Which patients do I treat? An experimental study with economists and physicians

Ahlert, Marlies; Felder, Stefan; Vogt, Bodo Article Which patients do I treat? An experimental study with economists and physicians Health Economics Review Provided in Cooperation with: Springer Nature Suggested Citation: Ahlert, Marlies; Felder, Stefan; Vogt, Bodo (2012) : Which patients do I treat? An experimental study with economists and physicians, Health Economics Review, ISSN 2191-1991, Springer, Heidelberg, Vol. 2, Iss. 1, pp. 1-11, https://doi.org/10.1186/2191-1991-2-1 This Version is available at: https://hdl.handle.net/10419/77829 Standard-Nutzungsbedingungen: Die Dokumente auf EconStor dürfen zu eigenen wissenschaftlichen Zwecken und zum Privatgebrauch gespeichert und kopiert werden. Sie dürfen die Dokumente nicht für öffentliche oder kommerzielle Zwecke vervielfältigen, öffentlich ausstellen, öffentlich zugänglich machen, vertreiben oder anderweitig nutzen. Sofern die Verfasser die Dokumente unter Open-Content-Lizenzen (insbesondere CC-Lizenzen) zur Verfügung gestellt haben sollten, gelten abweichend von diesen Nutzungsbedingungen die in der dort genannten Lizenz gewährten Nutzungsrechte. Terms of use: Documents in EconStor may be saved and copied for your personal and scholarly purposes. You are not to copy documents for public or commercial purposes, to exhibit the documents publicly, to make them publicly available on the internet, or to distribute or otherwise use the documents in public. If the documents have been made available under an Open Content Licence (especially Creative Commons Licences), you may exercise further usage rights as specified in the indicated licence. http://creativecommons.org/licenses/by/2.0/ RESEARCH Open Access Which patients do I treat? An experimental study with economists and physicians Marlies Ahlert 1 , Stefan Felder 2* and Bodo Vogt 3 Abstract This experiment investigates decisions made by prospective economists and physicians in an allocation problem which can be framed either medically or neutrally. The potential recipients differ with respect to their minimum needs as well as to how much they benefit from a treatment. We classify the allocators as either ‘selfish’,‘Rawlsian’, or ‘maximizing the number of recipients’. Economists tend to maximize their own payoff, whereas the physicians’ choices are more in line with maximizing the number of recipients and with Rawlsianism. Regarding the framing, we observe that professional norms surface more clearly in familiar settings. Finally, we scrutinize how the probability of being served and the allocated quantity depend on a recipient’s characteristics as well as on the allocator type. JEL Classification: A13, I19, C91, C72 Keywords: experimental economics, social orientation, individual choices, allocation of medical resources, principles of distribution 1 Introduction Prioritizing medical services and redefining access to health care are high on political agendas across the globe. Several countries have appointed commissions to define the rules for the health technology assessments and cost-benefit analyses which guide allocation decisions in health care. Experts on such panels, in particular health economists and health ethicists, tend to ignore the fact that not all medical allocation decisions can be made on the level of fixing general rules. If that were feasible, trade-off decisions behind a veil of uncertainty would involve only statistical lives. In fact, the allocation of scarce medical resources and the pursuant withholding of care cannot always be ‘pre-programmed’ by general rules. Not only will the individuals from whom care must be withheld have a face and an identity, the allocator himself will be a specific individual who will have to make an allocation choice in a specific situation. Consequently, specific individuals are affected by decisions over which the allocator has discretionary powers. These within rule-choices (as opposed to the choice of rules) are not determined by the rules. They must be made by an allocator according to his judgment. It is therefore important to analyze which allocation is chosen under which circumstances, and, in particular, to evaluate how medical care is allocated in the conflict between efficiency, selfish behavior, and the social orientation of decision makers. The experimental method has proven useful for testing theories on economic allocation. In particular, fairness ideals have been extensively scrutinized in the experimental laboratory (for a recent study see [1]). However, to our knowledge no experimental test has been carried out in the medical setting yet. We model the medical allocation problem and experimentally test the power of several theoretical concepts (ranging from utilitarianism to Rawlsian behavior) to predict subjects’choice behavior. The goal of this paper is to study allocation decisions by prospective physicians and econo-mists. The experiment is designed to reveal when and how individuals deviate from the self-regarding preferences induced by the embedded monetary reward function. Will strict payoff maximizers - individuals who conform to the preferences induced by the reward function - prevail, or * Correspondence: [email protected]h 2 Faculty of Business and Economics, University of Basel, (Peter Merian-Weg 6), (4002) Basel, Switzerland and Faculty of Economics and Business Administration, Duisburg-Essen University, (Univer-sitätsstraße 12), (45117) Essen, Germany Full list of author information is available at the end of the article Ahlert et al.Health Economics Review 2012, 2:1 http://www.healtheconomicsreview.com/content/2/1/1 © 2012 Ahlert et al; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. will we find deviations from such behavior that signify other relevant influences on the process of passing judgment? Are the choices made more in line with utilitarian principles or with an egalitarian rule a)? Do the principles applied depend on the framing of the problem, and do economists decide differently than physicians? The paper is organized as follows: Section 2 presents the allocation problem and the solutions for four different types of allocation rules. Based on these rules, we characterize four classes of allocators: two utilitarianoriented types (social utilitarian and purely selfish) and two types leaning towards egalitarianism (Rawlsian and maximizing the number of recipients). Section 3 describes details of the experimental design, including the characteristics of the potential recipients. We also calculate and compare the payoffs for ideal types of the four classes of allocators. In section 4, we classify the subjects who participated in the experiment based on their choices. We study the effects that arise from framing the allocation problem in a neutral and a medical fashion, where the allocator is described as a physician and the potential recipients as patients. Moreover, we compare the choices made by economists and physicians. In section 5, we investigate how the choices of different types of allocators depend on the minimum needs and productivity of potential recipients. In order to find out which subgroup of recipients is served and how much they receive, we first analyze the determinants of a positive recipient payoff using a logit model. Then we use ordinary least squares regression to analyze the determinants of the size of a recipient’s payoff, conditional on it being positive. Section 6 discusses and summarizes our findings. 2 The allocation problem and possible solutions In our experiment, an allocator distributes a resource among seven potential recipients. The individual recipients each require a specific minimum quantity of the resource in order to achieve a positive payoff. The potential recipients also vary in their productivity at transforming the quantity they receive into a payoff for themselves. The allocator’s payoff is a function of the sum of the recipients’payoffs. Moreover, the allocator faces a fixed fine for each individual he fails to serve (i.e. the individual does not receive his minimum quantity nor, therefore, a payoff). We do not set out to test the validity of the assumed other-regarding concerns in this paper. Such motives, however, appear to be prevalent in common medical allocation situations b). While a payoff maximizing allocator earns a maximal profit, an allocator following a rule not dictated by the preferences induced by the payoff function - for instance an egalitarian rule - loses out on profits. The experiment thus sheds light on the classic equity-efficiency tradeoff in a setting in which efficiency is not judged against purely selfish motives but relative to a complex evaluation. More specifically, the allocator (individual 0) allocates ration r i to nindividuals (i=1,2,..,n). With the endowment given by R, the allocator’s choice is restricted by  i ri≤R. The potential recipients are characterized by two parameters, m i and p i .m i is the minimum ration an individual needs to obtain a positive payoff, while p i is a productivity factor, transforming the allocated ration into a payoff for the recipient. In the medical setting, m i represents a physician’s minimal time or effort required to treat the patient and p i stands for the probability of treatment success or the effectivity of the treatment. The payoff of individual i= 1, 2,..,nis then πi=0, if ri<mi ri·piif ri≥mi .(2) The allocator incurs a fine equal to cfor every individual who does not receive the minimum ration m i c). In the medical setting, ccorresponds to the physician’s disutility of not treating a patient. One might interpret this as other-regarding preferences, typically due to empathy or internalized professional norms d). cisthesamefor all recipients who are not served and patients who are not treated. Finally, the allocator participates in the recipients’payoffs with the factor t. This design feature introduces the second element of other-regarding preferences on the part of the allocator. The allocator’s payoff π 0 is ttimes the sum of the recipients’payoffs, minus all fines e): π0=t· i πi− i|πi=0 c.(3) 2.1 The own payoff maximizing allocator and the social utilitarian allocator The own payoff maximizing allocator OPMA maximizes a target function WOPMA (π0,π1,π2, ..., πn)=π0,(4) where π 0 is determined according to (3). His optimal choice can be characterized as follows: He first ranks the individuals in decreasing order of their productivity factor p i , and then individuals with equal productivity in increasing order of their minimum required amount m i . Let Kbe the ranked set of possible recipients, with k= 1 as the most productive individual with the smallest m i , (or one of them, if there are several). The allocator Ahlert et al.Health Economics Review 2012, 2:1 http://www.healtheconomicsreview.com/content/2/1/1 Page 2 of 11 will serve k= 1 first, provided that m 1 ≤R. His remaining endowment then amounts to R-m 1 . Secondly, in consecutive order starting with k= 2, he will compare each individual to k= 1 and perform the following dominance test: t·mk·pk+c>t·mk·p1?(5) The test calculates the opportunity costs of allocating m k to the most productive individual. It consists of the foregone revenue t·m k ·p k and the fine c. If the opportunity costs are larger than the revenue from allocating m k to k= 1 (i.e. a positive test outcome), the OPMA will serve kto the extent permitted by the remaining endowment. This procedure is continued along the ranked set K,spendingm k if the individual kfulfills the test, and stops once the remaining endowment is too small to serve a further individual. The allocator will then give the remainder to the most productive individual, since this yields the maximal additional payoff. Note that if the fine were zero, no individual except k=1could pass the dominance test (p 1 ≥p k for all k), and the OPMA would spend the entire endowment on the most productive individual (or on a subset of the most productive individuals if this designation not unique). The utilitarian social welfare function sums the payoffs over all individuals, including the allocator. It attaches the same weight to the payoff of each and every individual and thus features other-regarding preferences more strongly than the OPMA target function: WUA (π0,π1,π2, ..., πn)=π0+ i πi.(6) When a social utilitarian allocator UA decides to serve individual kwith the minimal endowment m k ,hewill consider the corresponding payoff π k =m k ·p k as well as his own payoff π 0 =t·m k ·p k . The dominance test for the social utilitarian then changes to (t+1 )·mk·pk+c>(t+1 )·mk·p1.(7) Individuals that are not in position 1 (i.e. all but the most productive recipient or recipients) face a higher threshold for being served by the UA than by the OPMA. Hence, fewer potential recipients are included under the utilitarian social welfare regime than under the principle of maximizing own payoff. 2.2 The number maximizing allocator An allocator maximizing the number of recipients (NMA) has the following social target function: W NMA (π 0 ,π 1 ,π 2 , ..., π n )= i N i with N i =0, if π i =0 1, if π i >0. (8) This allocator first ranks the set of individuals according to increasing m i , the respective minimum ration required for a positive payoff. If two individuals need the same minimal amount, the one with higher productivity is ranked first. Let Lbe the correspondingly ranked set of individuals where l= 1 is the individual with the minimum m i . The number of recipients is maximized if the NMA follows the ranked individuals within Land allocates m i as long as the remaining endowment R− l mlis positive. Once the endowment becomes too small to serve a further individual, the allocator stops f). He will be indifferent as to how to allocate the remaining amount. To distinguish this type from the Rawlsian allocator that is discussed below, we assume that the remaining endowment is allocated along utilitarian principles, thus going to the most productive reci-pients. 2.3 The Rawlsian allocator with lexicographic maximin preferences The Rawlsian allocator’s (RA) preferences over two payoff distributions (π 0 ,π 1 ,π 2 ,....,π n )and π∗ 0,π∗ 1,π∗ 2, ..., π∗ nare represented by the lexicographical comparison of the payoff vectors for all individuals 0,1,2,...,n, arranged in increasing order. The RA prefers distributions which maximize the payoff of the individual 0 which is worst off. If there are several individuals 0, the RA compares the payoffs of the individuals with index 1 and again prefers the allocation with the higher payoff. If these, too, are equal, he proceeds to index 3, etc. Given that, generally, not every potential recipient can be served, the RA will first maximize the number of recipients. Next, rather than increasing the ration for one individual beyond m i , he will ‘save’ another individual the remaining endowment permitting. Similar to the NMA allocator, the RA will thereby favor individuals with low minimal needs. But the RA differs from the NMA when it comes to the allocation of the remaining endowment. Applying Rawls’principle [2] leads to a leximin solution with respect to the payoffs, firstly, of those recipients who received at least their minimum amount and, secondly, the allocator himself. It is important here that Rawls’criterion be applied to the payoffs of the allocator and the recipients simultaneously. The allocation resulting from Rawls’criterion differs strictly from a purely egalitarian allocation, which equalizes the allocated rations without incorporating the number of recipients and without taking into account the different productivities of the potential recipients. This (naïvely) non-consequentialistic egalitarian allocation is not considered here. Ahlert et al.Health Economics Review 2012, 2:1 http://www.healtheconomicsreview.com/content/2/1/1 Page 3 of 11 3 Experimental design and identification of ideal types of allocators In this section we report data from a series of experiments in which participants allocated a given amount of resources to seven potential recipients in ten different treatments. They knew that payments to themselves and to the recipients would be based on their choices in one out of the ten treatments, to be selected at random. A total of 17 experimental sessions were conducted at the Magdeburg Laboratory for Experimental Economics (MaXLab) between December 2007 and February 2008 using Urs Fischbacher’s [3] software tool z-tree. 136 students from the faculties of economics and medicine participated in the experiments g). No one was permitted to participate in more than one session. The allocators included 36 economics students and 22 medical students, whereas the recipients were almost all economics students. The sessions lasted between 45 and 90 minutes. Participants received a show-up fee of €3 and payoffs depending on their choices. Average earnings were €12.65 per person. In the allocation problems, payoffs were described in experimental currency units (ECU), with 100 ECU equaling €2. We used a purely economic frame with neutrally described allocation problems in 8 sessions and a medical frame in 9 sessions, where the potential recipients were described as patients and the allocator as physician. The framing did not change during the sessions, so that no individual acted under both framings. Experimental instructions are provided in Additional File 1: Appendix B. Eight individuals participated in each session. At the beginning of a session, we randomly chose one to be the allocator. The seven remaining participants were assigned to be recipients. Starting with session 6, we changed this aspect of the design and let all eight subjects allocate endowments among seven virtual recipients. In these sessions, only the allocators received actual payments; and they were informed that their decisions had no payoff consequences for other persons. The information about the characteristics of the recipients did not differ between the two design variants. The total endowment of the allocator was either 1000 ECU or 1600 ECU. The allocator’s par-ticipation rate in the recipients’payoffs was set at t= 0.2, and the fine for every individual not served at c= 50 ECU. From these parameter values, we can estimate the relative payoffs between the allocators and the recipients as follows: If the allocator chooses to serve all, his profit will exceed the average recipients’payoff by 40 percent since he receives one fifth of their total payoffs. In ten treatments, each representing one allocation problem, the allocator had to decide how many ECU to give to each of the seven individuals. The characteristics of the recipients differed across the treatments; see Table 1. Their minimum thresholds m i range from 10 ECU to 1000 ECU. The last column presents the sum of all the recipients’thresholds per treatment. When given an initial endowment of 1000 ECU, the allocator could serve all individuals only in treatment 2. In all other treatments, he is forced to forego at least one recipient and pay the fine of 50 ECU. When the total endowment was increased to 1600 ECU, the allocator could in principle serve all individuals in seven out of ten treatments, thus avoiding the fine completely. The productivity factor p i ranges from 1 to 5 and indicates the extent to which a recipient benefits from his allocation. For instance, in treatment 1 a ration of 300 ECU translates into a payoff of 1200 ECU for person 1, but only 600 for person 4. In a slight twist to the payoff function (2), we instructed the allocators to give each individual iwhom they wish to serve at least m i +1 ECU in order to secure a positive payoff h). Additional File 1: Appendix A shows the optimal solutions for all types of allocators (Additional File 1 Table 1 for total endowment = 1000 ECU and Addi-tional File 1 Table 2 for total endowment = 1600 ECU). Table 2 shows the average payoffs for the ideal types of allocators and their recipients in every treatment. Compared Table 1 The 10 allocation problems Treatment Person 123456 7 i mi 1m i 300 50 150 50 100 300 100 1050 p i 433235 4 2m i 200 100 10 50 50 10 100 520 p i 421232 3 3m i 300 200 500 100 100 200 300 1700 p i 421234 1 4m i 500 100 50 50 50 100 600 1450 p i 411232 4 5m i 300 100 200 100 300 200 1000 2200 p i 232232 3 6m i 100 50 100 50 500 100 500 1400 p i 331232 3 7m i 200 200 200 200 200 400 400 1800 p i 222222 2 8m i 200 200 200 200 200 200 200 1400 p i 123132 2 9m i 400 50 10 10 50 600 50 1170 p i 211223 3 10 m i 100 100 100 100 100 500 500 1500 p i 222223 3 mi: minimum ration individual ineeds to obtain a positive payoffpiproductivity factor, transforming an allocated ration into a payoff for the recipient Ahlert et al.Health Economics Review 2012, 2:1 http://www.healtheconomicsreview.com/content/2/1/1 Page 4 of 11 to the utilitarian type, the OPMA reduces the total payoff. The NMA, and particularly the RA, reduce total payoff even further. They both choose a lower payoff for themselves than the OPMA does. Moreover, they allocate lower average payoffs to the recipients than the OPMA. These results, of course, reflect the target functions of the NMA and the RA, because both types give primary consideration to maximizing the number of recipients and consider the recipients’ payoffs only as a secondary criterion. When the total endowment is higher, allocator and recipient payoffs differ more - in percent - under number maximizing or Rawlsian social orientation than under the OPMA principle. This effect is reversed for the utilitarian type: Here the percentage difference between the allocator’s payoffs and the total payoff is bigger if R= 1000 ECU, but smaller for the sum of the recipients’payoffs. Being a utilitarian (and maximizing total payoff) rather than being selfish is more “costly”to the allocator if the available amount to be distributed is smaller, i.e. when scarcity is more severe. For the other types, the costs of being utilitarian as compared to being selfish are higher when the shadow price of the resource constraint increases. The 58 allocators made a total of 3,948 allocation decisions i. Table 3 provides information on the number of treatments, allocators and observations in the two framings, for high and low budgets, real and virtual recipients, and the allocator’s profession. A treatment is defined as a decision problem in which a given endowment is allocated among 7 potential recipients. A treatment thus provides 7 observations of allocations. With ten treatments in a session, each allocator takes 70 decisions in total. 4 Results 4.1 Classification of allocator Based on their actual choices, we classified the allocatorsubjects according to their proximity to one of the four ideal types described above using a variance test j). An allocator was classified as belonging to one type if this mean Euclidian distance from the respective optimal choice was significantly smaller than his mean Euclidian distance from the optimal choices of the three alternatives. Table 4 shows the classification results for both the economic and the medical settings. In addition to ‘pure’types, we also observe individuals which are in Table 2 The average treatment payoffs of the allocators and their recipients for the four ideal types of social orientation (in parentheses: in percentage of the OPMA type) Allocator’s payoff π 0 Sum of recipients’ Payoffs  i πi Total payoff π0+ i πi Type R= 1000 R= 1600 R= 1000 R= 1600 R= 1000 R= 1600 UA 443.24 (86.8) 881.24 (92.7) 3391.20 (112.0) 5431.20 (107.0) 3834.44 (108.4) 6312.44 (104.7) OPMA 510.42 (100) 950.42 (100) 3027.10 (100) 5077.10 (100) 3537.52 (100) 6027.52 (100) NMA 505.30 (99.0) 918.04 (96.6) 2926.50 (96.7) 4665.20 (91.9) 3431.80 (97.0) 5583.24 (92.6) RA 433.03 (84.8) 781.76 (82.3) 2565.13 (84.7) 3983.81 (78.5) 2998.15 (84.8) 4765.57 (79.1) UA: utilitarian allocator, OPMA: own payoff-maximizing allocator, NMA: number maximizing allocator, RA: Rawlsian allocator Table 3 Number of treatments, allocators, and observations Framing Budget Recipients No. of treatments No. of allocators No. of observations Economists Physicians Economists Physicians Medical high real 0 0 0 0 0 virtual 134 7 8 378 560 low real 60 0 6 0 420 virtual 80 8 0 560 0 Neutral high real 0 0 0 0 0 virtual 160 8 8 560 560 low real 50 5 0 350 0 virtual 80 8 0 560 0 Total 564 36 22 2408 1540 Ahlert et al.Health Economics Review 2012, 2:1 http://www.healtheconomicsreview.com/content/2/1/1 Page 5 of 11 between two types. If these tests were inconclusive for an allocator - so neither significantly close to one type nor in between two types - (at the 10 percent significance level), this individual was not classified. Not one of the 58 allocators was classified as being a social utilitarian who maximizes the total payoff. Among the economists, 19 percent were classified as OPMA, compared to 9 percent among the physicians. At 44 percent, the share of NMA among economists is higher than that among physicians by a factor of 1.6. Only 3 of the 36 economists were classified as RA, while as many as 7 out of 22 physicians appear to lean towards Rawlsian leximin allocations. The mixed types confirm this tendency: 5 physicians and only 1 economist were classified as the mixed NMA/RA type. Unclassified allocators made up around 14 percent among econo-mists and only 5 percent among physicians. 4.2 Framing and professional effects In this section, we want to shed light on the effects of the medical and neutral framing as well as on possible differences in the choices made by economists and physicians. Table 5 shows the mean Euclidean distances of the decisions made by the three allocator types in the two professional populations, based on the allocated payoffs including the allocator’s. While economists lean towards maximizing their own payoff, the physicians are closest to the allocator type that maximizes the number of recipients. Regarding framing, the results suggest that economists move further away from the allocator types - most accentuated in the case of the OPMA - when the setting is medical as compared to the neutral, purely economic framing. By contrast, physicians are closer to one of the three types when the framing is medical. Thus, it appears that the classification becomes clearer when the setting corresponds to the allocator’s own professional background. When the setting is unfamiliar, the categories of the classification system prove less powerful. This seems to indicate that “professional norms”guide these decisions. This holds even for the OP-MA, for the corresponding motivations become more forceful in the setting in which it is considered legitimate to maximize one’s own payoff. Table6givestheresultsofvariancetestsfortheprofessional and framing effects described above. The last row shows that physicians are significantly further away from the OPMA payoff than economists. The opposite goes for the Rawlsian types. The framing effects are surprising, since they work in different directions for economists and physicians. When the setting is medical, economists allocate in less own-payoff maximizing ways, while physicians move towards payoff maximization. Economists seem to get cold feet in the medical setting and move away from their professional focus on maximizing a given objective function. It holds well for the physicians, too, that their professional norms emerge more clearly in the setting familiar to them k). 4.3 Efficiency costs The efficiency cost of an allocator’s choices corresponds to the deviation from the social utilitarian welfare π0+ i πi. Table 7 shows these costs by profession and framing. While framing effects are more or less absent, Table 4 The classification of the allocators in the two settings Economists Physicians Framing Type neutral medical Total neutral medical total UA 0 0 0 0 0 0 UA/OPMA 0 1 1 0 0 0 OPMA 6 1 7 1 1 2 OPMA/NMA 1 2 3 0 1 1 NMA 10 6 16 2 4 6 NMA/RA 0 1 1 2 3 5 RA 2 1 3 3 4 7 Not classified 2 3 5 0 1 1 Total 21 15 36 8 14 22 UA: utilitarian allocator, OPMA: own payoff-maximizing allocator, NMA: number maximizing allocator, RA: Rawlsian allocator Table 5 Euclidian distance from types of allocators; economists and physicians Economists Physicians Framing Type neutral medical neutral medical OPMA 444 643 789 612 NMA 399 502 552 462 RA 622 682 569 528 OPMA: own payoff-maximizing allocator, NMA: number maximizing allocator, RA: Rawlsian allocator Table 6 Framing and professional effects (relative squared Euclidian distance to the types of allocator) - results from a variance test Type OPMA NMA RA Faculty Framing effect (medical vs. neutral setting) Economists 1.45*** 1.26* 1.10 Physicians (1/1.29)** (1/1.20) (1/1.08) Faculty effect (physicians vs. economists) 1.28** 1.12 (1/1.19)** OPMA: own payoff-maximizing allocator, NMA: number maximizing allocator, RA: Rawlsian allocator ***, (**), [*] significant at the 99% (95%) [90%] confidence level, resp. Ahlert et al.Health Economics Review 2012, 2:1 http://www.healtheconomicsreview.com/content/2/1/1 Page 6 of 11 the difference between economists and physicians is considerable and statistically significant. While the economists’choices lead to an efficiency cost of between 9 and 12 percent, the choices made by the physicians involve an efficiency loss of 16 to 20 percent. The size of the efficiency costs in percentage terms appears not to depend on the total endowment. Interestingly, although we find many OPMA, the second part of Table 7 indicates that the average willingness to sacrifice one’s own profits to pursue other goals is large. It compares the allocators’own payoffs to those of an ideal-type OPMA. Economists choose a payoff that is between 4.5 percent and 10.6 percent lower than that of the OPMA. For physicians, the payoff is as much as 11.7 percent to 17.3 percent lower. The allocators’ sacrifice of their own profits decreases when total endowment increases. As with the efficiency costs, framing effects are absent and professional effects are statistically significant regarding the chosen amount of own profit. 5 Who is served and how much do they receive? Hypotheses and tests The way an allocator distributes the endowment depends on his target function, the budget and the characteristics of the potential recipients. As the latter parameters are the same for all allocators, differences among them will arise due to differences in their target functions. 5.1 The determinants of a positive payoff Let us first consider the OPMA. His criterion for serving an individual is the dominance test (5), which can be rewritten as pk+c t·mk >p1.(9) It follows that the OPMA is more likely to choose individuals who are very productive or need only a small minimum ration. An NMA and an RA will not, as a first criterion, consider the individuals’productivity when deciding whom to serve. The decisive parameter in their first move is the individuals’minimum need. They will first choose individuals with a small minimum need, allowing them to maximize the number of recipients with positive payoffs. If, for example, there are two individuals with the same minimum need but different productivities and only one of them can be served, we assume that the NMA then allocates the minimum amount to the individ-ual with the higher productivity (and the rest to the served recipient with the highest productivity). The RA will also choose this individual, but allocate the remaining amount according to the leximin criterion. The distributive problems often feature several recipients with the same minimum amount but different productivities. If not all of them can be served, productivity will play a role. We therefore also expect productivity to have a small influence on the probability of being served by these types. Under certain parameter constellations (different from those in our experiments), the own-payoff maximizing selection of recipients could even equal that of the other allocators. If the participation factor twere very low or the fine cwere very high, every individual would pass the dominance test. In this case, an OPMA will also maximize the number of recipients with positive payoffs and serve the same individuals as the other allocators. We can thus derive the following propositions concerning the allocation of positive rations to individuals: Proposition set 1: The OPMA, the NMA, and the RA are more likely to serve an individual whose minimum need is low. Only the OPMA has a strong concern regarding an individual’s productivity. His willingness to Table 7 Social utilitarian welfare of the allocators’choices and their own profit (in parentheses: in percentage of the respective reference) Economists Physicians Framing Reference neutral medical neutral medical UA a) Social utilitarian welfare: π0+ i πi R= 1000 3834.44 (100) 3375.97 (88.0) 3417.36 (89.1) —(—) 3087.71 (80.5) R= 1600 6312.44 (100) 5695.27 (90.2) 5787.23 (91.7) 5289.35 (83.8) 5255.80 (83.3) OPMA a) Own profit π 0 R= 1000 510.42 (100) 456.57 (89.4) 460.71 (90.3) —(—) 422.26 (82.7) R= 1600 950.42 (100) 908.07 (95.5) 910.53 (95.8) 838.85 (88.3) 838.99 (88.3) UA: utilitarian allocator, OPMA: own payoff-maximizing allocatora) for the values see Table 2 Ahlert et al.Health Economics Review 2012, 2:1 http://www.healtheconomicsreview.com/content/2/1/1 Page 7 of 11 allocate a positive ration increases the more productive the potential recipient is. In order to test proposition set 1, we ran a logit regression that exploits the panel structure of the data. We applied a random effects model to account for the possibility of specific correlation between the error terms of an allocator’s choices. Then we included dummy variables for the treatments, since the shadow price of the total endowment depends on the sum of the minimum thresholds, which differs from treatment to treatment. Treatment 2 has the lowest sum  i mi=520 ECU and served as benchmark treatment. Total endowment is included with a dummy variable which takes on the value 1 for R= 1600 ECU and 0 for R= 1000 ECU. Moreover, absolute and slope dummies for the different types were incorporated, with the unclassified allocators serving as benchmark. For the potential recipients, the mean probability of a positive payoff was 75.3 percent. Table 8 presents the results of the logit model. The likelihood ratio test for rho = 0 indicates that the joint hypothesis of zero slope coefficient can be rejected. Regarding the treatment effect on the probability of being served, the coefficient shows the expected sign. As the resources were always scarcer than in treatment 2, the probability of a positive payoff was significantly lower in those treatments. In the neutral framing of the allocation problem fewer potential recipients were served than in the medical setting, though the difference is not significant. The dummy for sessions where only one individual acted as allocator is positive, indicating that the presence of actual recipients in the laboratory positively affects the allocator’s willingness to serve them. This result, however, is not significant. The endowment dummy shows the expected sign. When the endowment rises to 1600 ECU, significantly more potential recipients are served. The intercept dummies differentiate between pure and mixed types of allocators. NMA, RA, and their mixed type served more recipients than OPMA on average. The probability of a positive payoff falls significantly with an increase in a potential recipient’s minimum threshold. This effect is significant and stronger for NMA, but not for OPMA and RA. Finally, the productivity of potential recipients has a large effect on the probability of being served. The effects are significant and larger for NMA, but not for OPMA. 5.2 Explaining the conditioned positive allocation The second choice refers to the size of the allocated ration, conditional on the payoff being positive. The experimental design implies that the ration must be higher than the minimum threshold. Therefore, explaining the conditioned positive allocation means explaining the extra ration r i -m i . We already know that OPMA allocates the minimum ration to all individuals he chooses to serve, except for the most productive, who receives the entire remaining endowment. A general relationship between the allocated extra ration r i -m i and a potential recipient’s productivity cannot be Table 8 Explaining the probability of positive payoffs: a logit model Variables Coefficient Std. err. Variable Coefficient Std. err. Constant -0.790 0.543 OPMA 0.024 0.652 treatment_1 -2.557*** 0.313 OPMA . NMA 0.457 0.643 treatment_3 -1.376*** 0.306 NMA 1.149** 0.554 treatment_4 -1.078*** 0.295 NMA . RA 1.748*** 0.556 treatment_5 -1.326*** 0.297 RA 2.252*** 0.708 treatment_6 -0.776*** 0.302 treatment_7 -1.002*** 0.290 Minimum need -0.007*** 0.001 treatment_8 -1.377*** 0.289 OPMA . minimum need 0.001 0.001 treatment_9 0.039 0.315 NMA . minimum need -0.002*** 0.001 treatment_10 -0.920*** 0.299 RA . minimum need 0.000 0.001 Economic framing -0.349 0.273 Productivity 1.314*** 0.117 1 allocator only 0.398 0.380 OPMA . Productivity 0.282 0.193 Endowment 1.568*** 0.312 NMA . productivity 0.422*** 0.166 RA . productivity 0.043 0.224 Number of observations: 3,948 Number of groups: 58 Pseudo R 2 : 0.26 Rho: 0.188*** 0.042 UA: utilitarian allocator, OPMA: own payoff-maximizing allocator, NMA: number maximizing allocator, RA: Rawlsian allocator ***, (**), [*] significant at the 99% (95%) [90%] confidence level, resp. Ahlert et al.Health Economics Review 2012, 2:1 http://www.healtheconomicsreview.com/content/2/1/1 Page 8 of 11