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### Outline

I. Introduction

II. Definitions and Basic Terminology

A. Game, Players, and Preferences

B. Game Representation and Solution Concepts

1. Normal Form

2. Extensive Form

3. Extensions

III. Game Theory in Political Science

A. A Brief History

B. Game Theory in American Politics

C. Game Theory in International Relations

D. Game Theory in Comparative Politics

IV. Conclusion

## I. Introduction

Game theory is a branch of applied mathematics that is used to model multiactor interdependent decision making. Game theory is widely used in many social science disciplines, including political science, economics, sociology, and anthropology, where researchers are interested in outcomes when at least two actors interact with certain purposes.

Game theory is a method of modeling. A usual game theoretic model specifies some essential aspects of a situation of interest and tries to make logical inferences about ensuing outcomes given the initial setup. There can be a simple election model, for instance, where there are two candidates who want to win the election and n voters who want to elect the candidate who is going to make policies that are beneficial for the voters. Two candidates announce their respective policy platforms, and voters vote. Whoever gets the majority of votes wins and makes policies. Given the initial setting, the solution to the game provides logically deduced inferences about outcomes of interest, such as who can win under which conditions and which policies should follow.

In modeling a situation, a game theoretic model captures only essentials and inevitably leaves out unnecessary details. Thus, a game theoretic model does not and cannot perfectly reflect the reality. Thus, a game theoretic model may seem too abstract. Indeed, one of the common criticisms of game theory is that game theoretic models are too unrealistic. Yet abstraction is common for any kind of modeling. For instance, a model of an airplane or a model of an automobile usually does not feature every nut and bolt of an actual airplane or automobile. Instead, the models would probably have a cockpits and wings for airplanes and doors, tires, and wheels for cars, yet they would probably not have emergency oxygen masks, all cockpit buttons, and smoke detectors in the restrooms of airplane models and cup holders and detailed electrical lines connecting batteries to different parts of car models. Yet these models may still be useful for certain purposes. Similarly, a game theoretic model capturing the relationship between Congress and bureaucrats may feature only two actors even though Congress and bureaucrats are not unitary actors but rather composed of groups of individuals in reality. Congress and bureaucrats in the model would also be assumed to have a few primary goals, such as reelection and budget maximization even though there are many other potential motivations for each actor. Yet as is the case for the automobile and airplane models, the model can still be proven to be useful to study the relationship between Congress and bureaucrats.

There is no golden rule as to how abstract or realistic a model should be. In addition, there are a number of ways to model a situation by emphasizing certain aspects of the situation at the expense of other aspects being bracketed. Thus, it is hardly possible to tell if a model in itself is either right or wrong. Instead, a model can be judged by how useful and applicable it is to a modeled situation. Generally speaking, a researcher can make a model resemble the reality more faithfully with many details, but in doing so, the researcher has to face the usual trade-off between details and generalizability. That is, a detailed model may capture a particular situation more accurately but not be generalizable beyond the particular situation. A more abstract model, in contrast, may be more general and applicable to a broader set of situations but may seem too unrealistic to approximate a particular case. In addition, each addition of details would make the model more complicated and make the math difficult or even analytically intractable. Ultimately, the decision for the initial setup and how complicated a model should be depends on the researcher’s purpose.

Game theory is often called a method. But it is important to acknowledge the difference between methods of modeling like game theory and methods of empirical testing, which usually refers to research methods. Essentially, game theory is a method of theorizing or modeling, as opposed to a method of empirical testing such as regression analysis, factor analysis, longitudinal analysis, and other qualitative and quantitative research methods introduced in other research papers. The critical difference is that a method of theorizing is used to generate hypotheses or expected relations between important and interesting factors under research investigation while a method of empirical testing examines if and how well the generated hypotheses match real data. Thus, the two methods are complementary. Often, scholars use a method of theorizing such as game theory in the theoretical section of a research study to generate hypotheses and then use an appropriate research method to test the hypotheses in an empirical section in a single study. For instance, a game theoretic model in a study may generate the hypothesis that states the following: As trade between Country A and Country B increases, the probability of an interstate conflict between A and B increases. Then an appropriate empirical method such as binomial logit or probit analysis in the study can be installed to see if the stated hypothesis holds in reality with actual trade and conflict data sets.

Game theory is used to capture multiactor interdependent decision making processes. Naturally, then, there should be more than one actor making decisions in the models. This differentiates game theory from decision theory, which models a single-actor decision-making process. Also, the ensuing outcome (and thus the ensuing payoff) of the multiactor decision making should be interdependent in a game theoretic model. In other words, the final outcome needs to be jointly determined by actors involved in the model. The rock-paper-scissors game is a good example. More than one player is needed to play this game, and the outcome is jointly determined by the decisions of both actors. Player A can take one of three actions: rock, paper, or scissors. Likewise, Player B can take one of the same three actions. And the outcomes, A wins and B loses, the game is tied, or A loses and B wins, are jointly determined by the actions that Player A and Player B take. By assuming that both players want to win, one can model the rock-paper-scissors game, solve it, and make predictions about each player’s optimal actions in the game.

Since many social scientific research questions are about outcomes that result when multiple actors interact, game theory can be very useful in making inferences about potential outcomes in multiactor decision-making situations. Indeed, game theory has become increasingly popular in many social science disciplines for the past half century or so. In economics, where the use of game theory had been accepted earlier than in political science, game theory has been applied to model interactions between different sets of economic actors. For instance, economists have used game theory to model behaviors of competing firms, wage bargaining between a labor union and management, behaviors of producers and consumers, and competition among bidders at an auction. In political science, scholars have applied game theory to model behaviors of competing candidates in an election; the interaction between candidates and voters; the interactions between a bureaucratic agency and Congress; the interactions between the executive and the legislative branches in American politics; behaviors of states involved in interstate militarized disputes; behaviors of states in trade disputes; alliance behaviors; the role of mediators in conflicts; negotiations among states over design of international organizations in international politics; negotiations between parties to form, continue, and dissolve a coalition government; intrastate conflicts between factions in a country; and interactions between a government and an opposition in comparative politics.

This research paper introduces the basics of game theory and reviews the use of game theory in political science. In the next section, a few basic components of game theory and important terminology are introduced. Then a few representative examples of the use of game theory in various political science contexts are discussed. The discussion focuses particularly on three representative examples drawn from each of three subfields of political science: American, comparative, and international politics. These examples are among the most well known and widely cited and have made major contributions to the understanding of political phenomena. In the concluding section, a recap of the research paper is provided, and the ongoing effort of moving game theoretic models forward and the future of game theory are briefly discussed.

## II. Definitions and Basic Terminology

### A. Game, Players, and Preferences

A game refers to a strategic situation that involves at least two rational individuals called players. A rational player is one who engages in goal-directed behavior—one who has well-defined goals, such as vote maximization or profit maximization, can order her or his preferences over alternative outcomes given a set of alternatives, and chooses the best alternative(s) for the realization of the given goals. For instance, when we model a Congressional election, we would probably assume that any serious candidates want to maximize their vote shares and that winning the election is the candidates’ goal. Then each would have a set of alternatives that he or she needs to make a choice over, such as where to spend time and energy during the campaign, given district A, B, C, and D. A rational player then would choose the best alternative that would allow him or her to increase the vote share the most. This does not necessarily mean that a rational player is greedy or pursues only materialistic benefits. Contrary to a popular misunderstanding, game theory is agnostic about the origin of goals; hence, goals for players in a game may well be altruistic or emotional.

Players are assumed to have clear goals when they are involved in a strategic situation. Given the goal, they are able to arrange their preference ordering over every possible outcome. For instance, players involved in a rock-paper-scissors game are assumed to have a goal of winning. Then their preference ordering would be presumably to win over to tie and to tie over to lose (hence, win over lose). Usually, preferences are translated by some utility functions that assign payoffs (real numbers) to each outcome when outcomes are determined by the combination of actions by all players. Payoffs for the outcomes are assigned such that the preference relations are maintained, given a set of outcomes and actors’ preference relations among the outcomes. Following the rock-paper-scissors example, the utility function may assign a real number for each outcome so that each player gets a payoff of 5 for winning, 0 for tying, and –5 for losing. Since 5 is greater than 0 and 0 is greater than –5, the preference relationship still holds.

We play many games in our everyday lives; games are often being played when people interact. For instance, if an individual drives a car in a busy street, that individual plays a game with the drivers of the other cars. Most drivers have clear goals: They want to spend the least amount of time on a road without being involved in a car accident, and each driver’s decisions to change lanes, to stop at a light, and to choose one road over another all affect others’ driving time on a road. Similarly, when an individual makes a bid for a pair of concert tickets on eBay, he or she is playing a game with other bidders. The individual would want to win an auction with a minimum bid, others would do the same, and one bidder’s bid affects others’ willingness and the price of bidding since each bid would affect the others.

Many political situations in real life can be thought of as games. The decisions of the Soviet Union and the United States about developing, stockpiling, and locating nuclear weapons during the cold war era can be modeled as a game between the two superpowers. Similarly, the decision making by Nikita Khrushchev and the Soviet government during the Cuban missile crisis to build and remove a missile base on Cuban soil and the decision-making process by John F. Kennedy and the United States government to respond to the attempted construction of the missile base can be modeled as a game between two players pursuing their own security interests and interacting with various policy alternatives.

### B. Game Representation and Solution Concepts

#### 1. Normal Form

Formally speaking, a game consists of (a) a set of players, (b) a set of actions (or combinations of actions called strategies) for each player, and (c) preferences over the set of action (or strategy) profiles for each player. And a game is usually represented in one of two ways: normal form or extensive form.

The normal-form representation of a game specifies the players, the actions or strategies—the combinations of actions—for each player, and the payoff received by each player in a matrix. This is useful to represent situations where players make a strategic decision without knowing other players’ decisions. A well-known example, the prisoner’s dilemma game, is presented in the normal form in Table 1.

**Table 1. Prisoner’s Dilemma**

The background story of the prisoner’s dilemma game is as follows. Two prisoners have been caught and are being interrogated by the police. The crime that they have committed and been caught for is relatively minor, but they have also committed a more serious crime in the past, and the police interrogate the prisoners to prosecute them for the serious crime as well. The prisoners are interrogated separately, without a way to communicate or collude with each other. The deal that the police propose to each prisoner is that if both prisoners remain quiet for the serious crime, both prisoners will serve only 1 year each in prison for the light crime without being convicted for the serious crime; if one prisoner remains quiet for the serious crime but the other one confesses the serious crime, the one who confesses is set free while the one who remains silent serves a 9-year prison term; and if both prisoners confess, both are prosecuted for the serious crime and serve 6 years each in prison.

Here, two players, Prisoner 1 and Prisoner 2, are playing the game. They need to decide whether to keep quiet or confess, without being sure about the other player’s decision. In the normal-form representation, players are usually listed on the top and the left side of the payoff matrix. Available actions are defined in columns and rows, and respective payoffs are listed in each cell where actions by players intersect. For instance, if Prisoner 1 remains quiet while Prisoner 2 confesses, Prisoner 1’s payoff is −9 and Prisoner 2’s payoff is 0.

Given the setup, then, what is the best strategy available for each player? There are possibly many solution criteria, but one intuitive strategy would be a simple procedure of elimination of dominated strategies. The simple procedure of elimination of dominated strategies relies on the reasoning that a rational player should not choose a strategy if there exists an alternative strategy that raises his or her payoffs against all possible strategies of his or her opponent (McCarty & Meirowitz, 2007). If we apply the procedure to the prisoner’s dilemma, we can indeed obtain a solution. Suppose that Prisoner 1 believes that Prisoner 2 will confess. If that is the case, then it is better for Prisoner 1 to confess as well, because Prisoner 1 gets −6 by confessing as opposed to getting −9 by remaining silent. Similarly, suppose that Prisoner 1 believes that Prisoner 2 will remain silent. If that is the case, then it is better for Prisoner 1 to confess, because Prisoner 1 can get 0 by confessing as opposed to getting −1 by remaining silent. Thus, remaining silent is always dominated by confessing. Thus, the strategy of remaining silent can be safely eliminated from Prisoner 1’s choice set. Since the game is symmetric and both players reason the same way, Prisoner 2 can also eliminate the option of remaining quiet. As a result, the only combination that remains possible is confession for both players. In general, if we repeat the procedure of elimination of dominated strategies, we may get a solution or at least tighten our predictions by eliminating a few strategies.

A more formal solution concept that is commonly used is the Nash equilibrium. Formally, Nash equilibrium is defined as “an action profile a* with the property that no player i can do better by choosing an action different from ai *, given that every other player j adheres to aj *” (Osborne, 2004, p. 22). In the prisoner’s dilemma game, the action profile of the confess–confess combination is one (and only one) Nash equilibrium since there is no incentive to deviate from the confess–confess action profile for either player given that the other player sticks with confessing. That is, one becomes worse off only by deviating because if one player decides to deviate and remain silent, he or she will receive –9 as opposed to –6 in the confess–confess case. In all other action profiles, however, each player can be better off by deviating from the profiles, given the other player sticking with the action; hence, these action profiles are not Nash equilibria.

#### 2. Extensive Form

An extensive-form representation of a game involves the same set of elements in a normal-form representation— a set of players, a set of actions or strategies, and preferences for each player over a set of possible outcomes—with the addition of sequences. Thus, an extensive-form representation of a game is especially useful when one needs to explicitly take into account the sequence of actions by players. A game tree is commonly used to graphically represent a game (see Figure 1).

**Figure 1. A Game Tree**

The game is a simple legislation game among three legislators. The specific context of the game is as follows. Three legislators vote for the legislation to raise the pay for legislators. All legislators are assumed to want to get a pay raise, but they also do not want to be seen as pursuing their own self-interests by their respective voters. Thus, although all want to get a pay raise, each wants the bill to pass with others’ votes but not with his or her own vote. Assuming a simple majority rule, this would imply the preference ordering of (a) bill passage with voting no, (b) bill passage with voting yes, (c) bill nonpassage with voting no, and (d) bill nonpassage with voting yes, the least favorite for all legislators. Now let’s further assume the utility function that assigns a value of 2 for bill passage and –1 for voting yes, which is consistent with the description above. As a result, the first option gives 2, the second option gives 2 − 1 = 1, the third option gives 0, and the fourth option gives –1.

The players, actions, and payoffs are all represented in the game tree. Players, Legislators 1, 2, and 3, are listed at each node. Although there seem to be multiple Legislator 2s and 3s, it is just the way it is represented, and there is actually only one Legislator 2 and one Legislator 3 playing at every possible node. Actions, Yay and Nay, are listed above the branches of the game tree. Outcomes, Pass or No, are followed from the combinations of actions taken by all players, and the payoffs are listed such that the top one is the payoff for the first legislator, the middle one is for the second legislator, and the bottom one is for the third legislator.

Again, there are potentially many solution criteria, yet the most intuitive one is to move backward and find solutions. This procedure is called backward induction, since it starts at the bottom and moves backward to find a solution. The rationale for the procedure is that when a player has to make a decision, the player will predict actions that the players in the game will subsequently take if they all act rationally, and the player will choose the action that maximizes the payoff for him or her.

Applying the backward induction to the game drawn in Figure 1 yields an equilibrium. Starting at the bottom nodes where Legislator 3 has to make a decision, Legislator 3 will choose “Nay” at the left node since it gives 2 compared with 1, choose “Yay” at the second-from-the-left node since it gives 1 compared with 0, choose “Yay” at the second-from-the-right node since it gives 1 compared with 0, and choose “Nay” at the far-right node since it gives 0 instead of –1.Moving up the tree, now Legislator 2 makes the decision knowing how the rational Legislator 3 would make a decision. Since the legislator knows that Legislator 3 will choose “Nay” at the left node and “Yay” at the second-from-the-left node, if Legislator 2 votes “Yay,” Legislator 2 gets 1; if Legislator 2 votes “Nay,” Legislator 2 gets 2. Thus, at the left node, the rational choice for Legislator 2 is to vote “Nay.” Similarly, knowing that Legislator 3 will vote “Yay” at the second-from-the-right node and “Nay” at the far-right node, it is in Legislator 2’s best interest at the right node to vote “Yay,” receiving 1 instead of 0. Finally, moving another branch up, Legislator 1 has to make a decision. Now, Legislator 1 knows that Legislator 2 would choose “Nay” and Legislator 3 would choose “Yay” after Legislator 2 votes “Nay” if Legislator 1 votes “Yay,” and Legislator 1 knows that Legislator 2 chooses “Yay” and Legislator 3 chooses “Yay” after Legislator 2’s “Yay” vote if Legislator 1 votes “Nay,” so it is in Legislator 1’s best interest to vote “Nay” and receive 2 instead of voting “Yay” and receiving 1. Thus, the equilibrium for the simple voting game is that Legislator 1 votes “Nay” knowing that the rest of the legislators will still vote “Yay” and pass the bill to raise their salaries, and the rest indeed vote “Yay” since they still prefer to vote “Yay” and pass the bill. Thus, the first voter, Legislator 1, enjoys a clear advantage, often referred to as the first-mover advantage.

Formally, the solution concept for the extensive-form game with complete information is a subgame perfect equilibrium. A subgame is defined as follows: “For any nonterminal history, h, where history is defined as a sequence of actions taken thus far, the subgame following h is the part of the game that remains after h has occurred” (Osborne, 2004, p. 164). For instance, after the history of Legislator 1’s playing “Yay” and Legislator 2’s playing “Yay,” the subgame following “Yay–Yay” history is the game in which Legislator 3 decides to vote “Yay” or “Nay.” Then a subgame perfect equilibrium is defined as “a strategy profile s* with the property that in no subgame can any player i do better by choosing a strategy different from si *, given that every other player j adheres to sj *” (p. 165). In other words, a subgame perfect equilibrium is “a strategy profile that includes a Nash equilibrium in every subgame” (p. 166).

#### 3. Extensions

There are many possible extensions to the games that are presented in this research paper. One particular extension concerns the information assumed in the game. In the extensive-form game, the informational assumption is that every player involved in the game knows everything about the game; in particular, players know each other’s payoff structure. This is called complete information. In contrast, a game of incomplete information, a very common extension of a complete-information game, deals with a situation where at least one player is uncertain about others’ payoff functions and thus tries to learn the other players’ types.Ausual game of incomplete information posits that there is at least one player who can be one of two or more types, where each type corresponds to a different payoff function that the player might have. Then given the initial belief about the probability distribution over the types of the player, uninformed players update their beliefs about the probability distribution over the types of the player in question and make decisions based on their updated beliefs. Commonly, when an informed player moves first, the game is called a signaling game, and when an uninformed player moves first, the game is called a screening game.

Players are commonly assumed to follow the Bayesian rule when updating their beliefs about the probability distribution over the types of another player. A commonly administered solution concept is the perfect Bayesian Nash equilibrium. Essentially, it requires that a strategy-and-belief pair be consistent and mutually reinforcing.

Another common extension to the basic games is the repeated game. When players interact repeatedly, each player can condition his or her action on the other players’ prior actions. Thus, one may expect that different dynamics emerge when players engage in a strategic game repeatedly. For instance, when the prisoner’s dilemma game is played once, each player’s optimal strategy is to defect as shown previously.

## III. Game Theory in Political Science

### A. A Brief History

Game theory has become increasingly popular in political science since it was first introduced to the discipline in the 1950s and 1960s. An initial political science application of game theory was a group decision among a large number of actors or voters. From there, scholars have established the canonical theorems such as the median voter theorem and Arrow’s general possibility theorem, also known as Arrow’s impossibility theorem. These initial developments were modified and advanced to study the U.S. Congress and its committees’ decision making. Along with the development of theories about group decision making, often referred to as cooperative game theory or social choice theory, the development of bargaining games and noncooperative game theory found more applications across the subfield areas in political science. There are numerous applications of noncooperative game theory. For example, in American politics, scholars study campaign strategies of candidates in an election, or they study how Congress delegates some authority to an independent bureaucratic agency and controls it by monitoring and punishing if necessary. In comparative politics, scholars study how parties bargain over coalition-government building and ending coalition governing. In international relations, scholars study why and under what conditions states go to war and at what times states initiate trade disputes through the WTO’s dispute-settlement mechanism. As the number of studies using game theory to build theories increases, it is very common to find a few articles in any issue of the leading political science journals, such as the American Political Science Review or the American Journal of Political Science, that use game theory to explain political phenomena of interest.

### B. Game Theory in American Politics

In American politics, game theoretic models have been used in various political contexts. Examples include models of agenda setting, legislative bargaining, collective goods and particularistic goods provisions, lobbying, congressional committees, parties and elections, and bureaucratic agencies and legislature. Among these numerous models, one of the most well-developed research strands is the model of legislative bargaining. Since the seminal piece by Baron and Ferejohn (1989), there has been steady progress made in the use of game theory in legislative bargaining, with additional assumptions making models of congressional bargaining more accurately reflect institutional features of the U.S. Congress.

The Baron and Ferejohn (1989) model starts with an observation that the distribution of pork barrel projects is focused on a few states and localities, but the taxes used to fund pork barrel projects are widely spread throughout American legislative districts. To provide a logical answer to the question of why pork barrel projects are distributed in such way, Baron and Ferejohn present a legislative bargaining game.

The game features n members in the legislature, and each represents a legislative district. The task for the n member legislature is to decide the distribution of benefits across all legislative districts. Each member has an equal probability to be recognized to make a proposal at the beginning of a session. The proposal may be amended depending on the amendment rule. The amendment rule can be either open or closed. Finally, the proposal is voted on and is passed if the majority casts yes votes. In the following, only the game under a closed rule is discussed.

At the beginning of the game, a legislator is randomly chosen to make a proposal. A chosen legislator gets to propose a division of the benefits across all legislative districts, and the rest of the legislators get to vote yes or no for the proposal. When one makes a proposal, then it is brought to a vote, and following the simply majority rule, the proposal passes or fails. When the proposal passes, the legislature adjourns, the benefits are allocated as proposed, and the game ends. When the proposal fails to garner a majority of votes, another random draw is made, and another one out of n legislators is chosen to make a proposal. The game continues until a proposal is voted to pass. The utility for each legislator is defined such that each legislator enjoys the benefit allocated for his or her district since the legislator can bring the allocated money for the district, which increases the legislator’s probability of reelection. The benefit is discounted by a discount term, meaning that a legislator prefers the benefit in the present legislative session over the same benefit in the next legislative session.

Solving the game involves the backward induction procedure. Baron and Ferejohn (1989) find that there are multiple Nash equilibria and multiple subgame perfect Nash equilibria in this game, and they suggest a refinement. The refined solution concept that they provide to generate unique predictions for the game is called stationary sub game perfect Nash equilibrium. Essentially, the refinement provides a restriction such that only the strategies that are time independent remain.

The stationary subgame perfect Nash equilibrium suggests that under the closed rule, only a minimum winning coalition is formed. More specifically, the equilibrium for the game features that a recognized member proposes to receive 1 − δ(n − 1)/2n and offers δ/n to (n − 1)/2 other members selected at random, each member votes for any proposal in which at least δ/n is received, and the first proposal receives a majority vote, so the legislature completes its task in the first session. Thus, the randomly chosen proposer enjoys a huge advantage since he or she can bring approximately half of the entire benefits to his or her district. The chosen legislators, who vote yes, can bring some benefits to their districts. The remaining half of the legislators do not receive any allocation of benefits and vote no but fail to stop the bill from passing.

Since the publication of the Baron and Ferejohn (1989) model, a number of studies have built on the initial model. For instance, Baron (1991) considers a similar model with taxation coming into the utility function of each legislator. More recently, Volden and Wiseman (2007) consider legislative bargaining over division between collective goods and particularistic goods, then the distribution of particularistic goods. Overall, these series of studies of legislative bargaining have contributed to our understanding of legislative bargaining and the importance of legislative rules and institutional settings

### C. Game Theory in International Relations

In international politics, game theory has been used in illustrating the logic of anarchy and explaining the importance of relative and absolute gains, causes of war, problems of credible commitments and signals, the role of mediators in international conflicts, and the role of international organizations in various contexts. One of the pioneers in international relations is Bruce Bueno de Mesquita (1981, 1985). In a series of articles and books published since the late 1970s, he has made major contributions in the study of war and peace by modeling the decision making of a rational foreign policy maker to initiate a military conflict with an expected utility framework. Since then, there have been many game theoretic studies of interstate conflicts, and one of the most cited articles is James Fearon’s (1995) article on rational explanations for war.

Fearon (1995) questions why states go to a war even if waging a war is seemingly not an optimal choice for either state in most conflict cases. Capturing a war between states as a bargaining, he points out that agreeing on a division of a coveted good, such as a piece of territory, without fighting a war is often optimal for both states, rather than fighting a war and dividing the coveted good. For instance, if two states have a territorial dispute over a territory, π, and the two states can fight and divide π into p and π − p for State A and State B after the war, it is better for both states to not fight a war, and divide π into p and π − p, than to fight and divide π, as a war is costly for both states. Fearon then provides several conditions under which states might go to war even if there is a potential bargaining space where both are better off without fighting a war.

The game theoretic model of war by Fearon (1995) features two players, State A and State B, who have preferences over a set of issues, such as a disputed territory, represented by the interval between 0 and 1. State A prefers issue resolutions closer to 1 while State B prefers resolutions closer to 0. Let p be the probability of State A winning a war if two states fight a war and 1 − p be the probability of State B winning the war. And let c for each state be the cost of a war, due to battle casualties, revenue spent on a war, and destruction caused by a war, assuming c positive means that a war is costly for each state.

In its simplest form, the game flows as follows. At the beginning of the game, State A demands the division of 1, called x. State B receives the demand x from State A and chooses whether it wants to fight or back down. If State B chooses to fight, a war occurs and the payoffs are distributed for States A and B such that State A gets p − c and State B gets 1 − p − c. If State B chooses to back down, then a settlement occurs with State A getting x and State B getting 1 − x.

With slight modifications to the model, Fearon (1995) shows that there are three possible explanations for war. First, rational leaders may choose to go to war because they cannot locate a mutually acceptable settlement because of private information about the probability of winning and incentives to misrepresent private information. Second, rational leaders may decide to wage a war because of a commitment problem. Even if both states can agree on the terms of settlement, the division of 1, there is no enforcement mechanism in international politics that prohibits the state that becomes stronger after the settlement to demand more in the future. Anticipating this, the state on the losing side may prefer fighting a war to agreeing on the settlement. Finally, states may find it difficult to find a peaceful settlement because the issues under contest are indivisible.

Since the seminal article by Fearon (1995), many studies have built on Fearon’s model to study the dynamics of war. For instance, Alastair Smith (1998) offers a similar war model with each battle providing additional information about the probability of winning a war for both sides and both states making optimal decisions with updated beliefs about the probability of winning a war.

### D. Game Theory in Comparative Politics

There are a large number of studies in comparative politics that use game theoretical models to capture democratic policy making, with particular focus on comparisons between different institutional structures used across democracies. These institutional features include electoral systems such as specific voting rules and electoral district sizes, party systems, and relations between the legislature and bureaucrats. Other applications of game theory in comparative political settings include democratization and market reforms, especially how an incumbent government and an opposition interact in these political and economic transitions. Geddes (1991) provides one such model. She develops a game theoretic model of bureaucratic reform in Latin American democracies where politicians interact in deciding whether to support administrative reform. She derives propositions that reforms are more likely to occur when patronage is evenly distributed among the strongest parties and when the electoral weight of the strongest parties remains stable. She then tests the predictions with brief case studies of five Latin American countries.

One of the more well-known game theoretic models in comparative politics is the article by David Austen-Smith and Jeffrey Banks (1988) published in the American Political Science Review. The article, titled “Elections, Coalitions, and Legislative Outcomes,” brings together a model of election and a model of policy decision in a legislative setting and provides a comprehensive model of electoral and legislative behavior. As voters vote in anticipation of policy outcomes such as tax, education, and health policies and as policies are determined within a legislature where rational politicians condition their policy platforms on the prospect of electoral success, the two processes are clearly linked, and modeling both political processes simultaneously certainly advances a better understanding of elections and policy making in a legislature.

In their game, there are three parties competing for votes from n voters. At the onset of the game, the parties simultaneously declare their policy positions over a one-dimensional policy space. This one-dimensional policy space can conveniently be thought of as the left–right ideology spectrum. When the parties announce their respective positions, n voters cast their votes for parties. The electoral system is the proportional representation adopted in some European countries, and legislative seats are allocated to each party according to the proportion of votes that each party receives if a party receives more than s votes. In their model, Austen-Smith and Banks (1988) assume that every party receives at least s votes to reduce unnecessary complication of the model. When seats are allocated by the numbers of votes that each party receives, the parties try to form a government. The party with the largest number of seats first proposes a composition of the governing coalition, distribution of benefits among coalition members, and a policy to be implemented. On receiving the proposal, the members of the coalition either accept the proposal or reject the proposal. When the members of the coalition accept the coalition, the coalition government is constituted with the proposed policy subsequently implemented. If the members of the coalition reject the coalition proposal, then the party with the second- largest number of votes proposes a coalition, a policy, and a distribution of benefits. If the proposed package is accepted by the members of the coalition, the government coalition is formed, the benefits are distributed, and the policy is implemented accordingly. If the proposed package is not accepted, then the last party gets to make a proposal. If a coalition government is still not formed after the last party makes a proposal, then a caretaker government is implemented that makes equitable policy and benefit-distribution decisions.

Voters are assumed to be policy oriented, with their utility functions defined such that they prefer a policy closer to their own ideal policies. Parties are assumed to enjoy allocated benefits when they are included in a coalition government, and parties want to minimize the distance between their electoral platforms and the implemented policy.

Solving the game is complicated but logically straightforward since one needs only to follow the backward induction methods presented previously. In the equilibrium, it will always be the case that the majority party, if one exists, forms a government by itself, and if the party with the highest number of seats does not enjoy a majority, then the parties with the highest and the lowest number of seats form the governing coalition. This is because the party with the highest number of seats finds it cheaper to offer a coalition to the party with the lowest number of seats.

## IV. Conclusion

This research paper provides a gentle introduction to game theory. As a tool for researchers to deduce logically consistent hypotheses, game theory has been widely used in many different social science contexts. Basic terms and elements of game theory and the most important solution concepts are introduced with some sample applications. Then three representative examples in political science are provided in the latter part of the research paper. One can see that game theoretic models can be used to study many interesting political phenomena, including legislative bargaining in American politics, decisions to go to war in international relations, and formation of coalition governments in comparative politics. For its relatively short history in political science, the influence of game theory on the ways in which researchers approach research questions has been substantial.

There are also ongoing innovations in game theoretic applications in political science that look very promising. One such innovation is to incorporate insights from psychological research in specifications of utilities for players and their ways of processing information. Another innovation is to allow players in a model to make various errors. Often referred to as bounded rationality models, these models are often made to allow better reflection of reality in a model.

Also innovative in political science is an effort to test theoretically generated insights empirically. Often dubbed empirical implications of theoretical models, there have been impressive attempts to bridge the gaps between theoretical propositions and empirical testing, including statistical and qualitative research methods. It is well acknowledged in political science that a multimethod approach to a given research question often yields a better result, and the ongoing effort to bring these theoretical and empirical research tools to study a research topic will certainly help us better understand and analyze complex political phenomena.

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