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Social Image Concern and Reference Point Formation

Yosuke Hashidate ORCID logo

Abstract

This paper develops an axiomatic context-dependent model of social image concerns. Allowing for context-dependence based on choice sets, it examines how context-dependence impacts social image concerns, in particular how a decision maker exhibits various social emotions stemming from their intrinsic reference point, which may not be captured by conforming to social norms. To elicit the intrinsic reference point, this paper provides weaker versions of Strategic Rationality and Independence, in addition to the basic axioms, to characterize the model of the Reference-Dependent Image-Conscious utilitarian. This paper also examines how social emotions stemming from the intrinsic reference point are related to preference reversals as violations of the Weak Axiom of Revealed Preference (WARP). Finally, this paper considers the relationship between social image concerns and intrinsic reciprocity. The findings demonstrate that social image plays a large role and the model developed illustrates a condition in which cooperation is sustained in prisoner’s dilemma games.

JEL Classification Numbers: D01; D63; D64; D91

Corresponding author: Yosuke Hashidate, Graduate School of Business and Finance, Waseda University, Japan Society for the Promotion of Science, Tokyo, Japan, E-mail:

Article note: An earlier version of this paper was presented as “Reference-Dependent Fairness Attitudes: An Axiomatic Approach” and “A Cognitive Foundation for Social Image Concerns.” I am indebted to my adviser Akihiko Matsui for his unique guidance, constant support, and encouragement when I was a Ph.D. student at the University of Tokyo. I am very grateful to Yoichiro Fujii, Youichiro Higashi, Jay Lu, Kota Saito, and Norio Takeoka, and the anonymous referees for their invaluable suggestions and comments. I would also like to thank the participants at the UT Summer School in Economics 2017 (University of Tokyo), EEA-ESEM 2017 (ISCTE-IUL Campus, Lisbon), JEA 2017 Autumn Meeting (Aoyama Gakuin University, Tokyo), and Economics Research Meeting (Osaka Sangyo University). Part of this research was completed while I was a research associate at CIRJE, the Graduate School of Economics at the University of Tokyo. I am grateful for their hospitality. I would also like to express my gratitude to Editage (www.editage.jp) for their English language editing. Of course, all remaining errors are mine.


A Proof of Theorem 1

A.1 Sufficiency Part

We show the sufficiency part. Suppose that ¯ satisfies the axioms in the main theorem (Theorem 1).

Step 1.

In Step 1, we first show that the two induced binary relations j on ΔI(j{1,S}) are well-defined. Next, we show that j on ΔI(j{1,S}) satisfies the axiom of Independence in the vNM-type expected utility theorem. Finally, we represent the first term of the utility representation by showing that α1 is positive.

1andSare separable. Remember that, for each j{1,S}, we define j on ΔI as follows. The asymmetric and symmetric part of j are described by j and j, respectively. For all p1,q1Δ, p11q1 if {(p1,rS)}¯{(q1,rS)} for some rSΔS. For all pS,qSΔS, pSSqS if {(r1,pS)}¯{(r1,qS)} for some r1Δ.

We show that 1 is independent of any rSΔS.[36] We need to show that p11q1 if (p1,rS)1(q1,rS) for anyrSΔS. Notice that for any pj,qjΔj and λ[0,1], λpj+(1λ)qjΔj(j{1,S}).

Suppose {(p1,rS)}¯{(q1,rS)} and {(p1,lS)}{(q1,lS)}. Consider (p1,lS)ΔI and (p1,rS)ΔI. By Axiom 6 (Singleton Independence), letting λ=12, {(p1,12rS+12lS)}¯{(12p1+12q1,12rS+12lS)} and {(p1,12rS+12lS)}{(12p1+12q1,12rS+12lS)}. This is a contradiction. Hence, if there exists rSΔS such that {(p1,rS)}¯{(q1,rS)}, then for any rSΔS, {(p1,rS)}¯{(q1,rS)}.

In the same way, we can show that S is well-defined. We omit it.

1andSsatisfy Independence. We show that for each j{1,S}, j satisfies the axiom of Independence in the expected utility theorem (EUT). Consider 1. By Axiom 1 (Standard Preferences), it is easily shown that 1 satisfies Completeness, Transitivity, and Mixture Continuity. We show that 1 satisfies the axiom of Independence: For any p1,q1,r1 and λ[0,1],

p11q1λp1+(1λ)r11λq1+(1λ)r1.

Fix p1,q1,r1Δ and λ[0,1]. Then, for any pS,qSΔS,

p11q1{(p1,pS)¯{(q1,pS)}λ{(p1,pS)}+(1λ){(r1,q1)}¯λ(q1,pS)+(1λ)(r1,qS){(λp1+(1λ)l1,λpS+(1λ)qS)}¯{(λq1+(1λ)l1,λpS+(1λ)qS)}λp1+(1λ)r11λq1+(1λ)r1.

It is shown that 1 satisfies the axiom of Independence. In the same way, we can show that S satisfies the axiom of independence. We omit it.

By the von Neumann-Morgenstern’s Expected Utility Theorem (Kreps 1988), there exists a non-constant continuous and mixture linear function u:ΔR which represents 1. Furthermore, u is unique up to a positive affine transformation. Moreover, by Axiom 2 (Consistency), there exists a continuous and mixture linear utility function u^S:ΔSR with (αi)iS and a real number ε such that u^S(pS)=iSαiu(pi)+ε and iSαi=1. Define uS:=iSαiu(pi)=u^S(pS)ε. Hence, uS represents S.

α1is positive. Finally, we show α1>0. By Axiom 1 (Standard Preferences), define, for any pΔI, V({p}):=iIαiu(pi) that represents ¯ over As, i.e., the set of all singletons. Suppose α10. Take p,qΔI such that p11q1 and pSSqS. By Axiom 3 (Pareto), {p}{q}. If α10, V({p})V({q}){p}¯{q}. This is a contradiction.

Step 2.

In Step 2, we show that (u̅1,u̅S) in the second term of the RDIC represent (¯1,¯S), respectively. For each j{1,S}, we show that ¯j satisfies (i) completeness, (ii) transitivity, (iii) continuity, (iv) singleton independence, and (v) disjoint set-betweenness, by following from Olszewski (2007).

Properties on¯1and¯S. Remember the definition of the induced menu-preferences. For each j{1,S}, we say that A¯jB if for any qB and pA, pjjqj.

Transitivity. We show that, for each j{1,S}, ¯j is transitive. Take A,B,CA with A¯jB and B¯jC. By definition, it is obvious that A¯jC.

Completeness. We show that, for each j{1,S}, ¯j is complete. By the axioms of Standard Preferences, Singleton Independence of ¯, and Intrinsic Set-Betweenness of ¯j, for any AA, there exists a singleton equivalent pAΔI such that {pA}A. Consider arbitrary two menus A,BA. Take singleton equivalent, respectively, denoted by pA,pBΔI. Without loss of generality, assume {pA}¯j{pB}. We need to show that A¯jB{pA}¯j{pB}. The sufficiency part is obvious by definition. We show the necessity part. By the transitivity of ¯j, Aj{pA}¯j{pB}jB. Hence, A¯jB. This argument holds for any arbitrary menus.

Continuity. We show that, for each j{1,S}, ¯j is continuous. Take arbitrary two allocations p,qΔI with {p}¯{q}. Suppose that {p}¯j{q}. By definition, pjjqj. Since j is continuous, the continuity of ¯j holds on singletons. By taking singleton equivalent, we can show that the continuity of ¯j holds for any arbitrary menus.

Disjoint Set-betweenness. We show that, for each j{1,S}, ¯j satisfies Disjoint Set-Betweenness. This axiom is equivalent to Axiom 4 (Intrinsic Set-Betweenness).

Indifference to Randomization. We verify that ¯j satisfies the axiom of Indifference to Randomization. The axiom is stated as follows. For each menu AA, let co(A) be the convex hull of A.

Axiom 20.

(Indifference to Randomization): For any AA, Aco(A).

Take an arbitrary menu AA. A is compact, and there exist j-best and j-worst allocations of the menu A. The extreme points do not change by objective mixtures. Hence, for all AA, Aco(A).

We obtain the following result. For simplicity, let us introduce some notation: for each p1Δ and pSΔI, u1:=α1u(p1), uS:=iSαiu(pi).

Corollary 8.

¯j satisfies Completeness, Transitivity, Continuity, Disjoint Set-Betweenness, and Indifference to Randomization, if and only if there exist a pair (u,Wj) where u:ΔR and Wj:u(Δ)×u(Δ)R such that ¯j is represented by u̅j:AR, defined by

u̅j(A)=Wj(maxpAu(pj),minpAu(pj)).

Proof. The proof mainly follows from Dekel et al. (2001) and Kopylov (2009). By Step 1, we have already shown that there exists a non-constant linear u:ΔR. The remaining is to show the existence of Wj. To prove it, we use the result of Kopylov (2009), and especially, we show that ¯j satisfies Finiteness in Kopylov (2009).

Axiom 21.

(Finiteness): For any sequence {An} of A, there exists a positive integer N such that n=1NAnjn=1N+1An.

To show that ¯j satisfies Finiteness, take an arbitrary menu AA. A is compact, so there exist j-best and j-worst allocations denoted by p̅jΔj and p̅jΔj respectively. Take a menu BA such that for any qB, p̅jjqjjp̅j. Then, AjAB. Take A1,A2,A3A. Let p̅argmaxpA1A2A3u(pj), and p̅argminpA1A2A3u(pj). By construction, assume that p̅A1 and p̅A1. Suppose that A1A2¯jA1A3. Then, by definition, A1A2A1A2A3. Thus, Finiteness is satisfied with N=2. By Intrinsic Set-Betweenness in the above argument, we obtain the support of the sign that is maxuj and minuj. By Theorem 1.A (a weak EU representaion) in Dekel et al. (2001), there exists Wj:u(Δ)×u(Δ)R, which describes the desired representation.

Singleton Independence. We show that, for each ¯j satisfies Singleton Independence, to obtain the desired result. This follows from Axiom 6 (Singleton Independence) of ¯.

We obtain the following result.

Corollary 9.

¯j satisfies Completeness, Transitivity, Continuity, Disjoint Set-Betweenness, and Singleton Independence, if and only if there exist a pair (u,γj) where u:ΔR and γj[0,1] such that ¯j is represented by u̅j:AR, defined by

u̅j(A)=γjmaxpAu(pj)+(1γj)minpAu(pj).

Proof. The result follows from Theorem 1 in Olszewski (2007).

By putting u1:=α1u(p1) and uS:=iSαiu(pi) for each p1Δ and pSΔS, we obtain the desired utility representation on reference-point formation; for any A,BA, A¯jBu̅j(A)u̅j(B).

Step 3.

In Step 3, we introduce a binary relation ¯ on the set of menus A defined later. We show that V:AR represents ¯.

Utility Space. We consider a set of utilities of allocations on a utility space in each menu A. For any AA, define

u(A):={(u1(p1),uS(pS))R2|p=(p1,pS)A},

where u1(p1)=α1u(p1) and uS(pS)=iSαiu(pi). Let us A be denoted by {u(A)|AA}. Since each choice set AA is compact, u(A) is also compact, by the continuity of u:ΔR. By the uniqueness property of u, we can normalize u(Δ)=[0,1] and uS(ΔS)=[0,1]. A is a set of compact subsets of [0,1]2, endowed with the Hausdorff metric. Define ¯ on A in the following way.

Definition 14.

For any A,BA,

A¯BifA¯B,

where A=u(A) and B=u(B).

The asymmetric and symmetric parts of ¯ are denoted by and , respectively.

is well-defined. First of all, we show that ¯ is well-defined.

Lemma 1.

¯is well-defined.

Proof. Suppose A=B, i.e., u(A)=u(B). We need to show that AB. Then, for any pA there exists qB such that p11q1 and pSSqS. Hence, A1B and ASB hold. By Axiom 7 (Weak Dominance), we have AAB. In the same way, we have BAB. By Axiom 1 (Standard Preferences), in particular, the transitivity of ¯, we obtain AB.

Axioms.

Consider the axioms in Theorem 1 in the above utility space. We show that ¯ satisfies the following axioms. Let us introduce ¯j on A for each j{1,S}. We say that for each j{1,S}, A¯jB if for any vB and uA such that ujvj.

Axiom* 1.

(Pareto*): For any u,v[0,1]2, if uv, then {u}¯{v}.

Axiom* 2.

(Weak Dominance*): For any A,BA, if A1B, ASB, and A¯B, then AAB.

Axiom* 3.

(Intrinsic Set-Betweenness*): For each j{1,S}, if AB= and A¯jB, then A¯jAB¯jB.

Axiom* 4.

(Singleton Independence*): For any A,BA, w[0,1]2, and λ[0,1]

A¯BλA+(1λ){w}¯λB+(1λ){w}.

Axiom* 5.

(Weak Independence*): or any A,B,CA and λ[0,1],

(j{1,S})A¯jB[A¯BλA+(1λ)C¯λB+(1λ)C].

We obtain the following result.

Lemma 2.

¯ is a continuous weak order that satisfies Pareto*, Weak Dominance*, Intrinsic Set-Betweenness*, Singleton Independence*, and Weak Independence*.

Proof.

We omit the proof as we suppose that ¯ satisfies the axioms in Theorem 1. By the definition of ¯, it is easily verified that ¯ satisfies the axioms in the utility space.

Step 4.

In Step 4, we complete the utility representation of the RDIC utilitarian. First, we identify a finite subjective state space, by following from Kopylov (2009). Next, we show that V has a functional form of the subjective state space (Lemma 4).

A Utility Representation of a Subjective State Space. For each menu A in the utility space A, let u̅A be the reference point of the menu A. Let u̅1=γ1maxu1+(1γ1)minu1 where γ1[0,1], and u̅S=γSmaxuS+(1γS)minuS where γS[0,1].

Suppose that ¯ is a continuous weak order that satisfies Axioms* 1–5 in Step 3.

Definition 15.

There exists a unique function μ:[0,1]2×RR such that ¯ on A is represented by V:AR defined by (see Figure 4)

V(A)=λRμ(u̅A,λ)(maxuAλu̅1+(1λ)u̅S).

We have defined μ:[0,1]2×RR. The first argument is a reference point, so μ depends on reference points.[37] Even though we relax Independence, we still have an additive utility representation. The additivity follows from Axiom* 5 (Weak Independence*). By Axiom 5, the sign of μ depends on the reference point u̅.

To obtain the desired functional form, we show the following. First, we show that ¯ satisfies Finiteness* in Kopylov (2009).

Figure 4: Utility representation with a subjective state space (Definition 15).

Figure 4:

Utility representation with a subjective state space (Definition 15).

Axiom* 6.

(Finiteness*): For any sequence {An} of A, there exists a positive integer N such that n=1NAnn=1N+1An.

Lemma 3.

¯satisfies Finiteness (Axiom 6).

Proof.

We show the following: Take arbitrary two menus A,BA satisfying the following: for any A,BA and uA, u1v1 and uSvS. We obtain A¯B. This implies that, for any vAB, there exists uA such that u1=v1 and uS=vS. Then, by Axiom 2 (Weak Dominance*), we have AAB.

Take arbitrary menus A1,A2,A3,A4A. Let us denote the following:

  • u^argmaxuA1A2A3A4u1;

  • u^argminuA1A2A3A4u1 ;

  • v^argmaxvA1A2A3A4vS; and

  • v^argminvA1A2A3A4vS.

Without loss of generality, assume u^,u^A1 and v^,v^A2. Suppose A1A2A3¯A1A2A4. Then, by Axiom 2 (Weak Dominance*), we have A1A2A3A1A2A3A4. Hence, Axiom 6 (Finiteness*) is satisfied with N=3.

Identifying a Subjective State Space. We show the following lemma to obtain the desired representation.

Lemma 4.

Suppose that ¯ is a continuous weak order that satisfies Axioms* 2–5 in Step 3. Then, there exists at least one λ(0,1) such that ¯ is represented by (See Figure 5)

V(A)=μ(u̅A,1)maxuAu̅1+μ(u̅A,λ)maxuA(λu̅1+(1λ)u̅S)+μ(u̅A,0)maxuAu̅S.

Proof. Remember that we can normalize u(Δ)=[0,1] and uS(ΔI)=[0,1]. Fix ε<12. For all λR, define u(λ)=(u1(λ),uS(λ)) by

u1(λ)=12+ελ(λ,1λ),

and

uS(λ)=12+ε(1λ)(λ,1λ).

First, we show that for all λ[0,1], u1(1)u1(λ) and uS(0)uS(λ). For all λ[0,1], u1(λ)=12+εu1(λ). In the same way, for all λ[0,1], uS(λ)=12+εuS(λ).

Next, we show the following:

  • (i) for all λ[0,1], μ(u̅(A),λ)=0,

  • (ii) for all λ{0,1}, μ(u̅(A),λ)0, and

  • (iii) there exists a unique λ{0,1}, μ(u̅(A),λ)>0.

Figure 5: Identification of a subjective state space (Lemma 4).

Figure 5:

Identification of a subjective state space (Lemma 4).

Proof of (i).

We show (i) by the way of contradiction. Suppose that there exists λ[0,1] such that μ(u̅A,λ)0. Without loss of generality, suppose μ(u̅A,λ)<0. Consider the following menus:

  • A={u(λ)[0,1]2|λ{0,1,|λ||λ|+|1λ|}\{λ}};

  • A{u(λ)}.

Notice that the two menus are compact. Since u(|λ||λ|+|1λ|)A, and u1(|λ||λ|+|1λ|)u1(λ) and uS(|λ||λ|+|1λ|)uS(λ), by Axiom 2 (Weak Dominance), AA{u(λ)}.

If μ(u̅A,λ)<0 and μ(u̅A{u(λ)},λ)<0, then V(A)>V(A{u(λ)}), which represents AA{u(λ)}. This is a contradiction.

Proof of (ii).

We show (ii) by the way of contradiction. Suppose that there exists λ{0,1} such that μ(u̅A{u(λ)},λ)<0. Consider the following menus:

  • A={u(λ)[0,1]2|λ{0,12,1}};

  • A{u(λ)}.

The two menus are compact. Since u(12)A, by Axiom 2 (Weak Dominance), AA{u(λ)}. However, by the assumption of μ(,λ)<0, we have V(A)>V(A{u(λ)}), which represents AA{u(λ)}. This is a contradiction.

Proof of (iii).

We show (iii). To see this, fix a menu AA, and assume that there exists no λ(0,1) such that μ(u̅A,λ)=0. By (ii), for all λ(0,1), μ(u̅A,λ)=0. Take another menu BA such that for each j{1,S}, AjB, and AB. By definition, under the assumption, we have V(A)=V(B). Hence, to obtain V(A)>V(B), there exists λ(0,1) such that μ(u̅A,λ)>0. The uniqueness follows from the fact that V is mixture-linear: for any σ[0,1], V(σA+(1σ)B)=σV(A)+(1σ)V(B).

RDIC Utility Representation

By Lemma 4, we find three states, i.e., u̅1,λu̅1+(1λ)u̅S, and u̅S. We normalize μ in the following way. First, let (2γ11)μ(u̅A,λ)λ+μ(u̅A,1)=1. Define, for each menu AA, β1(u̅A):=μ(u̅A,1). Second, in the same way, normalize μ as follows: (2γS1)μ(u̅A,λ)(1λ)+μ(u̅A,0)=1. Let βS(u̅A):=μ(u̅A,0). Then, we have the utility representation in the following way (Figure 6): for any AA,

V(A)=maxuA(1+β1(u̅A))α1u1+(1βS(u̅A))uSβ1(u̅A)maxuAu̅1+βS(u̅A)maxuAu̅S.

By the axiom of Pareto, we obtain β1(u̅A)(1,) for any AA. In the same way, we obtain βS(u̅A)(,1) for any AA.

For any AA, define V(A)=V(A). Then, we have

A¯BA¯BV(A)V(B)V(A)V(B).

By letting u1=α1u(p1) and uS=iSαiu(pi), we have the desired representation; that is, for any AA,

V(A)=maxpA[iIαiu(pi)+β1(u̅A)(α1(u(p1)u̅1(A)))βS(u̅A)(iSαiu(pi)u̅S(A))].

We complete the proof of the sufficiency part.

Figure 6: Reference-dependent image-conscious utilitarian.

Figure 6:

Reference-dependent image-conscious utilitarian.

A.2 Necessity Part

We show the necessity part. We show that the utility representation satisfies the axiom of Weak Dominance and Intrinsic Set-betweenness. It is easy to prove the necessity of the other axioms.

A.2.1 Weak Dominance

First, to show Axiom 5 (Weak Dominance), take arbitrary two menus A,BA with A1B,ASB, and A¯B. Consider a RDIC with a four-tuple (u,α,β,γ); that is, for any AA,

V(A)=maxpA[iIαiu(pi)+β1(u̅A)(α1(u(p1)u̅1(A)))βS(u̅A)(iSαiu(pi)u̅S(A))].

Let iIαiu(pi), β1(u̅A)(α1(u(p1)u̅1(A))), βS(u̅A)(iSαiu(pi)u̅S(A)) be the first term, the second term, and the third term, respectively. For each AA, let Δ1(A):=α1(u(p1)u̅1(A)) where p is a maximizer in A, and ΔS(A):=u(pS)u̅S(A)) where u(pS):=iSαiu(pi) and p is a maximizer in A.

Consider the menu AB. Since A¯B, pA is a maximizer in AB. Then, Δ1(A)=Δ1(AB) and ΔS(A)=ΔS(AB) hold. Moreover, since A1B and ASB hold, u̅A=u̅AB. Then, β(A)=β(AB). Thus, the utility function has the same terms. Hence, V(A)=V(AB)AAB.

A.2.2 Intrinsic Set-betweenness

Next, we show that V represents ¯ that satisfies Axiom 4 (Intrinsic Set-Betweenness). Consider j{1,S}. Suppose that AB= and A¯jB. Then, by γj[0,1] and the definition of u̅j, we have u̅j(A)u̅j(AB) and u̅j(AB)u̅j(B). Hence, ¯j satisfies the axiom of Intrinsic Set-betweenness.

B Pure Altruism/Selfishness, and Inequity Aversion

We study the effect on the parameter α=(α1,(αi)iS). The component α1 captures the level of pure altruism/selfishness. The component αS=(αi)iS captures the level of other-regarding preferences. In Appdendix B.1, we study the comparative attitude toward pure altruism/selfishness. In Appendix B.2, by modifying the axioms in Theorem 1, we study inequity-averse preferences (Fehr and Schmidt 1999).

B.1 Comparative Statics on Pure Altruism/Selfishness

We study the comparative statics on pure altruism captured by the parameter α1 in the model.[38]α1 is charaterized by preferences over singletons, i.e., outcome-based utility.

Consider two decision makers X and Y. Both Mr.X and Mr.Y are denoted by decision maker 1. Assume that the set of other agents S is fixed.[39]

Definition 16.

For any ¯X and ¯Y on A such that Xj=Yj for each j{1,S}, ¯X is more purely altruistic than ¯Y if, for any p,qΔI with pSShqS for each h{X,Y},

{p}¯Y{q}{p}¯X{q}.

We consider the case that an allocation p is superior to q in other-regarding preference, i.e., pShSqS for each h{X,Y}. Under the assumption that Xj=Yj for each j{1,S}, the definition states that if Mr. Y weakly prefers p to q with pSSYqS, then another decision maker Mr. X also weakly prefers p to q with pSXSqS.

We obtain the following result.

Proposition 11.

Suppose that for eachh{X,Y},¯his represented by a four-tuple(u,αh,β,γ). Then, X is more purely altruistic than Y if and only ifα1Xα1Y.

Assume that β1(uA)=βS(uA)=0 for each AA. Moreover, assume that S={2}. As α1 is getting smaller, the resulting behavior is altruistic (see Figure 7; the case of 0<α<1). In the RDIC representation, the case of α1=1 corresponds to egalitarian.

Figure 7: Comparative statics on pure altruism.

Figure 7:

Comparative statics on pure altruism.

B.2 Inequity Aversion

We present a brief guideline of the axiomatization. To capture inequity aversion, we modify the axioms in Theorem 1. First, we modify Axiom 3 (Pareto). Instead, we introduce a weaker version of Monotonicity (monotonicity with respect to equal allocations). Second, we strengthen Axiom 5 (Weak Dominance). We require that ¯ satisfies Strategic Rationality (Kreps 1979); We study the case that there is no effect on social emotions stemming from image concerns. Third, we relax Independence to capture inequity-averse preferences.

Set-Up. Let XRn be the compact subset of Rn.

B.2.1 Outcome-Based Utility

Strategic Rationality is introduced in Kreps (1979).[40] This axiom states that the decision maker exhibits neither preferences for flexibility nor preferences for commitment.

Axiom 22.

(Strategic Rationality): For any A,BA,A¯BAAB .

We obtain the following result. This is the case that there is no effect on social image concerns. The corollary implies that social image concerns are axiomatically different from inequity aversion.

Corollary 10.

Suppose that¯is represented by a RDIC with a four-tuple(u,α,β,γ). Then, ¯satisfies Axiom 22 (Strategic Rationality) if and only if for anyAA, β1(u̅A)=0andβS(u̅A)=0.

B.2.2 Inequity-Averse Utility

We relax Axiom 3 (Pareto). The Pareto condition is not consistent with inequity-averse preferences. For each p ∈ Δ, let cp ∈ X be the certainty equivalent of p, i.e., p ~1δcp where δcp is the lottery that gives cp with certainty. The following weak monotone condition says that fairness is sustained in the case that every agent has the same allocation in terms of certainty equivalent of lotteries. We call such allocations equal allocations.

Axiom 23.

(Monotonicity with respect to Equal Allocation): For any p,qΔ with p ~1δcp and q ~1δcp, if cpcq, then {(δcp,,δcp)}¯{(δcq,,δcq)}.

We relax Axiom 6 and Axiom 7 in this paper. We define quasi-comonotonic allocations.

Definition 17.

For any p,qΔI, two allocations p and q are quasi-comonotonic if there exists no iS such that pi1p1 and qi1q1.

The following axiom is a modified version of Comotonic Independence introduced in Schmeidler (1989). This axiom states that if two allocations are quasi-comonotonic, then λ-mixture (λ(0,1)) with quasi-comonotonic allocations does not change the ranking of allocations.

Axiom 24.

(Quasi-Comonotonic Independence): For any p,q,rΔI that are pairwise comonotonic, and λ(0,1),

{p}¯{q}λ{p}+(1λ){r}¯λ{q}+(1λ){r}.

We provide an axiom of Inequity Aversion and obtain the desired result.

Axiom 25.

(Inequity Aversion): satisfies (i) envy and (ii) guilt: For any p,p,pΔ with p ~1δcp, p~1δcp, and p ~1δcp, if cp > cp > cp″,

  • (i) (Envy): {(δcp,(δcp)iS)}¯{(δcp,(δcp)iS)};

  • (ii) (Guilt): {(δcp,(δcp)iS)}¯{(δcp,(δcp)iS)}.

The first condition (i) states that the decision maker feels envy because the other agents obtain the higher expected payoff in terms of ther deicision maker’s risk preference than he does. The second condition (ii) states that the decision maker feels guilt because their expected payoff is larger than others’ payoff.

Remember that u is a linear function. In Appendix A, we show that ≿1 is represented by u : Δ → R, and ≿1 satisfies Independence. Thus, for each pi ∈ Δ, u(pi) ≔ Epi [ʋ] for some ʋ : XR (ʋ: vNM function). Let the certainty equivalent of pi is written by c(pi, ʋ) ≔ ʋ−1Epi [ʋ] ∈ X.

Corollary 11.

¯ satisfies Axiom 1 (Standard Preferences), Axiom 22 (Strategic Rationality), Axiom 23 (Monotonicity w.r.t. Equal Allocation), Axiom 24 (Quasi-Comonotonic Independence), and Axiom 25 (Inequity Aversion) if and only if there exists a tuple (v,α1,(αenvyi,αguilti)iS) where v:XR is a non-constant function, α1>0, and for each iS, αenvyi0, αguilti0, such that ¯ is represented by

V(A)=maxpA[α1c(p1,v)iS(αenvyimax{c(pi,v)c(p1,v),0}+αguiltimax{c(p1,v)c(pi,v),0})].
Remark 4.

The preferences-over-menus framework identifies social emotions such as pride, shame, and temptation stemming from social image concerns. On the other hand, inequity-averse preferences are identified by preferences over singletons, i.e., preference over allocations. The main finding is that we can identify image-conscious preferences and inequity-averse preference separately.[41]

C Proof of Propositions

C.1 Proof of Proposition 1

(i) and (ii). The uniqueness result follows from mainly EUT. We show (i) and (ii). By the standard uniqueness result of the EUT, we obtain αi=αi for all iI. Moreover, there exists a>0 and bR such that u=au+b.

(iv). Next, we show (iv). It is straightforward to prove γ=γ. Consider j{1,S}. By (i), u̅j=au̅j+b for some a>0 and bR. Let u^j:=maxpAuj, u^j:=minpAuj, u^j:=maxpAuj=uj, and u^j:=minpAuj for each j{1,S}. Notice that u1=α1u(p1) and uS=iSαiu(pi) for all pΔI. Then, u̅j=γju^j+(1γj)u^j=γj(au^j+b)+(1γj)(au^j+b)=a[γju^j+(1γj)u^j]+b. Since u̅j=au̅j+b, it must be γj=γj. Hence, γ=γ.

(iii). Finally, we show (iii). By (i) and the proof of Theorem 1, note that V=aV+bV=aV+b. The ex-post utility U is given by U=(1β1)α1u1+(1+βS)iSαiui. By (i), (ii), and (iv), U=aU+b. In the similar way, we have β()=β().

C.2 Proof of Proposition 2

Suppose that ¯ satisfies Axiom 8 (Weak Dominance I). Then, we show that βS(u̅A)=0 for all AA. Suppose that β1(u̅A)0 for some AA, and that the conditions in Axiom 8 hold. By the way of contradiction, suppose V(A)>V(AB) or V(A)<V(AB) occurs. Without loss of generality, assume ΔS(AB)<0. By Axiom 8, we must have AABV(A)=V(AB). However, if βS(u̅AB)<0, then V(A)>V(AB). This is a contradiction. Hence, βS(u̅A)=0 for all AA.

Fix p,qΔI such that the doubleton {p,q} is a socially conflicting menu. If {p,q}¯{p}, then V({p,q})V({p})V({p,q})V({q})0. Notice that Δ1=u(p1)u̅1({p,q})<0. Hence, {p,q}¯{p}V({p,q})V({q})0β1(u̅{p,q})α1Δ101<β1(u̅{p,q})0.

By Axiom 6 (Singleton Independence), we can show that this holds for any doubleton. Now, consider an arbitrary menu AA with p,q. If p is the maximizer in A, then 1<β1(u̅A)0 holds. Suppose not. Then, p is the maximizer in A for some pA. Since βS(u̅A)=0, by Axiom 10 (Pride), there exists qA such that the doubleton {p,q} is a socially conflicting menu. Moreover, by Axiom 9 (Self-interest), γ1=1 holds (Corollary 1). Δ1=u(p1)u̅1(A)<0 holds for all AA. Thus, 1<β1(u̅A)0 holds. Hence, 1<β1(u(Δ)2)0.

The necessity part is easily verified. We omit it.

C.3 Proof of Proposition 3

In Proposition 2, we have βS(u̅A)=0 for all AA. By Corollary 1, γ1=1.

C.3.1 The Sufficiency Part

We show the sufficiency part. By Proposition 2, for each j{X,Y}, ¯j exhibits Pride if and only if 1<β1(u(Δ)2)0. Take p,qΔI such that {p,q} is a socially conflicting menu. And, take rΔ. Suppose that ¯X is more pride-seeking than ¯Y. Then, {p,q}¯Y{p}{p,q}¯X{p}. Then, VY({p,q})VY({p})=UY({p}). We also have VX({p,q})VX({p})=UX({p}). Since Mr. X and Mr. Y have the same u, α, and γ, we obtain VY({p})=UY({p})=VX({p})=UX({p})=U̅. Then, VY({p,q})U̅VX({p,q})U̅VY({p,q})VX({p,q}). Since Δ({p,q})0 and a1>0, we obtan 1<βX(u̅{p,q})βY(u̅{p,q})0. By using Corollary 1, we can show that this holds for any AA.

C.3.2 The Necessity Part

We show the necessity part. Take an arbitrary menu AA. Assume that 1<βX(u̅A)βY(u̅A)0. In the same way, take p,qΔI such that {p,q} is a socially conflicting menu. By βs(u̅A)=0 for all AA. By Corollary 1, γ1=1. We have VX({p,q})VX({p})=UX({p}). In the same way, we obtain VY({p,q})VX({p,q}). By the assumption of 1<βX(u̅A)βY(u̅A)0, we obtain the following property: ¯X exhibits more pride-seeking than¯Y.

C.4 Proof of Proposition 4

The proof of Proposition 4 is similar to that of Proposition 2. First, in Proposition 2, we have βS(u̅A)=0 for all AA.

Second, in the similar way of Corollary 1, we can show that γ1=0.

Take p,qΔI such that the doubleton {p,q} is a selfishly conflicting menu. If {p,q}¯{p}, then V({p,q})V({p})V({p,q})V({q})0. Notice that Δ1=u(p1)u̅1({p,q})0. Hence, {p,q}¯{p}V({p,q})V({q})0β1(u̅{p,q})α1Δ100β1(u̅{p,q})+. In the same way as Proposition 2, we can extend that this holds for any AA.

C.5 Proof of Proposition 5

The proof of Proposition 5 is similar to Proposition 3. In Proposition 2, we have βS(u̅A)=0 for all AA. Moreover, we have γ1=0 (Corollary 2).

C.5.1 The Sufficiency Part

We show the sufficiency part. By Proposition 4, for each j{X,Y}, ¯j exhibits Temptation if and only if 0β1(u(Δ)2)<+. Take p,qΔI such that {p,q} is a selfishly conflicting menu. Suppose that ¯X is more temptation-driven than ¯Y. Then, {p,q}¯Y{p}{p,q}¯X{p}. Then, VY({p,q})VY({p})=UY({p}). We also have VX({p,q})VX({p})=UX({p}). Since Mr. X and Mr. Y have the same u, α, and γ, we obtain VY({p})=UY({p})=VX({p})=UX({p})=U̅. Then, VY({p,q})U̅VX({p,q})U̅VY({p,q})VX({p,q}). Since Δ1({p,q})0 and a1>0, we obtan 0βY(u̅{p,q})βX(u̅{p,q})<+. By using γ1=0, we can show that this holds for any AA.

C.5.2 The Necessity Part

We show the necessity part. Take an arbitrary menu AA. Assume that 0βY(u̅A)βX(u̅A)<+. In the same way, take p,qΔI such that {p,q} is a selfishly conflicting menu. By Proposition 2, βS(u̅A)=0 for all AA. Furthermore, we have γ1=0 (Corollary 2). We have VX({p,q})VX({p})=UX({p}). In the same way, we obtain VY({p,q})VX({p,q}). By the assumption of 0βY(u̅A)βX(u̅A)<+, we obtain the following property: ¯X exhibits more temptation-driven than ¯Y.

C.6 Proof of Proposition 6

First, we show that β1(u̅A)=0 for all AA. Suppose that β1(u̅A)0 for some AA by the way of contradiction. And, suppose that the conditions in Axiom 15 (Shame-Based Dominance) holds. By the way of contradiction, for the social emotions of pride or temptation, V(A)>V(AB) or V(A)<V(AB) occurs. Without loss of generality, assume Δ1(AB)<0. By Axiom 15, we must have AABV(A)=V(AB). However, if β1(u̅AB)<0, then V(A)<V(AB). This is a contradiction. Hence, β1(u̅A)=0 for all AA.

By Corollary 3, we have γS=0.

Take p,qΔI such that the doubleton {p,q} is a shame-driven menu. If {p}¯{p,q}, then V({p})V({p,q})V({p})V({p,q})0. Notice that ΔS=u(pS)u̅S({p,q})0. Hence, {p}¯{p,q}V({p})V({p,q})0βS(u̅{p,q})α1Δ10βS(u̅{p,q})0. In the same way as Proposition 2, we can extend that this holds for any AA.

C.7 Proof of Proposition 7

The proof of Proposition 5 is similar to Proposition 3. In Proposition 4, we have β1(u̅A)=0 for all AA. Moreover, we have γS=0 (Corollary 3).

C.7.1 The Sufficiency Part

We show the sufficiency part. By Proposition 6, for each j{X,Y}, ¯j exhibits Shame if and only if βS(u(Δ)2)0. Take p,qΔI such that {p,q} is a shame-driven menu. And, take lΔ. Suppose that ¯X is more shame-averse than ¯Y. Then, {p}¯Y{p,q}{p}¯X{p,q}. Then, VY({p})=UY({p})VY({p,q}). We also have VX({p})=UX({p})VX({p,q}). Since Mr. X and Mr. Y have the same u, α, and γ, we obtain VY({p})=UY({p})=VX({p})=UX({p})=U̅. Then, U̅VY({p,q})U̅VX({p,q})VY({p,q})VX({p,q}). Since ΔS({p,q})0 and α1>0, we obtain βY(u̅{p,q})βX(u̅{p,q})0. By using γS=0, we can show that this holds for any AA.

C.7.2 The Necessity Part

We show the necessity part. Take an arbitrary menu AA. Assume that βY(u̅A)βX(u̅A)0. In the same way, take p,qΔI such that {p,q} is a shame-driven menu. And, take rΔ. We have β1(u̅A)=0 for all AA. Furthermore, we have γS=0 (Corollary 3). We have VX({p,q})VX({p})=UX({p}). In the same way, we obtain VY({p,q})VX({p,q}). By the assumption of βY(u̅A)βX(u̅A)0, we obtain the following property: ¯X exhibits more shame-averse than ¯Y.

C.8 Proof of Proposition 8

The proof of Lemma 8 follows from Corollary 2 in Olszewski (2007). We show the sufficiency part in (i) and (ii). The necessity part is immediately shown. Take A,BA with BA. Consider j{1,S}. Suppose that A¯jYBA¯jXB. By Step 2 in the proof of Theorem 1, u̅jY(A)u̅jY(B)u̅jX(A)u̅jX(B). Hence, we have

1u̅jY(A)u̅jY(B)u̅jX(A)u̅jX(B)0.

Since both X and Y has the same self-utility function u, the maximum and the minimum of j in each menu A or B are the same. Thus, the inequality implies γjXγjY.

C.9 Proof of Proposition 9

Take p,qΔI with (i) {q}1{p}, (ii) {p}S{q}, and (iii) {p,q}{q}. Fix the socially conflicting menu {p,q}.

Suppose that ¯ is represented by a RDIC with a four-tuple (u,α,β,γ). Consider the menu {p,q}, and the utility of the menu is described as follows.

V({p,q})=maxp{p,q}[iIαiu(pi)+β1(u̅{p,q})(α1(u(p1)u̅1({p,q})))βS(u̅{p,q})(iSαiu(pi)u̅S({p,q}))].

Let iIαiu(pi), β1(u̅{p,q})(α1(u(p1)u̅1({p,q}))), βS(u̅{p,q})(iSαiu(pi)u̅S({p,q})) be the first term, the second term, and the third term, respectively. For each menu AA, let Δ1(A):=α1(u(p1)u̅1(A)) and ΔS(A):=iSαiu(pi)u̅S(A)) where p is a maximizer in A.

Proof of (ii)

C.9.1 The Sufficiency Part

First, we show the sufficiency part. Take a selfish option r for the socially conflicting menu {p,q}. Suppose that ¯ exhibits more pride-seeking preferences. Then, we have 1<β1(u̅{p,q})0. We need to show 1<β1(u̅{p,q,r})β1(u̅{p,q})0.

By pride-seeking preferences, {p}{p,q}V({p})<V({p,q}). We prove it. Since γj[0,1] for each j{1,S}, we have Δ1({p,q})0 and ΔS({p,q})0. The condition (iii) {p,q}{q} implies that p is chosen from {p,q}. Without loss of generality, p is a maximizer in {p,q}.

Consider the second term in RDIC of V({p,q}). By pride-seeking preferences, 1<β1(u̅({p,q}))0. And, since Δ1({p,q})0, the second term is non-negative. If βS(u̅({p,q}))>0, then the third term is negative because Δ1({p,q})0. We must obtain (the second term) (the third term). On the other hand, if βS(u̅({p,q}))>0, then the third term is non-negative. Hence, V({p})V({p,q}){p}¯{p,q}.

Consider the menu {p,r} such that r is selfish than p. By pride-seeking preferences, without loss of generality, assume that p is chosen from {p,r}. By the RDIC representation, Δ1({p,r})0. In the same way, Δ1({p,r})0. By the way of contradiction, suppose that β1(u̅{p,r})0. Suppose also βS(u̅{p,r})0. Thus, we have the following. Both the second term and the third term are non-positive in the RDIC representation of V({p,r}). This implies that V({p})>V({p,r}). This is a contradiction. On the other hand, supoose <βS(u̅{p,r})0. By the definition of socially conflictingof doubletons,. Then, |(the second term)|<|(the third term)|, because p is chosen from {p,r} at the ex-post stage. We obtain V({p})>V({p,r}). This is a contradiction. Hence, we obtain β1(u̅{p,r})0.

Now, consider the menu {p,q,r}. By pride-seeking preferences, under the conditions, {p}¯{p,q}{p}¯{p,r}. By socially conflicting menus, we have β1(u̅{p,q})0, as shown in the above.

We need to show that β1(u̅{p,q,r})β1(u̅{p,q}). Consider the menu {p,r}. First, consider the case of the reference points of the two menus u̅{p,q,r} and u̅{p,r} with u̅{p,q,r}=u̅{p,r}. By the definition of β, we obtain β(u̅{p,q,r})=β(u̅{p,r}). As shown in the above, β1(u̅{p,r})0 and β1(u̅{p,q})0. By pride-seeking preferences, V({p,r})V({p,q}) implies that β1(u̅{p,r})β1(u̅{p,q}). Hence, β1(u̅{p,r})=β1(u̅{p,q,r})β1(u̅{p,q}).

Next, consider the case of u̅{p,q,r}u̅{p,r}. Remember that {p}¯{p,r}. In this case, Δ1({p,q,r})Δ1({p,q}), and ΔS({p,q,r})ΔS({p,q}). V({p,q})V({p,r}) implies that β1() must be 1<β1(u̅{p,q,r})β1(u̅{p,q})0. Hence, in the case that ¯ exhibits pride-seeking preferences, β1 is decreasing in the first argument.

C.9.2 The Necessity Part

Next, we show the necessity part. Take a selfish option r for the menu {p,q}. Suppose that 1<β1(u̅{p,q,r})β1(u̅{p,q})0. β1 is decreasing in the first argument on (1,0). {p}¯{p,q} holds with the three conditions implies that β1(u̅{p,q})0. Then, we have β1(u̅{p,q,r})β1(u̅{p,q})0.

First, consider the case of u̅{p,q,r}=u̅{p,r}. By the definition of β, we obtain β(u̅{p,q,r})=β(u̅{p,r}). Hence, {p,q,r}1{p,r} and {p,q,r}S{p,r}. This imply that in the menu {p,r}, ¯ exhibits pride, and that {p}¯{p,r}.

Second, consider the case of u̅{p,q,r}u̅{p,r}. Suppose that, by the way of contradiction, {p,r}{r}. This case says that ¯ exhibits shame of acting selfishly. Then, Δ1({p,r})<0 and ΔS({p,r})>0. Suppose that β1({p,r})>0. Then, The second term is non-negative, and the third term is positive.By the defnition of socially conicting of doubletons, |(the second term)|<|(the third term)| becausep is chosen from {p,r}at the ex-post stage Thus, V({p,r})<V({r}).This is a contradiction. Thus, we obtain {p,r}¯{r}. Hence, the three conditions are satisfied, and 1<β1(u̅{p,q,r})β1(u̅{p,q})0, so {p}¯{p,r}.

In the same way, we can show the first part of Proposition 9. We omit it.

C.10 Proof of Proposition 10

Take p,qΔI. Fix a menu {p,q}, and suppose the following. Take an arbitrary option rΔI. Suppose (i) {r}1{p}, (ii) {q}S{p}, and (iii) {p,q,r}{q,r}. Take an altruistic option r for the menu {p,q}.

First, we show the sufficiency part. Suppose that ¯ exhibits more shame-averse preferences. We need to show |βS(u̅{p,q,r})||βS(u̅{p,q})|; that is, βS is decreasing in the second argument on (,0).

Consider the menu {p,q}. By the three conditions and {p}¯{p,q}, <βS(u̅{p,q})0. Suppose that 1<β1(u̅{p,q})0. Since Δ1({p,q})>0 and ΔS({p,q})<0, the second term is negative, and the third term is negative. Hence, V({p})V({p,q}). Furthermore, consider the case of 0<β1(u̅{p,q})<+. Then, the second term is positive. By shame-averse preferences, {p}¯{p,q}. Then,the second term is positive. By shame-averse preferences, {p}¯{p,r}. which implies |(the second term)||(the third term)|. Hence, V({p})V({p,q}).

Consider the menu {p,r}. By shame-averse preferences, {p}¯{p,r}. By the similar argument above, we have <βS(u̅{p,r})0. We obtain V({p})V({p,r}).

Furthermore, consider the two menus {p,q,r} and {p,r}. First, without loss of generality, assume that u̅{p,q,r}=u̅{p,r}. By the definition of β, βS(u̅{p,q,r})=βS(u̅{p,r})βS(u̅{p,q}).

Second, consider the case of u̅{p,q,r}u̅{p,r}. Since ¯ exhibits more shame-averse preferences, we have the following: V({p})V({p,q})V({p})V({p,r}). Then, V({p,q})V({p,r}). Moreover, ΔS({p,q,r})ΔS({p,q})<0. βS is decreasing in the second argument, i.e., βS(u̅{p,q,r})βS(u̅{p,q}).

Next, we show the necessity part. Suppose |βS(u̅{p,q,r})||βS(u̅{p,q})|; that is, βS is decreasing in the second argument on (,0). ¯ is shame-averse, i.e., {p}¯{p,q}. We need to show {p}¯{p,r}. By the way of contradiction, suppose that {p}{p,r}. Suppose that, at the ex-post stage, p is chosen from {p,r}. Then, Δ1({p,r})>0 and ΔS({p,r})<0. {p}{p,r} implies that ¯ exhibits pride of acting altruistically. Then, β1(u̅{p,r})0. The second term is negative, and the third term is negative, so V({p})>V({p,r}). This is a contradiction.

C.11 Proof of Proposition 11

C.11.1 The Necessity Part

First, we show the necessity part. We show that the decision maker X is more altruistic than the decision maker Y. Suppose α1Xα1Y. Take two allocations p,qΔI such that pSShqS for each h{X,Y}. By definition, suppose {p}¯Y{q}. Since pSShqS, we have iSαiu(pi)>iSαiu(qi). We consider the two cases: (i) u(p1)u(q1), and (ii) u(p1)<u(q1).

Consider the case u(p1)u(q1). By definition, α1X>0. Then, we have α1Xu(p1)+iSαiu(pi)α1Xu(q1)+iSαiu(qi). Thus, {p}¯X{q}, which implies that X is more altruistic than Y.

Consider the case u(p1)<u(q1). Remember that {p}¯Y{q}. Then, the following must hold: α1Y(u(p1)u(q1))iSαiu(qi)iSαiu(pi). We have u(p1)u(q1)<0 and we suppose α1Xα1Y, so α1X(u(p1)u(q1))iSαiu(qi)iSαiu(pi). Hence, α1Xu(p1)+iSαiu(pi)α1Xu(q1)+iSαiu(qi). Thus, {p}¯Y{q}, which implies that X is more altruistic than Y.

C.11.2 The Sufficiency Part

Next, we show the sufficiency part. Take two allocations p,qΔI such that pSShqS and p11hq1, for each h{X,Y}. Without loss of generality, suppose {p}Y{q}. Then, we have α1Yu(p1)+iSαiu(pi)=α1Yu(q1)+iSαiu(qi). We obtain

α1Y=iSαiu(pi)iSαiu(qi)u(q1)u(p1).

Suppose that X is more altruistic than Y. Then, {p}¯X{q}. We obtain α1Xu(p1)+iSαiu(pi)α1Xu(q1)+iSαiu(qi). Hence,

iSαiu(pi)iSαiu(qi)u(q1)u(p1)α1X.

We have α1Xα1Y, which completes the proof.

D Proofs of Corollaries

D.1 Proof of Corollary 1

Suppose that ¯ is represented by a RDIC with a four-tuple (u,α,β,γ).

We show the sufficiency part. Suppose that ¯1 exhibits Axiom 9 (Self-Interest). Remember that ¯1 is represented by u̅1:AR. Take arbitrary two menus A,BA with A¯1B. By Axiom 9, u̅1(A)=u̅1(AB). By the definition of ¯1, u̅1(A)=u̅1(AB) implies that maxpAu(p1)=maxpABu(p1). Hence, γ1=1.

We show the necessity part. Suppose γ1=1. Take arbitrary two menus A,BA with A¯1B. Since γ1=1, u̅1(A)u̅1(B)maxpAu(p1)maxpBu(p1). Thus, u̅1(A)=u̅1(AB) holds, because of maxpAu(p1)=maxpABu(p1). Therefore, ¯1 satisfies Axiom 9.

D.2 Proof of Corollary 2

The proof is similar to the proof of Corollary 1. We omit it.

D.3 Proof of Corollary 3

The proof is similar to the proof of Corollary 1. We omit it.

D.4 Proof of Corollary 4

The proof follows from Lemma 1 in Segal and Sobel (2007). Define the correspondence Φ:ΣΣ by, for each AA and σΣ,

Φi(σ):={σiAi|{σi}¯{σi}forallσiAi}.

Since, given σΣ, ¯i,σ satisfies the axioms of the vNM-type EUT. Ai is nonempty. The vNM-type EUT guarantees that Φ() is convex. Ai is compact. Hence, Φ() satisfies the conditions in the Kakutani’s fixed point theorem. Hence, a Nash equilibrium exists.

D.5 Proof of Corollary 5

The necessity part is easily shown. We omit it.

We show the sufficiency part. Suppose that ¯ satisfies Axioms 1, 2, 3, 6, 13, 15, and 16. By Axiom 16 (Shame-Based Independence), we slightly modify the Step 4 in the sufficiency part of Theorem 1. We replace the signed measure μ with μS(u̅S(A),λ), for all AA and λ[0,1]. Since we normalize the utility space, formally let μS:[0,1]×[0,1]R. In the same way as the proof in Theorem 1, we can obtain βS:u(Δ)(,1).

Moreover, we show that β1(u̅S(A))=0 for all AA. By the way of contradiction, suppose that β1(u̅S(A))0 for some AA, and that the conditions in Axiom 15 (Weak Dominance II) holds. Then, by the social emotions of pride or temptation, V(A)>V(AB) or V(A)<V(AB) occurs. Hence, β1(u̅S(A))=0 for all AA.

By Corollary 3, ¯ satisfying Axiom 13 (Social-interest’) implies that γS=1. Hence, there exists a four-tuple (u,α,β,γ) with β1()=0, such that ¯ is represented by

V(A)=maxpA[iIαiu(pi)βS(maxqAiSαiu(qi))(iSαiu(pi)maxqAiSαiu(qi))].

D.6 Proof of Corollary 6

The necessity part is easily shown. We omit it. We show the sufficiency part.

¯ satisfies Axioms 1, 2, 3, 6, 9, 8, and 17. The proof step is similar to the proof of Corollary 5. The difference is to replace the signed measure μ with μ1(u̅1(A),λ), for all AA and λ[0,1]. By normalizing the utility space, formally let μ1:[0,1]×[0,1]R.

By Corollary 1, ¯ satisfying Axiom 9 (Self-Interest) implies that γ1=1. Thus, we obtain a four-tuple (u,α,β,γ) with βS(u(Δ))=0, such that ¯ is represented by

V(A)=maxpA[iIαiu(pi)+β1(maxqAu(q1))(α1(u(p1)maxqAu(q1)))].

D.7 Proof of Corollary 7

The necessity part is easily shown. We omit it. We show the sufficiency part. By Corollary 1, ¯ satisfying Axiom 9 (Self-Interest) implies that γ1=1. By Corollary 3, ¯ satisfying Axiom 13 (Social-interest’) implies that γS=1. By Corollary 5, βS:u(Δ)(,1). By Corollary 6, β1:u(Δ)(1,+). Thus, we obtain the desired utility representation: For any AA, there exists a four-tuple (u,α,β,γ) such that ¯ is represented by

V(A)=maxpA[iIαiu(pi)+β1(maxqAu(q1))(α1(u(p1)maxqAu(q1)))βS(maxqAiSαiu(qi))(iSαiu(pi)maxqAiSαiu(qi))],

D.8 Proof of Corollary 10

Suppose that ¯ is represented by a RDIC with a four-tuple (u,α,β,γ).

First, we show the sufficiency part. ¯ satisfies Axiom 22, i.e., Strategic Rationality: A¯BAAB. Take arbitrary two menus, A,BA with A¯B. By Axiom 22, AABV(A)=V(AB). In the way of contradiction, suppose β1(u̅AB)>0. Without loss of generality, A1B. Then, the second term in the RDIC is not equal to zero. In the case of β1(u̅AB)>0, without u1(p1)u̅AB1=0, V(A)<V(AB) or V(A)>V(AB) holds. This is a contradiciton. Hence, β1(u̅A)=0 holds for any AA. We can show that βS(u̅A)=0 in the same way. Thus, for any AA, β1(u̅A)=βS(u̅A)=0.

Second, we show the necessity part. Suppose that for any AA, β1(u̅A)=βS(u̅A)=0. Then, for any AA,

V(A)=maxpAα1u(p1)+iSαiu(pi).

Take arbitrary two menus A,BA with A¯B. Then, V(A)V(B), which says that for any qB, there exists pA such that {p}¯{q}. This implies that V(A)=V(AB). Thus, we obtain A¯BAAB.

D.9 Proof of Corollary 11

Suppose that ¯ is represented by a RDIC with a four-tuple (u,α,β,γ).

We show the sufficiency part. Suppose that ¯ satisfies Axioms 22 (Strategic Rationality), 23 (Monotonicity w.r.t. Equal Allocations), 24 (Quasi-Comonotonic Independence), and 25 (Inequity Aversion). By Axiom 22, β1(u̅A)=βS(u̅A)=0, for each AA (Corollary 10).

In the same way as the step 3 in Theorem 1, we consider the utility space of allocations by using the concept of certainty equivalent of lotteries. For any pΔI, define

u(p):={((c(p1,v),,c(pn,v))RI|pΔI}

where u(pi)=Epi[v]andc(pi,v)=v1Epi[v]forsomev:XR.

We consider a binary relation on RI, and then verify that on RI satisfies the axioms in Corollary 11. This is straightforward, so we omit it.

Let U:RIR that represents . We can show that (i) U is mixture-linear for quasi-comonotonic allocations, (ii) U is homothetic and that (iii) U is unique up to positive affine transformation.

By (iii), we can normalize U(1,,1)=1, and U(0,,0)=0. Let αenvyi:=U(1,(0)i). By Axiom 25 (Inequity Aversion), take p,pΔ with cp>cp. Define

(c(pi,v)c(p1,v))U(1,(0)iS):=αenvyimax{c(pi,v)c(p1,v),0}.

By (ii), we can prove that this is well-defined. In the same way, the guilt part is shown. Take p,pΔ with cp>cp. Define

(c(p1,v)c(pi,v))u(1,(0)iS):=αguiltimax{c(p1,v)c(pi,v),0}.

Hence, there exists a tuple (v,α1,(αenvyi,αguilti)iS) where v:XR is a non-constant function, α1>0, and for each iS, αenvyi0, αguilti0, such that ¯ is represented by

V(A)=maxpA[α1u(p1,v)(iSαenvyimax{u(pi,v)c(p1,v),0}+αguiltimax{c(p1,v)c(pi,v),0})].

E Examples

E.1 Example 9

Fix a player iI={1,2}. Let j be the opponent of the player i. The following table (Table 5) is the payoffs of player i. Let the reference point of the game G denoted by u̅G.

Table 5:

A prisoner’s dilemma game with image-conscious preferences.

Player i\Player jCooperation
Cooperation(1+βi(u¯G))+(1βj(u¯G))
Defection(1+βi(u¯G))(1+g)+(1βj(u¯G))(l)

Suppose that the opponent j takes C (Cooperation) (Table 5). Then,

(1+βi(u̅G))+(1βj(u̅G))(1+βi(u̅G))(1+g)+(1βj(u̅G))(l)(1βj(u̅G))(1+l)(1+βi(u̅G))g(1βj(u̅G))(1+βi(u̅G))g1+l.

Hence,

βj(u̅G)1(1+βi(u̅G))g1+l.

E.2 Example 10

Fix a player iI={1,2}. Let j be the opponent of the player i. The following table (Table 6) is the payoffs of player i. Let the reference point of the game G denoted by u̅G.

Table 6:

A prisoner’s dilemma game with image-conscious preferences.

Player i\Player jDefection
Cooperation(1+βi(u¯G))(l)+(1βj(u¯G))(1+g)
Defection0

Suppose that the opponent j takes D. Let the reference point of the game G by u̅G. Then,

(1+βi(u̅G))(l)+(1βj(u̅G))(1+g)0(1βj(u̅G))(1+g)(1+βi(u̅G))l(1βj(u̅G))(1+βi(u̅G))l1+g.

Hence, βj(u̅G)1(1+βi(u̅G))l1+g leads to take D.

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Received: 2018-05-21
Accepted: 2020-03-27
Published Online: 2020-08-20

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