Lloyd Shapley, Distinguished Fellow 2007

Lloyd Shapley has made fundamental contributions to economic theory and game theory. His work is central to the modern understanding of competitive behavior and its game theoretic underpinnings. The introduction of the core as a general solution concept for cooperative games is commonly credited to Shapley and Donald Gillies. An allocation is in the core if no coalition can improve on its members’ utilities. The core provided an important alternative interpretation of competitive equilibrium for large numbers of agents, and is now one of the central notions of cooperative game theory. It is probably the most commonly used cooperative solution concept in economics. Its development played a major role in first convincing economists that game theory provides a useful set of tools for analyzing economic problems.

In 1967, Shapley published an important existence result for the core (the Bondareva-Shapley theorem). This work introduced the notion of “balanced,” a key notion to this day for understanding when games have nonempty cores.

Shapley wrote two seminal papers on two important classes of games. In their 1962 paper “College Admissions and the Stability of Marriage,” David Gale and Shapley studied matching games with nontransferable utility. The canonical example is the marriage game, where m women are to be matched with m men. A matching is “stable” if there is no woman and man who prefer each other to their current match. Gale and Shapley showed that the set of stable matching is precisely the set of core allocations, proved that there exists a stable matching, and provided an algorithm for finding it. In another 1962 paper, Martin Shubik and Shapley studied a transferable utility version of the matching game (called an assignment game). These two games have become workhorses in the study of labor markets and other two-sided markets.

In his 1953 paper, “A Value for N-Person Games,” Shapley provided a set of axioms that, in every cooperative game, uniquely identified a payoff for each agent. This payoff has come to be called the Shapley value. The paper has been enormously influential, both through the widespread use of the Shapley value, and by inspiring other values obtained through modifications to Shapley’s original axioms. The Shapley value has been used to quantify the impact of voting rules on the influence of individual voters, where it is often called the Shapley-Shubik index (after a 1954 Shapley and Shubik paper that analyzed the influence of different members of the United Nations Security Council). This measure is of more than academic interest: A related measure (the Banzhof index) has played a role in legal decisions concerning electoral districting and representation. The Shapley value has also been widely applied in accounting to problems of cost allocation.

The first attempts to provide learning foundations for Nash equilibrium focused on fictitious play. In 1964, Shapley provided a surprising example of a game with a unique Nash equilibrium, and yet behavior under fictitious play does not converge. In 1996, Shapley with Dov Moderer introduced a class of games, “potential” games, on which some adaptive dynamics do converge to Nash equilibria.

In 1974, Robert Aumann and Shapley published Values of Non-Atomic Games, a deep investigation of the Shapley value in settings with a large number of players, showing that in economies with a continuum of agents, the allocations implied by the Shapley value coincide with those of the core, and so with the competitive allocations, providing yet another novel insight into the nature of competition.

Lloyd Shapley is one of the giants of game theory and economic theory, and he deserves recognition as a Distinguished Fellow.