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# Game Theory

Aspects of game theory were first explored by the French mathematician Émile Borel (1871-1956), who wrote several papers on games of chance and theories of play. The acknowledged father of game theory, however, is the Hungarian-American mathematician John von Neumann, who in a series of papers in the 1920s and ’30s established the mathematical framework for all subsequent theoretical developments. During World War II military strategists in such areas as logistics, submarine warfare, and air defense drew on ideas that were directly related to game theory. Game theory thereafter developed within the context of the social sciences. Despite such empirically related interests, however, it is essentially a product of mathematicians.

The game theory was largely disseminated by John Von Newmann and Oskar Morgenstern (1944) in their works. The game theory has provided a means for analyzing many problems of interest in economics, management science, and other fields. In the International Relations, Game Theory is mostly related to study the conflict and consequences between two or more state and other actors of International Relations. Mostly the scholars Thomas Scheiling, William Riker and Martin Shubik used this theory on the study of international politics and relation. Among these scholars the contributions of Thomas Scheiling and Martin Shubik are significant to develop this theory than others.

Introduction:

As scholar Shubik says “Game Theory is a mathematical method for the study of some aspects of conscious decision making in the situation involving the possibilities of conflict and cooperation. It deals with the process in which the individual decision unit has only partial control over the strategic factors affecting its environment.”

Game Theory is a branch of applied mathematics that concerns itself with the study of multi-person interdependent decision-making in conflict situations. Such a situation exists when two or more decision makers, who have different objectives act on the same way and share the same resources. It is a set of techniques for analyzing strategic situations; that is, situations in which at least two agents make decisions that affect one another’s welfare. The perceptivities offered by the theory can be therefore applied across a wide range of disciplines including economics, political science, anthropology, sociology, biology etc.

Game Theory is a mathematical analysis of any situation involving a conflict of interest, with the intent of indicating the optimal choices that, under given conditions, will lead to a desired outcome. Although, game theory has roots in the study of such well-known enjoyment as checkers, tick-tack-toe, and poker also involves much more serious conflicts of interest arising in such fields as sociology, economics, and political and military science. There are two persons and multi person’s games. Game theory provides a mathematical process for selecting an Optimum Strategy (that is, an optimum decision or a sequence of decisions) in the face of an opponent who has a strategy of his own.

Game Theory is a theory of competition stated in terms of gains and losses among opposing players. In Game Theory a game refers to a strategic situation that involves at least two rational intelligent individuals called players. A rational player is one who consistently makes decisions tracking down of some well-defined objective and an intelligent player is one who knows everything we know about the game and who can make the same inferences as we do.

The game theory operates on the assumption that the international political process or international relation is the confrontation like a chess game, or a contrast between two merchants or brokers, or the movements of rival political candidates, or the counteractions of opposed diplomats, and within the game all institutions and countries are playing as individuals player. It is defined as a body of thought dealing strategies in situations of conflict and competition wherein each participant or player seeks to maximize his gains and minimize his losses. It is also a mathematical model in which the player is placed in a certain fixed situation and tires to make maximum gains out of his opponents.

There are two main branches of game theory: Cooperative (collaborative) and Non-cooperative (strategy) game theory. Non-cooperative game theory deals largely with how intelligent individuals interact with one another in an effort to achieve their own goals. But in cooperative both interact to achieve for their common goals.

Basic Concepts:

In game theory, the term game means a particular sort of conflict in which no of individuals or groups (known as players) participate. A list of rules stipulates the conditions under which the game begins, the possible legal “moves” at each stage of play, the total number of moves constituting the entirety of the game, and the terms of the outcome at the end of play.

a) Moves:

In game theory, a move is the way in which the game progresses from one stage to another, beginning with an initial state of the game through the final move. Moves may alternate between players in a specified fashion or may occur simultaneously. Moves are made either by personal/national choice or as counter the opponent for maximum benefits or achievements.

b) Payoff:

Payoff, or outcome, is a game-theory term referring to what happens at the end of a game. In such games as chess or checkers, payoff may be as simple as declaring a winner or a loser. In card or other gambling situations the payoff is usually money; its amount is predetermined by bets collected during the course of play, by percentages or by other fixed amounts calculated on the chances of winning. Moreover, in IR the payoff may be treaties determine role and responsibility, conventions for to do or not to do something collectively by players, agreements, compensation for something or some other outcomes as desire by the nation players with their opponent/s.

c) Extensive and Normal Form:

One of the most important distinctions made in characterizing different forms of games is that between extensive and normal. A game is said to be in extensive form if it is characterized by a set of rules that determines the possible moves at each step, indicating which player is to move, the probabilities at each point if a move is to be made by a chance determination, and the set of outcomes assigning a particular payoff or result to each possible conclusion of the game. The assumption is also made that each player has a set of preferences at each move in anticipation of possible outcomes that will maximize the player’s own payoff or minimize losses. A game in extensive form contains not only a list of rules governing the activity of each player, but also the preference patterns of each player.

Because of the enormous numbers of strategies involved in even the simplest extensive games, game theorists have developed so-called normalized forms of games for which computations (determining something by mathematical or logical methods) can be carried out completely. A game is said to be in normal form if the list of all expected outcomes or payoffs to each player for every possible combination of strategies is given for any sequence of choices in the game. This kind of theoretical game could be played by any neutral observer and does not depend on player choice of strategy.

d) Perfect Information:

A game is said to have perfect information if all moves are known to each of the players involved. Checkers and chess are two examples of games with perfect information; poker and bridge are games in which players have only partial information at their disposal. In IR role of UN can be groups as game of perfect information and any other role of nations with each other can grouped in partial or secreted information game.

e) Strategy:

A strategy is a list of the best possible choices for each player at every stage of a given game. A strategy, taking into account all possible moves, is a plan that cannot be upset, despite of what may occur in the game.

In the game theory the player built their strategy with the predication on the strategy chosen by his opponent, which gives information about all possible strategies, rules and payoffs or their possibilities that enables each player to maximize benefits while minimizing losses.

Scholar Davis B. Bobrow suggests that following properties of the situations with which game theory deals:

• They involve two or more ‘players’ whose interest conflict, at least in part.
• Each Player has two or more choices (Strategies) as to how to proceed in the game.
• The outcome of the game is determined once each player has chosen a strategy.
• And each possible outcome is associated with a particular payoff or return (positive or negative) to each player.

Commons Assumptions:

In game theory one usually makes the following assumptions:

1) Each decision maker [“PLAYER”] has available to him/her two or more well-specified choices or sequences of choices (called “PLAYS”).

2) Every possible combination of plays available to the players leads to a well-defined end-state (win, loss, or draw) that terminates the game.

3) A specified payoff for each player is associated with each end-state (a ZERO-SUM game means that the sum of payoffs to all players is zero in each end-state).

4) Each decision maker has perfect knowledge of the game and of his opposition; that is, he knows in full detail the rules of the game as well as the payoffs of all other players.

5) All decision makers are rational; that is, each player, given two alternatives, will select the one that give up him the greater payoff.

The last two assumptions, in particular, restrict the application of game theory in real-world conflict situations. On the other hand, game theory has provided a means for analyzing many problems of interest in economics, management science, and other fields.

Interests and Rules in Game:

As per the pro-scholars of the game theory, in the international politics the political game will be played with other players, which have these three characteristic with them when they are playing games:

Identical Interest: The player of the game will have same interest and want achieve maximum gains as per their interest. The strategy of both part of the game will be same and they only concern with their own interest. This will be the cooperative or collaborative game to achieve something common as per interests of players, where players negotiate binding contracts that allow them to plan joint strategies.

Game with opposite interest: On this players want to have maximum gains with minimum losses. It is based on “Loss & Gain Theory” of the game. Here one will get gains as per the losses of another player. This will be the non-cooperative or strategic game to achieve something opposite interests between players, where notiation and enforcement of a binding contract are not possible.

Game with mixed interest: This is the game, where the player will play game with common interest as helpful and with opposite interest as unhelpful. Here the losses and gains may be equal with each other. This will be some how cooperative- no-cooperative or collaborative- strategic game where sometime the interest are identical and sometime against each others.
There are certain assumption and rules of the game theory. The assumptions are that the players are guided by rational behaviour and choose the best course of action that brings them maximum gains. The rules are that the equation between the players is straight and the losses of one are the gains of another. The game theory is built up with the help of give importance conceptions: Strategy, Opponent, Pay-of, Rules and information.

Kinds of Game:

The situation of the game visualized are in the four kinds in the game theory for the study of the international relations and politics, these are:

Zero-sum two person game: Here the gains of one player are equal to the losses of the other player. In less formal terms, a zero-sum game is a game in which one player’s winnings equal the other player’s losses. Do notice that the definition requires a zero sum for every set of strategies. If there is even one strategy set for which the sum differs from zero, then the game is not zero sum.

Non-zero-sum two person game: In this kind of game, between two players the winner of the game will shared the outcome of the game and losses of one are not necessarily equal to the gains of another.

Zero-sum n-persons games: Here the player will be more than two, and the outcome is shared and the losses of the one are not necessarily equal to gains of another but the in accumulation the sum will become zero.
Non-zero-sum n-persons games: This is also a game between more than two player will play the games. In this game the situation in extremely complex and the gains and losses are shred by both sides of the game to some extent.

Prison’s Dilemma in Game Theory:

The prisoner’s dilemma was originally formulated by mathematician Albert W. Tucker and has since become the classic example of a “non-zero sum” game in economics, political science, evolutionary biology, and of course game theory.

A “zero sum” game is simply a win-lose game such as tic-tac-toe. For every winner, there’s a loser. If I win, you lose. Non-zero sum games allow for cooperation. There are moves that benefit both players, and this is what makes these games interesting.

Tucker began with a little story, like this: two robbers, Bob and Albert, are captured near the scene of a Bank Robbery and are given the “third degree” and interrogated in separate cells without a chance to communicate with each other by the police. For the purpose of this game, it makes no difference whether or not Bob or Albert actually committed the crime. Each has to choose whether or not to confess and implicate the other. If neither man confesses, then both will serve one year on a charge of carrying a concealed weapon. If each confesses and implicates the other, both will go to prison for 10 years. However, if one robber confesses and implicates the other, and the other robber does not confess, the one who has collaborated with the police will go free, while the other robber will go to prison for 20 years on the maximum charge.

Bob and Albert both told the same thing:

• If you both confess, you will both get 10 years in prison.
• If neither of you confesses, the police will be able to pin part of the crime on you, and you’ll both get 1 year.
• If one of you confesses but the other doesn’t, the confessor will make a deal with the police and will go free while the other one goes to jail for 20 years.

At first glance the correct strategy appears obvious. No matter what Albert does, bob will be better off “defecting” (confessing). Annoyingly, Albert realizes this as well, so Bob and Albert both end up getting 10 years. Surprisingly, if both “cooperated” (refused to confess) both be much comfortable.

And so the game becomes much more complicated than it first appeared. If Bob play time after time, the goal is to figure out Albert’s strategy and use it to minimize his total jail time. Albert will be doing the same. Remember, the object of the game is not to screw Albert over. The object is to minimize Bob’s jail time. If this means ruthlessly exploiting Albert’s generosity, then do so. If this means helping Albert out by cooperating, then do so.

The strategies in this case are: confess or don’t confess. The payoffs (penalties, actually) are the sentences served. We can express all this compactly in a “payoff table” of a kind that has become pretty standard in game theory. Here is the payoff table for the Prisoners’ Dilemma game:

Albert
confess don’t
Bob confess 10,10 0,20
don’t 20,0 1,1

The table is read like this: Each prisoner chooses one of the two strategies. In effect, Albert chooses a column and Bob chooses a row. The two numbers in each cell tell the outcomes for the two prisoners when the corresponding pair of strategies is chosen. The number to the left of the comma tells the payoff to the person who chooses the rows (Bob) while the number to the right of the column tells the payoff to the person who chooses the columns (Al). Thus (reading down the first column) if they both confess, each gets 10 years, but if Al confesses and Bob does not, Bob gets 20 and Al goes free.

So: how to solve this game? What strategies are “rational” if both men want to minimize the time they spend in jail? Al might reason as follows: “Two things can happen: Bob can confess or Bob can keep quiet. Suppose Bob confesses. Then I get 20 years if I don’t confess, 10 years if I do, so in that case it’s best to confess. On the other hand, if Bob doesn’t confess, and I don’t either, I get a year; but in that case, if I confess I can go free. Either way, it’s best if I confess. Therefore, I’ll confess.”

But Bob can and presumably will reason in the same way — so that they both confess and go to prison for 10 years each. Yet, if they had acted “irrationally,” and kept quiet, they each could have gotten off with one year each.

This game has fascinated game theorists for a variety of reasons. First, it is a simple representation of a variety of important situations. For example, instead of confess/not confess we could label the strategies “contribute to the common good” or “behave selfishly.” This captures a variety of situations economists describe as public goods problems. An example is the construction of a bridge. It is best for everyone if the bridge is built, but best for each individual if someone else builds the bridge. This is sometimes refered to in economics as an externality. Similarly this game could describe the alternative of two firms competing in the same market, and instead of confess/not confess we could label the strategies “set a high price” and “set a low price.” Naturally is best for both firms if they both set high prices, but best for each individual firm to set a low price while the opposition sets a high price.

A second feature of this game is that it is self-evident how an intelligent individual should behave. No matter what a suspect believes his partner is going to do, is is always best to confess. If the partner in the other cell is not confessing, it is possible to get 10 instead of 5. If the partner in the other cell is confessing, it is possible to get 1 instead of 0. Yet the pursuit of individually sensible behavior results in each player getting only 1 unit of utility, much less than the 5 units each that they would get if neither confessed. This conflict between the pursuit of individual goals and the common good is at the heart of many game theoretic problems.

A third feature of this game is that it changes in a very significant way if the game is repeated, or if the players will interact with each other again in the future. Suppose for example that after this game is over, and the suspects either are freed or are released from jail they will commit another crime and the game will be played again. In this case in the first period the suspects may reason that they should not confess because if they do not their partner will not confess in the second game. Strictly speaking, this conclusion is not valid, since in the second game both suspects will confess no matter what happened in the first game. However, repetition opens up the possibility of being rewarded or punished in the future for current behavior, and game theorists have provided a number of theories to explain the obvious intuition that if the game is repeated often enough, the suspects ought to cooperate.

Dominant Strategies

What has happened here is that the two prisoners have fallen into something called”dominant strategy equilibrium.”

DEFINITION Dominant Strategy: Let an individual player in a game evaluate separately each of the strategy combinations he may face, and, for each combination, choose from his own strategies the one that gives the best payoff. If the same strategy is chosen for each of the different combinations of strategies the player might face, that strategy is called a “dominant strategy” for that player in that game.

Definition of Dominant Strategy Equilibrium: If, in a game, each player has a dominant strategy, and each player plays the dominant strategy, then that combination of (dominant) strategies and the corresponding payoffs are said to constitute the dominant strategy equilibrium for that game.

In the Prisoners’ Dilemma game, to confess is a dominant strategy, and when both prisoners confess, that is dominant strategy equilibrium.

Issues With Respect to the Prisoners’ Dilemma

This remarkable result — that individually rational action results in both persons being made worse off in terms of their own self-interested purposes — is what has made the wide impact in modern social science. For there are many interactions in the modern world that seem very much like that, from arms races through road congestion and pollution to the depletion of fisheries and the overexploitation of some subsurface water resources. These are all quite different interactions in detail, but are interactions in which (we suppose) individually rational action leads to inferior results for each person, and the Prisoners’ Dilemma suggests something of what is going on in each of them. That is the source of its power.

Having said that, we must also admit candidly that the Prisoners’ Dilemma is a very simplified and abstract — if you will, “unrealistic” — conception of many of these interactions. A number of critical issues can be raised with the Prisoners’ Dilemma, and each of these issues has been the basis of a large scholarly literature:

• The Prisoners’ Dilemma is a two-person game, but many of the applications of the idea are really many-person interactions.
• We have assumed that there is no communication between the two prisoners. If they could communicate and commit themselves to coordinated strategies, we would expect a quite different outcome.
• In the Prisoners’ Dilemma, the two prisoners interact only once. Repetition of the interactions might lead to quite different results.
• Compelling as the reasoning that leads to the dominant strategy equilibrium may be, it is not the only way this problem might be reasoned out. Perhaps it is not really the most rational answer after all.

Several Nobel Prizes have been awarded to some of major figures of Game Theory: the Nobel was shared by John Nash, J.C. Harsanyi and R. Selten in 1994 and William Vickrey and James Mirrlees in 1996. Herbert Simon won the Nobel in 1979 for concepts (e.g. bounded rationality) which have since been incorporated into the corpus of (Evolutionary) Game Theory.

Criticism:

The theory is highly abstract and works only under assumed conditions. The players are rarely as rational as presumed by this theory. Game theory may be criticized on two important grounds. First, it stands on the premise of zero-sum match in which one side wins and the other side loses. But we may find instances in which the game ends without any side winning or losing. Second, most of the crucial issues of international relations go on endlessly. Because of that scholar Deutsch says: “Game theory usually assumes that most games have an end, but international relations resemble rather an unending game in which no great power can pick up its marbles and go home.”