A Theory of Medical Decision M aking underUncertaintyEdi Karni∗†December 21, 2008∗Johns Hopkins University, Baltimore, MD 21218, USA. e-mail address: [email protected]†Ibenefited from comments of Ani Guerdjikova, Mo she Leshno, Ma rzena Rostek, Marie-Louise Viero,Kip Viscusi, Peter Wakker and an anonymous referee.1Abstract: This paper presents an axiomatic model of medical decision making anddiscusses its potential applications. The m edical decisio n problems envisioned concern th ec h oice of a med ical treatm ent follo wing a diagnosis in situations in which data allow construc-tion of an empirical distributio n over the poten tial outcomes associated with the alternativ etreatments. In its descriptiv e in terp retation, the m odel is an hypothesis about the patient’sc ho ice beha vior. The theory also aims to aid ph y sicians recom m end treatme nts in a coheren tmanner.Keyw ords: Med ical decision making, medical treatment ch oice,JEL classification nu mbers: I19, D812For the purpose of this paper, the term medical decision makin g refers to the choice of acourse of action (action, for short) follo w ing a diagnosis of a patient’s co nd ition. An actionconsists of the medica l treatment itself; the facilit y in which it is to be admin istered; and,if perceived relevan t, the in dividu als who administer it. Con sider, for example, a patientdiagnosed with prostate cancer. G iven his specific personal c h ara cteristics (medical history,age, ph ysical condition, and so forth), t he patien t must c h oose among various treatments(e.g., radical prostatecto my, radiation therapy); the medical facilities in which he is to betreated (the local hospital, a medical cen ter in another cit y); and the ph ysician who performsthe surgery or administers the th erapy of choice. The consequences consist of the patient’spost-treatm ent state of health , including the side effects of treatmen t; the associated painand inconvenience; the direct monetary expenses; and the potential loss of income.1In many situations involving medica l decision making , the empirical probability (tha t is,therelativefrequency)ofthedifferent outcomes conditional on the treatmen ts, c haracteristicsof the patien t, and choice of hospital and ph ysicia n are known. The question is, ho w does(or how should) an informed patien t ch oose amo ng the possible courses of action?In this paper I propose an axiomatic theory of medical d ecision making in which thepatient’s preferences are represen ted by an outcome-dependent expected utility function.More formally, let a denote an action and denote b y c a v ector of the patient’s c h aracteristics1A recent study by Sommers et. al. (2007) underscores the importance of patients’ preferences forthe determination of the optimal treatment (defined as the expected QALY). See further discussion in theconcluding section.3(med ical history, age, gender, race, profession , fam ily situation, ph ysical state, and an yother personal attributes that may bear on the outcome of the m edical treatmen ts underconsideration). I examine the structure of a preference relation, <, on the set of actions thatis necessary and sufficien t for the follo w ing representa tion:(a, c) 7→ λ (a)Xω∈ΩU (f (ω; a, c) ,ω) p (ω | a, c)+v (a) ,where U is th e utilit y function ; ω denotes the post-treatment health state (or outcome); Ωis the set of all outcomes associated with a given diagno sis; f (ω; a, c) denotes the financialconsequence associated with the outcome ω conditional on the patien t’s characteristics andthe action; p (·|a, c) is the probability distribution on Ω conditional on the action and thepatient’s personal c h aracteristics; and λ and v represen t the “utility cost,” in clud ing thepain or discomfo rt associated w ith differ ent actions. N ote that the patient’s risk attitudes,captured by the utility functions of mon ey, U (·,ω) ,ω ∈ Ω, are outcome dependen t but notaction dependen t.2I assume throughout that, insofar as c hoosing the right treatmen t is concerned, there is noconflict of interests bet ween phy sicians and the patien ts. In other words, follow ing a diagnosistheobjectiveofbothpartiesistochoosethetreatmentthatisbestfromtheviewpointofthe patient’s w elfare in all its relevant aspects, health , financial, and the nonmonetary cost2Outcomes represent states of health, and the utility functions in this model are state-dependent functionsof the patient’s wealth. This is an important aspect of this model, which is missing from that of Sommers et.al. (2007). Empirical evidence suggests that there are significant variations both in the level and marginalutility of wealth across states of health (see Viscusi and Evans (1990)).4of the treatment. The decision ma king pro blem is in ter active, involving the integration ofthe private information of the two parties. Th e ph y sic ian input includes the diagnosis, theidentification of the alternative treatments, and the determin ation of the probabilities of theoutcomes associated with the alternativ e courses of action. The patient’s input consists ofhis personal c h aracteristics and preferences, includ in g his valuation of the health outcom es,his outcome-dependent risk attitudes, and his valuation of the nonfinancial aspect of thecost associated with the alternativ e treatments. T he decision m aker is sup posed to be ableto express his preferences over treatments for every conceivable probabilit y distribution overthe outcom es. This richn ess of the fram ework allo w s the identification of the ingredien ts ofthe re pre sentation, nam ely, the utilities an d the prob abilitie s.The analysis consists of two m a in parts: Th e firstpartisanapplicationofanexten-sion of the von Neu m an n-Morenstern expected u tilit y theory developed in Karni and Safra(2000) to obtain, for ever y giv en course of action, an outcom e-dependent expected utilityrepresentation. The second part, whic h is the no vel aspect of this work, in volv es linkingtherepresentationsacrossactions. Thisisdonebytwoaxioms,namely,action-independentrisk attitudes and action-independent preferences over lottery mixtures. Action independen trisk attitudes requires that the patient risk aversion is not altered b y the treatment itself.Action-in dependen t lottery m ixtures requires that mixing lotteries belonging to the sameindiffer en ce class under different treatments yield lotteries that are themselv es