Introduction to Non-Cooperative Game Theory in Software Paint code128b in Software Introduction to Non-Cooperative Game Theory bar code for .NET

Introduction to Non-Cooperative Game Theory using barcode development for none control to generate, create none image in none applications.generate pdf-417 Game theor none for none y deals with strategic interactions among multiple decision makers, called players. Each player s preference ordering among multiple alternatives is captured in an objective function for that player, which she tries to either maximize (in which case the objective function is a utility function or bene t function) or minimize (in which case we refer to the objective function as a cost function or a loss function). For a non-trivial game, the objective function of a player depends on the choices (actions, or equivalently decision variables) of at least one other player, and generally of all the players.

Hence, a player cannot simply optimize her own objective function independent of the choices of the other players. This results in a coupling between the actions of the players, and binds them together in decision making even in a noncooperative environment. If the players were able to enter into a cooperative agreement so that the selection of actions or decisions is done collectively and with full trust, so that all players would bene t to the extend possible, and no inef ciency would arise, then we would be in the realm of cooperative game theory, with issues of bargaining, coalition formation, excess utility distribution, etc.

of relevance there. Cooperative game theory will not be covered in this overview; see for example [67, 136, 188]. If no cooperation is allowed or possible among the players, then we are in the realm of non-cooperative game theory, where one has to introduce rst a satisfactory solution concept.

Leaving aside for the moment the issue of how the players can reach a satisfactory solution point, let us address the following question: if the players are at such a satisfactory solution point, what are the minimum features one would expect to see there To rst order, such a solution point should have the property that if all players. BIRT Reporting Tools Optimization, Game Theory, and Optimal & Robust Control but one st none none ay put, then the player who has the option of moving away from the solution point should not have any incentive to do so because she cannot improve her payoff. Note that we cannot allow two or more players to move collectively from the solution point, because such a collective move requires cooperation, which is not allowed in a non-cooperative game. Such a solution point where none of the players can improve her payoff by a unilateral move is known as a non-cooperative equilibrium or Nash equilibrium, named after John Nash, who introduced it and proved that it exists in nite games (that is games where each player has only a nite number of alternatives), some sixty years ago [124,125].

This is what we discuss below, following some terminology, a classi cation of non-cooperative games according to various attributes, and a mathematical formulation. We say that a non-cooperative game is nonzero sum if the sum of the players objective functions cannot be made zero even after appropriate positive scaling and/or translation that do not depend on the players decision variables. We say that a two-player game is zero sum if the sum of the objective functions of the two players is zero or can be made zero by appropriate positive scaling and translation that do not depend on the decision variables of the players.

If the two players objective functions add up to a constant (without scaling or translation), then the game is sometimes called constant sum, but according to our convention such games are also zero sum (since it can be converted to one). A game is a nite game if each player has only a nite number of alternatives, that is the players pick their actions out of nite sets (action sets); otherwise the game is an in nite game; nite games are also known as matrix games. An in nite game is said to be a continuous-kernel game if the action sets of the players are continua (continuums), and the players objective functions are continuous with respect to action variables of all players.

A game is said to be deterministic if the players actions uniquely determine the outcome, as captured in the objective functions, whereas if the objective function of at least one player depends on an additional variable (state of nature) with a known probability distribution, then we have a stochastic game. A game is a complete information game if the description of the game (that is, the players, the objective functions, and the underlying probability distributions if stochastic) is common information to all players; otherwise we have an incomplete information game. Finally, we say that a game is static if each player acts only once, and none of the players has access to information on the actions of any of the other players; otherwise what we have is a dynamic game.

A dynamic game is said to be a differential game if the evolution of the decision process (controlled by the players over time) takes place in continuous time, and generally involves a differential equation. In this section, we will be covering only static, deterministic, complete information non-cooperative games..


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