Social Image Concern and Reference Point Formation

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.

≿1and≿Sare 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,q1∈Δ, p1≿1q1 if {(p1, rS)}≻¯{(q1, rS)} for some rS∈ΔS. For all pS, qS∈ΔS, pS≿SqS if {(r1, pS)}≻¯{(r1, qS)} for some r1∈Δ.

We show that ≿1 is independent of any rS∈ΔS.^[36] We need to show that p1≿1q1 if (p1, rS)≿1(q1, rS) for anyrS∈ΔS. Notice that for any pj, qj∈Δj and λ∈[0, 1], λpj+(1−λ)qj∈Δj(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 rS∈ΔS such that {(p1, rS)}≻¯{(q1, rS)}, then for any rS∈ΔS, {(p1, rS)}≻¯{(q1, rS)}.

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

≿1and≿Ssatisfy 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],

Fix p1, q1, r1∈Δ and λ∈[0, 1]. Then, for any pS, qS∈ΔS,

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: Δ→R 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:ΔS→R with (αi)i∈S and a real number ε such that u^S(pS)=∑i∈Sαiu(pi)+ε and ∑i∈Sαi=1. Define uS:=∑i∈Sαiu(pi)=u^S(pS)−ε. Hence, uS represents ≿S.

α1is positive. Finally, we show α1>0. By Axiom 1 (Standard Preferences), define, for any p∈ΔI, V({p}):=∑i∈Iαiu(pi) that represents ≻¯ over As, i.e., the set of all singletons. Suppose α1≤0. Take p,q∈ΔI such that p1≻1q1 and pS∼SqS. By Axiom 3 (Pareto), {p}≻{q}. If α1≤0, 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 q∈B and p∈A, pj≿jqj.

Transitivity. We show that, for each j∈{1,S}, ≻¯j is transitive. Take A,B,C∈A 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 A∈A, there exists a singleton equivalent pA∈ΔI such that {pA}∼A. Consider arbitrary two menus A, B∈A. Take singleton equivalent, respectively, denoted by pA, pB∈ΔI. 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, A∼j{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, q∈ΔI with {p}≻¯{q}. Suppose that {p}≻¯j{q}. By definition, pj≿jqj. 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 A∈A, let co(A) be the convex hull of A.

Axiom 20.

(Indifference to Randomization): For any A∈A, A∼co(A).

Take an arbitrary menu A∈A. 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 A∈A, A∼co(A).

We obtain the following result. For simplicity, let us introduce some notation: for each p1∈Δ and pS∈ΔI, u1:=α1u(p1), uS:=∑i∈Sαiu(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:Δ→R and Wj:u(Δ)×u(Δ)→R such that ≻¯j is represented by u̅j:A→R, defined by

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:Δ→R. 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=1NAn∼j∪n=1N+1An.

To show that ≻¯j satisfies Finiteness, take an arbitrary menu A∈A. A is compact, so there exist ≻j-best and ≻j-worst allocations denoted by p̅j∈Δj and p̅j∈Δj respectively. Take a menu B∈A such that for any q∈B, p̅j≿jqj≿jp̅j. Then, A∼jA∪B. Take A1, A2, A3∈A. Let p̅∈argmaxp∈A1∪A2∪A3u(pj), and p̅∈argminp∈A1∪A2∪A3u(pj). By construction, assume that p̅∈A1 and p̅∈A1. Suppose that A1∪A2≻¯jA1∪A3. Then, by definition, A1∪A2∼A1∪A2∪A3. 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:Δ→R and γj∈[0, 1] such that ≻¯j is represented by u̅j:A→R, defined by

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

By putting u1:=α1u(p1) and uS: =∑i∈Sαiu(pi) for each p1∈Δ and pS∈ΔS, we obtain the desired utility representation on reference-point formation; for any A,B∈A, A ≻¯jB⇔u̅j(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∗:A∗→R represents ≻¯∗.

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

where u1(p1)=α1u(p1) and uS(pS)=∑i∈Sαiu(pi). Let us A∗ be denoted by {u(A)|A∈A}. Since each choice set A∈A is compact, u(A) is also compact, by the continuity of u: Δ→R. 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∗,B∗∈A∗,

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 A∼B. Then, for any p∈A there exists q∈B such that p1∼1q1 and pS∼SqS. Hence, A∼1B and A∼SB hold. By Axiom 7 (Weak Dominance), we have A∼A∪B. In the same way, we have B∼A∪B. By Axiom 1 (Standard Preferences), in particular, the transitivity of ≻¯, we obtain A∼B.

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∗≻¯j∗B∗ if for any v∈B∗ and u∈A∗ such that uj≥vj.

Axiom* 1.

(Pareto_^*): For any u, v∈[0, 1]2, if u≥v, then {u}≻¯{v}.

Axiom* 2.

(Weak Dominance_^*): For any A∗, B∗∈A∗, if A∗∼1∗B∗, A∗∼S∗B∗, and A∗≻¯∗B∗, then A∗∼∗A∗∪B∗.

Axiom* 3.

(Intrinsic Set-Betweenness_^*): For each j∈{1, S}, if A∗∩B∗=∅ and A∗≻¯j∗B∗, then A∗≻¯j∗A∗∪B∗≻¯j∗B.

Axiom* 4.

(Singleton Independence_^*): For any A∗, B∗∈A∗, w∈[0, 1]2, and λ∈[0, 1]

Axiom* 5.

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

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+(1−γ1)minu1 where γ1∈[0, 1], and u̅S=γSmaxuS+(1−γS)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×R→R such that ≻¯∗ on A∗ is represented by V∗:A∗→R defined by (see Figure 4)

We have defined μ:[0, 1]2×R→R. 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).

Axiom* 6.

(Finiteness_^*): For any sequence {An∗} of A∗, there exists a positive integer N such that ∪n=1NAn∗∼∪n=1N+1An∗.

Lemma 3.

≻¯∗satisfies Finiteness (Axiom 6).

Proof.

We show the following: Take arbitrary two menus A∗, B∗∈A satisfying the following: for any A∗, B∗∈A and u∈A∗, u1≥v1 and uS≥vS. We obtain A∗≻¯∗B∗. This implies that, for any v∈A∗∪B∗, there exists u∈A∗ such that u1=v1 and uS=vS. Then, by Axiom 2 (Weak Dominance^*), we have A∗∼A∗∪B∗.

Take arbitrary menus A1∗, A2∗, A3∗, A4∗∈A∗. Let us denote the following:

• u^∗∈argmaxu∈A1∗∪A2∗∪A3∗∪A4∗u1;

• u^∗∈argminu∈A1∗∪A2∗∪A3∗∪A4∗u1 ;

• v^∗∈argmaxv∈A1∗∪A2∗∪A3∗∪A4∗vS; and

• v^∗∈argminv∈A1∗∪A2∗∪A3∗∪A4∗vS.

Without loss of generality, assume u^∗, u^∗∈A1∗ and v^∗, v^∗∈A2∗. Suppose A1∗∪A2∗∪A3∗≻¯∗A1∗∪A2∗∪A4∗. Then, by Axiom 2 (Weak Dominance^*), we have A1∗∪A2∗∪A3∗∼∗A1∗∪A2∗∪A3∗∪A4∗. 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)

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

and

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+ε≥u1∗(λ). In the same way, for all λ∈[0, 1], uS∗(λ)=12+ε≥uS∗(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.

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′|+|1−λ′|}\{λ′}};

• – A∗∪{u∗(λ′)}.

Notice that the two menus are compact. Since u∗(|λ′||λ′|+|1−λ′|)∈A∗, and u1(|λ′||λ′|+|1−λ′|)≥u1(λ′) and uS(|λ′||λ′|+|1−λ′|)≥uS(λ′), by Axiom 2 (Weak Dominance∗), A∗∼∗A∗∪{u(λ′)}.

If μ(u̅A∗, λ′)<0 and μ(u̅A∗∪{u(λ′)},λ′)<0, then V∗(A∗)>V∗(A∗∪{u(λ′)}), which represents A∗≻∗A∗∪{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∗), A∗∼∗A∗∪{u(λ′)}. However, by the assumption of μ(⋅,λ′)<0, we have V∗(A∗)>V∗(A∗∪{u(λ′)}), which represents A∗≻∗A∗∪{u(λ′)}. This is a contradiction.

Proof of (iii).

We show (iii). To see this, fix a menu A∗∈A∗, 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 B∗∈A∗ such that for each j∈{1, S}, A∗∼j∗B∗, and A∗≻∗B∗. 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∗+(1−σ)B∗)=σV∗(A∗)+(1−σ)V∗(B∗).

RDIC Utility Representation

By Lemma 4, we find three states, i.e., u̅1,λu̅1+(1−λ)u̅S, and u̅S. We normalize μ in the following way. First, let (2γ1−1)μ(u̅A, λ∗)λ∗+μ(u̅A, 1)=1. Define, for each menu A∗∈A∗, β1(u̅A): =−μ(u̅A, 1). Second, in the same way, normalize μ as follows: (2γS−1)μ(u̅A, λ∗)(1−λ∗)+μ(u̅A, 0)=1. Let βS(u̅A): =μ(u̅A, 0). Then, we have the utility representation in the following way (Figure 6): for any A∗∈A∗,

By the axiom of Pareto∗, we obtain β1(u̅A∗)∈(−1, ∞) for any A∗∈A∗. In the same way, we obtain βS(u̅A∗)∈(−∞, 1) for any A∗∈A∗.

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

By letting u1=α1u(p1) and uS=∑i∈Sαiu(pi), we have the desired representation; that is, for any A∈A,

We complete the proof of the sufficiency part.