Abstract
The goal of this chapter is to illustrate two mathematical game theory concepts for allocating costs (savings) between cooperating participants, specifically in healthcare settings. These concepts are the nucleolus and the Shapley value. The focus of this chapter is on the practical application of the Shapley value for the cost sharing within the bundled payments model for the episodes of care mandated recently by the Center for Medicare Services (CMS). The general Shapley value methodology is illustrated, as well as an important particular case in which each participant uses only a portion of the largest participant's asset (the so-called airport game). The intended readers are primarily leaders of organizations and hospitals involved in the implementation of the CMS-mandated bundled payment model for the episodes of care.
TopBackground
By pooling resources and cooperating the participants usually reduce the total joint costs and realize savings. The question arises is how the reduced costs or the realized savings should be fairly allocated between them.
There could be different definitions of fair allocation. Some of them are:
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Equitable Allocation: Gives everyone the same satisfaction level, i.e. the proportion each player receives by their own valuation is the same for all of them. This is a difficult aim as players might not be truthful if asked their valuation.
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Proportional Allocation: Guarantees that each player gets his share. For instance, if three people divide up an asset then each gets at least a third by their own valuation.
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Envy-Free Allocation: Everyone prefers his own share to the others. No one is jealous of anyone else. No one would trade his share with anyone else’s.
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An Efficient or Pareto Optimal Allocation: Ensures that no other allocation would make someone better off without making someone else worse off. The term efficiency comes from the economics idea of the efficient market.
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Merit-Based Allocation: The more one brings to the coalition, the more one gets out of the division of the accumulated gains.
A concept of fairness is rather subjective. It depends on the participants’ socio-economic views and other factors. The fairness schemes described in the next sections form a basis of the two most popular cost allocation approaches: the nucleolus (Tijs and Driessen, 1986; Saad, 2009) and the Shapley value (Roth, 1988; Young, 1994).
Key Terms in this Chapter
Game Theory: A branch of applied mathematics that studies strategic situations in which participants (players) act rationally in order to maximize their returns (payoffs).
Cost Allocation: A problem that arises in many business situations that benefit from the effect of economy of scale or cooperating partners.
Core: A set of inequalities that meet the requirement that no participant or a group of participants pays more than their stand-alone cost.
Marginal Contribution: A value of the group with the player as a member minus the value of the group without the player minus the value created by the player working alone.
Shapley Value: A game theory concept aimed at the ‘fair’ allocation of the collective costs or profits (savings) between several collaborative participants. It is based on allocating the costs to the cooperating participants proportionally to the marginal contributions of each participant that is averaged over all possible combinations in which participants can cooperate.
Empty Core: A lack of unique cost allocation that satisfies all participants. If the core is empty, then unsatisfied participants have incentive to leave the cost sharing coalition.
Bundled Payment: A payment model to healthcare providers in which one single payment is disbursed to cover an episode of patient care through a contracting organization. The contracting organization is responsible for allocating the payments among all providers.
Coalition: A group of k cooperating partners.
Nucleolus: A game theory concept defined as minimizing the maximum “unhappiness” of a coalition. “Unhappiness” (or “excess”) of a coalition is defined as the difference between what the members of the coalition could get by themselves and what they are actually getting if they accept the allocations suggested by the nucleolus.