# Social Preferences and Deliberately Stochastic Behavior††thanks: An earlier
version of this study was presented under the title “Stochastic Choice and
Social Preferences: Inequity Aversion versus Shame Aversion.” We thank Ryota
Iijima, Tetsuya Kawamura, Fabrice Le Lec, Yusuke Osaki, and the participants
at the CTW Summer Camp (Hirosaki University), ABEF (Nagoya University of
Commerce and Business), the Game Theory Workshop (Komazawa University and
Online), ASSA 2021 (Virtual), Kansai University, Hitotsubashi University, and
EWMES 2021(virtual) for their helpful comments and suggestions. This research
was supported by a grant-in-aid from the Tokyo Center for Economic Research
(TCER) and JSPS (KAKENHI Grant Number JP19J01049 and JP20K13457) (Hashidate).
Part of this research was completed while Yoshihara was a research associate
at the Center for Mathematics and Data Science, Gunma University. Yoshihara is
grateful for their hospitality. All remaining errors are ours.

Yosuke Hashidate JSPS Overseas Research Fellow, Department of Economics,
Boston University; Email<EMAIL_ADDRESS>Address: 270 Bay State
Road, Boston, MA 02215. Keisuke Yoshihara Email<EMAIL_ADDRESS>
(First Draft: April 27, 2020; Current Draft: April 27, 2023)

###### Abstract

This study proposes a tractable stochastic choice model to identify
motivations for prosocial behavior, and to explore alternative motivations of
deliberate randomization beyond ex-ante fairness concerns. To represent social
preferences, we employ an additively perturbed utility model consisting of the
sum of expected utility and a nonlinear cost function, where the utility
function is purely selfish while the cost function depends on social
preferences. Using the cost function, we study stochastic choice patterns to
distinguish between stochastic inequity-averse behavior and stochastic shame-
mitigating behavior. Moreover, we discuss how our model can complement recent
experimental evidence of ex-post and ex-ante fairness concerns.

_Keywords_ : Preference for Randomization; Perturbed Utility; Personal Norms;
Social Preferences; Ex-Ante Fairness.

_JEL Classification Numbers_ : D63; D64; D81; D91.

##  1 Introduction

One of the most significant and difficult issues concerning understanding
social preferences is identifying the motivations behind “seemingly”
altruistic and prosocial actions. As various models of social preferences can
explain altruistic and prosocial behavior, it is difficult to identify which
motivation is important in various contexts.111There are three main topics of
social preferences: (i) outcome-based social preferences (See, e.g., Fehr and
Schmidt (1999).), (ii) intention-based social preferences (See, e.g., Rabin
(1993).), and (iii) belief-based social preferences (See, e.g., Bénabou and
Tirole (2006).). In particular, we cannot easily identify the motivations
behind deterministic behavior. Consider a simple dictator game as a motivating
example, where the dictator chooses one allocation from the choice set
$\\{(10,0),(5,5)\\}$.222The allocation $(10,0)$ means that the dictator
obtains a payoff of 10, while the recipient obtains nothing. Assuming that the
decision-maker chooses $(10,0)$ or $(5,5)$ in a deterministic manner, the
former choice exhibits selfishness, whereas the latter exhibits altruism.
However, we do not have sufficient information to judge the motivation behind
this altruistic behavior.333On the one hand, the decision maker is purely
altruistic in the sense of outcome-based social preferences. On the other
hand, the experimental evidence shows that, in the presence of recipients, it
is psychologically costly to choose the former allocation due to social
pressure (DellaVigna et al., 2012). In Dana et al. (2006), the decision-maker
acts altruistically out of concern for their social image. In other words,
they may feel social pressure to behave generously.

One way to identify motivations is to consider stochastic behavior. Indeed,
the observed choice behavior in many settings is generally stochastic. One of
the reasons why this behavior is stochastic is deliberate randomization, which
was experimentally studied in Agranov and Ortoleva (2017) (See also Strzalecki
(2022).). Decision makers deliberately choose allocations randomly because
they are optimal for them to do so for various reasons, such as trembling
hands or errors with implementation costs (Fudenberg et al., 2015), Allais-
style lottery preferences (Machina, 1985; Cerreia-Vioglio et al., 2019),
hedging against ambiguity (Saito, 2015a), and regret minimization (Dwenger et
al., 2018).

Even in social preferences, deliberate randomization can occur. For example,
ex-ante fairness concern can lead to stochastic behavior (Hashidate and
Yoshihara, 2021). This taste captures equality of “opportunity,” i.e.,
equality of ex-ante expected payoffs (Saito, 2013). As in Miao and Zhong
(2018), who experimentally study combinational preferences of ex-post and ex-
ante fairness, a substantial proportion of subjects is inconsistent with this
type of taste.444In inequity-averse preferences, decision-makers dislike the
difference between their payoffs and their recipients’ payoffs, which can lead
to (ex-post) fairness concerns. Such a taste can provoke ex-ante fairness
concerns. This data suggests other possibilities for decision-makers to
randomize allocations in social contexts.

The trade-off between selfishness and altruism can also provoke deliberate
randomization. Consider a simple dictator game with $\\{(10,0),(5,5)\\}$ as a
thought experiment. By allowing stochastic behavior, impurely altruistic
behavior can be captured by deliberate randomization. Let $\rho$ be the choice
probability of choosing $(10,0)$ and $1-\rho$ be the choice probability of
choosing $(5,5)$. Suppose that the decision maker has some motivation for
social preferences. How does the decision maker determine the choice
probability? If $\rho=0$, the decision maker obtains a payoff of 5 with
certainty. In this case, the decision maker may feel the temptation to act
selfishly.555The publicity of behavior does not cause the temptation, but by
the immediacy of the timing that the decision maker obtains payoffs (Saito,
2015b). See also Noor (2011) and Noor and Ren (2023). Thus, they have the
incentive to increase $\rho$. Conversely, if $\rho=1$, there are two main
motives for decreasing $\rho$. First, the realized allocation $(10,0)$ is
unfair, provoking the decision maker to engage in guilt avoidance because of
payoff differences. Second, the decision-maker may be perceived as selfish if
he/she chooses $\rho=1$. Hence, the decision-maker is averse to choosing
$\rho=1$.

We consider two explanations for the deliberately stochastic behavior through
the trade-off. First, the decision maker may be inequity-averse; that is, the
decision maker dislikes “unfair” allocations.666An allocation is fair if every
agent obtains the same payoff. However, the decision maker is partially
selfish, as proven by substantial experimental evidence.777In the model of
Fehr and Schmidt (1999), empirical finding shows that the parameter of envy
$\alpha$ is larger than the parameter of guilt $\beta$ (Eckel and Gintis,
2010). Thus, they can choose the former even if they have inequity-averse
preferences. These tastes may exhibit deliberate randomization because there
is a trade-off between selfishness and fairness.888This type of deliberate
randomization is different from behavior stemming from ex-ante fairness in
Saito (2013). See Hashidate and Yoshihara (2021) for stochastic behavior due
to ex-ante fairness concerns. Second, the decision maker may be perceived as
selfish by other agents if he/she engages deterministically in selfish
behavior because the realized allocation has the role of public motives
mentioned above.999Decision makers pay considerable attention to social
concerns, as they care about how other agents perceive their decision-making.
Recent experimental evidence, such as Dana et al. (2006), has suggested a
possible explanation: the decision maker acts altruistically out of concern
for their social image. In particular, the decision maker can feel shame by
acting selfishly. This perception makes the image-conscious decision maker
averse to openly displaying such deterministic selfish behavior. Such concerns
affect the trade-off between selfish motivations and personal norms, impacting
behavior.101010Personal norms might differ from social norms. Hashidate (2021)
studies such social decision-making with reference-dependent
preferences.,111111Personal norms can be affected by social norms. In this
paper, we do not distinguish personal norms from social norms. In addition,
there is a difference between self-image and social image in personal norms.
There is a crucial distinction between shame and guilt. Generally, the
difference between them is captured by the publicity of behavior; shame is
caused by the publicity of behavior, but guilt can occur even in private
(Buss, 1980; Gehm and Scherer, 1988). Personal norms can be affected by both
shame and guilt. Therefore, to mitigate the feeling of shame, he/she may
engage in deliberately stochastic behavior. We thus need to examine whether
stochastic behavior is due to impure altruism, even if we observe that a
stochastic prosocial action has been taken.

In this study, we provide a unified framework for identifying the motivations
behind prosocial stochastic behavior.121212In general, intrinsic reciprocity
does not provoke deliberate randomization because deliberately stochastic
behavior may give opponents misperceptions of each other. In this paper, we do
not consider intrinsic reciprocity. We use stochastic choice functions as
primitives and axiomatically characterize the deliberate randomization model
stemming from inequity aversion and shame.131313The two main theories of
behavior adopt different approaches. On the one hand, the axiomatic study of
inequity aversion regards preferences over allocations as primitives (Rohde,
2010). On the other hand, the axiomatic study of social concerns considers
preferences over choice sets, that is, the menus of allocations, as primitives
(Saito, 2015b; Dillenberger and Sadowski, 2012). One of the objectives of this
study is to distinguish between inequity-averse behavior and shame-mitigating
behavior. To do so, we study stochastic choice functions in a unified manner.

The contributions of this study are threefold. First, we characterize the
deliberate randomization model in social preferences, following the additive
perturbed utility (henceforth, APU) models of Fudenberg et al. (2015),
consisting of the sum of expected utility and a nonlinear cost function. The
characterization gives us the additively separable preference structure
between selfishness and personal norms. We introduce the following four
axioms. The first axiom of Continuity is standard. The second axiom of Menu
Acyclicity requires that the rankings between menus are acyclic. Cycles
between allocations can occur as a stochastic behavior. The third axiom of
Selfishness requires the utility function to be _purely selfish_ , in the
sense that the decision-maker maximizes only his/her payoff. The fourth axiom
of Personal Norm Ranking requires stochastic behavior to be characterized by a
norm utility representation, which is related to the cost of randomization.

Second, we characterize the costs of deliberate randomization. The
tractability of our model stems from the additive separability between
selfishness and altruistic personal norms. As the private ranking represented
by the utility function is purely selfish, the altruistic aspects of the
decision-maker are captured by the cost function of randomization. In
particular, two cases have been considered. One is that personal norm utility
is inequity-averse. The other is that personal norm utility is altruistic. As
a benchmark, we consider a special case in which personal norms are selfish.
We then study both stochastic and deterministic behavior. The latter
corresponds to the expected utility of the risk preferences.

Third, we provide a method for identification based on stochastic behavior. In
particular, we study stochastic choice patterns stemming from ex-post fairness
concerns and shame-mitigating behavior. On the one hand, we identify the ex-
post fairness concern by observing that the decision-maker chooses fair
allocation with a higher probability. On the other hand, we identify shame-
driven behavior by observing that the decision-maker stochastically reveals
the trade-off between selfishness and altruism.

Moreover, we consider the relationship between our model and the experimental
data in Miao and Zhong (2018). Miao and Zhong (2018) discuss alternative
motives of pure selfishness, efficiency concerns, i.e., utilitarian
preferences, and inequality aversion. This study can capture a trade-off
between selfishness and personal norms, such as efficiency and fairness
concerns. Their experimental data is based on objective randomization;
Subjects’ probabilistic allocation choices are implemented through a
randomization device. This study’s target is rather subjective randomization,
i.e., a “mental” coin toss in one’s mind (Strzalecki, 2022). However, we
contribute to the field of social preferences by proposing an alternative
stochastic model, which is different from ex-ante fairness. By doing so, we
seek to gain a deeper understanding of the motivations behind altruistic or
prosocial behavior.

The remainder of this paper is organized as follows. Section 2 reviews the
results of this study and describes the model. Section 3 introduces the axioms
that characterize the model and states the main results. Section 4 presents a
literature review. Section 5 concludes the paper. All proofs are provided in
the Appendix.

##  2 The Model

This section presents a stochastic choice model of social preferences that
captures the following aspects for which we apply the perturbed utility models
of Fudenberg et al. (2015) to social preferences.

We investigate a decision maker who chooses the “optimal” probability over
allocations in menus; that is, deliberate randomization is
beneficial.141414Given a decision context or situation, deliberate
randomization can be interpreted as a mixed strategy in terms of the best
response for decision problems. For example, in an experiment, if the
dictator’s choices are private, he/she may choose the most selfish allocation
with certainty. If the dictator’s choices are public, his/her behavior can be
stochastic to avoid being perceived as selfish.

  1. (i)

The decision-maker is privately selfish.

  2. (ii)

The decision-maker can have a personal norm, leading to a trade-off between
selfishness and altruism.

  3. (iii)

The decision-maker deliberately randomizes allocations to conceal selfishness.

First, we assume that the utility function is selfish. Although this setting
can be seen as an extreme assumption, substantial evidence shows that a
decision-maker’s personal ranking is selfish (Dana et al., 2006, 2007).

Second, we consider personal norms to study stochastic behavior stemming from
the trade-off between selfishness and altruism. Personal norms include
fairness, social/image concerns, and selfishness. Deliberate randomization can
occur not only when the choice deviates from fairness but also when the choice
deviates from socially acceptable behavior. If selfishness is the personal
norm, his/her behavior can correspond to the standard additively perturbed
utility model (Fudenberg et al., 2015).

Third, to model deliberately stochastic behavior, we construct the cost
function of deliberate randomization, which imposes penalties for choosing
“selfish” allocations. In general, the choice of allocation is publicly
recognized. The key is that choosing the most selfish allocation with
certainty may be psychologically costly. Thus, fairness concerns and deviation
from socially acceptable behavior can lead to deliberate randomization. The
model captures this aspect by using a cost function, and tractability is based
on perturbed utility models.

#####  Setup.

Let $I=\\{1,2\\}$ be a set of individuals, where $1$ is the decision-maker,
and $2$ is the other (passive) agent.151515We can extend the $n$-th agents’
case. We assume that the set of payoffs is $\mathbb{R}$: A vector
$\bm{x}=(x_{1},x_{2})\in\mathbb{R}^{2}$ is called an allocation of payoffs
among individuals, yielding a payoff $x_{i}\in\mathbb{R}$ for each $i\in I$.
Let $X\subseteq\mathbb{R}^{2}$ be a compact set of allocations. A choice set,
the menu, is a nonempty subset of $X$. Let $\mathcal{A}$ be a collection of
all nonempty finite subsets of $X$ (See Fudenberg et al. (2014).). The
elements in $\mathcal{A}$ are denoted by $A,B,C\in\mathcal{A}$. Let
$\mathcal{D}=\\{({\bm{x}},A)\in X\times A\mid{\bm{x}}\in A\\}$.

We study a stochastic choice rule $\rho$ that maps a menu $A$ to a probability
distribution over the allocations in menu $A$, denoted by $\rho(A)$. Formally,
we denote the stochastic choice rule by
$\rho:\mathcal{A}\rightarrow\Delta(X)$, where $\Delta(X)$ is the set of
probability distributions over $X$ with finite support. Given a menu
$A\in\mathcal{A}$ with ${\bm{x}}\in A$, we denote the probability that an
allocation ${\bm{x}}$ is chosen from menu $A$ by a non-negative number
$\rho({\bm{x}},A)\geq 0$ and $\sum_{{\bm{x}}\in A}\rho({\bm{x}},A)=1$. In
other words, $\rho(A)$ defines the probability distribution over menu $A$. For
example, consider menu $A=\\{{\bm{x}},{\bm{y}}\\}$. Then,
$\rho(A)=({\bm{x}},\rho({\bm{x}},A);{\bm{y}},\rho({\bm{y}},A))$, where the
allocation ${\bm{x}}$ is realized with probability $\rho({\bm{x}},A)$ and the
allocation ${\bm{y}}$ is realized with probability $\rho({\bm{y}},A)$.

#####  The Model.

Before describing the model, we state the additional notation. First, let
$u:\mathbb{R}\rightarrow\mathbb{R}$ be a continuous and monotonic function. We
call $u$ a selfish utility function if, for each $x_{1},y_{1}\in\mathbb{R}$,
$x_{1}>y_{1}\Leftrightarrow u(x_{1})>u(y_{1})$. The utility function $u$
captures the decision maker’s personal ranking. Second, let
$\succsim_{n}^{\rho}$ over $X$ be a personal norm ranking over allocations,
formally introduced in Definition 2 (Section 3.1). The binary relation
captures the decision maker’s personal norm ranking, being generally more
altruistic than the decision maker’s personal ranking. Personal norm ranking
$\succsim_{n}^{\rho}$ is represented by a continuous and weakly monotone
function $\varphi:X\rightarrow\mathbb{R}$. We call $\varphi$ the personal norm
utility function if $\varphi$ represents $\succsim_{n}^{\rho}$.

We introduce an additive perturbed utility representation (Fudenberg et al.,
2015). We say that a function $c:[0,1]\rightarrow\mathbb{R}\cup\\{+\infty\\}$
is a cost function if $c$ is strictly convex, $C^{1}$ over $(0,1)$, and
$\lim_{q\rightarrow 0}c^{\prime}(q)=-\infty$. If the property of steep cost,
$\lim_{q\rightarrow 0}c^{\prime}(q)=-\infty$, is ruled out, we call $c$ a weak
cost function. Hence, an APU with a weak cost function is called a weak APU.
An APU representation has the form:

$\rho(A)=\arg\max_{\rho\in\Delta(A)}\sum_{{\bm{x}}\in
A}\Bigl{(}u({\bm{x}})\rho({\bm{x}})-c(\rho({\bm{x}},A))\Bigr{)},$ (1)

for some utility function $u:X\rightarrow\mathbb{R}$ and cost function $c$.

Now, we are ready to define the model in this paper.

######  Definition 1.

$\rho$ is an additive perturbed utility with social preferences (APU(SP)) if
$\rho$ has an additive perturbed utility form with a pair $(u,\varphi)$ where
$u:\mathbb{R}\rightarrow\mathbb{R}$ is a selfish utility function and
$\varphi:X\rightarrow\mathbb{R}$ is a personal norm utility function:

$\rho(A)=\arg\max_{\rho\in\Delta(A)}\sum_{{\bm{x}}\in
A}\Bigl{(}u(x_{1})\rho({\bm{x}})-c_{\varphi({\bm{x}})}(\rho({\bm{x}},A))\Bigr{)},$
(2)

where $c_{\varphi(\cdot)}:[0,1]\rightarrow\mathbb{R}\cup\\{+\infty\\}$ is a
strictly convex function that satisfies

  1. (i)

$C^{1}_{\varphi(\cdot)}$ over $(0,1)$; and

  2. (ii)

$c^{\prime}_{\varphi({\bm{x}})}(p)>c^{\prime}_{\varphi({\bm{y}})}(p)$ for each
$p\in(0,1)$, if

$\displaystyle\varphi({\bm{x}})<\varphi({\bm{y}}).$

Here, we explain the interpretation of the model. First, regarding the model’s
components, the selfish utility function $u$ represents a personal ranking on
$X$ that maximizes only the decision maker’s payoff. Second, the cost function
of randomization $c$ depends on personal norm utility $\varphi$. By
definition, the marginal cost increases if allocations are not personally
normatively better (Property (ii)).

Because $c_{\varphi(\cdot)}$ is strictly convex, deliberate randomization
benefits the decision-maker.161616Formally, we can also write down
$c_{\varphi(\cdot)}$ in the following manner:
$\mathcal{C}:\varphi(X)\times[0,1]\rightarrow\mathbb{R}\cup\\{+\infty\\}$. For
example, property (ii) implies that
$\displaystyle\dfrac{\partial^{2}\mathcal{C}(a,p)}{\partial a\partial p}<0$
where $a\in\varphi(X)$. If there is no social concern, the decision maker
adopts deterministic behavior by choosing the most selfish allocation in the
menu (See Corollary 3.).

#####  FOC for $\rho$.

We consider the first-order condition (FOC) for $\rho$. Let the Lagrange
multiplier be denoted as $\lambda:\mathcal{A}\rightarrow\mathbb{R}$. For each
menu $A\in\mathcal{A}$, the optimal randomization has constraint
$\sum_{{\bm{x}}\in A}\rho({\bm{x}},A)=1$. We immediately obtain the FOC for
$\rho$ as follows:

$u(x_{1})-c^{\prime}_{\varphi({\bm{x}})}(\rho({\bm{x}},A))+\lambda(A)\begin{cases}\geq
0&\text{if $\rho({\bm{x}},A)=1$,}\\\ =0&\text{if
$\rho({\bm{x}},A)\in(0,1)$,}\\\ \leq 0&\text{if
$\rho({\bm{x}},A)=0$.}\end{cases}$ (3)

What captures the Lagrange multiplier, $\lambda$? As the decision maker might
bear a moral cost due to acting selfishly with certainty, APU(SP) deliberately
describes stochastic prosocial behavior as optimal randomization. Constraint
level $\lambda$ corresponds to the psychological cost of prosocial motives,
which can be interpreted as the decision-maker’s social preferences. This cost
includes not only guilt and shame but also inequity aversion. Therefore,
deliberate stochastic behavior can occur.

######  Example 1.

Consider the menu $\\{(10,0),(5,5)\\}$ mentioned in Section 1 and an APU(SP)
with a tuple $(u,\varphi_{1})$ where $u(x_{1})=x_{1}$ and
$\varphi_{1}({\bm{x}})=(x_{1}+1)(x_{2}+1)$ for each ${\bm{x}}\in X$. The
maximizer of $\varphi_{1}$ in $A$ is the latter allocation $(5,5)$; that is,
we obtain $11=\varphi(10,0))<\varphi(5,5))=36$. By definition, we have

$\displaystyle
c^{\prime}_{\varphi_{1}(10,0)}(\rho((10,0),\\{(10,0),(5,5)\\})>c^{\prime}_{\varphi(5,5)}(\rho((5,5),\\{(10,0),(5,5)\\}).$

As $u(10)>u(5)$, there is a trade-off between private ranking ($u$) and
personal norm ranking ($\varphi_{1}$). Thus, deliberate stochastic behavior
can occur.

Similarly,
$\varphi_{2}({\bm{x}})=x_{1}-\alpha\max\\{x_{2}-x_{1},0\\}-\beta\max\\{x_{1}-x_{2},0\\}$
for each ${\bm{x}}\in X$ with $\alpha>0$ and $\beta\in(0,1)$. Thus, the
maximizer of $\varphi_{2}$ in $A$ is $(5,5)$. A trade-off exists between $u$
and $\varphi_{2}$. Hence, deliberate stochastic behavior can occur.

##  3 Axiomatization

This section describes the axioms, results, and testable implications. First,
to characterize the APU(SP), we introduce five axioms: (i) positivity, (ii)
continuity, (iii) menu acyclicity, (iv) selfishness, and (v) personal norm
ranking. Next, we state the representation theorem and uniqueness result.
Finally, we consider the testable implications of the model.

###  3.1 Axioms

First, we state the basic axiom of continuity, which guarantees that the
utility functions are continuous.

######  Axiom 1.

(Continuity): For any menu $\\{{\bm{x}}^{1},\cdots,{\bm{x}}^{m}\\}$ with
allocation sequences $\lim_{n\rightarrow\infty}{\bm{x}}^{k}_{n}={\bm{x}}^{k}$
for each $k=1,\cdots,m$,

$\displaystyle\lim_{n\rightarrow\infty}\rho({\bm{x}}^{k}_{n},\\{{\bm{x}}^{1}_{n},\cdots,{\bm{x}}^{m}_{n}\\})=\rho({\bm{x}}^{k},\\{{\bm{x}}^{1},\cdots,{\bm{x}}^{m}\\}).$

Next, we introduce the acyclic condition introduced by Fudenberg et al. (2014,
2015), which is the axiom of menu acyclicity. This axiom requires that menu
rankings do not have cycles, whereas allocation rankings can have
cycles.171717Allocation rankings are induced as follows. For any
${\bm{x}},{\bm{y}}\in X$, ${\bm{x}}$ is preferred to ${\bm{y}}$,
${\bm{x}}\succ^{\rho}{\bm{y}}$ if $\rho({\bm{x}},A)>\rho({\bm{y}},A)$ for some
$A\ni{\bm{x}},{\bm{y}}$. Similarly, for any ${\bm{x}},{\bm{y}}\in X$,
${\bm{x}}$ is indifferent to ${\bm{y}}$, ${\bm{x}}\sim^{\rho}{\bm{y}}$ if
$\rho({\bm{x}},A)=\rho({\bm{y}},A)\in(0,1)$ for some $A\ni{\bm{x}},{\bm{y}}$.
Then, define $\succsim^{\rho}:=\succ^{\rho}\cup\sim^{\rho}$. In the same way,
menu rankings are induced as follows. For any $A,B\in\mathcal{A}$, $A$ is
weaker than $B$, $A\succ_{m}^{\rho}B$ if $\rho({\bm{x}},A)>\rho({\bm{x}},B)$
for some ${\bm{x}}\in A\cap B$. Similarly, for any $A,B\in\mathcal{A}$, $A$ is
tied with $B$, $A\sim_{m}^{\rho}B$ if
$\rho({\bm{x}},A)=\rho({\bm{x}},B)\in(0,1)$ for some ${\bm{x}}\in A\cap B$.
Then, define $\succsim_{m}^{\rho}:=\succ_{m}^{\rho}\cup\sim_{m}^{\rho}$. This
axiom characterizes a menu-invariant APU.

######  Axiom 2.

(Menu Acyclicity): For any finite sequences of pairs
$\\{({\bm{x}}_{k},A_{k})\\}_{k=1}^{n}$ with ${\bm{x}}_{k}\in A_{k}\cap
A_{k+1}$ and ${\bm{x}_{n}}\in A_{1}$,

$\displaystyle\begin{rcases}\rho({\bm{x}}_{1},A_{1})>\rho({\bm{x}}_{1},A_{2})\\\
\rho({\bm{x}}_{k},A_{k})\geq\rho({\bm{x}}_{k},A_{k+1})&\text{for
$1<k<m$}\end{rcases}\Rightarrow\rho({\bm{x}}_{n},A_{n})\ngeq\rho({\bm{x}}_{n},A_{1}).$

Moreover, we introduce an axiom of selfishness. This axiom requires the
decision maker to be selfish at the private stage of decision making. We
excluded the behavior of overwhelming norms. For example, considering a menu
$\\{(10,0),(5,5)\\}$, if the decision maker adopts deterministic behavior, the
axiom requires the decision maker to choose $(10,0)$ with certainty. Owing to
public recognition, the decision-maker may randomize this, but we do not
assume that the decision-maker is purely altruistic.

######  Axiom 3.

(Selfishness): If there exists $A\in\mathcal{A}$ such that
$\rho({\bm{x}},A)=1$, then $x_{1}>y_{1}$ for all ${\bm{y}}\in
A\setminus\\{{\bm{x}}\\}$.

Finally, we introduce the axiom of the personal norm ranking. First, we induce
a personal norm ranking, that is, a binary relation on $X$, as follows:

######  Definition 2.

${\bm{y}}$ is normatively better than ${\bm{x}}$, that is,

${\bm{y}}\succ_{n}^{\rho}{\bm{x}}\ \text{if}\
\rho({\bm{x}},A)>\rho({\bm{x}},A\cup\\{{\bm{y}}\\})$

for some $A\in\mathcal{A}$ with ${\bm{x}}\in A$ and ${\bm{y}}\in X\setminus
A$.

Before stating the axiom in detail, we state the following remark. To study
personal norms, we pay attention to Remark 1, focus on the strict part
$\succ_{n}^{\rho}$, and then define a binary relation.

######  Remark 1.

Menu Acyclicity implies Regularity (Fudenberg et al., 2015).

######  Axiom 4.

(Regularity): For all $A,B\in\mathcal{A}$ and ${\bm{x}}\in A\subseteq B$,
$\rho({\bm{x}},A)\geq\rho({\bm{x}},B)$.

Now, we define $\succsim_{n}^{\rho}$ and $\sim_{n}^{\rho}$ as follows: For
each ${\bm{x}},{\bm{y}}\in X$, ${\bm{y}}\succsim_{n}^{\rho}{\bm{x}}$ if
${\bm{x}}\nsucc_{n}^{\rho}{\bm{y}}$. Similarly, for each ${\bm{x}},{\bm{y}}\in
X$, ${\bm{y}}\sim_{n}^{\rho}{\bm{x}}$ if ${\bm{x}}\nsucc_{n}^{\rho}{\bm{y}}$
and ${\bm{y}}\nsucc_{n}^{\rho}{\bm{x}}$. Thus, we obtain a utility function
that represents $\succsim_{n}^{\rho}$.

Personal norm ranking is interpreted as follows. Given a menu $A$ with
${\bm{x}}\in A$, suppose that allocation ${\bm{y}}$ is added to menu $A$.
Suppose that $\rho({\bm{x}},A)>\rho({\bm{x}},A\cup\\{{\bm{y}}\\})$, that is,
the added allocation decreases the choice probability of ${\bm{x}}$.
Subsequently, the added allocation ${\bm{y}}$ reveals that it is normatively
better than ${\bm{x}}$ in the sense that the choice probability of ${\bm{x}}$
decreases in the presence of ${\bm{y}}$. However, the monotone behavioral
pattern can occur through choice-set size effects like regularity. We need to
restrict it to distinguish peronal norms from regularity.

We consider the properties of personal norm ranking $\succsim_{n}^{\rho}$,
summarized in the following axiom: The first two axioms guarantee that
$\succ_{n}^{\rho}$ is represented by the utility function (Kreps, 1988). The
third axiom is the weaker version of monotonicity. We relax the monotone
condition as personal norm rankings include inequity-averse preferences, such
as Fehr and Schmidt (1999).

######  Axiom 5.

(Personal Norm Ranking): $\succ_{n}^{\rho}$ satisfies the following
statements:

  1. (i)

$\succ_{n}^{\rho}$ is asymmetric: For any ${\bm{x}},{\bm{y}}\in X$,
${\bm{x}}\succ_{n}^{\rho}{\bm{y}}$ implies
${\bm{y}}\nsucc_{n}^{\rho}{\bm{x}}$.

  2. (ii)

$\succ_{n}^{\rho}$ is negatively transitive: For any
${\bm{x}},{\bm{y}},{\bm{z}}\in X$, ${\bm{x}}\nsucc_{n}^{\rho}{\bm{y}}$ and
${\bm{y}}\nsucc_{n}^{\rho}{\bm{z}}$ implies
${\bm{x}}\nsucc_{n}^{\rho}{\bm{z}}$.

  3. (iii)

$\succ_{n}^{\rho}$ is monotone with respect to fair allocations: For any
$x,y\in\mathbb{R}$ with $x>y$, $(x,x)\succ_{n}^{\rho}(y,y)$.

###  3.2 Results

We state the representation theorem (Theorem 1) and uniqueness result
(Proposition 1). We also study the case where pure selfishness is a personal
norm (Proposition 2). Moreover, we find that the expected utility
representation is a special case of an APU. We restrict the case to which the
utility function is purely selfish (Corollaries 1, 2, and 3).

#####  Representation Result.

First, we present the main result of this study.

######  Theorem 1.

The following statements are equivalent:

  1. (a)

$\rho$ satisfies Continuity, Menu Acyclicity, Selfishness, and Personal Norm
Ranking.

  2. (b)

$\rho$ is an additive perturbed utility with social preferences (APU(SP)).

We call the model in Theorem 1 additive perturbed utility with social
preferences (APU(SP)) if the axioms in Theorem 1 are satisfied.

#####  Proof Outline.

We provide the proof outline of the sufficiency part in Theorem 1 (See
Appendix A.1 in detail.). There are four steps. In Step 1 (Appendix A.1.1), we
show that we obtain a menu-invariant APU (Lemma 1). We apply the result of
Fudenberg et al. (2014), a version of Farkas lemma (Lemma 2). In Step 2
(Appendix A.1.2), we show that we obtain a utility representation of personal
norm ranking $\succsim^{\rho}_{n}$ (Lemma 3). To do so, we have the three
claims. To begin, we show that the binary relation $\succsim^{\rho}_{n}$ has a
utility representation $\varphi$ (Claim 1). Next, we show that $\varphi$ is
continuous (Claim 2). Finally, we show that $\varphi$ is monotone with respect
to fair allocations. In Step 3 (Appendix A.1.3), we show that the selfish
utility function is selfish (Claim 4) and continuous (Claim 5). In Step 4
(Appendix A.1.4), we complete the representation of APU(SP). We construct the
cost function based on the personal norm utility function $\varphi$. We
construct the cost function using Definition 2 (Claim 6).

#####  Uniqueness Result.

Next, we present the uniqueness result. To obtain the uniqueness result, we
introduce Positivity. This axiom states that all probabilities are positive.
Even though Positivity seems to be an extreme assumption in social
preferences, this axiom is helpful for the identification of the model.

######  Axiom 6.

(Positivity): For any menu $A\in\mathcal{A}$ and allocation ${\bm{x}}\in A$,
$\rho({\bm{x}},A)>0$.

We state the uniqueness result. The proof is in the Appendix (See Appendix
B.1).

######  Proposition 1.

Suppose that $\rho$ satisfies Positivity in addition to the axioms in Theorem
1. If $(u,\varphi)$ and $(\widehat{u},\widehat{\varphi})$ represent the same
$\rho$, then the following holds: There exsit real numbers $\alpha>0$ and
$\beta\in\mathbb{R}$ such that

  1. (i)

$\widehat{u}=\alpha u+\beta$;

  2. (ii)

$\widehat{\varphi}=\alpha\varphi$.

One of the motivations of deliberate randomization stems from the “payoff
differences” $u(x_{1})-u(y_{1})$ between allocations in the menu (See, in
detail, Definition 4 of a Fenchnerian utility representation in Appendix
B.1.). In addition, the personal norm utility matters. The key point is that
the two utility functions have the same unit $\alpha$. Indeed, the absolute
level of the cost function $c_{\varphi}$ does not matter. Thus, we are free to
shift it by constants.

######  Remark 2.

Generally, a menu-invariant APU (Fudenberg et al., 2014), characterized by
Menu Acyclicity, is not uniquely identified. Any utility function $u$ can be
transformed into a cost function $c_{\bm{x}}$ with ${\bm{x}}\in X$. However,
we can recover the uniqueness result by considering specific properties of the
cost function.

#####  Selfishness.

We study a particular case in which pure selfishness is a personal norm. We
can imagine simplistic moral maxims such as “selfishness is good.” Again,
consider a simplified dictator game, $\\{(9,1),(5,5)\\}$. If many people
predict that dictators choose the former allocation, they may not incur moral
costs such as shame by choosing it from the menu.

######  Axiom 7.

(Selfishness as a Norm): For any ${\bm{x}},{\bm{y}}\in X$, $x_{1}>y_{1}$ if
and only if ${\bm{x}}\succ_{n}^{\rho}{\bm{y}}$.

We obtain the following result. The proof is in the Appendix (Appendix B.2).

######  Proposition 2.

Suppose that $\rho$ satisfies the axioms in Theorem 1. Then, $\rho$ satisfies
Selfishness as a Norm if and only if there exists a selfish utility
$u:\mathbb{R}\rightarrow\mathbb{R}$ such that $\rho$ has an additive perturbed
utility form in the following way:

$\rho(A)=\arg\max_{\rho\in\Delta(A)}\sum_{{\bm{x}}\in
A}\Bigl{(}u(x_{1})\rho({\bm{x}})-c_{u(x_{1})}(\rho({\bm{x}},A))\Bigr{)},$ (4)

where $c_{u(\cdot)}:[0,1]\rightarrow\mathbb{R}\cup\\{+\infty\\}$ is a strictly
convex function that satisfies

  1. (i)

$C^{1}_{u(\cdot)}$ over $(0,1)$; and

  2. (ii)

$c^{\prime}_{u(x_{1})}(p)>c^{\prime}_{u(y_{1})}(p)$ for each $p\in(0,1)$, if

$\displaystyle u(x_{1})>u(y_{1}).$

Proposition 2 implies that the decision-maker chooses a selfish allocation
with a high probability through the selfish norm. This result is a special
case of weak APU if a utility function $u$ is purely selfish, and a cost
function $c$ is independent of $u$ (See Example 2 and Corollary 2.).

To explore this type of stochastic behavior, we define the selfishness-based
FOSD as follows:

######  Definition 3.

(Selfishness-Based FOSD): For each $A\in\mathcal{A}$,
$\rho({\bm{x}},A)>\rho({\bm{y}},A)$ if $x_{1}>y_{1}$.

We obtain the following corollary. The proof is in the Appendix (Appendix
C.1).

######  Corollary 1.

Suppose that $\rho$ has an APU form in Proposition 2; then, $\rho$ satisfies
selfishness-based FOSD.

We provide a numerical example. Consider three allocations ${\bm{x}}=(4,4)$,
${\bm{y}}=(5,2)$, and ${\bm{z}}=(6,1)$. We consider two menus
$\\{(4,4),(5,2)\\}$ and $\\{(4,4),(5,2),(6,1)\\}$.

######  Example 2.

Suppose that APU(SP) has an entropy-type cost function. Given menu $A$ with
${\bm{x}}\in A$,

$c_{u(x_{1}))}(\rho({\bm{x}},A))=\eta^{\frac{u(x_{1})}{\gamma}}\rho({\bm{x}},A)\log\rho({\bm{x}},A),$
(5)

where $\eta=2$ and $\gamma=100$. Then, we obtain choice probabilities as
follows. We verify that $\rho$ satisfies Regularity, and that no preference
reversals occur under $c_{u(\cdot)}$.

Table 1: An Example of Stochastic Behavior when Personal Norm is Selfish Menus | Choice Probabilities
---|---
| $\rho({\bm{x}})$ | $\rho({\bm{y}})$ | $\rho({\bm{z}})$
$\\{(4,4),(5,2)\\}$ | $0.2753$ | $0.7247$ |
$\\{(4,4),(5,2),(6,1)\\}$ | $0.0944$ | $0.2504$ | $0.6552$

In Example 2, as the parameter $\gamma\rightarrow+\infty$, we obtain an
entropic-type cost function independent of $u$. This case corresponds to
Fudenberg et al. (2015) (See Equation (1).).

Formally, we characterize the case that APU(SP) with $c_{u}$ corresponds to
APU (Fudenberg et al., 2015). Following Fudenberg et al. (2014), we consider a
condition due to Tversky (1972):

######  Axiom 8.

(Order Independence): For any $A,B\in\mathcal{A}$, ${\bm{x}},{\bm{y}}\in
A\setminus B$, and ${\bm{z}}\in B$,

$\displaystyle\rho({\bm{z}},B\cup\\{{\bm{x}}\\})\leq\rho({\bm{z}},B\cup\\{{\bm{y}}\\})\Leftrightarrow\rho({\bm{x}},A)\geq\rho({\bm{y}},A).$

Fudenberg et al. (2014) show that Positivity and Order Independence implies
Acyclitiy (See Proposition 10 in Fudenberg et al. (2014). To describe the
axiom, let us introduce additional notation. We say that a finite sequence of
quadruples $\\{(x_{k},A_{k}),(y_{k},B_{k})\\}^{k}_{n=1}$ is admissible if

  1. (i)

$x_{k}\in A_{k}$ and $y_{k}\in B_{k}$ for all $k\in\mathbb{N}$

  2. (ii)

$(y_{k})^{n}_{k=1}$ is a permutation of the $(x_{k})^{n}_{k=1}$, and

  3. (iii)

$(B_{k})^{n}_{k=1}$ is a permutation of the
$(A_{k})^{n}_{k=1}$.181818$y_{k}=x_{f(k)}$ for some permutation
$f:\\{1,\cdots,n\\}\rightarrow\\{1,\cdots,n\\}$ and $B_{k}=A_{g(k)}$ for some
permutation $g:\\{1,\cdots,n\\}\rightarrow\\{1,\cdots,n\\}$.

######  Axiom 9.

(Acyclicity):For any finite sequence of quadruples
$\\{(x_{k},A_{k}),(y_{k},B_{k})\\}^{k}_{n=1}$, there exists no admissible
sequence such that $\rho({\bm{x}}_{1},A_{1})>\rho({\bm{y}},B_{1})$,
$\rho({\bm{x}}_{k},A_{k})\geq\rho({\bm{y}}_{k},A_{k})$, and
$\rho({\bm{x}}_{n},A_{n})\geq\rho({\bm{y}}_{n},B_{n})$.

The axiom of Acyclicity implies the following ordinal properties:

  1. (a)

$\rho({\bm{x}},A)>\rho({\bm{y}},A)\Leftrightarrow\rho({\bm{x}},B)>\rho({\bm{y}},B)$
for any ${\bm{x}},{\bm{y}}\in A$ with $x_{1}>y_{1}$;

  2. (b)

$\rho({\bm{x}},A)>\rho({\bm{x}},B)\Leftrightarrow\rho({\bm{y}},A)>\rho({\bm{y}},B)$
for any ${\bm{x}},{\bm{y}}\in A$ with $x_{1}>y_{1}$.

Thus, we obtain the following corollary. The proof is in the Appendix
(Appendix C.2).

######  Corollary 2.

Suppose that $\rho$ satisfies the axioms in Theorem 1. Then, $\rho$ satisfies
Positivity, Selfishness as a Norm, and Order Independence if and only if there
exists a selfish utility $u:\mathbb{R}\rightarrow\mathbb{R}$ such that $\rho$
has an additive perturbed utility form in the following way:

$\rho(A)=\arg\max_{\rho\in\Delta(A)}\sum_{{\bm{x}}\in
A}\Bigl{(}u(x_{1})\rho({\bm{x}})-c(\rho({\bm{x}},A))\Bigr{)},$ (6)

where $c:[0,1]\rightarrow\mathbb{R}\cup\\{+\infty\\}$ is a cost function.

Finally, as a benchmark case, we consider the case that $\rho$ is
deterministic. We rule out positivity. Then, we can obtain an expected utility
representation where $u$ is purely selfish. Generally, we can consider the
following axiom for deterministic choice behavior:

######  Axiom 10.

(Deterministic Choice): For any $A\in\mathcal{A}$, there exists ${\bm{x}}\in
A$ such that $\rho({\bm{x}},A)=1$.

We obtain the following corollary. The proof is in the Appendix (Appendix
C.3).

######  Corollary 3.

Suppose that $\rho$ has an APU form. Then, $\rho$ satisfies Deterministic
Choice if and only if

$\rho(A)=\arg\max_{\rho\in\Delta(A)}\sum_{{\bm{x}}\in
A}u(x_{1})\rho({\bm{x}}).$ (7)

This result is equivalent to “deterministic” choice behavior. Let $C$ be a
choice correspondence; that is, for each $A\in\mathcal{A}$, $C(A)\subset A$ is
a nonempty subset of $A$. Suppose that $u$ is selfish utility. Thus, we have

$\displaystyle C(A)=\arg\max_{{\bm{x}}\in A}u(x_{1}).$

Let $V:\Delta(X)\rightarrow\mathbb{R}$. Then, $V(\rho)=\sum_{{\bm{x}}\in{\rm
supp}(\rho)}u(x_{1})\rho({\bm{x}})$.

###  3.3 A Testable Implication

We study the testable implications of the APU(SP). In the APU(SP), we consider
a case in which the utility function is selfish. We can consider various
personal norms, from selfishness and altruism to inequity-aversion. If
personal norms differ, the resulting stochastic behavior also differs.

In this study, we provide simple and testable implications. We fix a doubleton
$\\{{\bm{x}},{\bm{y}}\\}$, where ${\bm{x}}=(4,4)$ and ${\bm{y}}=(5,2)$ with

$\displaystyle\rho({\bm{x}},\\{{\bm{x}},{\bm{y}}\\})>\rho({\bm{y}},\\{{\bm{x}},{\bm{y}}\\}),$

irrespective of their social preferences.191919We can design such a menu
irrespective of the preference type: inequity-averse or shame-averse. For
example, we can collect data on $(\alpha,\beta)$ in pilot experiments. We
consider a case in which a new allocation ${\bm{z}}=(6,1)$ is added to the
doubleton.

In fairness concerns, ${\bm{x}}$ is fairer than ${\bm{y}}$. Thus,

$\displaystyle\rho({\bm{x}},\\{{\bm{x}},{\bm{y}},{\bm{z}}\\})>\rho({\bm{y}},\\{{\bm{x}},{\bm{y}},{\bm{z}}\\})$

holds. Conversely, for social concerns, that is, due to the trade-off between
selfishness and personal norm,

$\displaystyle\rho({\bm{x}},\\{{\bm{x}},{\bm{y}},{\bm{z}}\\})\leq\rho({\bm{y}},\\{{\bm{x}},{\bm{y}},{\bm{z}}\\})$

can occur. Due to the addition of the third allocation ${\bm{z}}$, if the
decision maker is socially-conscious, the trade-off between selfishness and
altruism can occur. As a result, the “intermidiate” allocation ${\bm{y}}$ can
be chosen with a higher probability. This type of behavioral pattern is known
as the Compromise effect or Extremeness Aversion.

In empirical studies, it is common to focus on parametric models. Here, we
suggest examples of APU(SP) compatible with fairness and altruistic concerns.

Throughout the empirical test, suppose that the observed data is explained by
APU(SP), and that the utility function $u$ is selfish. Let
$A=\\{{\bm{x}},{\bm{y}}\\}$, and $B=\\{{\bm{x}},{\bm{y}},{\bm{z}}\\}$. Suppose
that we observe the data of
$\rho({\bm{x}},\\{{\bm{x}},{\bm{y}}\\})>\rho({\bm{y}},\\{{\bm{x}},{\bm{y}}\\})$.
Then, we perform a one-sided test using the following hypotheses:

$\displaystyle H_{0}:\rho({\bm{x}},B)>\rho({\bm{y}},B),$

and

$\displaystyle H_{1}:\rho({\bm{x}},B)\leq\rho({\bm{y}},B).$

$H_{0}$ implies that the third allocation ${\bm{z}}=(6,1)$ does not affect the
magnitude relationship between $\rho({\bm{x}},B)$ and $\rho({\bm{y}},B)$; That
is, the choice probability $\rho({\bm{y}},A)>\rho({\bm{y}},B)$ holds. In other
words, if the cost function is fairness-driven, adding ${\bm{z}}$ to $A$ is
irrelevant.

If $H_{0}$ is rejected, the observed behavior can stem from impurely
altruistic behavior. However, such a preference reversal phenomenon can occur
if the cost function is based on social concerns.

##### Fairness Concerns: The Case of Inequity Aversion

First, we consider the case in which stochastic choice behavior exhibits
inequity-averse preferences. As stated above, the allocation ${\bm{z}}=(6,1)$
is the most selfish allocation in the menu $\\{{\bm{x}},{\bm{y}},{\bm{z}}\\}$,
whereas ${\bm{z}}$ can also be interpreted as the most “unfair” allocation in
the menu. Let $\succsim^{\rho}_{l}$ be a binary relation over $\Delta(X)$ that
satisfies inequity-averse preferences. We obtain
${\bm{x}}\succ^{\rho}_{l}{\bm{y}}\succ^{\rho}_{l}{\bm{z}}$. By definition, we
also obtain
$\rho({\bm{x}},\\{{\bm{x}},{\bm{y}},{\bm{z}}\\})>\rho({\bm{y}},\\{{\bm{x}},{\bm{y}},{\bm{z}}\\})>\rho({\bm{z}},\\{{\bm{x}},{\bm{y}},{\bm{z}}\\})$;
that is, we have that

$\displaystyle\rho({\bm{x}},\\{{\bm{x}},{\bm{y}},{\bm{z}}\\})>\rho({\bm{y}},\\{{\bm{x}},{\bm{y}},{\bm{z}}\\}).$

Suppose that APU(SP) is represented by a quadratic-type cost function as
follows:

$\displaystyle\varphi({\bm{x}})=\alpha\max\\{x_{2}-x_{1},0\\}+\beta\max\\{x_{1}-x_{2},0\\}$

for each ${\bm{x}}\in X$. Given menu $A$ with ${\bm{x}}\in A$, consider the
following cost function:

$c_{\varphi({\bm{x}})}(\rho({\bm{x}},A))=\eta^{\alpha\max\\{x_{2}-x_{1},0\\}+\beta\max\\{x_{1}-x_{2},0\\}}\rho({\bm{x}},A)^{2},$
(8)

where $\eta=2$, $\alpha=1$ and $\beta=1$. Then, we have
$\rho({\bm{x}},\\{{\bm{x}},{\bm{y}}\\})\approx 0.8333$ and
$\rho({\bm{y}},\\{{\bm{x}},{\bm{y}}\\})\approx 0.1667$. Thus, we infer that
the decision maker prefers allocation ${\bm{x}}$ over allocation ${\bm{y}}$.
In addition, because the third allocation ${\bm{z}}$ is selfish, it does not
affect fairness-driven behavior. Furthermore, we have
$\rho({\bm{x}},\\{{\bm{x}},{\bm{y}},{\bm{z}}\\})\approx 0.7838$ and
$\rho({\bm{y}},\\{{\bm{x}},{\bm{y}},{\bm{z}}\\})\approx 0.1605$. Thus, we have
$\rho({\bm{x}},\\{{\bm{x}},{\bm{y}},{\bm{z}}\\})>\rho({\bm{y}},\\{{\bm{x}},{\bm{y}},{\bm{z}}\\})$.

##### Social Concerns: The Case of Impure Altruism

Next, we consider the case in which stochastic choice behavior exhibits
altruism. On the one hand, because the added allocation ${\bm{z}}=(6,1)$ is
more selfish than ${\bm{x}}=(4,4)$ and ${\bm{y}}=(5,2)$, this allocation
${\bm{z}}$ achieves a trade-off between personal ranking (selfishness) and
personal norms (altruism). On the other hand, by adding the allocation
${\bm{z}}=(6,1)$ to the doubleton, the decision maker may be incentivized to
engage in selfish behavior. This type of motivation can also be interpreted as
the temptation to act selfishly. Therefore, we can observe the following:

$\displaystyle\rho({\bm{x}},\\{{\bm{x}},{\bm{y}},{\bm{z}}\\})\leq\rho({\bm{y}},\\{{\bm{x}},{\bm{y}},{\bm{z}}\\}),$

which is a different stochastic behavior stemming from inequity-averse
preferences. For example, suppose that APU(SP) is represented by an entropy-
type cost function as follows:

$\displaystyle\varphi({\bm{x}})=(x_{1}+1)(x_{2}+1)$

for each ${\bm{x}}\in X$. Given menu $A$ with ${\bm{x}}\in A$,

$c_{\varphi({\bm{x}})}(\rho({\bm{x}},A))=\eta^{\frac{\gamma}{\varphi({\bm{x}})}}\rho({\bm{x}},A)\log\rho({\bm{x}},A),$
(9)

where $\eta=2$ and $\gamma=100$. Then, we have
$\rho({\bm{x}},\\{{\bm{x}},{\bm{y}}\\})\approx 0.5651$ and
$\rho({\bm{y}},\\{{\bm{x}},{\bm{y}}\\})\approx 0.4379$. In addition, because
the third allocation ${\bm{z}}$ is more selfish than the other two
allocations, it affects shame-mitigating behavior. Moreover, we have
$\rho({\bm{x}},\\{{\bm{x}},{\bm{y}},{\bm{z}}\\})\approx 0.2901$ and
$\rho({\bm{y}},\\{{\bm{x}},{\bm{y}},{\bm{z}}\\})\approx 0.3466$. Hence,
preference reversal can occur, that is,
$\rho({\bm{x}},\\{{\bm{x}},{\bm{y}},{\bm{z}}\\})<\rho({\bm{y}},\\{{\bm{x}},{\bm{y}},{\bm{z}}\\})$.

###  3.4 Discussion through Experimental Data

We discuss the explanatory power of APU(SP). We focus on experimental data
reported in Miao and Zhong (2018), who experimentally investigate preferences
for randomization in social decision-making. In their version of dictator
games, dictators determine choice probabilities between two allocations. Miao
and Zhong (2018) observed that a substantial proportion of subjects exhibit
preferences for randomization.

According to individual-level analysis in Miao and Zhong (2018), more than
half of the subjects’ behavior can be explained by ex-post fairness, ex-ante
fairness, or the combination. However, a substantial proportion of subjects is
incompatible with ex-post and ex-ante fairness preferences. These choice
patterns reflect different tastes, such as self-interest, equality and
efficiency motives underlying preference for randomization.

######  Remark 3.

(Objective Randomization vs. Subjective Randomization): Rigorously, subjective
randomization, i.e., deliberate randomization like a “mental” coin toss in
one’s mind is different from objective randomization like a randomization
device.

On the one hand, the choice of $p\in[0,1]$ over two allocations can be
interpreted as objective randomization. On the other hand, subjective
randomization may not be directly observable because analysts cannot observe
the “mental coin” directly. Here, there is no room to distinguish objective
randomization from subjective randomization.

However, the data suggests that subjects are willing to pay for an “objective”
coin toss. Here, we suppose that preferences for randomization in the
experimental data can lead to deliberately stochastic behavior.

We consider APU(SP) to complement incompatible choice patterns with a
combination of ex-post and ex-ante fairness. We present a numerical example.
We study the three cases of cost functions in deliberate randomization: (i)
inequity aversion, (ii) altruism, and (iii) selfishness.

Table 2: Experimental Data (Miao and Zhong, 2018) and APU(SP) | Miao & Zhong (2018) | Selfishness | Inequity Aversion | Shame
---|---|---|---|---
Menus | Interior Prob. | Mean Prob. | Choice Prob. of the Left Allocation
$\\{(20,0),(0,20)\\}$ | 0.10 | 0.67 | 1 | 0.5 | 0.676
$\\{(16,4),(4,16)\\}$ | 0.26 | 0.74 | 1 | 0.5 | 0.995
$\\{(0,20),(0,0)\\}$ | 0.18 | 0.41 | 0.5 | 0 | 0.5
$\\{(4,16),(0,0)\\}$ | 0.12 | 0.52 | 0.9815 | 0.0007 | 0.5
$\\{(0,20),(0,10)\\}$ | 0.25 | 0.49 | 1 | 0.001 | 0.622
$\\{(4,16),(2,8)\\}$ | 0.27 | 0.56 | 1 | 0.156 | 0.6516
$\\{(20,0),(0,0)\\}$ | 0 | 0 | 0.5 | 0 | 0.5
$\\{(16,4),(8,2)\\}$ | 0.11 | 0.76 | 0.8766 | 0.163 | 0.7735
$\\{(20,0),(10,0)\\}$ | 0.07 | 0.74 | 0.9995 | 0.001 | 0.629
$\\{(16,4),(0,0)\\}$ | 0.03 | 0.55 | 0.9999 | 0.0022 | 0.5
$\\{(10,10),(0,0)\\}$ | 0.03 | 0.66 | 1 | 1 | 0.5

In Table 2, to begin with, the data on Miao and Zhong (2018) represents the
percentage of choosing interior probability ($p\in(0,1)$) for the left
allocation in menus, and the mean probability for interior probabilities
implementing the left allocation, respectively (See Table 2 of Appendix 1 in
Miao and Zhong (2018).). Next, Selfishness (Column 4) captures the choice
probability of choosing the left allocation when the cost function is given in
Equation (5). Moreover, Inequity Aversion (Column 5) represents the choice
probability of choosing the left allocation when the cost function is given in
Equation (8). Finally, Shame (Column 6) represents the choice probability of
choosing the left allocation when the cost function is given in Equation (9).

Miao and Zhong (2018) report that a substantial proportion of subjects is
inconsistent with ex-post or ex-ante fairness concerns. For example, in the
two menus $\\{(20,0),(0,20)\\}$ and $\\{(16,4),(4,16)\\}$, compared with other
menus, we may guess a substantial proportion of subjects chooses interior
probabilites if ex-ante fairness matters. However, only 10 % and 26 % of
subjects choose interior porbabilities, respectively. Rather, observing
behavior in other menus, different motivations can matter. Indeed, beyond ex-
ante fairness concerns, various motives can provoke deliberately stochastic
behavior in social contexts. APU(SP) can capture these behavioral patterns.
Such subjects can be explained by APU(SP).

##  4 Literature Review

We review the literature on deliberate randomization in stochastic choices,
inequality aversion, and shame/guilt.

#####  Preference for Randomization and Deliberate Randomization.

Our model is related to the additive perturbed utility (APU) in Fudenberg et
al. (2014, 2015) because we apply the concept of APU to social contexts. We
apply a weaker version of the acyclic condition in the stochastic choice, that
is, Menu Acyclicity. We consider a specific model of a menu-invariant APU in
which the uniqueness result is recovered through additional conditions. Our
model allows for violations of Luce’s IIA, a well-known property of stochastic
choice. In Corollary 2, deliberate stochastic behavior can correspond to Weak
APU, in which the personal norm is purely selfish.

Cerreia-Vioglio et al. (2019) characterize a deliberately stochastic choice
model in risky choices inspired by Machina (1985), and provide a new acyclic
condition with first-order stochastic dominance (FOSD) called Rational Mixing.
Hashidate and Yoshihara (2021) verifies that social preferences like inequity
aversion might deviate from FOSD.

To begin with, in terms of the axiom of Regularity (Axiom 4), APU(SP)
satisfies this axiom, whereas Cerreia-Vioglio et al. (2019) do not. Their
notion of deliberate randomization can occur through the violations of
Regularity. Hence, APU(SP) does not nest the deliberate stochastic model of
Cerreia-Vioglio et al. (2019).

Next, we consider the property of stochastic dominance. In Corollary 1,
deliberately stochastic behavior satisfies the selfishness-based FOSD
(Definition 3); thus, it is a special case of Cerreia-Vioglio et al. (2019).
Indeed, Machina (1985)’s model can be interpreted as APU, where the decision
maker maximizes expected utility with a convex function (Strzalecki, 2022).
The non-linearity can capture non-expected utility in risky environments. This
flavor is related to Cerreia-Vioglio et al. (2019). Our setup is not based on
risk preferences, but we can consider preferences over lotteries of
allocations. We then consider both objective and subjective randomization
together.

In sum, our approach can have a flavor of Random Utility (henceforth, RU). In
this paper, we have two multiple utility functions. One is a selfish utility
function. The other is a personal norm function. APU(SP) captures the trade-
off. Fudenberg et al. (2015) consider a relationship between random utility
and APU. In this sense, APU(SP) has a more specific structure of the trade-off
between selfishness and personal norms than RU does when we apply social
preferences to RU.

#####  Inequity Aversion.

Saito (2013) developed an inequity-averse model under risk, a convex
combination of ex-ante and ex-post fairness, called the expected inequity-
averse model (EIA).202020See, for the axiomatic study of ex-post fairness,
Rohde (2010), and for the experimental evidence on ex-ante fairness, Brock et
al. (2013) and Miao and Zhong (2018). In this study, we do not consider ex-
ante fairness because we have not considered the stochastic behavior stemming
from ex-ante fairness. Hashidate and Yoshihara (2021) complements this aspect
by assuming that the utility function is based on Saito (2013), and that
objective randomization can lead to deliberately stochastic behavior.

Cheung (2023) axiomatically studies a model of guilt moderation. In the Cheung
(2023)’s utility representation, both “envy” and “guilt,” stemming from payoff
differences between dictator’s payoffs and recipient’s payoffs, are evaluated
by Choquet expected utility (Schmeidler, 1989). Such a convex preference can
also lead to deliberately stochastic behavior.

Our approach differs from the previous studies. The primitive of the model
differs in the related studies. In addition, the related studies modify or
extend a selfish utility function to capture ex-ante, ex-post fairness, or
their extensions. We do not modify a utility function; Rather, we consider the
case that the randomization cost function in APU is based on inequity-averse
preferences (See Subsection 3.3.).

#####  Shame/Guilt.

Dillenberger and Sadowski (2012) is a seminal axiomatic study on
shame.212121In related studies, see, for example, Saito (2015b) and Hashidate
(2021). They investigate preferences over menus and introduce a preference for
commitment (preferring smaller menus) to capture the shame of acting
selfishly. To identify shame, they study an asymmetric two-stage decision
problem; In the first stage of choosing a menu, other agents, such as
recipients, can not publicly observe the decision-maker’s choice. However,
their behavior is publicly observed in the second stage of the choice from the
menu chosen in the first stage.

We take a stochastic function as a primitive and construct a stochastic choice
model considering randomization costs. Our approach differs in the
preferences-over-menus framework. Even though the publicity of choice behavior
is the same as the related studies, we try to capture shame-driven behavior by
avoiding choosing the most selfish allocation with certainty, which leads to
deliberate randomization. For example, the level of shame can differ in the
timing of decision-making. Some image-conscious decision-makers can feel shame
once they take selfish actions. Others can feel shame after they take selfish
actions repeatedly. In both cases, shame can lead to deliberate randomization.

Noor and Ren (2023) axiomatically characterize a model of “guilt.” They
consider three-stage decision problems and investigate preferences over menus
of menus. For example, they consider a choice between
$\\{\\{{\bm{x}}\\},\\{{\bm{y}}\\}\\}$ and $\\{\\{{\bm{x}},{\bm{y}}\\}\\}$. The
menu of menus $\\{\\{{\bm{x}}\\},\\{{\bm{y}}\\}\\}$ is similar to the menu
$\\{{\bm{x}},{\bm{y}}\\}$, if the timing of decision-making does not matter.
In reality, people may have time lags in each stage of decision problems. If
so, the decision maker is tempted to be selfish when he/she faces the menu
$\\{{\bm{x}},{\bm{y}}\\}$, even though his/her personal norm can support that
he/she chooses a more altruistic allocation. Such a self-control cost can be
interpreted as “guilt.” To avoid such guilt, At the first stage of decision
problems, the decision maker can commit to the singleton of the selfish
allocation.

Our approach is different from Noor and Ren (2023). First, for simplicity, we
assume that the decision-maker’s personal ranking is selfish, unlike the
foundation of Noor and Ren (2023). They consider such selfishness stemming
from “guilt-avoidance” at the choice of menus of menus. Next, the role of
personal norms is considered differently. In APU(SP), the trade-off between
selfishness and personal norms can lead to stochastic behavior. On the other
hand, in Noor and Ren (2023), the existence of personal norms can lead to
preferences for commitment to selfish allocations.

##  5 Conclusion

In this study, we examined stochastic prosocial behavior stemming from impure
altruism, such as shame and inequity-aversion. By applying APU (Fudenberg et
al., 2014, 2015) in social contexts, we characterized APU(SP), where the
utility function is purely selfish, and the randomization cost function
captures psychological costs such as shame and guilt-avoidance.

There has been limited research on social preferences in the stochastic choice
literature. Thus, this study contributes to a better understanding of human
decision-making in social contexts. Our results suggest that fairness and
social concerns can lead to deliberate stochastic behavior. It is also worth
noting that different motivations for deliberate randomization in social
contexts have different behavioral patterns in choice probabilities. The
psychological costs of randomization characterize these behavioral patterns.
Thus, axiomatization in this study facilitates the identification of
motivations behind stochastic prosocial behavior.

This study has some limitations, which offer opportunities for future
research. Theoretically, we do not consider ex-ante fairness (Fudenberg and
Levine, 2012; Saito, 2013). Therefore, future studies should consider this
aspect. In particular, our model is within the framework of perturbed utility
models. We can consider a more general framework (Cerreia-Vioglio et al.,
2019), beyond Positivity and Regularity.

Moreover, we can consider an axiomatic foundation for the distinction between
objective and subjective randomization. Objective randomization can occur due
to convex preferences in decision-making under risk/uncertainty. To capture
subjective randomization, we can consider implementation costs as in APU.

Additionally, it may be beneficial to conduct experimental studies on social
preferences and stochastic or risky choices. The personal norms of decision-
makers are diverse. Some may be based on social norms, conventions, and
cultures, while others may differ. This difference can be captured and
identified through behavior.

##  Appendix A Proof of Theorem 1

###  A.1 Sufficiency Part

####  A.1.1 Step 1

In Step 1, we show Lemma 1. We say that a function
$c:[0,1]\rightarrow\mathbb{R}\cup\\{\infty\\}$ is a cost function if it is
strictly convex, and differentiable, i.e., $C^{1}$ over $(0,1)$. We say that
$\rho$ has a menu-invariant APU if there is a pair
$(u,(c_{\bm{x}})_{{\bm{x}}\in X})$ where $u:X\rightarrow\mathbb{R}$ is a
utility function, and $c_{\bm{x}}:[0,1]\rightarrow\mathbb{R}\cup\\{+\infty\\}$
is a cost function for each ${\bm{x}}\in X$, such that $\rho$ has a form

$\displaystyle\rho(A)=\arg\max_{\rho\in\Delta(A)}\sum_{{\bm{x}}\in
A}\Bigl{(}u({\bm{x}})\rho({\bm{x}},A)-c_{\bm{x}}(\rho({\bm{x}},A))\Bigr{)}.$

We show the following lemma, following Fudenberg et al. (2014). Suppose that
$\rho$ satisfies positivity.

######  Lemma 1.

The following statements are equivalent:

  1. (i)

$\rho$ is a menu-invariant APU.

  2. (ii)

There exists $\lambda:\mathcal{A}\rightarrow\mathbb{R}$ such that
$\lambda(A)>\lambda(B)$ if $\rho({\bm{x}},A)>\rho({\bm{x}},B)$, and
$\lambda(A)=\lambda(B)$ if $\rho({\bm{x}},A)=\rho({\bm{x}},B)$.

  3. (iii)

$\rho$ satisfies Menu Acyclicity.

###### Proof.

We show that $(i)\Rightarrow(ii)$, $(ii)\Rightarrow(i)$, and
$(ii)\Leftrightarrow(iii)$.

#####  $(i)\Rightarrow(ii)$:

Suppose that $\rho$ has a menu-invariant APU. Take arbitrary menus
$A,B\in\mathcal{A}$ with ${\bm{x}}\in A\cap B$.

Let $\lambda:\mathcal{A}\rightarrow\mathbb{R}$ be the Lagrangean multiplier
associated with each menu. The first-order condition (FOC) for $\rho$ is as
follows.

$u(x_{1})-c^{\prime}_{{\bm{x}}}(\rho({\bm{x}},A))+\lambda(A)\begin{cases}\geq
0&\text{if $\rho({\bm{x}},A)=1$,}\\\ =0&\text{if
$\rho({\bm{x}},A)\in(0,1)$,}\\\ \leq 0&\text{if
$\rho({\bm{x}},A)=0$.}\end{cases}$ (10)

By the FOC for $\rho$ and the strict convexity of $c_{\bm{x}}$,

$\displaystyle\rho({\bm{x}},A)>\rho({\bm{x}},B)$ $\displaystyle\Leftrightarrow
c^{\prime}_{\bm{x}}(\rho({\bm{x}},A))>c^{\prime}_{\bm{x}}(\rho({\bm{x}},B))$
$\displaystyle\Rightarrow\lambda(A)>\lambda(B).$

Hence, we have $\lambda(A)>\lambda(B)$ if $\rho({\bm{x}},A)>\rho({\bm{x}},B)$.
And, in the same way, we have $\lambda(A)=\lambda(B)$ if
$\rho({\bm{x}},A)=\rho({\bm{x}},B)\in(0,1)$.

#####  $(ii)\Rightarrow(i)$:

Suppose that $\lambda:\mathcal{A}\rightarrow\mathbb{R}$ satisfies
$\lambda(A)>\lambda(B)$ if $\rho({\bm{z}},A)>\rho({\bm{z}},B)$, and
$\lambda(A)=\lambda(B)$ if $\rho({\bm{z}},A)=\rho({\bm{z}},B)$, for all
$A,B\in\mathcal{A}$ and ${\bm{z}}\in X$.

We construct $(c_{{\bm{x}}})_{{\bm{x}}\in X}$ using $\lambda$. Without loss of
generality, assume that $\lambda(\cdot)\in(0,1)$. Take an arbitrary allocation
${\bm{z}}\in X$. Let

$\displaystyle\overline{w}({\bm{z}}):=\begin{cases}1&\text{if
$\rho({\bm{z}},A)>0$, for all $A\ni{\bm{z}}$,}\\\ \min\\{\lambda(A)\mid
A\in\mathcal{A},\rho({\bm{z}},A)=1\\}&\text{otherwise.}\end{cases}$

And, let

$\displaystyle\underline{w}({\bm{z}}):=\begin{cases}0&\text{if
$\rho({\bm{z}},A)>0$, for all $A\ni{\bm{z}}$,}\\\ \max\\{\lambda(A)\mid
A\in\mathcal{A},\rho({\bm{z}},A)=1\\}&\text{otherwise.}\end{cases}$

We construct a function $g_{\bm{z}}:[0,1]\rightarrow\mathbb{R}$ for each
${\bm{z}}\in X$ such that

  1. (i)

$g_{\bm{z}}(0)=\underline{w}({\bm{z}})$;

  2. (ii)

$g_{\bm{z}}(\rho({\bm{z}},A))=\lambda(A)$ if $\rho({\bm{z}},A)\in(0,1)$; and

  3. (iii)

$g_{\bm{z}}(1)=\overline{w}({\bm{z}})$.

Define a cost function $c_{\bm{z}}:[0,1]\rightarrow\mathbb{R}$ by

$\displaystyle c_{\bm{z}}(q):=\int_{0}^{q}g_{\bm{z}}(p)dp.$

Take ${\bm{z}}\in A\cap B$ for some $A,B\in\mathcal{A}$. Suppose
$\rho({\bm{z}},A),\rho({\bm{z}},B)\in(0,1)$ with
$\rho({\bm{z}},A)>\rho({\bm{z}},B)$. Then, we have

$\displaystyle\lambda(A)>\lambda(B)$ $\displaystyle\Leftrightarrow
g_{\bm{z}}(\rho({\bm{z}},A))>g_{\bm{z}}(\rho({\bm{z}},B))$
$\displaystyle\Leftrightarrow\int_{0}^{\rho({\bm{z}},A)}g_{\bm{z}}(p)dp>\int_{0}^{\rho({\bm{z}},B)}g_{\bm{z}}(p)dp$
$\displaystyle\Leftrightarrow
u({\bm{z}})-g_{\bm{z}}(\rho({\bm{z}},A))>u({\bm{z}})-g_{\bm{z}}(\rho({\bm{z}},B)).$

The last inequality is the FOC for $\rho$ in menu-invariant APU.

We show $C^{1}_{{\bm{z}}}$ over $(0,1)({\bm{z}}\in X)$. By definition, we
obtain

$\displaystyle\frac{d}{dq}\int_{0}^{q}g_{\bm{z}}(p)dp=g_{\bm{z}}(q).$

By definition and the assumption of $\lambda(\cdot)\in(0,1)$, the strict
convexity of $c_{\bm{z}}$ follows from $g^{\prime}_{\bm{z}}(q)>0$ for all
$q\in(0,1)$.

#####  $(ii)\Leftrightarrow(iii)$:

We apply the following lemma in Fudenberg et al. (2014).

######  Lemma 2.

Let $\mathcal{X}$ be a finite set. Suppose that
$\succ,\sim\subset\mathcal{X}\times\mathcal{X}$ such that the $\succ$ is
asymmetric and $\sim$ is symmetric. Then, the following conditions are
equivalent.

  1. (i)

There is no cycle meaning that there is no sequence

$\displaystyle(\chi_{1},\chi_{2}),(\chi_{2},\chi_{3}),\cdots,(\chi_{m-1},\chi_{m}),(\chi_{m},\chi_{1})$

in $\succ\cup\sim$ where at least one of them belongs to $\succ$.

  2. (ii)

There exists a function $v:\mathcal{X}\rightarrow\mathbb{R}$ such that
$v(\chi)>v(\chi^{\prime})$ if $\chi\succ\chi^{\prime}$, and
$v(\chi)=v(\chi^{\prime})$ if $\chi\sim\chi^{\prime}$.

Define a binary relation $\succsim_{m}$ over $\mathcal{A}$, i.e., a menu
ranking, as follows. We say that a menu $A$ is revealed weaker than a menu
$B$, i.e., $A\succ_{m}B$, if $\rho({\bm{x}},A)>\rho({\bm{x}},B)$ for some
${\bm{x}}\in A\cap B$. In the similar way, we say that a menu $A$ is revealed
tied with a menu $B$, i.e., $A\sim_{m}B$, if
$\rho({\bm{x}},A)=\rho({\bm{x}},B)\in(0,1)$ for some ${\bm{x}}\in A\cap B$.
Let $\succsim_{m}:=\succ_{m}\cup\sim_{m}$.

Let $\succsim$ be a menu ranking $\succsim_{m}$ with
$\mathcal{X}=\mathcal{A}$. Then, condition (i) can be rewritten as follows.
There does not a sequence of menus $\\{A_{k}\\}_{k=1}^{m}$ such that

$\displaystyle
A_{1}\succsim_{m}A_{2}\succsim_{m}\cdots\succsim_{m}A_{m}\succ_{m}A_{1}.$

By the definition of $\succsim_{m}$, this condition satisfies Menu Acyclicity.

Let $v=\lambda$. Then, the desired result is obtained. $\Box$

####  A.1.2 Step 2

In Step 2, we show Lemma 3, which describes the characterization of personal
norm rankings $\succsim_{n}^{\rho}$.

######  Lemma 3.

$\succsim^{\rho}_{n}$ on $X$ is represented by a function
$\varphi:X\rightarrow\mathbb{R}$ that is continuous and monotone with respect
to fair allocations, provided that

$\displaystyle{\bm{x}}\succsim_{n}^{\rho}{\bm{y}}\ \text{if}\ \text{only}\
\text{if}\ \varphi({\bm{x}})\geq\varphi({\bm{y}}).$

To prove the lemma, we show the following three claims.

#####  Utility Representation.

First, we show that $\succsim_{n}^{\rho}$ is represented by
$\varphi:X\rightarrow\mathbb{R}$.

######  Claim 1.

There exists $\varphi:X\rightarrow\mathbb{R}$ that represents
$\succsim_{n}^{\rho}$.

The proof of Claim 1 is standard in decision theory. We apply the result
(Theorem 1.4.8) in Bridges and Mehta (1995).

##### ($\Rightarrow$):

The so-called Birkhoff’s order separability is necessary and sufficient for
the existence of an order isomorphism; That is, for any ${\bm{x}},{\bm{y}}\in
X$, if ${\bm{x}}\succsim_{n}^{\rho}{\bm{y}}$, then
$\varphi({\bm{x}})\geq\varphi({\bm{y}})$. Such a function $\varphi$ is called
an order homomorphism. Birkhoff’s order separability in our setting is as
follows.

######  AXIOM 1.

(Birkhoff’s Order Separability): There exists a countable subset $Z\subset X$
such that for any ${\bm{x}},{\bm{y}}\in X\setminus Z$ there exists
${\bm{z}}\in Z$ with
${\bm{x}}\succ_{n}^{\rho}{\bm{z}}\succ_{n}^{\rho}{\bm{y}}$.

Remember that $X\subseteq\mathbb{R}^{2}$ is a compact set of allocations.
Suppose that $X$ has a countable order-dense subset $Z^{\prime}$ in the sense
of Birkhoff. Let $Z^{\prime\prime}$ be the set of end point of all the jumps
of $X$. Let $({\bm{x}},{\bm{y}})$ be the open interval denoted by

$\displaystyle({\bm{x}},{\bm{y}}):=\\{{\bm{z}}\in
X\mid{\bm{x}}\prec_{n}^{\rho}{\bm{z}}\prec_{n}^{\rho}{\bm{y}}\\}.$

We say that the open interval $({\bm{x}},{\bm{y}})$ with ${\bm{x}},{\bm{y}}\in
X$ is called a jump with end points ${\bm{x}}$ and ${\bm{y}}$, if it is empty.

The set $Z^{\prime\prime}$ is countable. By the axiom of Personal Norm
Ranking, $\succsim_{n}^{\rho}$ is a total preorder; That is, it is (i)
reflexive, (ii) transitive, and (iii) connected. The binary relation
$\succsim_{n}^{\rho}$ of personal norm rankings satisfies the three
conditions. Hence, $(X,\succsim_{n}^{\rho})$ is a totally preordered set.
There are countably many jumps in a totally preordered set
$(X,\succsim_{n}^{\rho})$ (Bridges and Mehta, 1995).

Let $Z:=Z^{\prime}\cup
Z^{\prime\prime}=\\{{\bm{z}}^{1},{\bm{z}}^{2},\cdots\\}$. Define a function
$r:X\times X\rightarrow\\{0,1\\}$ by

$\displaystyle r({\bm{x}},{\bm{y}})=\begin{cases}1&\text{if}\
{\bm{y}}\succ_{n}^{\rho}{\bm{x}}\\\ 0&\text{otherwise.}\end{cases}$

Then, define a real-valued function $\varphi:X\rightarrow\mathbb{R}$ by

$\displaystyle\varphi({\bm{x}}):=\sum_{n=1}^{\infty}2^{-n}r({\bm{z}}^{n},{\bm{x}}).$

We show that $\varphi$ is order embedding; That is, for any
${\bm{x}},{\bm{y}}\in X$, ${\bm{x}}\succsim_{n}^{\rho}{\bm{y}}$ if and only if
$\varphi({\bm{x}})\geq\varphi({\bm{y}})$. Take arbitrary ${\bm{x}},{\bm{y}}\in
X$.

Suppose that ${\bm{x}}\succsim_{n}^{\rho}{\bm{y}}$. Since
$(X,\succsim_{n}^{\rho})$ is a totally preordered set, there exists a natural
number $n$ such that ${\bm{y}}\succ_{n}^{\rho}{\bm{z}}^{n}$. Then, since
$\succsim_{n}^{\rho}$ is transitive, we have
${\bm{x}}\succ_{n}^{\rho}{\bm{z}}^{n}$. Hence, by definition, we obtain
$\varphi({\bm{x}})\geq\varphi({\bm{y}})$.

Suppose that ${\bm{x}}\succ_{n}^{\rho}{\bm{y}}$. We show
$\varphi({\bm{x}})>\varphi({\bm{y}})$. To prove it, we consider the two cases.
First, the open interval $({\bm{y}},{\bm{x}})$ is a jump. Second, the open
interval $({\bm{y}},{\bm{x}})$ is not a jump.

Consider the first case. Then, we have ${\bm{y}}\in Z^{\prime\prime}\subset
Z$. There exists a natural number $k$ such that ${\bm{y}}={\bm{z}}^{k}$, but
$\neg({\bm{y}}\succ_{n}^{\rho}{\bm{z}}^{k})$, i.e.,
${\bm{y}}\nsucc_{n}^{\rho}{\bm{z}}^{k}$. By the definition of
$\succsim_{n}^{\rho}$, we have ${\bm{z}}^{k}\succsim_{n}^{\rho}{\bm{y}}$. By
the above argument, we have $\varphi({\bm{z}}^{k})\geq\varphi({\bm{y}})$.
Since ${\bm{x}}\succ_{n}^{\rho}{\bm{y}}$, ${\bm{x}}\neq{\bm{y}}$. Hence,
${\bm{x}}\neq{\bm{z}}^{k}$. If ${\bm{z}}^{k}\succsim_{n}^{\rho}{\bm{x}}$, we
have $\varphi({\bm{z}}^{k})\geq\varphi({\bm{x}})$. By the definition of
$\varphi$, it is a contradiction. Thus,
${\bm{x}}\succ_{n}^{\rho}{\bm{z}}^{k}$. We obtain
$\varphi({\bm{x}})>\varphi({\bm{z}}^{k})\geq\varphi({\bm{y}})$. Therefore,
$\varphi({\bm{x}})>\varphi({\bm{y}})$.

Consider the second case. There exists ${\bm{\nu}}\in X$ such that
${\bm{x}}\succ_{n}^{\rho}{\bm{\nu}}$ and ${\bm{\nu}}\succ_{n}^{\rho}{\bm{y}}$.
Suppose that ${\bm{y}}\in Z$. Then, we obtain
$\varphi({\bm{x}}>\varphi({\bm{y}})$. Suppose that ${\bm{y}}\notin Z$. If
${\bm{\nu}}\in Z$, then there exists a natural number $j$ such that
${\bm{\nu}}={\bm{z}}^{j}$, ${\bm{x}}\succ_{n}^{\rho}{\bm{z}}^{j}$, and
${\bm{y}}\nsucc_{n}^{\rho}{\bm{z}}^{j}$. Then, by the above argument, we have
$\varphi({\bm{x}})>\varphi({\bm{y}})$. Suppose ${\bm{\nu}}\notin Z$. Then, by
the Birkhoff’s order separability, there exists ${\bm{x}}^{m}\in Z^{\prime}$
such that ${\bm{x}}\succ_{n}^{\rho}{\bm{z}}^{m}\succ_{n}^{\rho}{\bm{y}}$.
Hence, we obtain ${\bm{x}}\succ_{n}^{\rho}{\bm{z}}^{m}$ and
${\bm{y}}\nsucc_{n}^{\rho}{\bm{z}}^{m}$. By the above argument, we have
$\varphi({\bm{x}})>\varphi({\bm{y}})$.

##### ($\Leftarrow$):

Suppose that there exists $\varphi:X\rightarrow\mathbb{R}$. Take an arbitrary
pair of rational numbers $r,r^{\prime}\in\mathbb{Q}$ with $r<r^{\prime}$. Let

$\displaystyle A_{r,r^{\prime}}:=\\{{\bm{x}}\in X\mid
r<\varphi({\bm{x}})<r^{\prime}\\}.$

If the set $A_{r,r^{\prime}}$ is non-empty, we can choose an element
${\bm{a}}=(r,r^{\prime})\in A_{r,r^{\prime}}$. We construct

$\displaystyle A:=\cup_{r,r^{\prime}}\\{{\bm{a}}_{r,r^{\prime}}\\},$

which is the union of the sets $\\{{\bm{a}}_{r,r^{\prime}}\\}$ with
$A_{r,r^{\prime}}\neq\\{\varnothing\\}$. Then, the set $A$ is countable.

Let $K$ be the set of end points of all jumps in $X$. Suppose that the open
interval $({\bm{x}},{\bm{y}})$ is a jump in $X$. Then, the interval
$(\varphi({\bm{x}}),\varphi({\bm{y}}))$ in $\mathbb{R}$ contains a rational
number $r_{{\bm{x}},{\bm{y}}}$. Hence, there is an injection from the set of
all jumps in $X$ into the countable set of rational numbers. It follows that
the set $K$ is countable.

We have the following fact.

######  Fact 1.

The set $Z:=A\cup K$ is a Birkhoff’s order-dense subset.

###### Proof.

Observe that if there exist ${\bm{x}},{\bm{y}}\in X\setminus Z$ with
${\bm{y}}\succ_{n}^{\rho}$, then the open interval $({\bm{x}},{\bm{y}})$ is
not a jump. Hence, there exists ${\bm{\nu}}\in X$ such that
${\bm{y}}\succ_{n}^{\rho}{\bm{\nu}}\succ_{n}^{\rho}{\bm{x}}$. Since the
function $\varphi$ is order embedding, there exist rational numbers
$r,r^{\prime}\in\mathbb{Q}$ such that

$\displaystyle\varphi({\bm{y}})>r>\varphi({\bm{\nu}})>r^{\prime}>\varphi({\bm{x}}).$

This implies that the set $A_{r,r^{\prime}}$ is non-empty. Moreover, we have
${\bm{y}}\succ_{n}^{\rho}{\bm{a}}_{r,r^{\prime}}\succ_{n}^{\rho}{\bm{x}}$.
Therefore, $X$ is order separable in the sense of Birkhoff; That is, it is
shown that there exists a Birkhoff order-dense subset. $\Box$

#####  Continuity.

Next, we show that $\varphi:X\rightarrow\mathbb{R}$ is continuous. The proof
of Claim 2 is based on the procedure of Claim 3. We show that the defined
$\varphi$ is continuous.

######  Claim 2.

$\varphi:X\rightarrow\mathbb{R}$ is continuous.

###### Proof.

It suffices to show that $\varphi^{-1}((a,b))$ is open for all
$a,b\in\mathbb{R}$, where $(a,b)$ is an open interval. Without loss of
generality, for any $a\in\mathbb{R}$, let $\varphi(a,a)=\alpha(a)=a$. And,

$\displaystyle\varphi^{-1}((a,b))=\varphi^{-1}((a,\infty)\cap(-\infty,b))=\varphi^{-1}((a,\infty))\cap\varphi^{-1}((-\infty,b)).$

We have $\varphi(a,a)=a$, so

$\displaystyle\varphi^{-1}((a,b))=\varphi^{-1}((\varphi(a,a),\infty))=\\{{\bm{x}}\in
X_{+}|{\bm{x}}\succ_{n}^{\rho}(a,a)\\}.$

The set $\\{{\bm{x}}\in X_{+}|{\bm{x}}\succ_{n}^{\rho}(a,a)\\}$ is open. The
strict upper contour set of $(a,a)$ is open if $\succsim_{n}^{\rho}$ is
continuous. In the same way, the strict lower contour set of $(b,b)$ is also
open. $\varphi^{-1}((a,b))$ is, therefore, open since it is the intersection
of two open sets. $\Box$

#####  Monotonicity with respect to Fair Allocations.

Finally, we show that $\varphi\rightarrow\mathbb{R}$ is monotone with respect
to fair allocations. Remember that we say that
$\varphi:X\rightarrow\mathbb{R}$ is monotone with respect to fair allocations
if for any ${\bm{x}},{\bm{y}}\in X_{+}$ (or $X_{-}$),
${\bm{x}}\geq{\bm{y}}\Rightarrow\varphi({\bm{x}})\geq\varphi({\bm{y}})$.

######  Claim 3.

$\varphi:X\rightarrow\mathbb{R}$ is monotone with respect to fair allocations.

###### Proof.

Consider the case of $X_{+}$. Take arbitrary ${\bm{x}}\in X_{+}$. By the
definition of $\succ_{n}^{\rho}$, ${\bm{x}}\succsim_{n}^{\rho}(0,0)$. Then
there exists $\overline{\alpha}\in\mathbb{R}_{+}$ such that
$(\overline{\alpha},\overline{\alpha})\succsim_{n}^{\rho}{\bm{x}}$.

We show the following: For any ${\bm{x}}\in X_{+}$, there exists
$\alpha({\bm{x}})\in\mathbb{R}_{+}$ such that
${\bm{x}}\sim_{n}^{\rho}(\alpha({\bm{x}}),\alpha({\bm{x}}))$. Let
$A^{-}:=\\{\alpha\in\mathbb{R}|{\bm{x}}\succsim_{n}^{\rho}(\alpha,\alpha)\\}$
and
$A^{+}:=\\{\alpha\in\mathbb{R}|(\alpha,\alpha)\succsim_{n}^{\rho}{\bm{x}}\\}$.
By the weak monotonicity of $\succsim_{n}^{\rho}$ in definition, we have
$A^{-}\neq\varnothing$ and $A^{+}\neq\varnothing$. Since $\succsim_{n}^{\rho}$
is a closed preorder, both $A^{-}$ and $A^{+}$ are closed sets. By the
connectedness of $\mathbb{R}_{+}$, $A^{-}\cap A^{+}\neq\varnothing$.
$A^{-}\cap A^{+}$ is a singleton set.

Define $\alpha({\bm{x}})$ to be the element of $A^{-}\cap A^{+}$. Now, let us
define $\varphi:X\rightarrow\mathbb{R}$ by
$\varphi({\bm{x}})=\alpha({\bm{x}})$ for each ${\bm{x}}\in X_{+}$. For any
${\bm{x}},{\bm{y}}\in X_{+}$,

$\displaystyle{\bm{x}}\succsim_{n}^{\rho}{\bm{y}}$
$\displaystyle\Leftrightarrow(\alpha({\bm{x}}),\alpha({\bm{x}}))\succsim_{n}^{\rho}(\alpha({\bm{y}}),\alpha({\bm{y}}))$
$\displaystyle\Leftrightarrow\alpha({\bm{x}})\geq\alpha({\bm{y}})$
$\displaystyle\Leftrightarrow\varphi({\bm{x}})\geq\varphi({\bm{y}}).$

Hence, for any ${\bm{x}},{\bm{y}}\in X_{+}$ if ${\bm{x}}\geq{\bm{y}}$, then
$\varphi({\bm{x}})\geq\varphi({\bm{y}})$. In the same way, we can show the
case of $X_{-}$, so we skip the part. $\Box$

####  A.1.3 Step 3

In this step, we show that $u:\mathbb{R}\rightarrow\mathbb{R}$ is selfish and
continuous (Lemma 4).

######  Lemma 4.

There exists a continuous function $u:\mathbb{R}\rightarrow\mathbb{R}$ by
$u(x_{1})$ for each ${\bm{x}}\in X$.

We show the two claims.

######  Claim 4.

$u:\mathbb{R}\rightarrow\mathbb{R}$ is selfish, i.e., for each ${\bm{x}}\in
X$, $u:=u(x_{1})$.

###### Proof.

Suppose not. Let $u({\bm{x}})=x_{1}+x_{2}$. Take ${\bm{x}},{\bm{y}}\in X$ such
that $x_{1}>y_{1}$ and $x_{1}+x_{2}<y_{1}+y_{2}$. Without loss of generality,
assume that $\varphi({\bm{x}})<\varphi({\bm{y}})$. Consider an arbitrary menu
$A\in\mathcal{A}$ with ${\bm{x}},{\bm{y}}\in A$. Then,
$u({\bm{x}})=x_{1}+x_{2}<y_{1}+y_{2}=u({\bm{y}})$ and
$\varphi({\bm{x}})<\varphi({\bm{y}})$ hold. By construction, we have
$\rho({\bm{x}},A)<\rho({\bm{y}},A)$. Without loss of generality, assume
$\rho({\bm{y}},A)=1$. The axiom of Selfishness requires that $y_{1}>x_{1}$.
However, $x_{1}>y_{1}$. This is a contradiction. $\Box$

######  Claim 5.

$u:\mathbb{R}\rightarrow\mathbb{R}$ is continuous.

###### Proof.

To show the claim, we show that for any open set $Y\subset\mathbb{R}$,
$u^{-1}(Y)$ is open. Let $Y=(a,b)$ be the open interval where
$a,b\in\mathbb{R}$ with $a<b$. Consider
$u^{-1}(Y)=\\{x_{1}\in\mathbb{R}|u(x_{1})\in Y\\}$. Take $x_{1}\in u^{-1}(Y)$.
By the continuity of $\rho$, both
$E=\\{x_{1}\in\mathbb{R}|u(x_{1})>u(y_{1})\\}$ and
$F=\\{x_{1}\in\mathbb{R}|u(z_{1})>u(x_{1})\\}$ are open. Without loss of
generality, let $b=u(y_{1})$, and $a=u(z_{1})$. By the continuity of $\rho$,
the set $u^{-1}(Y)\cup E\cup F$ is open. Hence, $u^{-1}(Y)$ is open. $\Box$

####  A.1.4 Step 4

In this step, we complete the proof and obtain the desired utility
representation (APU(SP)). We incorporate the personal-norm ranking
($\succsim_{n}^{\rho}$) into menu-invariant APU. In the first-order optimality
condition (FOC) for $\rho$, we construct the cost function by replacing
$c_{{\bm{x}}}$ with $c_{\varphi({\bm{x}})}$ for each ${\bm{x}}\in X$.

Given an arbitrary menu $A$ with ${\bm{x}}\in A$, take an allocation
${\bm{y}}\in X\setminus A$ such that ${\bm{y}}\succ_{n}^{\rho}{\bm{x}}$. Fix
${\bm{x}}$ and ${\bm{y}}$. For notational convenience, let
$B:=A\cup\\{{\bm{y}}\\}$. By the definition of the personal-norm ranking
($\succsim_{n}^{\rho}$), we obtain $\rho({\bm{x}},A)>\rho({\bm{x}},B)$.
Without loss of generality, assume that
$\rho({\bm{x}},A),\rho({\bm{x}},B)\in(0,1)$. Then, by the menu-invariant
utility representation with a purely selfish utility $u$, we have

$\displaystyle\rho({\bm{x}},A)>\rho({\bm{x}},B)$
$\displaystyle\Leftrightarrow\lambda(A)>\lambda(B)$
$\displaystyle\Leftrightarrow-\lambda(A)<-\lambda(B)$
$\displaystyle\Leftrightarrow
u(x_{1})-c^{\prime}_{{\bm{x}}}(\rho({\bm{x}},A))<u(x_{1})-c^{\prime}_{{\bm{y}}}(\rho({\bm{x}},B))$
$\displaystyle\Leftrightarrow
c^{\prime}_{{\bm{x}}}(\rho({\bm{x}},A))>c^{\prime}_{{\bm{y}}}(\rho({\bm{x}},B)).$

As in Step 3, we can restrict the utility function $u$ to the selfish utility,
i.e., $u({\bm{x}}):=u(x_{1})$. Remember that, by the FOC for $\rho$, we have

$\displaystyle
u(x_{1})-c^{\prime}_{{\bm{x}}}(\rho({\bm{x}},A))=u(y_{1})-c^{\prime}_{{\bm{y}}}(\rho({\bm{y}},A)).$

Without loss of generality, assume that $u(x_{1})>u(y_{1})$. Then, we have
$c^{\prime}_{{\bm{x}}}(\rho({\bm{x}},A))>c^{\prime}_{{\bm{y}}}(\rho({\bm{y}},A))$.
By the strict convexity of $c_{{\bm{x}}}$ for each ${\bm{x}}\in X$, we have
$c^{\prime\prime}_{{\bm{x}}}(\rho({\bm{x}},A))>c^{\prime\prime}_{{\bm{y}}}(\rho({\bm{y}},A))$.
Hence, it is verified that $\rho({\bm{x}},A)>\rho({\bm{y}},A)$. Let
$p=\rho({\bm{x}},A)$ and $q=\rho({\bm{y}},A)$. Since
$c^{\prime}_{{\bm{x}}}(\cdot)$ is monotone, we have
$c^{\prime}_{{\bm{x}}}(p)>c^{\prime}_{{\bm{y}}}(q)$.

Finally, we construct $c_{\varphi(\cdot)}(\cdot)$. Let $p=\rho({\bm{x}},A)$
and $q=\rho({\bm{y}},A)$ with $p>q$. Define $c_{\varphi(\cdot)}(\cdot)$ with
$c^{\prime}_{\varphi({\bm{x}})}(p)>c^{\prime}_{\varphi({\bm{y}})}(q)$ if
$c^{\prime}_{{\bm{x}}}(p)>c^{\prime}_{{\bm{y}}}(q)$.

######  Claim 6.

$c_{\varphi(\cdot)}(\cdot)$ is well-defined.

###### Proof.

Consider ${\bm{x}}$ and ${\bm{y}}$ with a menu $A$ with ${\bm{x}}\in A$, and
${\bm{y}}\in X\setminus A$.

To begin with, suppose ${\bm{y}}\succ_{n}^{\rho}{\bm{x}}$, as in Step 4. By
the way of contradiction, suppose that $c^{\prime}_{\varphi({\bm{x}})}(p)\leq
c^{\prime}_{\varphi({\bm{y}})}(p)$. Assume that
$\rho({\bm{x}},A),\rho({\bm{y}},A)\in(0,1)$. By FOC for $\rho$ (See Equation
(10).), we have
$u(x_{1})-c^{\prime}_{\varphi({\bm{x}})}(\rho({\bm{x}},A))=u(y_{1})-c^{\prime}_{\varphi({\bm{y}})}(\rho({\bm{y}},A))$.
By Step 4, $u(x_{1})>u(y_{1})$. We must have
$c^{\prime}_{\varphi({\bm{x}})}(p)>c^{\prime}_{\varphi({\bm{y}})}(p)$. Hence,
the assumption of $c^{\prime}_{\varphi({\bm{x}})}(p)\leq
c^{\prime}_{\varphi({\bm{y}})}(p)$ is a contradiction.

Next, in the case of $x_{1}>y_{1}$ and ${\bm{x}}\succ_{n}^{\rho}{\bm{y}}$
satisfying the monotonicity with respect to fair allocations, it can be a case
of $\rho({\bm{y}},A)=0$. By the FOC for $\rho$,
$u(x_{1})-c^{\prime}_{\varphi({\bm{x}})}(\rho({\bm{x}},A))\geq
u(y_{1})-c^{\prime}_{\varphi({\bm{y}})}(\rho({\bm{y}},A))$ holds. Notice that
$C_{\varphi(\cdot)}^{1}$ over $(0,1)$. We thus obtain
$u(x_{1})-c^{\prime}_{\varphi({\bm{x}})}(\rho({\bm{x}},A))\geq u(y_{1})$.

Finally, if $\rho({\bm{x}},A)=1$, by FOC for $\rho$, we obtain $\rho$ through
$u(x_{1})>u(y_{1})$. $\Box$

To complete the proof, we verify that $c_{\varphi(\cdot)}(\cdot)$ is a cost
function. By definition, the strict convexity of $c_{\varphi(\cdot)}(\cdot)$
stems from that of $c_{\cdot}(\cdot)$.

We can obtain the conditions (i) and (ii) in Definition 4. The first condition
stems from the differentiability of $c_{\cdot}(\cdot)$. The second condition
is as follows:
$c^{\prime}_{\varphi({\bm{x}})}(p)>c^{\prime}_{\varphi({\bm{y}})}(p)$ for each
$p\in(0,1)$, if $\varphi({\bm{x}})<\varphi({\bm{y}})$. Hence, we have the
desired utility representation. $\Box$

###  A.2 Necessity Part

Suppose that $\rho$ is represented by a pair $(u,\varphi)$.

#####  Continuity.

Let $u:\mathbb{R}\rightarrow\mathbb{R}$ and $\varphi:X\rightarrow\mathbb{R}$
be continuous functions. $\rho$ is represented by

$\rho_{u,\varphi}(A)=\arg\max_{\rho\in\Delta(A)}\sum_{{\bm{x}}\in
A}\Bigl{(}u(x_{1})\rho({\bm{x}})-c_{\varphi({\bm{x}})}(\rho({\bm{x}})\Bigr{)}.$

Remember that $X$ is compact. The inverse function $u^{-1}(\cdot)$ is bounded.
$u^{-1}(\cdot)$ is a subset of the menu $A$, i.e., $u^{-1}(\cdot)\subseteq A$,
and $A$ is bounded and closed. Since $u$ and $c_{\varphi(\cdot)}$ are
continuous, the inverse image of a closed set is closed. Hence,
$\rho_{u,\varphi}$ is closed.

#####  Selfishness.

Take an arbitrary menu $A$ with $\rho({\bm{x}},A)=1$. For each
$x_{1},y_{1}\in\mathbb{R}$, $x_{1}>y_{1}$ if and only if $u(x_{1})>u(y_{1})$
holds. Hence, $x_{1}>y_{1}$ for all ${\bm{y}}\in A\setminus\\{{\bm{x}}\\}$.

#####  Menu Acyclicity.

Since $c_{\varphi({\bm{x}})}$ is a special case of $c_{\bm{x}}$ for each
${\bm{x}}\in X$, $\rho$ satisfies Menu Acyclicity (Fudenberg et al., 2014).

#####  Personal Norm Ranking.

By definition, we have $C^{1}_{\varphi(\cdot)}$ over $(0,1)$. For each
${\bm{x}},{\bm{y}}\in X$, if $\varphi({\bm{x}})<\varphi({\bm{y}})$, i.e.,
${\bm{x}}\prec_{n}^{\rho}{\bm{y}}$, then
$c^{\prime}_{\varphi({\bm{x}})}(p)>c^{\prime}_{\varphi({\bm{y}})}(p)$ for all
$p\in(0,1)$.

$\varphi:X\rightarrow\mathbb{R}$ represents $\succsim_{n}^{\rho}$; That is,
for each ${\bm{x}},{\bm{y}}\in X$, $\varphi({\bm{x}})\geq\varphi({\bm{y}})$ if
and only if ${\bm{x}}\succsim_{n}^{\rho}{\bm{y}}$. Hence,
$\succsim_{n}^{\rho}$ is a weak order. Moreover, since $\varphi$ is monotone
with respect to fair allocations, it is straightforward that
$\succsim_{n}^{\rho}$ satisfies the monotonicity with respect to fair
allocations. $\Box$

##  Appendix B Proofs of Propositions

###  B.1 Proof of Proposition 1

We prove Proposition 1. We exploit the fact that under Positivity, APU can be
seen as an extension of the “Fechnerian” utility representation (Definition 4)
of stochastic choice from doubletons to general menus.

Suppose that $\rho$ satisfies the axioms in Theorem 1. Both $u$ and $\varphi$
are unique up to positive affine transformations with the same unit. That is,
there exist $\alpha>0$, $\beta_{u},\beta_{\varphi}\in\mathbb{R}$ such that

  1. (i)

$\widehat{u}=\alpha u+\beta_{u}$, and

  2. (ii)

$\widehat{\varphi}=\alpha\varphi+\beta_{\varphi}$.

The first result (i) follows from Fudenberg et al. (2015)’s Corollary 1, and
the second result (ii) is a standard uniqueness result. By using this result,
we show the following statement:

######  Lemma 5.

Suppose that $(u,c_{\varphi})$ and $(\widehat{u},\widehat{c}_{\varphi})$
represent the same $\rho$. Then, there exists real numbers $\alpha>0$ and
$\beta_{u},\beta_{\varphi},\gamma,\delta\in\mathbb{R}$ such that
$\widehat{u}=\alpha u+\beta_{u}$ and
$\widehat{\varphi}=\alpha\varphi+\beta_{\varphi}$ with

$\displaystyle\widehat{c}_{\widehat{\varphi}}(p)=\alpha c_{\varphi}(p)+\gamma
p+\delta$

for all $p\in(0,1)$.

###### Proof.

First, we show (i). By Corollary 1 in Fudenberg et al. (2015), $\rho$ on
binary menus has a “Fechnerian utility” representation.

######  Definition 4.

(Fenchnerian Utility): We say that a stochastic choice rule $\rho$ on binary
menus has a Fechenerian utility representation if there exists a utility
function $u:X\rightarrow\mathbb{R}$ and a strictly increasing transformation
function $g$ such that

$\displaystyle\rho({\bm{x}},\\{{\bm{x}},{\bm{y}}\\})=g(u({\bm{x}})-u({\bm{y}})).$

######  Lemma 6.

(Fenchnerian Utility Representation): Suppose that $\rho$ is defined on binary
menus (doubletons) and satisfies Positivity and Continuity hold. Then $\rho$
satisfies Menu Acyclicity if and only if $\rho$ on binary menus has a
Fechnerian utility representation.

###### Proof.

($\Leftarrow$): We show the necessity part. Take arbitrary allocations
${\bm{x}},{\bm{y}},{\bm{y}}^{\prime}\in X$. Suppose that
$\rho({\bm{x}},\\{{\bm{x}},{\bm{y}}\\})\geq\rho({\bm{x}},\\{{\bm{x}},{\bm{y}}^{\prime}\\})$.
Since $\rho$ has a Fenchnerian utility representation on binary menus,

$\displaystyle g(u(x_{1})-u(y_{1}))\geq g(u(x_{1})-u(y^{\prime}_{1}))$
$\displaystyle\Leftrightarrow u(x_{1})-u(y_{1})\geq
u(x_{1})-u^{\prime}y^{\prime}_{1})$ $\displaystyle\Leftrightarrow
u(x_{1})-u(y_{1})\geq u(x_{1})-u(y^{\prime}_{1}).$

For each binary menu, let us define

$\displaystyle\lambda(\\{{\bm{x}},{\bm{y}}\\}):=-\frac{u(x_{1})+u(y_{1})}{2}.$

Since $u(y_{1})=-u(x_{1})-2\lambda(\\{{\bm{x}},{\bm{y}}\\})$,

$\displaystyle\rho({\bm{x}},\\{{\bm{x}},{\bm{y}}\\})\geq\rho({\bm{x}},\\{{\bm{x}},{\bm{y}}^{\prime}\\})\Leftrightarrow
u(x_{1})+\lambda(\\{{\bm{x}},{\bm{y}}\\})\geq
u(x_{1})+\lambda(\\{{\bm{x}},{\bm{y}}^{\prime}\\}).$

Moreover, we obtain $u(y_{1})\leq u(y^{\prime}_{1})$. By definition, we obtain

$\displaystyle-\frac{u(x_{1})+u(y_{1})}{2}>-\frac{u(x_{1})+u(y^{\prime}_{1})}{2}\Leftrightarrow\lambda(\\{{\bm{x}},{\bm{y}}\\})>\lambda(\\{{\bm{x}},{\bm{y}}^{\prime}\\}).$

And,

$\displaystyle-\frac{u(x_{1})+u(y_{1})}{2}=-\frac{u(x_{1})+u(y^{\prime}_{1})}{2}\Leftrightarrow\lambda(\\{{\bm{x}},{\bm{y}}\\})=\lambda(\\{{\bm{x}},{\bm{y}}^{\prime}\\}).$

Hence, we have
$\lambda(\\{{\bm{x}},{\bm{y}}\\})>\lambda(\\{{\bm{x}},{\bm{y}}^{\prime}\\})$
if
$\rho({\bm{x}},\\{{\bm{x}},{\bm{y}}\\})>\rho({\bm{x}},\\{{\bm{x}},{\bm{y}}^{\prime}\\})$,
and
$\lambda(\\{{\bm{x}},{\bm{y}}\\})=\lambda(\\{{\bm{x}},{\bm{y}}^{\prime}\\})$
if
$\rho({\bm{x}},\\{{\bm{x}},{\bm{y}}\\})=\rho({\bm{x}},\\{{\bm{x}},{\bm{y}}^{\prime}\\})$.
Thus, this separable utility representation ensures that $\rho$ satisfies Menu
Acyclicity.

($\Rightarrow$): We show the sufficiency part. Suppose that $\rho$ satisfies
Menu Acyclicity. Then, $\rho$ has a Menu-Invariant APU representation. By the
first order condition (FOC) for $\rho$, for any allocations
${\bm{x}},{\bm{y}}\in X$,

$\displaystyle
u(x_{1})-u(y_{1})=c^{\prime}_{\bm{x}}(\rho({\bm{x}},\\{{\bm{x}},{\bm{y}}\\})-c^{\prime}_{\bm{y}}(1-\rho({\bm{x}},\\{{\bm{x}},{\bm{y}}\\})).$

Without loss of generality, assume that
$\rho({\bm{x}},\\{{\bm{x}},{\bm{y}}\\})>\rho({\bm{y}},\\{{\bm{x}},{\bm{y}}\\})$.
For each ${\bm{x}}\in X$, $c^{\prime}_{{\bm{x}}}$ is strictly increasing, if
$\rho({\bm{x}},\\{{\bm{x}},{\bm{y}}\\})>\rho({\bm{y}},\\{{\bm{x}},{\bm{y}}\\})$
holds, then we have

$\displaystyle u(x_{1})-u(y_{1})>u(x_{1})-u(y^{\prime}_{1})$

for some ${\bm{y}}^{\prime}\in X$. Then, we obtain

$\displaystyle\rho({\bm{x}},\\{{\bm{x}},{\bm{y}}\\})>\rho({\bm{x}},\\{{\bm{x}},{\bm{y}}^{\prime}\\}),$

which is a desired result. $\Box$

We apply the result of Lemma 6 to (i). Take
${\bm{x}},{\bm{y}},{\bm{z}},{\bm{t}}\in X$. Suppose
$\rho({\bm{x}},\\{{\bm{x}},{\bm{y}}\\})\geq\rho({\bm{z}},\\{{\bm{z}},{\bm{t}}\\})$.
Then,

$\displaystyle\rho({\bm{x}},\\{{\bm{x}},{\bm{y}}\\})\geq\rho({\bm{z}},\\{{\bm{z}},{\bm{t}}\\})$
$\displaystyle\Leftrightarrow u(x_{1})-u(y_{1})\geq u(z_{1})-u(t_{1})$
$\displaystyle\Leftrightarrow\widehat{u}(x_{1})-\widehat{u}(y_{1})\geq\widehat{u}(z_{1})-\widehat{u}(t_{1}).$

By the uniqueness result of $u$, there exist $\alpha>0$ and
$\beta_{u}\in\mathbb{R}$ such that $\widehat{u}=\alpha u+\beta$.

Next, we show (ii). In Fudenberg et al. (2015), to obtain the uniqueness
result, we need a richer technical condition. Intuitively, the range of
observed stochastic behavior is rich enough.

######  Axiom 11.

(Richness): For any ${\bm{x}}\in X$ and $p,q\in(0,1)$ such that $p+q\leq 1$,
there exist ${\bm{y}},{\bm{z}}\in X$ (not necessary distinct) such that

  * •

$\rho({\bm{x}},\\{{\bm{x}},{\bm{y}}\\})=p$, and

  * •

$\rho({\bm{y}},\\{{\bm{x}},{\bm{y}}\\})=q$.

The axiom of Richness implies that the range of the utility function $u$
equals $\mathbb{R}$, i.e., $u(X)=\mathbb{R}$, and that there are at least
three allocations with each utility level. However, APU(SP) restricts $u$ to
be purely selfish, i.e., $u(x_{1})$ for each ${\bm{x}}\in X$, so without loss
of generality, suppose $u(X)=\mathbb{R}$.

Take $p,p^{\prime}\in(0,1)$. For all $q<1-p,1-p^{\prime}$, we have

  * •

$\rho({\bm{x}},A)=p$,

  * •

$\rho({\bm{x}}^{\prime},A^{\prime})=p^{\prime}$,

  * •

$\rho({\bm{y}},A)=q$, and

  * •

$\rho({\bm{y}}^{\prime},A^{\prime})=q$,

for some ${\bm{x}},{\bm{x}}^{\prime},{\bm{y}},{\bm{y}}^{\prime}\in X$ with
$y_{1}=y^{\prime}_{1}$, and $A,A^{\prime}\in\mathcal{A}$ with
${\bm{x}},{\bm{y}}\in A$ and ${\bm{x}}^{\prime},{\bm{y}}^{\prime}\in
A^{\prime}$. By the first order condition (FOC) for $\rho$,

$\displaystyle\widehat{c}^{\prime}_{\widehat{\varphi}({\bm{x}})}(p)-\widehat{c}^{\prime}_{\widehat{\varphi}({\bm{x}}^{\prime})}(p^{\prime})$
$\displaystyle=\widehat{c}^{\prime}_{\widehat{\varphi}({\bm{x}})}(p)-\widehat{c}^{\prime}_{\widehat{\varphi}({\bm{y}}^{\prime})}(q)+\widehat{c}^{\prime}_{\widehat{\varphi}({\bm{y}}^{\prime})}(q)-\widehat{c}^{\prime}_{\widehat{\varphi}({\bm{x}}^{\prime})}(p^{\prime})$
$\displaystyle=\widehat{u}(x_{1})-\widehat{u}(y_{1})+\widehat{u}(y^{\prime}_{1})-\widehat{u}(x^{\prime}_{1})$
$\displaystyle=\alpha(u(x_{1})-u(y_{1})+u(y^{\prime}_{1})-u(x^{\prime}_{1}))$
$\displaystyle=\alpha(c^{\prime}_{\varphi({\bm{x}})}(p)-c^{\prime}_{\varphi({\bm{x}}^{\prime})}(p^{\prime})).$

Fix two allocations ${\bm{x}},{\bm{x}}^{\prime}\in X$ and let $p=\frac{1}{2}$.
First, let
$\gamma:=\widehat{c}^{\prime}_{\widehat{\varphi}({\bm{x}})}(\frac{1}{2})-\alpha
c^{\prime}_{\varphi({\bm{x}}^{\prime})}(\frac{1}{2})$. For all $p\in(0,1)$,

$\displaystyle\widehat{c}^{\prime}_{\widehat{\varphi}({\bm{x}})}(p)-\widehat{c}^{\prime}_{\widehat{\varphi}({\bm{x}}^{\prime})}\Big{(}\frac{1}{2}\Bigr{)}$
$\displaystyle=\alpha\Bigl{(}c^{\prime}_{\varphi({\bm{x}})}(p)-c^{\prime}_{\varphi({\bm{x}}^{\prime})}\Big{(}\frac{1}{2}\Bigr{)}\Bigr{)}$
$\displaystyle\Leftrightarrow\widehat{c}^{\prime}_{\widehat{\varphi}({\bm{x}})}(p)$
$\displaystyle=\alpha
c^{\prime}_{\varphi({\bm{x}})}(p)+\widehat{c}^{\prime}_{\widehat{\varphi}({\bm{x}}^{\prime})}\Big{(}\frac{1}{2}\Bigr{)}-\alpha
c^{\prime}_{\varphi({\bm{x}}^{\prime})}\Big{(}\frac{1}{2}\Bigr{)}.$

Then, we obtain

$\displaystyle\widehat{c}^{\prime}_{\widehat{\varphi}({\bm{x}})}(p)=\alpha
c^{\prime}_{\varphi({\bm{x}}^{\prime})}(p)+\gamma.$

Next, define
$\delta:=\widehat{c}_{\widehat{\varphi}({\bm{x}})}(\frac{1}{2})-\alpha
c_{\varphi({\bm{x}}^{\prime})}(\frac{1}{2})-\frac{\gamma}{2}$. For all
$p\in(0,1)$,

$\displaystyle\widehat{c}_{\widehat{\varphi}({\bm{x}})}(p)-\widehat{c}_{\widehat{\varphi}({\bm{x}})}\Bigl{(}\frac{1}{2}\Bigr{)}$
$\displaystyle=\int^{p}_{\frac{1}{2}}\widehat{c}^{\prime}_{\widehat{\varphi}({\bm{x}})}(q)dq$
$\displaystyle=\int^{p}_{\frac{1}{2}}(\alpha
c^{\prime}_{\varphi({\bm{x}})}(q)+\gamma)dq$
$\displaystyle=\alpha\Bigl{(}c_{\varphi({\bm{x}})}(p)-c_{\varphi({\bm{x}})}\Big{(}\frac{1}{2}\Bigr{)}\Bigr{)}+\Bigl{(}p-\frac{1}{2}\Bigr{)}\gamma.$

Thus, we obtain

$\displaystyle\widehat{c}_{\widehat{\varphi}({\bm{x}})}(p)=\alpha
c_{\varphi({\bm{x}})}(p)+\delta+\gamma p.$

This result restricts the uniqueness of $c_{\varphi(\cdot)}$ to the same unit
$\alpha$ of $u$. Let us write down the cost function in the following way:
$c_{\varphi(\cdot)}:=\mathcal{C}(\varphi(\cdot),p)$ for each $p\in(0,1)$. The
uniqueness result requires that the cost function is affine in the first
argument, i.e.,

$\displaystyle\mathcal{C}(\alpha\varphi(\cdot),p)$
$\displaystyle=\alpha\mathcal{C}(\varphi(\cdot),p)$
$\displaystyle=\widehat{\mathcal{C}}(\varphi(\cdot),p).$

By re-writing $c_{\varphi(\cdot)}$, the desired result is obtained. $\Box$

###  B.2 Proof of Proposition 2

We prove Proposition 2.

#####  the Necessity part ($\Leftarrow$):

We show the necessity part. We show that the statement in Axiom 7:
$x_{1}>y_{1}$ if and only if ${\bm{x}}\succ_{n}^{\rho}{\bm{y}}$.

Suppose that there exists a pair $(u,c_{u})$. Given an arbitrary menu
$A\in\mathcal{A}$, fix ${\bm{x}},{\bm{y}}\in A$ with $x_{1}>y_{1}$. By the
first order condition (FOC) for $\rho$

$\displaystyle
u(x_{1})-c^{\prime}_{u(x_{1})}(\rho({\bm{x}},A))=u(y_{1})-c^{\prime}_{u(y_{1})}(\rho({\bm{y}},A)).$

Since $x_{1}>y_{1}$, we have $u(x_{1})>u(y_{1})$. Then, we obtain
$c^{\prime}_{u(x_{1})}(\rho({\bm{x}},A))>c^{\prime}_{u(y_{1})}(\rho({\bm{y}},A))$.
Since $c^{\prime}_{u(\cdot)}$ is a strictly increasing function, we obtain
$\rho({\bm{x}},A)>\rho({\bm{y}},A)$.

The property (ii) states that
$c^{\prime}_{u(x_{1})}(p)>c^{\prime}_{u(y_{1})}(p)$ for each $p\in(0,1)$, if

$\displaystyle u(x_{1})>u(y_{1}).$

Consider another menu $B:=A\cup\\{{\bm{z}}\\}$ such that $x_{1}>z_{1}$. Since
APU(SP) satisfies Regularity, suppose $\rho({\bm{x}},A)>\rho({\bm{x}},B)$.
Then, we verify that it works.

$\displaystyle
c^{\prime}_{u(x_{1})}(\rho({\bm{x}},A))>c^{\prime}_{u(x_{1})}(\rho({\bm{x}},B)).$

Consider the FOC for $\rho$:

$\displaystyle
u(x_{1})-c^{\prime}_{u(x_{1})}(\rho({\bm{x}},A))>u(y_{1})-c^{\prime}_{u(x_{1})}(\rho({\bm{x}},B))\Leftrightarrow\lambda(A)=\lambda(B).$

By definition, we thus obtain $\rho({\bm{x}},A)>\rho({\bm{x}},B)$.

Suppose $\rho({\bm{x}},B)=\rho({\bm{z}},B)$. Then, by the FOC for $\rho$, we
obtain

$\displaystyle
c^{\prime}_{u(x_{1})}(\rho({\bm{x}},B))>c^{\prime}_{u(z_{1})}(\rho({\bm{z}},B)).$

In this argument, we obtain $x_{1}>z_{1}$ if and only if
${\bm{x}}\succ_{n}^{\rho}{\bm{z}}$.

#####  the Sufficiency part ($\Rightarrow$):

We show the sufficiency part. Suppose that $\rho$ satisfies Selfishness as a
Norm. Then, we have $\varphi=u$. Given a menu $A,B\in\mathcal{A}$ with
${\bm{x}}\in A\cap B$, suppose that $\rho({\bm{x}},A)>\rho({\bm{x}},B)$. We
can write it down as follows. By the FOC for $\rho$,

$\displaystyle
c^{\prime}_{u(x_{1})}(\rho({\bm{x}},A))<c^{\prime}_{u(x_{1})}(\rho({\bm{x}},B)).$

Moreover, take arbitrary ${\bm{x}},{\bm{y}}\in X$ with $x_{1}>y_{1}$. Suppose
that $p=\rho({\bm{x}},A)=\rho({\bm{y}},A^{\prime})\in(0,1)$. Such two pairs
$({\bm{x}},A)$ and $({\bm{y}},A^{\prime})$ exist due to the rich our setting,
as in the proof of Proposition B.1. Then, by the FOC for $\rho$,

$\displaystyle
u(x_{1})-c^{\prime}_{u(x_{1})}(\rho({\bm{x}},A))=u(y_{1})-c^{\prime}_{u(y_{1})}(\rho({\bm{y}},A^{\prime})).$

Since $u(x_{1})>u(y_{1})$, we obtain
$c^{\prime}_{u(x_{1})}(\rho({\bm{x}},A))>c^{\prime}_{u(y_{1})}(\rho({\bm{y}},A^{\prime}))$.
Hence, condition (ii) in $c_{u(\cdot)}$ is satisfied. $\Box$

##  Appendix C Proofs of Corollaries

###  C.1 Proof of Corollary 1

The definition of selfishness-based FOSD is as follows: For each
$A\in\mathcal{A}$, if $x_{1}>y_{1}$, then $\rho({\bm{x}},A)>\rho({\bm{y}},A)$.

Suppose that $\rho$ has an APU form in Proposition 2. Take two allocations
${\bm{x}},{\bm{y}}\in X$ with $x_{1}>y_{1}$. By the first-order condition
(FOC) for $\rho$, we obtain

$\displaystyle
u(x_{1})-c^{\prime}_{u(x_{1})}(\rho({\bm{x}},A))=u(y_{1})-c^{\prime}_{u(y_{1})}(\rho({\bm{y}},A)).$

Since $u(x_{1})>u(y_{1})$, we obtain

$\displaystyle
c^{\prime}_{u(x_{1})}(\rho({\bm{x}},A))>c^{\prime}_{u(y_{1})}(\rho({\bm{y}},A)).$

By the strict convexity of $c^{\prime}_{u(\cdot)}$, we have
$\rho({\bm{x}},A)>\rho({\bm{y}},A)$. $\Box$

###  C.2 Proof of Corollary 2

($\Leftarrow$): First, we show the necessity part. Suppose that $\rho$
satisfies the axioms in Theorem 1. To begin, Positivity is satified due to the
definition of $c$, which guarantees that $c$ is steep, i.e.,
$\lim_{q\rightarrow 0}c^{\prime}(q)=-\infty$.

Next, to show Selfishness as a Norm, we apply the result of Fudenberg et al.
(2015). APU is characterized by Acyclicity. The axiom implies the ordinal
property:
$\rho({\bm{x}},A)>\rho({\bm{y}},A)\Leftrightarrow\rho({\bm{x}},B)>\rho({\bm{y}},B)$
for any ${\bm{x}},{\bm{y}}\in A$ with $x_{1}>y_{1}$. Consider arbitrary two
menus $A\in\mathcal{A}$ with ${\bm{x}},{\bm{y}}\in A$ with $x_{1}>y_{1}$.
Without loss of generality, suppose $\rho({\bm{x}},A)>\rho({\bm{y}},A)$. By
the first-order condition (FOC) for $\rho$,

$\displaystyle
u(x_{1})-c^{\prime}(\rho({\bm{x}},A))=u(y_{1})-c^{\prime}(\rho({\bm{y}},A)).$

Since $u(x_{1})>u(y_{1})$,
$c^{\prime}(\rho({\bm{x}},A))>c^{\prime}(\rho({\bm{y}},A))$. Since
$c^{\prime}$ is monotone, the property (ii) of the cost function in
Proposition 2 is satisfied. This property holds for any menus becasuse of the
ordinal propery. Thus, it is shown that Selfishness as a Norm is satisfied.

Finally, we verify that APU satisfies Order Independence. Take arbitrary two
menus $A,B\in\mathcal{A}$ with ${\bm{x}},{\bm{y}}\in A\setminus B$. And, take
an arbitrary allocation ${\bm{z}}\in B$. Suppose that
$\rho({\bm{z}},B\cup\\{{\bm{x}}\\})\leq\rho({\bm{z}},B\cup\\{{\bm{y}}\\})$.
Then, by the FOC for $\rho$,

$\displaystyle u(z_{1})-c^{\prime}(\rho({\bm{z}},B\cup\\{{\bm{x}}\\}))$
$\displaystyle\leq u(z_{1})-c^{\prime}(\rho({\bm{z}},B\cup\\{{\bm{y}}\\}))$
$\displaystyle\Leftrightarrow c^{\prime}(\rho({\bm{z}},B\cup\\{{\bm{x}}\\}))$
$\displaystyle\geq c^{\prime}(\rho({\bm{z}},B\cup\\{{\bm{y}}\\}))$
$\displaystyle\Leftrightarrow\lambda(B\cup\\{{\bm{x}}\\})$
$\displaystyle\leq\lambda(B\cup\\{{\bm{y}}\\})).$

Without loss of generality, suppose $x_{1}>y_{1}$. By the way of
contradiction, suppose

$\displaystyle u(x_{1})-c^{\prime}(\rho({\bm{x}},B\cup\\{{\bm{x}}\\}))$
$\displaystyle>u(y_{1})-c^{\prime}(\rho({\bm{y}},B\cup\\{{\bm{y}}\\}))$
$\displaystyle\Leftrightarrow c^{\prime}(\rho({\bm{x}},B\cup\\{{\bm{x}}\\}))$
$\displaystyle<c^{\prime}(\rho({\bm{y}},B\cup\\{{\bm{y}}\\})).$

Since $c^{\prime}$ is monotone, we have
$\rho({\bm{x}},B\cup\\{{\bm{x}}\\})<\rho({\bm{y}},B\cup\\{{\bm{y}}\\})$. This
is consistent with the assumption of $x_{1}>y_{1}$. Thus,

$\displaystyle u(x_{1})-c^{\prime}(\rho({\bm{x}},B\cup\\{{\bm{x}}\\}))$
$\displaystyle\leq u(y_{1})-c^{\prime}(\rho({\bm{y}},B\cup\\{{\bm{y}}\\}))$
$\displaystyle\Leftrightarrow c^{\prime}(\rho({\bm{x}},B\cup\\{{\bm{x}}\\}))$
$\displaystyle\geq c^{\prime}(\rho({\bm{z}},B\cup\\{{\bm{y}}\\}))$
$\displaystyle\Leftrightarrow\rho({\bm{x}},B\cup\\{{\bm{x}}\\})$
$\displaystyle\geq\rho({\bm{y}},B\cup\\{{\bm{y}}\\}).$

Hence, we obtain $\rho({\bm{x}},A)\geq\rho({\bm{y}},A)$ because, by the FOC
for $\rho$,
$u(x_{1})-c^{\prime}(\rho({\bm{x}},A)=u(y_{1})-c^{\prime}(\rho({\bm{y}},A)$.
Thus, it is shown that APU satisfies Order Independence.

($\Rightarrow$): Next, we show the sufficiency part. Under APU(SP), we show
that Positivity , Selfishness as a Norm, and Order Independence imply
Acyclicity.

Under APU(SP), Acyclicity is equivalent to the ordinal properties in the
following manner.

  1. (a)

$\rho({\bm{x}},A)>\rho({\bm{y}},A)\Leftrightarrow\rho({\bm{x}},B)>\rho({\bm{y}},B)$
for any ${\bm{x}},{\bm{y}}\in A$;

  2. (b)

$\rho({\bm{x}},A)>\rho({\bm{x}},B)\Leftrightarrow\rho({\bm{y}},A)>\rho({\bm{y}},B)$
for any ${\bm{x}},{\bm{y}}\in A$.

First, we consider the property (a), whichis straightforward. Suppose that
$x_{1}>y_{1}$. Then,

$\displaystyle\rho({\bm{x}},A)>\rho({\bm{y}},A)$ $\displaystyle\Leftrightarrow
c^{\prime}_{u(x_{1})}(\rho({\bm{x}},A))>c^{\prime}_{u(y_{1})}(\rho({\bm{y}},A))$
$\displaystyle\Leftrightarrow
u(x_{1})-c^{\prime}_{u(x_{1})}(\rho({\bm{x}},A))=u(y_{1})-c^{\prime}_{u(y_{1})}(\rho({\bm{y}},A))$
$\displaystyle\Leftrightarrow
c^{\prime}_{u(x_{1})}(\rho({\bm{x}},B))>c^{\prime}_{u(y_{1})}(\rho({\bm{y}},B))$
$\displaystyle\Leftrightarrow\rho({\bm{x}},B)>\rho({\bm{y}},B).$

Second, we consider the property (b). Given a menu $B^{\prime}\in\mathcal{A}$,
take ${\bm{z}}\in B^{\prime}$. By Order Independence, we have

$\displaystyle\rho({\bm{z}},B^{\prime}\cup\\{{\bm{x}}\\})>\rho({\bm{z}},B^{\prime}\cup\\{{\bm{y}}\\})$
$\displaystyle\Leftrightarrow\rho({\bm{x}},A)>\rho({\bm{y}},A).$

Hence, we have $x_{1}>y_{1}$. Let $\widehat{B}:=B^{\prime}\cup\\{{\bm{x}}\\}$.
By the FOC for $\rho$,

$\displaystyle\rho({\bm{x}},A)>\rho({\bm{x}},\widehat{B})$
$\displaystyle\Leftrightarrow
c^{\prime}_{u(x_{1})}(\rho({\bm{x}},A))<c^{\prime}_{u(y_{1})}(\rho({\bm{y}},\widehat{B}))$
$\displaystyle\Leftrightarrow
u(x_{1})-c^{\prime}_{u(x_{1})}(\rho({\bm{x}},A))>u(y_{1})-c^{\prime}_{u(y_{1})}(\rho({\bm{y}},\widehat{B}))$
$\displaystyle\Leftrightarrow\lambda(A)>\lambda(\widehat{B}).$

We thus obtain $\rho({\bm{y}},A)>\rho({\bm{y}},B^{\prime}\cup\\{{\bm{y}}\\})$.
Letting $B:=B^{\prime}\cup\\{{\bm{y}}\\}$, the desired result is obtained.
Therefore, $\rho$ satisfies Acyclicity, so let $c_{u(\cdot)}=c$. $\Box$

###  C.3 Proof of Corollary 3

($\Leftarrow$): First, we show the necessity part. For each $A\in\mathcal{A}$,
suppose that

$\displaystyle\rho(A)=\arg\max_{\rho\in\Delta(A)}\sum_{{\bm{x}}\in
A}u(x_{1})\rho({\bm{x}},A).$

This utility representation states that the decision maker chooses the most
selfish allocation with certainty; That is, for any $A\in\mathcal{A}$, there
exists ${\bm{x}}\in A$ such that $x_{1}>y_{1}\in A\setminus\\{{\bm{x}}\\}$,
$\rho({\bm{x}},A)=1$. It is shown that Deterministic Choice is satisfied.

($\Rightarrow$): Next, we show the sufficiency part. Suppose that $\rho$
satisfies Deterministic Choice. By Selfishness, for each $A\in\mathcal{A}$,
the allocation ${\bm{x}}\in A$ such that $\rho({\bm{x}},A)=1$ satisfies
$x_{1}>y_{1}$ for any ${\bm{y}}\in A\setminus\\{{\bm{x}}\\}$. Hence,
$\rho(A)=\arg\max_{\rho\in\Delta(A)}\sum_{{\bm{x}}\in
A}u(x_{1})\rho({\bm{x}},A)$. $\Box$

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