CSCE475/875 Multiagent Systems

Handout 13: Coalitional Game Payoffs and The Shapley Value

November 3, 2011

(Based on Shoham and Leyton-Brown 2011)

 

Payoffs

First, let denote a mapping from a coalitional game (that is, a set of agents  and a value function ) to a vector of  real values, and let  denote the th such real value. Denote such a vector of  real values as . Each  denotes the share of the grand coalition’s payoff that agent  receives. When the coalitional game  is understood from context, we write  as a shorthand for .

 

Definition 12.2.1 (Feasible payoff) Given a coalitional game (N, v), the feasible payoff set is defined as

 

In other words, the feasible payoff set contains all payoff vectors that do not distribute more than the worth of the grand coalition. We can view this as requiring the payoffs to be weakly budget balanced.

 

Definition 12.2.2 (Pre-imputation) Given a coalitional game , the pre-imputation set, denoted , is defined as

 

We can view the pre-imputation set as the set of feasible payoffs that are efficient, that is, they distribute the entire worth of the grand coalition. Looked at another way, the pre-imputation set is the set of feasible payoffs that are strictly budget balanced. (In this setting these two concepts are equivalent; do you see why?)

 

Definition 12.2.3 (Imputation) Given a coalitional game , the imputation set, , is defined as

 

Imputations are payoff vectors that are not only efficient but individually rational.  That is, each agent is guaranteed a payoff of at least the amount that he could achieve by forming a singleton coalition.

 

The Shapley Value

 

Perhaps the most straightforward answer to the question of how payoffs should be divided is that the division should be fair.

 

First, say that agents  and  are interchangeable if they always contribute the same amount to every coalition of the other agents. That is, for all  that contains neither  nor ,  The symmetry axiom states that such agents should receive the same payments.

 

Axiom 12.2.4 (Symmetry) For any , if  and  are interchangeable then

 

Second, say that an agent  is a dummy player if the amount that  contributes to any coalition is exactly the amount that  is able to achieve alone. That is, for all  such that  The dummy player axiom states that dummy players should receive a payment equal to exactly the amount that they achieve on their own.

 

Axiom 12.2.5 (Dummy player) For any , if  is a dummy player then .

 

Finally, consider two different coalitional game theory problems, defined by two different characteristic functions and , involving the same set of agents. The additivity axiom states that if we re-model the setting as a single game in which each coalition  achieves a payoff of , the agents’ payments in each coalition should be the sum of the payments they would have achieved for that coalition under the two separate games.

 

Axiom 12.2.6 (Additivity) For any two  and , we have for any player  that , where the game  is defined by  for every coalition .

 

 

Theorem 12.2.7 Given a coalitional game , there is a unique pre-imputation  that satisfies the Symmetry, Dummy player, Additivity axioms.

 

 

 

Definition 12.2.8 (Shapley value) Given a coalitional game , the Shapley value of player  is given by

 

This expression can be viewed as capturing the “average marginal contribution” of agent , where we average over all the different sequences according to which the grand coalition could be built up from the empty coalition. More specifically, imagine that the coalition is assembled by starting with the empty set and adding one agent at a time, with the agent to be added chosen uniformly at random. Within any such sequence of additions, look at agent s marginal contribution at the time it is added to the set , its contribution is Now multiply this quantity by the different ways the set  could have been formed prior to agent ’s addition and by  different ways the remaining agents could be added afterward. Finally, sum over all possible sets  and obtain an average by dividing by  the number of possible orderings of all the agents.

 

Example of the Shapley value in action …

 

Consider the voting game.  Recall that the four political parties  and  have 45, 25, 15, and 15 representatives, respectively, and a simple majority (51 votes) is required to pass the $100 million spending bill.

 

If we want to analyze how much money it is fair for each party to demand, we can calculate the Shapley values of the game.

 

Note that every coalition with 51 or more members has a value of $100 million, and others have $0. In this game, therefore, the parties , and  are interchangeable, since they add the same value to any coalition. (They add $100 million to the coalitions  that do not include them already and to ; they add $0 to all other coalitions.)

 

The Shapley value of  is given by:

 

The Shapley value for  (and, by symmetry, also for  and ) is given by:

 

Thus the Shapley values are , which add up to the entire $100 million.