Social interaction effects: The impact of distributional preferences on risky choices

This paper identifies convex distributional preferences as a possible cause for the empirical observation that agents belonging to the same group tend to behave similarly in risky environments. We first show theoretically that convex distributional preferences imply social interaction effects in risky choices in the sense that observing a peer choose a risky (safe) option increases the agent’s incentive to choose the risky (safe) option as well, even when lotteries are stochastically independent and the agent can only observe the lottery chosen by the peer but not the corresponding outcome. We then confirm our theoretical predictions experimentally. Electronic supplementary material The online version of this article (10.1007/s11166-018-9275-5) contains supplementary material, which is available to authorized users.

Definition 1. Suppose that in the two-player case lottery L n , with n = 1, 2, ..., N, yields outcome x n with probability p n and outcome zero with probability 1 p n . Let 0 < p 1 < p 2 < ... < p N 1 < p N  1 and 0 < x N < x N 1 < ... < x 2 < x 1 , i.e. a lottery with a lower index is riskier than a lottery with a higher index. Let u ln denote the expected utility of the DM if he chooses lottery L l while his peer chooses L n .
If the DM has a utility function as given in Equation (1), u ln can be expressed as u ln = p l x l (1 ⇢) + p n x n + p l p n (⇢ ) min{x l , x n } (A1) For the (K > 2) player case, that is, for a social environment in which there are ⇤ Corresponding author. Address: University of Innsbruck, Department of Economics, Universitätsstrasse 15, A-6020 Innsbruck, Austria. Email: anita.gantner@uibk.ac.at.
K 1 > 1 peers, the Charness and Rabin function extends to: where max{⇢, } < 1 and where o k is the material payo↵ of peer k 2 {1, ..., K 1}. 1 Definition 2. Suppose that in the K > 2 player case we have N 2 lotteries as in Definition 1. Let c n be the fraction of peers that choose lottery n. Let u l (c) denote the expected utility of the DM if he chooses lottery L l while his peers choose according to c = (c 1 , c 2 , ..., c N 1 , c N ).
Then u l (c) = P N n=1 c n u ln , and using the utility function as given in Equation (A2) Proof. First note that regardless of whether the DM has convex, linear, or concave distributional preferences, we have u l (c) = u s (c) for l s. To see this, note 1 Here we use the convention that the DM is agent K. Note that dividing through K 1 ensures that the relative impact of distributional concerns on the DM's utility payo↵ is independent of the number of players in his reference group. See Fehr and Schmidt (1999) for a similar normalization.
2 that u s (c) = p s x s (1 ⇢) + P N n=1 c n p n x n + p s (⇢ )x s P s n=1 p n c n (when the DM also chooses L s ), and u l (c) = p l x l (1 ⇢) + P N n=1 c n p n x n + p l (⇢ )x l P s n=1 p n c n (when the DM chooses L l with l > s). Then u l (c) u s (c) = (p s x s p l x l )[1 ⇢ + P s n=1 p n c n (⇢ )] = 0, since p s x s = p l x l . Next, we show that for s > l > r we have u s (c) > u l (c) > u r (c) if the DM has convex distributional preferences, u s (c) = u l (c) = u r (c) if the DM has linear distributional preferences, and u s (c) < u l (c) < u r (c) if the DM has concave distributional preferences. This is easily seen by noting that Thus, for s > l and p n x n = , 8n we have For a DM with convex (linear; concave) distributional preferences this equation is strictly positive (zero; strictly negative) since ⇢ > 0 (⇢ = 0; ⇢ < 0) and p s x s = p n x n > p l x n for p n > p l . The argument for l > r is similar.
Proposition A1 tells us that in a world with N binary lotteries with equal expected values and K agents, a DM with convex (concave) distributional preferences is risk-averse (risk-loving) when comparing lotteries that are more risky than the safest lottery chosen by a strictly positive fraction of peers, but risk-neutral when comparing lotteries that are less risky.
Proposition A2. (Distributional Preferences and Social Interaction Effects with N Lotteries and K Agents). Suppose a DM with a utility function as given in Equation (A2) is indi↵erent between two lotteries L r and L s with r < s when he observes that the peers choose according to c = (c 1 , ..., c r , ..., c s , ..., c N ).
Then observing at least one of the peers switch from L s to L r (L r to L s ) and no peer switch in the opposite direction implies the following orderings over the two lotteries for the DM: If his distributional preferences are (i) convex then L r L s (L s L r ); (ii) linear then L r ⇠ L s ; (iii) concave then L s L r (L r L s ).
Proof. If at least one peer switches from L s to L r (L r to L s ) and none switches in the opposite direction, let this be denoted byĉ = (ĉ 1 , ...,ĉ r , ...,ĉ s , ...,ĉ N ), wherê c n = c n for n 6 = r, s, andĉ r > c r (ĉ r < c r , respectively), andĉ s = c s ĉ r + c r . We show that forĉ r > c r the di↵erence [u r (ĉ) u s (ĉ)] [u r (c) u s (c)] is strictly positive for convex, strictly negative for concave, and zero for linear distributional preferences. This is easily seen by noting that Thus, we have Since for s > r we have x r > x s and p s > p r , the sign of (⇢ ) yields the desired results.
A.2 Proof for the Remark following Proposition 2 (Common Gambles) Proof. To see that Proposition 2 extends to the case of common gambles, note that with perfect correlation u rr = p r x r and u ss = p s x s , while u rs and u sr remain as given in (3) and (4). Then, for case (i) of Proposition 2 we have to show that (a) and (b) above hold for common gambles. For (a), we have u rr = u sr () , which in particular implies that p r x r < p s x s , since 1 > 1 ⇢ + p r (⇢ ) for ⇢ > . Thus, while being risk-neutral with independent gambles when the peer chooses the riskier lottery, a DM with convex distributional preferences is risk-loving with common gambles. For (a) to hold, note that u rs < u ss () p s x s [1 (1 p r ) p r ⇢] > p r x r (1 ⇢), which is true since the DM was shown to be risk-loving, i.e. p r x r < p s x s and it was also shown that 1 The DM is then risk-averse in his choice between lotteries if the peer chooses the safer lottery, just as in the independent gambles case. To complete the argument for the case of convex distributional preferences, we note that condition u rr > u sr is equivalent to , which is satisfied since 1 > 1 ⇢+p r (⇢ ) for ⇢ > , and p r x r > p s x s by risk aversion. The argument for the other two cases of Proposition 2 follows similar lines.

Appendix B:
Extended Model (Departure from Risk-Neutrality) Suppose the DM values his own monetary payo↵ m according to the von-Neumann-Morgenstern utility function v(m) when acting in isolation. In order to avoid imposing unnecessary structure on v(.), we exploit the fact that in the basic model the DM decides between lotteries whose support consists of only three monetary payo↵s: x r , x s and 0. We can thus normalize the utility associated with two payo↵s and use the number representing the utility of the third payo↵ as an index for the DM's risk attitude. Let v(m) = m for m 2 {0, x r } and v(x s ) = v. Then for v < x s the DM is risk-loving, and a larger di↵erence x s v means more risk-loving; for v = x s the agent is risk-neutral; and for v > x s the agent is risk-averse, and a larger di↵erence v x s means more risk-averse. In a social comparison context we assume that the DM is risk-neutral regarding the di↵erence between his own and his peer's payo↵, but may display other risk attitudes regarding his own material payo↵s, as described above. Then the DM's preferences can be represented by the function where To preserve monotonicity of the agent's utility with regard to his own material payo↵ we impose the following parameter restrictions: 3 Given the functional form from (A4), the DM's expected utility u ln from choosing lottery L l , with l = r, s, while his peer chooses lottery L n , with n = r, s, is Given that the peer chooses L r , the DM now prefers L s over L r if and only if u sr > u rr . Using (A7) and (A5), this simplifies to Given that the peer chooses L s , the DM now prefers L s over L r if and only if 2 There are other plausible ways of extending the functional form (1) to lotteries -see Fudenberg and Levine (2012), or Saito (2013), for instance. We chose this simple form mainly for parsimony and transparency, and to stay as close as possible to our baseline functional form (1). With this functional form, a DM's trade-o↵ between the first component of the utility function, which accounts for his preferences over the own monetary payo↵, and the second component, which accounts for social preferences, is distorted by the DM's risk attitude regarding his own payo↵, but this distortion remains within narrow bounds, as we keep the highest and lowest possible payo↵ untransformed. u ss > u rs . Using (A6) and (A8), this simplifies to: Conditions (A9) and (A10) extend those found earlier for an agent who is riskneutral in isolation -it is easy to see that for v = x s they correspond precisely to our earlier conditions (6) and (7). To identify the impact of a change in the choice of the peer on the incentives of a DM behaving in accordance with the functional form (A4) with preference parameters ⇢, and v, we calculate -for given values of x r , x s and p s and given choice of the peer -the critical probability p ⇤ r for which the DM is just indi↵erent between lotteries L s and L r . We shall refer to p ⇤ r as the DM's "indi↵erence probability". Let p ⇤ r (L r ) denote the value of the DM's indi↵erence probability when the peer chooses L r , and p ⇤ r (L s ) the corresponding value when the peer chooses L s . From (A9) and (A10) it is immediately seen that for ⇢ , implying that the indi↵erence probability of a DM with linear distributional preferences is not a↵ected by the peer's choice. For ⇢ 6 = , condition (A9) yields the following equation: .

(A11)
Thus, for ⇢ 6 = the value for p ⇤ r (L r ) is the solution to a quadratic equation, which, in general, is not unique. However, taking the parameter restrictions R1 and R2 for monotonicity of utility in own material payo↵ into account, it can be shown that only the solution with the positive root is admissible.
Turning to the case where the peer chooses L s , condition (A10) yields the following equation: To evaluate the impact of a change in the peer's choice on the incentives of a DM with given distributional parameters ⇢ and and given risk attitude v in isolation, we define the di↵erence in indi↵erence probabilities as d(⇢, , v) = p ⇤ r (L s ) p ⇤ r (L r ).

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Then d(⇢, , v) > 0 indicates that if the peer chooses L s , the DM will require a larger p r to prefer L r over L s , compared to the situation where the peer chooses L r , i.e., he will choose L s for a larger range of probabilities. Thus, with d(⇢, , v) > 0 the DM has the tendency to follow the peer, with d(⇢, , v) < 0 the DM has the tendency to deviate from the behavior of the peer, and with d(⇢, , v) = 0 the DM's choice is not a↵ected by the decision of the peer. Furthermore, the absolute value of d(⇢, , v) is a measure for the size of the impact of a change in the peer's choice on the DM's indi↵erence probability.
In the proof below, we show that for ⇢ > the di↵erence in indi↵erence probabilities d(⇢, , v) is positive for each admissible value of v; furthermore, d(⇢, , v) is convex and it obtains its (unique) minimum at v = x s . By contrast, for ⇢ < the di↵erence in indi↵erence probabilities is negative for each admissible value of v; furthermore, d(⇢, , v) is concave in this case and it obtains its (unique) maximum at v = x s . Together with the observation that for ⇢ = we have p ⇤ r (L s ) = p ⇤ r (L r ), these findings imply the following result: Proposition A3. (Distributional Preferences and Social Interaction Effect with General Risk Attitudes in Isolation) Suppose a DM whose preferences can be represented by a utility function as in Eq. (A4) is indi↵erent between lotteries L r and L s when he observes the peer choose L s (L r ). Then observing the peer choose L r (L s ) instead implies the following orderings over the two lotteries for the DM: If his distributional preferences are (i) convex then L r L s (L s L r ); (ii) linear then L r ⇠ L s ; (iii) concave then L s L r (L r L s ).
Furthermore, for given distributional preference parameters ⇢ and , the size of the impact of a change in the peer's choice on the DM's indi↵erence probability is smaller for a DM who is risk-neutral in isolation than for a DM who is risk-averse or risk-loving in isolation.
Proof. To show that for ⇢ > the di↵erence in indi↵erence probabilities d(⇢, , v) is positive for each admissible value of v, we first take the derivative of d(⇢, , v) with respect to v. We find the derivative to be Setting this derivative equal to zero, we find a unique solution at v = x s . Now we just need to evaluate d(⇢, , v) at v = x s . If its value is positive, we know that d(⇢, , v) has a minimum at v = x s , and thus d(⇢, , v) is always positive.
Inspection of the numerator in the above equation shows that it can be rewritten as 2p 2 s x s (x r x s )(⇢ ) 2 , which is positive for any ⇢, . Inspection of the denominator shows that if ⇢ > , then the expression in the denominator is positive. Thus, for ⇢ > , the term d(⇢, , v) is positive. If, instead, ⇢ < , then the first term in the denominator is negative, and for the second term (in brackets) it must be that ⇢ < 1 < 1 ⇢ since ⇢ < 1. Then x r (1 ⇢) > p s x s (⇢ ), since x r > p s x s and 1 ⇢ > ⇢ . In this case, the denominator is negative and thus, for ⇢ < , the term d(⇢, , v) is negative.  Figure A1 illustrates the results for convex distributional preferences using the lottery parameters of the experiment. 4 Figure A1 displays the shape of d(⇢, , v) for various types of convex distributional preferences. For each curve, ⇢ and are kept fixed and d(⇢, , v) is plotted as a function of v, where we allow for arbitrary values of v within the admissible range. On the left graph of Figure A1, we fix ⇢ at some positive value and vary , allowing for positive values (such parameter combinations correspond to convex altruism) as well as negative values (such parameter combinations correspond to inequality aversionà la Fehr and Schmidt 1999). On the right graph we fix ⇢ at some negative value and vary , allowing only for values < ⇢ to ensure convexity (such parameter combinations correspond to spiteful preferences). Each curve shows the change in the DM's indi↵erence probability when the peer's decision moves from risky to safe. Note first that for all cases of convex distributional preferences, the level of the curve d(⇢, , v) is positive, i.e. the DM's indi↵erence probability is larger if the peer chooses the safe alternative, and smaller if the peer chooses the risky alternative. In other words, for a risky choice of the peer the DM requires a smaller probability for the favorable outcome x r to occur in order to prefer the risky alternative. The DM thus follows the peer's behavior. Note further that, as stated in Proposition A3, the U-shaped curves all have their minimum for a risk-neutral DM, i.e. the social interaction e↵ect is smallest here, while it is larger the more risk-averse or riskloving a DM is.

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Remember that in the experiment subjects are exposed to choices where the safer lottery contains no risk at all. Here it is important to note that all our results hold in particular for p s = 1, and we prefer to use a secure payo↵ in the experiment since it makes the decision less complex for subjects.  Figure A2: Presentation of a Decision Pair in Part 2 of the Experiment 11 Appendix D: Experimental Instructions

General Information
You are participating in two experiments on decision making. A research foundation has provided the funds for conducting the experiment. You can earn a considerable amount of money by participating. The text below describes exactly how your total earnings will be determined. For better comprehension only male pronouns are used; they should be understood as gender neutral.

Anonymity:
Your decisions remain anonymous. Neither the experimenters nor other subjects will be able to link you to any of your decisions. In order to keep your decisions private, please do not reveal your choices to any other participant. Each participant will only be informed about his own earnings, but not about the earnings of other participants.

Procedure:
First, Experiment 1 will be described and completed, and then we proceed to Experiment 2. At the beginning of each experiment you will receive precise instructions. We will read the instructions aloud and you will have time for questions. Please do not hesitate to ask questions if there is need for clarification. Upon completion of both experiments you will be paid out your total earnings.

Experimental Currency:
You earnings will be given in Talers in both experiments. At the end of both experiments, Talers will be converted into Euros, and you will be paid the Euro amount in cash. A Taler corresponds to 10 Cents, that is 10 Talers correspond to 1 Euro.

No Private Communication:
Please do not talk to other participants. If you have questions after reading the instructions or during the experiment, please raise your hand and one of the experimenters will come to clarify your question.

Experiment 1
In Experiment 1 another participant is assigned to you as your passive partner. Your decisions in this experiment have consequences for you own earnings as well as for the earnings of your passive partner. Your passive partner, however, cannot affect your earnings. A random decision of the computer will determine who your passive partner is, and you will not know his identity at any point in time. Your passive partner will also never be informed about your identity.
Decisions and Pairs of Alternatives: You will make a total of 10 decisions in Experiment 1. Each of your decisions is a choice between alternative LEFT and alternative RIGHT. Each alternative is a distribution of Talers between you and your passive partner.
Example: You may be asked whether you prefer alternative LEFT, in which you receive 15 Talers and your passive partner receives 30 Talers, or alternative RIGHT, in which you receive 20 Talers and your passive partner also 20 Talers. You have to make a choice between these two alternatives. This decision problem would be represented in the following way: Your earnings from Experiment 1 will be determined in the following way: Earnings as active participant: At the end of the experiment one of the 10 decision problems is selected by a random draw made separately for each participant, and the alternative chosen in this decision problem is paid out in real money. Each random draw is made publicly from a bingo cage with ten numbered balls (with numbers from 1-10). All numbers are equally likely to be drawn. The number on the ball drawn for you determines the decision problem from which you receive your payoff as active participant. This number is entered in the computer, and the computer assigns the corresponding payoff to you. For example, if the decision problem above were chosen for you, and if you had chosen alternative LEFT, then you would receive 15 Talers as active participant, while your passive partner would receive 30 Talers from his role as a passive partner.
Earnings as passive partner: Just like your passive partner receives money from your decision without doing anything, you receive money from another participant without doing anything, i.e. you are the passive partner of some other participant. The computer program ensures that you are not assigned the same person as active participant and passive partner. That is, if your decision determines the payoff of participant x, then the decision of participant x will not determine your payoff, but that of another participant.
To summarize: In Experiment 1 you will make 10 decisions, one of which will determine your actual payoff as active participant. At the time when you make your decisions, you will not know which of the 10 decisions will determine your payoff from the role of an active participant. All of your decisions are equally likely to be drawn for determining the payoff. In addition to your earnings in the role of an active participant you will also receive earnings from the role of a passive partner. You cannot affect the amount of your earnings in the role of a passive partner; it depends exclusively on the decisions of another participant. The computer program ensures that you are assigned different partners in both roles.
The draws for determining the earnings from Experiment 1 will be made after Experiment 2 is completed.

Experiment 2
Your decisions in Experiment 2 have consequences for your own earnings only; they do not affect the earnings of other participants. This is also true for all other participants: they can only affect their own payoffs but not the payoffs of any other participant.
Decision rounds and pairs of alternatives: Experiment 2 consists of 30 decision rounds. In each round, you will see a pair of alternatives on the screen. All participants see the same pair of alternatives in each round. Each pair of alternatives consists of Alternative A and Alternative B. Alternative A is always a sure payoff, Alternative B is always a lottery.
Active and passive role: In some rounds, you will see pairs of alternatives which you have already seen in previous rounds. In such rounds, some participants have an active role, and others have a passive role. Participants in an active role have to make a new decision. Participants in a passive role do not make a new decision; the computer automatically implements the decision they made the first time they saw this pair of alternatives. In rounds with active and passive roles you will be informed at the beginning of a round which role is assigned to you.

Information:
When you see a pair of alternatives for which you have already made a decision in a previous round, the screen will display which decision you made the first time you saw this pair of alternatives. If you are in an active role, you will also be informed about the decision some other participant made the first time he saw this pair of alternatives. This information will be displayed on your screen. The other participant is now in the passive role, that is, the computer automatically implements the same decision that he made the first time he saw this pair of alternatives. Thus, you know precisely how this other participant decides in the current round. This other participant remains anonymous for you, that is, you will not get to know his identity at any point in time. As mentioned above, your earnings in Experiment 2 do not depend on the decisions of other participants, but only on your own decisions.
Your task in a decision round: If you are in an active role in a given round, you will be asked to make a decision between the two alternatives. You may change your decision as long as you have not clicked the "confirm" button. If you click "confirm", you will go to the next round with a new pair of alternatives. If you are in a passive role in a given round, the computer will implement your previous decision for this pair of alternatives. You are only asked to click "confirm" in order to get to the next round.
Your earnings from Experiment 2 will be determined at the end by using the following two-stage procedure: Stage 1: From a bingo cage with 30 numbered balls (numbers from 1-30) a ball will be drawn, visible to anyone in this room. All balls are equally likely to be drawn. The number on the drawn ball determines the round that will be paid out. Therefore, the round that determines earnings from Experiment 2 is the same for all participants.
Stage 2: A separate random draw follows for each participant. The bingo cage now contains 20 numbered balls (numbers from 1-20) for each draw. Again, all balls are equally likely to be drawn. The number on the ball is the personal lucky number of the respective participant. If this participant has chosen Alternative A in the round that is determined to be paid out (in Stage 1), then he receives the corresponding sure payoff independent of his personal lucky number (since Alternative A is always a sure payoff). If, instead, the participant has chosen Alternative B in that round, then his personal lucky number determines his payoff as shown in the following example: Example: Suppose that in Stage 1 the number 7 is drawn. This means that for all participants round 7 will be paid out. Suppose that in this round the following pair of alternatives was presented: Alternative Your earnings from Experiment 2 depend on how you decided in this round. If you chose Alternative A, then you receive a total of 50 Talers from Experiment 2 independent of your personal lucky number. If you chose Alternative B, then your earnings from Experiment 2 depend on your personal lucky number. Suppose that in Stage 2 the number 3 was drawn as your personal lucky number. In this case you receive 100 Talers if you chose Alternative B, since your personal lucky number is between 1 and 6. Suppose another participant also chose Alternative B in this round, but the number 8 was drawn as his personal lucky number. Then he receives 0 Talers.