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Being a branch of applied mathematics, game theory is used in the social sciences, mostly in economics, but also in engineering, biology, political science, international relations, psychology, and philosophy. Game theory studies behavior in strategic situations, in which an individual’s success in making choices depends on the choices of others. Game theory may be defined as the study of mathematical models of conflict and cooperation between intelligent rational decision-makers. It provides general mathematical techniques for analyzing situations in which two or more individuals make decisions that will influence one another’s welfare. Game theory offers important fundamental insights for scholars in all branches of social sciences. An important distinction exists between the disciplines of individual and interactive decision making. Individual decision making, whether under certainty or uncertainty, leads to well-defined problems. This problems may be difficult to solve, but they don’t involve conceptual issues. As soon as the objective function is specified, the meaning of optimal decision is clear. In interactive decision making, the meaning of the optimal decision is unclear, because no player completely controls the final outcome of the interaction. The decision making process has to start from defining the problem before providing procedures for solving it. Game theory defines solution concepts, to various classes of interactive decision making situations and then provides procedures for their computation. Game theory may be applied not only in games like chess, but also in many other social situations which are commonly not regarded as games. Classical models fail to deal with interdependent decision making because they treat players as inanimate subjects. A game theory model is constructed around the strategic choices available to players, where the preferred outcomes are clearly defined and known. Game theory purpose is to find optimal solutions to situations of conflict and cooperation, under the assumptions that players are rational and act in their own best interests._x000D_
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In some situations solutions can be found, in others, although formal attempts at a solution may fail, the analytical synthesis itself can reveal different aspects of the problem. Game theory offers an interesting perspective on the nature of strategic selection in familiar as well as unusual circumstances. On the basic level of the assumption of rationality, it can be argued that players behave rationally by instinct, although experience shows, that this is not always true, since decision makers frequently adopt simplistic algorithms, which lead to sub-optimal solutions. In business, organizations that select sub-optimal strategies shut down in the face of competition from optimizing organizations. The assumption of rationality is not an attempt to describe how players actually make decisions, but merely that they behave as if they were not irrational. All theories and models should not be dismissed simply because they fail to represent all realistic possibilities. A model should be discarded only if its predictions are wrong or useless. _x000D_
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The game theory was introduced as the scientific discipline by von Neumann and Morgenstern in the book “Theory of games and Economic Behavior”. Much of the early work of the game theory was done during World War II. Much of the success of game theory is derived from its position in the mathematical foundation of the social sciences. Real proof of the power of game theory has come from development of economics. In order to understand conflict and cooperation theorists study models and hypothetical examples. It is easier to understand real competitive situations by studying hypothetical examples. Game theory represents a model of decision making, but not the social reality of decision making. While game theory ensures that a result follows logically from a model, it cannot ensure that the result itself represents reality, except in case when the model is an accurate one._x000D_
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In the language of game theory, a ‘game’ refers to any social situation involving two or more individuals. The individuals involved in a game may be called the ‘players’. Players may be individual persons, organizations or, in some cases, nature itself. When nature is designated as one of the players it is assumed, that it moves according to the laws of chance. Basically, a game must have two or more players, one of which may be nature. Each player must have more than one choice, because a player with one way of strategy cannot alter the outcome of the game. There are two basic assumptions that game theorists generally make about players: they are rational and intelligent. A decision – maker is rational if he makes decisions consistently in pursuit of his own objectives. In game theory, each player’s objective is to maximize the expected value of his own payoff. For any rational decision-maker there must exist some way of assigning utility numbers to the various possible outcomes that he cares about, such that he would always choose the option that maximizes his expected utility. “ This result is called the ‘expected- utility maximization theorem’, according to which if decision-maker would prefer option 1 over option 2 when event A occurs, and he would prefer option 1 over option 2 when event A does not occur, then he should prefer option 1 over option 2 even before he learns whether even a will occur or not.” ( Myerson, 1991) Utility payoff is not necessarily the same as monetary payoff. A utility values are not necessarily measured in dollars. For many decision-makers, utility may be nonlinear function of monetary worth. The utility payoff of an individual may depend on many variables besides his own monetary worth. When there are two or more decision makers involved, a special difficulty arise in the assessment of subjective probabilities._x000D_
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When rational decision- makers interact, their decision problems must be analyzed together. The player of the game is intelligent if he understands the game theory and its predictions. An outcome of the game is the result of strategic selections by all the players in a game. It is assumed, that in some order of preference individuals are capable of identifying possible outcomes. In case if player is indifferent to the difference between two or more outcomes, then those outcomes are equal. There are games where an ordinal scale is sufficient, in others it is necessary to have interval scales where preferences are set out in proportional games. A so-called ‘pure strategy’ for player is a plan for entire game. “If a player selects a strategy without knowing which strategies were chosen by the other players, then the player’s pure strategies are simply equivalent to his or her choices. If player’s strategy is selected subsequent to those of other players and knowing what they are, then there will be more pure strategies than choices.” ( Kelly, 2003) In a game of ‘complete information’, players know their own strategies and are aware of what other players have already chosen. A ‘game of imperfect information’ is one in which players are ignorant of one another’s moves, and can only anticipate what the other player will do. _x000D_
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Usually games are classified into three categories: games of skill, games of chance and games of strategy. Games of skill are one-player games where single player has complete control over the outcomes. It is a game in which outcome is determined by skill rather than by chance, as chess. Games of chance are one-player games against nature. The player does not control the outcomes completely and strategic selections do not lead to certain outcomes. In the games of chance the outcome of a game depends partly on the player’s choice and partly on nature._x000D_
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Games of chance are involving either risk or uncertainty. In the games of chance the player’s outcomes are uncertain and the success’ probability is unknown. Games of strategy are games involving two or more players, not including nature. Each of the players has some control over the outcomes. It is very important to understand the behavior of all the players in the game assuming that they are all rational and intelligent individuals. Player’s optimal strategy should maximize his expected payoff with respect of his subjective probability distribution over the possible strategies of the other players. First player has to assess some subjective probability distribution to summarize beliefs about what strategies will be used by the other players and then to select strategy that maximizes his expected payoff with respect of these beliefs. The decision-analytic approach to the problem is to try to predict the behavior of the players and then to solve the decision problem. Basically, games of chance are one-player games against nature. The player is not making decisions under the conditions of complete certainty in the games of nature. Nature affects the outcomes resulting from the player’s choice in an unpredictable way. It is very important to get acquainted with the probability theory in order to understand games of risk._x000D_
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Another category of games of chance is the category of games involving uncertainty. It is a game, in which player is opposed by nature, but unlike games of risk the player is not able to predict nature’s moves. Usually human behavior results from intentional decision, and decisions typically involve some judgment of the potential rewards and risk associated with each action. In many complex decisions, the risk is associated with the unpredictability of the decisions of other people. Understanding how people predict each others’ behavior and make choices on the basis of these predictions and the available opportunities and rewards, is a central question for psychology.