induced equilibrium or stability such as different types of voting rules, committee types, and norms of reciprocity, among others.

A subset of these types of spatial games is the spatial theory of elections. In 1957 Anthony Downs proposed the famous median voter theorem in which candidates seeking political office will take a position on the dimension at the median preference of voters (Duncan Black contributed similar findings for committees in 1958).The implication of this theorem is that in a two-candidate election the position of both candidates will be similar or close to the median voter. In a single-dimension environment, equilibrium is ensured as long as there are an odd number of players. James Enelow and Melvin Hinich (1984) introduced the ideal of salience and separability for spatial preferences that allowed voter choices to be more realistic.

Individual Decision Making

As opposed to the previous discussion that involved two or more players, as the name implies, individual decision making concerns the study of an individual player and whether that player’s behavior conforms to the tenets of expected utility theory. The classic example that illustrates the problems of expected utility theory is the Allais paradox, named after Maurice Allais (1953). Consider two sets of gambles: A and B, and C and D. An individual has a choice between, for A, getting one million dollars with certainty and, for B, an 89 percent chance of getting one million dollars, a 10 percent chance of getting five million dollars, and a 1 percent chance of getting nothing. When experiments are conducted, participants overwhelmingly pick A. Now consider gamble C and D. An individual has a choice between, for C, a 10 percent chance of getting five million dollars and a 90 percent chance of getting nothing and, for D, an 11 percent chance of getting one million dollars and an 89 percent chance of getting nothing. In this case participants generally choose C. However, it can be shown that selecting A in the first lottery and C in the second lottery violates the independent axiom since both gambles yield the same expected payoffs. Instead, to satisfy this condition players should select A and D or B and C. (See also Daniel Ellsberg’s 1961 paradox for similar results.)

Daniel Kahneman and Amos Tversky (1979) proposed prospect theory, which examines the way people frame decisions. They argue that individuals do not view probabilities in a linear manner; rather, they view probabilities according to a nonlinear weighting function that alters their perceptions of losses and gains. Other developments in this area have included George Ainslie and Nick Haslam’s (1992) hyperbolic discounting and Robert Sugden’s (1993) regret theory.

Behavioral Game Theory

One of the most recent developments in formal theory is the advent of behavioral game theory. It is argued that people are generally nonrational and do not make decisions according to the dictates of expected utility theory. Behavioral game theory attempts to predict how people actually behave by incorporating psychological (or stochastic) elements and learning into formal models. A critical part of behavioral game theory is laboratory experiments to test the behavior of players within various game theory models. Richard McKelvey and Thomas Palfrey’s (1995, 1998) quantal response equilibrium is an example of an application of behavioral game theory since this concept allows players to make errors and learn during game play.

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