DP2020-23

Risks on Others

Takaak i HAMADA Tomoh i ro HARA

August 20, 2020

Risks on Others∗

Takaaki Hamada† Tomohiro Hara‡

August 20, 2020

Abstract

We investigate other regarding preferences when others are involved in some risks. We introduce

two concepts to examine risk attitudes towards others: “absolute level of risk attitudes toward others”

(ARAO) which capture general risk attitudes toward recipients’ risky payoffs, and “relative level of

risk attitudes toward others” (RRAO) which capture behavioral difference depending on types of risks

conditional on same expected outcomes. For RRAO, we compare two different types of risks: “state risk”

where recipients’ initial endowments are risky and “non-state risk” where recipients’ states are not risky

but transfers are risky. We provide a novel experimental design to measure both ARAO and RRAO,

and theoretically explore potential mechanisms behind behaviors based on these concepts by extending

representative inequality aversion models under risks. In our experiment, we find decision makers exhibit

robust risk averse behaviors on ARAO, which cannot be predicted by existing theories. Yet, we find no

strong evidence on RRAO.

Keywords: Other-regarding preferences, charity, risk preferences, cognitive bias

∗This project is funded by JSPS Grand-in-Aid for JSPS Research Fellows (17J07819). We thank Emel Filiz-Ozbay, ErkutOzbay, Akihiko Matsui for helpful suggestions. All errors are our own.

†Research Institute for Economics and Business Administration, Kobe University, 2-1 Rokkodai, Nada, Kobe 657-8501,Japan/Faculty of Management and Administration, Shumei university, 1-1 Daigakucho, Yachiyo-shi, Chiba 276-0003, Japan.Email: t.hamada.econ@gmail.com

‡Department of Economics, University of Maryland at College Park. email: tomohara@umd.edu.

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1 Introduction

Suppose you are asked to donate some cash to poor people from an unknown charitable organization. Even if

you are nice person and willing to contribute to poor people, you may become skeptical about that unknown

organization, thinking “does this organization actually give my donation to poor people?” People are likely

to be unwilling to give to unfavorable group, and these phenomena are well-documented in economic and

psychological literature. To name a few recent works, Gneezy, Keenan and Gneezy (2014) document that

people are less likely to donate when they find their donation is used for administration rather than directly

helping the poor, and Karlan and Wood (2016) find people would like to give more charity if they know the

charity organization is more effective. These findings imply people care about receivers’ quality when they

donate. In the real world, however, people cannot observe full information on others’ quality. Instead, we

are frequently uncertain on whether others are worth giving or not. Research in other-regarding preferences

does not provide clear views on how people incorporate these sorts of uncertainty.

In this paper, we interpret such a situation as “risks on others” (hereafter ROO), and attempt to uncover

how people evaluate these types of risks. Not only charities, ROO appears wide range of decisions. A

college professor may not supervise a student well when he is skeptical on the student’s performance. People

may provide less efforts for volunteering activities when they have suspicions that they are working for

those who are not suffering. In addition, we regard wide range of racial/ethnic/gender discrimination as

ROO. For instance, when an employer cannot observe full information on an employees productivity, they

infer statistical properties of the productivity from their group identities (e.g gender and ethnicity) and treat

employees differently by groups. This is a classical story of statistical discrimination which is well-documented

in labor economics (see surveys such as Neumark (2018)). Here, we can regard unobservable productivity

that is attached to others’ group affiliations as ROO. It is also well-known that statistical discrimination

and taste-based discrimination are closely related (e.g. Bertrand and Mullainathan (2004)), and taste-based

discrimination can be seen as one pertern of other-regarding preferences. However, previous literature could

not theoretically analyize interaction between taste and statistical discrimination. Understanding ROO can

highlight potential mechanism bridging those concepts.

Though there are enormous numbers of studies conducted to investigate behavior on charitable giving,

only a few recent papers investigate the relationship between charitable giving and risk attitudes (e.g. Brock,

Lange and Ozbay (2013), Saito (2013) and Lopez-Vargas (2014)). Moreover, those papers analyze situations

where transfers are risky, and very little attention has been given to the risk and uncertainty on those

receiving transfers. The main theme of this paper is to provide a novel experimental setting to answer

following questions. How does a decision maker react to ROO in a situation where she wants to transfer

some amount to a risky recipient? How do we theoretically conceptualize ROO, and how do we measure it?

To answer above questions, we firstly define and characterize risk preferences toward others. By doing

so, we are able to to evaluate decision makers’ attitudes (i.e. risk averse for others, risk loving for other, and

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risk neutral for others).

Second, we design a new experiment to identify risk attitude. It is a modified version of a standard

dictator game summarized in Figure 1. Here, we consider a dictator game with three players, a dictator

(denoted D in the figure), a certain recipient (denoted C), and a risky recipient (denoted R). We have two

treatment arms, but let’s just consider the first treatment (“A: State risk treatment”) at this stage. The

lottery p is a safe lottery, where a dictator is going to give $5 to a certain recipient who has ec as an initial

endowment. In the lottery q, the risky recipient has no endowment with probability 0.9, but has $5 with

probability 0.1.

D𝑒" = 10

𝑥 C𝑒' = 0.5 VS D

𝑒" = 10

R 90%𝑒*+ = 0

R 10%𝑒*, = 5

Lottery : p

D𝑒" = 10

C𝑒' =0.5 VS D

𝑒" = 10

Third party

R𝑒* = 0

Gives 5 in 10%

A: State risk treatment

Lottery : p

Lottery : q

Lottery : r

B: Non-state risk treatment

𝑥

𝑥

𝑥

Figure 1: Simplified design of experiment

Fixing the parameters on lottery q, we ask dictators to find an endowment ec under which lottery p and q

are indifferent. We apply the Holt and Laury (2002) type of price list to elicit the point which is indifferent on

giving to the certain recipient (C) and giving to the risky recipient (R). Our list starts with ec = 0 and ends

at ec = 5 with some increments in the example of Figure 1. As we discuss carefully on theoretical background

later, we expect to observe that people generally prefer to give to a certain recipient in the beginnings of the

list and then switch to give to a risky recipient at some point. We refer this point as “switching point” in

the entire paper. This list elicitation enables us to understand RRO.

Theoretically analyzing the feature of our experiment, we predict that dictators behave as if they are risk

lovers in five out eight decisions under standard axiomatized models on inequality aversion (Fehr and Schmidt

(1999) and Saito (2013)). That is, switching points from “giving to a certain recipient” to “giving to a risky

recipient” are predicted to be lower than the point where the risky recipient’s expected initial allocation

is equal to the certain recipient’s. However, in the actual experiment that we conducted at the computer

laboratory at the University of Maryland at College Park in 2018, we find strong risk aversion toward others.

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Results suggest that well-used Fehr and Schmidt (1999) and its applications cannot capture people’s decisions

on other’s allocation when recipients surface risks. Surprisingly, there is no theoretical nor experimental paper

evaluate ROO in general: as far as we know, all the previous models rely on linear specification in Fehr and

Schmidt (1999), but our results cannot be explained without allowing some curvatures on the utility function.

Bellemare, Kroger and Van Soest (2008) is an exception using ultimatum game to estimate inequality aversion

with more general form of utility function allowing curvature. Yet, they analyze subjective expectation under

ultimatum game, and do not explicitly examine and conceptualize risks on others.

In addition to the above main experiment, we also attempt to measure what we call “relative level of risk

attitude toward others” (RRAO). Given same level of risks, decision makers may behave differently toward

different types of risks. We are going to explicitly distinguish risks attached to others’ states from risks which

attached to transfers but not others’ states1.

To investigate RRAO, we use the second treatment arm in Figure 1. Dictators are asked to compare the

lottery p to the lottery q in the “state-risk treatment”, and the lottery p to the lottery r in the “non-srtate risk

treatment”. Those two treatments are randomly assigned between subjects. In the lottery q (i.e. state-risk

treatment), the risky recipient has no endowment with probability 0.9, but has $5 with probability 0.1. In

the lottery r (i.e. non-state risk treatment), the risky recipient has no initial endowment for sure, but has

a chance to receive $5 with probability 0.1 from a third party (the experimenter). Therefore, for those two

treatment arms, decision makers confront exactly same level of ROO, but may behave differently due to the

diffence of others’ states.

Theoretically speaking, we expect to observe that dictators behave in same manner for two treatments

under existing theories. This is because all the existing models only consider final outcomes, but not initial

states. In the theory section, we provide two potential mechanisms where people behave differently in those

two treatments: preferences for others’ initial endowment, and subjective probabilities on others’ states.

Yet, as a result from our experiment, we could not find differences between two treatment. That is, at

least for our experiment, we cannot find any evidence that people care other’s risky initial endowment to

change their behavior. However, this cannot be the conclusion that ROO does not matter for people’s decision

making in general. There are a lot of existing studies showing people care whether others belong to unfa-

vorabale group or not: one potential interpretation is that interacting with risks might reduce attention to

unfavorable group. Identifying how and when people care RRAO may be interesting topic for future research.

The rest of the paper proceeds as follows. Section 2 explains the detail design of the experiment. In

Section 3, we presents the theoretical model to analyze our experimental environment, as well as differences

with other existing literature. Section 4 provides the results from our experiment as well as interpretation of

the results, and Section 5 provides the concluding remarks.

1ROO can take various forms, and we just compare two examples. One can think of different types of ROO and evaluateRRAO separetely.

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2 Expermental design

The experiment has 2 treatments. For each treatment, subject complete 8 tasks to elicit their preferences.

After completing the main task, subjects solve another task for another study (that is not reported in this

paper), and then answer simple demographic questions.

The main tasks are the modified version of a dictator game played by three subjects, one dictator and

two recipients. Summarized design of the game is reported in the introduction (Figure 1). A dictator always

endows $10. One of the recipients is always involved in uncertainty (call her a risky recipient), while the

other is never involved in uncertainty (call her a certain recipient). We randomly assigned one out of those

three roles in the beginnings of the game without telling to the subjects, and subjects play all the games as

if they are the dictator. In the end of the experiment, subjects are notified of their roles.

In each task, subjects make decisions on 11 binary choices allocating a portion of his endowment to either

the certain or the risky recipient. The screen shot of the actual experiment (for the state-risk treatment,

which is explained soon) is shown in Figure 2. We use the list elicitation method which is similar to Holt

and Laury (2002) or Exley (2016): fixing the amount of transfers and settings of the risky recipient, we

increase the initial endowment of the certain recipient (ec) as we go down the list. In the first half of the

decision page, we provide the experimental environment. We instructed that numbers highlighted by pink

changes over tasks. Fixing the amount the dictator gives to recipients ($4 in the example), we ask dictators

to which recipients they prefer to give that amount. In the example in Figure 2, the list starts from dictator

decides to give to either the certain recipient (person B) with no endowment or the risky recipient (person

C)2 who has 6 with 10% chance and has 0 with 90% chance. In the other words, dictators confront a trade-off

between giving to risky and certain recipients, which enable us to identify dictators’ absolute risk attitudes

toward others (ARAO). As the dictator keeps making decisions and going down the list, the endowment of

the certain recipient increases. We show in the Appendix A.2 that under the standard inequality aversion

theories, dictators prefer to give to the certain recipient in the beginnings of the list, and then switch to the

risky recipient at some point. We also theoretically show that there exists at most one switching point from

certain to risky recipient. This property, as well as the results from online pilot experiment ensure the note

in the middle in Figure 2 saying “Many people prefer to give to Person B in the beginnings of the list and

then switch to give to Person C at some point. Therefore, one way to complete this list is to explore the best

row to switch one option to another”.

To understand the relative risk attitudes toward others (RRAO: if risks depend on other’s states or not),

we use a framing. There are two treatment arms in this experiment: “state-risk treatment” and “non-state

risk treatment”. Making all the other conditions equal, the state-risk treatment imposes a risky recipient to

endow $erH with probability 1 − p and $0 with probability p. In contrast, in the non-state risk treatment,

a risky recipient has no initial endowment for sure, but has a chance to receive $erH from a third party

2Note that when we explain our experimental settings in Figure 1, we denote a dictator as person D, a certain recipient asperson C, and risky recipient as person R.

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Figure 2: The screen shot of the actual experiment

(the experimenter). A certain recipient has different initial endowment ec for sure. As Eckel, Grossman

and Johnston (2005), people prefer to give poor people (i.e. low endowment) than rich people (with high

endowment). For the state-risk treatment, dictators may think that they strongly want to avoid giving to

rich recipient, while for the non-state risk treatment, dictators may think that they do not have to strongly

avoid those who are poor but receieve some transfer from outside by luck.

Table 1 summarizes the initial endowments and payoffs in both treatments, and Table 2 reports the list

of parameters in our experiment. We vary (i) the amount of fixed transfer, (ii) endowment of risky-rich, and

(iii) probability of being rich.

As you may notice, both ex-ante and ex-post earnings are same in those treatments. However, if there

exists some preferences for other’s state (in this case, initial endowment), the amount a dictator wants to

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Table 1: Summary of risky task

Dictator (D) Certain Recipient (C) Risky Recipient(R)Panel A: State-risk treatmentEndowment 10 ec 0 with prob. p

erH with prob. 1− p

Transfer from third party 0 0 0

Final allocation when 10−X ec +X 0 with prob. pD gives X to C. erH with prob. 1− p

Final allocation when 10−X ec X with prob. pD gives X to R. X + erH with prob. 1− p

Panel B: Non-state risk treatmentEndowment 10 ec 0

Transfer from third party 0 0 0 with prob. perH with prob. 1− p

Final allocation when 10−X ec +X 0 with prob. pD gives X to C. erH with prob. 1− p

Final allocation when 10−X ec X with prob. pD gives X to R. X + erH with prob. 1− p

Table 2: Summary of parameters in the experiment

Pair Transfer (x) Endowment of risky-rich (erH) probability of risky-poor (p)Pair 1 5 6 0.9Pair 2 5 6 0.8Pair 3 5 12 0.9Pair 4 3 6 0.9Pair 5 3 6 0.8Pair 6 3 12 0.9Pair 7 5 6 0.95Pair 8 5 12 0.95

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give to a risky recipient may change between those treatments. While we do not explicitly mention the

context, the treatment mimic the example in the introduction: let’s consider you are going to give to donate

to unknown charitable organization. In the state-risk treatment, you think that the organization is actually

rich and do not have to gather money with (say) 10% chance, and that makes you feel that you probably do

not want to donate to that organization. In the non-state risk treatment, on the other hand, you know the

organization is poor, but have 10% chance to receive good amount of money by winning lottery. In terms

of both final outcomes in each realization of event and expected earnings are same in those two situations,

people may behave differently.

To understand how people evaluate ROO, one can think of simpler designs. Alternative designs are

summarized in Figure 3. First, consider the simplest dictator game with risks in Figure 3-(1), where a

dictator decide to give some of his endowment to risky recipient. We may be able to directly compare how

much dictators give in each treatment. While this design seems to be reasonable, it involves motivation of

an excuse not to give to others which may interact with risks. As Exley (2016) shows in her experiment,

people use others’ risks for excuses with self-serving motivation. It is unclear in this design to determine how

excuse interact with state-risk and non-state risk, and becomes obstacle to identify risk preferences in each

treatment. In contrast, our experiment fixes amounts that dictator can give for each round, so that we can

exclude that the excuse to become the driving factor3.

Second, we can consider another experiment with a simpler lottery in Figure 3-(2) based on outcomes.

That is, we may ask a dictator to decide lottery p′ with allocation (me, other) = (6, 4.5) for sure versus a

lottery q′ with probabilistic allocations, (me, other) = (6, 4) with 90% chance and (me, other) = (6, 9) with

10% chance. While this choice seems to be much simpler to analyze because there are only two agents,

interpreting the decision makers choice does not make sense when we want to investigate the importance

of ROO: when other’s state is risky, choosing between above two lotteries means the decision maker is also

choosing other’s state itself. This can be interesting topic for future research, while it is out of scope for our

research question. Our experimental design and theoretical model, on the other hand, explicitly prepare a

safe agent and risky agent, and asks a decision maker to decide who to give. That is, DM does not have to

decide other’s state because of the presence of both safe and risky agents. We explicitly model trade off such

that giving to one recipient means not giving to the other recipient, and how that helps to identify the risk

preferences.

3For selfish dictators, fixing the amount that they can give to recipients implies dictators are indifferent for any choices inour experiment. Our results suggest this is not the case, because dictators change the switching points by risky recipients’ risks.

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D𝑒" = 10

R 90%𝑒&' = 0

R 10%𝑒&( = 5

(1) Simple dictator game with state-risk

Lottery : p Lottery : q’

(2) Choice by outcome-base

𝑥+,-./&

(Me, Other)=(6, 4.5) 90%: (Me, Other)=(6, 4)10%: (Me, Other)=(6, 9) VS

Figure 3: Potential alternative designs

3 Hypothesis

3.1 Set up

In this section, we theoretically examine our experimental environment, discuss how we evaluate ARAO

and RRAO, and explore potential mechanisms driving the results. Let a = (aD, aC , aR) be final allocations

of three subjects including each initial endowment, where subject D is the dictator, and C and R are the

certain and risky recipients. Let e = (eD, eC , eR) be the vector of initial endowments of three subjects. For

example when a dictator allocates X to recipient C, the final allocation a = (eD −X, eC +X, eR). Dictator

has preference over lotteries for (a, e), generally denoting a lottery as F (a, e). In our experiment we ask

participants to choose one of followings in each treatment:

State risk :

FSC (ec) ≡ {p : ((eD −X, ec +X, 0)︸ ︷︷ ︸

final allocation

, (eD, ec, 0)︸ ︷︷ ︸initial

), 1− p : ((eD −X, ec +X, erH)︸ ︷︷ ︸final allocation

, (eD, ec, erH)︸ ︷︷ ︸initial

)}

FSR(ec) ≡ {p : ((eD −X, ec, X), (eD, ec, 0)), 1− p : ((eD −X, ec, erH +X), (eD, ec, erH))}

(1)

Non− state risk :

FNC (ec) ≡ {p : ((eD −X, ec +X, 0), (eD, ec, 0)), 1− p : ((eD −X, ec +X, erH), (eD, ec, 0))}

FNR (ec) ≡ {p : ((eD −X, ec, X), (eD, ec, 0)), 1− p : ((eD −X, ec, erH +X), (eD, ec, 0))}

(2)

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For each T ∈ {S,N}, where S means state risk treatment and N non-state risk treatment, FTC (ec) and

FTR (ec) are lotteries conditional on giving recipient C and R respectively when the initial endowment of

recipient C is ec. A dictator is said to have a standard social preference if and only if there exists a unique

cutoff eTc ∈ (0, erH) such that FTR (eTc ) ∼ FT

C (eTc ), FTC (ec) � FT

R (ec) for any ec ∈ [0, eTc ), and FTR (ec) � FT

C (ec)

for any ec ∈ (eTc , erH ]. This means dictator’s preference reverses from FTC (ec) � FT

R (ec) to FTR (ec) � FT

C (ec)

as ec increases from 0 to erH with just one switching point eTc ∈ (0, erH). As we see in the experimental

design this may be intuitive, and also supported by representative social preference models. We prove

dictators following Saito (2013) have a standard social preference in Appendix A.2.

Now we formalize two concepts on ROO in our experimental environment: absolute level of risk attitude

toward others (ARAO) and relative level of risk attitude toward others (RRAO). First, let us consider the

ARAO in each treatment. Consider two lotteries FTC (er) and FT

R (er) in each T , where er is an expected

earning of a risky recipient without transfer from a dictator (i.e. er = p0 + (1 − p)erH). Then in both

treatments a recipient C has er for sure, and a recipient R is expected to have er, so that their (expected)

earnings before transfer are same between two lotteries. Fixing a decision problem between FTC (er) versus

FTR (er), a dictator is said to be (i) risk averse, (ii) risk neutral, (iii) risk loving if and only if (i)FT

C (er) �

FTR (er), (ii)F

TC (er) ∼ FT

R (er), (iii)FTR (er) � FT

C (er). Then under the assumption of the standard social

preference, eTc > er is equivalent to FTC (er) � FT

R (er), and eTc < er is equivalent to FTR (er) � FT

C (er).

Prediction 1 Suppose dictators have a standard social preferences. Dictators are (i)risk averse, (ii)risk

neutral, (iii)risk loving, if and only if (i) eTc > er, (ii) eTc = er, (iii) e

Tc < er for each T ∈ {S,N}.

Second we consider the RRAO by comparing two treatments, which only differ by a state of a risky recip-

ient (i.e. if they have initial endowment in small probability or not). A dictator is said to have independent

preference of recipient’s initial endowment if and only if F (a, e) ∼ F (a, e′) for any a, e and e′ (e 6= e′). In

the definition note that final allocations are same between two lotteries, and the only difference is the initial

endowment. Dictators who have this type of preference make the same choice between our two treatments.

As a consequence, we obtain following for RRAO.

Prediction 2 Suppose dictators have a standard social preferences. If dictators have independent preference

of recipient’s initial endowment, then eSc = eNc . Equivalently if eSc 6= eNc , then there exist some dictators who

do not have independent preference of recipient’s initial endowment.

When we observe eSc > (<) eNc , we can intuitively say dictators are more risk averse (loving) in state risk

treatment than in non-state risk treatment because dictators are more (less) likely to prefer giving to a certain

recipient in state risk treatment. Outcome-based social preference models (Fudenberg and Levine (2011),

Saito (2013), Lopez-Vargas (2014)) can not predict eSc 6= eNc . The potential mechanisms behind that will be

discussed in section 3.3.

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3.2 Relation to previous literature

There are some theoretical and experimental studies on the intersection of social preferences and risk. Ex-

perimental evidence by Brock et al. (2013) shows that when we take into account the risk and uncertainty

toward other people, we consider both ex-ante and ex-post fairness. This implies it might not be sufficient

just to consider the expected utilities on formally proposed other-regarding preferences, such as Fehr and

Schmidt (1999), Bolton and Ockenfels (2000), and Charness and Rabin (2002). This problem has been the-

oretically studied in Fudenberg and Levine (2011), axiomatized by Saito (2013), and generalized to more

flexible functional form by Lopez-Vargas (2014), Rau and Muller (2017).

While many useful models on the social preference and risk have been developed recently, they may not

make effective predictions for our experiment. First, for the prediction 1, the most representative model

Saito (2013) predicts dictators are risk loving in 5 out of 8 treatments, which is not intuitive and we shall see

later is not observed in our experiment. So far less attention has been given to the risk attitude on “other’s

earnings” in existing studies while some studies focus on the relation between DM’s risk attitude for “own”

payoff and attitude for inequality (Lopez-Vargas (2014), Rau and Muller (2017)). We need to extend existing

models to describe people’s risk attitudes for other’s earnings, and in Section 3.3.1, we theoretically challenge

to capture them.

Second, for the prediction 2, existing models predict eSc = eNc since they all are outcome-based theory.4

On the other hand, we expect eSc 6= eNc to be observed through other’s initial endowment effect. Many

experiments of non-risky decision environments show people have a preference for other’s initial endowment,

states or types; for people prefer to give to known charitable organizations over anonymous others (Eckel

and Grossman (1996)), knowing others’ social status than not knowing (Charness and Gneezy (2008)), poor

than rich (Eckel et al. (2005)), and even artificially created “minimal group” members in lab than non-

members(Chen and Li (2009)). From these papers it’s naturally inferred that ROO has some peculiar effects

on people’s behavior. To our best knowledge, Fong and Oberholzer-Gee (2011) is the only exception that

incorporate risks for unfavorable agents. Recipients of their dictator game was a disabled group (preferred

to give) and a drug users group (not preferred to give). They asked dictators how much they want to give

with and without uncertainty. When confronting an uncertain situation, they provide an option to get rid

of uncertainty, and they find a significant amount of dictators paid for that option. However, Fong and

Oberholzer-Gee (2011) did not explicitly state the degree of uncertainty (i.e. dictators did not know the

proportion of favorable group is in the uncertain treatment). Consequently, they also could not identify if

the composition of the unfavorable group matters or risk itself matters when people were willing to pay to

avoid risks. Our experimental design can clearly identity the effect of other’s states, and we also explore the

potential mechanisms behind the behavioral difference toward each risk from the view points of preference

and bias for other’s initial endowment in section 3.3.2.

4Note that in our experiments final outcomes are exactly same in both treatments.

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3.3 Model Analysis: Potential Mechanisms

Using a theoretical model, we analyze our experimental environment and explore the potential mechanisms

behind the behavior. Let u(a, e) be the utility of a dictator for final allocations and initial endowments (a, e),

and assume the following additive separable form:5

u(a, e) = v(aD) +∑

j∈{C,R}

gj(aD − aj), (3)

where v : R+ → R+, gj : R → R−. We assume gj(0) = 0, g′j(x) < 0 if x > 0 and g′j(x) > 0 if x < 0, which

are basic properties of inequality aversion. One of the most representative form of u is the following function

suggested by Fehr and Schmidt (1999):

uFS(a, e) = aD +∑

j∈{C,R}

[− αj max[aD − aj , 0]− βj max[aj − aD, 0]

], (4)

where αj ≥ 0, βj ≥ αj and βj ≤ 1 for each j. Following Fudenberg and Levine (2011), Brock et al. (2013)

and Saito (2013), assume the utility for lotteries depend on both ex-ante and ex-post fairness concerns. We

assume that there exists a function U(y, z) with ∂U(y,z)∂y > 0, ∂U(y,z)

∂z > 0, and decision maker’s utility for

lottery F is described as follows:

U(F ) = U(u (EF [a, e]) ,EF [u(a, e)]), (5)

where EF is an expectation operator by F (a, e). One special case of (5) is Saito (2013)’s axiomatized form:

USaito(F ) = δuFS(EF [a, e]) + (1− δ)EF [uFS(a, e)], (6)

where δ ∈ (0, 1), which captures a relative weighting for ex-ante fairness.

3.3.1 Discussion for prediction 1: ARAO

In this section we apply the general model (3) and (5). And assume that gC = gR, which may be reasonable

since dictators are matched with anonymous recipients.6 First we show the following property.

Lemma 1 Consider lotteries FTC (er) and FT

R (er) in each treatment T . Suppose a dictator follows the form

(3) and (5), and gC = gR. Then FTC (er) % FT

R (er) if and only if EFTC (er)[u(a, e)] ≥ EFT

R (er)[u(a, e)].

This means that a preference for lotteries FTC (er) and FT

R (er) are characterized by expected utility, which

comes from the equivalency of FTC (er) and FT

R (er) w.r.t. ex-ante fairness under the assumption of gC = gR,

5While function u depends on the initial endowments e, it does not appear in the form explicitly. We denote u as a functionof e because the final allocation a depends on e, and e itself may affect the function form g (See section 3.3.2).

6Note that this may not hold in state risk treatment if dictators has a preference for other’s initial endowment, but at leastcan be applied to non-state risk treatment.

12

and the monotonicity of U .7 This is one notable property of our experimental design. Lemma 1 implies the

following.

Lemma 2 Suppose a dictator follows the form (3) and (5), and gC = gR ≡ g. Then the dictator is (i)risk

averse, (ii)risk neutral, (iii)risk loving if and only if (i)G > 0, (ii)G = 0, (iii)G < 0, where

G ≡ EFTC (er)[u(a, e)]− EFT

R (er)[u(a, e)] = g(eD − 2X − er) + pg(eD −X) + (1− p)g(eD −X − erH)

−g(eD −X − er)− pg(eD − 2X)− (1− p)g(eD − 2X − erH) (7)

We can see function G in lemma 2 is rewritten as follows:

G ≡ EFTC (er)[u(a, e)]− EFT

R (er)[u(a, e)] = g(eD − 2X − er)− pg(eD − 2X)− (1− p)g(eD − 2X − erH)︸ ︷︷ ︸net expected utility of giving C

−{g(eD −X − er)− pg(eD −X)− (1− p)g(eD −X − erH)︸ ︷︷ ︸net expected utility of not giving C

} (8)

Dictator’s risk attitude is essentially characterized by functional form of g. To see this, as examples, let us

take two functional form of g and draw figures under a parameters’ pair in our experiment, (eD, X, erH , p) =

(10, 5, 6, 0.9).

! (= !$ = !%)

'( − 2+ − ',-(= −6)

/( − /0

1

2

3

4

5 67

'( − 2+ − 8',(= −0.6)

'( − + − ',-(= −1)

'( − +(= 5)

'( − + − 8',(= 4.4)'( − 2+

(= 0)

!('( − + − 8',) > ?!('( − +) + (1 − ?)!('( − + − ',-)

!('( − 2+ − 8',) = ?!('( − 2+) + (1 − ?)!('( − 2+ − ',-)

Figure 4: (Case 1) linear assumption in each domain

Case 1 is Fehr and Schmidt (1999)’s linear form. The first term of G in (8) is the difference between the

value of g when aD − aj is surely eD − 2X − er and the expected value of g when aD − aj is eD − 2X with

7Let us remind you that the expected final outcomes of FTC (er) and FT

R (er) are (eD−X, er+X, er) and (eD−X, er, er+X),

respectively. Then gC = gR implies u(EFT

C(er)

[a, e])= u

(EFT

R(er)

[a, e]).

13

prob p and eD − 2X − erH with prob 1− p. Since expected values of gap aD − aj are same, when g is Fehr

and Schmidt (1999)’s form, we can see both of them are expressed at the level of point M in figure 4, so

that the first term of G is 0. Similarly the second term of G in (8) is the difference between the value of g

when the gap is surely eD −X − er and the expected value of g when the gap is eD −X with prob p and

eD −X − erH with prob 1 − p. When g is Fehr and Schmidt (1999)’s form, the farmer one is expressed at

the level of point J and later one is at the point K in figure 4, so that the second term of G is positive.

Therefore G is negative, and we can see the dictator is risk loving by Lemma 2. In this case, the linearity of

g in envy domain makes the net expected utility of giving certain recipient C small (zero),8 which results in

the dictator’s risk loving behavior.

! (= !$ = !%)

'( − 2+ − ',-(= −6)

/( − /01 23

'( − 2+ − 4',(= −0.6)

'( − + − ',-(= −1)

'( − +(= 5)

'( − + − 4',(= 4.4)'( − 2+

(= 0)

!('( − + − 4',) > ;!('( − +) + (1 − ;)!('( − + − ',-)

!('( − 2+ − 4',) > ;!('( − 2+) + (1 − ;)!('( − 2+ − ',-)

=

>

?

@A

Figure 5: (Case 2) concave in envy domain, and linear in guilt domain

In case 2, g is exactly same as case 1 in guilt domain, that is linear form, and so is the utility level of g

at the point N in envy domain. The difference is the curvature of g in envy domain; g is concave form. The

first term of G in (8) is the difference between values of g at the point M and O, which is positive. That

comes from the concavity of g in envy domain. The second term of G in (8) is the difference between values

at the point J and K, which is positive but smaller than in case 1. Since by figure 5 the first term of G is

larger than the second one, the dictator is risk averse in this example. We can see more concave form of g

in envy domain, intuitively more risk averse in envy domain, makes the net expected utility of giving certain

recipient C larger, which makes dictators more risk averse.9 Bellemare et al. (2008) find in their experiment

that their subjects have a increasing and concave utility function in envy domain using ultimatum game

8More precisely, the more convex in the envy domain, the smaller net expected utility of giving C is.9The functional form in guilt domain also affects the risk attitude. Especially the more convex form in guilt domain, the

value of G becomes larger, which results the dictator becomes more risk averse.

14

under certain decision environment. Risk averse behaviors in our experiment may provide an evidence for

the importance of this result when we treat people’s risk attitude toward other’s allocation.

Now let us uncover the interaction of Saito (2013) and ARAO. The following is a characterization result

on the risk attitude by Saito (2013)’s form.

Proposition 1 Suppose eD −X ≥ X and eD −X < erH +X. And suppose a dictator follows the form (6),

and gC = gR(anonymity). Then the dictator is (i)risk averse, (ii)risk neutral, (iii)risk loving if and only if

(a)er < eD − (1 + p)X, (b)er = eD − (1 + p)X, (c)er > eD − (1 + p)X.

Proof: See Appendix A.3. �

Proposition 1 implies the risk attitude of dictators following the form (6), in our experiment is as follows:

Pairs eD X erH p risk attitudePair 1 10 5 6 0.9 risk lovingPair 2 10 5 6 0.8 risk lovingPair 3 10 5 12 0.9 risk lovingPair 4 10 3 6 0.9 risk aversePair 5 10 3 6 0.8 risk aversePair 6 10 3 12 0.9 risk aversePair 7 10 5 6 0.95 risk lovingPair 8 10 5 12 0.95 risk loving

Table 3: Prediction of risk attitude from the form (6)

We can see under the assumption of Fehr and Schmidt (1999)’s linear form dictators are predicted to be

risk loving in 5 pairs out of 8 in our experiment. From these, we obtain the following.

Prediction 1-1 Suppose a dictator follows the form (5), and gC = gR. Then Fehr and Schmidt (1999)’s

form predicts Table 3. If they are not observed, that implies dictators do not follow Fehr and Schmidt (1999)’s

form.

3.3.2 Discussion for prediction 2: RRAO

To have prediction 2, we consider two potential mechanisms: preference for other’s state and subjective

probability caused by differences in states. As a benchmark, dictators who follow the form (5) care only the

final outcomes a = (aD, aC , aR). Thus, as we see in previous section, dictators choose the same action in

both treatments, which implies eSc = eNc . Now we discuss the potential effect of recipient’s initial endowment

which realizes eSc 6= eNc . First let us introduce the concept of preferences for other’s state using Fehr and

Schmidt (1999)’s form.

Definition 1 Suppose a dictator’s utility for certain allocations and initial endowments is represented by

15

(4). Dictator has a preference for other’s initial endowment if there exists some ej and e′j (ej 6= e′j) such that

αj(ej) 6= αj(e′j) and/or βj(ej) 6= βj(e

′j). (9)

As we show multiple examples in the introduction, people may prefer giving others whose initial endow-

ment is low to giving others whose initial endowment is high even if final allocations are same. This situation

can be described by the definition 1, in particular when αj(ej) > αj(e′j) and βj(ej) < βj(e

′j) for e′j > ej .

10

This means guilt concern is weakened for others who have higher initial endowments, while envy concern is

strengthened. If a dictator has this kind of preference, he/she wants to avoid giving a recipient whose initial

endowment is high, which implies they are less likely to prefer risky recipient in the state-risk treatment than

in the non-state risk treatment. If many dictators in our experiment have this kind of preference, we can

observe eSc > eNc .

Prediction 2-1 Suppose a dictator has a preference for other’s initial endowment. Then eSc 6= eNc can be

observed. Especially, suppose αj(ej) > αj(e′j) and βj(ej) < βj(e

′j) for e

′j > ej, then eSc > eNc can be observed.

Proof: See Appendix A.4 for the proof of second statement. �

Another potential mechanism is subjective probability. Though each dictator knows the objective probabil-

ity, he/she may form her own subjective probabilities which differ by other’s states. This captures a decision

maker’s psychological evaluation for the true objective probability, which concept comes from probability

weighting in prospect theory (Kahneman and Tversky (1979)).

Definition 2 Let π(s, ej) be dictator’s subjective probability of recipient’s initial endowment being ej, where

s is an objective probability. Dictator’s subjective probability depends on other’s initial endowments if there

exists ej , e′j (ej 6= e′j) such that

π(s, ej) 6= π(s, e′j).

As one possible example, dictators may think rich one is more likely to be their recipients, that is π(s, e′j) >

π(s, ej) for e′j > ej . This probability weighting follows the idea of a rank-dependent utility developed by

Quiggin (1982) and Schmeidler (1989). 11 If a dictator forms this kind of subjective probability, he/she

wants to avoid giving a risky recipient in other treatment than in comp treatment since he/she perceives

larger chance to feeling envy in other treatment.

Prediction 2-2 Suppose dictator’s subjective probability depends on other’s initial endowments. Then eSc 6=

eNc can be observed. Especially, if π(s, e′j) > π(s, ej) for e′j > ej, then eSc > eNc can be observed.

Proof: See Appendix A.5 for the proof of second statement. �

10Cox, Friedman and Sadiraj (2008) introduce this formulation as an example of a concept “more altruistic than(MAT)”.11However note that we cannot naturally extend their arguments to inequality aversion models, and this is one of the theoretical

challenges.

16

3.4 Summary of predictions

Sum up above predictions as follows. First two predictions presented here state general features of individuals’

behaviors, while the later three predictions state potential mechanisms behind dictators’ behaviors.

Prediction 1 Suppose dictators have a standard social preferences. Dictators are (i)risk averse, (ii)risk

neutral, (iii)risk loving, if and only if (i) eTc > er, (ii) eTc = er, (iii) e

Tc < er for each T ∈ {S,N}.

Prediction 1 predicts dictators’ bahavior for ARAO. Elicited cutoffs (eTc ) directly map to risk preferences for

others in absolute term in our experiment.

Prediction 2 Suppose dictators have a standard social preferences. If dictators have independent preference

of recipient’s initial endowment, then eSc = eNc . Equivalently if eSc 6= eNc , then there exist some dictators who

do not have independent preference of recipient’s initial endowment.

Prediction 2 predicts behviors for RRAO. That is, we expect to observe same outcome for two different

treatments (state-risk and non-state-risk treatments) under standard social preferences in previous literature,

and we expect to observe behavioural differences if ROO matters.

Prediction 1-1 Suppose a dictator follows the form (5), and gC = gR. Then Fehr and Schmidt (1999)’s

form predicts Table 3. If they are not observed, that implies dictators do not follow Fehr and Schmidt (1999)’s

form.

Above predition considers potential mechanism behind Prediction 1 for ARAO. Analyzing potential behavior

expected from Fehr and Schmidt (1999) and its applications, we expect to observe risk loving behavior for 5

out of 8 tasks. If we cannot observe such behaviors, it implies that people do not follow Fehr and Schmidt

(1999). We presented alternative modification which predict risk averse behaviors by allowing curvatures for

the functional form.

Prediction 2-1 Suppose a dictator has a preference for other’s initial endowment. Then eSc 6= eNc can be

observed. Especially, suppose αj(ej) > αj(e′j) and βj(ej) < βj(e

′j) for e

′j > ej, then eSc > eNc can be observed.

Prediction 2-2 Suppose dictator’s subjective probability depends on other’s initial endowments. Then eSc 6=

eNc can be observed. Especially, if π(s, e′j) > π(s, ej) for e′j > ej, then eSc > eNc can be observed.

Those two predictions show potential mechanisms behind Prediction 2. The first prediction analyzes the case

when dictators have preference on other’s initial endowment. The second prediction considers the case when

dictators have subjective risk attitudes toward others, and the attitudes differ by others’ initial endowments.

17

4 Results

The experiment was conducted in October and November in 2018 at the computer laboratory in the Uni-

versity of Maryland at College Park. We had 16 sessions in total with 192 subjects in total (12 participants

by session and 96 participants by treatment), and all the participants were the undergraduate students at

the University of Maryland. Including the instruction, participants spent 30 to 45 minutes at the laboratory

earning $18.11 on average which include the participation fee of $7. The maximum earnings in our experi-

ment was $25.25, and the minimum earnings was $9.512.

Main results are presented in Figure 6. For each small figure, we present the turning points of decisions (i.e.

when DMs switch who to give, from a safe receiver to a risky receiver). Clear (white) bars represent choices

in the non-state risk treatment, and green bars represent choices in the state-risk treatment. Vertical red

dashed lines represent the expected value of a risky recipient’s expected endowment (er = perL+(1− p)erH)

for each lottery. We exclude those who exhibit inconsistent behavior predicted from our theory. That is, we

exclude those who have more than two switching points and those who give to risky recipient in the first

choice in a list and switch to risky recipient at some point. Because of the instruction which repeatedly

mentions people are likely to start from certain recipient and switch the recipient only once throughout each

list, 85% of DMs are consistent with our theory.

As we can clearly see from the graphs, there are two main findings. First, for ARAO, we find strong risk

aversion for all sets of parameters. More precisely, turning points switching from certain recipients to risky

recipients are higher than the “expected value” which is calculated from er = p ∗ erH . As we discussed in

Section 3.3.1 and summarized as Prediction 1-1, standard theories on the inequality aversion involving risks

cannot explain this phenomenon: strong risk aversion observed from the results suggest that we need to modify

existing theories such as Saito (2013). As we investigate the feature of the utility function, one potential

solution is to allow curvature for the utility function as suggested in Bellemare et al. (2008)13. Note that this

result is not likely to be observed from the “random” choices given by subjects. If subjects are completely

selfish and do not care other’s allocation, choices in our experiment do not matter for subjects because we fix

the amount of transfer (and therefore amounts for dictator’s final allocation). Conditional on final allocation

and the proportion of high-type risky recipients, we examined differences between distributions. That is, we

compare distributional differences between (i) (a) vs (b) vs (c), (ii) (d) vs (e) vs (f), and (iii) (g) vs (h),

which suggests that decision makers behave differently, and thus they care and try to avoid risks on others.

Second, for RRAO, we could not observe clear differences between two treatments. Not only from those

12To fully control the environment, we restrict the number of participants to 12 per each session. To guarantee this number,we invited more than 12 subjects per session, and paid the participation fee of $7 for those who came to the lab later than the12th subjects

13We cannot back-up the functional form of the utility from this experiment, because there is no variation in own allocation.That is, we fixed own allocation as well as the amount of transfer in our experiment. This is going to be a trade-off betweenallowing potential to excuse (Exley (2016)) not to give to others using risks on others. We believe this is going to be an interestingresearch agenda for future.

18

(a) (x, erH , 1− p) = (3, 6, 10) (b) (x, erH , 1− p) = (3, 6, 20) (c) (x, erH , 1− p) = (3, 12, 10)

(d) (x, erH , 1− p) = (5, 6, 5) (e) (x, erH , 1− p) = (5, 6, 10) (f) (x, erH , 1− p) = (5, 6, 20)

(g) (x, erH , 1− p) = (5, 12, 5) (h) (x, erH , 1− p) = (5, 12, 10)

Figure 6: Main results: DMs’ decisions by each parameter

graphs, regression analyses in Table 4 and simple t-test checking differences of means in Table 5 all suggest

that we cannot reject the hypothesis that DM behave differently in those two treatments. These results imply

that at least in our experiment, we could not observe particular effect on RRAO, when we regard high initial

endowment as unfavorable situation. This may be because people only care about final outcome as currently

assume in all the existing literature, or “high initial endowment” maybe not unfavorable enough to avoid.

Therefore, it is too early to conclude that RRAO does not matter for people’s decision making. We believe

that dictating and quantifying importance of RRAO will be future research topic.

19

Table 4: Regression analysesDependent variable: turning point of DM’s choice

(1) (2) (3) (4)Other Risk 0.128 0.00700 -0.0640 -0.155

(0.222) (0.430) (0.212) (0.345)

Endowment of Rich 0.462∗∗∗ 0.453∗∗∗ 0.464∗∗∗ 0.461∗∗∗

(0.0244) (0.0361) (0.0225) (0.0321)

Prob. being Rich 0.0529∗∗∗ 0.0544∗∗∗ 0.0565∗∗∗ 0.0550∗∗∗

(0.00777) (0.0118) (0.00571) (0.00801)

Endowment of Rich× Other-Risk 0.0187 0.00722(0.0489) (0.0450)

Prob. Rich× Other-Risk -0.00296 0.00279(0.0155) (0.0114)

[1em] Constant -0.657∗∗ -0.596 -0.473∗ -0.426(0.241) (0.348) (0.206) (0.264)

Observations 1217 1217 1133 1133R2 0.275 0.275 0.335 0.335

Standard errors in parentheses∗ p < 0.05, ∗∗ p < 0.01, ∗∗∗ p < 0.001

Table 5: Means and differences of each experiment

Parameters (x, er, p) State risk Non-State risk Difference(3, 6, 10) 3.03 (0.134) 2.68 (0.164) 0.349 (0.211)N 81 73(3, 6, 20) 3.01 (0.134) 3.02 (0.156) -0.007 (0.206)N 78 74(3, 12, 10) 5.96 (0.330) 5.48 (0.334) 0.49 (0.463)N 78 73(5, 6, 5) 2.24 (0.178) 2.28 (0.164) -0.04 (0.242)N 77 81(5, 6, 10) 3.04 (0.168) 2.96 (0.183) 0.079 (0.249)N 76 76(5, 6, 20) 3.18 (0.158) 3.20 (0.155) -0.008 (0.221)N 73 73(5, 12, 5) 4.92 (0.325) 4.81 (0.301) 0.11 (0.464)N 77 73(5, 12, 10) 5.62 (0.283) 5.568 (0.327) 0.048 (0.432)N 75 75

Standard errors are in parentheses.

20

5 Concluding remarks

In this paper, we provide an experiment to capture risk on others (ROO). While ROO appears in the

wide range of economic activities, such as charitable giving and discrimination, previous literature has not

investigated theoretical and empirical properties of ROO. We introduce two concepts to examine risk attitudes

towards others: “absolute level of risk attitudes toward others” (ARAO) which captures general risk attitudes

toward recipients’ risky payoffs, and “relative level of risk attitudes toward others” (RRAO) which captures

behavioral difference depending on types of risks conditional on same expected outcomes.

We introduce the modified dictator game with three agents (a dictator, safe and risky recipients) to

identify both ARAO and RRAO. In this environment, the dictator confronts trade off between giving to safe

and risky recipients. Our goal in this experiment is to find the point that dictators switch from preferring to

give to certain recipient to risky recipient (so that dictators are indifferent to give to either recipients). From

this structure of the experiment, we can identify risk attitudes in general toward others, which is ARAO.

We compare this value between two treatments to evaluate ROO: “state-risk treatment”, where the risky

recipient has high initial endowment with small probability (which is generally unfavorable under inequality

averse preference) and “non-state risk treatment”, where the risky recipient does not have any endowment for

sure but has some small chance to receive some transfer from a third party. Comparing those two treatments,

we can evaluate how people react to different types of risks, and identify RRAO.

There are two findings from our experiment conducted at the University of Maryland at College Park

in 2018. First, for ARAO, we find that people are consistently risk averse that they do not want to give

to others with risk. The results are inconsistent with existing theoretical model such as Fehr and Schmidt

(1999) and Saito (2013). We suggest to allow curvature on the utility function as suggested in Bellemare et

al. (2008), instead of the linear specification in Fehr and Schmidt (1999) so that we can address the issue.

Second, For RRAO, we did not find any difference between two treatments. That is, under our experimental

condition, it is likely that people evaluate “state-risk treatment” and “non-state risk treatment” in same

manner. This result stems from two potential reasons. First, people may just care outcomes, and do not

care initial condition. This is consistent with the previous theories investigating the interaction between

risk preferences and other-regarding preferences, such as Saito (2013). Second, people may not regard “high

initial endowment of other” as unpreforable. While some experiment (e.g. Eckel and Grossman (1996), Eckel

et al. (2005)) find that people dislike to give to those who are rich, some other research (e.g. Chen and Li

(2009) ) shows the parameter “envy” is much smaller than “guilt” for the Fehr-Schmidt utility function. To

understand the importance of RRAO, one may conduct some new experiment which has more specific context

on unpreferable groups. While we fail to detect difference between treatments, our experimental design has

a strength that we can evaluate unpreferable group in monetary term.

21

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A Proofs of Propositions

A.1 Preliminary

The form (4) is rewritten as follows:

uFS(a, e) = aD +∑

j∈{C,R}

(βj − σ(aD − aj)(αj + βj)

)(aD − aj), (10)

23

where σ(x) = 1l(x ≥ 0). Using (10) and (6), the utility for lottery FTj (ec) (j ∈ {C,R}, T ∈ {S,N}), which is

defined in section 3.1(1)(2), is rewritten as follows:

USaito(FTj (ec)) = (eD −X) +

(βC − σ(d(C, j))(αC + βC)

)d(C, j)

+ δ(βR − σ(EFT

j (ec)[d(R, j)])(αR + βR))EFT

j (ec)[d(R, j)] (11)

+ (1− δ)EFTj (ec)

[(βr − σ(d(R, j))(αr + βr)

)d(R, j)

],

where d(k, j) = (eD − X) − akj with akj being an final allocation of recipient k ∈ {C,R} when dictator

gives X to j ∈ {C,R}. The difference between αR and αr is whether they are random variable or not. αR

is dictator’s guilt parameter to risky recipient R, which is not random variable. On the other hand, αr is

αrL with prob p and αrH with prob 1− p, where rH means recipient R when he/she has erH initially or by

transfer, and rL means recipient R when he/she has erL(= 0) initially or by transfer. Thus αr is random

variable. βr and βR are defined similarly.

Now define V (ec|T ) ≡ USaito(FTC (ec))− USaito(FT

R (ec)) for each T ∈ {S,N}. Then we can see

V (ec|T ) = vC(ec|T )− δvR(er|T )− (1− δ)E[vr(er|T )], (12)

where

vC(ec|T ) ≡ −βCX +(d(C,R)σ(d(C,R))− d(C,C)σ(d(C,C))

)(αC + βC),

vR(er|T ) ≡ −βRX +(EFT

C (ec)[d(R,C)]σ(EFTC (ec)[d(R,C)])− EFT

R (ec)[d(R,R)]σ(EFTR (ec)[d(R,R)])

)(αR + βR),

vr(er|T ) ≡ −βrX +(d(R,C)σ(d(R,C)− d(R,R)σ(d(R,R)))

)(αr + βr),

equivalently,

vC(ec|T ) =

αCX if ec < eD − 2X

αC(eD −X − ec) + βC(eD − 2X − ec) if eD − 2X ≤ ec ≤ eD −X

−βCX if eD −X < ec

(13)

vR(er|T ) =

αRX if er < eD − 2X

αR(eD −X − er) + βR(eD − 2X − er) if eD − 2X ≤ er ≤ eD −X

−βRX if eD −X < er

(14)

24

vr(er|T ) =

αrX if er < eD − 2X

αr(eD −X − er) + βr(eD − 2X − er) if eD − 2X ≤ er ≤ eD −X

−βrX if eD −X < er

(15)

where er is erL with prob p, and erH with prob 1−p. vC(ec|T ) is a net utility from giving certain recipient

C who surely has ec, vR(er|T ) is a net utility from giving risky recipient R in the sense of ex-ante fairness,

and vr(er|T ) is a net utility from giving recipient R in the sense of ex-post fairness.

A.2 Existence and uniqueness of the cutoff

In the proof we use the notations and forms in subsection 3.3 and Appendix A.1. It is an essence to the proof

that vC(ec|T ), which is a net utility from giving certain recipient C who surely has ec, is a decreasing in ec.

Proposition 2 Suppose erL = 0, eD − 2X ≥ 0, eD − 2X < erH and eD −X > er. And suppose a dictator

follows Saito (2013)’s form (6), and αC = αR = αr ≡ α, βC = βR = βr ≡ β for any r ∈ {rL, rH}

(anonymity). Then there exists unique cutoff eTc ∈ (0, erH) such that FTR (eTc ) ∼ FT

C (eTc ).

Proof: By (12)(13)(14)(15) and anonymity, we can see

V (ec|T ) ≡ USaito(FTC (ec))− USaito(FT

R (ec)) = vC(ec|T )− δvR(er|T )− (1− δ)E[vr(er|T )], (16)

where

vC(ec|T ) =

αX if ec < eD − 2X

α(eD −X − ec) + β(eD − 2X − ec) if eD − 2X ≤ ec ≤ eD −X

−βX if eD −X < ec

(17)

vR(er|T ) =

αX if er < eD − 2X

α(eD −X − er) + β(eD − 2X − er) if eD − 2X ≤ er ≤ eD −X

−βX if eD −X < er

(18)

25

vr(er|T ) =

αX if er < eD − 2X

α(eD −X − er) + β(eD − 2X − er) if eD − 2X ≤ er ≤ eD −X

−βX if eD −X < er

(19)

Note that vC , vR and vr is weakly decreasing in ec, er and er respectively, so that V (ec|T ) is continuous

and weakly decreasing in ec. Thus it suffices to show that (i)V (0|T ) > 0, and (ii)V (erH |T ) < 0. Since

V (ec|T ) is independent of T , in what follows we will omit the term T .

(i)(ii): Using (16)(17)(18)(19), erL = 0, eD − 2X ≥ 0, eD − 2X < erH and eD −X > er,

V (0) = vC(0)− δvR(er)− (1− δ) {pvr(erL) + (1− p)vr(erH)}

= αX − δvR(er)− (1− δ) {pαX + (1− p)vr(erH)}

> αX − δαX − (1− δ) {pαX + (1− p)(α(eD −X − erH) + β(eD − 2X − erH))}

= (1− δ)(1− p) {αX − α(eD −X − erH)− β(eD − 2X − erH)}

= −(1− δ)(1− p)(α+ β)(eD − 2X − erH) > 0,

V (erH) = vC(erH)− δvR(er)− (1− δ) {pvr(erL) + (1− p)vr(erH)}

= (1− (1− δ)(1− p))vC(erH)− δvR(er)− (1− δ)pvr(erL)

< (1− (1− δ)(1− p)) {α(eD −X − erH) + β(eD − 2X − erH)}

− δ {α(eD −X − er) + β(eD − 2X − er)} − (1− δ)pαX

= (1− (1− δ)(1− p))(α+ β)(eD − 2X − erH) + (1− (1− δ)(1− p))αX − δ(α+ β)(eD −X − er)

+ δβX − (1− δ)pαX

= (1− (1− δ)(1− p))(α+ β)(eD − 2X − erH)− δ(α+ β)(eD − 2X − er)

= p(1− δ)(α+ β)(eD − 2X − erH)− δ(α+ β)(erH − er) < 0

�

26

𝑒"𝑂

𝑉(𝑒"|𝑇)

𝑒) − 2𝑋 𝑒) − 𝑋 𝑒-.

𝑉(0|𝑇)

𝑉(𝑒-.|𝑇)

𝑒"0

Figure 7: An example of unique cutoff in the case of eD −X < erH

A.3 Proof of proposition 1

Proof: When a dictator follows the form (4), and αC = αR = αr ≡ α, βC = βR = βr ≡ β for any

r ∈ {rL, rH}(anonymity), we can rewrite G in Lemma 2 as follows:

G = g(eD − 2X − er)− g(eD −X − er)

− p{g(eD − 2X)− g(eD −X)}+ (1− p){g(eD − 2X − erH)− (1− p)g(eD −X − erH)}

= vC(er|T )− E[vr(er|T )] (20)

where

vC(er|T ) =

αX if er < eD − 2X

α(eD −X − er) + β(eD − 2X − er) if eD − 2X ≤ er ≤ eD −X

−βX if eD −X < er

(21)

vr(er|T ) =

αX if er < eD − 2X

α(eD −X − er) + β(eD − 2X − er) if eD − 2X ≤ er ≤ eD −X

−βX if eD −X < er

(22)

Note that vr(0|T ) = αX by eD − 2X ≥ 0, andvr(erH |T ) = α(eD − X − erH) + β(eD − 2X − erH) or

−βX by eD − 2X ≤ erH . Now we consider the following three cases: (i)eD − X < er, (ii)er < eD − 2X,

(iii)eD − 2X ≤ er ≤ eD −X, and we shall check the sign of G.

27

(i): In this case, eD −X < erH holds since erH > er. Then G = −p(α+ β)X < 0.

(ii): In this case, we have G = αX − pαX − (1 − p)hr(erH |T ) = (1 − p){αX − vr(erH |T )}. Then we can

see G = −(1− p)(α+ β)(eD − 2X − erH) > 0 when erH ≤ eD −X, and also G = (1− p)(α+ β)X > 0

when eD −X < erH .

(iii): In this case, we have G = α(eD−X−er)+β(eD−2X−er)−pαX−(1−p)vr(erH |T ) = (α+β)(eD−2X−

er)+(1−p){αX−vr(erH |T )}. First when erH ≤ eD−X, G = (α+β)(eD−2X−er)+(1−p)(α+β)(eD−

2X − erH) = p(α+ β)(eD − 2X) ≥ 0.14 Next when eD −X < erH , G = (α+ β)(eD − (1 + p)X − er),

so that the sign of (eD − (1 + p)X − er) determines the sign of G. We can see (a)G > 0, (b)G = 0,

(c)G < 0 if and only if (a)er < eD − (1 + p)X, (b)er = eD − (1 + p)X, (c)er > eD − (1 + p)X.

Since eD −2X < eD − (1+p)X < eD −X, by (i)(ii)(iii) we can see (a)G > 0, (b)G = 0, (c)G < 0 if and only

if (a)er < eD − (1+p)X, (b)er = eD − (1+p)X, (c)er > eD − (1+p)X. By Lemma 2 the proof completes. �

!𝑒#𝑒$ − 2𝑋 𝑒$ − 𝑋𝑒$ − (1 + 𝑝)𝑋

𝐺 < 0𝐺 > 0 𝐺 < 0𝐺 = 0𝐺 > 0

(𝑖𝑖𝑖)(𝑖𝑖) (𝑖)

Figure 8: Characterization of G

A.4 Proof of prediction 2-1

Here let us denote preference parameters as αTj and βT

j to the recipient j when a dictator makes decisions in

treatment T ∈ {S,N}. On prediction 2-1, we prove the following.

Proposition 3 Suppose a dictator follows the form (6), and has a preference for other’s initial endowment

such that αNrH > αS

rH and βNrH < βS

rH , and αNR > αS

R and βNR < βS

R.15 Further suppose αS

C = αNC and

βSC = βN

C , and αSrL = αN

rL and βSrL = βN

rL.16 If eNc ∈ (0, erH) holds, then eSc > eNc .

Proof: Since a dictator follows the form (6), we can apply (12)(13)(14)(15). By assumptions we can attain

vc(eC |S) = vc(eC |N), vR(er|N) > vR(er|S), vrL(erL|S) = vrL(erL|N), and vrH(erH |N) > vrH(erH |S), so

that V (ec|S) > V (ec|N) for all ec ∈ [0, erH ]. Since both V (ec|S) and V (ec|N) are continuous and decreasing

in ec, eNc ∈ (0, erH) implies eSc > eNc . �

14Note that er = p0 + (1− p)erH .15it corresponds to αj(ej) > αj(e

′j) and βj(ej) < βj(e

′j) for e′j > ej in Prediction 2-1.

16This is reasonable since recipient C and rL’s initial endowment are same in both treatments, respectively.

28

𝑒"𝑂

𝑉(𝑒"|𝑇)

𝑒) − 2𝑋 𝑒) − 𝑋 𝑒-.

𝑉(0|𝑆)

𝑉(𝑒-.|𝑆)

𝑒"1

𝑉(𝑒-.|𝑁)

𝑒"3

𝑉(0|𝑁)

𝑉(𝑒"|𝑆)

𝑉(𝑒"|𝑁)

Figure 9: Effect of preference for other’s initial endowment in the case of eD −X < erH

A.5 Proof of prediction 2-2

Now we denote subjective probability as πT (s) to an objective probability s when a dictator makes decisions

in treatment T ∈ {S,N}. On prediction 2-2, we prove the following.

Proposition 4 Suppose a dictator follows the form (6), and has subjective probability such that πS(1− p) >

πN (1− p).17 Further suppose eD − 2X ≥ 0 and erH > eD − 2X + αrH

αrH+βrHX. Then it holds eSc > eNc .

Proof: Since a dictator follows the form (6), we can apply (12)(13)(14)(15). Denote subjective expected

value of recipient R’s earnings without transfer as erT = πT (1− p)erH for each T , and note that er

S > erN .

First it holds vC(ec|S) = vC(ec|N) since they does not depend on subjective probability πT (1 − p). Next

since erS > er

N , vR(·|S) = vR(·|N), and vR is non-increasing, it holds vR(erN |N) ≥ vR(er

S |S). Lastly,

erL ≤ eD − 2X implies vrL(erS |S) = vrL(er

N |N) = αrLX. And eD − 2X + αrH

αrH+βrHX < erH implies

vrH(erH |S) = vrH(erH |N) < 0, so that πN (1 − p)vrH(erH |N) > πS(1 − p)vrH(erH |S) since πS(1 − p) >

πN (1− p). Therefore by (12) we can see V (ec|S) > V (ec|N) for all ec ∈ [0, erH ], which completes the proof.

�

17It corresponds to π(s, e′j) > π(s, ej) for e′j > ej in Prediction 2-2.

29