# The ace game theorists

## While Robert John Aumann applied his mathematical skills to developing his works on repeated games, Thomas Crombie Schelling laid out simple and lucid models of conflict resolution — both expanding the frontiers of Game theory in a significant way

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The Nobel Prize in Economic Sciences in 2005 was jointly awarded to Robert J Aumann who was then at the Centre for Rationality, Hebrew University of Jerusalem, Israel; and Thomas C Schelling who was then at University of Maryland, *"for having enhanced our understanding of conflict and cooperation through game-theory analysis". *

Aumann did his BSc in mathematics from the City College of New York in 1950. Thereafter he did his MSc in 1952 and PhD in 1955 — both in mathematics, from MIT. As an undergraduate student, he was particularly interested in number theory and was influenced by the works of Edmund Landau. At MIT he studied modern branches of mathematics like algebraic topology. More specifically, he got interested in knot theory, a branch of algebraic topology that deals with the properties of knots (those tied in ropes). His doctoral dissertation, *'Asphericity of Alternating Linkages*', also concerned knot theory, for which his advisor was George Whitehead, Jr.

After finishing his PhD, he joined the mathematics faculty of the Hebrew University of Jerusalem in 1956 and has been a visiting professor at Stony Brook University since 1989. He has also held visiting professorship at the University of California, Berkeley (1971, 1985–1986), Stanford University (1975–1976, 1980–1981), and Université Catholique de Louvain (1972, 1978, 1984).

Schelling did his undergraduate studies in economics from the University of California, Berkeley in 1944 and received his PhD in economics from Harvard University in 1951. Schelling did not join the academia immediately after his PhD. He began his career working for federal agencies and programmes such as the US Bureau of the Budget (1945–46), the Marshall Plan in Europe (1948–50), and the Executive Office of the President (1951–53). He first joined the Economics faculty at Yale University only in 1953, where he stayed till 1958. Schelling was also a senior staff member of the RAND Corporation (1958–59), where his analysis of the nuclear arms race between the United States and the Soviet Union led to his publication of *'The Strategy of Conflict'* (1960). After Yale, he moved to Harvard University in 1958, where he stayed till 1990, before moving to University of Maryland thereafter. Schelling specialised in the application of game theory to cases in which adversaries must repeatedly interact, especially in international trade, treaties, and conflicts.

**Main works of Robert Aumann**

While Aumann was initially interested in pure mathematics and rather esoteric subjects such as 'knot theory', he got drawn to Game theory when he went to work for the Analytical Research Group (ARG) — a group affiliated with Princeton University's mathematics department. One of the problems assigned to him involved defending a city from attack by a squadron of aircraft, some armed with weapons and some sent as decoys. He recalled his conversations with John Nash, who had come to MIT as an instructor and thought that game theory would be appropriate to solve this problem.

Aumann is best known for his contribution to repeated games — the situation in which players encounter the same situation over and over again. He worked on conditions under which cooperation between parties arises; what is cooperation a function of? the number of participants, frequency of interaction etc. He also inquired into the reasons why cooperation is broken off — is it because of short time horizons or because the action of the other party can't be observed? Aumann's ideas were first presented in his 1959 paper *'Acceptable Points in General Cooperative n-Person Games'*. Aumann did a more thorough analysis of repeated games in his later work* 'Survey of Repeated Games',* published in 1981. Aumann's work found applications in varied areas such as price wars, international conflict, wage contracting etc.

Aumann was also the first to introduce the concept of correlated equilibrium, which takes off from the earlier concept of Nash equilibrium. We may recall that in a Nash equilibrium, no player wants to change his strategy, given the other player's strategy. Correlated equilibrium, on the other hand, describes a scene where players receive advice from a trusted mediator (or "correlating device") about what strategy to play. The mediator's advice forms a correlated equilibrium if no individual player has an incentive to deviate from the advice he has received, if he believes the other players are each following their own advice. Aumann showed that the set of correlated equilibria can contain more than just combinations of Nash equilibria: it can include forms of play that aren't Nash equilibria at all, but that sometimes result in a more positive societal outcome than any of the Nash equilibria.

Aumann collaborated with other game theorists such as Maschler and Lloyd Shapley on various aspects of repeated games. With Maschler, Aumann wrote a paper in 1964 titled *'The Bargaining Set for Cooperative Games'* on the issue of bargaining in cooperative games. The other work with Maschler was the 1985 paper *'Game-Theoretic Analysis of a Bankruptcy Problem from the Talmud',* which is an explanation of a difficult passage in the Babylonian Talmud. This is also referred to as the 'division problem', which is a dilemma of explaining the division of the heritage of a late husband between three surviving wives. Finally, an important joint work with Maschler was on repeated games of incomplete information. This began as a project to help shape United States policy in the arms control negotiations that were taking place with the Soviet Union at that time.

Aumann also collaborated with Lloyd Shapley on the Aumann–Shapley value, which is presented in the 1974 book titled *'Values of Non-Atomic Games'*. The 'Shapley value' of a finite multi-person game associates to each player the amount he should be willing to pay to participate. This book extended the value concept to certain classes of non-atomic games, which are infinite-person games in which no individual player has significance. Another collaboration with Shapley was on perfect equilibria in repeated games.

**Main works of ****Thomas Schelling**

While he was at RAND Corporation in 1957, and later in the government, he worked on issues such as arms race, nuclear deterrence and international negotiations, wherein he constructed game theoretic models. It was this work that led to his above-mentioned famous book '*The Strategy of Conflict'* in 1960. Schelling illustrated his arguments with the US–USSR conflict and explained in simple and clear terms using game theory. One of the areas that 'The Strategy of Conflict' proposed was that international negotiations or arms race were a non-zero-sum game and that cooperation could be a likely outcome. An important insight from his book was that the ability to retaliate is more of a deterrent than the ability to resist an attack—i.e., a country's best defense against nuclear war is the protection of its weapons rather than its people. Also, the uncertainty of an attack was far more potent to preserve peace as compared to the certainty of an attack. To quote from the Nobel website:

*Against the backdrop of the nuclear arms race in the late 1950s, Thomas Schelling's book The Strategy of Conflict set forth his vision of game theory as a unifying framework for the social sciences. Schelling showed that a party can strengthen its position by overtly worsening its own options, that the capability to retaliate can be more useful than the ability to resist an attack, and that uncertain retaliation is more credible and more efficient than certain retaliation. These insights have proven to be of great relevance for conflict resolution and efforts to avoid war. *

Another work for which Schelling became well known was his *'Essay on Bargaining'*, which was about the nuts and bolts of negotiations such as threats and promises, secret knowledge and strategies. Tobin remarked on Schelling works thus: *"His inspired work on non-zero-sum and cooperative games…We [Yale] had great seminars where he made us decide where [to meet], how we would divide a joint surplus or a mapped plot of land .... For weeks no one talked about anything else. Game theory has never been as much fun or as relevant." *

Schelling also proposed the idea of limited or graduated reprisals in his book *'Arms and Influence'* (1966). This was adopted by the United States in 1965 as *'Operation Rolling Thunder*', which involved the bombing of selected targets in North Vietnam in the expectation that it would deter the North Vietnamese from continuing the war. Even though Schelling had advised against escalating the bombing, the US went ahead with unhappy outcomes. While at Harvard, Schelling applied game theory to international trade negotiations, which led to two highly influential books — *Micromotives and Macrobehavior* (1978) and *Choice and Consequence (1984)*. The former work explained the tendency of urban neighbourhoods toward absolute segregation.

Schelling was much ahead of his times and had warned against global warming in 1992 and proposed a carbon tax to address it. However, he later became controversial in 2002, when he supported President Bush's withdrawal from the Kyoto Protocol on the grounds that there was no solid proof of a link between global warming and greenhouse gases.

**Conclusion **

Aumann was first a mathematician and then a game theorist. As noted above, he was interested in pure mathematics from the beginning; his PhD dissertation on an esoteric topic such as knot theory testifies this. He got attracted to game theory after a meeting with John Nash and the 1957 Luce–Raiffa book, *'Games and Decisions'*, which dealt with repeated games, made a big impression on him. But he basically remained a mathematician and brought his considerable skills to the field of game theory.

Schelling, on the other hand, was an economist with mathematical skills nowhere near those of Aumann. But Schelling had a knack of presenting complex problems in a simple and lucid style. However, his forte was conflict resolution and this was amply clear not only in his works reviewed above, but also in his Nobel lecture where he argues against the use of nuclear weapons and speaks in favour of international treaties to curb testing of such weapons. To quote from his lecture:

*I know of no argument in favour of the Comprehensive Test Ban Treaty, which the Senate rejected in 1999, more powerful than the potential of that treaty to enhance the nearly universal revulsion against nuclear weapons. The symbolic effect of nearly 200 nations ratifying the CTBT, which is nominally only about testing, should add enormously to the convention that nuclear weapons are not to be used and that any nations that do use nuclear weapons will be judged the violator of the legacy of Hiroshima. I never heard that argument (was) made on either side of the debate over the Treaty. When the Treaty is again before the Senate, as I hope it will be, this major potential benefit should not go unrecognized. The most critical question about nuclear weapons for the United States Government is whether the widespread taboo against nuclear weapons and its inhibition on their use is in our favour or against us. If it is in the American interest, as I believe obvious, advertising a continued dependence on nuclear weapons, i.e., a US readiness to use them, a US need for new nuclear capabilities (and new nuclear tests) – let alone ever using them against an enemy – has to be weighed against the corrosive effect on a nearly universal attitude that has been cultivated through universal abstinence of sixty years. *

On the whole, both Aumann and Schelling's work led to new developments in game theory and a more widespread use of game theoretic models in social sciences.

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