John Forbes Nash Jr (1928-2015)
The mathematician John Forbes Nash Jr, shared the Royal Bank of Sweden Prize in Economic Science in Memory of Alfred Nobel with John Harsanyi and Reinhard Selten in 1994 for his contributions in the early 1950s to the theory of strategic games: his proof of the existence of Nash equilibrium in non-cooperative games with compact strategy spaces, the Nash bargaining solution for cooperative games (games in which binding agreements are possible among players, with costless enforcement of contracts), and the Nash programme linking non-cooperative and cooperative games.
Nash equilibrium as a solution concept for non-cooperative games has been fundamental to the expansion of game theory from a small sub-field within mathematical economics to increasingly pervasive influence throughout economics, the other social sciences, law, business strategy, and evolutionary biology. In the words of Nobel laureate Roger Myerson (1991: 105), “Nash’s (1951) concept of equilibrium is probably the most important solution concept in game theory” (see also Myerson 1999).John Forbes Nash Jr was born on 13 June 1928, in Bluefield, West Virginia, to an electrical engineer and his wife, a retired school teacher. Nash showed an early aptitude for mathematics and enrolled in advanced mathematics courses at Bluefield College while still in high school. He continued on to the Carnegie Institute of Technology in Pittsburgh (now Carnegie Mellon University) on a full scholarship, initially majoring in chemical engineering but switching to mathematics during his first year. He completed his studies in only three years, graduating in 1948 with both a BSc. and an MSc. in mathematics, making such an impression in the Mathematics Department that he continued on to Princeton University for a PhD in mathematics with a one-sentence reference letter from his adviser: “This man is a genius.” Nash submitted his doctoral dissertation in 1950, and was academically active for ten years, teaching at the Massachusetts Institute of Technology (MIT) and doing research at the RAND Corporation.
During that decade he published five seminal papers in game theory and ten papers, viewed by many mathematicians as being of perhaps even greater significance, in topology and differential equations (Kuhn and Nasar 2002 reprint Nash’s most important mathematical articles on the imbedding problem for Riemannian manifolds and on continuity of solutions of parabolic and elliptic equations, as well as his papers in game theory). His decade of productivity was followed by an illness-induced absence from research of some 25 years, during which he was diagnosed as paranoid schizophrenic, and then had a remarkable recovery (see Nasar 1998). A movie about his life, A Beautiful Mind (Howard 2001), won the Academy Award for Best Picture.In 1928, John von Neumann proved the minimax theorem that, as long as mixed strategies are allowed (a linear combination of each player’s possible actions, attaching a probability to each possible pure strategy), at least one equilibrium point exists for any two-person, zero-sum game: a proof simplified by Jean Ville in 1938 (see von Neumann and Morgenstern 1944). This result, together with von Neumann and Morgenstern’s Theory of Games and Economic Behavior, is widely credited with laying the foundations of game theory. However, von Neumann and Morgenstern (1944) did not have any existence proof for the stable set, their solution concept for n-person games (and in the 1960s William Lucas demonstrated that the stable set solution does not always exist). John Nash (1950b, 1950c, 1951) proved that any non-cooperative, normal form game has at least one equilibrium point, regardless of the number of players or whether the sum of the payoffs is constant, provided only that mixed strategies are allowed and that the space of strategies available to the players is compact (bounded and closed). Nash used the Brouwer fixed point theorem (any continuous function mapped from a compact and convex subset of Euclidean space to the same subset maps at least one point to itself) to prove existence of equilibrium in his thesis (1950b, 1951).
At David Gale’s suggestion, Nash (1950c) used the Kakutani fixed point theorem (extending Brouwer’s result to multi-valued mappings) for a more elegant and more general proof in a published note. Nash (1950b: 1, 3) defined a non-cooperative game as one in which “each participant acts independently, without collaboration or communication with any of the others” and the equilibrium of a non-cooperative game as “an n-tuple S such that each player’s mixed strategy maximizes his pay-off if the strategies of the others are held fixed”. For the outcome of a game to not be a Nash equilibrium would violate the assumption of individual rationality, since at least one player could increase his or her payoff by acting differently. A second interpretation of Nash equilibrium in Nash’s unpublished dissertation (but not in his articles), in terms of statistical populations (“mass action”) rather than individual rationality, later proved useful in evolutionary biology, in the words of the Nobel Prize citation, “in order to understand how the principles of natural selection operate in strategic interaction within and among species”.After Nash published, Martin Shubik and other economists reinterpreted A.A. Cournot’s 1838 analysis of equilibrium among oligopolists, each maximizing profit taking the output of the others as given, as a special case of Nash equilibrium. When Nash told von Neumann (a fellow at the Institute for Advanced Study in Princeton) about his stronger and more general proof of existence of equilibrium, von Neumann dismissed the young graduate student’s result as “just another fixed point theorem” and as “trivial” which it isn’t (quoted by Kuhn and Nasar 2002: xix and caption to photograph no. 4). The preface to the 1953 third edition of von Neumann and Morgenstern (1944) made a bare mention of Nash (1951) in connection with “further developments” of n-person game theory in the direction of non-cooperative games. Not distinguishing between cooperative and non-cooperative games, and hence not ruling out coalitions with binding agreements, was the stumbling block that kept von Neumann and Morgenstern from proving existence of equilibrium for n-person games.
Nash (1950a, 1953) proposed the Nash bargaining solution for cooperative games. His first published article, “The bargaining problem” (1950a), was written while taking Albert Tucker’s game theory seminar at Princeton, but was based on an idea Nash had while taking an international trade course at Carnegie Institute of Technology, the only economics course he ever took. Nash suggested four conditions that two players might plausibly wish an arbitrator to follow in dividing the gains from cooperation between them: Pareto efficiency, symmetry (it does not matter if the identities of the two players are switched), independence of a positive linear transformation of a player’s utility function, and independence of irrelevant alternatives (that is, if a point that is neither the threat point or the solution becomes infeasible, the solution is not altered). He then proved that the only solution that satisfies all four conditions is to maximize the product of each player’s gain in utility relative to the threat point (the threat point being the fallback utilities that the players would receive if no agreement was reached). Of the four conditions, independence of irrelevant alternatives is the one usually omitted by proponents of other solution concepts for cooperative games. The Nash programme linked cooperative and non-cooperative games by identifying the threat point of a cooperative game as the Nash equilibrium of the non-cooperative game that would be played if no agreement was reached. One other Nash contribution has been relevant for economics: methods developed in a 1958 article by Nash (reprinted in Kuhn and Nasar 2002: ch. 12) are used to solve a class of parabolic partial differential equations that occur in certain finance problems. After his remarkable recovery from decades of illness, John Nash received a National Science Foundation grant in 2001 to work on a new evolutionary solution concept for cooperative games. He died on 23 May 2015, in an automobile accident.
Robert W. Dimand and Khalid Yahia
See also:
Formalization and mathematical modelling (III); Game theory (III); John von Neumann (I).
References and further reading
Howard, R. (director) (2001), A Beautiful Mind (movie), Hollywood, CA: Universal Studios.
Kuhn, H.W. and S. Nasar (eds) (2002), The Essential John Nash, Princeton, NJ: Princeton University Press. Myerson, R.B. (1991), Game Theory: Analysis of Conflict, Cambridge, MA: Harvard University Press. Myerson, R.B. (1999), ‘Nash equilibrium in the history of economic theory’, Journal of Economic Literature, 37 (3), 1067-82.
Nasar, S. (1998), A Beautiful Mind, New York: Simon and Schuster.
Nash, J.F. Jr (1950a), ‘The bargaining problem’, Econometrica, 18 (2), 155-62.
Nash, J.F. Jr (1950b), ‘Non-cooperative games’, PhD dissertation, Princeton University, facsimile reprint in H.W. Kuhn and S. Nasar (eds) (2002), The Essential John Nash, Princeton, NJ: Princeton University Press, pp. 53-84.
Nash, J.F. Jr (1950c), ‘Equilibrium points in n-person games’, Proceedings of the National Academy of Sciences, 36 (1), 48-9.
Nash, J.F. Jr (1951), ‘Non-cooperative games’, Annals of Mathematics, 54 (2), 286-95.
Nash, J.F. Jr (1953), ‘Two-person cooperative games’, Econometrica, 21 (1), 128-40.
Nash, J.F. Jr (2008), ‘The agencies method for modeling coalitions and cooperation in games’, International Game Theory Review, 10 (4), 539-64.
Nash, J.F. Jr (2014), ‘Research studies approaching cooperative games with new methods,’ in R.M. Solow and J. Murray (eds), Economics for the Curious: Inside the Minds of 12 Nobel Laureates, Basingstoke: Palgrave Macmillan, pp. 112-22.
Rubinstein, A. (1995), ‘John Nash: the master of economic modeling’, Scandinavian Journal of Economics, 97 (1), 9-13.
Von Neumann, J. and O. Morgenstern (1944), Theory of Games and Economic Behavior, Princeton, NJ: Princeton University Press, 3rd edn 1953, sixtieth anniversary edn with introduction by H.W. Kuhn and afterword by A. Rubinstein 2004.