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Aggregative Game
In game theory, an aggregative game is a game in which every player’s payoff is a function of the player’s own strategy and the aggregate of all players’ strategies. The concept was first proposed by Nobel laureate Reinhard Selten in 1970 who considered the case where the aggregate is the sum of the players' strategies. Definition Consider a standard non-cooperative game with ''n'' players, where S_i \subseteq \mathbb is the strategy set of player ''i'', S=S_1 \times S_2 \times \ldots \times S_n is the joint strategy set, and f_i:S \to \mathbb is the payoff function of player ''i''. The game is then called an ''aggregative game'' if for each player ''i'' there exists a function \tilde_i:S_i \times \mathbb \to \mathbb such that for all s \in S : : f_i(s)=\tilde_i \left( s_i,\sum_^n s_j \right) In words, payoff functions in aggregative games depend on players' ''own strategies'' and the ''aggregate'' \sum s_j. As an example, consider the Cournot model where firm ' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Game Theory
Game theory is the study of mathematical models of strategic interactions among rational agents. Myerson, Roger B. (1991). ''Game Theory: Analysis of Conflict,'' Harvard University Press, p.&nbs1 Chapter-preview links, ppvii–xi It has applications in all fields of social science, as well as in logic, systems science and computer science. Originally, it addressed two-person zero-sum games, in which each participant's gains or losses are exactly balanced by those of other participants. In the 21st century, game theory applies to a wide range of behavioral relations; it is now an umbrella term for the science of logical decision making in humans, animals, as well as computers. Modern game theory began with the idea of mixed-strategy equilibria in two-person zero-sum game 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 mathema ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Comparative Statics
In economics, comparative statics is the comparison of two different economic outcomes, before and after a change in some underlying exogenous variable, exogenous parameter. As a type of ''static analysis'' it compares two different economic equilibrium, equilibrium states, after the process of adjustment (if any). It does not study the motion towards equilibrium, nor the process of the change itself. Comparative statics is commonly used to study changes in supply and demand when analyzing a single Market (economics), market, and to study changes in monetary policy, monetary or fiscal policy when analyzing the whole macroeconomics, economy. Comparative statics is a tool of analysis in microeconomics (including general equilibrium analysis) and macroeconomics. Comparative statics was formalized by Sir John Richard Hicks, John R. Hicks (1939) and Paul A. Samuelson (1947) (Kehoe, 1987, p. 517) but was presented graphically from at least the 1870s. For models of stable equili ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cournot Competition
Cournot competition is an economic model used to describe an industry structure in which companies compete on the amount of output they will produce, which they decide on independently of each other and at the same time. It is named after Antoine Augustin Cournot (1801–1877) who was inspired by observing competition in a spring water duopoly. It has the following features: * There is more than one firm and all firms produce a homogeneous product, i.e., there is no product differentiation; * Firms do not cooperate, i.e., there is no collusion; * Firms have market power, i.e., each firm's output decision affects the good's price; * The number of firms is fixed; * Firms compete in quantities rather than prices; and * The firms are economically rational and act strategically, usually seeking to maximize profit given their competitors' decisions. An essential assumption of this model is the "not conjecture" that each firm aims to maximize profits, based on the expectation that its own ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bertrand Competition
Bertrand competition is a model of competition used in economics, named after Joseph Louis François Bertrand (1822–1900). It describes interactions among firms (sellers) that set prices and their customers (buyers) that choose quantities at the prices set. The model was formulated in 1883 by Bertrand in a review of Antoine Augustin Cournot's book ''Recherches sur les Principes Mathématiques de la Théorie des Richesses'' (1838) in which Cournot had put forward the Cournot model. Cournot's model argued that each firm should maximise its profit by selecting a quantity level and then adjusting price level to sell that quantity. The outcome of the model equilibrium involved firms pricing above marginal cost; hence, the competitive price. In his review, Bertrand argued that each firm should instead maximise its profits by selecting a price level that undercuts its competitors' prices, when their prices exceed marginal cost. The model was not formalized by Bertrand; however, the idea ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Nash Equilibria
In game theory, the Nash equilibrium, named after the mathematician John Nash, is the most common way to define the solution of a non-cooperative game involving two or more players. In a Nash equilibrium, each player is assumed to know the equilibrium strategies of the other players, and no one has anything to gain by changing only one's own strategy. The principle of Nash equilibrium dates back to the time of Cournot, who in 1838 applied it to competing firms choosing outputs. If each player has chosen a strategy an action plan based on what has happened so far in the game and no one can increase one's own expected payoff by changing one's strategy while the other players keep their's unchanged, then the current set of strategy choices constitutes a Nash equilibrium. If two players Alice and Bob choose strategies A and B, (A, B) is a Nash equilibrium if Alice has no other strategy available that does better than A at maximizing her payoff in response to Bob choosing B, and Bob ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Convex Set
In geometry, a subset of a Euclidean space, or more generally an affine space over the reals, is convex if, given any two points in the subset, the subset contains the whole line segment that joins them. Equivalently, a convex set or a convex region is a subset that intersects every line into a single line segment (possibly empty). For example, a solid cube is a convex set, but anything that is hollow or has an indent, for example, a crescent shape, is not convex. The boundary of a convex set is always a convex curve. The intersection of all the convex sets that contain a given subset of Euclidean space is called the convex hull of . It is the smallest convex set containing . A convex function is a real-valued function defined on an interval with the property that its epigraph (the set of points on or above the graph of the function) is a convex set. Convex minimization is a subfield of optimization that studies the problem of minimizing convex functions over convex se ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Quasiconcave
In mathematics, a quasiconvex function is a real-valued function defined on an interval or on a convex subset of a real vector space such that the inverse image of any set of the form (-\infty,a) is a convex set. For a function of a single variable, along any stretch of the curve the highest point is one of the endpoints. The negative of a quasiconvex function is said to be quasiconcave. All convex functions are also quasiconvex, but not all quasiconvex functions are convex, so quasiconvexity is a generalization of convexity. ''Univariate'' unimodal functions are quasiconvex or quasiconcave, however this is not necessarily the case for functions with multiple arguments. For example, the 2-dimensional Rosenbrock function is unimodal but not quasiconvex and functions with star-convex sublevel sets can be unimodal without being quasiconvex. Definition and properties A function f:S \to \mathbb defined on a convex subset S of a real vector space is quasiconvex if for all x, y \i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Strategic Complementarities
In economics and game theory, the decisions of two or more players are called strategic complements if they mutually reinforce one another, and they are called strategic substitutes if they mutually offset one another. These terms were originally coined by Bulow, Geanakoplos, and Klemperer (1985). To see what is meant by 'reinforce' or 'offset', consider a situation in which the players all have similar choices to make, as in the paper of Bulow et al., where the players are all imperfectly competitive firms that must each decide how much to produce. Then the production decisions are strategic complements if an increase in the production of one firm increases the marginal revenues of the others, because that gives the others an incentive to produce more too. This tends to be the case if there are sufficiently strong aggregate increasing returns to scale and/or the demand curves for the firms' products have a sufficiently low own-price elasticity. On the other hand, the production de ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Quasiconcavity
In mathematics, a quasiconvex function is a real number, real-valued function (mathematics), function defined on an interval (mathematics), interval or on a convex set, convex subset of a real vector space such that the inverse image of any set of the form (-\infty,a) is a convex set. For a function of a single variable, along any stretch of the curve the highest point is one of the endpoints. The negative of a quasiconvex function is said to be quasiconcave. All convex functions are also quasiconvex, but not all quasiconvex functions are convex, so quasiconvexity is a generalization of convexity. ''Univariate'' Unimodality, unimodal functions are quasiconvex or quasiconcave, however this is not necessarily the case for functions with multiple argument of a function, arguments. For example, the 2-dimensional Rosenbrock function is unimodal but not quasiconvex and functions with Star_domain, star-convex sublevel sets can be unimodal without being quasiconvex. Definition and pr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Reinhard Selten
Reinhard Justus Reginald Selten (; 5 October 1930 – 23 August 2016) was a German economist, who won the 1994 Nobel Memorial Prize in Economic Sciences (shared with John Harsanyi and John Nash). He is also well known for his work in bounded rationality and can be considered one of the founding fathers of experimental economics. Biography Selten was born in Breslau (Wrocław) in Lower Silesia, now in Poland, to a Jewish father, Adolf Selten (blind bookseller; d. 1942Roberts, Sam"Reinhard Selten, Whose Strides in Game Theory Led to a Nobel, Dies at 85" New York ''Times'', September 2, 2016. Retrieved 2016-09-03.), and Protestant mother, Käthe Luther.O'Connor, J J, and E F Robertson"Reinhard Selten" ''www-history.mcs.st-and.ac.uk'', November 2010. Retrieved 2016-09-03. Reinhard Selten was raised as Protestant. After a brief family exile in Saxony and Austria, Selten returned to Hesse, Germany after the war and, in high school, read an article in Fortune magazine about ga ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Nash Equilibrium
In game theory, the Nash equilibrium, named after the mathematician John Nash, is the most common way to define the solution of a non-cooperative game involving two or more players. In a Nash equilibrium, each player is assumed to know the equilibrium strategies of the other players, and no one has anything to gain by changing only one's own strategy. The principle of Nash equilibrium dates back to the time of Cournot, who in 1838 applied it to competing firms choosing outputs. If each player has chosen a strategy an action plan based on what has happened so far in the game and no one can increase one's own expected payoff by changing one's strategy while the other players keep their's unchanged, then the current set of strategy choices constitutes a Nash equilibrium. If two players Alice and Bob choose strategies A and B, (A, B) is a Nash equilibrium if Alice has no other strategy available that does better than A at maximizing her payoff in response to Bob choosing B, and Bob ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mean Field Game Theory
Mean-field game theory is the study of strategic decision making by small interacting agents in very large populations. It lies at the intersection of game theory with stochastic analysis and control theory. The use of the term "mean field" is inspired by mean-field theory in physics, which considers the behavior of systems of large numbers of particles where individual particles have negligible impacts upon the system. In other words, each agent acts according to his minimization or maximization problem taking into account other agents’ decisions and because their population is large we can assume the number of agents goes to infinity and a representative agent exists. In traditional game theory, the subject of study is usually a game with two players and discrete time space, and extends the results to more complex situations by induction. However, for games in continuous time with continuous states (differential games or stochastic differential games) this strategy cannot be us ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
