Transcript The early applications of game theory to economics from

Eshet 2010, Amsterdam, March 25-28
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“The Holy Grail is a solution concept which, for every
game, picks out one and only one combination of
strategies as the solution. The Holy Grail of game
theory is the uniqueness of game solution. This
objective was not pursued by Nash, who was interested
only to prove that all games have a solution (the
existence problem). This objective was pursued by
Shapley before and Harsanyi after, who find an unique
solution to cooperative games and then by HarsanyiSelten (1988) with equilibrium refinements.”
Sudgen (2001)
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Von Neumann and Morgenstern assessed the
multiplicity of solutions “not something to run from but
rather to embrace” (Schotter 1992)
Nash before - and Harsanyi after – aimed at finding a
unique solution to cooperative and to non-cooperative
games through the axiom of symmetry
Shubik was a key actor in moving economic applications
of game theory from TGEB towards Nash-Harsanyi’s
approach
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Game theory was moulded such as “to make it accessible
as a research vehicle only to mathematicians” (Luce and
Raiffa 1957)
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interdisciplinary vs. economic-oriented method
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neutral tool vs. supportive/critical of neoclassical
economics
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cooperative vs. non-cooperative
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equilibrium-based vs. disequilibrium analysis
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solution multiplicity vs. uniqueness
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All these issues were already patently pointed out in the
way vision of the co-founder of game theory, Oskar
Morgenstern, was received and then abandoned by his
student Martin Shubik
Shubik took the first steps on the path leading the
application of game theory
 from cooperative to non-cooperative approach
 from disequilibrium to equilibrium analysis
 from solutions multiplicity to uniqueness
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The inadequacy of the assumption of maximizing
behaviour, whose removal undermined the metaphysics of
neoclassical economics
No straightforward principle of social rationality in games
as constrained maximization is for individual decisionmaking
Game theory as the foundation of a social and
interdisciplinary theory of strategic interaction based on
“live” variables and disequilibrium analysis (not on Hicks’
“dead” variables and perfect foresight)
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To solve a game did not mean to determine how
opposing maximizing choices could be balanced by
means of assumptions on others’ behaviour
Solutions are multiple and often indeterminate as the
outcome of the process of coalition formation
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The word equilibrium is practically expunged from TGEB
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Cooperative games were offered as a new foundation
for a social science such as economics
“All these considerations illustrate once more what a
complexity of theoretical forms must be expected in social
theory. Our static analysis alone necessitated the creation
of a conceptual and formal mechanism which is very
different from anything used, for instance, in mathematical
physics. Thus the conventional view of a solution as a
uniquely defined number or aggregate of numbers was
seen to be too narrow for our purposes, in spite of its
success in other fields. The emphasis on mathematical
methods seems to be shifted more towards combinatorics
and set theory - and away from the algorithm of
differential equations which dominate mathematical
physics.” (von Neumann and Morgenstern 1947, p. 45)
“It is my belief that von Neumann was even more committed than
Morgenstern to the idea of a solution as a set of imputations. He
felt that it was premature to consider solutions which picked out
a single point and he did not like noncooperative equilibrium
solution. In a personal conversation with von Neumann (on the
train from New York) to Princeton in 1952) I recall suggesting
that I thought that Nash’s non-cooperative
equilibrium solution
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theory might be of considerable value in applications to
economics. He indicated that he did not particularly like the
Nash solution and that a cooperative theory made more social
sense. Albert Tucker, in a personal conversation, informed me
that in his conversations with von Neumann, he had displayed
somewhat the same attitude to the single point solution, the
value, proposed by Lloyd Shapley.” (Shubik 1992)
“We give two independent derivations of our solution to the
two-person cooperative game. In the first, the cooperative
game is reduced to a non-cooperative game. To do this, one
makes the players’ steps of negotiation in the cooperative
game become moves in non-cooperative model. (…) The
second approach is by the axiomatic
method. One states as
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axioms several properties that it would seem natural for the
solution to have and then one discovers that the axioms
actually determine the solution uniquely. The two approaches
to the problem, via the negotiation model or via the axioms,
are complementary; each helps to justify and clarify the other”
(Nash 1953, p. 129)
What is a “rational” prediction for Nash in playing a game?
“Starting from the principle that a rational prediction should
be unique, that the players should be able to deduce and
make use of it, and that such knowledge on the part of each
player of what to expect the others to do should not lead him
to act out of conformity with the
prediction, one is led to the
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concept of a solution defined before.” [Nash Ph.D.Th. 1952]
The axiomatic method is described as follows: “One states as
axioms several properties that it would seem natural for the
solution to have and then one discovers that the axioms
actually determine the solution uniquely.” (Nash 1953)
Nash (1950) and (1953) give the axiomatic foundation of
bargaining theory: 1) Independence of utility scales’
calibration, 2) Maximization of individual utility, 3) Pareto
efficiency, 4) Independence of Irrelevant Alternatives, 5)
Symmetry
The symmetry axiom: The bargaining solution does not
depend on who is labelled player I and player II. If the players’
labels are reversed, each will still get the same payoff
The axiom “expresses equality of bargaining skill” (Nash 1950)
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“The symmetry axiom, Axiom IV, says that the only significant
(in determining the value of the game) differences between the
players are those which are included in the mathematical
description of the game, which includes their different sets of
strategies and utility functions. One may think of Axiom IV as
requiring the players to be intelligent
and rational beings. But
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we think it is a mistake to regard this as expressing ‘equal
bargaining ability’ of the players, in spite of a statement to
this effect in ‘The Bargaining Problem’. With people who are
sufficiently intelligent and rational there should not be any
question of ‘bargaining ability’, a term which suggests
something like skill in duping the other fellow.” (pp. 137-8)
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“The bargaining parties will follow identical (symmetric)
rules of behaviour (whether because they follow the
same principles of rational behaviour or because they
are subject to the same psychological laws).”
“This postulate is here equivalent to the assumption
which we would express on the common-sense level by
saying that each party will make a concession at a given
stage of the negotiations if and only if he thinks he has
at least as much “reason” as his opponent has to yield
ground at that point.”
Harsanyi (1956)
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Harsanyi supports Nash’s model because it
determines a unique solution to the bargaining game.
The symmetry axiom makes “possible to define a
unique rational solution in terms of the directly
relevant independent variables alone. Consequently,
rational players can have no reasonable ground for
making their bargaining strategies dependent on
other variables intrinsically irrelevant for utility
maximization.” (Harsanyi 1961, p. 190)
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To emphasize the early achievement of unique solutions
as primary goal
To introduce various devices designed to suppress
strategic interaction in the basic assumptions of their
models.
The symmetry assumption permit each player to make
fully accurate predictions of the (probable) choices of
the others
The consequence of symmetry is to give players perfect
information along all dimensions, even though perfect
information is not introduced as an explicit assumption.
1949 (Fall) Arrival to Princeton
1952 GT applications to business cycle Econometrica
1952 GT to information theory JPE
1953 (June) Ph.D. discussion
1953 GT to duopoly (with Nash-Mayberry) Econometrica
1954 GT to information theory
QJE
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1955 (August) Center for Advanced Study in Behavioral
Sciences (CASBS), Stanford
1956 GT to management science Management Science
1959 “Strategy and Market Structure”
1959 Edgeworth market games (equivalence between
Edgeworth’s contract curve and the core)
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1953 Ph.D. Thesis was largely representative of
Morgenstern’s view
Cooperative games are more useful than noncooperative games for economics
Solution are settled by the “standards of society” based
on some form of cooperation (explicit or implicit)
The core of Shubik’s thesis is the analysis of
disequilibrium processes with no perfect foresight and
non-homogeneous beliefs (Morgenstern 1935)
“The nature of a non-cooperative equilibrium point as a
static phenomenon which has an objective existence of
its own, but which may never be attained by the
individuals in the economy is stressed. The possibility
that many economic situations may not inherently
possess any equilibrium state is discussed.” (Shubik’s
PhD Thesis 1953 p. 151)
“How can the player maximize his own ends? He is not
forced with a simple maximization problem of the
variety that appears so often in mechanics. He must
make his way as best he is able in the face of the
countervailing power. An examination of his action
must result in an analysis of coalitions, the possibility
of collusion, the analysis of threats” (Shubik 1954 p. 7)
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A Comparison of Treatments of the Duopoly Situation,
discusses various duopoly analyses of Cournot,
Edgeworth and others from the perspective of game
theory.
There is no mention of Nash equilibrium: Nash's work
on non-cooperation does not even appear in the
bibliography.
Cournot solution is very briefly described and the main
reference is Fellner's (1949) book on oligopoly, where
the concept of equilibrium in Cournot is dismantled
“In these two early papers, therefore, Shubik displays a
certain scepticism towards the Nash equilibrium, on the
grounds that the hypothesis of non-cooperation is of
questionable relevance, and towards the Cournot
solution.” (Leonard 1995)
This shows that the connection between Cournot and
Nash was not perceived instantly by Shubik, but
involved a reinterpretation of the work of Cournot from
Fellner's critique to the implausibility of the reaction
curve dynamics to the static interpretation in which the
dynamics are suppressed, and the equilibrium point
preserved.
In 1955 Shubik moved from Princeton to Stanford
Shubik acknowledged that when he was writing
“Edgeworth Market Games” in 1955, “except for some
assistance by Howard Raiffa, I had no one to check my
abominal [sic] mathematics, and it appeared with
several false theorems.” (Shubik diaries)
Just like his mentor Morgenstern, Shubik’s mathematics
were limited, a restriction that would not only hinder
the influence of “Edgeworth Market Games”
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The paper relates a neoclassical economic concept, the
contract curve, to a game theoretic concept, the core
Proof was largely incomplete (convergence to Walrasian
general equilibirum was shown by Debreu and Scarf
1963)
This notwithstanding, “From that time on, economics
has remained by far the largest area of application of
game theory (…) core theory was extensively developed
and applied to market economies.” (Aumann 1985)
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Edgeworth market games do not converge to a single
point as the number of players increases to infinity, but
under certain conditions, the core of these games does
approach a single imputation in the limit.
Shubik noted that this “single
imputation has economic
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meaning either in the theory of monopoly or the theory
of pure competition.“
In one particular class of games, the core approaches a
single imputation where a single player will act as a
perfectly discriminating monopolist
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While the Von Neumann-Morgenstern stable set
contains numerous imputations, the core delineates the
true solution
As the number of players became large, the core shrunk
to one point that is the competitive equilibrium
The core is an effective solution concept for market
games because as Shubik discovered if the competitive
equilibrium point exists, then it must be in the core.
In this economy there is only one imputation that is not
dominated
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Cooperative games are restricted to the case of
collusive behaviour
Nash bargaining solution picks out the efficient solution
in bilateral monopoly (vs. von Neumann Morgenstern’s
solution)
Emphasis on standards of society is replaced with the
static analysis of threats and bargaining process as in
Nash’s bargaining model
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Nash equilibrium is defined as the only reasonable
solution in oligopolistic models (strictly enforceable)
Dynamic processes are described in terms of the static
adjustment process to equilibrium
The outcome of the Cournot duopoly model is
interpreted as a Nash equilibrium, i.e. a static fixed
point
This interpretation of Cournot “marks therefore the true
landmark for the application of game-theoretic,static
equilibrium theory to economic analysis” (Giocoli 2004)
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Shubik’s definition of external symmetry
Games are based on the assumption that players are
alike for everything is not explicitly included in the
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formal description of the game
This assumption is the corollary of the axiom of
symmetry postulated by Nash and Harsanyi
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TGEB “in general does not yield determinate solutions
for two-person nonzero-sum games and for n-person
games” (Harsanyi 1976)
Many economists, and especially applied economists,
are troubled that games had “too many equilibria and
no way to choose, Kreps (1990)
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If considerations of rationality only narrowed the range
of possible equilibria in game-theoretic models without
providing unique, determinate results and economists
were even more disturbed by the variety of solution
concepts proposed in cooperative game theory.
Non cooperative game theory received increasing
attention in the 1970s for this reasons which
differentiate it from cooperative game theory
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The central question of orthodox pre-war microtheory
is how market equilibrium actually attained and it has
been shunted aside ever since the Formalist Revolution
of the 1950s. In general equilibrium theory, the
question of whether it exists at all dominated the issue
of convergence to equilibrium so successfully as to
swallow it up entirely (Blaug 2005)
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Conditions guaranteeing the uniqueness of solutions
are crucial to their applications in comparative statcis
models based on equilibrium state. “Lacking conditions
that guarantee uniqueness, economists must resort to
considerations of historical conditions and dynamic
stability, which greatly complicate the analysis”
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In the 1950s Shubik changed his view from that
intellectual indebted to his advisor Oskar Morgenstern
to that promoting the next-to-come mainstream view
of the application of game theory to economics.
An explanation is given by the necessity of making
game theory acceptable for economists. In this way
game theory was acknowledged as fully integrated in
the body of economics but it weakens its empirical
meaning.