Shapley value
The Shapley value is a solution concept in cooperative game theory. It was named in honor of Lloyd Shapley, who introduced it in 1951 and won the Nobel Prize in Economics for it in 2012. To each cooperative game it assigns a unique distribution of a total surplus generated by the coalition of all players. The Shapley value is characterized by a collection of desirable properties. Hart provides a survey of the subject.
The setup is as follows: a coalition of players cooperates, and obtains a certain overall gain from that cooperation. Since some players may contribute more to the coalition than others or may possess different bargaining power, what final distribution of generated surplus among the players should arise in any particular game? Or phrased differently: how important is each player to the overall cooperation, and what payoff can he or she reasonably expect? The Shapley value provides one possible answer to this question.
For cost-sharing games with concave cost functions, the optimal cost-sharing rule that optimizes the price of anarchy, followed by the price of stability, is precisely the Shapley value cost-sharing rule.
Formal definition
Formally, a coalitional game is defined as:There is a set N and a function that maps subsets of players to the real numbers: , with, where denotes the empty set. The function is called a characteristic function.
The function has the following meaning: if S is a coalition of players, then, called the worth of coalition S, describes the total expected sum of payoffs the members of can obtain by cooperation.
The Shapley value is one way to distribute the total gains to the players, assuming that they all collaborate. It is a "fair" distribution in the sense that it is the only distribution with certain desirable properties listed below. According to the Shapley value, the amount that player i gets given a coalitional game is
where n is the total number of players and the sum extends over all subsets S of N not containing player i. The formula can be interpreted as follows: imagine the coalition being formed one actor at a time, with each actor demanding their contribution − as a fair compensation, and then for each actor take the average of this contribution over the possible different permutations in which the coalition can be formed.
An alternative equivalent formula for the Shapley value is:
where the sum ranges over all orders of the players and is the set of players in which precede in the order. Finally, it can also be expressed as
which can be interpreted as
Examples
Business example
Consider a simplified description of a business. An owner, o, provides crucial capital in the sense that without him no gains can be obtained. There are k workers w1,...,wk, each of whom contributes an amount p to the total profit. LetThe value function for this coalitional game is
where m is the cardinality of. Computing the Shapley value for this coalition game leads to a value of for the owner and for each worker.
Glove game
The glove game is a coalitional game where the players have left- and right-hand gloves and the goal is to form pairs. Letwhere players 1 and 2 have right-hand gloves and player 3 has a left-hand glove.
The value function for this coalitional game is
The formula for calculating the Shapley value is
where is an ordering of the players and is the set of players in which precede in the order.
The following table displays the marginal contributions of Player 1.
Observe
By a symmetry argument it can be shown that
Due to the efficiency axiom, the sum of all the Shapley values is equal to 1, which means that
Properties
The Shapley value has many desirable properties.Efficiency
The sum of the Shapley values of all agents equals the value of the grand coalition, so that all the gain is distributed among the agents:Proof:
since is a telescoping sum and there are |N|! different orderings R.
Symmetry
If and are two actors who are equivalent in the sense thatfor every subset of which contains neither nor, then.
This property is also called equal treatment of equals.
Linearity
If two coalition games described by gain functions and are combined, then the distributed gains should correspond to the gains derived from and the gains derived from :for every in . Also, for any real number,
for every in .
Null player
The Shapley value of a null player in a game is zero. A player is null in if for all coalitions that do not contain.Given a player set, the Shapley value is the only map from the set of all games to payoff vectors that satisfies all four properties: Efficiency, Symmetry, Linearity, Null player.
Stand-alone test
If v is a subadditive set function, i.e.,, then for each agent i:.Similarly, if v is a superadditive set function, i.e.,, then for each agent i: .
So, if the cooperation has positive externalities, all agents gain, and if it has negative externalities, all agents lose.
Anonymity
If i and j are two agents, and w is a gain function that is identical to v except that the roles of i and j have been exchanged, then. This means that the labeling of the agents doesn't play a role in the assignment of their gains.Marginalism
The Shapley value can be defined as a function which uses only the marginal contributions of player i as the arguments.Characterization
The Shapley value not only has desirable properties, it is also the only payment rule satisfying some subset of these properties. For example, it is the only payment rule satisfying the four properties of Efficiency, Symmetry, Linearity and Null player. See for more characterizations.Aumann–Shapley value
In their 1974 book, Lloyd Shapley and Robert Aumann extended the concept of the Shapley value to infinite games, creating the diagonal formula. This was later extended by Jean-François Mertens and Abraham Neyman.As seen above, the value of an n-person game associates to each player the expectation of his contribution to the worth or the coalition or players before him in a random ordering of all the players. When there are many players and each individual plays only a minor role, the set of all players preceding a given one is heuristically thought as a good sample of the players so that the value of a given infinitesimal player around as "his" contribution to the worth of a "perfect" sample of the population of all players.
Symbolically, if is the coalitional worth function associating to each coalition measured subset of a measurable set that can be thought as without loss of generality.
where denotes the Shapley value of the infinitesimal player in the game, is a perfect sample of the all-player set containing a proportion of all the players, and is the coalition obtained after joins. This is the heuristic form of the diagonal formula.
Assuming some regularity of the worth function, for example assuming can be represented as differentiable function of a non-atomic measure on ,, with density function, with . Under such conditions
as can be shown by approximating the density by a step function and keeping the proportion for each level of the density function, and
The diagonal formula has then the form developed by Aumann and Shapley
Above can be vector valued.
In the argument above if the measure contains atoms is no longer true—this is why the diagonal formula mostly applies to non-atomic games.
Two approaches were deployed to extend this diagonal formula when the function is no longer differentiable. Mertens goes back to the original formula and takes the derivative after the integral thereby benefiting from the smoothing effect. Neyman took a different approach. Going back to an elementary application of Mertens's approach from Mertens :
This works for example for majority games—while the original diagonal formula cannot be used directly. How Mertens further extends this by identifying symmetries that the Shapley value should be invariant upon, and averaging over such symmetries to create further smoothing effect commuting averages with the derivative operation as above. A survey for non atomic value is found in Neyman