Prospect Relativity 1 Running Head: PROSPECT RELATIVITY Prospect Relativity: How Choice Options Influence Decision Under Risk

In many theories of decision under risk (e.g., expected utility theory, rank dependent utility theory, and prospect theory) the utility or value of a prospect is independent of other prospects or options in the choice set. The experiments presented here show a large effect of the available options set, suggesting instead that prospects are valued relative to one another. The judged certainty equivalent is strongly influenced by the options available. Similarly, the selection of a preferred option from a set of prospects is strongly influenced by the prospects available. Alternative theories of decision under risk (e.g., the stochastic difference model, multialternative decision field theory, and range frequency theory), where prospects themselves or prospect attributes are valued relative to one another, can provide an account of these context effects. Prospect Relativity 3 Prospect Relativity: How Choice Options Influence Decision Under Risk Human behavior, as distinct from mere unintentional movement, results from decision. And decisions almost always involve trading-off risk and reward. In crossing the road, we balance the risk of accident against the reward of saving time; in choosing a shot in tennis, we balance the risk of an unforced error against the reward of a winner. Choosing a career, a life-partner, or whether to have children, involves trading off different balances between the risks and returns of the prospects available. Understanding how people decide between different levels of risk and return is, therefore, a central question for psychology. Understanding how people trade off risk and return is also a central issue for economics. The foundations of economic theory are rooted in models of individual decision making. For example, to explain the behavior of markets we need a model of the decision making behavior of buyers and sellers in those markets. Most interesting economic decisions involve risk. Thus, an economic understanding of markets for insurance, of risky assets such as stocks and shares, of the lending and borrowing of money itself, and indeed of the economy at large, requires understanding how people trade risk and reward. In both psychology and economics, the starting point for investigating how people make decisions involving risk has not been empirical data on human behavior. Instead, the starting point has been a normative theory of decision making, expected utility theory (first axiomatized by von Neumann & Morgenstern, 1947) which specifies how people ought to make decisions and plays a key role in theories of rational choice (for a review see Shafir & LeBoeuf, 2002). The assumption has then been that, to an approximation, people do make decisions as they ought to, that is, expected utility theory can be viewed as a descriptive, as well as a normative, theory of human behavior. At the core of expected utility theory is the assumption that people make choices that maximize their utility, and they value a risky option by the expected utility (in a probabilistic sense of expectation) that it will provide. In general, the prospect (x1, p1; x2, p2; ... ; xn, pn), where outcome xi occurs with probability pi, Prospect Relativity 4 and p1+p2+...+pn=1, has expected utility (hereafter EU), U x 1 , p 1 ; x 2 , p 2 ; ... ; x n , p n = p 1 u x 1 p 2 u x 2 ... p n u x n (1) (The function U gives the utility of a risky prospect. The function u is reserved for the utility of certain outcomes only.) In psychology and experimental economics, there has been considerable interest in probing the limits of this approximation, that is, finding divergence, or agreement, between EU theory and actual behavior (e.g., Kagel & Roth, 1995; Kahneman, Slovic & Tversky, 1982; Kahneman & Tversky, 2000). In economics more broadly, there has been interest in how robust economic theory is to anomalies between EU theory and observed behavior (for a range of views, see, e.g., Akerlof & Yellen, 1985; de Canio, 1979; Cyert & de Groot, 1974; Friedman, 1953; Nelson & Winter, 1982; Simon, 1959, 1992). The present paper demonstrates a new and large anomaly for EU theory in decision making under risk. Specifically, we report results that seem to indicate that people do not possess a well-defined notion of the utility of a risky prospect and hence, a fortiori, they do not view such utilities in terms of EU. Instead, people's perceived utility for a risky prospect appears highly context sensitive. We call this phenomenon prospect relativity. Considerable further work is, of course, required to establish the generality and scope of our results across the vast array of decision making domains of psychology and economic interest. But we believe that these anomalies are sufficiently striking to motivate the attempt to explore them further. Motivation from Psychophysics In judging risky prospects, people must assess the magnitudes of risk and return that they comprise. The motivation for the experiments presented here was the idea that some of the factors that determine how people assess these magnitudes might be analogous to factors underlying assessment of psychophysical magnitudes, such as loudness or weight. Specifically, people appear poor at providing stable absolute judgments of such magnitudes Prospect Relativity 5 and are heavily influenced by the options available to them. For example, Garner (1954) asked participants to judge whether tones were more or less than half as loud as a 90 dB reference loudness. Participants' judgments were entirely determined by the range of tones played to them. Participants played tones in the range 55-65 dB had a half-loudness point, where their judgments were more than half as loud 50% of the time and less than half as loud 50% of the time, of about 60 dB. Another group, who received tones in the range 65-75 dB had a half-loudness point of about 70 dB. A final group, who heard tones in the range 75-85 dB, had a half-loudness point of about 80 dB. Laming (1997) provides an extensive discussion of other similar findings. Context effects, like those found by Garner (1954), are consistent with participants making perceptual judgments on the basis of relative magnitude information, rather than absolute magnitude information (Laming, 1984, 1997; Stewart, Brown, & Chater, 2002a, 2002b). If the attributes of risky prospects behave like those of perceptual stimuli, then similar context effects should be expected in risky decision making. This hypothesis motivated the experiments in this paper, which are loosely based on Garner's (1954) procedure. Existing Experimental Investigations A small number of experiments has already investigated the effect of the set of available options in decision under risk. Mellers, Ordóñez and Birnbaum (1992) measured participants' attractiveness ratings and buying prices (i.e., the price that a participant would pay to obtain a single chance to play the prospect and receive the outcome) of simple binary prospects of the form "p chance of x" presented within two different sets of prospects. In one context, the distribution of expected values of accompanying prospects was positively skewed, and in the other context, the distribution of expected values was negatively skewed. Attractiveness ratings were significantly influenced by this manipulation, with higher ratings for prospects in the positive skew condition than for the same prospects in the negative skew condition. However, context had a minimal effect on buying price. With more complicated Prospect Relativity 6 prospects of the form "p chance of x otherwise y", the effect of skew on buying price was slightly larger. The large effect that the set of options available had on attractiveness ratings and much smaller effect on buying price is consistent with a similar demonstration by Janiszewski and Lichtenstein (1999). They gave participants a set of prices for different brands of the same product to study. The prices varied in range. The range had an effect on judgments of the attractiveness of a new price, but not on the amount participants would expect to pay for a new product. The set of options available as potential certainty equivalents (hereafter CEs) has been shown to affect the choice of CE for risky prospects. In making a CE judgment, a participant suggests, or selects from a set of options, the amount of money for certain that is worth the same to him or her as a single chance to play the prospect. We shall consider CE judgments extensively in our experiments. Birnbaum (1992) demonstrated that skewing the distribution of options offered as CEs for simple prospects, whilst holding the maximum and minimum constant, influenced the selection of a CE. When the options were positively skewed (i.e., most values were small) prospects were under-valued compared to when the options were negatively skewed (i.e., most values were large). MacCrimmon, Stanbury and Wehrung (1980) present some evidence that the set in which a prospect is embedded can affect judgments about the prospect. They presented participants with two sets of five prospects to be ranked in order of attractiveness. The expected value of each prospect was constant across all prospects and both sets. There were two prospects in common between the two sets. If context provided by the other prospects in a set did not affect the attractiveness of a prospect, each participant should have consistently ranked one prospect as more attractive than the other, in both sets. MacCrimmon et al. found that 9 of a total of 40 participants had a different ordering of the two prospects in the two sets. They argue that this is not merely inconsistency because these participants made consistent rankings within a set, and instead reflects the influence of the other prospects in the choice Prospect Relativity 7 set. Following this logic, however, it would take only one participant to have a different ordering of the two prospects, but who otherwise behaved consistently, to conclude that there was an effect of choice set. A random fluctuation in risk aversion between sets might produce this result. With such a small number of data points, and no concrete null hypothesis allowing a significance test to be made, any conclusion based on this results must be very tentative. In summary, there is an effect of previously-considered prospects on the attractiveness rating assigned to a current prospect, and also a small effect on buying price. Moreover, the context provided by a set of values from which a CE is to be chosen affects CE judgments. Finally, in choosing between prospects there is a suggestion that the wider choice may affect preferences between identical pairs of prospects. In the experiments below we find large and systematic effects of choice set (both potential CEs and accompanying prospects) on the valuation of individual prospects. These effects are not compatible with EU theory or some of its most influential variants, according to which the value of a prospect is independent of other available options. These results are, though, compatible with a variety of models that discard this "independence" assumption concerning the value of prospects. Summary of Experiments As indicated above, in this paper we adapt methodologies from psychophysics to investigate the possibility of that context effects influence decision under risk. The aim of Experiments 1A-1C was to determine whether options offered as potential CEs influence estimates of a prospect's CE. We consistently found substantial effects. In Experiment 2, we introduce a new procedure to investigate these effects in which, under certain assumptions, it is optimal for participants to provide truthful CEs. In Experiment 3, we examine whether these effects are similar to those observed in magnitude estimation tasks. In the remaining two experiments, Experiments 4 and 5, we investigate whether the effect of available options arises in choices between prospects as well as in CE judgments about prospects. Experiment 1A Prospect Relativity 8 Following a similar logic to Garner's (1954) loudness judgment experiment described above, participants were given a set of options as possible CEs for each prospect. Participants were asked to decide on a CE for the prospect, and then select the option closest to their estimate. For each prospect, options were either all lower in value than the mean free choice CE (given by another group of participants) or all higher, as illustrated in Figure 1. If a participant is not influenced by the set of options, then his or her choice of option should be that nearest to his or her free choice CE. Under this hypothesis, the expected proportion of times each option will be chosen can be calculated by integrating the free choice distribution between appropriate bounds. The key prediction is that the highest option in the low condition (L4) and the lowest option in the high condition (H1) should be chosen more than half of the time. This prediction holds for any symmetrical distribution of free choice CEs. Alternatively, if participants' responses are solely determined by the set of options presented to them, then the distribution of responses across options should be the same for both the low and the high value range of option CEs.