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Convex Cooperative Games
In game theory, a cooperative game (or coalitional game) is a game with competition between groups of players ("coalitions") due to the possibility of external enforcement of cooperative behavior (e.g. through contract law). Those are opposed to non-cooperative games in which there is either no possibility to forge alliances or all agreements need to be self-enforcing (e.g. through credible threats). Cooperative games are often analysed through the framework of cooperative game theory, which focuses on predicting which coalitions will form, the joint actions that groups take and the resulting collective payoffs. It is opposed to the traditional non-cooperative game theory which focuses on predicting individual players' actions and payoffs and analyzing Nash equilibria. Cooperative game theory provides a high-level approach as it only describes the structure, strategies and payoffs of coalitions, whereas non-cooperative game theory also looks at how bargaining procedures will ... [...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 mathem ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Superadditive
In mathematics, a function f is superadditive if f(x+y) \geq f(x) + f(y) for all x and y in the domain of f. Similarly, a sequence \left\, n \geq 1, is called superadditive if it satisfies the inequality a_ \geq a_n + a_m for all m and n. The term "superadditive" is also applied to functions from a boolean algebra to the real numbers where P(X \lor Y) \geq P(X) + P(Y), such as lower probabilities. Properties If f is a superadditive function, and if 0 is in its domain, then f(0) \leq 0. To see this, take the inequality at the top: f(x) \leq f(x+y) - f(y). Hence f(0) \leq f(0+y) - f(y) = 0. The negative of a superadditive function is subadditive. Fekete's lemma The major reason for the use of superadditive sequences is the following lemma due to Michael Fekete. :Lemma: (Fekete) For every superadditive sequence \left\, n \geq 1, the limit \lim a_n/n is equal to \sup a_n/n. (The limit may be positive infinity, for instance, for the sequence a_n = \log n!.) For example ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Superadditivity
In mathematics, a function f is superadditive if f(x+y) \geq f(x) + f(y) for all x and y in the domain of f. Similarly, a sequence \left\, n \geq 1, is called superadditive if it satisfies the inequality a_ \geq a_n + a_m for all m and n. The term "superadditive" is also applied to functions from a boolean algebra to the real numbers where P(X \lor Y) \geq P(X) + P(Y), such as lower probabilities. Properties If f is a superadditive function, and if 0 is in its domain, then f(0) \leq 0. To see this, take the inequality at the top: f(x) \leq f(x+y) - f(y). Hence f(0) \leq f(0+y) - f(y) = 0. The negative of a superadditive function is subadditive. Fekete's lemma The major reason for the use of superadditive sequences is the following lemma due to Michael Fekete. :Lemma: (Fekete) For every superadditive sequence \left\, n \geq 1, the limit \lim a_n/n is equal to \sup a_n/n. (The limit may be positive infinity, for instance, for the sequence a_n = \log n!.) For example, f ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Supermodularity
In mathematics, a function :f\colon \mathbb^k \to \mathbb is supermodular if : f(x \uparrow y) + f(x \downarrow y) \geq f(x) + f(y) for all x, y \isin \mathbb^, where x \uparrow y denotes the componentwise maximum and x \downarrow y the componentwise minimum of x and y. If −''f'' is supermodular then ''f'' is called submodular, and if the inequality is changed to an equality the function is modular. If ''f'' is twice continuously differentiable, then supermodularity is equivalent to the condition : \frac \geq 0 \mbox i \neq j. Supermodularity in economics and game theory The concept of supermodularity is used in the social sciences to analyze how one agent's decision affects the incentives of others. Consider a symmetric game with a smooth payoff function \,f defined over actions \,z_i of two or more players i \in . Suppose the action space is continuous; for simplicity, suppose each action is chosen from an interval: z_i \in ,b/math>. In this context, supermodularity ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Submodular
In mathematics, a submodular set function (also known as a submodular function) is a set function whose value, informally, has the property that the difference in the incremental value of the function that a single element makes when added to an input set decreases as the size of the input set increases. Submodular functions have a natural diminishing returns property which makes them suitable for many applications, including approximation algorithms, game theory (as functions modeling user preferences) and electrical networks. Recently, submodular functions have also found immense utility in several real world problems in machine learning and artificial intelligence, including automatic summarization, multi-document summarization, feature selection, active learning, sensor placement, image collection summarization and many other domains. Definition If \Omega is a finite set, a submodular function is a set function f:2^\rightarrow \mathbb, where 2^\Omega denotes the power set o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Supermodular
In mathematics, a function :f\colon \mathbb^k \to \mathbb is supermodular if : f(x \uparrow y) + f(x \downarrow y) \geq f(x) + f(y) for all x, y \isin \mathbb^, where x \uparrow y denotes the componentwise maximum and x \downarrow y the componentwise minimum of x and y. If −''f'' is supermodular then ''f'' is called submodular, and if the inequality is changed to an equality the function is modular. If ''f'' is twice continuously differentiable, then supermodularity is equivalent to the condition : \frac \geq 0 \mbox i \neq j. Supermodularity in economics and game theory The concept of supermodularity is used in the social sciences to analyze how one agent's decision affects the incentives of others. Consider a symmetric game with a smooth payoff function \,f defined over actions \,z_i of two or more players i \in . Suppose the action space is continuous; for simplicity, suppose each action is chosen from an interval: z_i \in ,b/math>. In this context, supermodularity ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lloyd Shapley
Lloyd Stowell Shapley (; June 2, 1923 – March 12, 2016) was an American mathematician and Nobel Prize-winning economist. He contributed to the fields of mathematical economics and especially game theory. Shapley is generally considered one of the most important contributors to the development of game theory since the work of von Neumann and Morgenstern. With Alvin E. Roth, Shapley won the 2012 Nobel Memorial Prize in Economic Sciences "for the theory of stable allocations and the practice of market design." Life and career Lloyd Shapley was born on June 2, 1923, in Cambridge, Massachusetts, one of the sons of astronomers Harlow Shapley and Martha Betz Shapley, both from Missouri. He attended Phillips Exeter Academy and was a student at Harvard when he was drafted in 1943. He served in the United States Army Air Corps in Chengdu, China and received the Bronze Star decoration for breaking the Soviet weather code. After the war, Shapley returned to Harvard and graduated wi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bondareva–Shapley Theorem
The Bondareva–Shapley theorem, in game theory, describes a necessary and sufficient condition for the non-emptiness of the core of a cooperative game in characteristic function form. Specifically, the game's core is non-empty if and only if the game is ''balanced''. The Bondareva–Shapley theorem implies that market games and convex games have non-empty cores. The theorem was formulated independently by Olga Bondareva and Lloyd Shapley in the 1960s. Theorem Let the pair \langle N, v\rangle be a cooperative game in characteristic function form, where N is the set of players and where the ''value function'' v: 2^N \to \mathbb is defined on N's power set In mathematics, the power set (or powerset) of a set is the set of all subsets of , including the empty set and itself. In axiomatic set theory (as developed, for example, in the ZFC axioms), the existence of the power set of any set is post ... (the set of all subsets of N). The core of \langle N, v \rangle is n ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Imputation (game Theory)
In fully cooperative games players will opt to form coalitions when the value of the payoff is equal to or greater than if they were to work alone. The focus of the game is to find acceptable distributions of the payoff of the grand coalition. Distributions where a player receives less than it could obtain on its own, without cooperating with anyone else, are unacceptable - a condition known as ''individual rationality''. Imputations are distributions that are efficient and are individually rational. Theory For 2-player games the set of imputations coincides with the core, a popularly studied concept due to its stability against group deviations. The core is a solution concept of cooperative games Cooperative game may refer to: * Cooperative board game, board games in which players work together to achieve a common goal * Cooperative game theory In game theory, a cooperative game (or coalitional game) is a game with competition between gro ... and consists of multiple imputatio ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Rice's Theorem
In computability theory, Rice's theorem states that all non-trivial semantic properties of programs are undecidable. A semantic property is one about the program's behavior (for instance, does the program terminate for all inputs), unlike a syntactic property (for instance, does the program contain an if-then-else statement). A property is ''non-trivial'' if it is neither true for every partial computable function, nor false for every partial computable function. Rice's theorem can also be put in terms of functions: for any non-trivial property of partial functions, no general and effective method can decide whether an algorithm computes a partial function with that property. Here, a property of partial functions is called ''trivial'' if it holds for all partial computable functions or for none, and an effective decision method is called ''general'' if it decides correctly for every algorithm. The theorem is named after Henry Gordon Rice, who proved it in his doctoral dissertation ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Nakamura Number
In cooperative game theory and social choice theory, the Nakamura number measures the degree of rationality of preference aggregation rules (collective decision rules), such as voting rules. It is an indicator of the extent to which an aggregation rule can yield well-defined choices. *If the number of alternatives (candidates; options) to choose from is less than this number, then the rule in question will identify "best" alternatives without any problem. In contrast, *if the number of alternatives is greater than or equal to this number, the rule will fail to identify "best" alternatives for some pattern of voting (i.e., for some profile (tuple) of individual preferences), because a voting paradox will arise (a ''cycle'' generated such as alternative a socially preferred to alternative b, b to c, and c to a). The larger the Nakamura number a rule has, the greater the number of alternatives the rule can rationally deal with. For example, since (except in the case of four individuals ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dictator Game
The dictator game is a popular experimental instrument in social psychology and economics, a derivative of the ultimatum game. The term "game" is a misnomer because it captures a decision by a single player: to send money to another or not. Thus, the dictator has the most power and holds the preferred position in this “game.” Although the “dictator” has the most power and presents a take it or leave it offer, the game has mixed results based on different behavioral attributes. The results – where most "dictators" choose to send money – evidence the role of fairness and norms in economic behavior, and undermine the assumption of narrow self-interest when given the opportunity to maximise one's own profits. Description The dictator game is a derivative of the ultimatum game, in which one player (the proposer) provides a one-time offer to the other (the responder). The responder can choose to either accept or reject the proposer's bid, but rejecting the bid would result i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
