A Brief History of Game Theory

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Transcript A Brief History of Game Theory

A Brief History of Game Theory
From various sources
The first known
solution to such a twoperson zero-sum game
is contained in
a letter dated 13
November 1713 from
James Waldegrave to
Pierre-Remond de
Montmort concerning a
two player version of
the card game le Her.
De Montmort
wrote to the
mathematician
Nicolas Bernoulli
(perhaps best known,
at least among
economists, for his
formulation of the St.
Petersburg Paradox).
The great French
mathematician and
probabilist, Emile
Borel published four
notes on strategic
games between 1921
and 1927.
Borel gave the first formal definition of what
we shall call a mixed strategy and
demonstrated the existence of the minimax
solution to two player zero-sum games with
either three or five possible
strategies. He initially conjectured that games
with more strategies would not have such a
solution but, being unable to find a counter
example, he later considered that to be an
open question.
That question was
answered by the
Hungarian (and later
American)
mathematician John
von Neumann in 1928.
Wikipedia describes von Neumann as having
“made major contributions to a vast range of
fields, including set theory, functional analysis,
quantum mechanics, ergodic theory,
continuous geometry, economics and game
theory, computer science, numerical analysis,
hydrodynamics (of explosions), and statistics, as
well as many other mathematical fields.”
In 1913 the German
mathematician Ernst
Zermelo, most famous for his
axiomatisation of set theory,
gave a statement concerning
a result about what we
would now call extensive
form games. In a game like
chess either white can
guarantee that he wins or
black can guarantee that he
wins or both players can
guarantee at least a draw.
There were a number of
substantial advances in the
book by von Neumann and
Morgenstern: the axiomatic
development of the theory of
expected utility; the formal
definition of normal form
games and extensive form
games; the elaboration of the
minmax theorem for twoperson zero sum games; and
the definition of what
are now called cooperative or
coalitional games.
In the early 1950s John
Nash [1950,1951]
proposed a definition of
equilibrium, that we
now call the Nash
equilibrium, that has
become the central
solution concept for
noncooperative game
theory.
Nash as an older
man, after he won
the Nobel Prize in
Economics.
That’s not Nash.
That’s Russel
Crowe in one of his
best performances
in a film about
Nash’s life.
1994 Nobel Prize in Economics was award to
John Nash, John C. Harsanyi and Reinhard Selten
"for their pioneering analysis of equilibria in the
theory of non-cooperative games".
In two papers,
Extensive Games (1950)
and Extensive Games
and the Problem of
Information (1953), H.
W. Kuhn included the
formulation of
extensive form games
which is currently used,
and also some basic
theorems pertaining to
this class of games.
Lloyd Shapley in his
paper A Value for NPerson Games
characterised, by a set
of axioms, a solution
concept that associates
with each coalitional
game,v, a unique outcome, v. This solution in
now known as the
Shapley Value.
In the same year, 1953, Shapley published a
paper on “Stochastic Games,” introducing a
new and important model of games.
In 2012, Shapley
won the Nobel Prize
in Economics with Al
Roth "for the theory
of stable allocations
and the practice of
market design".
Robert J. Aumann's
greatest contribution
was in the realm of
repeated games, which
are situations in which
players encounter the
same situation over
and over again.
Aumann was the first to define the concept of
correlated equilibrium in game theory, which
is a type of equilibrium in non-cooperative
games that is more flexible than the classical
Nash equilibrium. Furthermore, Aumann has
introduced the first purely formal account of
the notion of common knowledge in game
theory. He collaborated with Lloyd Shapley on
the Aumann-Shapley value. He is also known
for his agreement theorem, in which he
argues that under his given conditions, two
Bayesian rationalists with common prior
beliefs cannot agree to disagree.
In 2005 Aumann won
the Nobel Prize in
Economics with
Thomas C. Schelling
"for having enhanced
our understanding of
conflict and
cooperation through
game-theory analysis".