This is a mathematical book on game theory and, thus, starts with a def-
inition of game theory. This introduction also provides some preliminary
considerations about this field, which mainly lies in the boundary between
mathematics and economics. Remarkably, in recent years, the interest for
game theory has grown in utterly disparate disciplines such as psychology,
computer science, biology, and political science.
A definition of game theory. Game theory is the mathematical theory of
interactive decision situations. These situations are characterized by the
following elements: (a) there is a group of agents, (b) each agent has to
make a decision, (c) an outcome results as a function of the decisions of
all agents, and (d) each agent has his own preferences on the set of possible
outcomes. Robert J. Aumann, one of the most active researchers in the field,
said in an interview with Hart (2005): “game theory is optimal decision
making in the presence of others with different objectives”.
A particular collection of interactive decision situations are the so-called
parlor games. Game theory borrows the terminology used in parlor games
to designate the various elements that are involved in interactive decision
situations: the situations themselves are called games, the agents are called
players, their available decisions are called strategies, etc.
Classical game theory is an ideal normative theory, in the sense that it
prescribes, for every particular game, how rational players should behave.
By rational player we mean one who (a) knows what he wants, (b) has
the only objective of getting what he wants, and (c) is able to identify the
strategies that best fit his objective. More recently, a normative game the-
ory for bounded-rational players and even a pure descriptive game theory