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# Social preferences under uncertainty. An experiment

In his short story “The Lottery in Babylon”, Borges imagines a city where a lottery, managed by a secret society, mandates the fate of its citizens. Every sixty nights, drawings are made and determine each citizen’s fate until the next drawing, and this with “incalculable consequences”. Like the narrator, one might end up a proconsul, or one might end up a slave… until the next drawing, unless of course the draw mandates death!

This is one extreme example of enforced social mobility of course, but the author, intriguingly, ends the story by saying that the lottery has become so much part of life in Babylon that one is not any more sure it even ever existed. Could our lives also be determined by a secret society of Fates?

In our more prosaic lives as well, chance, or misfortune, can befall any of us without warning. While one may think one’s in control of one’s destiny, a lot of our decisions have unpredictable consequences. Yet, one must decide one way or the other. In most of the literature on decisions in a risky setting, chance only affects oneself. However, more recently, some authors have started to investigate the social dimension of risk, that is, whether perceptions of a risk may be affected not only by how that risk affects oneself, but also by how it affects others (Bolton and Ockenfels, 2010Brennan, González, Güth and Levati, 2008Charness and Jackson, 2009Güth, Levati and Ploner, 2008Bradler, 2009Harrison, Lau, Rutström and Tarazona-Gómez, 2013Linde and Sonnemans, 2012Rohde and Rohde, 2011)

This matters because people are sensitive to how their wealth compares with that of others, so that a catastrophe that affects all in the same way, such as a tsunami in a coastal city in Japan, will have a different impact than a risk that affects only oneself, such as when one’s house burns down.

In a new working paper, Social preferences under uncertainty, I decided to investigate how individuals consider different types of social lotteries by using the simplest experimental design I could think of. I got a number of experimental subjects (humans of the student variety) to choose between different allocations of wealth between themselves and an anonymous other, some of those allocations being subject to chance.

The main variables I was interested in were as follows:

• If the allocation is subject to chance, does a bad outcome for me (low payoff) also mean a bad outcome for the others?
• If there is an alternative to risk, such that I can get a payoff for sure and the other as well, does that safe alternative guarantee me as much as the other, less, or more?

I was expecting, based on earlier literature, that correlation in payoffs (the first aspect) would not influence choice that much. That actually did not fit with my intuition, as I thought people would dislike negatively correlated social payoffs most. I also expected that people would be more ready to take risk if that meant avoiding a safe but subordinate alternative. Because many people do not like inequality even if it favours themselves, I also expected people would be more ready to take risk to avoid dominant safe alternative, maybe out of a sense of fairness.

In the event, I did find that people did not like their payoffs to be correlated with that of others, though the effect was small, and that they indeed did not like inequality in terms of the safe alternative. This means they were more likely to choose a risky but fair lottery giving expected payoff of 45 ECU for both when the alternative was a safe but unequal payoff than when it was a safe and equal payoff.

The issue is that while dislike for correlated lotteries can be explained through altruism (people do not like others to bear risk even if themselves do bear risk), dislike for unequal safe payoffs can be explained through inequality aversion, which goes against altruism in some cases. Indeed, altruism mandates preferring higher payoffs for the other no matter what, while inequality aversion means that one prefers the other to have less if the other has more than oneself!

This generates some issues when generalizing the social preferences as modeled in Fehr and Schmidt (1999) or Bolton and Ockenfels (2000) and in a lot of the literature on choice in social settings, as one cannot be at the same time altruistic and inequality averse. Why were people not happy for the other to get a higher payoff than themselves in the safe lottery? Among others, Brock, Lange and Ozbay (2013)Cappelen, Konow, Sørensen and Tungodden (2010) and Krawczyk and Le Lec, (2010) in particular suggested fairness considerations had to play a role. I used a suggestion in Fudenberg and Levine (2012) whereby utility would be a mix of utility for ex-ante, expected payoffs and ex-post social outcomes from the lottery. I later found out this idea was used in recent theoretical papers by Krawczyck (2011) and Saito (2012).

The formula for utility of lottery $L=(a,\frac{1}{2};b,\frac{1}{2})$ takes the form $U_{L}=\lambda(\frac{1}{2}u(a)+\frac{1}{2}u(b))+(1-\lambda)u(\frac{1}{2}a+\frac{1}{2}b)$

that is, utility is $\lambda$ times the expected utility of the lottery, with the utility of different outcomes possibly influenced by the utility of the other in that situation, so $u(me,you)=u(me)-\alpha|u(you)-u(me)|$ for example, plus $1- \lambda$ times the utility of the expected value of the lottery, maybe influenced by the expected value of the lottery for the other.

An individual would thus judge a lottery along two dimensions: Is it fair? Are the outcomes correlated or not with those of the other? I did find that people’s choices were consistent with them putting a high weight on ex-ante fairness ($1-\lambda$ high), but also taking account of the type of risk borne in the lottery. However, I was still not able to reconcile subjects’ preferences among lotteries with my modelling, suggesting that maybe individuals do indeed have different social preferences under risk than under certainty.

In terms of experimental methods, one of my issue with standard ways to elicit preferences is that, when offered the choice between two lotteries, subjects can generally only say: “I prefer this one”, or “I prefer this other”, but not by how much. Some experiments allow people to express indifference, and some, such as Connolly and Butler (2006) ask people to express the strength of their preferences or emotions associated with a lottery, though not in an incentive compatible way. Peter Moffatt in his excellent book Experimetrics (forthcoming at Palgrave Macmillan ) explains how far expressed strength of preferences can indeed be used to refine estimates, but as he mentions along Grether and Plott (1979), “task related incentives are the bedrock of theories under test”, so that tests of theories “cannot be taken seriously in the absence of non negligible task-related incentives”. I had the idea to make people pay to increase the probability to obtain their preferred lottery. In principle, they would pay only up to the point where their preferred lottery minus the price would be indifferent vs. the less preferred lottery. I think I am the first to implement such a system, which ended up working nicely, and allowed me to have more precise and robust estimates of subjects’ preferences (see the econometric section). However, this made the experiment more difficult to understand for subjects.

Another issue I had with other experiments on the topic was how the choices offered to the subjects looked too much like each other so that subjects could quite easily apply decisions made in one setting to another setting. For example, if they keep being offered a lottery that gives either 20 or 80 with probability ½ and ½, while the payoff of the other varies between choice instances, then they might just economize on the thinking cost and always make the same decisions without considering the payoff of the other. I therefore randomized payoffs in my lotteries, at the cost however of not being able to compare decisions directly across different situations, and therefore having to adopt a parametric approach to identifying subjects’ preferences (i.e. postulating a model of choice, estimating its parameters and looking at its consequences in terms of choice among different types of lotteries).

For more details, the paper is available at SSRN and at RePEC.