GEORGIA STATE UNIVERSITY ANDREW YOUNG SCHOOL OF POLICY STUDIES

ECONOMICS DEPARTMENT

EC 8500 History of Economic Thought

Prof. Bruce Kaufman

John Forbes. Nash Jr. Interaction of Rationality and Beauty of

Reasoning

Prithvijit Mukherjee

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“Adam Smith needs revision!” he declares triumphantly. To his baffled

classmates, he explains: "Adam Smith said the best result comes from

everyone in the group doing what's best for himself, right? Adam Smith was

wrong!" The message: Sometimes it's better to cooperate!

- A quote from Beautiful Mind

I. Introduction

The quote from Beautiful Mind lays probably the best summary of Nash’s contribution to

economics, non-cooperative games (Invisible Hand at work), bargaining games (or

cooperative games) and Nash Program (non-cooperative reduction of cooperative games).

Nash’s mathematical analysis of game theory bringed on the boundaries of economics which

widened the horizon, the rational (homoeconomicus) man can be studied in various settings

and economics contained in the realm of production and allocation of material goods.

Game theory started more formally with mathematicians attempting to find scientific solution

to human conflicts in a simple setting or games. Christian Huygens and Gottfried Leibniz

were the first to understand the importance of this approach. “A pivotal development in game

theory was proven by Ernst Zermelo in 1912 that finite games such as tic-tac-toe, chckers,

and chess have an optimal solution, or strategy. His theorem was although not very universal

since the proof required perfect information about past movies and all possible future

moves.” (Karier 2010). Followed by the research by Emile Borel publishing four notes in

Game Theory between 1921-1927, he was the first to define the mixed strategy and

demonstrated the existence of a minimax solution to a two-player zero-sum game. But the

biggest breakthrough came in 1928 with the von Neumann publication “On the Theory of

Social Games” in Mathematische Annalen and later refined along with an expected utility

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theory in von Neumann and Oscar Morgenstern book “Theory of Economic Games and

Behavior”.

In 1944, with the publication of von Neumann’s and Oskar Morgenstern’s

Theory of Games and Economic Behavior, the scattered contributions to game

theory were synthesised and extended. It also set an agenda that took

considerable hold of the discipline for years to come.

There are two feature of this book are noteworthy. First, it contains essentially

no new important game-theoretic results. Its theoretical centrepiece remains

the Minimax Theorem. In particular, the authors’ analysis of non-zero-sum

games, and games with more than two players, produced a lot of interesting

ideas, but no firm conclusion. Second, Morgenstern an economist at Princeton,

had perceived the potential value of game theory to economics.” (Ryan 2002)

There was stream of change in the nature of research tools which were evolving in economic

analysis, Cournot, Walras,Pareto Jevon, Edgeworth, Marshall among other infused the use of

mathematics in economics which got formalized with the with culminating in the publication

of Paul Samuelson’s Foundation of Economic Analysis in 1947, along with the development

of game theory as a field of mathematical study. The two fields of study converged with The

Theory of Games and Economic Behavior along Nash’s seminal work on existence of

equilibrium in non-cooperative games only expanded the scope of applicability of game

theory to economics and other social sciences.

The popularisation of game theory as a tool is attributed to Albert Tucker a mathematician

from Princeton and Nash’s supervisor while giving a presentation to an audience of

psychologist on game theory invented the example of Prisoner’s Dilemma showing for a

rational person the dominant strategy was to betray his partner. Prisoner’s Dilemma

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illustrates the basic and very intuitively the process of reach a (Nash) equilibrium in the

game. This simple example of prisoner’s dilemma without the oodles of math caught the

attention of academics and soon game theory started being applied to various disciplines.

II. Short Biography

There are probably only a handful of mathematicians in the world to become famous for their

contribution to economics and even less who became famous for their life stories (Karier

2010) Nash’s story comes around a full circle, from a genius mathematician, working in

military intelligence, devastating mental illness and redemption winning the Nobel Prize in

1994.

John Forbes Nash Jr. was born or as he writes in his biography “beginning of a legally

recognized individual occurred on June 13, 1928.” His father John Forbes Nash was WWI

veteran was an electrical engineer from Texas who came to work for Appalachian Electric

Power Company in Bluefield Virginia. His mother Margaret Virginia Martin, was school

teacher but her life was considerably affected by a partial loss of hearing due to scarlet fever

infection as a student.

Nash writes in his biography writes that Bluefield was a small city which was a centre of

trade hardly had any scholars so from the intellectual viewpoint Nash relied on extra-

curriculum reading like Men of Mathematics, by E.T. Bell In high school he Nash displayed

his talent in mathematics by proving one of Fermat’s Theorem. In his last year of high school

his parents arranged for classes in mathematics at the Bluefield college which gave him a

head start, as he writes he did not learn much from the first maths courses at Carnegie Tech.

He joined Carnegie Tech. to study chemical engineering switched his major and he graduated

with an M.S .in mathematics in addition to his B.S. For his graduate studies Nash got through

both Harvard and Princeton with his more than celebrated one line recommendation letter by

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R.L. Duffin, “This man is a genius”. Soon after he graduated he started working at MIT in

1951 but his academic career came to a standstill in 1959, his classes became completely

chaotic and start his tragic mental illness. It is only 1980s that Nash began to recover; today

he continues to explore mathematics at Princeton University.

III. Contribution

Non Cooperative Games

When young John Nash came into Princeton he studied mathematics and was influenced by

von Neumann’s research on economic application of game theory.

“Von Neumann and Morgenstern have developed a very fruitful theory of

two-person zero-sum games in their book Theory of Games and Economic

Behavior. This book also contains a theory of n-person games of a type which

we would call cooperative. This theory is based on an analysis of the

interrelationship of various coalitions which can be formed by the players of

the game.” (Pg 5 Nash 1950a)

In essence his dissertation is an extension of von Neumann idea of a zero-sum cooperative

game but with the following important changes:

a) Nash extended the analysis to non-cooperative games, “it is assumed that each

participants acts independently, without collaboration or communication with any of

the others” (Nash 1950a).

b) The most important contribution of Nash which he called the “basic ingredient in our

theory” was the idea of a general proof of the existence of equilibrium in n-player

case ( which is now called Nash Equilibrium) which von Neumann provided a proof

with certain rules.

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Nash in his dissertation used the Brouwer fixed-point theorem to prove the existence of the

equilibrium. Ever since then many others inspired from Nash used the Brouwer and Kakutani

fixed point theory, for example the early proofs of the existence of Walrasian Equilibrium

was based on Nash’s idea. (Myerson 1999)

Nash in his dissertation gives the following interpretation of his work;

“We proceed by investigation the question: what would be “rational”

prediction of the behaviour to be expected of rational playing the game in

question? By using the principles of 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 other to do should

not lead him to act out of conformity with the prediction, one is led to the

concept of a solution. In his interpretation we need to assume that players

know the full structure of the game in order to able to deduce the prediction

for themselves. It is quite strongly a rationalistic and idealizing interpretation

(pg 23, Nash 1950a.)

The notion of the rational individual plays a key role in Nash’s equilibrium and in current

research in economics. Nash seminal work broadly expanded the horizon of applicability of

game theory to economics. Situations analyzed in economics are usually non-zero sum

games, this extended the applicability of von-Neumann and Morgenstern work in Theory of

Games and Economic Behavior. The notion of a Nash Equilibrium has been the mainstay in

analysis of conflicts of interests and social outcomes. This has become the testing ground for

the validity of the theory’s prediction via a huge body of growing literature in experimental

economics.

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Bargaining

John Nash took only one course in Economics at Carnegie Tech is that of International

Economics that made analyse the possibility of modelling bargaining. Nash followed an

axiomatic approach toward the solution to the bargaining problem which it does not assume

transferable utility in a two person cause. In his paper “The Bargaining Problem” he develops

an axiomatic approach to solving the bargaining problem, extending the analysing of von

Neumann The Theory of Economic Games and Behaviour.

“Nash’s Bargaining theory builds on the insight that individuals’ utility scales

can be defined up to spate increasing linear transformations, but this result

follows only from von Neumann and Morgenstern’s 1947 derivation of utility.

Thus, Nash’s bargaining solution could not have been appreciated before

1947” (Myerson, 1999)

Nash Bargaining solution satisfies the following axioms:

1) Invariant to affine transformation or scale invariance

2) Pareto optimality

3) Independence of irrelevant alternatives

4) Symmetry

The only function which satisfies if the players choose the maximize (ui - d)(uj – d) in a two

player scenario where U is the utility function and d is the disagreement point (this is referred

to as Nash Bargaining Solution). Although in this approach the axiom abstracts away from

reality, like the symmetry axiom equal bargaining power, in real world hardly this is the case.

But the model provides a benchmark to be considered in an ideal world.

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“The modern theory of bargaining, which is based on Nash’s formulation of the bargaining

problem and incorporates the specific solution that he proposed, lies at the heart of studies of

many real-life negotiations. In particular, Ariel Rubinstien’s celebrated model of bargaining

provides a firm foundation for Nash’s bargaining solution. As such, Nash’s bargaining

solution is the concept often used in applications of bargaining theory, such as in studies

involving firm-union wage negotiations and international negotiations”

Nash Program

In 1950 and 1953 paper, Nash offered an application of his program for reducing cooperative

game theory to non-cooperative equilibrium analysis.

“This meaning of this term (Nash Program) can be understood by studying the

two papers together: The bargaining problem is a description of the utilities

that can be achieved. Nash in the first paper considers an abstract rule that

picks a solution from any possible bargaining problem. His simple and

reasonable axioms yield an unique solution now called the Nash bargaining

solution. The second paper takes as its starting point a normal form game,

views the set of possible outcomes of this cgame as the attainable payoffs of a

bargaining problem, describes a new noncooperative game to model the

negotiation procedures, computes a Nash equilibrium of the negotiation game

and observes that his Nash equilibrium results in the payoffs associated with

the bargaining solution. Thus, the axiomatic solution is justified showing that

it is the Nash equilibrium outcome of some noncooperative game.” (Gul 1997)

The Nash program draws an important equivalence between cooperative and non-cooperative

games which highlight the role of a rational human reasoning.

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IV. Discussion of Nash’s contribution

It is merely not enough to laud the wide array of application and further research in game

theory and bargaining theory which was inspired by Nash’s work. But in essence understand

the inter-twining of Nash’s work and economic theory.

Rubinstein (1995) points an interesting aspect of Nash’s 1950 paper, he discusses if the paper

evaluated by current standards, the referee would make the following complaints:

1) The paper lack economic example which demonstrate the usefulness of the model.

The paper provides only one “example” namely a “Three-man Poker Game” which is

unrealistic

2) The model is unrealistic: it is difficult to think of any strategic interaction in which

each player chooses a single action and all players move simultaneously. Exceptions

are mostly from zero –sum games which have already analyzed by von Neumann and

Morgenstern. Thus difficult to see the value addition to zero-sum games

3) The concept of equilibrium is too weak to be interesting. In economics we need

powerful tools and this equilibrium is usually uninformative

4) The notion of mixed strategy which has some appeal in the context of zero-sum

games is not realistic in the context of non zero sum games.

These are all valid complaints but still do not undermine the importance of Nash’s

contribution to economics. The beauty of Nash’s work is the simple structure to represent the

process of reason about a situation. What Nash achieved in his work a model of interactive

rationality which is key to economics, at the end of the day through various models, schools

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of thought what we are trying to understand and synthesize is how economic agents interact

given priors. Even the simplest model of perfect competition taught in Econ 101 models the

interactive rationality of individual firms vis-à-vis the market structure. What Nash’s

contribution expanded our horizon beyond market structure to every aspect of human

interaction which involves a pay-off to an individual. It is the generality of definition of

equilibrium which has aided future research is applying game theory to social sciences.

Nash’s in his interpretations of his dissertation and even his bargaining paper (Nash 1950b)

highlights the most critical aspect on his theory hinges on notion of rationality. Nash writes

“… we idealize the bargaining problem by assuming that the two individuals are highly

rational, that each can accurately compare his desires for various things, that they are equal in

bargaining skill…” Nash made no tall claims of the practical applicability of his theory, then

the question arises what makes Nash’s work so pivotal to the progress of economic theory.

The common denominator which binds the current exposition in research in economics and

Nash’s contribution is the notion of a rational agent or the homoeconomicus. The notion of

perfectly rational although far from the reality has yield tractable results in economic analysis

than any other theory of human behaviour.

“… the functional goal of social science is not just to predict human behavior

in the abstract, but analyze social institutions and evaluated proposals for

institutional reforms” (Myerson 1999)

One of the most important research questions have been studying the social structure and

institutions in which human behaviour is nested. The assumption of perfect rationality allows

us to clearly distinguish the effects which are due to irrational behaviour and that from flaws

in the institutional structure. Economic theory in essence will collapse if we don’t endorse to

the notion that individuals are self utility maximizers, after all economics as a social science

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is trying to measure and improve a notion of welfare, as understood and analysed by the

father of modern economics, Adam Smith in The Wealth of Nation. When analysing

economic problems related to welfare or social outcomes if analysis reveals the deviations

from optimal solution is due to flaws in the design or some agents donot completely

understand their scenario gives us direct policy prescription one can work on. This notion of

rationality although vague and abstracted from reality is the key to economic analysis and

Nash’s solutions.

“The Nash equilibrium is useful not just when it is itself an accurate predictor

of how people will behave in a game but also when it is not, because then it

identifies situations in which there is a tension between individual incentives

and other motivations. A class of problems that have received a good deal of

study from this point of view is the family of “social dilemmas”, in which

there is a socially desirable action that is not a Nash Equilibrium.” (Holt and

Roth 2004)

One of the more recent and robust strand of economics taking shape today is experimental

economics which was started as field of enquiry more formally by Nobel Laureate Vernon

Smith. “It is worth mentioning that Nash both commented on and participated in early

experiments in economics” (Holt and Roth 2004), it was a natural development from terse

mathematical structures and assumption to apply testing these notions in the laboratory on

people actually making these decisions. Game theory moved to the frontier of economic

research due to it is strongly testable prediction, which laid the foundation for experimental

economics. This extension is possibly more exciting avenues of developing tractable theories

of irrational behaviour to make theories in economics more “real”. “Before Smith’s

experiments it was widely believe that the competitive predictions of supply/demand

intersections required very large numbers of well-informed traders. Smith showed that

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competitive efficient outcomes could be observed with surprisingly small number of traders,

each with no direct knowledge of others’ costs or value. An important developing area of

game theory is to explain these and other experimental results in the context of well specified

dynamic models of interactions.” (Holt and Roth 2004). Also experimental studies have

shown abstract models have predictive power which only reinforces the idea of the rational

agent might not although unreal can predict with some degree of accuracy the outcomes in

the economy. One of the most important aspect which has been highlighted by experimental

economics in the dynamic evolution of interaction which eventually converge towards

learning and better understanding of the situation, even papers written about use of heuristics

in analysing economic scenarios converge on average to a Nash Equilibrium on average.

In conclusion it is hard to refute the importance of Nash’s theory to economics as a field of

study, probably he is the most celebrated Mathematician in economics and his ideas are so

integrated with the study of economics, today hardly it is required to quote Nash’s paper

when talking about Nash equilibrium. Also, Nash’s work has not only inspired work in

further in game theory and bargaining theory but also in various other application and newer

avenues of research like experimental economics. It often takes a lot of words to express

something very simple, a model of interactive rationality fundamentally shifted the frontier of

research in economics.

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Reference:

1) Damme, E and Weibull, J (1995), “Equilibrium in Strategic Interaction: The

Contributions of John C. Harsanyi, John F. Nash and Reinhard Selten” Scandinavian

Journal of Economics, 97 15-40

Ryan, Matthew J. (2002) "Mathematicians as Great Economists: John Forbes Nash,

Jr." Agenda 9, no. 2: 121-134.

2) Holt, C. A. and Roth, A. E., (2004) “The Nash equilibrium: A perspective,” in

Proceedings of the National Academy of Sciences, vol. 101, no. 12,.

3) Karier, T. (2010) “Intellectual Capital: Forty Years of the Nobel Prize in Economics”,

Cambridge University Press

4) Muthoo, A, (2002) “On John Nash's Scientific Contributions, in Game Theory: A

Festschrift in honor of John Nash,” edited by C. Kottaridi and G. Siourounis, Eurasia

Publications, Athens, Greece, 2002, 134-137.

5) Myerson, R.B. (1999) "NASH Equilibrium and The History of Economic Theory",

Journal of Economic Literature 36:1067-1082.

6) Nash, Jr., John F. 1950a. “Noncooperative Games” Dissertation, Princeton University

7) Nash, Jr., John F. 1950b. "The bargaining problem." Econometrica 18:155-162.

8) Nash, Jr., John F. 1951. "Noncooperative games." Annals of Mathematics 54:289-

295.

9) Nash, Jr., John F. 1953. "Two-person cooperative games." Econometrica 21:128-140.

10) Nash, Jr., John F.. - Biographical". Nobelprize.org. Nobel Media AB 2013. Web. 17

Apr 2014. <http://www.nobelprize.org/nobel_prizes/economic-

sciences/laureates/1994/nash-bio.html>

11) Rubinstein, Ariel. 1995. "John Nash: the master of economic modeling."

Scandinavian Journal of Economics 97:9-13.