Informatica 17 (1993) 175-182 175 MORAL HAZARD PROBLEM SOLVING BY MEANS OF PREFERENCE RANKING METHODS Ines Saražin Lovrečič, Health Care Institution of Slovenia, Miklošičeva 24, 61000 Ljubljana, Slovenia AND Janez Grad, Department of Economics, University of Ljubljana, Kardeljeva pl. 17, 61000 Ljubljana, Slovenia Keywords: moral hazard, preference ranking, pseudo-model Edited by: Matjaž Gams Received: February 17, 1993 Revised: May 18, 1993 Accepted: July 21, 1993 Moral hazard problems in the field of humanitarian health aid delivery can be difficult to solve, especially in outstanding circumstances caused by human or natural factors. in this paper, we present a solution to this problem by means of preference-ranking methods. The idea of a pseudo-model is also included, where standard input is considered as well as subjective elements. 1 Presentation of the problem The treatment of refugees from Bosnia-Herzegovina and Croatia in 1992 presents a problem which the Slovenian health care system has to solve on the macroeconomic level. The problems which occur are as follows: — shortage of financial resources, — shortage of sanitary and pharmaceutical material, — daily variation of data which depends both on the domestic and foreign political environment. Since the media inform us daily about the lack of financial resources, we will not follow this topic any further. Let us address the issue of how much demand can be covered by the available state budget and how much help we can expect from various humanitarian organisations (domestic and foreign). Simultaneously, we raise the question, which risk group has priority at delivery. Therefore, our task is moral hazard problem solving. With regard to available facilities of the Slovenian health care system (supply) and requests (demand), we defined criteria which can be considered in various optimization models, such as: rationalisation of sanitary material, medicines, maximisation of preventive medicine etc. This can be formalised as a vector of criteria [Jfci,..., jfc„]T. Along with standard criteria k\,...,kn they are the so-called subjective criteria, representing the impact on the final decision of subjective reasoning (see Figure 1) based on the estimated help from unreliable sources. The result of such a model is a set of optimal solutions of the preference functions under given conditions such that the space of optimal development of health aid is bounded by this optimal set. Example. Suppose that we have two vaccination programmes for war refugees. The first one makes use of only reliable domestic resources, while the second one anticipates only financial and material support from abroad and charitable organisations. In the current situation, we can hardly judge which of the two programmes is more realistic. 2 Modern preference-ranking methods The multicriteria nature of moral-hazard problems requires a suitable solving method. On the 176 Informatica 17 (1993) 175-182 Ines Saražin Lovrečič, Janez Grad h ¿n V s ki ÏCp v + s V ...results from the optimisation and simulation model S .. .suitably formalised subjective elements Figure 1: Combination of matrices V and S basis of already-known advantages [3, 1] of up-to-date methods of multicriteria decision making, we decided to use the preference-ranking method as a tool for problem solving. PROMETHEE (Preference Ranking Organization Method for Enrichment Evaluations) is a group of general-purpose methods, developed in Europe and also used elsewhere in the world. Their purpose is to help the decision maker in alternative evaluations using preference functions. For detailed discussion of the methods, see [1], and [3] for a specialised version for health care system. Here we only devise the necessary theoretical basis for PROMETHEE. Let A be the the set of feasible decisions (actions). Suppose that criteria c\,...,cm are applied by the decision maker to evaluate individual actions; in short, Cj are numeric functions defined on A. The decision maker defines a generalised criterion Qj(a, b), also called the preference function (PF) for every Cj. Actually, it is a function of the difference Cj(a) — Cj(b), where a, b € A. There are six standard types of PF [1] and three types specialised for health-care system problem-solving [3]. In addition, most types have some parameters to determine. The choice of type of PF will be shown later by an example. Define preference index II as the average of all generalised criteria: m n(a,b) = Wj<3j(a, 6), 3=1 where Wj are weights (wj > 0, for all j and Wj = 1), a and b are arbitrary actions. The basis for action ranking is given by the so-called flows (leaving, entering, and net flow): *+(a) = £ n(«,6), b€A 4T(a) = J)n(M), beA §(a) = $+(«)- Since the argument of PF is the difference Cj{a) — Cj(6), the choice of parameters depends greatly on the distribution of differences for all a, b £ A. The use of PF is sensible only if the ranking can be influenced by their parameters. The accurate determination is left to the decision maker for the concrete problem. But the interval from the smallest to the biggest difference is recommended. 3 Formalisation of the pseudo-model Given a situation where both standard and subjective elements are to be considered, we combine both matrices V and S into one matrix denoted by T (Figure 1). The entries of T represent the input into the PROMETHEE model The procedure where the subjective elements are taken into account is called pseudo-modelling. In our case, by delivering health aid, the risk groups are ranked according to the results of pseudo-modelling. MORAL HAZARD PROBLEM SOLVING ... Informática 17 (1993) 175-182 177 Table 1: The model standard input criteria min/max At A2 As A4 type parameters children women elder rest p.f. a max 19.81 2.62 2.10 0.26 I - c2 max 6.93 1.98 0.80 0.20 I - Ca min 1.15 0.16 27.50 1.28 III p = 25 ct min 96.25 27.50 11.00 2.75 III p = 65 a min 23.45 6.70 2.68 0.67 V p = 18, q = 0.60 û(«0 1. C(<0 1 1 1 1 1 1 -*— 1 1 1 1 1 —t— 1 1 1 1 1 —1 ■ 1 1 I 1 1 1 1 1 1 I --é 1- 1 1 1 1 1 -1 ■ 05 10 15 d 0246 d Figure 2: PF for criterion C\ 4 Numerical example For preventive action, we take four health programmés (HP). The first one makes use only of reliable domestic resources, while the others foresee financial and material support from abroad and charitable organisations. Programmes differ in the costs which have to be covered for the same target i.e. the most suitable scheduling of risk groups versus different preventive programmes. 4.1 Under the first programme, all costs are covered by domestic resources (100%). In table 1 only standard input is taken into account in the PR.OMETHEE model. From table 1 it is clear that there are five criteria altogether which refer to the material costs of preventive vaccination. Criterion C\ measures preventive examination costs, C2 vaccination costs (labour, vaccine), C3 sanitary material costs, C4 laboratory material costs and C5 medical costs. The first two are maximised on the 'better to prevent than to cure' principle, the other three are minimised. Figure 3: PF for criterion C2 The actions are represented as risk groups: children women (^2), elder persons (-A3) and others (>14). In table 1 the average values for each criterion and action are also shown. The types of PF with adequate parameters are determined according to the rules in [1]. For the first criterion, we stick to the usual argument that high-quality preventive examination is particularly important, regardless of the costs. Accordingly, we choose the type of PF which treats every minimal difference d, as strict preference. The type I suits these requirements and it has no parameters to determine (see Figure 2). For tlie second criterion, we still do not rationalise the imunisation and vaccine costs. Both are necessary for preventing infections and deseases. Again, the most suitable choice is PF of type I. The difference between the costs of various immunisation programmes are illustrated in Figure 3. 178 Informatica 17 (1993) 175-182 Ines Saražin Lovrečič, Janez Grad Figure 4: PF for criterion C$ Figure 5: PF for criterion Cs Criterion C3 represents the costs of sanitary materia!, which are linearly dependent on its prices. The same is true for stored quantities. Here we choose PF of type III, i.e. PF with linear preferences. It is shown in Figure 4. Type III of PF is also chosen for the fourth criterion and is justified by the same argument as for C3. For criterion C5 the principle of rationalisation is used again. However, in contrast to tlie last two criteria, we introduce the so-called indifference threshold q. It stands for nonsensitivity to differences between costs of medicines to a certain extent. We pay attention to them only when the differences exceed the threshold. Such a situation can be dealt with using PF of type V with parameters q and p (Figure 5). The first parameter is the indifference threshold and the second denotes the strict preference threshold. The results of the computer-solved problem are presented in Table 2. The preference outranking list is defined by net flows. We see that the highest priority for delivering humanitarian aid has the risk group A2 (women), followed by ay (children), a4 (others) and A3 (elder persons). 4.2 The second programme includes an additional two criteria Oi and O2, which determine implementability of C\ and C1. 1.2865 1.4240 -0.1375 2 a2 1.5732 0.7347 0.8385 1 As 1.0643 1.3154 -0.2511 3 At 0.8430 1.2929 -0.4499 4 Table 5: Input data for the pesimistic HP criteria min/max Ar A2 A* A4 tip parameters children women elder rest p.f. c, max 19.81 2.62 2.10 0.26 I - c3 max 6.93 1.98 0.80 0.20 I - C3 min 1.15 0.16 27.50 1.28 III p = 25 Ci min 96.25 27.50 11.00 2.75 III p = 65 a min 23.45 6.70 2.68 0.67 V p = 18, q = 0.6 Pi max 0.60 0.70 0.65 0.67 I - P2 max 0.62 0.63 0.68 0.74 I - 180 Informatica 17 (1993) 175-182 Ines Saražin Lovrečič, Janez Grad Table 6: Results of analysis of the pesimistic HP action leaving flow enter.flow net flow outranking list at 1.0007 1.7097 -0.7089 4 a2 1.5732 0.7347 0.8385 1 a3 1.0643 1.4583 -0.3939 3 a4 1.2715 1.0071 0.2644 2 Table 7: Input data for the HP of compromise criteria min/max ¿1 A2 A3 Aa tip parameters children women eldest rest p.f. Ci max 19.81 2.62 2.10 0.26 I - c2 max 6.93 1.98 0.80 0.20 I - cz min 1.15 0.16 27.50 1.28 III p = 25 c4 min 96.25 27.50 11.00 2.75 III p = 65 c5 min 23.45 6.70 2.68 0.67 V p= 18, g = 0.6 Ki max 0,75 0.80 0.85 0.90 I _ K2 max 0.80 0.79 0.81 0,78 I - Table 8: Results of analysis for the HP of compromise action leaving flow enter.flow net flow outranking list a\ 1.2865 1.4240 -0.1375 3 a2 1.2875 1.0205 0.2670 1 a3 1.3501 1.1726 0.1775 2 a4 0.9858 1.2929 -0.3070 4 MORAL HAZARD PROBLEM SOLVING ... Informática 17 (1993) 175-182 181 Table 9: Net flow value analysis of the standard and optimistic HP Table 11: Net flow value analysis of the standard and HP of compromise action D I&TW Ai 0.2075 -0.1375 -0.3450 -166.27 Ai 0.7739 0.8385 0.0646 8.35 A3 -0.5515 -0.2511 0.3004 54.47 A* -0.4298 -0.4499 -0.0201 -4.68 action D TÄlt*) Ai 0.2075 -0.1375 -0.3450 -166.27 Ä2 0.7739 0.2670 -0.5069 -65.50 a3 -0.5515 1.1770 1.7285 313.42 A4 -0.4298 -0.3070 0.1228 28.57 Table 10: Net flow value analysis of the standard and pessimistic HP action D Ax 0.2075 -0.7089 -0.9164 -441.64 Ai 0.7739 0.8385 0.0646 8.35 A3 -0.5515 -0.3939 0.1576 28.58 At -0.4298 0.2644 0.6942 161,52 the essential ascertaining is as follows. The addition of subjective elements to the standard input is the cause of change in the preference structure, i.e. the rankings of alternative risk groups. It can be deduced from the comparison of results that the smallest discrepancy is found between the standard and optimistic HP. The cause of this phenomenon lies in the high percentage of realis-ability of criteria C\ and C2. In the case where we decide to apply pseudo-modelling, moral-hazard problem solving depends on the input data of the subjective characters. In the follow-up, we have to examine the changes of net flows which are due to the addition of subjective elements. Table 9 shows the values of net flows of the standard input as well as the optimistic programme the dif- ferences between net flows of the standard input — 4>v| = D for all actions, and changes relative to the net flows of the standard input (j^y)-The comparison of results between the standard and pessimistic HP is found in Table 10, while Table 11 refers to the standard programme and the programme of compromise. The relative changes for particular actions are again minimal when comparing the standard and optimistic HP. Surely this is a consequence of the smallest discrepancy between the optimistic and standard HP in view of their inputs. In practi- cal terms, with the optimistic HP the health-care system is able to cover almost all costs of health aid. In other words, with at most 9% reduction in certainty of the cost coverage, only two (already adjacent) actions swapped their places in the preference structure. Net flow analysis shows that their absolute values change with the addition of subjective elements and they do not change uniformly for each action. Therefore, the preference structure changes if: — we add subjective elements and — we change their values. From this point the analysis can be continued, for instance with varying implement ability intervals of criteria, and studying stability of preference structure. We can also consider more criteria of implement ability. Finally, we can observe the behaviour of particular actions according to the varying implement ability intervals of criteria or the addition of new criteria. 6 Summary In this paper, we have exposed the moral-hazard problem in the field of humanitarian health aid delivery in outstanding circumstances. In the practical example, we have dealt with four various preventive health programmes. For the case when both objective and subjective elements are included, we constructed a pseudo-model. The PROMETHEE method is the basic tool for risk-group ranking. Both subjective and objective elements are treated equally, so we can avoid over and under estimation of either group of factors. 182 Informatica 17 (1993) 175-182 Ines Saražin Lovrečič, Janez Grad References [1] Brans J.P., Mareschal B. (1986): How to select and how to rank projects. North Holland, European Journal of Operational Research (24), pages 228-238. [2] Drummond Michael F., Stoddard Greg. L,, Torrance George W. (1990): Methods for the Economic Evaluation of Health Care Programmes. Oxford, University Press, 182 pages. [3] Sarazin L. Ines, Grad Janez (1992): Multicri-teria decision making in health care system (in Slovene language). Ljubljana, Slovene economic review, (43/5), pages 334-347.