2 7 CONJUNTO DE ACRIVIDADES FISICO-RECREATIVAS.

Most research where game theory is applied in healthcare has mainly concentrated on Emergency Departments (EDs) and how to deal with diversions of patients and ambulances. Hagtvedt et al., 2009 [78] considered cooperative strategies for hospitals, in order to reduce occurrences when am- bulances are turned away due to the ED being full. These strategies lie in between the extreme of individualism and a central planner approach. The game theory approach was used to show that without some form of cooperative scheme, the incentives to defect are strong and will often lead

Chapter 7 A GAMETHEORETICALCONSIDERATION OFCCU INTERACTION 165 to system-wide pre-emptive diversion. The hospitals operate under a Prisoners’ Dilemma when the patient load is sufficient. They claimed that the penalty must be large enough to balance out the variance of the overflow before there is a an optimal threshold. Having examined these aspects of cooperative solutions to ambulance diversion, they believed the incentives require a centralised agent to route patients, at least when the patient load is high.

Deo and Gurvich, 2011 [41] proposed a queueing network model of two EDs to study the network effect of ambulance diversion. Each ED aims to minimise the expected waiting time of its patients (walk-ins and ambulances) and chooses its diversion threshold based on the number of patients at its location. They modelled the decentralised decision making in the network as a non-cooperative game. Analysis of the game reveals that, at equilibrium, EDs declare diversion status defensively to avoid getting arrivals from each other. This equilibrium undermines all potential pooling benefits of ambulance diversion, a phenomenon labelled as the depooling effect. These results provided one potential explanation for the evidence regarding defensive diversion and the impact of cancelling ambulance diversion in Massachusetts in January 2009. They proposed and analysed an alternative solution to the social planner’s problem in which the diversion thresholds are set to be equal to the EDs’ respective capacities. When there are available beds in one ED simultaneously with queued patients at the other, this policy routes all the “refutable” patients to the ED with available beds and thus recovers most of the pooling benefits. In addition to its being easier to implement than the true social optimum, it reduced the expected waiting times of both EDs.

Knight, 2012 [102] in his working paper considered a non-cooperative game, modelling a system of two hospitals and two interacting services: the Emergency Medical Vehicle and the Emergency Department. Using the PoA measure he showed that high levels of inefficiencies can be obtained due to the hospitals acting selfishly.

Some other work that has not concentrated on EDs, but has healthcare implications includes: Knight and Harper, 2012 [103] work, where results concerning the congestion related implica- tions of decisions made by patients when choosing between healthcare facilities were presented. Using theoretical results from routing game theory the following conclusions regarding the PoA were proved analytically: the PoA increases with worth of service, up to a point; in a system with insufficient capacity the PoA is low; and choice causes the highest level of inefficiency when the capacity of the system matches the perceived worth of service.

Game theory may be useful in modelling patient and doctor arrivals by considering the conflicting interests of both parties. It is likely that patients arrive early to beat the system or arrive late know- ing that they will have to wait anyway. Similarly, doctors may arrive late, assuming that the first patient will be late. There should be either some sort of mechanisms to enforce punctuality, or the

Chapter 7 A GAMETHEORETICALCONSIDERATION OFCCU INTERACTION 166 appointment systems should be designed to account for all parties behaviour (Van Ackere, 1990 [159]). One might expect that when clinics are run under more credible appointment systems, both patients and doctors will become more punctual.

Howard, 2002 [88] developed a model of the accept / reject decision for transplant organs and he showed how game theory might be used in making risk / benefit decisions in diagnostic radiology and other areas where risk / benefit needs to be considered. Howard believed that physicians would reject a low-quality organ if the patient is relatively healthy and can wait for a better quality organ. Roth has made significant contributions to the fields of game theory and is known for his empha- sis on applying his economic theory to solutions for “real-world” problems, including healthcare. Roth developed a very successful clearinghouse to facilitate the matching of doctors to residence programs (Roth, 1984 [142]). Today this clearinghouse is called the National Resident Matching Program. Roth along with Sonmez and Unver is a founder of the New England Program for Kidney Exchange that pairs compatible kidney donors and recipients. Roth in 2012 won the Nobel Memo- rial Prize in Economic Sciences jointly with Shapley “for the theory of stable allocations and the practice of market design”.

Most of existing game-theoretic queueing models focused on a setting in which the firms’ decision is either price / or capacity (Levhari and Luski, 1978 [111]; Cachon and Harker, 2002 [20]; Kalai et al., 1992 [96]; Cachon and Zhang, 2007 [21]; Allon and Federgruen, 2007 [4]). In these models, the choice of price / or capacity determines the arrival rate for each firm. The work by: Tezcan, 2008 [156], Stolyar, 2005 [153], and Adan et al., 1994 [1] is also not directly applied in healthcare settings; however the authors studied settings in which routing decisions are made upon customer / patient arrival, and once the customers are assigned to a queue they cannot be rerouted, therefore their models can be easily applied to different healthcare models.

7.3

Introduction

As mentioned before, this part of the thesis will concentrate on a non-cooperative game theoretical models of two CCUs. It is assumed that both CCUs act selfishly and the impact of lack of collab- oration will be studied. If CCUs are overcrowded they can declare being in “transfer” status and patients are diverted to the other CCU if they have available beds to accept extra patients.

It is assumed that the players are the CCU managers, who assess crowding in terms of the total number of patients in the CCU and request transfer when this crowding measure exceeds a prede- termined cut-off. The other CCU will accept the transfer request if their bed occupancy is below their predetermined cut-off. Otherwise, the transfer request is cancelled and depending on the

Chapter 7 A GAMETHEORETICALCONSIDERATION OFCCU INTERACTION 167 model, either each CCU is forced to accept its own patients or patients are refused admission to a CCU and are admitted to an ordinary ward within the hospital. Each player chooses a transfer cut-off with the objective of maintaining the utilisation rate as close to, but below, 80%. The 80% utilisation rate will be referred to as the target.

A Markov chain model of the two CCUs, the Nevill Hall (NH) and the Royal Gwent (RG) will be adapted to investigate the impact of patient’s transfers. Each CCU will be admitting two arrival streams: their own patients and transfers from the other CCU.