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Game theory is a branch of mathematical
analysis developed to study decision making in conflict situations. Such a
situation exists when two or more decision makers who have different
objectives act on the same system or share the same resources. Game theory
provides a mathematical process for selecting an OPTIMUM STRATEGY
(that is, an optimum decision or a sequence of decisions) in the face of an
opponent who has a strategy of his own. The following assumptions hold true
in game theory:
(1) Each decision maker has available to him two or more
well-specified choices or sequences of choices
(2) Every possible combination of plays available to the
players leads to a well-defined end-state (win, loss, or draw) that
terminates the game.
(3) A specified payoff for each player is associated with
each end-state; a ZERO-SUM game means that the sum of payoffs to all
players is zero in each end-state.
(4) Each decision maker has perfect knowledge of the game
and of his opposition; that is, he knows in full detail the rules of the
game as well as the payoffs of all other players.
(5) All decision makers are rational; that is, each
player, given two alternatives, will select the one that yields him the
greater payoff.
To use the governing dynamics scene to
motivate an analysis of Nash equilibria in a non-cooperative game, consider
John's four friends as the only players in the bar. Assuming that each
player does what is best for himself regardless of what is best for the
group, John says: "If we all go for the blonde, we block each other and not
a single one of us is going to get her.” This is an example of competitive
behavior resulting in a non-optimal outcome.
John then suggests that "no one goes for
the blonde" so that they will all end up with the brunettes—behavior based
on cooperation rather than competition. Altruistic behavior can therefore
be shown as resulting in the optimal outcome.
However, Martin suspects that John is
only saying this in order to sidetrack his friends and pair himself with the
blonde, thus the friends suspect each other of cheating and will all once
again engage in non-cooperative behavior leading to a suboptimal outcome
(the ‘prisoners’ dilemma).
Using a Nash
box, present the friends’ problem in visual terms.
Thanks to Lynn Butler
Department of Mathematics
Haverford College, Pennsylvania
http://www.haverford.edu/math/lbutler/maths-illustrated.html |