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In game theory, the Nash equilibrium is the most commonly-used solution concept for non-cooperative games. A Nash equilibrium is a situation where no player could gain by changing their own strategy. The idea of Nash equilibrium dates back to the time of Cournot, who in 1838 applied it to his model of competition in an oligopoly.
In the mathematical areas of order and lattice theory, the Knaster–Tarski theorem, named after Bronisław Knaster and Alfred Tarski, states the following:
In graph theory, a vertex cover of a graph is a set of vertices that includes at least one endpoint of every edge of the graph.
The Gödel Prize is an annual prize for outstanding papers in the area of theoretical computer science, given jointly by the European Association for Theoretical Computer Science (EATCS) and the Association for Computing Machinery Special Interest Group on Algorithms and Computational Theory. The award is named in honor of Kurt Gödel. Gödel's connection to theoretical computer science is that he was the first to mention the "P versus NP" question, in a 1956 letter to John von Neumann in which Gödel asked whether a certain NP-complete problem could be solved in quadratic or linear time.
In game theory, folk theorems are a class of theorems describing an abundance of Nash equilibrium payoff profiles in repeated games. The original Folk Theorem concerned the payoffs of all the Nash equilibria of an infinitely repeated game. This result was called the Folk Theorem because it was widely known among game theorists in the 1950s, even though no one had published it. Friedman's (1971) Theorem concerns the payoffs of certain subgame-perfect Nash equilibria (SPE) of an infinitely repeated game, and so strengthens the original Folk Theorem by using a stronger equilibrium concept: subgame-perfect Nash equilibria rather than Nash equilibria.
In game theory, a correlated equilibrium is a solution concept that is more general than the well known Nash equilibrium. It was first discussed by mathematician Robert Aumann in 1974. The idea is that each player chooses their action according to their private observation of the value of the same public signal. A strategy assigns an action to every possible observation a player can make. If no player would want to deviate from their strategy, the distribution from which the signals are drawn is called a correlated equilibrium.
In computational complexity theory, Polynomial Local Search (PLS) is a complexity class that models the difficulty of finding a locally optimal solution to an optimization problem. The main characteristics pls take it
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In game theory, the purification theorem was contributed by Nobel laureate John Harsanyi in 1973. The theorem justifies a puzzling aspect of mixed strategy Nash equilibria: each player is wholly indifferent between each of the actions he puts non-zero weight on, yet he mixes them so as to make every other player also indifferent.
Quantal response equilibrium (QRE) is a solution concept in game theory. First introduced by Richard McKelvey and Thomas Palfrey, it provides an equilibrium notion with bounded rationality. QRE is not an equilibrium refinement, and it can give significantly different results from Nash equilibrium. QRE is only defined for games with discrete strategies, although there are continuous-strategy analogues.
In computer science, PPAD is a complexity class introduced by Christos Papadimitriou in 1994. PPAD is a subclass of TFNP based on functions that can be shown to be total by a parity argument. The class attracted significant attention in the field of algorithmic game theory because it contains the problem of computing a Nash equilibrium: this problem was shown to be complete for PPAD by Daskalakis, Goldberg and Papadimitriou with at least 3 players and later extended by Chen and Deng to 2 players.
In game theory, an epsilon-equilibrium, or near-Nash equilibrium, is a strategy profile that approximately satisfies the condition of Nash equilibrium. In a Nash equilibrium, no player has an incentive to change his behavior. In an approximate Nash equilibrium, this requirement is weakened to allow the possibility that a player may have a small incentive to do something different. This may still be considered an adequate solution concept, assuming for example status quo bias. This solution concept may be preferred to Nash equilibrium due to being easier to compute, or alternatively due to the possibility that in games of more than 2 players, the probabilities involved in an exact Nash equilibrium need not be rational numbers.
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Congestion games (CG) are a class of games in game theory. They represent situations which commonly occur in roads, communication networks, oligopoly markets and natural habitats. There is a set of resources ; there are several players who need resources ; each player chooses a subset of these resources ; the delay in each resource is determined by the number of players choosing a subset that contains this resource. The cost of each player is the sum of delays among all resources he chooses. Naturally, each player wants to minimize his own delay; however, each player's choices impose a negative externality on the other players, which may lead to inefficient outcomes.
In algorithmic game theory, a succinct game or a succinctly representable game is a game which may be represented in a size much smaller than its normal form representation. Without placing constraints on player utilities, describing a game of players, each facing strategies, requires listing utility values. Even trivial algorithms are capable of finding a Nash equilibrium in a time polynomial in the length of such a large input. A succinct game is of polynomial type if in a game represented by a string of length n the number of players, as well as the number of strategies of each player, is bounded by a polynomial in n.
In theoretical computer science, the term isolation lemma refers to randomized algorithms that reduce the number of solutions to a problem to one, should a solution exist. This is achieved by constructing random constraints such that, with non-negligible probability, exactly one solution satisfies these additional constraints if the solution space is not empty. Isolation lemmas have important applications in computer science, such as the Valiant–Vazirani theorem and Toda's theorem in computational complexity theory.
The Lemke–Howson algorithm is an algorithm that computes a Nash equilibrium of a bimatrix game, named after its inventors, Carlton E. Lemke and J. T. Howson. It is said to be "the best known among the combinatorial algorithms for finding a Nash equilibrium", although more recently the Porter-Nudelman-Shoham algorithm has outperformed on a number of benchmarks.
Fisher market is an economic model attributed to Irving Fisher. It has the following ingredients:
Guoqiang Tian is a Chinese-American economist. He is the Alfred F. Chalk Professor of Economics at Texas A&M University. He is Honorary Dean of Institute for Advanced Research at Shanghai University of Finance and Economics.
In graph theory, a vertex cover in a hypergraph is a set of vertices, such that every hyperedge of the hypergraph contains at least one vertex of that set. It is an extension of the notion of vertex cover in a graph.
References
- Papadimitriou, Christos (1994). "On the Complexity of the Parity Argument and Other Inefficient Proofs of Existence". Journal of Computer and System Sciences. 48 (3): 498–532. CiteSeerX 10.1.1.321.7008 . doi: 10.1016/S0022-0000(05)80063-7 . Paper available online at Papadimitriou's Homepage.
- C. Daskalakis, P. W. Goldberg and C.H. Papadimitriou (2009). "The Complexity of Computing a Nash Equilibrium". SIAM Journal on Computing . 39 (3): 195–259. CiteSeerX 10.1.1.68.6111 . doi:10.1137/070699652.
- Xi Chen and Xiaotie Deng (2006). "Settling the complexity of two-player Nash equilibrium". Proc. 47th FOCS. pp. 261–272.