A BRIEF INCOMPLETE HISTORY OF GAME THEORY

Asrar Chowdhury Handout 1: History of Game Theory in Economics
asrarul@juniv.edu; asrarul@gmail.com 01819 219050; 01534 229000
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A BRIEF HISTORY OF GAME THEORY IN ECONOMICS
1.1: Introduction
Since the dawn of history, people have been playing games. When individuals play
games, they express their incentives and motives. The nature and rules of the game
determines if players will play together or against each other. This latter feature forms
the basis of social evolution: a combination of both cooperation and conflict. From the
times of the Chinese military strategist, Sun Tzu (500 BCE) in The Art of War to social
scientists like Thomas Hobbes (1588-1679), John Locke (1632-1704), David Hume (1711-
1776), Jean-Jacques Rousseau (1712-1778), Adam Smith (1723-1790), sages have been
analysing how and why people play games. However, it was not until 1944 that the
branch of applied mathematics Game Theory formally launched. The first application of
Game Theory was in economic theory. It came from the limitation of mainstream
economic theory in providing a rigorous interpretation to decision-making under
strategic situations.
2.1: Rationality in Model Building
The analytical foundation of mainstream economic theory is the neoclassical school.
Rationality forms the basis of decision-making in this school of thought. Rationality
assumes given the constraints they find themselves in, decision-makers are optimizers.
They maximize things that give them happiness and minimise things that give them
pain. In economic theory, rationality has important implications in model building.
First: rationality narrows down the set of possible outcomes. The analysis focuses on
what the model wants to build on. If a rational individual is a clever individual, it
makes sense to focus attention on rational behaviour in model building. Second:
rationality helps predict the outcome of an economic system. It therefore lays the
foundations of efficiency for an economic system. When all decision-makers in an
economic system are rational; they are optimizing their net benefit functions within the
constraint they are facing; then, the final outcome is such that no decision-maker would
find an incentive to deviate from their optimal position. Once economic agents have
optimised their respective functions and reached a situation from where they do not
want to deviate, the economic system reaches a stable outcome. This stable outcome is
also known as ‘equilibrium’.
2.2: Perfect Competition as a Benchmark
The neoclassical school analyzes decision-making within a market-setting. The
benchmark market is perfect competition. There are two crucial features of perfect
competition. First: the existence of many buyers and many sellers. Second: decision-
making is made independently and individually. The last feature has two implications:
(a) decisions are not made in coalitions (together/jointly); and (b) decisions are not
inter-dependent. Logically, then, the decision of one agent is neither influenced by
another agent, nor does it influence that of another agent.

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Independent and individual decision-making under perfect competition implies
each decision-maker tries to do the best they can irrespective of what other decision-
makers are doing. This leads to an impersonal relationship between decision-makers in
the economic system. Based on this impersonal relationship, Adam Smith introduced
his famous metaphor the invisible hand in Moral Sentiments (1759). This metaphor
became part of folklore in economic literature. Based on the invisible hand of free
markets, when an economic system consists of rational decision-makers whose
decisions are independent of each other, the agents are guided by an “invisible hand to
promote an end which was no part of [their] intention”. The economic system reaches a
stable situation from which the system has no incentive to deviate. The system reaches
equilibrium.
Perfect competition is a theoretical extreme. Like the ideal human body
temperature of 98.4 degrees Fahrenheit it almost never exists. It is used as a benchmark
to explain deviations from this ‘perfect’ world. Based on the optimum condition of
MR=MC for the competitive firm and MV=ME for the competitive buyer, the
neoclassical analysis of monopoly, monopsony, and monopolistic competition exhibit
one common feature. In none of the markets does the economic agent calculate the
possible response of other agents in the market system due to their own chosen actions.
The monopolist and the monopsonist do not need to calculate the response of any other
agent because they are the only producer or buyer in their markets. Monopolistic
competition assumes the presence of many buyers and many sellers each selling a
slightly differentiated product, thus giving them some degree of monopoly power.
However, the presence of many buyers and sellers implies that the decisions of the
agents are independent of each other. The absence of inter-dependent decision making
allows MR=MC and/or MV=ME to be the basis of analysing equilibrium in the above
markets.
The above principle of MR=MC and/or MV=ME is based on independent and no
interdependent decision-making between firms and buyers in a market. This principle
fails to analyze equilibrium in markets that have few sellers and/or few buyers. Such
markets include duopoly and oligopoly in product markets; duopsony and oligopsony
in factor markets; and the special case of a bilateral monopoly that involves one seller
and one buyer. The breaking up of independent decision making leads to strategic
decision-making. This arises when each agent calculates the possible responses of other
agents due to their own actions.
3.1: Pre-Cursors to Game Theory
The first formal model of strategic behaviour in markets was in oligopoly markets
where there are few sellers. In 1838. The French economist Antoine Augustine Cournot
(1801-1877) discussed a duopoly where the two duopolists set their output based on
residual demand. The next development came in 1883. The French economist Joseph
Bertrand (1822-1900) analyzed a duopoly where the two duopolists set prices at the

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same time (simultaneous game). In 1934 the German economist Heinrich Stackelberg
(1905-1946) analyzed a duopoly where the two duopolists make output decisions one
after the other in a sequence (sequential game).
3.2: The Birth of Game Theory: It all happened at Princeton
Although the Cournot, Betrand, and Stackelberg models were significant
improvements, they failed to clearly explain how equilibrium is reached in the presence
of inter-dependent decision-making in oligopoly markets. The formal birth of Game
Theory had to wait till almost the end of World War II. It happened in 1944 at the
Department of Mathematics at Princeton University, USA. It was an outcome of two
intellectual debates. The first was between the French Mathematician Emile Borel (1871-
1956) and the Hungarian Mathematician John Von Neumann (1903-1957) that started in
1921. The question was: do zero-sum games have a solution? In 1928, von Neumann
solved the problem. It is now famously known as the mini-max theorem. The second
debate was between John Von Neumann and the Austrian economist Oskar
Morgenstern (1902-1977). They asked the question: can utility be quantified? The
solution to this question is now famously known as the Von Neumann- Morgenstern
Expected Utility Theory. Combining these two debates, Von Neumann and
Morgenstern launched their magnum opus in 1944 while both were at Princeton
University, USA, Theory of Games and Economic Behavior.
Von Neumann and Morgenstern formally laid the foundations of Game Theory
as a branch of applied mathematics. However, their effort failed to generate stir for two
reasons. The Theory of Games and Economic Behavior was based on the notion of zero-sum
games. These are known as ‘Games of Conflict’ or ‘Non-Cooperative Games’. Most
games in social sciences are non-zero-sum games. The Theory of Games and Economic
Behavior had a second limitation. It did not establish how equilibrium in games of
interdependent decision-making would arise. The world did not have to wait too long
for this solution. The final thread came from the doctoral dissertation of another
mathematician from Princeton.
A young John Nash (b 1928) arrived at Princeton in the Fall of 1948 to start a
PhD. Nash had come from what is now known as the Carnegie Mellon University. He
had rejected Harvard because they did not offer him a generous stipend. He came to
Princeton with what remains one of the shortest reference letters ever for a PhD
candidate. His professor from Carnegie Mellon, Richard Duffin (1909-1996), wrote just
one sentence: This man is a genius. The rest was indeed, history. Just eighteen months
later, Nash submitted a 28 page doctoral dissertation. The number 28 went on to
become a superstitious number at Princeton. Von Neumann solved the mini-max
theorem in 1928. Nash was born in 1928. Nash’s doctoral dissertation was 28 pages.
In his dissertation, Nash showed how equilibrium in n-person non-cooperative
games is achieved. This was the missing link to Von Neumann and Morgenstern’s

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magnum opus. Nash published two papers from his doctoral dissertation: Equilibrium
Points in n-Person Games in 1950 and Non Cooperative Games in 1951. These two papers
established equilibrium where all players calculate their best strategy based on what
they assume is the best strategy of the other players. Under this assumption, every finite
game must have at least one solution such that once reached, no player within the game
will have an incentive to deviate from their chosen actions given the actions of the other
players in the game. A second hypothesis was: every finite game has at least one
solution; and solutions will exist in odd numbers.
The intuition behind Nash’s equilibrium is very simple. If all rational economic
agents in a system are trying to do the best they can assuming the others are doing the
same, the economic system must be in equilibrium such that no single agent will want
to unilaterally deviate from their position. Although appealing, it was not as simple as
that. A Nash equilibrium does not necessarily imply that a game will reach a solution
that is the best possible solution for all players in the game. All it says, a finite game will
have at least one solution. This became evident through the works of Merrill Flood
(1908-1991), Melvin Dresher (1911-1992) and Nash’s PhD supervisor was another great
mathematician from Princeton, Albert Tucker (1905-1995).
Merrill Flood and Melvin Dresher were working at RAND in 1949. RAND was
the research wing of the US Air Force. RAND was instrumental in the proliferation of
Game Theory during the Cold War between the USA and today’s Russia (then the
USSR). Flood and Dresher identified a game where it is in the best interest for players
to cooperate, but individual self-interest invokes them to not cooperate. By doing so,
they reach a Nash equilibrium that is not in the best interest for all players in the game.
Differently, the game reaches a bad equilibrium that is inferior to a superior outcome
that could have been reached and was available. This phenomenon was canonized by
Albert Tucker. In the summer of 1950 Tucker was at Stanford University. He was
working on a problem in his room when a graduate student of psychology knocked and
asked what he was doing. The answer was short: game theory. ‘Why don’t you explain to
us in a seminar’? Tucker used his now famous example of two thieves who were put
into separate cells and asked the same question by the judge. Tucker christened the
phenomenon as The Prisoners’ Dilemma. The term has remained so in the annals of Game
Theory.
3.3: Game Theory moves away from Princeton
Identification and formalization of the Prisoners’ Dilemma by Flood, Dresher & Tucker
had important implications in economic theory and the social sciences. It provided a
rigorous explanation to why intervention into markets can be justified in the presence of
public goods that are jointly consumed once provided, but almost impossible to finance
through voluntary contribution. The Prisoners’ Dilemma game played more than once
formed the basis to why it is beneficial for players to sacrifice individual benefits in the
short-run for greater benefits in repeated encounters. This phenomenon formed the

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basis to why players in a game find a mutual interest to cooperate if a Prisoners’
Dilemma game is played more than once. This lead to what is known as Trigger
Strategies. Anatol Rapoport (1911-2007) proposed a simple strategy called Tit For Tat in
a repeated Prisoners’ Dilemma tournament. The Tournament was held by the political
scientist from Michigan University, Robert Axelrod (b 1943). Rapoport proposed, each
player cooperates with the other player in a repeated Prisoners’ Dilemma as long as the
other player does the same. If one player defects in one round, the game ends with the
other player applying the Trigger of Tit For Tat by no cooperation and ending the game.
Variants of Tit for Tat are found in the Grim Trigger introduced by James Friedman
(1971); the Trembling Hand Grim Trigger introduced by Michael Taylor in 1987.
The Prisoners’ Dilemma found more applications in more variant forms. The Prisoners’
Dilemma in n-person games provided a basis to why individual optimization with a
fixed common resource may lead to depletion of the resource over time. This is now
known as the Tragedy of the Commons after Garrett Hardin in 1968. The intuition is
very simple. The Tragedy of the Commons formed the basis of analyzing global
phenomena like Global Warming. Elinor Ostrom (1933-2012) and Oliver Williamson (b
1932) extended the Tragedy of the Commons to discuss the commons and economic
governance.
3.4: Refinement of the Nash Equilibrium
The influence of Game Theory in economics and the social sciences was just the tip of
the iceberg. Inter-dependent decision-making meant there were more avenues waiting
to be explored once the necessary theoretical development was made. The development
of Game Theory from the 1950s was based on the limitations of the Nash Equilibrium.
The Nash Equilibrium argued that a game can have multiple equilibria (solutions). It
did not provide how to identify the set of multiple equilibria and which one to choose
as the more probable equilibrium. While working at the RAND, Thomas Schelling (b
1921) introduced two terms Focal Point (the logical outcome from information from
outside a game) and Credible Commitment (sending a signal that a player will commit to
a certain actions). The Focal Point provided a basis for behavioural sciences why players
prefer one set of equilibrium to another. It also helped narrow down probable solutions
from a larger set. The Credible Commitment (Threat) gave an added edge to explain
previous models like the Stackelberg model in a sequential game setting.
The Nash Equilibrium analyzed non-cooperative static games. Problems arose
with finding solutions to dynamic games that were sequential or multi stage in nature.
To overcome this problem, the German Game Theorist Reinhard Selten (b 1930)
introduced sub-game perfect equilibrium. This equilibrium concept identifies the Nash
equilibrium that is consistent with each sub-game in the whole game. This intuition
identified it as the solution to the game. Another solution with games that have many
decision nodes was introduced by the Israeli-American Game Theorist Robert Aumann
(b 1930). The intuition is simple. It looks at where the game ended. Then it works

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backward to the beginning of the game. This process enabled Game Theorists to focus
on only that part of the game that lead to the solution. This solution method was termed
Backward Induction. A further development in the Nash Equilibrium came through the
works of the Hungarian-American Game Theorist John Harsanyi (b 1920).
The Nash equilibrium was based on games of complete information. In economic
theory, this is known as games under symmetric information when all players in a
game make decisions based on having the same set of information. By the 1970s,
economists like George Akerlof (b 1940), Michael Spence (b 1943) and Joseph Stiglitz (b
1943) started analyzing decision-making under asymmetric information or games under
incomplete information. Harsanyi had by that time refined the Nash Equilibrium under
Bayesian Probability. This enabled the analysis of games where different players have
different sets of information about themselves. Harsanyi’s refinement was a blessing for
economists who were battling with asymmetric information.
3.5: Goodbye to Rationality
By the 1960s and the 1970s, psychologists studying decision-making, started to
challenge the rationality assumption upon which the neoclassical school of economic
theory and Game Theory was based. One of the pioneers was Herbert Simon (1916-
2001). He introduced the notion of Bounded Rationality. When individuals make
decisions they posses limited information, limited cognitive ability, and limited time
within which to make decisions. With the development of bounded rationality,
economic theory slowly started to become a behavioural science. Economists started
analyzing biases and heuristics.
Behavioral game theory has three ingredients. First: mathematical theories of how
ethics and morality affect decision-making. This then affects how individuals or groups
bargain and trust each other. Second: how cognitive limitations in the brain restrict the
number of steps needed to calculate an optimum decision. Third: how individuals learn
from own experience and evolutionary experience and adjust to their decision-making.
The Centipede Game and the Ultimatum Game were one of the first beavioural
games that emerged in the 1980s. Robert Rosenthal (b 1933) first introduced the
Centipede Game in 1981. Two players take turns choosing either to take a slightly larger
share of a slowly increasing pot, or to pass the pot to the other player. Based on sub-
game perfect equilibrium and backward induction, the rational outcome of the game
would be for the player making the first decision to take the entire pot in the first
round. However, empirical evidence suggested otherwise. Players tend to partially
cooperate so the pot becomes larger as the game proceeds. This empirical outcome
challenged the rationality assumption of Game Theory.
The second behavioural game, the Ultimatum Game was introduced in 1982 by
Werner Güth, Rolf Schmittberger, and Bernd Schwarze. It was later developed by

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Martin Nowak, Karen Page, Karl Sigmund in 2000 to explain fairness and reasoning in
decision-making. In the Ultimatum Game, two players decide how to divide a sum of
money. The first player proposes how to divide the sum between the two players. The
second player can either accept or reject this proposal. If the second player rejects,
neither player receives anything. If the second player accepts, the money is split
according to the proposal. The game is played only once. This game has also been
termed as the Dictator Game since the first player acts as a dictator declaring their terms
under which the sum of money shall be split between the players. This game challenges
the rationality assumption. Rationality would suggest the proposer keeps all the money
and does not offer anything to the other player. Empirical evidence suggests ethics and
morality and evolutionary behaviour leads to some degree of fairness in the splitting of
the money.
4.1: And The Game Ends
Princeton University was the place to be during 1944-1950. Game Theory was formally
born through John Von Neumann and Oskar Morgenstern in 1944. John Nash solved
the multi-agent inter-dependent decision-making problem in 1950. In that same year,
Albert Tucker formalized Merrill Flood and Melvin Dresher’s problem and christened it
the Prisoners’ Dilemma. Since then, Game Theory may have moved away from
Princeton, but there was no turning back. In less than seven decades since 1950, Game
Theory has established itself as one of the fundamental discoveries and developments
in the Twentieth Century that ranks alongside that of the Genome. In just seven decades
since 1944, Game Theory has established itself as one of the most influential branches of
applied mathematics since Calculus and Probability.
The beauty of Game Theory remains within itself. As long as humans are around, there
will be inter-dependent decision-making somewhere in this universe. What was started
by mathematicians and economists has now attracted the attention of military
strategists, marketing gurus, evolutionary scientists, forensic experts. Game Theory has
even attracted the attention of linguists. As this story of the history of Game Theory
ends, another starts somewhere else encompassing branches of human knowledge this
brief historical anecdote failed to identify. As humans progress, new problems and new
challenges will appear. Game Theory will be refined and updated like Calculus and
Probability have, to face these new challenges.
Further Readings
1: Sun Tzu. BCE 500. The Art of War.
2: Thomas Hobbes. 1651. Leviathan.
3: John Locke. 1689. Two Treatises of Government.
4: David Hume. 1738. A Treatise on Human Nature.
5: Adam Smith. 1759. Theory of Moral Sentiments.
6: Jean-Jacques Rousseau. 1762. The Social Contract.
7: Adam Smith. 1776. The Wealth of Nations.

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8: Augustine Cournot. 1838. Researches into the Mathematical Principles of the Theory
of Wealth.
9: Joseph Bertrand. 1883. Recherches sur les Principles Mathematiques de la Theorie des Richesses.
Journal de Savants.
10: Heinrich Stackelberg. 1934. Marktform und Gleichgewicht. J Springer.
11: John von Neumann and Oskar Morgenstern. 1944. Theory of Games and Economic
Behavior. Princeton University Press.
12: John Nash. 1950. Non Cooperative Games. PhD Thesis. Princeton University.
13: John Nash. 1951. The Bargaining Problem. Econometrica.
14: Thomas Schelling. 1960. The Strategy of Conflict. Harvard University Press.
15: John Harsanyi. 1977. Rational Behavior and Bargaining Equilibrium in Games and Social
Situations. Cambridge University Press.
16: Herbert Simon. 1982. Model of Bounded Rationality. MIT Press.
17: Robert Axelrod. 1984. The Evolution of Cooperation. Basic Books.
18: Reinhard Selten. 1988. Models of Strategic Rationality. MIT Press.
19: Robert Aumann. 1989. Lectures on Game Theory. Underground Classics.
20: Robert Aumann. 1995. Repeated Games with Incomplete Information. MIT Press.
21: Sylvia Nasar. 1998. A Beautiful Mind. Simon and Schuster.
22: Tom Siegfried. 2006. A Beautiful Math: John Nash, Game Theory, and a Modern Quest for
Nature. Joseph Henry Press.
23: Lawrence Freedman. 2013. Strategy: A History. Oxford University Press.
 The End 

A BRIEF INCOMPLETE HISTORY OF GAME THEORY

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