Discussion Papers In Economics
And Business

Mixture Attitudes of Expectation-Based Loss Aversion

Daijiro Kawanaka

Discussion Paper 23-02

February 2023

Graduate School of Economics
Osaka University, Toyonaka, Osaka 560-0043, JAPANMixture Attitudes of Expectation-Based Loss Aversion∗
Daijiro Kawanaka†
AbstractKöszegi and Rabin (2006, 2007) formulate loss-aversion models called Preferred Personal
Equilibrium (PPE) and Choice-acclimating Personal Equilibrium (CPE) so successfully thatmany papers have been applied their models to a variety of economic ﬁelds in this decade. In
this paper, without assuming any speciﬁc functional form, I show that these two loss-aversion
models satisfy strong mixture aversion in the sense that a non-degenerate lottery is strictlypreferred to any mixture between the lottery and its certainty equivalent. This property distinguishes these loss-aversion models from many other standard models such as the expectedutility theory and disappointment aversion so that this result allows experimental economists
to test a class of these models in a simple setting.

JEL Classiﬁcation Numbers: D11, D81, D91
Keywords: Prospect theory, Loss aversion, Reference point, Mixture aversion, Betweenness∗

The author would like to thank Nobuyuki Hanaki, Taisuke Imai, Keigo Inukai, Takeshi Murooka, and Yosuke
Yasuda. This research was supported by Grant-in-Aid for JSPS Fellows from the Ministry of Education,
Culture, Sports, Science and Technology of Japan.
†

Graduate school of Osaka University. Email: kawanaka.daijiro@gmail.com

1

Introduction

Kahneman and Tversky (1979) proposed “prospect theory” to explain decisions under risk
deviating from the expected utility theory of Neumann and Morgenstern (1944). In theprospect theory, it is often assumes that a decision maker has an exogenous reference pointthat causes loss aversion. Literature has often criticized that theorists can assume arbitraryreference points and explain any decisions. To overcome this problem, Köszegi and Rabin (KR) have developed models of endogenous reference-dependent utility, called Preferred
Personal Equilibrium (PPE, Köszegi and Rabin 2006) and Choice-acclimating Personal Equilibrium (CPE, Köszegi and Rabin 2007). In their models, the reference point is assumed asa stochastic distribution, not a deterministic point.
In this decade, the KR loss aversion models have been applied to a variety of ﬁelds suchas wage sheme (Herweg et al. 2010; Daido et al. 2013), pricing (Rosato 2016), ﬁnancial
decisions (Ai et al. 2018; Meng and Weng 2018), market design (Balzer and Rosato 2021;
Meisner and Wangenheim 2022; Muramoto and Sogo 2022), and team incentives (Daido andMurooka 2016). Most applications of KR assume a linear utility function for simplicity.
Few theoretical papers analyze the KR models due to their complexity. As one of the few
theoretical papers, Masatlioglu and Raymond (2016) provide an axiomatic foundation of thelinear case of CPE.
However, behavioral economists should pay more attention to the nonlinear model of KR.
This is because the linear models of KR cannot solve Rabin’s (2000) critique of the standard
expected utility theory. On this issue, Masatlioglu and Raymond (2016) state in their ﬁnal
paragraph that “Linearity of CPE is crucial for the calibration result of Safra and Segal(2005). If the assumption that µ (the gain-loss function) is linear is relaxed, it is possible togenerate plausible small- and large-stakes risk aversion. Köszegi and Rabin’s (2007) Table 1
does exactly this. However, the most tractable form of Köszegi and Rabin’s (2007) model,that with linear gain-loss utility, cannot avoid an extension of the Rabin critique. Thus, inorder to model individuals who exhibit plausible behavior over both small- and large-stakes
lotteries, we must turn to nonlinear gain-loss functionals.”
In this paper, I analyze the both of PPE and CPE without assuming any speciﬁc functional
form, and show that they satisfy strong mixture aversion: a decision maker strictly prefers anon-degenerate lottery p to any mixture θp⊕(1−θ)δ of p and its certainty equivalent δ. This is
in contrast to many standard models as the expected utility theory, the betweenness literature1

(Dekel 1986; Gul 1991), and the negative certainty independence models (Dillenberger 2010;
Cerreia-Vioglio et al. 2015). That is, in these later models, a decision maker must be
indiﬀerent between the two options p and θp ⊕ (1 − θ)δ for any mixture θ. Hence, theseproperties will allow experimental economists to diﬀerentiate KR from other models.
Masatlioglu and Raymond (2016) show that a class of CPE satisﬁes a weak version of mixture aversion but this paper generalize their results in the following senses: First, Masatlioglu
and Raymond focus solely on CPE, whereas I study both CPE and PPE. 1 Second, Masatlioglu and Raymond assume speciﬁc functional forms such as lineality, whereas I assume only
very mild conditions. Hence, my results suggest that the mixture aversion property can beapplied to a wider range of real-world scenarios. Third, Masatlioglu and Raymond’s results
claim a weak mixture aversion that a decision maker weakly avoids a mixture alternative,
while my results claim a strong one. This improves experimental testability of CPE andPPE.
The rest part of this paper is organized as follows: Section 2 deﬁnes a model. Section 3
introduces Choice-acclimating Personal Equilibrium (CPE) proposed by Köszegi and Rabin(2007), and shows its behavioral implications. Section 4 introduces Preferred Personal Equilibrium (PPE) proposed by Köszegi and Rabin (2006), and shows its behavioral implications.
Section 5 concludes.

2

Preliminaries

Let X be an arbitrary set of deterministic prizes. For any probability distribution p overX, denote supp(p) := {x ∈ X | p(x) > 0}. I refer a probability distribution p with a ﬁnite
supp(p) to a lottery over X, and denote the set of all lotteries by ∆X . I denote a degeneratelottery yielding a z ∈ X by δz . I consider a binary relation ⪰ over ∆X and a choice function
over ∆X . As usual, I deﬁne a lottery θp ⊕ (1 − θ)q for each p, q ∈ ∆X and all θ ∈ (0, 1).
∑∑
Given a lottery p ∈ ∆X , I will abbreviate x∈supp(p) p(x) as x p(x) for short.
In this section, I introduce a general model of endogenous reference dependent utility, andshow its property as a lemma. To do so, I introduce a gain-loss function denoted by µ : R →R. On the function, Bowman, Minehart, and Rabin (1999) postulate and Köszegi and Rabin(2006, 2007) employ the following assumptions (A1)-(A4) corresponding to Kahneman and1

As one of the few theoretical papers analyzing PPE, Freeman (2019) shows that PPE violations of the
independence axiom and can also lead to violations of the weak axiom of revealed preference.

2

Tversky’s (1979) explicit or implicit assumptions about their “value function:”(A1) µ is strictly increasing in x ∈ R.
(A2) µ(y) + µ(−y) < µ(x) + µ(−x) for all x, y such that y > x ≥ 0.
(A3) f (x) is concave for all x > 0 and convex for all x < 0.
(A4) limx→0 µ′ (−x)/µ′ (x) > 1 if x > 0.
In words, these assumptions say that the graph of µ is an S-shaped curve as Kahneman andTversky draw. In addition, Köszegi and Rabin (2006, 2007) assume on µ that µ(0) = 0 as(A0) which says that the curve passes through the origin. However, this (A0) is not essential
for the present analysis since µ(0) is just a constant term for total utility which we will deﬁne
the below. I do not need (A1), (A3) and (A4) for the present analysis of this paper. This
study needs (A2) and the following additional condition:(A5) µ(α) + µ(−α) is convex for α ∈ R+ .
Note that a “general CPE” called by Masatlioglu and Raymond (2016) satisﬁes (A2) and(A5). Thus, a functional of our analysis is more general than their called general CPE.
In this paper, I consider a functional UKR : ∆X × ∆X → R given by
UKR (p | q) :=∑u(x)p(x) −

∑∑
x

x

µ(u(x) − u(r))p(x)q(r),(1)

rwhere u : X → R is a Bernoulli function and µ : R → R is gain-loss function. In (1), theﬁrst term represents a material utility and the second represents a psychological utility. In
the later term, an r represents a reference point, and the deﬁnition of UKR says that the r is
distributed according to distribution q.
Let (u, µ) be given Bernoulli function and gain-loss function. For notational simplicity, I
will denote the term of psychological loss byΛpq := −∑∑xµ(u(x) − u(r))p(x)q(r)(2)

rfor each p, q ∈ ∆X . Then, Λpq satisﬁes a triangle inequality as follows:
Lemma 1 (Triangle inequality). Let a function Λ : ∆X ×∆X → R reduced from (u, µ) satisfy(A2) and (A5). Then,Λpq + Λqr ≥ Λpr3(3)

for each p, q, r ∈ ∆X . Moreover,Λpq + Λqr = Λpr(4)binds only if p = q = r.
■

Proof of Lemma 1. See Appendix A.1.

This triangle inequality is important for my analysis, and to my knowledge no paper claim
this property of psychological loss with stochastic reference points.

3

Choice acclimating personal equilibrium

In this section, I will show that a choice acclimating personal equilibrium (CPE) developed
by Koszegi and Rabin (2007) satisﬁes a strong version of mixture aversion. Masatlioglu andRaymond (2016) have shown that a relation ⪰ with a CPE of a gain-loss function µ given by α if α ≥ 0 µ(α) :=
where 0 ≤ λ ≤ 1(5) λα if α < 0 
satisﬁes weak mixture aversion: if p ∼ q, then θp ⪰ p ⊕ (1 − θ)q for all θ ∈ (0, 1). This section
will show that such a mixture aversion holds even when µ is more general, and moreover, it
strictly holds when q is degenerate. (i.e., q is a certainty equivalent of p.)
By a function UKR introduced in section 2, I deﬁne CPE as follows:
Deﬁnition 1. For a function UKR reduced from given (u, µ), a binarly relation ⪰ has a CPE
model ifp ⪰ q ⇔ UKR (p | p) ≥ UKR (q | q)for each p, q ∈ ∆X . I will often abbreviate UKR (p | p) to UKR (p).
The result of this section is summarized as follows:
Theorem 2. Let ⪰ have a CPE model satisfying (A2) and (A5). For any θ ∈ (0, 1), anyp, q ∈ ∆X and z ∈ X such that p ∼ q ∼ δz and p ̸= δz ̸= q,p ⪰ θp ⊕ (1 − θ)q ≻ θp ⊕ (1 − θ)δz
holds.

4(6)

■

Proof of Theorem 2. See Appendix A.2.

In words, the ﬁrst ⪰ of (6) claims weak mixture aversion holds, and the second ≻ of (6)claims strong mixture aversion holds for the certainty equivalent. An example is as follows:
Example 1. Suppose a CPE with µ given by (5) and u(z) := z. Consider a lottery p :=()($8, 50%; $0, 50%). Then, UKR (p) = UKR (δ3 ) = 3 > 4/11 = UKR 12 p ⊕ 21 δ3 . Hence, thestrong mixture aversion holds.
I emphasize that Theorem 2 holds even if µ is neither linear in a sense of (5) nor of othersymmetric property such as Masatlioglu and Raymond (2016) imposed in their section IV.
Hence, according to Theorem 2, experimental economists can test CPE model by assuming
very mild conditions (A2) and (A5).

4Preferred personal equilibrium

A preferred personal equilibrium (PPE) developed by Koszegi and Rabin (2006) is also afamous model of endogenous reference-dependence loss aversion deﬁned by functional UKR .
In this section, I will show that PPE also satisﬁes a strong version of mixture aversion forcertainty equivalents.
To introduce PPE model, I deﬁne unacclimating personal equilibrium (UPE) model by achoice function:
Deﬁnition 2. For given UKR , a choice function CU P E : 2∆X → 2∆X has a UPE model
whenever for all D ⊆ ∆X , p ∈ CU P E (D) if and only if UKR (p | p) ≥ UKR (q | p) for all q ∈ D.
Deﬁnition 3. For given UKR , a choice function CP P E : 2∆X → 2∆X has a PPE model
whenever for all D ⊆ ∆X , p ∈ CP P E (D) if and only if p ∈ CU P E (D) and UKR (p | p) ≥UKR (q | q) for all q ∈ CU P E (D).
Deﬁnition 4. For given choice function C, z ∈ X is a certainty equivalent of given lotteryp ∈ ∆X if C({p, δz }) = {p, δz }.
Now, I can extend the ﬁrst part of Theorem 2 of CPE to PPE:
Theorem 3. Let CP P E be reduced from UKR satisfying (A2) and (A5). Take any p, q ∈ ∆X ,any D ⊆ ∆X and any θ ∈ (0, 1). Then, p, q ∈ CP P E (D ∪ {θp ⊕ (1 − θ)q}) if p, q ∈ CP P E (D).

5■

Proof of Theorem 3. See Appendix A.3.

Theorem 3 claims that a decision maker weakly prefers p (∼ q) to θp ⊕ (1 − θ)q in theperspective of PPE. That is, PPE model also exhibits the weak mixture aversion.
From Lemma 1 and Theorem 3, I can also obtain the following claim:
Collolary 4. Take any p ∈ ∆X , any D ⊆ ∆X and any θ ∈ (0, 1). Then, θp ⊕ (1 − θ)δz ̸∈
CP P E (D ∪ {θp ⊕ (1 − θ)δz }) if p, δz ∈ CP P E (D).
This collolary corresponds to the second part of Theorem 2: a decision maker strictlyprefers p (∼ δz ) to θp ⊕ (1 − θ)δz in the perspective of PPE. That is, PPE model also exhibits
the strong mixture aversion for certainty equivalents.

5

Conclusion

This paper shows that both CPE (Köszegi and Rabin 2007) and PPE (Köszegi and Rabin
2006) both exhibit weak mixture aversion under milder conditions than known in the literature. Moreover, the both models exhibit strong mixture aversion for certainty equivalents.
This behavioral implication identiﬁes these models of loss aversion from other decision models
such as disappointment aversion (Gul 1991). This insight will encourage both of theoreticaland experimental research applying Köszegi-Rabin type loss aversion.

A

ProofsA.1

Proof of Lemma 1

Given a gain-loss function µ : R → R, I deﬁne ν(α) := −µ(α) − µ(−α) for each α ∈ R. By
construction,
Λpq + Λqp =∑∑x

ν(x − y)p(x)q(y)(7)

yholds for any p, q ∈ ∆X .
Take arbitrary a, b, c ∈ X. From (A2), ν(α) is strictly increasing in α ∈ R+ . Thus, by
the geometric triangle inequality, I have
ν(|a − b| + |b − c|) ≥ ν(|a − c|).

6(8)

From (A5), ν(|α|) is concave. Thus, I have
ν(|a − b|) + ν(|b − c|) ≥ ν(|a − b| + |b − c|).

(9)

From (8) and (9), I have
ν(|a − b|) + ν(|b − c|) ≥ ν(|a − c|).

(10)

Because ν(α) = ν(−α), I obtain from (10) thatν(a − b) + ν(b − a) + ν(b − c) + ν(c − b) ≥ ν(a − c) + ν(c − a).

(11)

Take arbitrary p, q, r ∈ ∆X . Multiplying the both hand sides of (11) by p(a)q(b)r(c) andtaking summation of them for each (a, b, c) ∈ supp(p) × supp(q) × supp(r), I have2Λpq + 2Λqr ≥ 2Λpr .

(12)

Hence, I obtain (3). Moreover, inequalities (8), and thus (12), bind only if p = q = r. Hence,■

now I obtain (4).

A.2

Proof of Theorem 2

To show Theorem 2, I show the following Lemma 5:
Lemma 5. Let ⪰ have a CPE model. Take any p, q ∈ ∆X , z ∈ X and θ ∈ (0, 1) such thatp ∼ q ∼ δz . Then, p ≻ θp ⊕ (1 − θ)q if and only if
Λpq + Λqp > Λpp + Λqq .

(13)

Especially, p ≻ θp ⊕ (1 − θ)δz if and only ifΛpδz + Λδz p > Λpp .

(14)

Proof of Lemma 5. Suppose p ∼ q, i.e.,∑u(x)p(x) − Λpp =x∑u(x)q(x) − Λqq .

(15)

x

Take an arbitrary θ ∈ (0, 1). Then, UKR (θp ⊕ (1 − θ)q) is calculated asθ∑xu(x)p(x) + (1 − θ)∑u(x)q(x) − θ2 Λpp − (1 − θ)θ(Λpq + Λqp ) − (1 − θ)2 Λqq .

x

7

(16)

Hence, UKR (p) − UKR (θp ⊕ (1 − θ)q) is calculated as()∑∑(1 − θ)u(x)p(x) − Λpp −u(x)q(x) + Λqq + (1 − θ)θ(Λpq + Λqp − Λpp − Λqq ), (17)xxwhich is larger than zero if and only if p ≻ θp ⊕ (1 − θ)q. From (15) and (17), I can say thatp ≻ θp ⊕ (1 − θ)q if and only if (13) holds. BecauseΛ δz δz = 0(18)

by construction of (2), I obtain (14) by asigning q = δz to (13).

■

From Lemmas 1 and 5, Theorem 2 is shown.

A.3■

Proof of Theorem 3

To show Theorem 3, I show the following Lemmas 6 and 7
Lemma 6. Take an arbitrary D ⊆ ∆X . If p, q ∈ CU P E (D), then Λpq + Λqp ≥ Λpp + Λqq .
Proof of Lemma 6. Suppose p, q ∈ CU P E ({p, q}). Then, UKR (p | p) ≥ UKR (q | p) andUKR (q | q) ≥ UKR (p | q) hold. That is,∑u(x)p(x) − Λpp ≥

∑

x

and

∑

u(x)q(x) − Λqp ,(19)

u(x)p(x) − Λpq .

(20)

x

u(x)q(x) − Λqq ≥

∑

x

x

From (19) and (20), I have Λpq + Λqp ≥ Λpp + Λqq .

■Lemma 7. Take any p, q ∈ ∆X and any D ⊆ ∆X . Let m := θp ⊕ (1 − θ)q for given θ ∈ (0, 1).
Then, p, q ∈ CU P E (D ∪ {m}) if p, q ∈ CP P E (D).
Proof of Lemma 7. Suppose p, q ∈ CP P E (D). Then, I have UKR (p | p) = UKR (q | q), i.e.,∑∑
x u(x)p(x) − Λpp =x u(x)q(x) − Λqq . Therefore, I obtain from (19) and (20) that
Λqp ≥ Λqq ,(21)

Λpq ≥ Λpp

(22)

and

8

respectively. Note that UKR (p | p) = UKR (q | q) = θUKR (p | p) + (1 − θ)UKR (q | q) by
Deﬁnition 4. Therefore, I obtain that
UKR (q | q) ≥ UKR (m | q)(23)

from (21). Similarly, UKR (p | p) ≥ UKR (m | p) holds from (22). Hence, I havep, q ∈ CU P E (D ∪ {m}).

(24)

From Theorem 2 and (24), I have p, q ∈ CP P E (D ∪ {m}).

■From Lemmas 6 and 7, Theorem 3 is shown.

■

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