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I try to give simple definitions and intuitive examples of the basic kinds of games and their solution concepts. There are four kinds of games: static or dynamic, and complete or incomplete information. Complete information means there is no private information. The corresponding solution concepts are: Nash equilibrium in static games of complete information; backwards induction or subgame-perfect Nash equilibrium in dynamic games of complete information; Bayesian Nash equilibrium in static games with incomplete information; and perfect Bayesian or sequential equilibrium in dynamic games with incomplete information.

The main theme of the paper is that these solution concepts are closely linked. As we consider progressively richer games, we progressively strengthen the solution concept, to rule out implausible equilibria in the richer games that would survive if we applied solution concepts available for simpler games. In each case, the stronger solution concept differs from the weaker concept only for the richer games, not for the simpler games.

Download Citation Data. Share Twitter LinkedIn Email. Technical Working Paper In defence of inclusive fitness theory. Mild cognitive impairment in a chess player. We report the case of a chess player with superior premorbid cognitive function who presented to the Cognitive Disorders clinic at the National Hospital for Neurology and Neurosurgery with a 2-year history of symptoms of possible memory We report the case of a chess player with superior premorbid cognitive function who presented to the Cognitive Disorders clinic at the National Hospital for Neurology and Neurosurgery with a 2-year history of symptoms of possible memory loss.

Initially the MRI scan appearance was within normal limits and his cognitive scores inside the normal range; subsequently his cognitive function deteriorated and he fulfilled criteria for Mild Cognitive Impairment MCI two years later. Knowing one's future preferences: A correlated agent model with Bayesian updating. Analysis of the effect of InfoRanking on content pollution in peer-to-peer systems.

Cognitive load in the multi-player prisoner's dilemma game: Are there brains in games? Abstract: We find that differences in the ability to devote cognitive resources to a strategic interaction imply differences in strategic behavior. In our experiment, we manipulate the availability of cognitive resources by applying a In our experiment, we manipulate the availability of cognitive resources by applying a differential cognitive load.

In cognitive load experiments, subjects are directed to perform a task which occupies cognitive resources, in addition to making a choice in another domain. The greater the cognitive resources required for the task implies that fewer such resources will be available for deliberation on the Evolutionary game theory meets social science: Is there a unifying rule for human cooperation?

Modeling agent negotiation is of key importance in building multiagent systems. Negotiation provides the basis for managing the expectations of the individual negotiating agents, and it enables selecting solutions that satisfy the agents Negotiation provides the basis for managing the expectations of the individual negotiating agents, and it enables selecting solutions that satisfy the agents as much as possible.

Thus far, most of the negotiation models have limitations when employed in large scale multiagent systems. This paper presents a negotiation model for. MALL: A multi-agent learning language for competitive and uncertain environments. Gradual learning and the evolution of cooperation in the spatial Continuous Prisoner's Dilemma.

Gradual learning and the evolution of cooperation in the spatial Lugo 2 , and M. San Miguel Incentives in dynamic value networks — Challenges and a theoretical research proposal. We specify the characteristics of We specify the characteristics of dynamic value networks and view them from the perspective of different compensation theories and then ask what challenges they pose to these theories and models.

Self-fulfilling Mechanisms in Bayesian Games. Equilibrium analysis of coexisting IEEE Related Topics.

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. 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.

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.

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. 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.

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.

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. 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.

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.

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. 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.

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. Suppose that Prisoner 1 believes that Prisoner 2 will confess.

Similarly, suppose that Prisoner 1 believes that Prisoner 2 will remain silent. Thus, remaining silent is always dominated by confessing. 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. 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. 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. 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. 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.

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. Moving up the tree, now Legislator 2 makes the decision knowing how the rational Legislator 3 would make a decision. Finally, moving another branch up, Legislator 1 has to make a decision. 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. 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.

This is called complete information. 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. Thus, one may expect that different dynamics emerge when players engage in a strategic game repeatedly. Game theory has become increasingly popular in political science since it was first introduced to the discipline in the s and s. An initial political science application of game theory was a group decision among a large number of actors or voters.

These initial developments were modified and advanced to study the U. 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. 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. 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 , 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.

The Baron and Ferejohn 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. The main theme of the paper is that these solution concepts are closely linked. As we consider progressively richer games, we progressively strengthen the solution concept, to rule out implausible equilibria in the richer games that would survive if we applied solution concepts available for simpler games.

In each case, the stronger solution concept differs from the weaker concept only for the richer games, not for the simpler games. Download Citation Data. Share Twitter LinkedIn Email. Technical Working Paper DOI Issue Date July Acknowledgements and Disclosures. Published Versions Journal of Economic Perspectives, vol.