Journal of Economic Literature Vol. XXXVIII (June 2000) pp. 332–382 Starmer: Developments in Non-Expected Utility Theory Developments in Non-Expected Utility Theory: The Hunt for a Descriptive Theory of Choice under Risk CHRIS STARMER1 1. Introduction stimulated developments in non-EU is surely of widespread concern: put bluntly, OW MANY THEORIES of decision the standard theory did not fit the facts. Hmaking under risk and uncertainty As the standard theory of individual can you think of? Readers of this article decision making, and as a core component will no doubt be familiar with Expected of game theory, EUT constitutes a key Utility Theory (EUT), the standard the- building block of a vast range of eco- ory of individual choice in economics. nomic theory. It should be no surprise, Many, I expect, will know of a few alter- therefore, that developing a better un- natives to this model. But how many, I derstanding of the determinants of indi- wonder, will be aware that these so- vidual choice behavior seemed a natural called non-expected utility models now research priority to many theorists. number well into double figures? An Around two decades of quite intensive enormous amount of theoretical effort research on the topic has generated a has been devoted towards developing al- great deal of theoretical innovation plus ternatives to EUT, and this has run a much richer body of evidence against hand-in-hand with an ongoing experi- which models can be judged. There can mental program aimed at testing those be few areas in economics that could theories. The good and proper division claim to have sustained such a rich in- of labor suggests that a relatively small teraction between theory and evidence group of specialists will be fully aware of in an ongoing effort to develop theories the details of this literature. At the same in closer conformity with the facts. time, the implications of developments Considered together, the accumulated in this field are of more than passing in- theory and evidence present an oppor- terest to the general economist, since what tunity to reflect on what has been 1 University of East Anglia, Norwich. I owe achieved. Perhaps the most obvious thanks to Colin Camerer, Robin Cubitt, Mark question to address to this literature is Machina, John Quiggin, Uzi Segal, Robert Sug- this: has it generated, or does it show den, Peter Wakker, and George Wu, plus an anonymous referee for extremely helpful com- the prospect of generating, a serious ments on and discussions around this paper. I am contender for replacing EUT, at least also grateful for support from the Economic and for certain purposes? If the question Social Research Council of the UK (Risk and Hu- man Behaviour Research Programme: Award No. seems disarmingly straightforward, pro- L211252053). viding a clear-cut answer will not be. 332 Starmer: Developments in Non-Expected Utility Theory 333 Identifying a “best theory” naturally re- experimental testing, (2) how new theo- quires judgements about the relative ries may help us to explain a range of importance of predictive accuracy, sim- phenomena “in the field,” and (3) plicity, tractability, and so on. Such whether non-expected utility theory of- judgments are complicated by the fact fers a viable alternative to EUT for that the evidence, much of which de- everyday theoretical use. In the penulti- rives from the experimental paradigm, mate Section 6, I discuss two emerging is open to different interpretations. lines of enquiry which I see as particu- In what follows, my aim will be to set larly exciting paths for future research. out what I take to have been key A final section offers some concluding theoretical developments in the area, to reflections. review the related evidence and draw conclusions about the current state of 2. Where It Began play and the prospects for the future. In Although the primary purpose of this doing so, rather than simply to present paper is to review alternatives to EUT, an exhaustive list of models, my aim will that theory provides the natural point of be to identify and discuss different departure, since most of the theories I modeling strategies picking specific will be discussing can be understood as models as illustrations. I also intend to generalizations of this base theory.2 narrow my sights in two significant re- EUT was first proposed by Daniel Ber- spects. First, my focus will be on de- noulli (1738) in response to an apparent scriptive as opposed to normative is- puzzle surrounding what price a reason- sues. Second, I will concentrate on the able person should be prepared to pay problem of modeling choices under risk to enter a gamble. It was the conven- as opposed to the more general cate- tional wisdom at the time that it would gory of uncertainty (the distinction is be reasonable to pay anything up to the explained in the next section). Clearing expected value of a gamble, but Ber- the ground in this way will, I hope, noulli presents this counterexample. A sharpen the focus on one central re- coin is flipped repeatedly until a head is search problem which continues to mo- produced; if you enter the game, you tivate much of the research in this receive a payoff of, say, $2n where n is arena: the endeavor to develop a “satis- the number of the throw producing the factory” account of actual decision be- first head. This is the so-called St. Pe- havior in situations of risk. It will be a tersburg game. It is easy to see that its personal view, but one which I hope expected monetary payoff is infinite, yet will help the interested nonspecialist Bernoulli believed most people would find a trail through this expansive and only be prepared to pay a relatively quite detailed literature. small amount to enter it, and he took this The paper is organized as follows. intuition as evidence that the “value” of Sections 2 and 3 set the scene with dis- a gamble to an individual is not, in gen- cussions of the standard theory and the eral, equal to its expected monetary evidence that prompted theorists to value. He proposed a theory in which look for alternatives. Section 4 provides individuals place subjective values, or the core overview of non-expected util- “utilities,” on monetary outcomes and ity theories. Section 5 seeks to evaluate what has been achieved so far, and in 2 I shall not dwell on this account of EUT. For those interested in further discussion an excellent three subsections I discuss (1) how new starting place is Paul Schoemaker’s (1982) review theories have fared in a second phase of in this journal. 334 Journal of Economic Literature, Vol. XXXVIII (June 2000) the value of a gamble is the expectation ternative models of choice, it will be of these utilities. While Bernoulli’s the- useful to present one set of axioms from ory—the first statement of EUT— which EUT can be derived. In the ap- solved the St. Petersburg puzzle, it did proach I adopt, at least to begin with, not find much favor with modern preferences are defined over prospects economists until the 1950s. This is where a prospect is to be understood as partly explained by the fact that, in the a list of consequences with associated form presented by Bernoulli, the theory probabilities. I will assume throughout presupposes the existence of a cardinal that all consequences and probabilities utility scale; an assumption that did not are known to the agent, and hence, in sit well with the drive towards ordinaliza- choosing among prospects, the agent tion during the first half of the twentieth can be said to confront a situation of century. risk (in contrast to situations of uncer- Interest in the theory was revived tainty in which at least some of the out- when John von Neumann and Oskar comes or probabilities are unknown). I Morgenstern (1947) showed that the ex- will use lowercase letters in bold (e.g. pected utility hypothesis could be de- q, r, s) to represent prospects, and the rived from a set of apparently appealing letter p to represent probabilities (take axioms on preference. Since then, nu- it that p always lies in the interval merous alternative axiomatizations have [0,1]). A given prospect may contain been developed, some of which seem other prospects as consequences, but highly appealing, some might even say assuming that such compound prospects compelling, from a normative point of can be reduced to simple prospects fol- view (see for example Peter Hammond lowing the conventional rules of prob- 1988).3 To the extent that its axioms can ability, any prospect q can be repre- be justified as sound principles of ra- sented by a probability distribution q = tional choice to which any reasonable (p1, . , pn) over a fixed set of pure person would subscribe, they provide consequences X = (x1, . , xn) where grounds for interpreting EUT norma- pi is the probability of xi, pi ≥ 0 for all i, tively (as a model of how people ought and Σipi = 1. Hence, the elements of X to choose) and prescriptively (as a prac- are to be understood as an exhaustive tical aid to choice). My concern, how- and mutually exclusive list of possible ever, is with how people actually choose, consequences which may follow from a whether or not such choices conform particular course of action. While this with a priori notions of rationality. Con- notation allows a prospect to be written sequently, I will not be delayed by simply as vector of probabilities (as q questions about whether particular axi- above) it will sometimes be useful to be oms can or cannot be defended as explicit about the consequences too sound principles of rational choice, and (e.g.