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Game theory studies the decision making of interacting players and has
applications in economics, political science, finance, international
relations, and sociology. In a typical problem of game theory two or
more participants, called players, make decisions (choose strategies)
in a conflicting or competitive situation and receive a payoff, which
not only depends on his own decision, but also on those of the other
players. Non-cooperative game theory discusses the behaviour of a
player seeking to maximise his or her own gain. In certain
models there are strategies which are optimal or represent
equilibria in some well-defined sense. Mathematical game theory
explores the nature of such solutions to the game, give proofs of
existence, universal bounds, algorithms, etc. for any size and
realisation of the game.
In many situations of interest the game is characterised by a large number of possible strategies and complicated relationships between the strategic choices of the players and the resulting payoff to each player. It is thus tempting to model the payoffs by a random function and try to find typical properties of a solution of a game, which in the limit of an infinitely large game are realized with probability one. In this context, the properties of the game are encoded not in the payoff matrices but in the probability distribution of the payoff matrices. |