Game Theory and Economic Analysis by [edited by] Christian Schmidt [Schmidt Christian]

Author:[edited by] Christian Schmidt [Schmidt, Christian]
Language: eng
Format: epub
Tags: General, Mathematics, Economics, Business & Economics, Theory, Electronic books, Game Theory, Wirtschaftswissenschaften
ISBN: 9780415259873
Publisher: London ; Routledge, 2002.
Published: 2002-01-15T05:00:00+00:00

Coalition and the Shapley value

Shapley (1953) solves axiomatically the problem of finding a unique solution for any game in coalitional form (N, v). Three (or four, depending on the presentation) axioms are needed to find this solution, called the Shapley value. The first axiom, or symmetry axiom, states that any permutation of players in N leaves the Shapley value unchanged (as long as values taken by function v are changed properly, the position of player i in N does not matter). Then one introduces the concept of a “carrier” of a game in coalitional form. A coalition R is said to be a carrier of the game if and only if the worth of any coalition S remains unchanged when one restricts S to its intersection with R (∀S⊆N, v(S∩R) = v(S) ). A “null player” or “dummy player” is a player who stays out of all carriers of the game. The Shapley value of a null player is zero. The second axiom, or axiom of the carrier, states that players in a carrier must share the total value of the carrier v(R) among them without allocating anything to null players (one has v(R) = v(N) ). (One sometimes divides this axiom into a null player axiom and an efficiency axiom, which amounts globally to four axioms.) The third axiom, or additivity axiom, states that the Shapley value of game (N, v + w) equals the Shapley value of game (N, v) plus the Shapley value of game (N,w).

The remarkable result of Shapley lies in the proof that there exists only one vector function (one component per player), called the Shapley value, defined on the set of games in coalitional form whose set of players is N, that satisfies the axioms presented above. One should however note that the strongest requirement comes from the second axiom, since players in a carrier R (R⊆N, the complement, if it exists, consisting in null players) must share the total available worth for N, v(N). The Shapley value for player i, noted Sh (v), is i

given by the following formula:

| S|!(| N|−| S|−1)!

Sh ( v) = Α

( v( S∪{ i})− v( S) )

i

| N|!

S⊆ N\ i

where |X| denotes the cardinal of X.

In order to compute the Shapley value for player i, it is sufficient to consider coalitions S to which i does not belong, counting for each of these coalitions the number of its members and knowing its worth v(S). (There are numerous tricks to short circuit this fastidious calculus and compute more quickly the Shapley value when faced with games presenting some kind of symmetry in coalitions’ worth or in players’ situations in N; see Aumann 1989, for instance.) One can also say that the above formula expresses the fact that the Shapley value can be seen as the weighted sum of marginal contributions (the v(S∪{i})−v(S) terms) of player i to each coalition in N.

From the point of view of the player computing its Shapley value, there is a kind of “statistical” analysis

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