The Nobel series: Masters of game theory

The Nobel Prize in Economic Sciences in 1994 was awarded jointly to John F Nash, Princeton University, USA; John C Harsanyi, University of California, Berkeley, USA; and Reinhard Selten, Rheinische Friedrich-Wilhelms-Universität, Bonn, Germany, "for their pioneering analysis of equilibria in the theory of non-cooperative games". The three awardees took forward the legacy of the pioneer game theorists — Von Neumann and Oskar Morgenstern.

Nash attended Carnegie Mellon University for his undergraduate studies. He initially began majoring chemical engineering but switched to chemistry later, and eventually to mathematics. After graduating in 1948 with a MS in mathematics, Nash went to Princeton University for his PhD in mathematics and science, which he completed in 1950. His PhD dissertation was of mere 26 pages with only two citations — one of the shortest in graduate study in the USA. In 1951, he joined the faculty of the Massachusetts Institute of Technology (MIT) where he pursued research into partial differential equations. He resigned in the late 1950s after bouts of mental illnesses. Nash then began an informal association with Princeton where he became a senior research mathematician in 1995. The 2001 film 'A Beautiful Mind' has captured the life and works of John Nash very well, with Russell Crowe essaying the role of Nash.

Harsanyi first wanted to study chemical engineering in Lyon, France. But owing to the outbreak of the war, he returned home to Hungary and enrolled in the University of Budapest to study pharmacology. Harsanyi was pushed into forced labour for seven months because of his Jewish descent but he escaped after that. After the war, he completed his PhD from the University of Budapest in philosophy and sociology in 1947. Later, he attended Sydney University (MA, 1953) to study economics; he then immigrated to the United States and attended Stanford University from where did PhD in economics in 1959. From 1964, he was a professor at the Haas School of Business in the University of California, Berkeley.

Selten studied mathematics at Goethe University, Frankfurt and obtained his diploma in 1957. He then worked as scientific assistant to Heinz Sauermann until 1967. In 1961, he received his doctorate at Frankfurt in mathematics with a thesis on the evaluation of n-person games. He was a visiting professor at Berkeley where he taught from 1969 to 1972 at the Free University of Berlin and, from 1972 to 1984, at the University of Bielefeld. He then accepted a professorship at the University of Bonn.

In this paper, we will discuss the main works of the three Nobel laureates and the application of their works.

Main works of John Nash

Nash was an outstanding mathematician and is best known for making a distinction between cooperative games (where binding agreements can be made) and non-cooperative games (where binding agreements are not feasible).

Nash was noticed for the first time when he was still at graduate school and wrote his first paper 'The Bargaining Problem' in the journal Econometrica. He expanded on this paper in his doctoral thesis 'Non-Cooperative Games' which appeared in September 1951 in the journal 'Annals of Mathematics'. Nash thus established the mathematical principles of game theory, a branch of mathematics that examines the rivalries between competitors with mixed interests. Nash showed that for any finite game (meaning that there are finite 'moves' in the game), all the players can arrive at an optimal outcome, known as the Nash equilibrium. This is nothing but an outcome which includes actions of players; such action for a player remains the same, no matter what action the other player chooses. There are two interpretations that Nash gave to Nash equilibrium — one based on rationality and the other on statistical populations. As the Nobel website tells us:

This solution concept later came to be called Nash equilibrium. In a Nash equilibrium, all of the players' expectations are fulfilled and their chosen strategies are optimal. Nash proposed two interpretations of the equilibrium concept: one based on rationality and the other on statistical populations. According to the rationalistic interpretation, the players are perceived as rational and they have complete information about the structure of the game, including all of the players' preferences regarding possible outcomes, where this information is common knowledge. Since all players have complete information about each others' strategic alternatives and preferences, they can also compute each others' optimal choice of strategy for each set of expectations. If all of the players expect the same Nash equilibrium, then there are no incentives for anyone to change his strategy. Nash's second interpretation – in terms of statistical populations – is useful in so-called evolutionary games. This type of game has also been developed in biology in order to understand how the principles of natural selection operate in strategic interaction within and among species. Moreover, Nash showed that for every game with a finite number of players, there exists an equilibrium in mixed strategies.

From 1952, Nash taught at the Massachusetts Institute where his research on the theory of real algebraic varieties, Riemannian geometry, parabolic and elliptic equations was extremely deep and significant. In fact, his work 'The imbedding problem for Riemannian manifolds' published in 1956 is considered a pathbreaking study in mathematics. After this, Nash worked on ideas that would appear in his paper 'Continuity of solutions of parabolic and elliptic equations' which was published in the 'American Journal of Mathematics' in 1958.

Nash got glowing tributes from his colleagues and students. P Ordeshook wrote: The concept of a Nash equilibrium n-tuple is perhaps the most important idea in non-cooperative game theory. ... Whether we are analysing candidates' election strategies, the causes of war, agenda manipulation in legislatures, or the actions of interest groups, predictions about events reduce to a search for and description of equilibria. Put simply, equilibrium strategies are the things that we predict about people.

Three comments from fellow students throw light on the genius of Nash:

❃Nash was out of the ordinary. If he was in a room with twenty people, and they were talking, if you asked an observer who struck you as odd it would have been Nash. It was not anything he consciously did. It was his bearing. His aloofness.

❃Nash was totally spooky. He wouldn't look at you. He'd take a lot of time answering a question. If he thought the question was foolish, he wouldn't answer at all. He had no effect. It was a mixture of pride and something else. He was so isolated but there really was underneath it all a warmth and appreciation of people.

❃A lot of us would discount what Nash said. ... I wouldn't want to listen. You didn't feel comfortable with the person.

Main works of Harsanyi

Harsanyi worked on games of incomplete information in which players don't know what other players' strategies and objectives are. His work found application in the economics of information as we shall see below.

Basically, Harsanyi extended Nash's work. While Nash had focused on games in which each player knew the other players' preferences, Harsanyi introduced the assumption that players have incomplete information about other players. Harsanyi provided the framework for analysing games with incomplete information and his work was published in 1967-68 in three papers. These articles were published under the title 'Games with Incomplete Information Played by Bayesian Players' (Management Science 14, 159-82, 320-34 and 486-502).

He assumed that each player is one of several "types". Each type represents a set of possible preferences for the player and a set of subjective probabilities that the player places on the other players' type. Each player then chooses a strategy for each of his types. Harsanyi showed that for every game with incomplete information, there is an equivalent game with complete information.

Harsanyi's contributions to moral philosophy are also well known. As early as 1955, Harsanyi had pioneered the "veil of ignorance" concept (though not by that name) that philosopher John Rawls made famous in his 1971 book, 'A Theory of Justice'. Harsanyi was a strong defender of the "rule of utilitarianism", the idea that the most ethical act is to follow the rule that will yield the most happiness.

While Harsanyi's work has found many applications, it has perhaps been most applied to auctions, where players don't know each other's objectives and each player has to place some subjective probability on other players' types. Other areas where Harsanyi's theory has been applied is in the analysis of oligopolies, international negotiations on trade policy or environmental policy, and wage negotiations in the labour market.

Main works of Reinhard Selten

The main work for which Selten is known is the concept of subgame perfection. This is essentially a refinement of the Nash equilibria concept. While Nash set the tone for non-cooperative games, his framework gave rise to a number of equilibria, all of which were not relevant or interesting. Selten refined this by introducing stricter conditions that eliminated equilibria that were economically infeasible. Selten's work can be found in the publication 'Spieltheoretische Behandlung eines Oligopolmodells mit Nachfrageträgheit' (Zeitschrift für die Gesamte Staatswissenschaft 121, 301-24 and 667-89, 1965).

In Selten's world, competitors react only to threats and promises that they perceive as credible. The winning players are those who learn to distinguish credible threats and promises from bluffing, and this process of bluffing, counter-bluffing, and discernment becomes a feature of the ongoing game. Selten's work has been applied to topics in industrial organisation such as analysis of predatory pricing, entry deterrence, oligopolies, and similar competitive situations. It has even been applied to models in other social sciences, and also in biology and computer science.

Selten (1975) made another seminal contribution to game theory, one that shaped the research agenda for decades to come. There, he defined a new, finer notion of perfectness, called 'trembling hand perfection' which better captured the idea of forward–looking rationality and credibility in all conceivable extensive form games. The idea was to focus only on those equilibria where agents could make some mistake with a small probability. In particular, this definition suggests that a robust definition of rationality must also deal with some deviations from rationality by the other players.

Selten has also contributed to the theory of "bounded rationality", which was first posited by Herbert A Simon, another Nobel Prize winner. It builds on the observation that human perception, judgment, and memory are nowhere near perfect as the rationality assumption of neoclassical framework would have us believe. The objective was to make economic theory more realistic and move away from the rationality assumption of the neoclassical framework.

Conclusion

Nash, Harsanyi and Selten took game theory to new frontiers. Their work has found application in all situations where there is strategic interaction between players. Hence, such areas of application have included industrial organisation (analysis of oligopolies and regulatory behaviour), international relations (the most famous application is the face-off between the US and the then USSR in the Cuban missile crisis, which I have covered in these columns earlier) and auctions (analysis of various types of auctions and concepts such as 'winners curse').

All the three have made other contributions as well. For example, Nash has developed a basic solution for cooperative games, usually referred to as Nash's bargaining solution. (In a transaction when the seller and the buyer value a product differently, a surplus is created. A bargaining solution is then a way in which buyers and sellers agree to divide the surplus). Nash also initiated the 'Nash programme', a research programme designed to base cooperative game theory on results from non-cooperative game theory. John Harsanyi has also made significant contributions to the foundations of welfare economics and to the area of morality and justice.

Selten also contributed to the development of models of nuclear deterrence under a contract with the US Arms Control and Disarmament Agency (which later became the US State Department's Bureau of Arms Control). More recently, he worked with a group of military experts to apply game theory to the strategic analysis of the Kosovo conflict. The results were published in 'Zur Lösung des Kosovo-Konfliktes' (Toward a Resolution of the Kosovo Conflict), a volume he edited with Erich Reiter, director-general of the Austrian Ministry of Defense (Nomos Verlag, 2003).

Hence, Nash gave the basic foundations of the game theoretic analysis, Harsanyi applied this to incomplete information and Selten developed it with respect to strategy and dynamics.

The writer is an IAS officer, working as Principal Resident Commissioner, Government of West Bengal. Views expressed are personal