Simple Models of Competition between Two Hospitals

In the NHS internal market, hospital trusts compete with each other to sell patient care to health purchasers. We are interested in modelling the patterns of health provision resulting from the competition process; these can be compared with the optimal patterns of provision under a centrally planned system. As a starting point it is instructive to model simplified aspects of the competition process To this end, I consider a system in which there are just two trusts T1 and T2 competing for patients. In the most general form of this model, the trusts have a fnite capacities, limiting the number of patients they can treat, and the patients live in a range of locations, some closer to T1 and others closer to T2. Each trust has overheads, and also faces a marginal cost for treating each patient. Under the centrally planned system, it will be optimal to run a hospital at just one site if overheads are relatively high and patients travel costs are low, subject to the chosen hospital having su±cient capacity. Otherwise it will be optimal to run hospitals on both sites. In the internal market, each hospital will be assumed to make a charge per-patient for treatment. The purchaser(s), acting on the patients' behalf, allocate(s) each patient to one or other of the trusts: the choice is determined by the charge for treatment at each trust and the distance of the patient to each trust. Each trust sets its own treatment price. If it raises its price it will make more money from each patient, but at the cost of losing market share to the other trust. Two alternative game-theoretical scenarios may be considered: Scenario 1: Each trust seeks to maximise its profit. Scenario 2: Each trust seeks a fixed profit, corresponding to fixed return on its assets, as specified by the Department of Health [1]. Where more than one price will achieve this return, it will be assumed that the trust will set the lower or lowest price. Whichever scenario operates, the price one trust should set will in general depend on the price charged by the other. More generally, the pricing strategy of one trust will depend on the strategy of the other. Under rational bargaining, each trust's strategy will be an optimal response to the strategy of the other, and in this sense any resulting solution can be described as a Nash equilibrium. The form of the solution will depend on the bargaining dynamics of the interaction between the trusts. A resulting solution may be grossly defined by a configuration which specifies which hospital(s) are operational. The possible configurations may be denoted by S1 (T1 only), S2 (T2 only), and S12 (both hospitals). Within configuration S12, different patterns of patient allocation between the trusts may be possible. It is of interest to characterise the properties (uniqueness, stability) of the solutions.