Game Theory for Cost Allocation in Healthcare

Introduction

Game theory is a branch of applied mathematics that studies strategic situations in which participants (players) act rationally in order to maximize their returns (payoffs). As such, game theory provides models of rational behavior (decision-making) for strategic interactions.

Many types of problems that involve decision strategies for cooperating or non-cooperating participants present a fruitful ground for application of mathematical game theory (Dowd, 2004; Cachon & Netessine, 2004).

In particular, cost allocation problems arise in many situations in which participants work together, such as healthcare providers who have to coordinate patient care in order to reduce the cost and improve quality of care. It was demonstrated that a natural framework for developing methodology for cost allocation problems could be based on game theoretical concepts (Tijs & Driessen, 1986; Roth, 1988; Young, 1994; Moulin, 2003). About a dozen of alternate concepts have been proposed to determine the ‘fair’ allocation but only a few of these concepts have received wide attention: the nucleolus and the Shapley value.

In this chapter these two concepts are compared. The focus is on demonstration of the practical application of the Shapley value for the cost allocation for cooperating providers. Two cases are illustrated:

• 1.

  The general application of the Shapley value methodology, and

• 2.

  An important particular case, in which each participant uses only a portion of the largest participant’s asset (the so-called airport game).

Background

By pooling resources and cooperating the participants usually reduce the total joint costs and realize savings. The question arises is how the reduced cost or the realized saving should be allocated fairly between them.

The simplest approach is dividing the cost reduction (savings) equally between all participants. However, this does not seem fair because the different contribution of each participant to the total gain. Another approach that looks fair is sharing the savings proportionally to the participants’ own costs. However, the savings for some participants can be too low to keep them in voluntary cooperation with the bigger participants.

There could be different definitions of fair division. Some of them are:

• •

  Equitable Division: 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.

• •

  Proportional Division: 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.

• •

  Envy-Free Division: 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.

• •

  An Efficient or Pareto Optimal Division: 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.

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 section form a basis of the two most popular cost allocation approaches: the nucleolus (Tijs & Driessen, 1986; Saad, 2009) and the Shapley value (Roth, 1988; Yong, 1994).