1 Other People s Money 1 Other People s Money: Self-Other Differences in Financial Decision-Making
2 Other People s Money 2 Abstract Many important financial decisions are taken not by the person who will ultimately gain or lose from the outcome, but on their behalf by people such as policy makers, doctors, and managers. Previous studies that aim to understand such surrogate decision-making have certain limitations, such as using between-subject designs or decision outcomes that are nonmonetary, hypothetical, or which do not entail the possibility of losses. We examined decision-making about risk and time when deciding for others compared to for oneself in a study that circumvents previous limitations. We also exploit the rigours of a modelling framework, that has been shown to account for both psychological and neural processes, in order to gain insight into the underlying decision processes. Our behavioural and modelling findings reveal that individuals are both less patient and more loss-averse with other peoples money, while modelling results reveal that choices are more random when deciding for others for both risk and time. Thus, our study reveals unique biases in both the preferences and decision rules used in surrogate decision-making. We suggest the pattern of results are best explained by a preference for self-enhancement in the context of decision-making for others. Keywords: decision making; risk preferences; time preferences; self other differences; principal-agent.
3 Other People s Money 3 You can spend your own money on yourself. When you do that, why then you really watch out what you re doing, and you try to get the most for your money. Finally, I can spend somebody else s money on somebody else. And if I spend somebody else s money on somebody else, I m not concerned about how much it is, and I m not concerned about what I get. And that s government. And that s close to 40% of our national income. Milton Friedman, Fox News interview (May 2004) Introduction The fact that decisions are often taken on behalf of someone else highlights an asymmetry in research where the focus has been on decisions on behalf of oneself. Decisions on behalf of others are ubiquitous across much human social and economic activity, from doctors taking decisions on behalf of patients (Wonderling, Gruen, & Black, 2005) to City traders deciding on behalf of investors. In political science and economics, the problem of motivating a party to act on behalf of another is known as the principal agent problem (Stiglitz, 1987). The principal agent problem arises when a principal compensates an agent for performing certain acts that are useful to the principal and costly to the agent, and where elements of the performance are costly to observe. For example, investors (the principal) delegate to traders (the agent) the responsibility to manage their money, which the former cannot continuously monitor and control. This is the case, to some extent, for all contracts that are written in a world of information asymmetry, uncertainty and risk. It is crucial therefore to understand how people actually make decisions on behalf of others and how this differs from when they make decisions on their own behalf. Taking a doctor as an example, subject to resource constraints, he/ she might act in order to maximise a patient s health (Williams, 1988; Culyer, 1989), to maximise the economic utility of the patient i.e.
4 Other People s Money 4 minimising the financial cost (Evans, 1984), or to maximise social welfare i.e. minimising the cost to society (Mooney, 1994). Previous studies have reported how people decide on behalf of others. However, with respect to monetary decisions, these have limitations such as: using non-monetary decision outcomes (Beisswanger, Stone, Hupp, & Allgaier, 2003; Garcia-Retamero & Galesic, 2012; Kray, 2000; Kray & Gonzalez, 1999; Polman, 2012a; Stone et al., 2013; Wray & Stone, 2005; Zikmund-Fisher, Sarr, Fagerlin, & Ubel, 2006); hypothetical monetary outcomes (Bartels & Rips 2010; Borresen, 1987; Hsee & Weber, 1997; Takahashi, 2007; Ziegler & Tunney, 2012); between-subject designs that do not directly compare the within-person decisions made for oneself with the same decisions made for somebody else (Fernandez- Duque & Wifall, 2007; Pronin, Olivola, & Kennedy, 2008); and outcomes that are only in the domain of risky gains without the possibility for losses (Stone, Yates, & Caruthers, 2002; Teger & Kogan, 1975). Additionally, studies that have avoided such limitations (Cvetkovich, 1972; Polman, 2012a) have dealt only with decision-making in the domain of risk, thereby missing important determinants of decision-making, including how far in the future outcomes will be received (i.e., none of the previous studies have avoided those limitations in decisions about time). Our study aims to fill those gaps in asking how people make decisions about real money for self and others in the two most fundamental decision domains risk and time. We were also interested in the insights one can gain regarding the underlying decision processes by using modelling frameworks. In particular, it is not yet clear whether decisions on behalf of others reveal one s own true preferences, one s expectations about the preferences of the other person, or another (non-preference-based) decision strategies (see DellaVigna, 2009). This is important because simply observing behavioural choices does not tell us what the underlying mechanisms of such decision-making may be, or whether such behaviour reveals true preferences or other decision strategies. Specifically, according to
5 Other People s Money 5 popular economic and behavioural theories (DellaVigna, 2009), there are three stages through which decisions are made: (1) start with underlying preference (desires, wants and needs ); (2) then form beliefs about likelihood/probability of different outcomes; (3) and use a decision making rule to assess the available information and make a choice that satisfies the preferences, given the beliefs. Psychological factors can affect each step. For example, preferences are affected by the desire to avoid unpleasant emotions such as regret (e.g., Loomes & Sugden, 1982). Moreover, urges for immediate gratification create self-control problems and challenge the assumption that the discount factor is time-consistent (Laibson, 1997), and reference points challenge the assumption that decision utility depends on total lifetime wealth (Kahneman & Tversky, 1979). Beliefs are often formed following intuitions incapable of assessing actual risk, which lead to overconfidence about one s abilities (Tversky & Kahneman, 1992). Different decision rules or heuristics are adopted to simplify decisions, under the constraint that rationality is bounded (Simon, 1997). Recent developments in decision neuroscience have offered a range of computational approaches to model such phenomena (Glimcher et al., 2009; Rangel, Camerer, & Montague, 2008). The parameters of these models correlate with neural activity in specific brain structures, which suggest,, that those models might provide a mechanistic level of explanation of overt decision behaviours. We employ such computational models to test specific hypotheses about how individuals make financial decisions involving risk and time for self and for others. Such models are designed to reveal whether specific preferences and decision rules account for observed behavioural choices. 1 Hypotheses 1 Because payoff probabilities and delays will be explicitly presented in our decision tasks, we will model only changes in preferences and decision strategies.
6 Other People s Money 6 Inspired by Hsee and Weber s (1997) research on self-other discrepancies in decision making, we aimed to test five hypotheses four concerning preferences and one concerning decision rules, which predict unique behavioural results in choices: default preference, selfenhancement preference, preference-as-feeling, stereotypical preference, and choice randomness. 2 Default Preference Hypothesis. The default preference hypothesis, analogous to the false consensus in social psychology (Marks & Miller, 1987), assumes that people use their own preference to predict that of others and, as such, assume others have the same preference as themselves. This hypothesis predicts similar choices for self and other. Self-Enhancement Preference Hypothesis. The self-enhancement preference is based on the better-than-average effect in social psychology (Alicke et al., 1995), according to which people are motivated to preserve positive self-image and to defend against negative self-image. This self-enhancement motivation biases individuals to perceive themselves as better than others and as more likely to possess admirable characteristics (Guenther & Alicke, 2010). For example, individuals tend to see themselves as more rational than others (Alicke et al., 1995). In decision making specifically, a well-known finding is that people see others as more susceptible to a host of cognitive and motivational biases (Pronin, Gilovich, & Ross, 2004). Therefore, this predicts more deliberate or reflective choices for self than for others, which should cause specific patterns of behaviours in decisions about time and risk. In decisions about time, it predicts more impulsive choices for others compared to self. This prediction is supported by a patience-as-virtue notion wherein people consider patience a worthy characteristic given it is associated with self-restraint and endurance that are important virtues in all the major religions, psychology, political science, and philosophy 2 Hsee and Weber (1997) only examined whether people can accurately predict the risk preferences of others, i.e. the authors did not ask the participants to actually make risky decisions on behalf of others. Also, Hsee and Weber did not test the choice randomness hypothesis.
7 Other People s Money 7 (Callan, 1993; Dahl, 1971; Schnitker & Emmons, 2007). Patience is considered a more or less stable propensity to react to situational demands with a willingness to defer immediate (or short-term) gratification for future rewards (Fowler & Kam, 2006: 115). For example, in psychology, willingness to delay gratification predicts behaviours related to educational attainment and drug addiction (Funder, Block, & Block, 1983; Wulfert et al., 2002). In labor markets, impatient workers search less intensively for work and accept lower wages (DellaVigna & Paserman, 2005). Impatience has a negative impact on personal finances such that impatient people are most likely to default on debt (Meier & Sprenger, 2012) and have higher levels of debt on revolving accounts such as credit cards (Meier & Sprenger, 2010). In decisions about risk, the self-enhancement hypothesis predicts people perceive others to be less risk seeking, based on the assumption that risk seeking is an admirable characteristic (Shapira, 1995). Hsee and Weber (1997) did not find such self-enhancement effect on risk taking though rejecting this hypothesis might be premature, because other selfenhancement needs may also affect risky choices. DeMartino et al. (2006) defined in neuromechanistic terms what 'rational' means in risky choice the rationality index is the between-subjects measure of susceptibility to the framing manipulation in risky choices, which is the degree to which risk aversion changes when risky payoffs (gambles) are presented as gains or losses respectively. This model implies that when deciding for others, self-enhancing individuals may assume others are less able to control their emotions, and as a result they are more influenced by framing effects and would make more loss-averse choices. Preference-As-Feeling Hypothesis. This hypothesis is based on a dual-process model assumption that one s preference is an expression of one's feelings toward each choice option. Dual-process models propose that reflective or experiential systems differentially dominate choices for self and others. Specifically, when people make choices that go against their goals, such choices often are based on affective evaluations, an account known as the
8 Other People s Money 8 risk-as-feeling hypothesis (Hsee & Weber, 1997; Loewenstein et al., 2001; Stone, Yates, & Caruthers, 2002). This model suggests experiential processes weigh more heavily in decisions for self than for others (Beisswanger, Stone, Hupp, & Allgaier, 2003). This is usually attributed to decision-makers affective evaluation system is more directly impacted by rewards and punishments (Rolls, 2005), while decisions for others usually are either not reinforced or the impact of rewards and punishments is diminished by being decoupled from self-initiated actions that realize an outcome (see also Nicolle, Symmonds, & Dolan, 2011). In decisions about time, this model predicts more impulsive or impatient (temporally discounted) choices for self, due to feelings driving the person to choose more immediate payoffs. This is because the extent to which our time preferences depart from consistent or exponential time discounting is a function of our liking immediate rewards (McClure et al., 2004). When people make intertemporal decisions for another person, they may base their preferences on their own feelings; but also because of a difficulty fully empathizing with that person or imagining the other person having as strong feelings (Fernandez-Duque & Wifall, 2007; Hsee & Weber, 1997), their preference might regress toward more exponential, or less hyperbolic, discounting. Therefore, this hypothesis would predict less impulsive choices for other compared to self. In decisions about risk, the preference-as-feeling hypothesis predicts more risk-averse choices for self. This is because people would decide for others partly on their own emotions, which often favour risk avoidance (Loewenstein et al., 2001), but due to difficulty considering others as having feelings as strong as one s own, this preference should regress toward risk neutrality. Hence why we should make relatively riskier, or more risk neutral, choices for others. Indeed, linking behaviour to the real world, it is conceivable people are more risky with others money (e.g., the archetypal City Traders), or might just become more
9 Other People s Money 9 realistic or risk-neutral (e.g., the expert paternalistic doctor), when compared to the riskaversion displayed when we make such choices for ourselves. Stereotypical Preference Hypothesis. Stereotypical preferences emerge when people base their decisions for others on some stereotype about the group to which the recipient belongs (see Hsee & Weber, 1997). In the risk domain, social psychology studies have demonstrated that women are less likely to support high-risk decisions, and only for mixedsex pairs (men deciding for women and women deciding for men) both males and females are found to select more risky gambles for others than for self (e.g., Cvetkovich, 1972; Stone & Allgaier, 2008; Stone et al., 2002). In intertemporal choice, it is still an open question as to whether men and women change their decisions depending on whether the other person is male or female respectively. Therefore we included a gender variable in the design of our intertemporal choice study by informing the decision maker whether the choice for other is on behalf of a man or a woman. The stereotypical preference hypothesis would predict that choices for others would vary depending on whether the other is man or woman. Choice Randomness Hypothesis. This hypothesis concerns the decision rule and predicts that people are less motivated to exert effort, pay attention and deliberate when deciding on behalf of someone else (as in Milton Freidman s famous quote). We test this hypothesis by assessing the noise in participants choices by fitting a stochastic decision rule with a noise parameter and estimating both the rule and the parameter for each design condition. The choice randomness hypothesis would predict more noisy choices for other than for self. Experiment 1: Intertemporal Choice for Self and Others To examine preferences and decision rules for self and others, we designed an intertemporal choice task, which asked participants to make choices between larger rewards delivered later, and smaller rewards delivered sooner, sometimes for themselves and other
10 Other People s Money 10 times on behalf of an anonymous other participant. It is known that different individuals display significant variability in their preferences, with relatively low-discounters preferring to wait for a later higher-reward option and relatively high-discounters preferring the more immediate smaller reward (Kable & Glimcher, 2007; Pine et al 2009). This behavioural variability can be quantified by estimating an individual s discount rate, a parameter which quantifies an individual s disposition to discount the value of delayed relative to more immediate rewards. We were also interested in model parameters that could distinguish individuals both in their valuations and choice preferences. In previous studies of value comparison, ventromedial prefrontal cortex (vmpfc) activity has been found to correlate with the subjective value of a chosen option or with the value difference between chosen and unchosen options (Basten, Biele, Heekeren, & Fiebach, 2010; Fitzgerald et al 2009 check date) both when deciding for oneself and for others (Nicolle et al., 2012). Methods Participants. Forty-five participants (23 male and 22 female) were randomly allocated to two groups of 22 and 23 subjects respectively. All subjects provided informed consent and the study was covered by approval granted by University College London Research Ethics Committee. All subjects were students at London universities (the vast majority were undergraduates and from University College London). Design. Subjects were invited to the laboratory and seated in separate cubicles to ensure privacy in decision making. To reinforce the social nature of the task, they were given written instructions which emphasised that their payoff may be determined by another subject s choices (in which case the other subject s payoff would be determined by their own choices). The study was conducted using the z-tree software (Fischbacher, 2007). The study consisted of 240 trials where, in each trial, a subject chose between two amounts to be received at different dates in the future for a beneficiary. The main treatment
11 Other People s Money 11 variable was a within-subject variation in the beneficiary of the decisions, which could be either Me (i.e. the subject themselves), Other (Female) (i.e. another subject in the lab that was female) and Other Male. In addition to being displayed on the screen, the background of the screen was colour coded depending on the beneficiary: black for self, light blue for another male and pink for another female. In each trial, the subject was shown a screen with the following information: the beneficiary of the trial (i.e. if that trial is chosen, this determines who receives the payoff), the trial number (from 1 to 240), and two payment options, which are framed as Option i: x in y weeks ) together with two buttons. Subjects could press only one button, corresponding to their chosen Option. To prevent accidental double clicks and to encourage subjects to think, the buttons did not appear until a second after the options were displayed. The options always had a higher amount of money at a later date (e.g. 17 in 20 weeks) in one option (the LATER option) and a lower amount of money at a sooner date (e.g. 14 in 1 weeks) in the other option (the SOONER option), but which Option was labelled Option 1 (and presented on the left of the screen) and which was labelled Option 2 (and presented on the right) was chosen at random; this enabled us to identify subjects that may have been consistent in their behaviour only because they clicked the same button each time. A screenshot of a specific trial is shown in Figure 1. The SOONER option payment date was always either 0, 1, or 2 weeks (0 weeks being the day of the experiment) and the LATER date varied between 1 and 25 weeks; the SOONER payment amount varied between zero and 24 and the LATER payment amount varied between 6 and 49. A number of attention checks were included in the parameter choices: i.e. options where the payment size was the same but the dates were different, or options where the dates were the same but the payment size was different and options where the payment size was zero.
12 Other People s Money 12 There were 80 trials containing different possible combinations of magnitudes and delays reflecting a wide range of implied discount factors (for more information how the options were generated see Nicolle et al., 2012) and subjects faced the entire set once for each beneficiary. Trials were split up into blocks of 40 choices for the same beneficiary (all participants performed all three beneficiary conditions); the blocks were ordered at random as were the different trials within a block. There was a 30 second time between blocks for rest. Although subjects could make decisions at their own pace (there was no time limit hence also no incentive to rush decisions), a session could not be completed until all subjects had completed their decision-making (as payoffs may depend on other subject s decisions). After all subjects had completed all of their choices, subjects were randomly assigned to be paid based either on their own choice or the choice of another subject and one choice was randomly selected for each subject to be paid. In total, each session lasted about 65 minutes including reading instructions and dealing with payment administration. In order to completely isolate the effect of time discounting on decision making, there was no show up fee in this experiment, and subjects earned on average Subjects were paid into their bank account or via Paypal at the time specified in the payoff-relevant choice (e.g. if the payoff-relevant choice was to be paid 17 in 20 weeks, they were paid 17 exactly 20 weeks after the experiment). Behavioural modelling. A function which describes the pattern of discounting can be estimated by observing choices between delayed outcomes. Economic theories of rational behaviour posit that goods ought to be discounted exponentially with delay (Samuelson, 1937). Formally, an outcome which has value or utility M if received immediately (t = 0) is worth γ M T if delayed t periods into the future. The present-time Utility, U of receiving M at time is thus given by:
13 Other People s Money 13 U T M (1) Here the discount rate, γ, represents the constant proportional decrease in value with each added time period of delay. Contrary to predictions of exponential discounting a large body of evidence with both human and animal subjects has demonstrated that discount rates appear to decrease with increasing delays (Green & Myerson, 2004; Rubinstein, 2003). Again, each participant s unique discount rate was estimated by fitting a discount function to their choices, whereby the discounted utility/value of an option (U) varies hyperbolically as a function of reward or payoff magnitude (M) and delay (T): U M 1 T (2) A proposed alternative is the quasi-hyperbolic approximation to hyperbolic discounting (Laibson, 1997), which is formalized as exponential discounting, with an additional preference for immediate rewards (present bias), expressed as a second discount factor, δ, and applied to all time-periods except the first: U T M T ( M 0 T ) (3) Where MT0 is the immediate payoff, MT is the payoff in period T, δ is an individual s present bias parameter and γ is the individual s long-run discount factor. When δ = 1, individuals are not present biased and the quasi-hyperbolic model reduces to standard exponential discounting. Such present biased time preferences may be the result of the interplay between two separate decision-making systems: the affective system which values immediate gratification and the deliberative system which makes long-run plans and displays higher discount factors (McClure et al., 2004). Given a subjective (i.e. temporally discounted) value associated with each of the two (sooner vs. later) options in the pair, the associated probability of making each choice is
14 Other People s Money 14 estimated with a softmax (logistic) decision rule, where β > 0 determines the randomness of the decision higher numbers indicate more deterministic, less random choice (in our simulations β was constrained between 0 and 10): P ( Later) ( U ( Later) U ( Sooner)) 1 e 1 (4) This is a standard stochastic decision rule that calculates the probability of taking one of two actions according to their relative subjective values (see Glimcher et al., 2009). Recent neuroimaging work has shown that this model describes neural signals in medial prefrontal cortex, which compute the choice preferences of another individual as well as oneself both in terms of subjective valuations of options and in terms of choices (Nicolle et al., 2012). Critically, it is necessary to optimise the choice pairs such that they give the most efficient estimate of potential subjects discount rates. To optimise these choices, we generated a random 100 choice pairs, each comprising one smaller, sooner reward and one larger, more delayed reward, but with the magnitudes and delays varying across the pairs. We then computed the decisions predicted to be made by simulated subjects with discount rates ranging from 0 to 1. When plotted those simulated discount rates against the predicted number of choices of the delayed option, the closer this graph is to the diagonal, the better different discount rates are reflected in different subject choices, and therefore the lower the error introduced by the model estimation process (Nicolle et al., 212). We generated 10,000 such choice sets and chose the set whose curve was closest to the diagonal (in terms of enclosed area). In this choice set, magnitudes ranged between 0 and 24 for sooner options, with their delays ranging from 0 to 2 weeks. For the more delayed options of each pair, the magnitudes ranged from 6 to 49, and the delays from 1 to 25 weeks, since these provided the lowest correlations in self and other choices and values.
15 Other People s Money 15 The three choice models (Equations 1-3) were separately fit to participants choices in the pre-screen questionnaire, optimising their free parameters (γ), (δ), and (β) to maximise the likelihood of the choices (Equation 4) given the parameters. This was realised through standard Matlab functions used to compute maximum likelihood estimation. Model comparison was performed at the group level (fixed effects), by summation of log likelihoods across participants. Selection between models proceeded using the Bayesian information criterion (BIC) (Schwarz, 1978), where L is the maximized group level log likelihood, k is the number of free parameters in the model and n the number of independent observations. BIC = 2 L + 2k ln(n) (5) For BIC, Kass and Raftery (1995, p. 777) propose the following taxonomy of what constitutes a substantial difference in criterion values: 0 2 Weak; 2 6 Positive; 6 10 Strong; >10 Very Strong. We used BIC to compare between three versions of the model. Finally, after selecting the winning model, we included in the analysis only the participants for whom the model performed above certain minimum goodness-of-fit measure. We applied McFadden's Pseudo R-Squared (McFadden, 1979) to measure the improvement from null model to fitted model: R ln Lˆ( M 1 ln Lˆ( M 0 ) ) 2 Full (6) where MFull is the model with predictors, M0 is the model without predictors which assumes random choice, and Lˆ is the estimated likelihood. The likelihood for the null model is defined as follows: Lˆ( M ) 1 k N 0 (7) where N is the number of choice options (each choice is equally likely so with two options 1/N = 0.5) and k is the number of observations or choices. Note that the theoretical
16 Other People s Money 16 range is 0 < McFadden Pseudo R 2 < 1, but as a rule of thumb, the model has an excellent fit when 0.20 < McFadden Pseudo R 2 < 0.40 (McFadden, 1979, p. 307). Results Table 1 presents the behavioural results in each condition, according to beneficiary (self, other male, other female) and gender of the decision maker. The differences between the conditions were examined with a mixed-model two-way (Beneficiary x Gender) univariate analysis of variance (General Linear Model), with Beneficiary as within-subject variable, participant Gender as between-subject variable, and the mean proportion of choices of the later (delayed) option as the dependent measure. There was a significant main effect of Beneficiary, F(2, 86) = 6.93, p =.002 ( 2 =.14). Post-hoc analysis revealed that, after Bonferroni adjustment for multiple comparisons, the proportion of later picks for Self was significantly higher than the proportions for both Male (p =.010) and Female (p =.043) others, while Male was not significantly different from Female others (p = 1.000). There was no significant main effect of the participant s own Gender, F(1, 43) = 0.54, p =.467 ( 2 =.01), suggesting that male and female participants did not exhibit different levels of impatience (i.e. delay discounting). The Beneficiary x Gender interaction was also not significant, F(2, 86) = 0.30, p =.739 ( 2 =.01), which means that male and female participants did not respond differently when choosing for a same- or an opposite-sex beneficiary (i.e., the degree of time discounting, indicated by the proportions of later choices, did not vary depending on the gender of the decision maker and the beneficiary respectively). Behavioural Modelling. The computational modelling reveals additional interesting variance in the data, which the overall behavioural data does not reveal because averaging across all choices misses much of the interesting variance. The modelling also allows us separate preferences from choice randomness.
17 Other People s Money 17 The impact of delay depends on the participant s unique discount rate (γ), which was constrained to fall between 0 and 1. In our task, this was a reasonable constraint, as a subject with a discount rate of 1 would always choose the smaller sooner option on trials where the delay and magnitude favoured opposite choices. The hyperbolic discount model produced the best model fit (BIC = 6273) compared to the exponential model (BIC = 6283) and the quasihyperbolic model (BIC = 6854) (note that smaller numbers reflect a better fit). With our winning hyperbolic model, replacing the separate parameters for Other decisions for males and females (βmale and βfemale; γmale and γfemale) with the single gender-independent free parameter (βother and γother) resulted in even better BIC performance: 5956 vs respectively. Here we present the statistical analysis of the parameters of this best-fitting selfother hyperbolic model (although similarly significant results are obtained when analysing the parameters of the model with separate male and female parameters). We excluded from the analysis 4 participants (2 male and 2 female) for whom the model showed unsatisfactory McFadden's Pseudo R-Squared goodness-of-fit measure (mean R-Squared =.12), thus analysing the data from the 41 remaining participants (mean R- Squared =.72). Table 2 presents the modelling results in each condition, according to beneficiary (Self and Other). The discounting parameters (γ) from this winning model and the simpler metric (proportion delayed choices) were very highly correlated for Self (r = -0.81) and Other (r = -0.83). We analysed the distribution of the inferred individual time discounting parameters for Self and Other. The differences between the Self and Other conditions were examined with nonparametric statistical analysis, because this variable was not normally distributed according to the Kolmogorov-Smirnov test of normality for Self (D = 0.33, p <.001) and Other (D = 0.31, p <.001). The Wilcoxon Signed Ranks Test revealed that the discounting parameter (γ) was significantly higher in Other beneficiary condition
18 Other People s Money 18 compared to Self, Z = 2.78, p = 006. We also re-analysed the behavioural data from those participants (i.e. without the 4 excluded participants), which revealed the same pattern. The proportion of delayed choices was significantly higher in the Self beneficiary condition compared to Other, Z = 3.31, p = 001 (t(40) = 3.02, p = 004, using paired-samples t-test). Finally, we analysed how the choice randomness parameter (β) varied across the two conditions. We used nonparametric statistical analysis, because this variable was not normally distributed according to the Kolmogorov-Smirnov test of normality for Self (D = 0.34, p <.001) and Other (D = 0.44, p <.001). The Wilcoxon Signed Ranks Test revealed that the average β across participants was significantly higher when deciding for Self beneficiary compared to Other, Z = 3.52, p < 001 (i.e., decisions were less deterministic, or more random, for Others). Discussion The behavioural results and the modelling parameters for time discounting showed that individuals are more impulsive (less patient) with other peoples money, which supports the self-enhancement hypothesis. 3 Our modelling results also revealed that choices are more random when making time-money trade-offs for others, which supports the choice randomness hypothesis. Thus, the study revealed unique biases in both the preferences and also in the decision rules that individuals employ when making a choice that satisfies their time preferences (in line with the decision framework proposed at the beginning, DellaVigna, 2009). We did not find support for the default preference, preference-as-feeling, and the stereotypical preference hypotheses. In this respect, note that the preference-as-feelings 3 In a similar task, Ziegler and Tunney (2012) asked their participants to make hypothetical decisions in terms of what the other person "should" choose, and found that they were less impulsive for others, compared to self. This result illustrates how subtle differences in the task design can affect what factors influence self-other decisions. Specifically, asking what the other person should choose, especially in a hypothetical task, most likely primes normative thinking (i.e. what is good for the other person). In contrast, our task asked the participants what they want/prefer to choose for the other individual (which obviously only works when the choices are real and not hypothetical).
19 Other People s Money 19 hypothesis (Hsee & Weber, 1997; Loewenstein, Weber, Hsee, & Welch, 2001) also predicts self-other differences should be observed in opposite-sex pairs. That is individuals should make similar decisions for self and other when genders are the same because the automatic feelings provoked by the choice options are easier to imagine and project upon the other person if the other is more similar to oneself. This pattern was not observed in our data, which implies that different underlying decision processes drive self-other choices involving time and money. Finally, there may be alternative reasons why decision-makers are more impulsive for other people, over and above the hypothesis that they are self-enhancing. One possibility is that we do not know as much about other people's future plans as we do about our own future plans (especially when the other people are anonymous strangers). This might make us concerned about choosing delayed rewards for other people because, for example, we do not know if the other person might be leaving the country, or closing their bank account, or might otherwise not be able to receive the delayed reward. In formal terms, this would amount to less precise estimate of another person s future states compared to one s own future states. Therefore, if our participants are risk averse, they may judge providing the other person with sooner rewards to be the favourable ( kinder ) option (i.e., they may predict it to be the less risky, or more loss-averse option given the uncertainty about the recipient s future circumstances). A potential direction for future research would be to explicitly model the uncertainty (aka belief distributions) about the future for self and other. However, this uncertainty-kindness explanation may be less likely to have played a strong part in our study, because our participants should have known that others individual circumstances are unlikely to change within 6 months (the later choice date varied between 1 and 25 weeks), given that the vast majority were undergraduates from the same university. Experiment 2: Risk Taking for Self and Others
20 Other People s Money 20 We also investigated whether decision making behaviour under risk is different when people use their own money rather than somebody else s money. This is an important test, because our hypotheses have not been tested with decisions about risk. Similarly to the intertemporal choice experiment, we tested the following hypotheses (three about preferences and one about decision rules) in decision making under risk: default preference, self-enhancement preference, preference-as-feeling, and choice randomness. We did not test the stereotype hypothesis in this study, because, as we discussed before, the effect of gender has already been established in risky choice for self and others (e.g., Cvetkovich, 1972; Stone & Allgaier, 2008; Stone et al., 2002), and removing the factor of beneficiary gender from our design provided enhanced power for assessing the effect of decision frame (important for testing the self-enhancement preference hypothesis, see introduction). Methods Participants. Fifty individuals participated in the study; the study was conducted in three groups of 18, 18 and 16 subjects (33 females and 17 males). All subjects had previously provided informed consent and the study was covered by approval granted to the Department of Economics from the University College London Research Ethics Committee. All subjects were students at London universities (the vast majority were undergraduates and from University College London). Design. Subjects were invited to the laboratory and seated in separate cubicles to ensure privacy in decision making. To reinforce the social nature of the task, they were again given written instructions which emphasised that their payoff would be partly determined by another subject s choices (and their partner s payoff would depend partly on their payoffs). The study was conducted using the z-tree software (Fischbacher, 2007). Screenshots of two trials are shown in Figure 2; the task was adapted from Wright et al. (2012). In each trial, participants needed to choose between a Gamble option and a Sure
21 Other People s Money 21 option. First, the participants saw a blank screen with fixation cross; second, they were shown the pie chart with 4 sectors for the gamble option and the value of the Sure option; after the pie chart had been displayed for four seconds, two buttons appeared, one of which they pressed to register their choice which they needed to do within two seconds. If the Gamble option is selected, subjects win one of the four specified amounts with a probability given by the corresponding area of the pie chart. If the Sure option is chosen, subjects win the specified Sure amount for certain. The colours (chosen from a set of neutral colours) and the order and orientation of the pie sectors is determined at random. Wright et al. (2012) showed that the degree of risk in those lotteries positively correlated with neural activity in posterior parietal cortex, a region strongly associated with risk; while the manipulation of valence was expressed in greater activity for gains than losses in valuerelated areas of orbitofrontal cortex and bilateral striatum. The study investigated two main factors. The first factor was the effect of beneficiary, whereby subjects played half the trials on behalf of themselves and half on behalf of another subject in the laboratory. The beneficiary was displayed on the screen in each individual trial. The second factor was the effect of winning versus losing money in a trial. Half the trials were in the Gain frame where the gamble amounts and sure amount were positive and the other half were in the Loss frame where they were negative. This provided a 2x2 factorial design in this basic design with within-subject factors for frame (gain or loss); and beneficiary (self or other). For each subject, the first half of the session was played for one beneficiary and the second half for the other beneficiary; within each half, gain and loss frame trials were played in a random order. The colour of the background screen was specific to the beneficiary type. There were 49 distinct gamble options and they were played 4 times each by each subject (once for each combination or gain/loss and self/other) for a total of 196 decisions per
22 Other People s Money 22 subject. The sure option was always a gain or loss of 3 and the gamble options ranged from 0 to 6. Thus, across the set of 49 gain trials, we parametrically manipulated the degree of risk in the lottery (using 7 levels of variance) and orthogonally manipulated its EV (7 levels). One-half of the lotteries had an EV above the sure amount and one-half below (mean EV across all 49 lotteries was equal to the sure option), which provided a simple metric of risk preference indexed as the proportion of riskier choices made (proportion risky choices: riskneutral, 0.5; risk-averse, <0.5; risk-seeking, >0.5). To manipulate valence, we created 49 perfectly matched loss trials by multiplying all amounts in our gain trials by -1 (see also Wright et al., 2012, for more information on how these trials were constructed). The full set of trial parameters was seen once in each cell of the design in every session, so the full set was shown 4 times in total gain and loss for self and other respectively. Participants payments comprised a 5 show up fee, plus a 12 initial endowment (allowing for any losses occurred in the loss frame trials), plus the outcome of 4 randomly chosen trials two where the subject chose on behalf of themselves (one gain and one loss frame) and two where another subject chose on their behalf (one gain and one loss frame). The 12 covered the worst possible outcome (where participants won 0 in the gain frame and lost 6 in the loss frame). Sessions lasted about an hour including reading instructions and processing payments and subjects earned an average of 17. Behavioural modelling. In our analysis, we assessed whether the participants simply become noisier in their choices for the other person, for example because they just do not care as much. Therefore, we fit a choice model to choices in each of the four cells of the factorial design (gain/loss, self/other) in which the utility of the sure and gamble options are computed in each trial (using a mean-variance utility function), with the softmax (logistic) decision rule used to choose between them. Therefore, we estimated a risk parameter and noise parameter in each condition for each subject. We modelled individual choices using a
23 Other People s Money 23 binary logistic regression utility model. On each trial the subjective values, or utilities (U), of both options (Gamble vs. Sure) were computed using a utility function; and then these values were compared with generated trial-by-trial probability of accepting the lottery, using a softmax function with a free parameter β (constrained between 0 and 10) that allows for noise in action selection as follows: P ( Gamble) ( U ( Gamble) U ( Sure)) 1 e Note that this randomness parameter β is also shown to correlate with prefrontal neural structures (anterior frontopolar cortex) that control for exploration versus exploitation strategies in decision making under uncertainty (Daw et al., 2006). Also, in line with prefrontal cortex role as the principal region implicated in behavioural control, Daw et al showed that activity in the orbitofrontal and ventro-medial prefrontal cortices correlates with the probability assigned by the model to the action chosen on a given trial (in the softmax model, this probability is a relative measure of the expected reward value of the chosen action). We then define the subjective value, or utility (U), of each lottery using the meanvariance model: U(Gamble) = EV + ρvar (9) EV is the expected (mean) value, Var is the variance of the gamble option. For a given lottery with N potential outcomes (m1, m2, mn), with probabilities p = p1, p2, pn, we define those statistical moments as follows: 1 (8) EV N n 1 m n p n (10) Var N n 1 2 ( mn EV ) p n (11)
24 Other People s Money 24 where ρ is an risk parameter (constrained between -1 and +1) reflecting the taste for risk, with zero indicating risk neutrality, a positive number risk seeking and a negative number risk aversion), and β reflects choice randomness. Equation 10 also represents the most basic mean model while Equation 9 is known as the classic mean-variance model (Markowitz, 1952; Bossaerts, 2010). In the mean variance valence model, on each trial the subjective values, or utilities (U), of both options (A and B) were computed using the above utility function, where ρ = ρ gain in gain trials and ρ = ρ loss in loss trials. Wright et al. (2012) show that this mean variance valence model best predicts behavioural data in this task compared to the basic expected value model (without variance term), the classic meanvariance model, the expected utility model (Camerer, 2003), the prospect theory model (Kahneman & Tversky, 1979), and the cumulative prospect theory model (Tversky & Kahneman, 1992). This winning model enables extending ideas derived from financial economics that individuals respond to risk as measured by the variance in potential outcomes (Markowitz, 1952; Bossaerts, 2010), by incorporating valence. Also, Wright et al. demonstrate that the components of this model correlate with activity in brain regions in the parietal and temporal cortices, anterior insula, and the ventral striatum, which are known to encode risk information and decision utility. In this model, there is no constant term as we assume a utility of 0 represents a point of subjective indifference between sure or gamble (i.e., rejection of offer has a utility of 0). We optimised subject-specific ρ and β parameters using nonlinear optimization implemented in Matlab for maximum likelihood estimation. In our simulations of the observed decisions for self and other, we extended the mean-variance model by replacing the parameters ρ and β with separate parameters for self-trials (ρself and βself) and other-trials (ρother and βother). Model fitting resulted in a set of maximum likelihood parameter estimates for each subject. Model comparison was performed at the group level (fixed effects), by summation of log likelihoods
25 Other People s Money 25 across participants. As in the previous study, selection between models proceeded using the Bayesian information criterion (BIC) (Schwarz, 1978). We used BIC to compare the following three models, which allowed us to ask whether decision behaviour for self and other was influenced by risk and valence (i.e., every model was fit to all the data for each subject). First, a simple mean model (see Equation 10) assumed that individuals only cared about the mean value of the options. Second, we asked whether choice was also influenced by risk, using a mean variance model (see Equation 9). Third, we asked whether both risk and valence influence choice, using a mean variance valence model with separate risk parameter that reflects risk preference in gain trials and loss trials respectively. 4 Again, after selecting the winning model, we included in the analysis only the participants for whom the model performed above the minimum McFadden's Pseudo R- Squared goodness-of-fit measure. Results Table 3 presents the behavioural and modelling results in each condition. For each individual, the proportion of riskier choices indexes risk preference, which was derived separately for each cell in the (Beneficiary x Domain) design: Self-Gain, Self-Loss, Other- Gain, Other-Loss. First we analysed the behavioural data across all participants before conducting the computational modelling. The differences between the conditions were examined with a repeated-measures two-way (Beneficiary x Domain) univariate analysis of variance (General Linear Model using probit link function), with the mean proportion of risky choices as the dependent measure. There was a significant main effect of Beneficiary, F(1, 49) = 7.64, p =.008 ( 2 =.14), which showed that the participants were more risk-averse when deciding for Self (vs. Other). There also was a significant main effect of Domain, F(1, 4 We did not fit any version of prospect theory, because the mean-variance model has been consistently shown to outperform prospect theory in this task (Wright et al., 2012; Wright, Symmonds, Morris, & Dolan, 2013).
26 Other People s Money 26 49) = 5.40, p =.024 ( 2 =.10), which is driven by the fact that the participants were more risk averse for Losses (vs. Gains). The Beneficiary x Domain interaction was not significant, F(1, 49) = 0.68, p =.415 ( 2 =.01), which means that the difference between gains and losses (i.e., the size of the valence effect) was not different (bigger or smaller) depending on the beneficiary. Behavioural Modelling. The mean variance valence model produced the best model fit (BIC = 11013) compared to the mean variance model (BIC = 11225) and the mean model (BIC = 12137). 5 With our winning model, replacing the single free parameter in our softmax decision rule (β) with separate parameters for gain trials (βgain) and loss trials (βloss) resulted in worse BIC performance: vs respectively. We included in the analysis only the 40 participants for whom the model showed satisfactory McFadden's Pseudo R-Squared goodness-of-fit measure (mean R-Squared =.35), while for the excluded 10 participants this measure indicated that their behaviour was random and inconsistent (mean R-Squared =.15) (McFadden, 1979). Risk-related parameters (ρ) from this winning model and the simpler metric (proportion risky choices) were very highly correlated for gain and loss trials in both conditions (self-gains, r = 0.80; self-loss, r = 0.95; other-gains, r = 78; other-losses, r = 0.82). Analysing the data from those participants alone, differences between conditions were examined with a repeated-measures two-way (Beneficiary x Domain) univariate analysis of variance (General Linear Model), with the mean risk aversion parameter (ρ) as the dependent measure. There was no significant main effect of Beneficiary, F(1, 39) = 0.09, p =.765 ( 2 = 5 Similarly to Wright et al. (2012), we also explored inter-individual differences to seek further evidence of behavioural independence between the influences of risk and valence. If these influences result from stable and independent processes, we can predict that knowing an individual s sensitivity to one influence would not predict sensitivity to the other. We conducted additional analysis, in which for each participant, the average proportion of riskier choices indexes risk preference, and the difference in riskier choices between (gain and loss) domains indexes the impact of valence. Crucially, these preferences were independent, with risk and valence effects showing no correlation across subjects for either Self (r = 0.03; p = 0.928) or Other (r = ; p = ).
27 Other People s Money 27.00), indicating that those participants were not more risk-averse when deciding for Self (vs. Other). There was still a significant main effect of Domain, F(1, 39) = 9.99, p =.003 ( 2 =.20), which means that the inferred risk aversion parameter was significantly lower (more risk averse) for Losses. The Beneficiary x Domain interaction was significant, F(1, 39) = 5.34, p =.026 ( 2 =.12), which confirms our hypothesis that the gain-loss asymmetry in choices is bigger when deciding for Others. The behavioural data (proportion of risky choices) from those participants showed the same pattern. There was no significant main effect of Beneficiary, F(1, 39) = 2.60, p =.115 ( 2 =.06), indicating that those participants were not more risk-averse when deciding for Self (vs. Other). There was significant main effect of Domain, F(1, 39) = 6.19, p =.017 ( 2 =.14), which means that the inferred risk aversion parameter was significantly lower (more risk averse) for Losses. The Beneficiary x Domain interaction was significant, F(1, 39) = 3.68, p =.063 ( 2 =.09), which confirms that the gain-loss asymmetry in choices is different between beneficiaries. Finally, the choice randomness parameter (β) was significantly higher in the Self beneficiary condition compared to Other, t(39) = 3.07, p = 004 (using paired-samples t-test). This shows that the choices for Other were more random (less deterministic) than for Self. Discussion There are several key findings. First, the novel finding from this experiment concerns differences in preferences and decision rules on behalf of other people when compared to making choices for oneself. Overall, there were more risk-averse choices made in the face of both gains and losses when participants were deciding for themselves compared to others. On face value, this result supports the risk-as-feeling hypothesis (Hsee & Weber, 1997). However, when we analysed the data only from the participants whose behaviour is explained by the computational choice model (and excluded 20% of the participants who were likely
28 Other People s Money 28 not choosing in line with expected value models), this behavioural pattern disappeared and the key behavioural finding was the bigger gain-loss asymmetry in risk preferences (framing effect) when deciding for others compared to oneself. This supports the self-enhancement preference hypothesis that people assume that others are less able to self-regulate their affective response to losses (and hence they may be more irrational and susceptible to the framing manipulation), which results in much more risk-averse choices in the domain of losses (relative to gains) when choosing on behalf of others. 6 This counterintuitive result was revealed only thanks to the use of mechanistic modelling that accounts for the underlying choice processes. 7 Secondly, the fact that choices are more random when deciding for others, irrespective of the underlying risk attitude, is other novel finding, which supports the choice randomness hypothesis. This was also revealed by the neurobiological computational model employed in the analysis. Given that the choice randomness parameter has a specific neural interpretation (Daw et al., 2006), this finding makes predictions for future brain imaging research in this area. This specific result could also have important implications for understanding how policy decisions are made in the real world. Finally, we replicate the greater gambling for gains than losses shown previously in this task (Wright et al., 2012; Wright et al., 2013; Wolf, Wright et al., 2013; Wright et al., 2013). Wright et al. explain these findings within a biologically grounded, process-based account of choice that evolves from option evaluation to action selection. Specifically, in the choice 6 Alternative explanations in terms of kindness and aversion to harmful action (Cushman et al., 2012) would predict the opposite results in the risky decisions for losses more risk seeking for losses because individuals would rather prefer to avoid the certain loss (harm) when deciding for others. 7 Note that we do not make any assumptions whether the self-enhancing belief that others are more impulsive (Experiment 1) is a conscious, deliberative thinking, while the belief that others are more affected by framing effects (Experiment 2) is unconscious. Future research should aim to determine whether various self-enhancing assumptions are conscious or unconscious thought processes.
29 Other People s Money 29 process, risk and loss independently engage approach avoidance mechanisms 8. Depending on context, this mechanism can produce either risk aversion for gains and risk seeking for losses or the exact opposite. In our design, the number of potential losses in the loss gamble are more salient than the single sure-loss option, thus triggering avoidance of the gamble; which leads to more risk aversion than in the gain frame (where the number of potential gains in the gamble can trigger approach tendencies instead). Other types of context or framing effects can reverse the gain-loss valence effect, such as when framing a sure option as a loss: e.g., you receive 50, and can either lose 30 or gamble for 50/50 chance to keep 50 or get 0 (i.e. the gamble contains only positive payoffs), which biases individuals to avoid that sure loss option and choose a gamble instead compared to when they are offered to keep 20 (De Martino et al., 2006). Note that in Kahneman and Tversky's (1979) classical task, individuals had to evaluate and select between two options and so could not express avoidance by withdrawal, but instead could potentially avoid losses by selecting/approaching the riskier (higher variance) option. In summary, those studies (including ours) illustrate the variety of intricate ways in which context determines how humans respond to aversive stimuli. In summary, consistent with a multi-stage decision process (DellaVigna, 2009), this experiment identified unique biases in both the risk preferences and also in the decision rules that individuals employ when making risky choices for self and others. General Discussion Many financial decisions are taken not by the person who will ultimately gain or lose from the outcome, but instead on behalf of someone else. This is a characteristic of much of the activity of policy makers, doctors, and managers. Indeed, there are also many non- 8 At a mechanistic level, in the brain it has been argued that there are a number of interacting valuation systems that together determine choice, including the Pavlovian system for approach-avoidance responses, habitual system for stimulus-response learning, and goal-directed system for reflective/model-based planning and decision-making (Rangel, Camerer, & Montague, 2008).
30 Other People s Money 30 financial situations that involve surrogate decision-making, e.g. carers must make very important medical decisions on behalf of those they care for, and next of kin must make life or death decisions on behalf of the patient. Previous studies have had limitations in comparing such surrogate decision-making with decisions for oneself, including betweensubject designs or outcomes that are nonmonetary, hypothetical, or do not involve losses. Our study avoids those limitations to explore such decisions about risk and temporal delay. We also employ computational models to test several hypotheses about the underlying decision processes driving choice behaviour. The behavioural results and the modelling parameters reveal that individuals are less patient and more loss-averse with other peoples money. Those results are best explained by preference for self-enhancement, which was derived from the better-than-average effect in social psychology (i.e., the observed behaviours reflect a favourable self-image caused by motivated self-enhancement needs) (Alicke et al., 1995; Guenther & Alicke, 2010). We are not aware of similar or related evidence in the self-other decision-making literature. This phenomenon might be more likely to occur when agents do not have information about the others particular goals, as in our design. This interpretation is corroborated by evidence that decisions for others are attuned to others preferences, and moreover that they draw on similar brain mechanisms as are used when decisions are made for the self, if the actors have the opportunity to learn what the other person would decide in a similar situation (Nicolle et al., 2012). In this respect, future research should study the neural mechanisms involved in decisions for others when their preferences for time and risk are not explicitly stated. Our modelling also revealed that choices are more random when deciding for others for both risk and time. This could reflect differences in motivation, as people are less random (more deliberate or conscientious) with their own money and accords with evidence that, with
31 Other People s Money 31 increasing personal involvement, decision makers pay more attention to the content of the information presented, process information more systematically, and are less likely to rely on simple heuristics (Beisswanger, Stone, Hupp, & Allgaier, 2003; Harkness, DeBono, & Borgida, 1985). This conclusion is also supported by evidence that the prefrontal cortex, the principal brain region implicated in behavioural control, is correlated with the probability assigned by the model to the chosen action (Daw et al., 2006). In summary, our two experiments reveal unique biases in both the preferences and also in the decision rules used to make a choice that satisfies one s own preferences compared to making choices on behalf of someone else; which is in line with the multi-stage general theoretical framework employed to understand decision making (DellaVigna, 2009). Those results also demonstrate how computational models can reveal the underlying processes in decision making. Our results inform the two alternative explanations in the literature for selfother decision differences: the dual-process models and the cognitive account but do not support predictions of a dual-process, preference-as-feeling, model (Hsee & Weber, 1997; Loewenstein et al., 2001). Therefore, our results are inconsistent with suggestions that different mental processes, rational versus experiential, weigh differently in intertemporal and risky decisions for self and others. A cognitive account postulates that self-other differences result from differential weighting of preference parameters, rather than from differences in motivation (Kray, 2000). Our modelling parameters are consistent with this interpretation, at least in the context of risky choice where the risk-aversion parameter in the risk-variance model can be interpreted as a weight on the variance dimension, which changes depending on the domain (gain vs. loss) and the recipient (self vs. other). However, this interpretation does not explain why those weights differ for self and other, i.e. what is the underlying psychological reason for this differential weighting. Therefore an additional explanatory
32 Other People s Money 32 variable, such as motivated self-enhancements needs, is still necessary to account for choice behaviour. From a practical perspective, the self-others discrepancy in intertemporal or risky choice is a serious bias. In politics, for example, policy makers need to understand the preferences, including risk and time preferences, of their constituents. The self-others bias may arise because those constituents are often detached and abstract agents or groups. In politics and business, being aware of this bias, helps lawmakers and managers introduce policies that better reflect people's willingness to trade off risk, time and benefits.
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40 Other People s Money 40 Table 1 Proportion of Delayed Later Choices According to Beneficiary (Self, Male, Female) and Gender of the Decision Maker Measure Proportion Later Choices Gender of Decision Maker Beneficiary Self Male Female Mean SE Mean SE Mean SE Male Female Total N
41 Other People s Money 41 Table 2 Modelling Results According to Beneficiary (Self and Other) Measure Beneficiary N Time Discounting (γ) Choice Randomness (β) Proportion Later Choices Self Other Mean SE Mean SE
42 Other People s Money 42 Table 3 Behavioural and Modelling Results Presented According to Beneficiary (Self vs. Other) and Payoff Domain (Gain vs. Loss) Participants Measure Beneficiary Payoff Domain Mean Statistic SE N All participants Participants showing good model fit (McFadden Pseudo R 2 > 0.20) Proportion Risky Choices Proportion Risky Choices Risk Aversion Parameter (ρ) Choice Randomness Parameter (β) Self Gain Loss Other Gain Loss Self Gain Loss Other Gain Loss Self Gain Loss Other Gain Loss Self Gain and Loss Other Gain and Loss
43 Other People s Money 43 Figure 1. An example trial screenshot for the delegated intertemporal choice task.
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