Discussion Papers In Economics
And Business
Inequality aversion with general payoff function
Daijiro Kawanaka

Discussion Paper 21-09

June 2021

Graduate School of Economics
Osaka University, Toyonaka, Osaka 560-0043, JAPANInequality aversion with general payoﬀ function ∗Daijiro Kawanaka †Abstract
In the existing axiomatic models of inequity aversion, players have linear payoﬀ functions, sothey can predict that a dictator chooses only a completely selfish or completely fair oﬀer in
dictator games. However, experimental literature documents that a significantly amount ofdictators oﬀers 20-30% of the total pie to the passive opponent. This note, in contrast, axiomatizes inequity averse representation with general payoﬀ function, so that we can explainsuch interior choices.
JEL codes: C72, D71, D81
Keyword: Inequality aversion under risk, Maxmin expected utility, Social preferences∗

The author would like to thank Yosuke Yasuda, his supervisor in Osaka University. This
research was supported by Grant-in-Aid for JSPS Fellows from the Ministry of Education,
Culture, Sports, Science and Technology of Japan.
†
Osaka University, Graduate School of Economics. Email: kawanaka.daijiro@gmail.com

1

Introduction

Experimental literature documents that a significantly amount of subjects oﬀers 20-30% of
the total pie to passive opponents in dictator games (Camerer, 2002). This can be explainedby inequilty aversion (Fehr and Schmidt, 1999) with concave payoﬀ functions. Nevertheless,existing axiomatic models of inequity aversion limit payoﬀ functions to be linear and cannot
describe the established experimental result.
In general, an inequity averse decision maker (denoted by 1, henceforth) chooses an allocation x ∈ Rn over n individuals, so that she maximizes
UF S (x) = u(x1 ) −n∑αi max{u(xi ) − u(x1 ), 0} −

i=2

n
∑

βi max{u(x1 ) − u(xi ), 0}

(1)

i=2

where u : R → R denotes each individual’s payoﬀ function, and αi , βi ∈ [0, 1] denote parameters of her envy and guilt for each i, respectively. Observe that the decision maker makes aninterior oﬀer x∗2 ∈ (0, 1/2) in a dictator game where n = 2 and x1 +x2 = 1 only if u is concave.
Rohde (2010) first axiomatizes representation (1) with linear payoﬀ functions (i.e., u(x) = x),
so her model can describe only the extreme cases x∗2 = 0 or x∗2 = 12 . Motivated by experimental studies of probablistic dictaor games (e.g., Brock et al. 2013), Saito (2013) considersa preference over risky allocations and axiomatizes a function combining V (p) = UF S (Ep [x])and V (p) = Ep [UF S (x)]. His model successfully distinguishes equality of opportunity andequality of outcome. Also in his axiomatic model, however, payoﬀ functions are limited to belinear, and thus, his model can describe only the extremely selfish or extremely fair oﬀer inprobabilistic dictator games. Moreover, such a model with linear payoﬀ functions is inconsist
with a fact that the decision makers are often risk averse.
This note considers a preference over risky allocations as assumed by Saito (2013), andaxiomatizes representation (1) with a general payoﬀ function u. We call it an Expected
Equality Representation (EER). Our result permits a decision maker to oﬀer an interior
donation between 0% and 50% in dictator games as experimental studies suggest. Moreover,
it is consistent with the fact that the decision maker is risk averse.
Theoretically, in contrast to Rohde and Saito, who employ strong versions of Comonotonic
Independence axiom (Schmeidler, 1989), we employ Certainty-Independent axiom of Gilboa
and Schmeidler (1989, GS). Because the Certainty-Independent is mathematically simpler as
GS claim, it permits us to analyze the general functional form. To characterize the EER,
we apply Anscombe and Aumann’s (1963, AA) framework of which domain is a set of acts
(i.e., functions from objective states to lotteries). Because a domain of our analysis is a setof risky allocations, a set of players corresponds to that of states in AA. GS weaken AA’s
independence axiom and obtain kinked indiﬀerence curves. Because indiﬀerence curves of
EER are also kinked in a similar way, we can obtain EER by applying GS’s method.
The rest of the paper is organized as follows. Section 2 sets up the model. Section 3
analyzes it. Section 4 concludes.

1

2

Model

Let I = {1, · · · , n} be a set of individuals and Z be a set of prizes. Each x = (x1 , · · · , xn ) ∈Z n is an allocation of payoﬀs among the individuals. A probability distribution pi on Z
with finite supports is called a lottery. The set of all lotteries is denoted by ∆Z, and write
D := (∆Z)n . The primitive of our model is a binary relation ≿ over D that describes apreference of the decision maker (denoted by 1 ∈ I). As usual, we reduce ≻ and ∼ from≿, define a probability mixture ⊕ over D, and say that a function f : ∆Z → R is linear iff (λx ⊕ (1 − λ)y) = λf (x) + (1 − λ)f (y) for each x, y ∈ D and λ ∈ [0, 1].
Definition 1. ≿ has an Expected Equality Representation∑(EER) if there exist a functionu : Z → R and (αi , βi ) ∈ R2 for each i ∈ I\{1} such that ni=2 βi ≤ 1 and ≿ is represented
by V (p) := UF S (Ep (x)).
We impose on ≿ four axioms. The first one is rationality:
Axiom 1. ≿ is weak order, continuious, and monotone in a sense that∀p, q ∈ D; e(p1 ) ≿ e(q1 ) ⇔ (p1 , p−1 ) ≿ (q1 , p−1 ).

(2)Note that an inequity averse individual often prefers (0, 10) ≻ (1, 9) in ultimatum gameswith n = 2. This violates Pareto eﬃciency:∀p, q ∈ D; [∀i ∈ I; e(pi ) ≿ e(qi )] ⇒ p ≿ q.

(3)

Therefore we impose (2) weaker than (3).
Although AA impose von Neumann-Morgenstern independence axiom, an other-regarding
individual may prefer randomization such as 21 (1, 0) ⊕ 21 (2, 4) ≻ (1, 0) ∼ (2, 4). This violates
the independence axiom as Diamond (1967) suggests this example in social choice literature.
Hence we weaken it in GS’s fassion as the following Axioms 2 and 3. To do so, let e(p) ∈ D
denote an equal allocation in which each individual receives the same lottery p ∈ ∆Z. Eache(p) corresponds to a “constant act” in GS.
Axiom 2. ∀p, q ∈ D, ∀r ∈ ∆Z; p ∼ q ⇒ 21 p ⊕ 12 e(r) ∼ 21 q ⊕ 12 e(r).
Axiom 3. ∀p, q ∈ D; p ∼ q ⇒ 12 p ⊕ 12 q ≿ p.
Fehr-Schmidt model compares only between the decision maker and each individual. Axiom 4 implies that no relation between two others aﬀects her ranking over allocations. Hence,it always holds if n = 2.
Axiom 4. ∀p, q, r, s ∈ D with e(p1 ) ∼ e(q1 ), and ∀i ̸= 1, (p−i , ri ) ≿ (q−i , ri ) if (p−i , si ) ≿(q−i , si ).

3

Representation

Under the four axioms, our main result holds as follows:2

Theorem 1. ≿ satisfies Axioms 1-4 iﬀ it has a EER.
Proof. First, we claim that ≿ is represented by some C-independent function I : D → R in
a sense that
I(av + b1) = aI(v) + b.
(4)
By Axiom 1, there exists a continuous function v : {e(p) ∈ D | p ∈ ∆Z} → R such thatv(p) ≥ v(q) ⇔ e(p) ≿ e(q).

(5)

for each p, q ∈ ∆Z. Moreover, the v can be linear by Axiom 2. By Axioms 5 and (2), there
exists a function f : (v(∆Z))n → R such thatf (v(p1 ), · · · , v(pn )) ≥ f (v(q1 ), · · · , v(qn )) ⇔ p ≿ q(6)for each p, q ∈ D. Given the f , define a function I : Rn → R such that
I(λv(p1 ), · · · , λv(pn )) = λf (v(q1 ), · · · , v(qn ))for each λ ∈ R. As shown in (iv) of GS’s Lemma 3.3, our Axioms 2 implies C-independenceof I.
Second, we show that ≿ is represented byn∑

U (p) = mink∈Kki v(pi )

(7)

i=1

for some unique, nonempty, closed, convex set K ⊊ Rn . Without loss of generality, we
consider a set given by
U := {v ∈ Rn | f (v) ≥ 0},
which is a closed convex cone whose vartix is 0. Therefore, we can apply supporting hyperplane theorem: there exists k ∈ Rn \{0} such that for all u ∈ U ,k · 0 ≤ k · u.

(8)

Let K denote a set of all vectors k satisfying (8). Note that the above result of hyperplane
theorem implies that K ̸= ∅. For k ∈ K, on the one hand, we have k · u ≥ 0. On the other
hand, if mink∈K k · v < 0, then u ̸∈ U , that is, f (u) < 0. Therefore now we havef (u) = 0 ⇐⇒ min k · u = 0.
k∈K

(9)

Since we can take u′ ∈ U such that f (u′ ) = 0 and u = u′ + c1 for given u ∈ Rn satisfyingthat f (u) = c, we havef (u) = f (u′ ) + c= min k · u′ + c

(10)
(11)

= min k · u,

(12)

k∈K
k∈K

3

where (10) is due to (4), and (11) is derived by (9). Combining (5) and (12) yields (7).
Similarly
to GS, K is unique, nonempty, closed and convex. For each k ∈ K, we assume∑nk=
1 without loss of generality.
i=1 i
Third, we show that (7) is rewritten by EER. Let v, v′ ∈ Rn satisfy that(v1 − vi )(v1′ − vi′ ) > 0(13)

for each i ∈ I. Then, there exists k ∈ K minimizing k · v and k · v′ both by Axiom 4 andthe properties of K. Therefore,
)
(
n
n
∑
∑
∗
U (p) = 1 −ki v(p1 ) +ki∗ v(pi ),
(14)
i=2

where k∗ ∈ arg minki ∈K(p1 , pi ), and∑ni=1 ki v(pi ).

i=2

Next we show that each ki∗ in k∗ depends only on{ki∗ (p1 , pi ) =k i if v(p1 ) ≥ v(pi ),k i if v(p1 ) ≤ v(pi )(15)where k i ≤ ki ≤ k i for each k ∈ K and each i ∈ I\{1}. Let v, v′ ∈ Rn satisfy (13) and
vj = vj′ for each j ∈ I\{1, i}. For any k, k′ ∈ K, we have k · v > k′ · v ⇔ (1 − ki )v1 + ki vi >(1 − ki′ )v1 + ki′ vi ⇔ (ki′ − ki )(v1 − vi ) > 0 ⇔ (ki′ − ki )(v1′ − vi′ ) > 0 ⇒ k · v′ > k′ · v′ , where thethird arrow holds by (13). Therefore, ki∗ (p1 , pi ) depends only on p1 − pi . Moreover,() by Axiom(15). Asigning k i , k i = (−αi , βi )4 and properties of K, it takes either k i or k i , so we have∑■to (14) and (15) yields EER. Condition (2) implies that ni=2 βi ≤ 1.
The proof mainly follows GS, whose maxmin expected utility (MEU) is characterized by
Axioms 1-3 and 5:
Axiom 5. ∀x, y ∈ D; [∀i ∈ I; e(xi ) ≥ e(yi )] ⇒ x ≿ y.
Axiom 5 is interpreted as Pareto eﬃciency in our setting. We have weakened Axiom
5 to (2) and impose Axiom 4 instead of it because an inequality averse individual reveals(0, 0) ≻ (0, 1) ineﬃciently in an ultimatum game, for example. When n = 2, Figures 1 and 2
depict indiﬀerence curves of MEU and EER, respectively. They diﬀer in signs of their slopes
above 45-degree line, and the diﬀerence is caused by Axioms 4 and 6.
Ep2 [u(x2 )]O

Ep2 [u(x2 )]45◦O

Ep1 [u(x1 )]Figure 1. Indiﬀerence curves of MEU

45◦

Ep1 [u(x1 )]Figure 2. Indiﬀerence curves of EER
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3.1

Parameter characterization

The representation (1) is said to exhibit inequity aversion if αi , βi > 0. To characterize theparameters, here consider the following two axioms for each i ∈ I\{1}.
Axiom 6. ∃x, y, z ∈ ∆Z; e(y) ≻ (xi , e−i (y)), e(y) ≻ (zi , e−i (y)) and e(y) ∼ 21 e(x) ⊕ 12 e(z).
Axiom 7. ∃x, y, z ∈ ∆Z; e(y) ≻ (xi , e−i (y)) ≻ (zi , e−i (y)) and e(y) ∼ 12 e(x) ⊕ 12 e(z).
As a colollary of Theorem 1, we can characterize it as follows:
Proposition 2. Let ≿ satisfy Axioms 1-4. Then, (αi , βi ) ∈ R2++ in EER iﬀ Axiom 6 holds.
Furthermore, αi > βi iﬀ Axiom 7 holds.
We can also characterize utilitarian (i.e., αi < 0 < βi ) and competitive (i.e., βi < 0 < αi ) in
similar ways.

4

Conclusion

We axiomatize Fehr-Schmidt type utility representation under risk. Contrast to existing
papers, our axiomatic model permits the decision maker to have a concave utility or berisk averse. Our analysis suggests that inequity aversion and uncertainty aversion have asimilar behavioral foundation (i.e., the same way of violation of independence axiom). Thisis consistent with Kameda et al (2016) finding that individual attitudes toward distributional
and uncertain tasks draw on common cognitiveneural processes.

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