What is Game Theory? What economists call game theory psychologists call the theory. Although game theory is relevant to parlor games such. There are two main branches of game theory. Noncooperative game theory. That is the branch of. I will discuss here. What economists call game theory psychologists call the theory of social situations, which is an accurate description of what game theory is about. Although game. Adams' Equity Theory on job motivation and free diagram, plus more free online training materials. In addition to game theory, economic theory has three other. The focus is on. preferences and the formation of beliefs. The most widely used form of. Decision theory. is often used in the form of decision analysis, which shows how best to. It is widely used in the macroeconomic analysis of broad. In recent years, political economy has emerged as a combination. Issues studied include tax policy, trade policy, and the role. European Union. Questions addressed by. An. One way to describe a game is by listing the players (or. Any time we have a situation with two or more players that involves known payouts or quantifiable consequences, we can use game theory to help determine the most. This is page i Printer: Opaque this Game Theory (W4210) Course Notes Macartan Humphreys September 2005. Game theory is "the study of mathematical models of conflict and cooperation between intelligent rational decision-makers." Game theory is mainly used in economics. Game Theory. This article sketches the basic concepts of the theory of games in order to discuss some of their philosophical implications and problems. Finite mathematics utility: game theory tool. Here is a little on-line Javascript utility for game theory (up to five strategies for the row and column player). Cooperative Game Theory Cooperative games are often de A model of optimality taking into consideration not only benefits less costs, but also the interaction between participants. Game theory attempts to look at the. ![]() In the case of a two- player game, the actions of the first. The entries in the matrix are two numbers representing the. A very. famous game is the Prisoner's Dilemma game. In this game the two. The game can be represented by the following. Note that higher numbers are better (more utility). If neither. suspect confesses, they go free, and split the proceeds of their crime. However, if. one prisoner confesses and the other does not, the prisoner who. If both prisoners confess. This game has fascinated game theorists for a variety of. First, it is a simple representation of a variety of important. For example, instead of confess/not confess we could label. An example is the construction of a bridge. It is best. for everyone if the bridge is built, but best for each individual if. This is sometimes refered to in. Similarly this game could describe the. No matter what a suspect. If. the partner in the other cell is not confessing, it is possible to get. If the partner in the other cell is confessing, it is. Yet the pursuit of individually. This conflict between the pursuit of individual goals and. A third feature of this game is that it changes in a very. Suppose for example that. In this case in the first period the suspects may reason. Strictly speaking, this conclusion is. However, repetition opens up. If. Some. of the power and meaning of game theory can be illustrated by assessing. Or you may. recognize that as a matter of logic this involves the fallacy of. Game theory can give precise meaning to. In fact the statement is false, and. Prisoner's Dilemma. Let us start with a variation on the Prisoner's Dilemma game we may call the Pride Game. The Pride Game is like the Prisoner's Dilemma game with the addition of the new. A proud individual is one who will not confess. In. other words, if I stand proud and you confess, I get 1. I can stand proud before your humiliation, but you get 0, because you stand humiliated. On the other hand, if we are both proud, then neither. It would be worse. In this case, I would get 3. The Pride Game is very different than the Prisoner's. Dilemma game. Suppose that we are both proud. In the face of your. I simply chose not to confess I would lose face, and my. To confess would be even worse as. I would be humiliated as well. In other words, if we are both proud, and we each. This type of situation - . Nash Equilibrium. Notice. that the original equilibrium of the Prisoner's Dilemma confess- confess. Pride game: if I think you are going. I would prefer to stand proud and humiliate you rather than. Now. suppose that we become . Specifically, let us imagine that. I am more generous and care more about you, I place a value. I receive in the . Not being completely altruistic, I. I do on yours. So, for. I get 3 units of utility, and you get. I am an altruist, I. I get 2/3 of the. Overall I get 4 units of utility instead of 3. Because I have. become a better more generous person, I am happy that you are getting 6. The new game with altruistic. This gives the payoff matrix of the Altruistic Pride Gameproudnot confessconfessproud. What. happens? If you are proud, I should choose not to confess: if I were to. I get a utility of 4, while if I choose not to confess I get. I do confess I get only 0. Looking at the. original game, it would be better for society at large if when you are. I were to choose not to confess. This avoids the confrontation of. However, as an. altruist, I recognize that the cost to me is small (I lose only. I prefer to ? If. I get 4. 8, if I choose not to confess I get 5, but if I. I get 5. 3. 3. So I should confess. Again, this is marked with an. Finally, if you confess, then I no longer wish to stand. If I choose not to confess I get only 0. So it. is best for me to. What do we conclude? It is no longer an. Each of us in the face of the. Of course it is. also not an equilibrium for us both to choose not to confess: each of. The only equilibrium is the box. So. far from making us better off, when we both become more altruist and. Notice how we can. It is true that if we. However: if we become more caring we will wish to. As this example shows, when we both try to do. To. put this in the context of day- to- day life: if we were all more. The behavior of criminals has a complication. More altruistic. criminals would choose to commit fewer crimes. However, as crime is not. If. in the balance more crimes are committed, the world could certainly be. The example shows how this might work. For. those of you who are interested in or already know more advanced game. Pride Game has only the one Nash equilibrium shown - it is. The Atruistic Pride Game, however, has. You can compute them using the fine. Gambit. written by Richard Mc. Kelvey, Andrew Mc. Lennan and Theodore Turocy. One. equilibrium involves randomizing between proud and confess, so is worse. Pride game. The other is. The. payoffs to that equilibrium gives each player 2. Pride Game. I'd. like to thank Jie Zheng for his help. They provide a much more. If you wish to learn more about game theory, there a variety. Game theory - Wikipedia. Game theory is . Today, game theory applies to a wide range of behavioral relations, and is now an umbrella term for the science of logical decision making in humans, animals, and computers. Modern game theory began with the idea regarding the existence of mixed- strategy equilibria in two- person zero- sum games and its proof by John von Neumann. Von Neumann's original proof used the Brouwer fixed- point theorem on continuous mappings into compact convex sets, which became a standard method in game theory and mathematical economics. His paper was followed by the 1. Theory of Games and Economic Behavior, co- written with Oskar Morgenstern, which considered cooperative games of several players. The second edition of this book provided an axiomatic theory of expected utility, which allowed mathematical statisticians and economists to treat decision- making under uncertainty. This theory was developed extensively in the 1. Game theory was later explicitly applied to biology in the 1. Game theory has been widely recognized as an important tool in many fields. With the Nobel Memorial Prize in Economic Sciences going to game theorist Jean Tirole in 2. Nobel Prize. John Maynard Smith was awarded the Crafoord Prize for his application of game theory to biology. History. The first known discussion of game theory occurred in a letter written by Charles Waldegrave, an active Jacobite, and uncle to James Waldegrave, a British diplomat, in 1. James Madison made what we now recognize as a game- theoretic analysis of the ways states can be expected to behave under different systems of taxation. It proved that the optimal chess strategy is strictly determined. This paved the way for more general theorems. Borel conjectured that non- existence of mixed- strategy equilibria in two- person zero- sum games would occur, a conjecture that was proved false. Game theory did not really exist as a unique field until John von Neumann published a paper in 1. His paper was followed by his 1. Theory of Games and Economic Behavior co- authored with Oskar Morgenstern. Von Neumann's work in game theory culminated in this 1. This foundational work contains the method for finding mutually consistent solutions for two- person zero- sum games. During the following time period, work on game theory was primarily focused on cooperative game theory, which analyzes optimal strategies for groups of individuals, presuming that they can enforce agreements between them about proper strategies. Flood and Melvin Dresher, as part of the RAND Corporation's investigations into game theory. RAND pursued the studies because of possible applications to global nuclear strategy. Nash proved that every n- player, non- zero- sum (not just 2- player zero- sum) non- cooperative game has what is now known as a Nash equilibrium. Game theory experienced a flurry of activity in the 1. Shapley value were developed. In addition, the first applications of game theory to philosophy and political science occurred during this time. Prize- winning achievements. In 1. 96. 7, John Harsanyi developed the concepts of complete information and Bayesian games. Nash, Selten and Harsanyi became Economics Nobel Laureates in 1. In the 1. 97. 0s, game theory was extensively applied in biology, largely as a result of the work of John Maynard Smith and his evolutionarily stable strategy. In addition, the concepts of correlated equilibrium, trembling hand perfection, and common knowledge. Schelling worked on dynamic models, early examples of evolutionary game theory. Aumann contributed more to the equilibrium school, introducing an equilibrium coarsening, correlated equilibrium, and developing an extensive formal analysis of the assumption of common knowledge and of its consequences. In 2. 00. 7, Leonid Hurwicz, together with Eric Maskin and Roger Myerson, was awarded the Nobel Prize in Economics . Roth and Lloyd S. Shapley were awarded the Nobel Prize in Economics . A game is non- cooperative if players cannot form alliances or if all agreements need to be self- enforcing (e. It is opposed to the traditional non- cooperative game theory which focuses on predicting individual players' actions and payoffs and analyzing Nash equilibria. As non- cooperative game theory is more general, cooperative games can be analyzed through the approach of non- cooperative game theory (the converse does not hold) provided that sufficient assumptions are made to encompass all the possible strategies available to players due to the possibility of external enforcement of cooperation. While it would thus be optimal to have all games expressed under a non- cooperative framework, in many instances insufficient information is available to accurately model the formal procedures available to the players during the strategic bargaining process, or the resulting model would be of too high complexity to offer a practical tool in the real world. In such cases, cooperative game theory provides a simplified approach that allows to analyze the game at large without having to make any assumption about bargaining powers. Symmetric / Asymmetric. If the identities of the players can be changed without changing the payoff to the strategies, then a game is symmetric. Many of the commonly studied 2. The standard representations of chicken, the prisoner's dilemma, and the stag hunt are all symmetric games. However, the most common payoffs for each of these games are symmetric. Most commonly studied asymmetric games are games where there are not identical strategy sets for both players. For instance, the ultimatum game and similarly the dictator game have different strategies for each player. It is possible, however, for a game to have identical strategies for both players, yet be asymmetric. For example, the game pictured to the right is asymmetric despite having identical strategy sets for both players. Zero- sum / Non- zero- sum. In zero- sum games the total benefit to all players in the game, for every combination of strategies, always adds to zero (more informally, a player benefits only at the equal expense of others). Other zero- sum games include matching pennies and most classical board games including Go and chess. Many games studied by game theorists (including the famed prisoner's dilemma) are non- zero- sum games, because the outcome has net results greater or less than zero. Informally, in non- zero- sum games, a gain by one player does not necessarily correspond with a loss by another. Constant- sum games correspond to activities like theft and gambling, but not to the fundamental economic situation in which there are potential gains from trade. It is possible to transform any game into a (possibly asymmetric) zero- sum game by adding a dummy player (often called . Sequential games (or dynamic games) are games where later players have some knowledge about earlier actions. This need not be perfect information about every action of earlier players; it might be very little knowledge. For instance, a player may know that an earlier player did not perform one particular action, while he does not know which of the other available actions the first player actually performed. The difference between simultaneous and sequential games is captured in the different representations discussed above. Often, normal form is used to represent simultaneous games, while extensive form is used to represent sequential ones. The transformation of extensive to normal form is one way, meaning that multiple extensive form games correspond to the same normal form. Consequently, notions of equilibrium for simultaneous games are insufficient for reasoning about sequential games; see subgame perfection. In short, the differences between sequential and simultaneous games are as follows: Perfect information and imperfect information. A game is one of perfect information if, in extensive form, all players know the moves previously made by all other players. Simultaneous games can not be games of perfect information, because the conversion to extensive form converts simultaneous moves into a sequence of moves with earlier moves being unknown. Most games studied in game theory are imperfect- information games. Interesting examples of perfect- information games include the ultimatum game and centipede game. Recreational games of perfect information games include chess and checkers. Many card games are games of imperfect information, such as poker or contract bridge. Complete information requires that every player know the strategies and payoffs available to the other players but not necessarily the actions taken. Games of incomplete information can be reduced, however, to games of imperfect information by introducing . Examples include chess and go. Games that involve imperfect or incomplete information may also have a strong combinatorial character, for instance backgammon. There is no unified theory addressing combinatorial elements in games. There are, however, mathematical tools that can solve particular problems and answer general questions. These methods address games with higher combinatorial complexity than those usually considered in traditional (or . A related field of study, drawing from computational complexity theory, is game complexity, which is concerned with estimating the computational difficulty of finding optimal strategies. The practical solutions involve computational heuristics, like alpha- beta pruning or use of artificial neural networks trained by reinforcement learning, which make games more tractable in computing practice. Pure mathematicians are not so constrained, and set theorists in particular study games that last for infinitely many moves, with the winner (or other payoff) not known until after all those moves are completed. The focus of attention is usually not so much on the best way to play such a game, but whether one player has a winning strategy. Many concepts can be extended, however. Continuous games allow players to choose a strategy from a continuous strategy set.
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