Value of a Game

The value of a game is a solution concept for cooperative games used in both forecasting (e.g., behavioral models) and normative contexts (e.g., fair division). Existence and uniqueness are guaranteed for this solution, which has two interpretations: the outcome of bilateral bargaining and the share-out resulting from bargaining based on the winnings of various possible coalitions.

The concept of value of the first type was proposed by Frederik Zeuthen in 1930 for bargaining problems involving two persons. It was further developed by John F. Nash in 1950 and 1953. These solutions were then extended to games with more than two players by John C. Harsanyi in 1959, among others.

The concept of value of the second type was introduced in 1951 by Lloyd S. Shapley and formalized by him in 1953. The Shapley value is characterized by four axioms: efficiency, symmetry, dummy player, and additivity. Its motivation was to overcome problems associated with the principal existing solution concept for n-person cooperative games, namely the stable sets introduced by John Von Neumann and Oskar Morgenstern in 1944. For such sets, neither uniqueness nor existence was guaranteed. The Shapley value, in contrast, guaranteed both existence and uniqueness.

In 1954, Shapley, working with Martin Shubik, elaborated a version of the Shapley value for simple games, that is, for games where the characteristic function can have values only of 1 (for winning coalitions) or 0 (for losing coalitions). Simple games are especially suited to representing voting situations, and value may be characterized as the voting power index. More generally, concepts proposed as power indices have been generalized as values and vice versa. The Penrose—Banzhaf—Coleman index and the Public Goods Index are examples of the former, the Tijs value of the latter. Some of these concepts have been axiomatically characterized, typically using axioms other than Shapley's axiom of additivity.

More recent studies have defined values for games with a priori unions between players. For the Shapley value, the earliest result was by Robert Aumann and Jacques Drèze in 1974, with later contributions by Roger Myerson. Guillermo Owen and others have proposed an analogous modification regarding the Banzhaf index.