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Game theory is a branch of applied mathematics that uses models
to study interactions with formalised incentive structures ("games").
Unlike decision theory, which also studies formalized incentive
structures, game theory encompasses decisions that are made in an
environment where various players interact strategically. In other
words, game theory studies choice of optimal behavior when costs
and benefits of each option are not fixed, but depend upon the future
choices of other individuals. It has applications in a variety of
fields, including economics, international relations, evolutionary
biology, political science, and military strategy. Game theorists
study the predicted and actual behaviour of individuals in games,
as well as optimal strategies. Seemingly different types of interactions
can exhibit similar incentive structures, thus all exemplifying
one particular game.
John von Neumann and Oskar Morgenstern first formalised the subject
in 1944 in their book Theory of Games and Economic Behavior. Game
theory has important applications in fields like operations research,
economics, collective action, political science, psychology, and
biology. It has close links with economics in that it seeks to find
rational strategies in situations where the outcome depends not
only on one's own strategy and "market conditions", but
upon the strategies chosen by other players with possibly different
or overlapping goals. Applications in military strategy drove some
of the early development of game theory.
Game theory has come to play an increasingly important role in
logic and in computer science. Several logical theories have a basis
in game semantics. And computer scientists have used games to model
interactive computations. Computability logic attempts to develop
a comprehensive formal theory (logic) of interactive computational
tasks and resources, formalising these entities as games between
a computing agent and its environment.
Game theoretic analysis can apply to simple games of entertainment
or to more significant aspects of life and society. The prisoner's
dilemma, as popularized by mathematician Albert W. Tucker, furnishes
an example of the application of game theory to real life; it has
many implications for the nature of human co-operation, and has
even been used as the basis of a game show called Friend or Foe?.
Biologists have used game theory to understand and predict certain
outcomes of evolution, such as the concept of evolutionarily stable
strategy introduced by John Maynard Smith and George R. Price in
a 1973 paper in Nature (See also Maynard Smith 1982). See also evolutionary
game theory and behavioral ecology. Analysts of games commonly use
other branches of mathematics, in particular probability, statistics
and linear programming, in conjunction with game theory.
Mathematical definitions:
There are a few alternative definitions of the notion of a 'game'.
Normal form game design:
A game in normal or strategic form combines the set of possible
strategies for each player and records the payoffs for each outcome.
Let N be a set of players. For each player there is given a set
of strategies
. The game is then a function:
So that, if one knows the tuple of strategies that were chosen by
the players, one is given the allocation payments, a real number
assignment. A further generalization can be achieved by splitting
the game into two functions: the normal form game, describing the
way in which strategies define outcomes, and a second function depicting
player's preferences on the set of outcomes. Hence:
Where is the outcome set of the game. And for each player there
is a preference function.
A reduced normal form exists as well. The reduced normal form combines
strategies for which are associated with the same payoffs.
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