A fair coin is about to be flipped.
A Heads voucher pays $10 if that coin lands heads.
A Tails voucher pays $10 if that coin lands tails.
What is the most you would pay for a Heads voucher? $______
If you owned a Heads voucher, what is the most you would pay for a Tails voucher? $______
1 Introduction
Analyses of choice under uncertainty typically treat risk aversion as a primitive and stylized fact. As frequently-cited evidence, the certainty equivalent of a gamble is nearly always below its expected value. For instance, a typical person finds a sure $3 about as attractive as a coin flip for $10.
From the perspective of Expected Utility Theory, the level of risk aversion implied by such small-stakes choices exceeds the level exhibited at larger stakes. For example, someone who’d always prefer a sure $3 over a coin flip for $10 must also prefer a sure $100 over a coin flip for a billion dollars (Reference RabinRabin 2000; Reference Rabin and ThalerRabin & Thaler, 2001). Prospect Theory (Reference Reinholtz, Fernbach and LangheKahneman & Tversky, 1979) was developed, in part, to accommodate such discrepancies.
In this project, we investigate a different sort of inconsistency in risk attitudes by comparing the valuation of a bet (e.g., a voucher which pays $10 if a coin lands heads) with the valuation of its perfect hedge (e.g., a second voucher which pays $10 if tails obtains). Logic requires that these two valuations sum to the amount their joint possession guarantees (e.g., $10) and, accordingly, correlate −1.0.Footnote 1 We find, instead, that they correlate positively. Furthermore, risk aversion implies that hedges should be worth more than their expected value – an implication many find counterintuitive.
Because lower bet valuations imply more risk aversion, whereas lower hedge valuations imply more risk tolerance, a positive correlation between them means that those who appear more risk-averse by one measure appear more risk-tolerant by the other. Although previous research has questioned the generalityFootnote 2 and predictive powerFootnote 3 of risk attitudes, our results are even more problematic, as these two measures come from the same domain (small stakes gambles involving money) and not only fail to cohere, but strongly contradict one another. Though levels of risk aversion are known to vary across elicitation procedures, many nevertheless assume that such procedures at least serve to rank individuals by their risk attitudes (see, e.g., Reference Charness, Gneezy and ImasCharness, Gneezy & Imas, 2013, p. 50). However, our results cast doubt upon even this more modest claim. They also raise the question of how bet and hedge valuations would be altered by an appreciation of this logic, if that could somehow be instilled – which is not so easy, as we will show.
2 Studies 1a to 1g
Our first set of experiments (summarized in Table 1) document this curious phenomenon. Though designs varied slightly (see Appendix B for methodological details), each participant was essentially asked the most they would pay for a 50% chance of $10 and the most they would pay to convert that to a certain $10 (by acquiring a perfect hedge). Logic requires that one’s valuation of the hedge equal $10 minus their valuation of the bet; that all responses in Figure 1 lie along the southeast diagonal (y=10-x). But as can be seen, few actually do, in any of the studies.Footnote 4
Table 1: Summary of studies 1a to 1g. Correct is % summing to prize; ugrads are Yale undergraduates; r is the correlation between bet and hedge WTP.
1 In study 1b, we separately analyzed results for 267 participants who answered $10 when asked: “What is the most you would pay for $10? In study 1g, we separately analyzed results for 240 participants who valued a pair of $5 Amazon gift certificates at $10 and scored perfectly on an eight-item test intended to assess their comprehension of bets and hedges (see Appendix B). In both subsets, results are consistent with the full sample. Since those who would pay $10 for $10 (or for two $5 gift certificates) were probably not understating their willingness to pay, this weighs against the idea that the positive relation between bet and hedge valuations was driven by heterogeneity in the degree to which participants shaded their true valuations downward.
Figure 1: Results of studies 1a-g. The figure plots bet valuations (x-axis) against hedge valuations (y-axis) for the seven experiments and two separately analyzed subsets summarized in Table 1. Dot area is proportional to number of participants at each coordinate. Note that 1f actually ranged from $0 to $100, but is re-scaled to match other studies in this figure. (see Appendix B for materials).
Studies 2a and 2b
Evaluating Heads and Tails vouchers separately and sequentially partially obscures their complementarity.Footnote 5 Thus, in our next two studies, MTurk workers chose between the pair of vouchers {Heads & Tails} and the package {Heads & $5} (see Table 2). With this more transparent formulation, most did prefer the voucher pair – and thus, at least implicitly, valued the Tails hedge above its expected value, as risk aversion dictates.Footnote 6, Footnote 7 However, even here, the explanatory power of risk preference remains in doubt, since those who preferred {Heads & Tails} to {Heads & $5} should also have preferred a sure $5 over a single Heads voucher, and vice versa. Yet we observed little relation between those two choices.Footnote 8
Table 2: Materials for Studies 2a and 2b.
Study 3
When considered separately, the Heads and Tails vouchers are symmetric: equivalent and interchangeable. Thus, it is easy to understand why many respondents value them similarly, even though the first voucher adds risk and the second removes it. To test whether respondents would explicitly endorse the symmetry argument over the normative argument, we presented them side by side (Table 3), with order counterbalanced, and simply asked respondents to choose one.
Table 3: Materials for Study 3
Only one in ten respondents (53/505) chose the correct argument ($7) over the symmetry argument ($3). Moreover, this small minority might actually have been even more confused, as those choosing $7 were less likely to pass an attention check and did significantly worse on a numeracy test appended to subsequent demographics questions (see Appendix D).Footnote 9
Study 4
To slightly generalize our basic paradigm, we next removed the symmetry between the bet and the hedge. We asked 182 MTurkers how much they would pay for four different bets: a 20% chance of $100, a 40% chance of $100, a 60% chance of $100, and an 80% chance of $100 as well as how much they would pay for the four corresponding hedges (which pay off in the remaining states). All four bet-hedge pairs were presented to each participant in random order. For each pair, respondents indicated their valuations of the bet, and then their valuations of the corresponding hedge. As before, possession of both guarantees the prize, and thus their two valuations should sum to $100 and correlate −1.0. However, unlike before, the bet and hedge are not symmetric, as they have different probabilities of delivering the prize. The results are summarized in Figure 2.
Figure 2: WTP for a P% chance at $100 and for its perfect hedge. The figure plots bet valuations (x-axis) against hedge valuations (y-axis) for each probability of $100 in Study 4. Dot area is proportional to number of participants at each coordinate.
With symmetry removed, valuations of the bet and the hedge correlate much more weakly, though still positively. Again, almost no data lie along the southeast diagonal, as the two valuations sum to well below $100 in all four cases. The mean valuations of the bet and hedge are reported in Table 4, with the subscript representing the asset’s valuation relative to its expected value.
Table 4: Mean bet and hedge valuations.
A comparison of the subscripts reveals that the bets and hedges generally deviate in the same direction from expected value (below it), rather than diverging to reveal a coherent attitude toward risk. However, though the two valuations do not cohere, hedges are, at least, valued at a higher fraction of their expected value. Moreover, respondents are willing to pay more than expected value to hedge the small residual risk of the 80% bet – providing at least one instance where a perfect hedge is priced above its expected value ($20), as should be universally expected for respondents who are risk averse.
The considerable difference between bet and hedge subscripts in Table 4 suggests that respondents are not simply ignoring possession of the bet when evaluating the hedge. For instance, they are willing to pay $6 for a 20% chance of $100 alone, but $38 for the same asset once they already own an 80% chance of $100. Although respondents clearly fail to fully appreciate the covariance between bets and hedges, the pattern remains distinct from complete covariance neglect, in which hedges and bets are treated as independent.Footnote 10
Nor are respondents in earlier “Heads and Tails” studies merely referencing their valuation of the bet to construct an equivalent valuation of the hedge. Although a substantial minority of hedge prices do equal bet prices in those studies, most don’t. And of the minority that do, many come from respondents who “simply” report the expected value for each asset ($5, $5). Though those are certainly reasonable valuations (and demanded by EUT), we strongly suspect that many of these respondents are treating these questions as math problems rather than an elicitation of their preferences (and would continue to just perform the math even if the numbers referenced millions of dollars). This suspicion draws some support from an analysis in Appendix C, which reports the data broken down by CRT score: those who answer ($5, $5) are no more reflective than off diagonal respondents and less reflective than the small minority of responses that lie elsewhere on the southeast diagonal.
In opposition to various “narrow framing” theories (e.g., Reference Tversky and KahnemanTversky & Kahneman, 1986) that assume respondents ignore possession of the bet when evaluating the hedge stand “multiple reference point” theories (e.g., Reference Kőszegi and RabinKoszegi & Rabin, 2006, Reference Kőszegi and Rabin2007) in which evaluation of hedge entails full consideration of both possible outcomes of the bet: the one in which the bet pays off (in which case money spent on the hedge is a waste and a loss) and the one in which the bet does not (but the hedge does, minus acquisition costs). By this multiple reference point formulation, acquisition of the hedge constitutes both a loss and a gain, which tends to make an unattractive combination to the extent that losses loom larger than gains. (See Appendix E for an examination of whether the multiple reference point perspective can make sense of our results. We conclude it cannot.Footnote 11)
3 Discussion
Valuations of bets and hedges are theoretically equivalent measures of risk attitudes, yet they often compel opposite conclusions. For instance, many who place a low value on risk-creating bets also place a low value on risk-reducing hedges. This marked departure from theoretical expectations seemingly impugns the construct validity of risk attitudes – even within the narrow domain of stylized monetary gambles. Of course, identifying discrepancies between a subset of measurement techniques isn’t usually regarded as sufficient cause to jettison a theoretically cherished construct – at least not within the social sciences. Nevertheless, to the extent that the two measures depart, it certainly raises the question about which, if either, better captures whatever we think we mean by risk aversion.
Adjudicating between potential measures of a putative construct requires the very thing that is lacking for constructs whose validity is still in question – agreement about the other thing(s) with which we should expect them to correlate. For instance, suppose Reference Holt and LauryHolt and Laury’s (2002) measure of risk preferences had corresponded much more highly with hedge valuations than with bet valuations. The conclusions drawn from this will still be conditioned by your prior faith in those measures. If you are confident that Holt and Laury’s measure captures the construct you care about, it would affirm hedge valuations and impugn bet valuations, but if you are confident that bet valuations best capture risk attitudes, the lack of relation with that other method would impugn that method.
In the experimental paradigm we use most commonly, the hedges are perfect, such that possession of one renders the coin irrelevant. This evokes the image of a very unusual transaction in which an owner of the bet pays $7 for the hedge and is then immediately handed $10. Since this is obviously equivalent to simply receiving $3, buying the hedge is equivalent to selling the bet. But the value of a perfect hedge also determines the aggregate value of the partial hedges from which it might be constructed. Consider an experiment in which chances to win a $100 prize can be purchased in cumulative increments of one percentage point each; a single point entitles its owner to a 1% chance of $100; fifteen points yields a 15% chance of $100, and so on. The expected value of each 1% chance is, of course, $1. Now consider a respondent for whom a 50% chance of winning a $100 prize is worth $30. That person values the first 50 points at 60 cents each, on average. However, since the next 50 points must be worth $70 in total, their average value must be $1.40. Moreover, if any of the incremental percentage points beyond 50% are also valued below $1.00, achieving that $1.40 average requires that valuation of later increments exceed $1.40. In other words, continuity demands that these partial hedges must eventually be worth much more than their expected value – and this point will typically come well before one has acquired a 100% chance of winning.
Note further that once a prize becomes more likely than not, additional increases in the chances of winning reduce variance at an accelerating rate. Since aversion to variance dominates every formal definition of risk aversion, those who are risk averse should often treat partial hedges like perfect hedges – as variance reducing assets that are worth even more than their expected value. Thus, assets that eliminate risk, like Tails vouchers, are an illustrative case, but not a special case.
The construct of risk aversion seemingly draws support from the popularity of insurance contracts, on which U.S. customers alone spend over a trillion dollars a year.Footnote 12 However, while people typically insure against their house catching fire, they rarely insure against a decline in house prices, though home equity comprises most of a household’s net worth at retirement.Footnote 13 Moreover, analogous contracts that extract a premium to reduce the variability of uncertain gains are also rare.Footnote 14 Thus, as Reference Friedman, Isaac, James and SunderFriedman et al. (2014) point out, while insurance contracts are typically invoked as evidence of risk aversion, customer behavior actually departs from textbook risk aversion; customers appear motivated to reduce the possibility of some types of harm, rather than reduce variance, per se. This distinction between downside risk and upside risk raises further questions about how subjects in our experiments interpret an actuarially unfair hedge: as an attractive premium to limit losses from unfavorable realizations of their risky asset (e.g., spending $700,000 on insurance to guard against the 50% chance their million dollar house will burn down) or as an unattractive censoring of the upside of their risky asset (e.g., as guaranteeing a mere $300,000 when they know their house will be worth a million if it does not burn down).
When we’ve presented this research, the most common objection is that respondents are just confused. We “concede” that, at some level, they are. For instance, consider the common response of someone who indicates they’d pay up to $3 for the bet and up to $3 for a hedge (if they had purchased or were endowed with the bet). This, in turn implies they’d pay up to $6 for both, but not, say, $7.50. But since the bet and hedge are worth $10 in combination, their responses imply they would decline receiving $2.50 if you attempted to hand it to them. This is obviously false and so their answer is, in an important sense, a mistake. And this mistake persists even when participants are explicitly given the normative explanation, as in Study 3. However, we see these mistakes as the phenomenon of interest. We don’t doubt that after a sufficiently intense and prolonged training session respondents could generate a normative pair of valuations, much as they could be taught to use Venn Diagrams or apply Bayes' Rule, or produce normative responses in many other contexts. But this doesn’t vitiate the phenomenon nor remove the challenges it poses to conceptions of risk attitudes.
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