Decision Are Preferences for Allocating Harm Rational? Alexander L

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Decision Are Preferences for Allocating Harm Rational? Alexander L. Davis, John H. Miller, and Sudeep Bhatia Online First Publication, April 3, 2017. http://dx.doi.org/10.1037/dec0000076

CITATION Davis, A. L., Miller, J. H., & Bhatia, S. (2017, April 3). Are Preferences for Allocating Harm Rational?. Decision. Advance online publication. http://dx.doi.org/10.1037/dec0000076 Decision © 2017 American Psychological Association 2017, Vol. 1, No. 999, 000 2325-9965/17/$12.00 http://dx.doi.org/10.1037/dec0000076 Are Preferences for Allocating Harm Rational?

Alexander L. Davis, John H. Miller, and Sudeep Bhatia Carnegie Mellon University

The allocation of nonmonetary harm is an important—yet understudied—domain of choice. Using a modiﬁed Dictator Game, we asked 27 participants to allocate a harmful event (time of putting their hand in ice water) between themselves and an anonymous stranger. We found substantially less coherent, and more egalitarian, preferences compared to other studies that ask participants to allocate monetary endowments. Speciﬁcally, 26% of participants made choices inconsistent with utility maximization, and 78% of participants behaved in an egalitarian manner. In comparable studies of monetary gains, only 2% were inconsistent and 30% egalitarian. The results suggest that the focus on monetary gains likely overestimates the rationality of other-regarding preferences and underestimates egalitarianism.

Keywords: altruism, prosocial behavior, rationality, revealed preference

Research in the last two decades has illumi- an individual going off to war, to the more nated our understanding of costly, other- mundane, like taking out the trash. regarding behavior (Andreoni & Vesterlund, We emphasize nonmonetary bads for a vari- 2001; Benabou & Tirole, 2006; Fehr & ety of reasons. First, as discussed above, it is a Schmidt, 2006; Fisman, Kariv, & Markovits, domain where decisions are frequently made, 2007; Forsythe, Horowitz, Savin, & Sefton, including examples that can be quite common 1994; Frey & Bohnet, 1995). Most of this work (like chores), to rather astounding, such as when has used experiments like the Dictator Game an individual risks his life to save the lives of to show that, in general, individuals are willing strangers (see, for just one example, Dovidio, to share a monetary windfall with an anony- Piliavin, Schroeder, and Penner’s (2006) dis- mous stranger in a way that is consistent with cussion of Paul Rusesabagina who helped Tut- some set of well-behaved (utility-maximizing) sis and Hutus escape the Rwandan genocide by preferences (Andreoni & Miller, 2002). This letting them stay in his hotel). Between these prior work focused almost exclusively on the extremes include the donation of blood and allocation of monetary gains. Here we examine organs, serving in the armed forces, participat- how individuals allocate nonmonetary bads, ex- ing in a revolution, or even becoming a suicide amples of which range from the heroic, such as bomber. Notwithstanding the importance and real-life consequences of nonmonetary bads, little systematic study has been devoted to understanding behavior in this domain. Second, there are rarely markets or prices for these bads, allowing us to test whether existing models of Alexander L. Davis, Department of Engineering and Pub- This document is copyrighted by the American Psychological Association or one of its allied publishers. lic Policy, Carnegie Mellon University; John H. Miller and other-regarding preferences generalize to a This article is intended solely for the personal use of the individualSudeep user and is not to be disseminated broadly. Bhatia, Department of Social and Decision Sci- novel context. Third, decisions about allocating ences, Carnegie Mellon University. harm likely invoke types of reasoning (e.g., Sudeep Bhatia is now at the Department of Psychology, moral-deontological) that may be inconsistent University of Pennsylvania. We would like to thank Terence Einhorn, John Sperger, with utility maximization. Finally, previous and April Jintao for their valuable assistance in helping work in psychology has found that people are conduct the research. willing to experience harm in place of another Correspondence concerning this article should be ad- person. For example, a study by Batson et al. dressed to Alexander L. Davis, Department of Engineering and Public Policy, Carnegie Mellon University, 5000 (1983) found that 65% of participants volun- Forbes Avenue, Pittsburgh, PA 15213. E-mail: alexander.l teered to take electric shocks in place of another [email protected] participant (who was actually an experimental

1 2 DAVIS, MILLER, AND BHATIA

confederate), and on average agreed to take to the pair. As a result, a participant with Le- more than 50% of the shocks (although none of ontief preferences tries to equalize the outcomes the shocks were actually implemented). In com- for herself and the other participant, behaving in parison with the typically more modest rates of an egalitarian manner. Finally, the utility func- generosity observed in Dictator Games using tion for a dictator with Perfect Substitute pref- ϭ ϩ monetary gains, these proportions raise the pos- erences is U(xs, xo) xs xo, meaning the sibility of greater generosity for allocations of outcomes for herself and the other person are nonmonetary harm. However, differences in ex- treated interchangeably. A participant with Per- perimental design, including the use of decep- fect Substitutes preferences will maximize the tion, lack of anonymity, and available informa- total amount of the good obtained by the pair. tion about the victim, limit comparability across these studies. In this work we provide a more direct comparison using a Dictator Game with Method allocations of nonmonetary harm. Our research strategy uses an approach cre- Participants ated by Andreoni and Miller (2002), where participants are asked to make a series of choices In advance we aimed to collect 25 pairs of that involve allocating a resource between dictators and receivers, a number we considered themselves and another person. In our modifi- large enough to detect interesting effects given cation, each choice involves a different tradeoff our budget constraints. Twenty-seven pairs of between harming another person and oneself. participants were ultimately recruited through Given these choices, we test whether the partic- the Center for Behavioral Decision Research ipant’s behavior is consistent with the General- website. Sessions were run in groups of four to ized Axiom of Revealed Preference (GARP; Af- eight participants (two to four pairs). The average age of the dictators was 24 years (with a riat, 1967; Varian, 1982). GARP captures the 1 intuition that inconsistency in observed choices range from 18–59), 14 were women. can reveal irrational decision making. For example, if a decision-maker chooses a bundle of Procedure goods A when an alternative bundle B could Participants were randomly assigned an iden- have been chosen at the same price, the choice tification number unknowable to others using reveals that A is preferred to B (or indifferent to shuffled opaque sealed envelopes. After random B). If we then observe the same decision-maker assignment, participants were told that they choosing bundle B when A is cheaper than B,an would be paid $10 in cash at the end of the inconsistency would arise. Remarkably, a for- experiment. They were informed that they malization of this simple idea captures all of the would be split into two rooms (based on observable consequences of the standard utility whether they had an even or odd identification theory used by economics. number), and that each odd-numbered partici- In addition to examining the consistency of pant (dictator) would make a decision that choices, we also explore the diversity of pref- would affect an even-numbered participant (re- erences that arise in this context. Specifically, ceiver), but that even-numbered participants Andreoni and Miller (2002) found that partici- would not make any choices that affected the pants’ preferences are well captured by three 2 This document is copyrighted by the American Psychological Association or one of its allied publishers. odd-numbered participants. All participants types of functions: Selfish, Leontief, and Perfect This article is intended solely for the personal use of the individual user and is not to be disseminated broadly. were assured that their choices throughout the Substitutes. If we call the dictator’s utility func- experiment would be anonymous. Before mak- tion U(x , x ), where x is the amount of good s o s ing their choices, participants came to the “Ice allocated to the dictator, and xo is the amount of good allocated to the other person, then a Self- 1 ish participant has the simple function U(xs, All de-identified data, R code for analyses, and materi- ϭ xo) xs, which is only dependent on the payoff als associated with the study are available through the Open to herself. The utility function for a dictator Science Framework at http://osf.io/82jpk. ϭ 2 We note that even though the receiver was anonymous, with Leontief preferences is U(xs, xo) min(xs, there was exactly one receiver, implying the existence of an xo); that is, the utility experienced is only as identifiable victim effect. However, such an effect is also large as the minimum amount of good allocated present in other Dictator Game studies. ALLOCATING HARM 3

Table 1 Parameters for the Eight Choices Made by Each Dictator, and Choice Patterns of Time Not in Water

(xs,xo) for Each of the Preference Types ´ Decision Tokens (M) Hold (ms) Pass (mo) M p Selﬁsh Leontief PerfectSub 1 60 1 1 60 1 (60, 0) (30, 30) (0–60, 0–60) 2 60 1.5 1 60 1.5 (60, 0) (24, 24) (60, 0) 3 60 2 1 60 2 (60, 0) (20, 20) (60, 0) 4 60 3 1 60 3 (60, 0) (15, 15) (60, 0) 5 60 1 1.5 40 .67 (40, 0) (24, 24) (0, 60) 6 60 1 2 30 .50 (30, 0) (20, 20) (0, 30) 7 60 1 3 20 .33 (20, 0) (15, 15) (0, 20) 8 80 1 1 40 1 (40, 0) (20, 20) (0–60, 0–60)

Water Experience” section of the questionnaire, Each dictator’s allocation of tokens was trans- where they were asked to raise their hand to formed into a period of time she and the re- notify the experimenter. The experimenter ceiver would have to submerge their hands in then walked to the participant with an insu- ice water if that particular decision was the one lated container of ice-water maintained at 3–5 randomly chosen to be executed at the end of °C, and had the participant submerge her the experiment. For each of the eight choices, hand in the water for 5 s timed with a digital each token was assigned a unique combination stopwatch that was observable by the partic- of hold and pass values. The hold value was the ipant. amount of time (in seconds) that the dictator Dictators then made eight allocation deci- 3 would have to keep her hand in ice water for sions for their single paired receiver. All par- each token so allocated, while the pass value ticipants then completed an auction to deter- was the amount of time (in seconds) that the mine how much they valued avoiding the receiver would have to keep his hand in ice ice-water experience. When all participants water for each token allocated to him. The eight were finished, a random number generator decisions are summarized in Table 1. For ex- picked one of the eight decisions to be implemented. Once this procedure was finished, the ample, one choice asked dictators to complete receivers were called into the payment room, the following task: one at a time, and received their payoffs, includ- Divide 60 tokens: Hold _____ tokens at 1 ing submerging their hand in ice-water. The second(s) each, and Pass _____ tokens participant’s auction decision was then checked at 1 second(s) each. to see if she had won the auction, and if so, her hand was submerged in ice water for 60 s and The choices were made via a computer inter- she was paid accordingly. Finally, they were face driven by the Qualtrics system (see Figure paid their $10 show-up fee. This process was 1 for a sample screen). Dictators were required repeated for dictators, who put their hand in the to press a “validate” button before submitting ice-water for the duration of the time specified their choices. Validation calculated the actual This document is copyrighted by the American Psychological Association or one of its allied publishers. by their randomly selected choice and then re- submersion times implied by the choices and This article is intended solely for the personal use of the individualceived user and is not to be disseminated broadly. the outcome of the auction. Additional details on the experimental procedure are pre- displayed these to the dictator to help her un- sented in the appendix. derstand the full consequences of her decision. Validation also verified that the choices met Choice Task three constraints that were known by the dicta-

The main task was a series of eight decisions, randomly ordered across subjects. Each deci- 3 Receivers also made predictions about the average behavior of dictators, but we do not report predictions sion required the dictator to make an allocation here to keep our discussion focused on the behavior of of a budget of either 60 or 80 tokens between the dictators. Data are available from the Open Science herself and her assigned, anonymous receiver. Framework. 4 DAVIS, MILLER, AND BHATIA

Figure 1. Sample screen of the eight choices presented to the dictators (each dictator received a randomly ordered table), with a constraint violation in the last choice. See the online article for the color version of this ﬁgure.

tors in advance: (a) that the total tokens allo- objects, that is, objects for which utility is cated in any given choice exactly equaled the monotonically increasing. Thus, we transform total number of tokens available in that choice,4 the outcome variable in the experiment—time (b) that allocations were nonnegative, integer in water—so that it represents a positive good— amounts, and (c) that the allocations would not time not in water. Let xs represent time for self have any participant receiving more than 60 s of not in water and xo time for other not in water. ϭ Ϫ ϭ Ϫ immersion time in the ice water, a constraint Therefore xs 60 ys and xo 60 yo imposed to avoid permanent harm. Participants because the maximum amount of time in ice were automatically notiﬁed with error messages water for either person is bounded above at 60 s. m and colored textboxes if they violated any of Furthermore, p ϭ s is the normalized price of these constraints. If all of the allocations over mo ´ the eight decisions were valid, a button became time not in water (price of giving) and M ϭ m enabled that allowed the dictators to submit 60 ϩ 60 ϫ s Ϫ m ϫ M is the normalized mo s their decisions, though they could still alter (and income, or how much time not in water the revalidate) their choices at that time before sub- dictator can allocate to herself. Figure 2 shows mission. all of the transformed budget sets across the eight choices used in the experiment. The Underlying Choice Design eight choices were chosen so that the multiple budget-set overlaps would provide many op- Each dictator’s choice task was to divide M tokens between self and other, with multipli- portunities for potential GARP violations. More information about the transformation, ers ms (hold value) and mo (pass value). The multipliers determined the ﬁnal amount of transformed budget set, and a discussion of time that the subject would have to endure the WARP/GARP violations are provided in the unpleasant task, such that if K tokens were appendix. kept, the dictator would immerse her hand in This document is copyrighted by the American Psychological Association or one of its allied publishers. ice water for y ϭ K ϫ m seconds and the 4

This article is intended solely for the personal use of the individual user and is not to be disseminated broadly. s s GARP allows choices on full budget set versus only the ϭ Ϫ ϫ receiver (other) for yo (M K) mo budget constraint. To allow this here, dictators would need seconds. For example, suppose the dictator to be given extra tokens (implying additional immersion was asked to divide 60 tokens (M ϭ 60), time) that they could allocate to either themselves or the receiver. Given the extra complication of including this where each token was worth2stooneself option, and our sense that it would not be exercised given ϭ ϭ (ms 2)and1stotheother person (mo 1). choices observed in related studies (see Fisman et al., 2007 If the dictator keeps 20 tokens (K ϭ 20), then and Andreoni & Miller, 2002), we constrained choices as the dictator gets 40 s (ϭ 20 ϫ 2) and the per the text. All of the dictators gave positive valuations for ϭ ϫ their Willingness to Accept additional harm based on the receiver gets 40 s ( 40 1). auction, and this is consistent with the notion that the option Standard GARP analysis and the estimation for immersion time beyond the minimum required would of utility functions requires “positive” choice not have been utilized. ALLOCATING HARM 5

Figure 2. Budget sets of the eight choices transformed to the space of time in not water for

self (xs) and other (xo).

Auction Choice Behavior

At the end of the experiment, participants First we consider the choices made when the completed a second-price auction in which the prices of giving time not in water were both lowest bidder in a session would receive a pay- equal to one, as in the standard Dictator Game. ment (determined by the amount bid by the In our experiment, dictators had to make two second-lowest bidder to encourage truthful rev- such allocations, one with 60 and one with 80 elation) to immerse her hand in ice water for tokens. With 60 tokens, the dictators keep a 60 s. We use the bids in this auction to deter- mean of 24.4s of putting their hand in ice water, mine each dictator’s minimum Willingness to thus, taking on 41% of the bad (time in water; Accept (WTA) for the ice-water-immersion ex- 95% conﬁdence interval, CI [32% to 49%]),5 or perience, and to control for possible heteroge- equivalently, 59% of the good (time not in neity in disutility from this experience. water; 95% CI [51% to 68%]). With 80 tokens, they keep a mean of 34.1 s or 43% of the bad

This document is copyrighted by the American Psychological Association or one of its allied publishers. Results (95% CI [38% to 47%]; given the constraint

This article is intended solely for the personal use of the individual user and is not to be disseminated broadly. preventing allocations of more than 60 s, at the Our results focus on three main questions: (a) what choices do dictators make in the domain of very least the dictator had to keep 25%). Be- cause the good is deﬁned in terms of time rel- harm, (b) are these choices rationalizable (con- ´ ϭ sistent with GARP), and (c) what do the implied ative to 40 s of time not in water (i.e., M 40), preferences look like? To begin the analysis, we this implies that the dictators took a mean of provide a brief summary of the choices made by 65% of the good in this latter case (95% CI dictators, and from there we explore the question of rationalizability. Finally, we take a more 5 Because the distributions of sharing were skewed, 95% in-depth look at the types of preferences that we conﬁdence intervals are calculated using a simple nonpara- observe across the dictators. metric bootstrap. 6 DAVIS, MILLER, AND BHATIA

[57% to 73%]).6 The generosity observed here sulted from the maximization of utility on a set is greater than observed in monetary good of well-behaved preferences (Afriat, 1967; Var- experiments, for example, in Andreoni and ian, 1982). GARP violations occur when a sub- Miller (2002), participants keep around 76% ject, directly or indirectly, alters her preferences and 83% of the good (money) under similar between bundles simultaneously available un- conditions. For further comparison, in regular der different budget sets. To benchmark the Dictator Games, Forsythe et al. (1994) find likelihood of GARP violations, we generate that participants keep an average of around random choices on the budget sets we used in 77–78% of the monetary endowment and our experiment and check for GARP (Bronars, Camerer (2003; chapter 2) reports an average 1987). Using this procedure, we find that on of 80%. Participants in our study kept about average 93% (N ϭ 1,000) of random choices 60% of the good, a result that is substantially result in at least one GARP violation. In our different from what is observed with mone- experiment, 48% (13/27) of the dictators had at tary gains. least one GARP violation. Of the 27 dictators, 16 of them allocated the Afriat (1972) developed the Critical Cost Ef- identical percentage of the bad under both of the ficiency Index (CCEI) as a measure of the de- above token levels (with 15 of them keeping 50%, gree of GARP violation. The CCEI for any and one keeping 25% of the time in water). Two choice is the proportionate reduction in spend- of the dictators (7%) took on more of the bad than ing needed to purchase a revealed-preferred they gave in at least one of these two choices, bundle, that is, how much less could the subject behavior that is almost never observed in the case spend to receive a bundle that is at least as good of the standard Dictator Game. Although 22 as what she bought. Thus, if the CCEI is 1.0, no (81%) of the dictators made at least one choice reduction is possible, but as the CCEI falls, the where they gave the receiver more harm than they GARP violation becomes more severe. Follow- kept for themselves, only 7 (26%) dictators did ing Varian (1994), we associate an individual’s this across all of their choices. We find that in 9% lowest CCEI as a measure of the degree of of all the total possible choices the dictators keep GARP violation. Slightly over half of our sub- more of the bad than they give, in 47% of the jects showed no violations of GARP. A CCEI of choices they give more of the bad than they keep, 0.95 has been used in the literature to mark and in the remaining 44% of choices the bad was significant GARP violations. Based on this equally distributed. Again, these results support threshold, we find that 26% (7/27) of our sub- the idea that participants were much more gener- jects had behavior that cannot be rationalized as ous in the context of nonmonetary harm than arising from utility-maximization on well- monetary gain. behaved preferences. By contrast, Andreoni and All of the dictators had a positive WTA for Miller (2002) explored monetary gains and 60 additional seconds in water, as determined found that less than 2% (3/176) of their partic- by the second-price auction, indicating they dis- ipants had such violations. Additional compar- liked the experience. The lowest WTA was isons with prior work are provided in the ap- $0.50 and the highest, an extreme outlier, was pendix. $1,000,000 (the second highest was $10). Re- moving the outlier, the mean WTA was $3.23 Preferences and the median was $2.75. WTA was uncorre- This document is copyrighted by the American Psychological Association or one of its allied publishers. In the prior work of Andreoni and Miller lated (Pearson’s r ϭϪ.03) with the mean of the This article is intended solely for the personal use of the individual user and is not to be disseminated broadly. (2002), a large amount of heterogeneity was ob- income-normalized amount of time not in water x served among the apparent preference types of the ͑ s ͒ kept by the dictator, suggesting that differ- ´M subjects. One simple way to classify preferences is ences in altruism across dictators cannot be to consider three prototypical forms of behavior: explained easily by their sensitivity to, or dis- (a) Leontief preferences, where the dictator seeks pleasure from, the ice-water task. to allocate the bad so that each person gets an equal amount, (b) Selfish preferences, where the Rationalizability

If choices satisfy GARP, then they are ratio- 6 For example, if the dictator kept 20 tokens and gave 60 ´ ϭ ϭ 60 Ϫ 20 nalizable in the sense that they could have re- with M 40, then she took all of the good (1 40 ). ALLOCATING HARM 7

Figure 3. Top panel: Euclidian distance of each dictator’s choices to prototypical pure Leontief and Selfish choices. Bottom panel: Dictators’ decisions to keep time not in water as a function of the price of giving time not in water for decisions 1–7 (with 60 tokens). The dashed red and blue lines indicate the behavior predicted by purely Selfish or Leontief preferences (respectively). See the online article for the color version of this figure.

dictator gives the other subject the maximum The distribution of apparent preference types amount of time in water possible, and (c) Perfect is quite different from what has been previously Substitutes where the dictator attempts to mini- observed in monetary-gain experiments. Here, mize the aggregate immersion time regardless of 22% of the dictators are Selfish, whereas under which person is subjected to the harm. Using these monetary-gains in Andreoni and Miller (2002) three prototypes, we can classify dictators as either about half were selfish; 78% are Leontief, ver- Strong, that is, the observed behavior is fully con- sus only 30% under monetary-gains; and, fi- sistent with the particular preference prototype, or nally, we observed no one behaving as if they Weak, that is, the observed behavior is best cap- had Perfect Substitutes, whereas 22% of sub- tured by the prototype in terms of mean Euclidean jects followed this prototype under monetary- distance (see Figure 3). Using this metric, Table 2 8 gains. In the realm of nonmonetary bads, a classifies the observed behavior of the 27 sub- 7 This document is copyrighted by the American Psychological Association or one of its allied publishers. jects. This article is intended solely for the personal use of the individual user and is not to be disseminated broadly. 7 If we admit the possibility of pure Leontief preferences, but with distribution ratios (for time not in water) other than 1/1 (self/other), our overall classification changes slightly. In Table 2 this case, counting both weak and strong types, we find 5 Distribution of the 27 Dictators by Closest Match Selfish, 13 Leontief at 1/1, 6 Leontief at 3/2, and 3 Leontief at to Prototypical Preference Forms 3/1. 8 Fisman et al. (2007) looked at only strong types, and found Leontief Selfish Perfect substitutes 20% of their participants being Selfish, 2.6% Leontief, and Strong 4 (15%) 1 (4%) 0 (0%) 2.6% having Perfect Substitutes. Andreoni and Vesterlund Weak 17 (63%) 5 (19%) 0 (0%) (2001) find strong types at 21.2% Selfish, 16.2% Leontief, and 0.07% Perfect Substitutes, and the total distribution of Total 21 (78%) 6 (22%) 0 (0%) types at 44%, 35%, and 21%, respectively. 8 DAVIS, MILLER, AND BHATIA

large majority (78%) of dictators make choices effect is still of interest given that individuals consistent with sharing the harm equally.9 must often make such choices. Our results present an interesting theoretical Discussion challenge. At the aggregate level, we know that some individuals must be harboring very differ- Individuals often make choices about how to ent sets of preferences over allocating monetary allocate bad events, sometimes in the mundane gains versus nonmonetary harm. It is not clear activities of daily household life, and at other that the existing literature on prosocial prefer- times over more substantial events like serving ences (Fehr & Schmidt, 2006) can easily ac- in the armed forces or rescuing someone in commodate such observations in a unified danger. Notwithstanding both the prevalence framework, because we would need to find a and importance of such decisions, they have single, metapreference function that displays received a surprising lack of attention in the such switching in these two contexts. More- experimental literature. Here, we focus on non- over, we observed a breakdown of well- monetary choices over bad events. We find behaved preference maximization in over a that subjects are much more generous in such quarter of our subjects in the context of non- choices—about three-quarters of them decid- monetary harm, and thus, even if we can solve ing to share the harm about equally—and that the first challenge, we may still have diffi- the behavior of about a quarter of the subjects culty explaining the behavior of one out of cannot be reconciled as being driven by the max- four subjects. imization of a well-behaved utility function— It may be the case that the imposition of harm versus the almost universal ability to do so in the on another causes a breakdown of our usual context of monetary gains. models, at least in some participants. There is A few psychologists have performed experi- some evidence that people prefer indirectly over ments on the allocation of harm in very different directly harming others (Royzman & Baron, domains to better understand the relationship 2002). That being said, even a simple rule like between empathy and altruism (Batson, 1987; “do no harm,” while perhaps being able to ex- Batson et al., 1988; Schaller & Cialdini, plain the asymmetry of preferences in the mon- 1988), and have also found that individuals etary-gain versus nonmonetary-loss contexts, tend to share the harm. These experiments would still lead to well-behaved preference were not designed to tease out preference maximization within each context. There is also types or violations of rationality, and relied evidence that moral motives, like justice and on very different methodologies (including harm, hold across cultures (Haidt & Graham, the use of deception), though the coincidence 2007), so perhaps the interplay of moral factors of results across radically different methodol- might provide some insight here. Alternatively, ogies suggests that such behavior may be perhaps monetary-gains versus nonmonetary- relatively robust. losses imply very different cognitive pro- In our experiment, we find preferences that cesses, resulting in both asymmetries of pref- are quite different than previous results using erences and the potential breakdown of the standard Dictator Game with monetary optimized choice. For example, moral judg- gains. Of course, our context differs from this ments tend to be the consequence of emo- prior work on two dimensions: we look at non- This document is copyrighted by the American Psychological Association or one of its allied publishers. tional responses, especially when directly monetary versus monetary and at losses versus This article is intended solely for the personal use of the individual user and is not to be disseminated broadly. gains. There is some evidence (Vohs, Mead, & Goode, 2006) that other-regarding preferences 9 The preferences across allocating harm to self and other in the context of money tend to be more selfish. are tied to preferences for harm to self. While we have data on each subject’s Willingness to Accept for 60 s of cold- The evidence about losses versus gains is less water immersion, we do not have enough additional infor- clear. Given these observations, we cannot eas- mation to derive an individual’s demand curve for the ily disentangle whether the behavior we are cold-water task. In general, subjects tend not to recognize observing here is tied purely to the nonmonetary how much they will adapt to harm (Ubel, Loewenstein, & Jepson, 2005), so we suspect that subjects will not discount nature of the choice or the imposition of harm additional time as much as they should. Even if they dis- on another. While teasing out these two effects counted such time radically, the equal sharing of time (vs. is an interesting question, exploring the joint either taking it or giving it all) is unexpected. ALLOCATING HARM 9

harming others is involved (Greene & Haidt, Andreoni, J., & Vesterlund, L. (2001). Which is the 2002), and there are examples of decreased fair sex? Gender differences in altruism. The Quar- rationality when high-intensity emotions are terly Journal of Economics, 116, 293–312. involved in decision making (Loewenstein & Bardsley, N. (2008). Dictator game giving: Altruism Lerner, 2003). or artefact? Experimental Economics, 11, 122– An alternative explanation involves demand 133. Batson, C. (1987). Prosocial motivation: Is it ever characteristics, where the researcher unknow- truly altruistic? In L. Berkowitz (Eds.), Advances ingly provides subtle cues about how partici- in experimental social psychology (pp. 65–122). pants are expected to respond (Bardsley, 2008; Cambridge, MA: Academic Press. Levitt & List, 2007; List, 2007; Orne, 1962; Batson, C. D., Dyck, J. L., Brandt, J. R., Batson, Zizzo, 2010). For example, Bardsley (2008) J. G., Powell, A. L., McMaster, M. R., & Griffitt, found that dictators are less likely to give to the C. (1988). Five studies testing two new egoistic receiver when taking from the receiver is pos- alternatives to the empathy-altruism hypothesis. sible. He argues that when one can only give to Journal of Personality and Social Psychology, 55, others, dictators see their task as being about 52–77. giving, but when one can take from the receiver, Batson, C., O’Quin, K., Fultz, J., Vanderplas, M., & the expected behavior (demand characteristic) Isen, A. (1983). Influence of self-reported distress is less clear and thus, giving is reduced. Indeed, and empathy on egoistic versus altruistic motiva- any participant in our experiment could have tion to help. Journal of Personality and Social Psychology, 45, chosen to leave the experiment with pay and 706–718. Benabou, R., & Tirole, J. (2006). Incentives and receive none of the pain from placing her hand prosocial behavior. The American Economic Re- in ice water or, assuming the receiver also with- view, 96, 1652–1678. drew, guilt from forcing another person to do Bronars, S. G. (1987). The power of nonparametric so, but no one did. Their right to do this was tests of preference maximization. Econometrica, explained in the informed consent document at 55, 693–698. the beginning of the experiment. Thus, even the Camerer, C. F. (2003). Behavioral game theory: Ex- mere participation in our experiment provides a periments in strategic interaction. Princeton, NJ: puzzle not easily resolved by standard social- Princeton University Press. preference theory. Dobell, A. R. (1965). A comment on AYC Koo’s Other-regarding behavior, across a wide va- “An Empirical Test of Revealed Preference The- riety of contexts, is an important element of our ory.” Econometrica, 33, 451–455. world. Prior work has had a very useful, though Dovidio, J. F., Piliavin, J. A., Schroeder, D. A., & near-exclusive, focus on one aspect of this do- Penner, L. (2006). The social psychology of prosocial behavior. Mahwah, NJ: Erlbaum Publishers. main, namely monetary gains. Here we ex- Fehr, E., & Schmidt, K. M. (2006). The economics of tend the exploration of other-regarding be- fairness, reciprocity and altruism: Experimental havior into an analysis of the allocation of evidence and new theories. In S. Kolm & J. Ythier harmful, nonmonetary events. We find that (Eds.), Handbook on the economics of giving, rec- individuals behave very differently in this iprocity and altruism (pp. 615–691). Amsterdam, new domain, both in terms of the preferences the Netherlands: Elsevier. that they reveal and their ability to behave in Fisman, R., Kariv, S., & Markovits, D. (2007). Indi- a rationalizable way. vidual preferences for giving. The American Eco- nomic Review, 97, 1858–1876. This document is copyrighted by the American Psychological Association or one of its allied publishers. Forsythe, R., Horowitz, J. L., Savin, N. E., & Sefton, M.

This article is intended solely for the personal use of the individual user and is not to be disseminated broadly. References (1994). Fairness in simple bargaining experiments. Afriat, S. (1967). The construction of utility functions Games and Economic Behavior, 6, 347–369. from expenditure data. International Economic Frey, B. S., & Bohnet, I. (1995). Institutions affect Review, 8, 67–77. fairness: Experimental investigations. Journal of Afriat, S. (1972). Efﬁciency estimates of production Institutional and Theoretical Economics, 151, functions. International Economic Review, 13, 286–303. 568–598. Greene, J., & Haidt, J. (2002). How (and where) does Andreoni, J., & Miller, J. (2002). Giving according to moral judgment work? Trends in Cognitive Sci- GARP: An experimental test of the consistency of ences, 6, 517–523. preferences for altruism. Econometrica, 70, 737– Haidt, J., & Graham, J. (2007). When morality op- 753. poses justice: Conservatives have moral intuitions 10 DAVIS, MILLER, AND BHATIA

that liberals may not recognize. Social Justice Re- Schaller, M., & Cialdini, R. B. (1988). The econom- search, 20, 98–116. ics of empathic helping: Support for a mood man- Levitt, S. D., & List, J. A. (2007). What do laboratory agement motive. Journal of Experimental Social experiments measuring social preferences reveal Psychology, 24, 163–181. about the real world? The Journal of Economic Ubel, P. A., Loewenstein, G., & Jepson, C. (2005). Perspectives, 21, 153–174. Disability and sunshine: Can hedonic predictions List, J. A. (2007). On the interpretation of giving in be improved by drawing attention to focusing il- dictator games. Journal of Political Economy, 115, lusions or emotional adaptation? Journal of Exper- 482–493. imental Psychology: Applied, 11, 111–123. Loewenstein, G., & Lerner, J. (2003). The role of Varian, H. R. (1982). The nonparametric approach to demand analysis. Econometrica, 50, 945–973. affect in decision making. In R. Davidson, K. Varian, H. R. (1994). Goodness-of-ﬁt for revealed Scherer, & H. Goldsmith (Eds.), Handbook of af- preference tests. University of Michigan CREST fective science (pp. 619–642). Oxford, United Working Paper 13. Kingdom: Oxford University Press. Vohs, K. D., Mead, N. L., & Goode, M. R. (2006). Orne, M. (1962). On the social psychology of the The psychological consequences of money. Sci- psychological experiment: With particular refer- ence, 314, 1154–1156. ence to demand characteristics and their implica- Warshall, S. (1962). A theorem on Boolean matrices. tions. American Psychologist, 17, 776–783. Journal of the ACM, 9, 11–12. Royzman, E. B., & Baron, J. (2002). The preference Zizzo, D. J. (2010). Experimenter demand effects in for indirect harm. Social Justice Research, 15, economic experiments. Experimental Economics, 165–184. 13, 75–98.

Appendix A

Budget Set Transformation

Let xs represent time for self not in water and number of tokens (weighted by the hold and ϭ xo time for other not in water. Therefore xs pass values), and that the time not in water Ϫ ϭ Ϫ 60 ys and xo 60 yo because the maximum allocated to oneself and the other person are amount of time in ice water for either person is both greater than or equal to zero (i.e., nobody bounded above at 60 s. Thus, the point (xs, can experience more than 60 s of ice water). ϭ xo) (0, 0) represents 60 s of time in water for Mathematically, a budget set is defined in this ϭ self and other, and (xs, xo) (60, 60) represents context as: 0 s of time in water for self and other. The set of ϭ ͕ ϩ ϫ ϭ ´ feasible choices (budget set), are defined by the B {xs, xo} |{xs xo p M} number of tokens and the hold and pass values. പ Ն Ն ͖ Specifically, the budget set includes the possible {xs, xo} | xs 0&xo 0 (1) pairs of time for self not in water (x ) and time s m for other not in water (x ) such that the total where p ϭ s is the normalized price of time not o mo amount of time in water allocated matches the in water and This document is copyrighted by the American Psychological Association or one of its allied publishers. This article is intended solely for the personal use of the individual user and is not to be disseminated broadly.

(Appendices continue) ALLOCATING HARM 11

Figure A1. A sample budget set in terms of time in water for self and other (upper), and its ϭ ϭ equivalent transformation for time not in water (lower). The budget set is M 60, ms 2, ϭ and mo 1, with the shaded area corresponding to the choices possible in the budget set.

m is not free to allocate all of the time not in water ´ ϭ ϩ ϫ s Ϫ ϫ ϭ ϭ M 60 60 ms M (2) to herself. For example, when m0 3 and ms 1, mo

is the normalized income.10 M´ can be inter- preted as the amount of time not in water that This document is copyrighted by the American Psychological Association or one of its allied publishers. the dictator has available to allocate to herself. 10 M´ is how much time not in water the dictator can This article is intended solely for the personal use of the individual user and is not to be disseminated broadly. ϭ ϭ ϭ allocate to herself. This comes from the constraint that, in For example, suppose ms mo 1 and M 60, then the dictator is free to allocate all 60 s of that situation, the maximum amount of time in water that can be allocated to the other person is 60 s. This gives y ϭ ´ 1 o time not in water (M ϭ 60 ϩ 60 ϫ Ϫ (M Ϫ K)m ϭ 60, and as a result, K ϭ M Ϫ 60. Because 1 o mo ϫ ϭ ϭ Ϫ 60 1 60) in any way desired. For the budget sets ys Kms,wegetys ͑M ͒ms, or the amount of bad ´ ϭ ϭ Ն mo we use, M 60 when M 60 and ms mo.On that the dictator can allocate to herself. The amount of good m the other hand, when m Ͼ m , then the dictator is then x ϭ 60 Ϫ y ϭ 60 ϩ 60 ϫ s Ϫ m ϫ M. o s s s mo s

(Appendices continue) 12 DAVIS, MILLER, AND BHATIA

then the dictator is only free to allocate 20 s of equivalent transformation of this budget set time not in water to herself (M´ ϭ 60 ϩ over positive goods. In the lower diagram, the 60 ϫ 1 Ϫ 1 ϫ 60). This is because each token normalized price of giving time not in water 3 m given to the other participant incurs3softime. to the other person, p, is equal to 2 (ϭ s ϭ mo 2 This constraint is necessary to ensure that no 1 ), and the shaded area corresponds to the participant is exposed to more than 60 s of time, transformed budget set over the two positive to avoid injury. goods, namely, time not in water for self and The upper diagram in Figure A1 represents other. Rotating the lower diagram around the the nontransformed space of possible choices intersection of the two dotted lines by 180 ϭ ϭ for a sample budget set M 60, ms 2, and degrees and using the dotted lines as axes ϭ mo 1, with the shaded area corresponding to with the transformed origin, shows how the the choices possible in the budget set. The nontransformed space is contained within the lower diagram in that ﬁgure represents the transformed one.

Appendix B

Checking WARP Formally, GARP relies on three notions of to x1. SARP and GARP both capture the famil- revealed preference (Varian, 1982). Consider iar concept of transitive preferences, but only allocations of time not in water to self and other GARP captures the necessary and sufﬁcient (xs, xo) with the price of xs normalized to 1 and conditions for choices to be consistent with ϭ ms utility maximization. the price of xo equal to p m . The total cost of ϭo ϩ purchasing this bundle is C xs pxo.Ifwe To check WARP, consider the budget set 1 1 1 ϭ ϩ Յ ´ observe a decision-maker choose x ϭ (xs, xo) B xs xop M, where xs is the amount of 2 ϭ 2 2 over x (xs, xo), then it must be the case that good (time not in water) kept for the dictator, xo ϭ 1 ϩ 1 Ն ϭ 2 ϩ 2 C1 xs pxo C2 xs pxo. When this is the amount of time not in water given to the 1 occurs we say that x is directly revealed pre- ϭ ms 2 receiver, p m is the ratio of hold and pass ferred to x , because the decision-maker is will- o ´ 1 values of the tokens, and M is the normalized ing to pay at least as much to get x as she is income. B can be written as the area beneath the willing to pay to get x2. If the inequality is strict, line x ϭ M´ Ϫ x p, with intercept M´ and slope such that C Ͼ C , then x1 is strictly directly s o 1 2 Ϫp. All points on this line or between it and (0, revealed preferred to x2. Finally, if x1 is directly 2 2 0) are within the budget set B. Thus, any point revealed preferred to x , and x is directly re- ϭ ´ Ϫ x3 x1 indirectly re- on the budget constraint xs M xop must be vealed preferred to , then is Ͻ ´ Ϫ vealed preferred to x3. The Weak Axiom of preferred to points xs M xop, as the latter Revealed Preference (WARP) states that if x1 is provide strictly less quantities of goods xs and xo 1 ϭ 1 1 directly revealed preferred to x2, then x2 is not than the former. Suppose the point x (xs, xo) 1 1 ϩ directly revealed preferred to x . The Strong lies on the budget constraint of B, such that xs This document is copyrighted by the American Psychological Association or one of its allied publishers. 1 ϭ ´ 2 ϭ 2 2 Axiom of Revealed Preference (SARP) states xop M. Then, the point x (xs, xo)isre- This article is intended solely for the personal use of the individual user and is not to be disseminated broadly. that if x1 is revealed preferred (directly or indi- vealed worse than x1 if it is within the budget set 2 2 2 2 rectly) to x , then x is not directly revealed B, such that xs ϩ xop Յ M´ . Substituting M´ ϭ 1 1 1 1 1 preferred to x . Finally, GARP states that if x is xs ϩ xop gives the condition that x is revealed 2 2 2 2 1 1 revealed preferred to x (directly or indirectly), preferred to x if xs ϩ xop Յ xs ϩ xop, unless the then x2 is not strictly directly revealed preferred two points are equal (x1 ϭ x2).

(Appendices continue) ALLOCATING HARM 13

ϭ To check WARP violations we follow Dobell Ao po xo 1 (1965). Recapitulating, x is directly revealed 1 1 1 2 1 3 1 4 1 5 1 6 1 7 1 8 preferred to good vector x2 if p1x1 Ն p1x2 and x1 poxo poxo poxo poxo poxo poxo poxo poxo 2 p2x1 p2x2 p2x3 p2x4 p2x5 p2x6 p2x7 p2x8 x . In our setup, each good vector consists of ϭ o o o o o o o o o o o o o o o o i ϭ i i ÉÉÉÉÉÉÉÉ the components xs and xo, such that x [xs, xo]. ΂ ΃ i i Similarly, the price vectors are p ϭ [1, po] 8 1 8 2 8 3 8 4 8 5 8 6 8 7 8 8 i poxo poxo poxo poxo poxo poxo poxo poxo i ϭ ms where po i . We begin by constructing the mo ϭ ϩ ϭ i j Then, define A As Ao: matrix A such that [aij] [p · q ], where (·) represents the inner product. Thus, the entry in 1 ϩ 1 1 2 ϩ 1 2 8 ϩ 1 8 the first row, first column is: xs poxo xs poxo ... xs poxo A ϭ É ÉÉÉ 1 1 1 1 1 ΂ ΃ ϭ ϭ ϩ a11 p · q xs p xo (3) 1 ϩ 8 1 2 ϩ 8 2 8 ϩ 8 8 xs poxo xs poxo ... xs poxo The entire matrix A can be constructed in the ϭ Thus, A has the desired property that [aij] following way. Consider the vector of hold [pi · qj]. We can then construct the matrix B that ϭ prices ps 1 and vector of good kept for the has the property [b ] ϭ [pi · qj Ϫ pi · qi]. This can ϭ R R ij self xs. Define As ps xs, where is the outer be done by subtracting a vector d with elements product of vectors. Then we have: d ϭ [p1 · x1, p2 · x2,...,p8 · x8] from A, that is B ϭ A Ϫ d. The vector d is just the diagonal ϫ ϭ elements of A as an 8 1 vector. Finally, we As ps xs ϭ Յ create a matrix C whose elements cij 1ifbij 1 2 3 4 5 6 7 8 0, and 0 otherwise. The matrix C represents the xs xs xs xs xs xs xs xs revealed preference relationship between each of x1 x2 x3 x4 x5 x6 x7 x8 ϭ s s s s s s s s the eight choices, where every entry that is equal ΂ÉÉÉÉÉÉÉÉ΃ to 1 implies piqj Յ piqi meaning the row choice is x1 x2 x3 x4 x5 x6 x7 x8 directly revealed preferred to the column choice. s s s s s s s s The matrix C can also be described as an adjacency matrix. We then set to zero any cells where ϭ R i ϭ j ij ij ϭ ji ji Similarly, define Ao po xo: x x , that is [xs, xo] [xs, xo].

(Appendices continue) This document is copyrighted by the American Psychological Association or one of its allied publishers. This article is intended solely for the personal use of the individual user and is not to be disseminated broadly. 14 DAVIS, MILLER, AND BHATIA

Appendix C

Checking SARP Consider the direct revealed preference rela- • R1[1, 1] ϭ 1ifR0[1, 1] ϭ R0[1, 1] ϭ 1 tions A Ͼ B, B Ͼ C, and C Ͼ A. Clearly this • R2[1, 2] ϭ 1ifR1[1, 1] ϭ R1[1, 2] ϭ 1 violates SARP. How do we construct an algo- • R3[1, 3] ϭ 1ifR2[1, 1] ϭ R2[1, 3] ϭ 1 rithm to figure this out? Consider the adjacency The first says that A is preferred to A if A is 0 matrix R that represents only direct revealed preferred to A and A is preferred to A. The preference: second says that A is preferred to B if A is preferred to A and A is preferred to B. The third ABC says that A is preferred to C if A is preferred to A and A is preferred to C. These updates make A 010 R0 ϭ no change to the matrix, giving: ΂B 001΃ ABC C 100 A 010 R3 ϭ For the adjacency matrix, there isa1incell ΂B 001΃ i, j if the option in row i is preferred to the C 100 option in column j. For example, cell i ϭ 1, j ϭ 2 is 1 because A is directly revealed preferred to Next, for i ϭ 2, k ϭ 1: B. We then take R0 and make a sequence of 4 ϭ 3 ϭ 3 ϭ adjustments that add indirect revealed prefer- • R [2, 1] 1ifR [2, 1] R [1, 1] 1 5 ϭ 4 ϭ 4 ϭ ence relationships to this matrix. As we iterate • R [2, 2] 1ifR [2, 1] R [1, 2] 1 6 ϭ 5 ϭ 5 ϭ through the algorithm, all of the indirect re- • R [2, 3] 1ifR [2, 1] R [1, 3] 1 vealed preference relationships are also repre- The first says that B is preferred to A if B is sented in the matrix. This is done, one at a time, preferred to A and A is preferred to A. The by asking whether each option is indirectly re- second says that B is preferred to B if B is vealed preferred to every other option, through preferred to A and A is preferred to B. The third each other option as an intermediary. This final says that B is preferred to C if B is preferred to matrix Rn, where n ϭ i ϫ j ϫ k, is called the A and A is preferred to C. Again, none of these transitive closure of R. change the matrix. The algorithm works using the following re- cursive update (Warshall, 1962): ABC A 010 R6 ϭ Rn[i, j] ΂B 001΃ Ϫ Ϫ C 1, if Rn 1[i, k] ϭ Rn 1[k, j] ϭ 1 100 ϭ ͭ (4) nϪ1 R [i, j], otherwise Next, for i ϭ 3, k ϭ 1: Applying this to our matrix R0, we start with • R7[3, 1] ϭ 1ifR6[3, 1] ϭ R6[1, 1] ϭ 1 This document is copyrighted by the American Psychological Association or one of its allied publishers. k ϭ 1, i ϭ 1, and make the following adjust- 8 ϭ 7 ϭ 7 ϭ This article is intended solely for the personal use of the individual user and is not to be disseminated broadly. • R [3, 2] 1ifR [3, 1] R [1, 2] 1 ments by looping through j ϭ {1, 2, 3}: • R9[3, 3] ϭ 1ifR8[3, 1] ϭ R8[1, 3] ϭ 1

(Appendices continue) ALLOCATING HARM 15

The first says C is preferred to A if C is connected to A. The second is whether there is preferred to A and A is preferred to A. The a path from B to B and B to B. The third is second says C is preferred to B if C is preferred whether there is a path from B to B and B to C. to A and A is preferred to B. This case is true so None of these are relevant because B is not we change R8[3, 2] ϭ 1. The third says C is preferred to B. preferred to C if C is preferred to A and A is preferred to C. ABC A 011 ABC R15 ϭ ΂B 001΃ A 010 R9 ϭ C 110 ΂B 001΃ C 110 Then for i ϭ 3, k ϭ 2: • R16[3, 1] ϭ 1ifR15[3, 2] ϭ R15[2, 1] ϭ 1 ϭ Thus, with k 1, we have found all prefer- • R17[3, 2] ϭ 1ifR16[3, 2] ϭ R16[2, 2] ϭ 1 ence relationships between A, B, and C through • R18[3, 3] ϭ 1ifR17[3, 2] ϭ R17[2, 3] ϭ 1 A. With k ϭ 2 we repeat this but try to find all The first says C is preferred to A if C is preference relationships between A, B, and C preferred to B and B is preferred to A. The through B. For i ϭ 1, k ϭ 2: second says C is preferred to B if C is preferred 10 ϭ 9 ϭ 9 ϭ • R [1, 1] 1ifR [1, 2] R [2, 1] 1 to B and B is preferred to B. The third says that 11 ϭ 10 ϭ 10 ϭ • R [1, 2] 1ifR [1, 2] R [2, 2] 1 C is preferred to C if C is preferred to B and B 12 ϭ 11 ϭ 11 ϭ • R [1, 3] 1ifR [1, 2] R [2, 3] 1 is preferred to C, both of which are true so The first says A is preferred to A if A is R18[3, 3] ϭ 1. preferred to B and B to A. The second says A is preferred to B if A is preferred to B and B ABC is preferred to B. The third says A is preferred A 011 to C if A is preferred to B and B is preferred to R18 ϭ C. The third one is true, so R12[1, 3] will be ΂B 001΃ changed to 1. C 111

ABC Then for i ϭ 1, k ϭ 3: A 011 • R19[1, 1] ϭ 1ifR18[1, 3] ϭ R18[3, 1] ϭ 1 12 ϭ R • R20[1, 2] ϭ 1ifR19[1, 3] ϭ R19[3, 2] ϭ 1 ΂B 001΃ 21 20 20 • R [1, 3] ϭ 1ifR [1, 3] ϭ R [3, 3] ϭ 1 C 110 The ﬁrst says A is preferred to A if A is preferred to C and C is preferred to A, which is Next, for i ϭ 2, k ϭ 2: true, so R19[1, 1] ϭ 1. The second says A 13 ϭ 12 ϭ 12 ϭ • R [2, 1] 1ifR [2, 2] R [2, 1] 1 is preferred to B if A is preferred to C and C is 14 ϭ 13 ϭ 13 ϭ • R [2, 2] 1ifR [2, 2] R [2, 2] 1 preferred to B, but A is already preferred to B, so 15 ϭ 14 ϭ 14 ϭ • R [2, 3] 1ifR [2, 2] R [2, 3] 1 no change is made. The third says A is preferred

This document is copyrighted by the American Psychological Association or one of its allied publishers. The ﬁrst asks whether B is preferred to B and to C if A is preferred to C and C is preferred to C,

This article is intended solely for the personal use of the individualB user and is not is to be disseminated broadly. preferred to A is connected to B and B is which again is already true.

(Appendices continue) 16 DAVIS, MILLER, AND BHATIA

ABC updates are possible, and we ﬁnish the algorithm. A 111 R21 ϭ ΂B 001΃ ABC C 111 A 111 R24 ϭ ϭ ϭ ΂B 111΃ Finally, for i 2, k 3: • R22[2, 1] ϭ 1ifR21[2, 3] ϭ R21[3, 1] ϭ 1 C 111 • R23[2, 2] ϭ 1ifR22[2, 3] ϭ R22[3, 2] ϭ 1 The resulting matrix R24 is the transitive clo- • R24[2, 3] ϭ 1ifR23[2, 3] ϭ R23[3, 3] ϭ 1 sure of R0, which shows all revealed preference The ﬁrst says B is preferred to A if B is relationships (direct and indirect). We can use preferred to C and C is preferred to A, which this to calculate the SARP violations by looking is true so R22[2, 1] ϭ 1. The second says B is at any case where R24[i, j] ϭ R24[j, i] ϭ 1 for preferred to B if B is preferred to C and C is i j. This reveals that there are three SARP preferred to B, which is true, so R23[2, 2] ϭ 1. violations, of 1) A Ͼ B and B Ͼ A,2)A Ͼ C and Because the matrix is now all ones, no further C Ͼ A, and 3) B Ͼ C and C Ͼ A.

Appendix D

Checking GARP GARP states that if x is revealed preferred to matrix Rn, we evaluate whether it is true that y (directly or indirectly) then py Յ px.Wecan Rn[i, j] ϭ Rn[j, i] ϭ 1, which would indicate check GARP by ﬁnding the transitive closure that alternative i is (directly or indirectly) of x, then if xRy and py Ͼ px, GARP is preferred to j and alternative j is preferred to violated. GARP requires only one additional i. We then construct a matrix Gn by modifying step above SARP. After computing the tran- Rn to Gn[i, j] ϭ 0ifGn[i, j] ϭ Gn[j, i] ϭ 1 and sitive closure of each choice to produce the pipj ϭ pipi.

Appendix E

Experimental Procedures Participants entered the lab, took a seat, and then randomly assigned an identiﬁcation num- were instructed to carefully read the informed ber unknowable to others using shufﬂed opaque consent document. The consent ensured that no sealed envelopes. Numbers were chosen such participant had any prior medical issues, such as that if there were an odd number of participants, This document is copyrighted by the American Psychological Association or one of its allied publishers. frostbite, that would interfere with their partic- the unpaired person would be assigned to be a This article is intended solely for the personal use of the individual user and is not to be disseminated broadly. ipation in the experiment. No potential partici- receiver, so that all dictators’ decisions would pants refused to participate. Participants were be real.

(Appendices continue) ALLOCATING HARM 17

After random assignment, initial instructions a random number generator picked one of the were read aloud to the participants. Participants eight decisions to be implemented. The out- were told that they would be paid $10 in cash at come of this decision was then recorded on a the end of the experiment. Participants were slip of paper for the associated dictator and informed that they would be split into two receiver and placed in individual envelopes, rooms (based on whether they had an even or though the experimenter could not attribute odd identification number), and that each odd- choices to any particular individual in the room. numbered participant (dictator) would make a These were given to the third experimenter in decision that would affect an even-numbered the payment room, in such a way that he could participant (receiver), but that even-numbered only know the amount of immersion time to participants would not make any choices that give each participant, but could not identify any affected the odd-numbered participants. They participant’s role in the experiment. were asked not to communicate with others and Once this procedure was finished, the receiv- to raise their hand if they had any questions ers were called into the payment room, one at a during the experiment. All participants were time, and received their payoffs. Receivers were assured that their choices throughout the exper- asked to submerge their hand in the ice-water iment would be anonymous. container for the duration of time specified by There were three experimenters, a female who the associated envelope. The participant’s auc- always stayed with the receivers, a male who tion decision was then checked to see if she had stayed with the dictators, and a third male in an won the auction, and if so, her hand was sub- additional room to deliver payoffs. The two ex- merged in ice water for 60 s and she was paid perimenters with the receivers and dictators accordingly. Finally, they were paid their $10 were both blind to the experimental hypotheses. show-up fee. This process was repeated for dic- Before making their choices, participants came tators, who put their hand in the ice-water for to the “Ice Water Experience” section of the the duration of the time specified by their ran- questionnaire, where they were asked to raise domly selected choice and then received the their hand to notify the experimenter. The ex- outcome of the auction. Although participants perimenter then walked to the participant with were told in the informed consent agreement an insulated container of ice-water maintained that they could stop at any point and that doing at 3–5 degrees Celsius, and had the participant so would not result in any penalty or loss of submerge her hand in the water for 5 s, timed benefits, no one failed to complete the task or with a digital stopwatch that was observable by dropped out.11 the participant. Dictators then made eight main allocation decisions for their paired receiver. All partici- 11 There is the potential that participants might have been pants then completed an auction to determine confused about whether discontinuing participation would how much they valued avoiding the ice-water forfeit their show-up fee (this was not the case), though we experience. When all participants were finished, have no evidence that this occurred.

This document is copyrighted by the American Psychological Association or one of its allied publishers. (Appendices continue) This article is intended solely for the personal use of the individual user and is not to be disseminated broadly. 18 DAVIS, MILLER, AND BHATIA

Appendix F

Price Elasticities As a final measure of overall choice behavior, Ϫ0.56, Ϫ0.64, and Ϫ0.71, with an overall mean we calculate the arc price elasticities of demand of Ϫ0.64.13 for the good, time for self not in water. The arc In our experiment, the observed mean elas- price elasticity measures the percent change in ticities for the three separate price increases and the amount of time for self not in water relative overall are given in Table F1. The average to a percent change in the price of time for self elasticity across all price changes is Ϫ0.40, not in water. In this case, the dictator pays this indicating that as the price of giving time for price by putting the other person’s hand in ice other not in water increases, the average amount water for some amount of time, so the price of of time for self not in water decreases, which is giving is captured in p.12 For example, consider the opposite of normal demand patterns. Of the two budget sets with M´ ϭ 60 and p ϭ {1, 2}. 27 dictators, 19 (70%) of them had nonpositive elasticities for all three price changes, and out of Here, p ϭ 1 indicates that for each second of the 81 (3 ϫ 27) elasticities observed, only 9 time for other not in water, the dictator must (11%) were positive. These results differ from lose1softime for self not in water. Similarly, ϭ prior work in the domain of positive, monetary for p 2, allocating1softime for other not in allocations where, for example, calculated water incursa2sloss of time for self not in equivalent arc elasticities were found to have water. Thus, if the dictator changes the amount the opposite sign (Andreoni & Vesterlund, of time for self not in water from 60 to 50 when 2001). The negative demand elasticities we ob- the price changes from 1 to 2, the proportionate serve in our participants are consistent with change in time for self not in water is Ϫ0.18 equality-seeking behavior, rather than effici- ϭ 50 Ϫ 60 ( ͑50 ϩ 60͒⁄2), indicating that the amount of time ency-seeking or social-welfare maximization. for self not in water dropped by 18%. The propor- Ϫ tionate change in price is 0.66 (ϭ 2 1 ). Thus, the ͑2 ϩ 1͒⁄2 12 Ϫ ϭϪ ϫ Technically p is the price of giving time not in water to arc price elasticity is 27% ( 100 the other person, paid by the dictator putting her hand in 0.18/0.66), indicating that as the price of giving water. However, for arc price elasticities the same result m time for other not in water increases, the amount obtains if p or its inverse 1 ϭ o (the price of keeping time p ms of time for self not in water decreases, which is not in water for self) is used. 13 Someone with pure Leontief preferences would allo- the opposite of what would normally be ex- ϭ cate tokens in these three situations such that ys yo. pected, where less of a good is purchased when ϭ ϭ Ϫ Because ys Kms, and yo (M K)mo, a little algebra its price increases. ϭ M shows that K pϩ1. As a result, the amount of bad kept is ϭ M ϭ Ϫ Arc price elasticities can be calculated using ys mspϩ1 and the mount of good kept is xs 60 M choices across budgets where prices vary but mspϩ1. Using this, a dictator with pure Leontief preferences will split the tokens such that both dictator and receiver get incomes do not. There are four such budget sets ϭ ϭ ϭ ϭ ´ ϭ ϭ 30s(p 1), 36 s (p 1.5), 40 s (p 2), and 45 s (p 3) in our experiment (with M 60, and p {1, of the bad. The equivalent amount of good kept by the 1.5, 2,3}). If our dictators had perfectly selfish dictator is 30, 24, 20, and 15 s, respectively. Thus, going preferences, these elasticities would all be 0 from p ϭ 1top ϭ 1.5 incurs a change in the quantity of

This document is copyrighted by the American Psychological Association or one of its allied publishers. Ϫ good kept (xs) from 30 to 24, giving a numerator of 0.22 (because, the numerator would always be zero 24Ϫ30 1.5 Ϫ 1

This article is intended solely for the personal use of the individual user and is not to be disseminated broadly. ϭ ϭ ͑ ͑30ϩ24͒ ⁄2͒ and a denominator of 0.40 ͑ ͑1.5 ϩ 1͒ ⁄2͒, for a selﬁsh person, who would keep all of the Ϫ resulting in an arc price elasticity of Ϫ56% ͑ϭ 100 ϫ 0.22͒. The time not in water regardless of the price of 0.40 other arc price elasticities for a dictator with Leontief pref- giving). If, instead, their preferences were pure erences would be Ϫ0.64 (p ϭ 1.5 to p ϭ 2), and Ϫ0.71 (p ϭ Leontief, the respective elasticities would equal 2top ϭ 3), with an overall mean of Ϫ0.64.

(Appendices continue) ALLOCATING HARM 19

Table F1 Average Arc Elasticities Given the Observed Choices of the Dictators (With M´ ϭ 60)

Price of giving 1 ¡ 1.5 1.5 ¡ 22¡ 3 Overall Average arc elasticity Ϫ.53 Ϫ.45 Ϫ.31 Ϫ.43 Choice pairs with arc elasticity Ͼ 0 0% 15% 19% 11%

Appendix G

Critical Cost Efﬁciency Index (CCEI)

Recall that if x1 is chosen given budget set account for this, we calibrated the two experi- 1 1 1 1 B1 : M´ ϭ xs ϩ xop , then x is revealed pre- ments by assuming that participants with the 2 2 2 1 1 1 1 ferred to x if xs ϩ xop Յ xsϩ xop , unless the two observed distributions of prototypical pref- two points are equal (x1 ϭ x2). If it is also the erences made choices with some error, on the ´ ϭ 2 ϩ 2 2 case that given B2 : M xs xop and a choice two different budget sets used. We found that, 2 2 2 2 1 of x it happens to be true that xs ϩ xop Ն xs ϩ conditional on the budget set, there was little 1 2 2 1 xop , then x is revealed preferred to x , violat- difference in the resulting mean CCEI for the ing WARP. If the inequality is strict in at least two distributions of participants, with the bud- one of the two choices, then the preferences get set used in our experiment leading to a mean violate GARP. CCEI of 0.95 and that used in Andreoni and To avoid this violation, we can multiply each Miller (2002) of 0.93, across either distribution 2 2 1 budget set by a number e such that xs ϩ xop Յ of preference types. Thus, the differences in 1 1 1 2 2 2 1 1 2 e ϫ (xs ϩ xop ) and e ϫ (xs ϩ xop ) Ն xs ϩ xop rationalizability we observe across the two ex- removes any WARP/GARP violations. e is periments are not likely driven by differences in called the critical cost efficiency index for a the underlying budget sets or observed prefer- participant, and measures how much of the par- ence distributions. ticipant’s endowments M´ i must be thrown away Fisman et al. (2007), also in the realm of to make the participant’s choices consistent. monetary gains, found 46% (35/76) of their Table G1 shows the distribution of CCEIs two-person-condition participants had such vi- across the 27 dictators. In terms of CCEI, the olations, however, their study involved fifty, most direct methodological comparison with randomly generated budget sets. If participants our experimental design is that of Andreoni and made random choices on the three respective Miller (2002). The actual budget sets, and ob- budget sets, we would predict 26% of partici- served heterogeneity of preferences (discussed pants would have severe CCEI in our study, below), differ between the two experiments. To 37% in Andreoni and Miller (2002), and 100% in Fisman et al. (2007). Finally, by using an individual’s CCEI mea- Table G1 sure, we can determine if the degree of GARP Distribution of the CCEI Across the 27 Dictators violation is related to observed behavior. We This document is copyrighted by the American Psychological Association or one of its allied publishers. Number of Cumulative Cumulative find no significant relationship between the This article is intended solely for the personal use of the individualCCEI user and is not to be disseminated broadly. subjects subjects % mean amount of time not in water kept and ϭϪ 1.000 14 14 52% CCEI (r .08). The mean WTA of the seven .999 5 19 70% dictators with CCEIs below 0.95 is $4.16, while .979 1 20 74% the mean for the 19 (eliminating the outlier) .917 1 21 78% with higher CCEIs is $2.89, though the differ- .833 1 22 81% ence here is not statistically significant. .749 3 25 93% .494 1 26 96% Received March 8, 2016 .333 1 27 100% Revision received December 13, 2016 Note. CCEI ϭ Critical Cost Efficiency Index. Accepted December 29, 2016 Ⅲ