Threecheers—psychological,theoretical,empirical—
Three cheers—psychological, theoretical, empirical—for lossaversionColin CamererCaltechcamerer@hss.caltech.eduAbstractThis note emphasizes the special role of prospect theory in drawing psychophysical considerations into theories of decision making under risk. One such consideration is the dependence of outcome value on a reference point, and the increased sensitivity of loss relative to gain (“lossaversion”). Lossaversion can explain the St. Petersburg Paradox without requiring concave utility, has the right psychology, is theoretically useful, and is a parsimonious principle that can explain many puzzles. An open question is whether aversion to loss is a stable feature of preference, is an expression of fear, or has some other properties. Keywords: Lossaversion, behavioral economics, experimental economics, information processing, consumer behaviorAuthor biographyProfessor Colin Camerer is the Rea and Lela Axline Professor of Business Economics at the California Institute of Technology (located in Pasadena), where he teaches both cognitive psychology and economics. Professor Camerer earned a BA degree in quantitative studies from Johns Hopkins in 1977, and an MBA in finance (1979) and a Ph.D. in decision theory (1981, at the age of 22) from the University of Chicago Graduate School of Business. Before coming to Caltech in 1994, he held positions at the Kellogg, Wharton, and University of Chicago business schools. His work uses evidence from psychology to inform economic models of risky choice, game theory, and market dynamics, with applications to bargaining, gambling, labor economics, and stock market bubbles. His latest work uses evidence from neuroscience to inform economics (“neuroeconomics”). Novemsky and Kahneman’s (2005) tour de force exploring the boundaries of lossaversion is a reminder of the power of prospect theory in explaining how people value risks, and also a reminder of the delicacy required to sharpen and apply the model. The deep contribution of prospect theory was not only to introduce a parsimonious and fruitful alternative to expected utility theory. More importantly, prospect theory introduced a perceptual and psychophysical perspective to thinking about money, goods, and risk. Previous approaches, stemming from Ramsey, DeFinetti, Savage, and von Neumann and Morgenstern, constrained the functional forms integrating probability and evaluation with logical axioms—most powerfully, the independence axiom—rather than with psychophysical principles. But why should principles from logic, which is evolutionarily new and effortful, necessarily trump highlyevolved principles from perception and psychophysical evaluation as a basis for theory? The rise of normative axiomatic foundations as description is good applied mathematics, but potentially leads to bad psychology and biology. Besides privileging axiomatic logic over biological mechanism, economic theory further imposes the prejudice that the value of money is derived from what it can buy, rather than being a “primary reinforcer” which directly produces psychophysical sensation, and is only loosely coupled to forecasts about utility from goods the money will buy. The psychophysical perspective immediately injects the observation, common in hedonic psychology since at least Helson (1964), that adaptation to current circumstances creates a natural (and evershifting) point of reference. In receiving money and goods, referencedependence naturally implies that people will not code receipts on an integrative scale (like the calculation of net worth across different wealth categories performed when you apply for a mortgage). Instead, receipts are naturally encoded, and responded to, as gains and losses from a reference point. So a pauper who wins $10 million in a lottery may be ecstatic with her staggering $10 million in net worth, while a dotcom tycoon whose $100 million paper fortune shrinks to only $10 million in a week may kill himself, distraught over having only a pathetic $10 million in net worth. The gainloss distinction derived from comparison to reference points is important in two ways: First, the psychophysical view strongly hints that a general principle of diminishing sensitivity to change works for both growing losses and growing gains, which implies convexity of disutility for losses, which in turn implies a preference for gambling in the domain of losses to “catch up” to the reference point. Behavioral ecologists are quick to point out that an animal which is starving should gamble on finding food (because the marginal utility of catching up is so great—live or die); and these instincts for switching riskpreference in the face of survival risk are probably deeply encoded in biological mechanism we share with other animals. Prospect theory suggests further that this instinct is overgeneralized to the case where the stakes are not death, but are merely a return to a reference point. Second, the psychophysical view that invites a gainloss distinction also allows the possibility that losses are more disliked than equalsized gains are liked (lossaversion). The purpose of this short essay is to remind readers of the power of lossaversion as an expression of psychological mechanism, a theoreticallyuseful modeling device, and an empirical phenomenon. I’ll remind readers how lossaversion upsets the orthodoxy of riskaversion in economic theory (as embodied solely in concavity of outcome utility), present some evidence, and suggest some unanswered researchable questions about how lossaversion works. Lossaversion can explain the St. Petersburg Paradox I’ll first show the power of lossaversion in resolving the St. Petersburg paradox, which is widely and falsely believed (from Bernoulli onward) to necessarily lead to a concept of locally concave utility for money (i.e., u’’(x)<0 everywhere). Please appreciate the radicalism of this claim: Any economics teacher who uses the St. Petersburg paradox as a “proof” that utility is concave (and gives students low grades for not agreeing) is confusing the sufficiency of an explanation for its necessity. Consider the classic St. Petersburg gamble: A fair memoryless coin is flipped until a “heads” turns up. If the head is flipped on the nth trial, the gamble pays $2n. The gamble therefore pays $2 with probability .5 (a head comes up right away), $4 with probability .25 (a tail, then the fateful head), $8 with probability .125, and so on. The expected value of the gamble is 1+1+1+.., repeated infinitely, so its expected value is infinite. But when asked how much they would pay to buy such a gamble, people routinely report sums around $20. (As Hacking (1980) wrote, "few of us would pay even $25 to enter such a game."). Bernoulli concluded from this (based on introspection) that the infinite sum of expected utilities of money outcomes could be finite—and reconcilable with a low buying price—if utility was sufficiently concave (i.e., u’’(x)<0). The first important feature of the problem is that very large sums simply can’t be paid out. Suppose the maximum payout is some sum X, and denote by M the maximum number of flips of tails which almost hits the payout limit (i.e., M=maxm ($2m<X), so $2m+1>X). Since $230 is around $1 billion, we will work with M around 30 or more. A statistical fact is that if you pay $20, the most likely outcome is that a head will come up quickly and you’ll lose money net of the price you paid (with probability .5+.25+.125+.0625=.9375; you have only a 7% chance of winning, though of course if you win, you might win big). This provides the germ of an explanation based on lossaversion. A little algebra shows how low buying prices can be if people are lossaverse. Call the buying price P. Consider a piecewiselinear model in which the utility of net loss, $2nP, is λ($2nP) is negative and the utility of net gain is $2nP if the net gain is positive. The factor λ is the coefficient of lossaversion, which is thought to be around 2 for money gambles of this sort (see Novemsky and Kahneman, 2005). Now we can solve for a price P which makes a person just willing to buy the gamble for price P. If $2n<P<$2n+1, then payouts less than $2n are encoded as losses and payouts greater than $2n+1 are encoded as gains. But losses loom larger than gains. This means that with piecewise linear utility, the lossaverse expected value is Σk=1→n λ($2kP)(1/2)k +Σk=n+1→M ($2kP)(1/2)k . The first term is negative and the second term is positive. Some algebra reduces this to a maximum buying price (assuming the inequality restriction $2n<P<$2n+1) of P=[Mn+nλ]/[λ(1(1/2)n)+(1/2)n(1(1/2)M)]. For a lossaversion coefficient of λ=2, if the maximum prize is around $1 billion (M around 30), then the maximum buying price is P*=$17.55; if the maximum is $1 trillion (M around 40) the price is P*=22.71. A higher lossaversion coefficient of λ=3 reduces these prices to $13.10 and $16.70. A lossaversion coefficient of λ=1 (riskneutrality) produces P*=M (expected value pricing). So the St. Petersburg paradox is not necessarily resolved uniquely by locally concave utility. Being more averse to losing than winning, and believing (sensibly) there is a maximum payout, can also explain the paradox without assuming concavity of utility. By not anticipating the relative distaste for loss, and the power of this lossaversion to explain his puzzle, Bernoulli made one error. He made another error by directing attention purely toward the evaluation of money consequences, rather than toward probabilities. My hunch is that statisticians of his time were enamored with the (relatively) new discovery of expectation. And so Bernoulli, too, was infatuated with the newlynormative appeal of attaching the correct probability to outcomes and did not seriously consider the possibility of nonlinear weighting of probabilities, as a behavioral principle. Yaari (1987) forcefully noted the duality of concavity of money utility and nonlinearity of probability weighting as equallyplausible explanations of riskaversion. His intuition was that not caring that much about the value of consequences if you win a lot (concave u(x)), and not thinking you were likely to win a lot (underweighting p(win)) were dual explanations. Why should the concave u(x) explanation be privileged over underweighting of p(win)? Prospect theory suggests further interesting implications of overweighting low probabilities (viz., explaining gambling on unlikely wins, like lottery tickets, and demand for insurance only for catastrophic risks). But Kahneman and Tversky were careful to leave a gap in the published graph showing the π(p) function. (Their graph is probably the only deliberatelydiscontinuous function published in Econometrica—and with good reason.) The incompleteness of their graph when p is low is a question mark, allowing the possibility that low probabilities are either dismissed (presumably due to inattention) and set to zero, or greatly overweighted (e.g., see Prelec’s beautiful 1998 axiomatization). Whether low probabilities are ignored or overweighted presumably depends on cognitive considerations that remained to be fully specified by a more nuanced cognitive theory, which requires a theory of attention. If people simply dismiss the chance of winning huge sums (or think they won’t get paid, as encoded by the maximum payout limit implicit in M) that also helps explain low buying prices for St. Petersburg gambles without concavity of utility. The key point—a blindingly simple one—is that it is simply false that low prices for St. Petersburg gambles necessarily imply that utility is locally concave (assuming a finite maximum payout). If the puzzle is explaining modest buying prices that are lower than expected values, concave utility is sufficient but not necessary. Piecewise linear utility with lossaversion is sufficient too. Decision isolation and lossaversionA crucial ingredient in empirical applications of lossaversion is “decision isolation”, or “focusing illusion”, in which single decisions loom large even though they are included in a portfolio of stream of similar decisions. If many small decisions are integrated into a portfolio of choices, or a broad temporal view—the way a gambler might view next year’s likely wins and losses, taken together—then the loss on any one gamble is likely to be offset by others, so aversion to losses is muted. Therefore, for lossaversion to be a powerful empirical force requires not only aversion to loss, but also a narrow focus so that local losses are not blended with global gains. This theme emerges in the 10 field studies discussed by Camerer (2000), which show the power of lossaversion (and other prospect theory features) for explaining substantial behaviors outside the lab. An example of lossaversion which illustrated the importance of decision isolation, and is also illustrative in a literal sense, comes from my work with Eric Zhikang Chua on savings decisions (2004). We did experiments in which subjects received simulated income in each of 30 periods drawn from a rightskewed lognormal distribution (which means most income was modest, but in some periods income was very large). In each period they could choose how much to consume from their savings, which equaled previous savings plus current income; whatever they didn’t consume went into savings. Their experimental earnings were a concave function of utility from spending in each period, added up across each of the 30 periods and across 7 “reincarnated” lifetimes (to study learning across when the entire 30period “life” was repeated over and over). Because income was random, optimal behavior required subjects to save a lot in early periods, to build up a “buffer stock” of savings in case future income was low. (Because utility was concave, having a very low consumption period if you ran short— akin to eating ramen noodles and sleeping in your car—was very bad in terms of experimental points.) To make the problem even more challenging, the utility of current consumption in a lifetime depended upon previous consumption, incorporating a kind of “habit formation”. This setup means there is an “internality” (an internal withinperson externality). Current consumption necessarily lowers the utility of future consumption, by raising the habit level. The basic finding is that subjects learned relatively quickly to save a lot in early periods to build up a buffer stock and to limit the negative internality from their habit. But lossaversion also shows up in a dramatic way. The utility function from current consumption in a period, adjusting for habit, is structured so that in lowincome periods optimal behavior requires subjects to consume so little that their experimental utility is sometimes slightly negative (depending on whethe

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