Fixation of inequality and emergence of the equal split norm:
Approach from behavioral bargaining theory
Yoshio Kamijo∗
2023-06-22
Abstract
Negotiation is at the heart of communication, social exchange, and economic transactions.
Using the bargaining model as the unit of analysis, this study aims to deepen our understanding of negotiation and economic behavior based on the behavioral bargaining theory(BBT) developed by Kamijo and Yokote (2022). We introduce a key concept to analyze abargaining situation: the stability of entitlements (people’s expectations of distribution orsense of ownership) for a bargaining pie. When a pair of entitlements initially formed isstable, negotiations are expected to end immediately; when unstable, negotiations are more
likely to end in delay or failure. We show the boundary condition for stable entitlements andfind that some unequal distribution between two players can be stable even for a symmetricbargaining problem. By seeking stable entitlements for all members of society, it is possible
to define a distribution norm mathematically. We show that the distribution norm that arises
in a symmetric situation is the golden rule of distribution: 50–50 split of the pie. Finally,
by examining the dynamic process of the formation of entitlements, we clarify the sufficientconditions under which the equal split norm emerges.
Keywords: Behavioral bargaining theory, Nash solution, Reference dependent utility, Stableentitlement, Inequality cap theorem, Distribution norm, Equal split norm∗

Waseda University, yoshio.kamijo@gmail.com

11 Introduction
Negotiation is at the heart of communication, social exchange, and economic transactions. It
occurs between families, friends, students and faculty, workers and employers, and businesses. A
standard tool to analyze the bargaining problem is Nash’s bargaining model, and it becomes thebasis for the analysis of negotiations, both theoretically and in applications.
Nash’s model are widely recognized for its generality and wealth of theoretical support.1 However,there is skepticism about it from both practical and experimental perspectives (Raiffa, 1982). It
is well known that the observed bargaining agreements in a laboratory are biased towards theequal split of the bargaining pie (Anbarci and Feltovich, 2013, 2018; Birkeland and Tungodden,
2014; Hoffman and Spitzer, 1982; Nydegger and Owen, 1974; Roth, 1995), but it is difficult to
explain such a phenomenon from the Nash solution. It has been reported that the influenceof disagreement point on the results of negotiations is smaller than one expected in the Nashsolution (Anbarci and Feltovich, 2013; Kamijo, 2023a). Moreover, many studies (Baranski,
2016, 2019; Cappelen et al., 2007; Luhan et al., 2019; Takeuchi et al., 2022) have shown thatexperimental results differ significantly between cases in which bargaining pies are produced
from the earnings of experiment participants and those in which they are given as gifts fromheaven. It is challenging to understand these results from the framework of standard bargaining
problems.
Kamijo and Yokote (2022) built a new bargaining theory, called the behavioral bargaining theory(BBT), by incorporating the reference-dependent utility a la Tversky and Kahneman (1979) into
Nash’s theory, and succeeded in explaining the deviation from the traditional model by considering
the variety of different feeling of entitlement caused by the experimental manipulations. For
example, the BBT can describe the bargaining outcome affected by an equal split norm, which
differs from the one caused by the preference for equality. In addition, the new theory can
explain how the bargaining results are different between the cases wherein the bargaining pie isgiven by the experimenter and produced from the participants’ efforts. They also examined theempirical validity of the theory by borrowing the experimental data from Takeuchi et al. (2022)
and confirmed the usefulness of BBT in explaining and interpreting the data.
This study aims to deepen our understanding of negotiation and economic behavior by using theframework of the BBT. While the BBT provides great applicability by incorporating entitlements
theoretically, essential questions remain as to how people form entitlements and what entitlements
are possible. We extend the BBT to shed light on these fundamental questions and address
situations in which players form entitlements endogenously.
To this end, we consider a situation in which bargainers participate in bargaining repeatedly
rather than only once. Suppose that, after the first round, a bargainer realizes that the amountof money she feels entitled to is largely different from the one the opponent proposes. Then, inthe next round, she would revise her entitlement closer to the proposed amount in pursuit of anagreement. If this repetition occurs sufficiently many times, then the process would result in a
steady state, or a fixed-point mathematically, where the subjective entitlements coincide with theobjective bargaining outcome. The notion of stability formalizes this steady state: we say thatan entitlement 𝐸𝑖 for two players 𝑖 = 1, 2 is stable if the Nash solution applied to the instancewith total bargaining pie 𝑀 = 𝐸1 + 𝐸2 returns back to the original entitlement (𝐸1 , 𝐸2 ).
A notion of stable entitlements is also helpful in understanding when negotiations begin andhow they proceed. If players are rational enough to expect the consequence of their bargaining,
Axiomatizations of the Nash solution are found in Nash (1950), Roth (1979) and Thomson (1994). Noncooperative games whose equilibrium outcome coincides with the Nash solution are discussed in Nash (1953),
Binmore et al. (1986), Anbarci and Feltovich (2013) and Kamijo (2023b). In addition to these theoretical supports
to the Nash solution, its several interpretations are provided by Young (1993), Rubinstein et al. (1992) andBastianello and LiCalzi (2019).
12

a negotiation might not begin when a pair of entitlements initially formed is stable because
they cannot change the outcome from the initial proposal through the bargaining. Alternatively,
even if the bargaining starts, it is expected to end immediately. In contrast, when the initialentitlements are unstable, the negotiation is more likely to end in delay or failure if there is atime limit.
We present three theorems concerning stable entitlement and its applications. First, we characterize stable outcomes in terms of the level of inequality. Our inequality cap theorem states
the following: (i) entitlements are stable if and only if the level of inequality at the bargaining
outcome is within a particular region, and (ii) the region is characterized by the parameter ofloss aversion. In light of this theorem, an unequal distribution becomes stable when a loss-averse
player in an advantageous position sticks to the distribution. It reveals an interesting relationshipbetween equality and loss aversion; while strong loss aversion seems to promotes equality, it issuggested that inequality, once established between two parties, is more likely to sustain by theloss aversion of the advantageous player. Furthermore, this theorem is useful to understand why
people often refrain from negotiation even though they seem to have a strategic advantage andwhy bargainers in the laboratory often fail to reach an agreement where they have contrasting
self-serving fairness (Roth and Murnighan, 1982).
Second, we define a distribution norm by using the concepts of stable entitlements. A distribution
norm in a society is one that is acceptable to all potentially existing members of the society.
Here, we will consider a variety of people as entities with different loss aversion parameters. A
distribution norm is one that, once accepted, will remain robust against bargaining between two
players from such a diverse group of people. We show that for a symmetric bargaining situationin the sense that both disagreement outcomes are the same, the distribution norm should be anequal split of the bargaining pie. Therefore, only the equal split norm satisfies the requirementsof a distribution norm in society.
Third, we investigate how the equal split norm emerges among players. We consider a society
consisting of a fixed number of players and formalize the aforementioned entitlement-revision
process: two members are matched randomly with some probability, conduct bargaining, andrevise their entitlements. The question then arises: when this matching process occurs sufficiently
many times, do the members of the society concur on the equal division of the total bargaining
pie? To put it differently, does the equal split norm emerge among the society members? Weprovide two sufficient conditions for this claim to hold: (i) the bargaining is conducted for a pie
that falls short of their requirements, or (ii) the society displays a certain form of homogeneity.
These findings are consistent with observations in real-life allocation problems.
The paper is organized as follows. In the next section, we briefly explain the setting of the BBT.
In Section 3, we introduce a notion of the stability of entitlements and show the inequality cap
theorem, one of our main results. We discuss a distribution norm of society in Section 4, and we
consider how an equal split norm arises in Section 5. Section 6 concludes the paper.

2 Model
2.1 Simple two-player bargaining game and Nash solution
A bargaining problem consists of players, a potential gain of agreement, and a disagreement
outcome. Two players 1 and 2 bargain over a fixed amount 𝑀 of some divisible good in pursuitof an agreed-upon outcome (𝑥1 , 𝑥2 ) with 𝑥1 + 𝑥2 = 𝑀. If the negotiation breaks down, player
𝑖 gets 𝑣𝑖 ≧ 0, of the divisible good for each 𝑖 = 1, 2. A disagreement outcome is denoted by
𝑣 = (𝑣1 , 𝑣2 ). As the context makes clear, we will denote the bargaining problem by (𝑀 , 𝑣)
without indicating the players.
One of the most prominent solutions to the bargaining problem is the Nash solution. The Nash3solution selects the utility pair that maximizes the product of the players’ utility differences
between the agreed and disagreeable outcomes (this product is called the Nash product).
Let 𝑢𝑖 (𝑖 = 1, 2) denote 𝑖’s utility function. Formally, the Nash solution chooses an allocation(𝑥∗1 , 𝑥∗2 ) that is a solution to the following maximization problem:max (𝑢1 (𝑥1 ) − 𝑢1 (𝑣1 )) × (𝑢2 (𝑥2 ) − 𝑢2 (𝑣2 ))𝑠.𝑡. 𝑥1 + 𝑥2 = 𝑀 , 𝑥1 ≧ 𝑣1 ,𝑥2 ≧ 𝑣2 .

We refer to the pair (𝑥∗1 , 𝑥∗2 ) as the Nash allocation and write𝑥∗ (𝑀 , 𝑣) = (𝑥∗1 (𝑀 , 𝑣), 𝑥∗2 (𝑀 , 𝑣)).
2.2 Bargaining model with a reference-dependent utility
As in Kamijo and Yokote (2022), we consider a bargaining between two players having entitlements
𝐸1 ≧ 0 and 𝐸2 ≧ 0. We assume that a player has a reference-dependent utility wherein theentitlement has a role of the reference. We model the utility function as a monotonicallyincreasing function considering asymmetry before and after the reference point. Let Ψ be afunction on [0, ∞) with Ψ(0) = 0, Ψ′ (𝑥) > 0, and Ψ″ (𝑥) < 0. Then, the utility function isdefined as the result of its simple transformation to model a loss aversion. Given a loss aversion
parameter 𝜆𝑖 ≧ 1, for any 𝑥𝑖 ≧ 0, the utility function of player 𝑖 is defined by𝑢𝑖 (𝑥𝑖 ; 𝐸𝑖 ) = {Ψ(𝑥𝑖 − 𝐸𝑖 )−𝜆𝑖 Ψ(𝐸𝑖 − 𝑥𝑖 )if 𝑥𝑖 ≧ 𝐸𝑖 ,if 𝑥𝑖 < 𝐸𝑖 .

(1)

If needed, it can be possible to add some positive constant in order to assure that 𝑢𝑖 (𝑥𝑖 ; 𝐸𝑖 ) ≧ 0.
In the behavioral bargaining theory (BBT) by Kamijo and Yokote (2022), the Nash solution is
calculated using the reference-dependent utility defined in (1). An allocation calculated in thisway depends not only the bargaining problem (𝑀 , 𝑣) but also the pair of entitlements (𝐸1 , 𝐸2 ).2
As a result, the Nash allocation becomes a function that associates (𝑀 , 𝑣, (𝐸1 , 𝐸2 )) with theefficient agreement, and is denoted by𝑥∗ (𝑀 , 𝑣, (𝐸1 , 𝐸2 )) = (𝑥∗1 (𝑀 , 𝑣, (𝐸1 , 𝐸2 )), 𝑥∗2 (𝑀 , 𝑣, (𝐸1 , 𝐸2 ))).
Kamijo and Yokote (2022) find that various experimental results that deviate from predictions
from conventional models can be explained by considering (𝐸1 , 𝐸2 ) that vary due to theexperimental manipulations. For example, setting 𝐸1 = 𝐸2 = 𝑀 /2 allows the BBT to describe
bargaining outcomes affected by an equal split norm. It can also explain how bargaining outcomesdiffer when the experimenter gives the bargaining pie or when the participants’ efforts generate
it. The former can be captured by setting 𝐸1 = 𝐸2 = 0, and the latter can be modeled using an
entitlement function that relates (𝑀 , 𝑣) and (𝐸1 , 𝐸2 ). They consider four types of entitlement
functions and test which type can better explain the data of Takeuchi et al. (2022). They
conclude that the type of entitlement function selected is consistent with Takeuchi et al. (2022)
experimental manipulations and participants’ responses in the post-experimental questionnaire.

3 Stability of entitlements and fixation of inequality
2
Kamijo and Yokote (2022) considered the following types of entitlement function 𝛾𝑖 for each 𝑖 = 1, 2: Equal
Split Norm is 𝛾𝑖 (𝑀, 𝑣) = 𝑀/2; Manna from Heaven is 𝛾𝑖 (𝑀, 𝑣) = 0; Disagreement is 𝛾𝑖 (𝑀, 𝑣) = 𝑣𝑖 ; Proportion𝑖
is 𝛾𝑖 (𝑀, 𝑣) = 𝑣 𝑣+𝑣
𝑀.
1

2

4

3.1 Stable entitlements
The concept of entitlement examined in Kamijo and Yokote (2022) and other literature is arelatively objective one with some basis in evidence (e.g., input earned in the production stage,
and the history-dependent status quo). However, the entitlement is essentially subjective one
and the feeling of that is very fragile and even varies in the process of the bargaining. Some study
attempts to analyze such a situation in a non-cooperative approach under a certain bargaining
protocol (Compte and Jehiel, 2003). In contrast, Shalev (2002) focus on the property that thebargaining agreement should satisfy under the situation that the reference point is endogeneously
formed, without specifying the bargaining protocol. Here, we follow the approach of Shalev(2002).
We introduce the notion of stable entitlements, which arise after the endogenous formation ofentitlements. Stable entitlements are those in which the subjective sense of entitlement of thetwo parties through negotiation is consistent with the outcome of the agreement. Therefore, nomatter the negotiation process, stable entitlements can be regarded as an agreement that bothparties will not withdraw later.
Formally, the stability of entitlements is defined as follows.
Definition 1. A pair of entitlements 𝐸 = (𝐸1 , 𝐸2 ) is stable to a simple bargaining problem(𝑀 , 𝑣) if and only if𝑥∗𝑖 (𝑀 , 𝑣, 𝐸) = 𝐸𝑖
holds for any 𝑖 = 1, 2.
Therefore, the pair of entitlements is stable when it is very the bargaining agreement under
the condition that they have such entitlements. Mathematically, this is the fixed point of thebargaining function 𝑥∗∗ (𝐸) ∶= 𝑥∗ (𝑀 , 𝑣, 𝐸). Since 𝑥∗∗ is a continuous function defined on thecompact set, the existence of a stable entitlement is guaranteed. By its definition, 𝐸1 + 𝐸2 = 𝑀
holds if (𝐸1 , 𝐸2 ) is stable to (𝑀 , 𝑣).
A notion of stability of entitlements is useful to understand when the bargaining begins or isdelayed. Suppose that negotiations needs transaction cost 𝜖 > 0 for both players, it starts when
one of them call for it, and the results of the bargaining are well described by the Nash bargaining
solution applied to the reference-dependent utility. Then, if an initial pair of entitlements (𝐸1 , 𝐸2 )is stable and this is an initial proposal chosen by some way (e.g., status quo, one proposed bythird party, etc.), they understand that the actual negotiation is useless and they immediately
agree on (𝐸1 , 𝐸2 ) without tough negotiation. In later subsections, we discuss which pair ofentitlements can be stable.
3.2 Inequality cap theorem
Some may think that if we admit the subjective assessment of an entitlement, we can explain
almost everything. In other words, does our theory of the subjective entitlement have falsifiability?
As the following theorem states, we can provide positive answer to this question by analyzing
the agreement that can be justifiable to the entitlements. In particular, an excessively unequal
outcome cannot be justified as a pair of stable entitlements. Thus, requiring the stability ofentitlements caps the possible inequality between the bargainers.
Theorem 1 (Inequality cap theorem). Suppose that 𝑣1 + 𝑣2 < 𝑀, 𝑣 < 𝐸 and Ψ′ (0) < ∞. Then,
𝐸 with 𝐸1 + 𝐸2 = 𝑀 is stable to (𝑀 , 𝑣) if and only if
1
𝑢 (𝐸 ; 𝐸 ) − 𝑢2 (𝑣2 ; 𝐸2 )≦ 2 2 2
≦ 𝜆2
𝜆1
𝑢1 (𝐸1 ; 𝐸1 ) − 𝑢1 (𝑣1 ; 𝐸1 )5holds true.
Proof. To simplify the notations, we write 𝑢𝑖 (𝑥𝑖 ) to denote 𝑢𝑖 (𝑥𝑖 ; 𝐸𝑖 ) for 𝑖 = 1, 2.
We first show ”only if” part. In this case, the Nash products at (𝐸1 , 𝐸2 ) is (𝑢1 (𝐸1 ) −𝑢1 (𝑣1 ))(𝑢(𝐸2 ) − 𝑢2 (𝑣2 )). Consider that the share of player 1 is slightly increased by 𝑥,
and then, the Nash product is 𝑓(𝑥) = (𝑢1 (𝐸1 + 𝑥) − 𝑢1 (𝑣1 ))(𝑢2 (𝐸2 − 𝑥) − 𝑢2 (𝑣2 )). Then,𝑓 ′ (𝑥) = 𝑢′1 (𝐸1 + 𝑥)(𝑢2 (𝐸2 − 𝑥) − 𝑢2 (𝑣2 )) − (𝑢1 (𝐸1 + 𝑥) − 𝑢1 (𝑣1 ))𝑢′2 (𝐸2 − 𝑥). Since 𝑓 is at least
locally maximized at (𝐸1 , 𝐸2 ),lim 𝑓 ′ (𝑥) ≦ 0 ⟺ Ψ′ (0)(𝑢2 (𝐸2 ) − 𝑢2 (𝑣2 )) − (𝑢1 (𝐸1 ) − 𝑢1 (𝑣1 ))𝜆2 Ψ′ (0) ≦ 0

𝑥𝑖 →+0which is equivalent to the right-hand inequality relationship of the condition specified in thistheorem since 𝐸 > 𝑣. We obtain the left-side inequality by considering the case that share ofplayer 1 is slightly decreased from the point 𝐸1 .
Next, we show the ”if” part. Consider again the situation that the share of player 1 increases by𝑥 from the point of 𝐸1 . Then, again, the Nash product is 𝑓(𝑥). By the definition of 𝑢𝑖 , for some
Δ𝑢 > 0, 𝑢1 (𝐸1 +𝑥) = 𝑢1 (𝐸1 )+Δ𝑢, 𝑢2 (𝐸2 −𝑥) = 𝑢2 (𝐸2 )−𝜆2 Δ𝑢, and 𝑢′2 (𝐸2 −𝑥) = 𝜆2 𝑢′1 (𝐸1 +𝑥).
Since 𝑓 ′ (𝑥) = 𝑢′1 (𝐸1 + 𝑥)(𝑢2 (𝐸2 − 𝑥) − 𝑢2 (𝑣2 )) − (𝑢1 (𝐸1 + 𝑥) − 𝑢1 (𝑣1 ))𝑢′2 (𝐸2 − 𝑥), we have𝑓 ′ (𝑥) = 𝑢′1 (𝐸1 + 𝑥)(𝑢2 (𝐸2 ) − 𝜆2 Δ𝑢 − 𝑢2 (𝑣2 )) − (𝑢1 (𝐸1 ) + Δ𝑢 − 𝑢1 (𝑣1 ))𝜆2 𝑢′1 (𝐸1 + 𝑥)
= 𝑢′1 (𝐸1 + 𝑥)(𝑢2 (𝐸2 ) − 𝑢2 (𝑣2 ) − 𝜆2 (𝑢1 (𝐸1 ) − 𝑢1 (𝑣1 )) − 2𝜆2 Δ𝑢) < 0.
The inequality holds by the right-side inequality of the condition of this theorem. Thus, theincrease of player 1’s share from 𝐸1 decreases the Nash product.
In contrast, let consider the situation that the share of player 1 decreases by 𝑥 from the point of
𝐸1 . The Nash product in this case is 𝑔(𝑥) = (𝑢1 (𝐸1 − 𝑥) − 𝑢1 (𝑣1 ))(𝑢2 (𝐸2 + 𝑥) − 𝑢2 (𝑣2 )). By
the definition of 𝑢𝑖 , for some Δ𝑢 > 0, 𝑢1 (𝐸1 − 𝑥) = 𝑢1 (𝐸1 ) − 𝜆1 Δ𝑢, 𝑢2 (𝐸2 + 𝑥) = 𝑢2 (𝐸2 ) + Δ𝑢,
and 𝑢′1 (𝐸1 − 𝑥) = 𝜆1 𝑢′2 (𝐸2 + 𝑥). Thus,𝑔′ (𝑥) = −𝑢′1 (𝐸1 − 𝑥)(𝑢2 (𝐸2 + 𝑥) − 𝑢2 (𝑣2 )) + (𝑢1 (𝐸1 − 𝑥) − 𝑢1 (𝑣1 ))𝑢′2 (𝐸2 + 𝑥)
= −𝜆1 𝑢′2 (𝐸2 + 𝑥)(𝑢2 (𝐸2 ) + Δ𝑢 − 𝑢2 (𝑣2 )) + (𝑢1 (𝐸1 ) − 𝜆1 Δ𝑢 − 𝑢1 (𝑣1 ))𝑢′2 (𝐸2 + 𝑥)
= 𝑢′2 (𝐸2 + 𝑥)(𝑢1 (𝐸1 ) − 𝑢1 (𝑣1 ) − 𝜆1 (𝑢2 (𝐸2 ) − 𝑢2 (𝑣1 )) − 2𝜆1 Δ𝑢) < 0,where the last inequality is because of the left-side inequality of the condition of this theorem.
Thus, the Nash product is maximized at 𝐸1 .
This theorem asserts that there is a reasonable range of the stable entitlements. Interestingly,
we can use the loss aversion parameter 𝜆 to describe that range. Clearly, when 𝑣1 = 𝑣2 = 0,
the equal division is only the stable entitlement in the absence of loss aversion (i.e., 𝜆𝑖 = 1
for 𝑖 = 1, 2). On the other hand, as the loss aversion parameter increases, more and more
entitlements become stable to (𝑀 , (0, 0)) (see Figure 1).
This theorem indicates that a certain unequal distribution among two people are sustained once
it happens to be established by some reasons even in a symmetric problem. This may explain
why some degree of unequal distribution of tasks and resources is ubiquitous even for small sizegroup without any strategic reason. In contrast, if a considerable size of inequality is kept, there
should exist some strategic reason behind it.

6

8

6

λ
x2 4

1.2
2.3

2

0
0

2

4

6

x1

8

Figure 1: Regions of Stable Entitlements. We assume Ψ(𝑥) = 𝑥0.8 .
The intuitive understanding of the theorem may seem easy at first glance, but it requires attention.
Simply put, loss aversion is related to the status quo bias, making it easier for inequality topersist once it is established. However, such an interpretation is incomplete because it does
not identify whether the status quo bias is at work for the strong or the weak. The theorem’s
statement arises because in cases where loss aversion is significant, the utility decrease of thestrong due to the correction of inequality is relatively larger than the utility increase of the weak.
Thus, the inequality prevails due to the status quo bias of the strong.
In the literature, Shalev (2002) discussed the similar property to stability (called the selfsupporting property) in Nash’s bargaining theory with endogenous formation of reference points,
and obtained the similar results (Theorem 3.1 in his paper) to our inequality cap theorem.
Moreover, he imposed an additional property, the robustness of the bargaining outcome to the
manipulations of the opponent’s reference point, to select the unique outcome from the region of
stable ones. While the mathematical setup of Shalev (2002) is slightly different from this study(e.g., in the definitions of a reference-dependent utility), the real difference lies in the positioning
and the goal. Shalev (2002) is positioning in the solution theory of bargaining. Thus, it sticks toobtaining the unique outcome and tries to provide axiomatic and non-cooperative foundations
for using an extension of the Nash solution. In contrast, this study comes from the behavioral
bargaining theory, which tries to understand the observed bargaining outcome in the laboratory
and provide useful insights on the economic activity related to the bargaining. Indeed, in thelater subsections we will show that the inequality cap theorem is quite useful for understanding
the diversity of bargaining outcomes. Moreover, in Sections 4 and 5, we will apply the conceptof stability beyond the level of two-person bargaining and discuss the distribution norm and theequal split norm, which are essential for understanding our economic phenomena.
3.3 Empirical support to the inequality cap theorem
In addition to providing some rationale to sustained inequality, the inequality cap theorem
helps us understand the limits of inequality that can be tolerated from the standpoint of thedisadvantaged party. To simplify the situation, assume that Ψ is close to linear within thebargaining range and therefore Ψ(𝑥) ≈ 𝑥. Then, the condition of the theorem is rearranged as7

follows,

𝐸𝑠≦ 𝜆𝑠 ,
𝐸𝑤

where 𝐸𝑠 , 𝐸𝑤 with 𝐸𝑠 ≧ 𝐸𝑤 are the shares of the strong and the weak, respectively, and 𝜆𝑠
is the loss aversion parameter of the strong. Thus, the theorem states that if the ratio of
inequality between the strong and the weak is greater than or equal to 𝜆𝑠 , then such inequality
is unacceptable from the weak perspective. In fact, in the extensive literature on experimental𝑥
ultimatum bargaining, offers of 20 ~ 30% of the total pie (i.e., inequality ratio 𝑥 proposer is aroundresponder
7/3 ~ 4) are often rejected by the responders (for review, see Güth and Kocher (2014)).
While an interesting consistency, this is not a good example of the application of our theorem
because the ultimatum bargaining game is a rule-based bargaining game. More direct evidence
is found from the experiment of Takeuchi et al. (2022). In their experiment, experimental
participants earn a disagreement payoff through their efforts in the first stage, and in thesecond stage, paired participants freely negotiate the distribution of the pie produced with theirdisagreement payoffs as input. The relation between a bargaining pie 𝑀 and their disagreement
payoffs (𝑣1 , 𝑣2 ) is manipulated, and in one treatment, termed the proportional surplus treatment,
it is explained as 𝑀 = 𝛽(𝑣1 + 𝑣2 ) where 𝛽 > 1 is changed across rounds according to certain
rules.
It is natural to think that experimental participants in the proportional surplus treatment will
form a proportional entitlement defined by
𝐸𝑖𝑝𝑟𝑜𝑝 = 𝛽𝑣𝑖 = 𝑀

𝑣𝑖
𝑣1 + 𝑣 2

for 𝑖 = 1, 2

at least before negotiations begin. Two pieces of evidence reinforce this conjecture. First,
after the experimental task, Takeuchi et al. (2022) ask participants to answer which way ofdivision is most preferred, the equal split division (𝑀 /2, 𝑀 /2), the equal surplus division1( 𝑀−𝑣21 −𝑣2 + 𝑣1 , 𝑀−𝑣21 −𝑣2 + 𝑣2 ), or the proportional surplus division ( 𝑣𝑀𝑣, 𝑀𝑣2 ). They answered
1 +𝑣2 𝑣1 +𝑣2
that the proportional surplus division is the most favorable in proportional surplus treatment.
Second, Kamijo and Yokote (2022) conducted the data fitting to the data of Takeuchi et al.
(2022), and showed that among the several types of entitlements, the proportional entitlements
explain the data in the proportional surplus treatment the best.
Even though they have proportional entitlements initially, their actual agreements show somedeviation from the proportional surplus division toward reducing inequality. To describe thispoint, we introduce two notions, the disagreement ratio 𝐷𝑅 and the agreement ratio 𝐴𝑅 definedby𝑣
𝑥 − 𝑣2𝐷𝑅 = 2 , 𝐴𝑅 = 2
𝑣1
𝑥1 − 𝑣 1
where player 2 is in stronger position than player 1 (i.e., 𝑣2 > 𝑣1 ) and (𝑥1 , 𝑥2 ) is their agreed
allocation. If they agree on the proportional surplus division, 𝐴𝑅 = 𝐷𝑅 should hold. Concerning
the inequality cap theorem, 𝐴𝑅 should be no more than 𝜆, under the assumption of the neutralcomponent function. The relation between 𝐷𝑅 and 𝐴𝑅 in the data of Takeuchi et al. (2022)
are summarized in Table 1. It is shown that even when 𝐷𝑅 is greater than or equal to 3, themeans and the medians of 𝐴𝑅 are about 2.3 Also, in this case, the value of 𝐴𝑅 is less than 3 formost of the data. Therefore, the majority of the agreed allocation belongs to the region that theinequality cap theorem suggests, even though they (probably) have very unequal entitlements atthe beginning of the negotiation.
3

Only the exception is mean of 𝐴𝑅 for 𝐷𝑅 = 8 where the agreements in this case show a considerable variance.

8𝐷𝑅

Num of data

1.5
2
3
4
6
8

34
80
11
53
17
20

Mean
1.26271.53102.09362.55812.37584.5965Median1.31551.62752.07692.75001.80001.9615SD
0.3977
0.52650.92131.45202.17554.5983𝐴𝑅% of ≦ 2
97.06
93.75
45.45
39.62
70.59
55.00

% of ≦ 3
100
100
81.82
56.60
76.47
60.00

% of ≦ 4
100
100
100
98.11
76.47
60.00

Table 1: Summary of the data in Takeuchi et al. (2022)

Note: 𝐷𝑅 = 𝑣2 /𝑣1 and 𝐴𝑅 = (𝑥2 − 𝑣2 )/(𝑥1 − 𝑣1 ) where (𝑣1 , 𝑣2 ) is pair of disagreement payoff satisfying 𝑣2 > 𝑣1
and (𝑥1 , 𝑥2 ) is the agreed allocation.

3.4 Some implications to the failure of the negotiation
The inequality cap theorem also gives us insight into whether bargainers can reach some agreement. Suppose an “unstable” proposal is being made, and one party (the disadvantaged party)
reluctantly tries to accept it. Then, immediately after subjectively accepting the distribution
proposal and it becoming the new status quo, they realize that there is still room for further
negotiation (because the proposed allocation is unstable). Such renegotiation is expected toincrease the likelihood of prolonged negotiations and a breakdown if there is a time limit.
An “unstable” proposal is likely to be made when two bargainers have different perspectives
on fairness. In a series of binary lottery bargaining experiments by Roth and Malouf (1979),
Malouf and Roth (1981), and Roth and Murnighan (1982), two subjects need to divide a certain
number of tokens by negotiation. The split tokens correspond to the probability of winning a
prize, and the amount of the prize varies between the two.4 . Here, there are two types of fairness.
One fairness criterion calls for the “equal probability agreement”, and the other is for the “equal
expected value agreement” (Murnighan et al., 1988). Indeed, the bargainer with a larger prize
prefers the former and one with a smaller prefers the latter.
In the experiment of Roth and Murnighan (1982), the number of tokens is 100, and the larger
and the smaller prizes are $20 and $5, respectively. Thus, the player with a larger prize calls
for a 50:50 split, and one with a smaller prize calls for a 20:80 split. If they negotiate theseas initial entitlements, it is expected that both sides will make certain compromises. However,when equity conflicts, often their bargaining becomes an argument about which equity is right,
and they stick to one outcome or the other. If one bargainer sticks to the 20:80 split of tokens,the difference in tokens between the two sides is almost four times, and such a disparity may
be more than the loss aversion parameter. Agreement on such a proposal is often tricky, anddisagreement happens more often. In fact, Roth and Murnighan (1982) observed that undersuch conditions, about 22% of negotiations were not completed in time, even though they had
more than 10 minutes for bargaining.5

4 Distribution norms
Stable entitlements can be considered as allocations that are reached in repeated negotiations forthe same amount of the pie because once they reach some agreement, it becomes the status quo
4
In this framework, the utility function of player 𝑖 is 𝑢𝑖 (𝑥𝑖 ; 𝐸𝑖 ) = 𝑃𝑖 × (𝑥𝑖 − 𝐸𝑖 ) when 𝑥𝑖 ≧ 𝐸𝑖 and𝑃𝑖 × 𝜆(𝑥𝑖 − 𝐸𝑖 ) when 𝑥𝑖 < 𝐸𝑖 , where 𝑥𝑖 is the probability of obtaining the prize and 𝑃𝑖 is the utility gained
from the prize. Due to the independence of the Nash solutions by a linear transformation, it is possible to ignore
the effect of the prize in this setting, and our inequality cap theorem holds.
5
From Table IV of Roth and Murnighan (1982). In the conditions that both players know both prizes, thefrequency of the disagreement is 5 out of 30 when their knowledge about their prizes becomes common knowledge
and 9 out of 35 when this is not common knowledge.

9and affects subsequent negotiations. When the agreement is stable, such a change in reference
point cannot change the negotiation outcome. In this light, it is possible to regard stability as aclosely related concept to establishing norms regarding distribution.
In the traditional textbook of von Neumann and Morgenstern (1944), they argue that theformation of norms (the standard of behavior in their terminology) occurs in the process of each
person modifying themselves in response to experience. They consider the standard of behaviors
as the results of such a process and define their solution concepts (called the stable set) as thesystem of allocations. In contrast, we model the process as adjusting the reference point or
status quo in repetitive bargaining environments and focus on the property of an allocation thatis invariant in the dynamic process.
Let’s define distribution norms by adding one more point to their considerations of the standardof behavior. The new perspective is that distribution norms in society need to be acceptable toall potentially existing members of society. Here, we will consider various people as entities withdifferent loss aversion parameters. A distribution norm is one that, once accepted, will remainrobust against bargaining between two players from such a diverse group of people.
To clarify the dependency of Nash allocation to the loss aversion parameters, we denote it by𝑥∗𝑖 (𝑀 , 𝑣, 𝐸; 𝜆1 , 𝜆2 ) for 𝑖 = 1, 2. Then, we define a distribution norm as follows.
Definition 2. A pair of entitlements 𝐸 = (𝐸1 , 𝐸2 ) is a distribution norm for a simple bargaining
problem (𝑀 , 𝑣) if and only if
𝐸𝑖 = 𝑥∗𝑖 (𝑀 , 𝑣, 𝐸; 𝜆1 , 𝜆2 )holds for any 𝑖 = 1, 2 and for any 𝜆1 ≧ 1 and 𝜆2 ≧ 1.
In other words, 𝐸 is a distribution norm when it is a stable entitlement for any two bargainers
in society. Thus, once some allocation 𝐸 is considered a distribution norm to (𝑀 , 𝑣), any two
players in the society facing the same bargaining problem agree on allocation 𝐸. In the definition,
by taking any value of 𝜆𝑖 ≧ 1, we model a situation where there are many people having different
values of loss aversion parameters.
It should be emphasized that in the definition of a distribution norm, players 𝑖 = 1, 2 are treatedas the role of a simple bargaining game, and any two individuals of the society are assigned oneof these roles before the bargaining. Therefore, we assume that a distribution norm is anonymousin the sense that it does not depend on the identity of the bargainer, but it is role-dependent.
Role dependency is an essential property of the norm that spreads consistently and widely (e.g.,
a norm regarding the division of housework between men and women). However, it may be
useful to consider identity-based entitlement when we try to answer how a norm arises. This
point is discussed in the next section.
By directly applying Theorem 1, we obtain the following theorem showing that an equal split is
the only distribution norm to a simple bargaining problem (𝑀 , (0, 0)).
Theorem 2. (𝑀 /2, 𝑀 /2) is the only distribution norm to a simple bargaining problem(𝑀 , (0, 0)).
Proof. Applying Theorem 1 to the case that 𝜆1 = 𝜆2 = 1, we have 𝑢2 (𝐸2 ; 𝐸2 ) ≦ 𝑢1 (𝐸1 ; 𝐸1 ),implying Ψ(𝐸2 ) ≦ Ψ(𝐸1 ). But since 𝐸1 ≦ 𝐸2 and Ψ is increasing, we should conclude that
𝐸1 = 𝐸 2 .
It is easily checked that (𝑀 /2, 𝑀 /2) is a stable entitlement for any 𝜆1 ≧ 1 and 𝜆2 ≧ 1.
One remark of this theorem is that it is easily extended to the statement that for any symmetricbargaining game (𝑀 , (𝑎, 𝑎)) with 0 ≦ 𝑎 ≦ 𝑀
2 , (𝑀 /2, 𝑀 /2) is the only distribution norm.
10This theorem provides a rationale for why an equal split norm plays such an essential role ineveryday life and the laboratory. This happens because the equal split allocation is a stableentitlement for any value of 𝜆𝑖 ≧ 1. Thus, this becomes the robust distribution by the negotiation
between any bargainers with different loss aversion parameters.
Once a distribution norm is established, one can form a rational expectation to the bargaining
result. Practically, this reduces the transaction cost regarding the bargaining, and they immediately agree on the one that the distribution norm suggests. In other words, they can avoid
negotiations to save time. Theorem 2 provides a good explanation of the phenomena in thelaboratory: experimental participants immediately agree on the equal split of the bargaining pie.

5 Emergence of the equal split norm
Theorem 2 states that once a norm of equal distribution is spread in society, this norm cannot
be overridden by any other distribution. In reality, information transmission processes suchas education, culture, and history play a role in sharing some norms among the members ofsociety—however, the myth of how these norms came to be remains.
To demonstrate the process of the birth of the equal split norm, we consider the dynamic processof revising the entitlements of players through the repetition of bargaining. Let 𝑁 = {1, 2, ..., 𝑛}with 𝑛 ≧ 2 be the finite set of people in a society. We consider the simple bargaining problem(𝑀 , (0, 0)) between two players in this society. For simplicity, we use the following notation(𝑥∗𝑖 (𝑀 , (0, 0), (𝐸𝑖 , 𝐸𝑗 ); 𝜆𝑖 , 𝜆𝑗 ), 𝑥∗𝑗 (𝑀 , (0, 0), (𝐸𝑖 , 𝐸𝑗 ); 𝜆𝑖 , 𝜆𝑗 ))to denote the Nash allocation of (𝑀 , (0, 0)) between 𝑖 ∈ 𝑁 and 𝑗 ∈ 𝑁 , 𝑗 ≠ 𝑖, where a pair of
their entitlements is (𝐸𝑖 , 𝐸𝑗 ) and their loss aversion parameters are 𝜆𝑖 and 𝜆𝑗 , respectively.
For any 𝑖, let 𝜆𝑖 ≧ 1 be the loss aversion parameter of 𝑖, which is assumed to be fixed in thisprocess. Let 𝐸𝑖1 be the initial entitlement of 𝑖 that will be updated through the bargaining
experience. For each period 𝑡 = 1, 2, ..., two players 𝑖 ∈ 𝑁 and 𝑗 ∈ 𝑁 , 𝑗 ≠ 𝑖 are randomly
chosen with fixed probability 𝑝𝑖𝑗 > 0, and they face the bargaining game (𝑀 , (0, 0)) with
their entitlements 𝐸𝑖𝑡 and 𝐸𝑗𝑡 at that time period. Through this bargaining, they agree on the
Nash bargaining allocation (𝑥∗𝑖 (𝑀 , (0, 0), (𝐸𝑖𝑡 , 𝐸𝑗𝑡 ); 𝜆𝑖 , 𝜆𝑗 ), 𝑥∗𝑗 (𝑀 , (0, 0), (𝐸𝑖𝑡 , 𝐸𝑗𝑡 ); 𝜆𝑖 , 𝜆𝑗 )) and thisbecomes their entitlements 𝐸𝑖𝑡+1 and 𝐸𝑗𝑡+1 in the next period. In contrast, if a player is notchosen at 𝑡, they keep the same entitlement to the next period. Let (𝐸𝑖𝑡 )𝑖∈𝑁 be the vector of thesociety members’ entitlements, which will be repeatedly updated due to the above dynamic rule
as time goes on.
We say that the equal split norm emerges in the society if and only if (𝐸𝑖𝑡 )𝑖∈𝑁 with 𝐸𝑖𝑡 = 𝑀 /2
for any 𝑖 ∈ 𝑁 is realized and is never overwritten by another entitlement vector. From thisframework of the dynamic adjustment of the entitlements, we can say that from Theorem 2, the“equal split” entitlement vector is a stable state of this dynamics in the sense that this vector
will not be changed forever if it is realized. Our question here is whether the process converges
to this state. The following theorem reveals the sufficient conditions of the convergence to theequal split norm.
Theorem 3 (Emergence of equal split norm). Suppose that Ψ(𝑥) = 𝑥 for any 𝑥 ≧ 0. The
dynamic process converges to the equal split norm in probability 1 if either of the following
conditions is satisfied;(i) for any 𝑖 ∈ 𝑁, 𝐸𝑖1 ≧ 𝑀 /2, and(ii) 𝑛 ≧ 3, and for any 𝑖 ∈ 𝑁, there exists another 𝑗 ∈ 𝑁 with 𝑗 ≠ 𝑖 such that 𝜆𝑖 = 𝜆𝑗 .

11

Proof. Let 𝜖 be defined by 𝜖 = min𝑖,𝑗∈𝑁,𝑖≠𝑗 𝑝𝑖𝑗 > 0.
(i) It suffices to show that for any state (the vector of entitlements) that can be reached from theinitial state (𝐸𝑖1 )𝑖∈𝑁 , there exists a finite sequence of bargaining pairs that leads to the equal
split norm.
We first show the following claim:
𝑀
Claim A. When 𝐸𝑖 ≧ 𝑀
2 and 𝐸𝑗 ≧ 2 , the Nash allocation is (𝑀 /2, 𝑀 /2).

Let 𝑥 be the share of player 𝑖. Then, the Nash product is one of the following threes:6
𝑓 (𝑥) = (−𝜆𝑖 (𝐸𝑖 − 𝑥) + 𝜆𝑖 𝐸𝑖 ) × ((𝑀 − 𝑥 − 𝐸𝑗 ) + 𝜆𝑗 𝐸𝑗 )if 𝑥 < 𝑀 − 𝐸𝑗 ,⎧{ 𝐿𝐺
𝑓𝐿𝐿 (𝑥) = (−𝜆𝑖 (𝐸𝑖 − 𝑥) + 𝜆𝑖 𝐸𝑖 ) × (−𝜆𝑗 (𝐸𝑗 − (𝑀 − 𝑥)) + 𝜆𝑗 𝐸𝑗 ) if 𝑀 − 𝐸𝑗 ≦ 𝑥 < 𝐸𝑖 ,⎨{if 𝑥 ≧ 𝐸𝑖 .
⎩ 𝑓𝐺𝐿 (𝑥) = ((𝑥 − 𝐸𝑖 ) + 𝜆𝑖 𝐸𝑖 ) × (−𝜆𝑗 (𝐸𝑗 − (𝑀 − 𝑥)) + 𝜆𝑗 𝐸𝑗 )Then, it is easily checked that 𝑓𝐿𝐺 (𝑥) is increasing in [0, 𝑀 − 𝐸𝑗 ), 𝑓𝐿𝐿 (𝑥) is maximized at
𝑥 = 𝑀 /2, and 𝑓𝐺𝐿 (𝑥) is decreasing in [𝐸𝑖 , 𝑀 ].7 Thus, it is maximized at 𝑥 = 𝑀 /2.
From this claim, we can derive two consequences. First, if 𝐸𝑖𝑡 ≧ 𝑀 /2 for any 𝑖 ∈ 𝑁 at someperiod 𝑡, the same condition still holds true at the next period 𝑡 + 1. Second, after some finite
sequence of bargaining pairs, the entitlement of every player 𝑖 becomes 𝑀 /2. If 𝑛 is even number,(1, 2), (3, 4), ..., (𝑛 − 1, 𝑛) is such sequence and if 𝑛 is odd number, (1, 2), (3, 4), ..., (𝑛 − 2, 𝑛 −1), (𝑛, 1) is the one. In both cases, the probability of obtaining such sequence is greater than orequal to 𝜖⌈𝑛/2⌉ .8(ii) To prove this, we need to show the following claim:
Claim B. Suppose 𝜆𝑖 = 𝜆𝑗 and 𝐸𝑖 < 𝐸𝑗 = 𝑀
2 . Then, the Nash allocation is (𝑀 /2, 𝑀 /2).
Let 𝑥 be the share of player 𝑖, and let 𝜆 = 𝜆𝑖 = 𝜆𝑗 . Then, the Nash product is one of thefollowing threes:𝑔 (𝑥) = (−𝜆(𝐸𝑖 − 𝑥) + 𝜆𝐸𝑖 ) × ((𝑀 − 𝑥 − 𝑀 /2) + 𝜆𝑀 /2)
if 𝑥 < 𝐸𝑖 ,⎧{ 𝐿𝐺
𝑔𝐺𝐺 (𝑥) = ((𝑥 − 𝐸𝑖 ) + 𝜆𝐸𝑖 ) × ((𝑀 − 𝑥 − 𝑀 /2) + 𝜆𝑀 /2)
if 𝐸𝑖 ≦ 𝑥 < 𝑀 /2,⎨{⎩ 𝑔𝐺𝐿 (𝑥) = ((𝑥 − 𝐸𝑖 ) + 𝜆𝐸𝑖 ) × (−𝜆(𝑀 /2 − (𝑀 − 𝑥)) + 𝜆𝑀 /2) if 𝑥 ≧ 𝑀 /2.
Then, it is easily checked that 𝑔𝐿𝐺 (𝑥) is increasing in [0, 𝐸𝑖 ), 𝑔𝐺𝐺 (𝑥) is increasing in [𝐸𝑖 , 𝑀 /2),and 𝑔𝐺𝐿 (𝑥) is decreasing in [𝑀 /2, 𝑀 ].9 Thus, it is maximized at 𝑥 = 𝑀 /2.
To compete the proof of (ii), with the aid of (i) of this theorem, it suffices to prove that for any
vector of the entitlements, there exists a finite sequence of bargaining pairs that leads to anentitlement vector (𝐸𝑖𝑡 )𝑖∈𝑁 such that 𝐸𝑖𝑡 ≧ 𝑀 /2 for any 𝑖 ∈ 𝑁. To describe such sequence, let
the set of strong players and the set of weak players at some period 𝑡 be defined as follows.
𝑆 = {𝑖 ∈ 𝑁 |𝐸𝑖𝑡 ≧ 𝑀 /2},and 𝑊 = {𝑖 ∈ 𝑁 |𝐸𝑖𝑡 < 𝑀 /2}.

We will show that after some repetitions of bargaining, 𝑆 = 𝑁 and 𝑊 = ∅ hold.
This classification is due to the domain to which the each bargainer belongs. For example, 𝑓𝐿𝐺 (𝑥) is the
Nash product when player 1 is in a loss domain and player 2 is in a gain domain and
7
These properties are proved as follows. Any of the three functions are concave and quadratic function. So, it isenough to say that 𝑓𝐿𝐺 (𝑥) is maximized at 𝑀/2+(𝜆𝑗 −1)𝐸𝑗 /2, which is greater than or equal to 𝑀/2 ≧ 𝑀 −𝐸𝑗 ;𝑓𝐺𝐿 (𝑥) is maximized at 𝑀/2 − (𝜆𝑖 − 1)𝐸𝑖 /2, which is less than or equal to 𝑀/2 ≦ 𝐸𝑖 .
8⌈⋅⌉ is a ceiling function defined by ⌈𝑥⌉ = min{𝑧 ∈ 𝑍|𝑥 ≦ 𝑧}.
9
These properties are proved as follows. Any of the three functions are concave and quadratic function. So, it
is enough to say that 𝑔𝐿𝐺 (𝑥) is maximized at (𝜆 + 1)𝑀/4, which is greater than or equal to 𝑀/2 > 𝐸𝑖 ; 𝑔𝐺𝐺 (𝑥)is maximized at (𝜆 + 1)𝑀/4 − (𝜆 − 1)𝐸𝑖 /2, which is greater than or equal to 𝑀/2; 𝑔𝐺𝐿 (𝑥) is maximized at
𝑀/2 − (𝜆 − 1)𝐸𝑖 /2, which is less than or equal to 𝑀/2.
6

12

Suppose 𝑆 ≠ 𝑁. We first show that there is a finite sequence of bargaining pairs that leads
to |𝑊 | = 1. To show this, it is important to notice that if two players in 𝑊 bargain over 𝑀,
their agreement should be efficient and thus, their updated entitlements should satisfy either (a)that both become strong players or (b) that one becomes strong and the other becomes weak.
Thus, after the bargaining between weak players, the size of 𝑊 should decrease at least by 1.
Therefore, at most |𝑊 | − 1 ≦ 𝑛 − 1 repetitions of bargaining, we obtain the entitlement vector
with |𝑆| = 𝑛 − 1.
Next, we explain how we can obtain the entitlement vector satisfying 𝑆 = 𝑁. Let 𝑊 = {𝑖}at some period 𝑡. Then, by the assumption of the theorem, there exists some 𝑗 ∈ 𝑆 such that
𝜆𝑗 = 𝜆𝑖 and some 𝑘 ∈ 𝑆, 𝑘 ≠ 𝑗. Consider the bargaining sequence such that (𝑗, 𝑘) first, and then(𝑗, 𝑖). The first bargaining leads to 𝐸𝑗𝑡+1 = 𝐸𝑘𝑡+1 = 𝑀 /2 by Claim A and the second implies
𝐸𝑗𝑡+2 = 𝐸𝑖𝑡+2 = 𝑀 /2 from Claim B. Hence, all players become the strong at period 𝑡 + 2.
Finally, from the proof of (i) of this theorem, there exists a finite sequence with length at
most ⌈𝑛/2⌉ that leads to the equal split norm from any entitlement vector satisfying 𝑆 = 𝑁.
In sum, for any entitlement vector, there exists a finite sequence of bargaining realizing theequal split norm, and the probability of obtaining such sequence is greater than or equal to𝜖(𝑛−1)+2+⌈𝑛/2⌉ .
Examining the sufficient conditions of the theorem is essential for understanding the emergenceof the equal split norm. The first sufficient condition implies that the members’ initial senseof entitlement is relatively large compared with the bargaining pie. This situation arises inloss-domain bargaining, where the bargaining pie is smaller than the expected level of theirshares. The norms established in shelters after a disaster are an example of this situation.10
Even in gain-domain bargaining, this sufficient condition is more likely to hold when people tend
to overestimate their contribution (Kahneman and Tversky, 1996).
On the other hand, the second condition requires that the population size be somewhat large,
and for any player, another player must have the same loss aversion parameter. Let us consider
the opposite cases to understand why this condition is important. For example, when the groupsize is two, and the initial entitlement is stable between them, it will continue to sustain even if
it is not an equal split. Also, even if there are three or more people, there can be cases where
only one person with different loss aversion parameters continues to be treated poorly, and theequal split norm is not realized.11 In other words, a certain amount of group size and a certain
degree of homogeneity concerning their characteristics are essential for establishing the norm ofequal division.
Finally, since the theorem assumes Ψ(𝑥) = 𝑥, it applies to distribution problems concerning
goods with small and medium stake sizes where the effects of diminishing marginal utility are
negligible. Also, since this theorem considers the establishment of an equal-split norm throughnegotiation, it assumes that the society size is not too large so that one person can interact andcommunicate with every other group member.
We can also think about another type of updating process of entitlements. If players learn the
feeling of entitlement from others, it is natural that they learn from the person who obtains most10
The implication that the norm of equal split arises in loss domain situations is consistent with the anthropological literature describing communal sharing systems. They explain that sharing systems exist in response toresource-poor environments (Kaplan et al., 1985; Blurton-Jones, 1984).
11
An example of this situation is three player society wherein 𝜆1 = 𝜆2 = 2 and 𝜆3 = 4, and their initialentitlements vector is 𝐸 0 = (0.5, 0.5, 0.4) with 𝑀 = 1. Then, the bargaining between players 1 and 3 gives the
allocation (0.6, 0.4), which is a stable entitlement for these players. Thus, player 3’s entitlement does not change
from the initial value even after repeated bargaining because his bargaining partner’s entitlement is either 0.5 or
0.6.

13

from the bargaining problem. Suppose that unmatched players in a certain period update theirentitlements to the highest value in society. Then, the situation described in (i) of Theorem 3
quickly occurs, and the equal split norm will be born immediately. Therefore, social learningfacilitates the emergence of the equal split norm.
The emergence of the equal split norm has been discussed in the literature on the evolutionary
bargaining theory (Young, 1993; Binmore et al., 2003). Ellingsen (1997) found that the equal
split norm is the stable state of the evolutionary dynamic process of the Nash demand game. Our
theorem offers an alternative explanation to the evolutionary model with at least two strengths.
First, the evolutionary models usually assume a perfectly homogeneous group of people, while
our model allows for a diverse group of people described by different loss aversion parametersand entitlements. Second, as Dawid and Dermietzel (2006) argue, the emergence of the equal
split norm depends on the type of evolutionary dynamic process and the parameters selection,
but their economic interpretation needs to be more obvious. In contrast, the sufficient conditionsof the theorem have clear interpretations and provide valuable insights into when the norm
emerges.

6 Concluding remarks
The analysis in this paper provides deep insights into social exchange and economic transactions
through negotiation. The discussion of stability provides perspective on the progress of thenegotiations: will they begin, be protracted, or be likely to break down? The inequality cap
theorem also reveals, on the one hand, that a certain degree of inequality can be maintained ina symmetric bilateral relationship (in terms of an objective bargaining problem) and, on theother hand, that equal outcomes can be accepted in an asymmetric bilateral relationship. The
latter point was discussed by Kamijo and Yokote (2022) as the bargaining outcome affected by
the equal split norm.
By applying the stability argument to groups, we formulated a distribution norm mathematicallyand showed that the distribution norm should be an equal split in a reasonably large group.
Furthermore, by considering a model in which entitlements of members in society are dynamically
updated, we found a sufficient condition for an equal split norm to be self-generating in society.
The results of these analyses provide a series of mechanisms by which equally splitting a pie, thegolden rule of distribution problems, is born and spread.
We end the paper with one remark about future extensions inspired by the inequality cap
theorem. The inequality cap theorem indicates that our fairness consideration is closely related
to our propensity for loss aversion. Reference dependence, the basis for the definition of loss
aversion, appears to be a more fundamental element in human systems than the pursuit offairness. Thus, it may be promising to construct a new theory of equity and fairness from this
aspect.

References
Anbarci, N. and Feltovich, N. (2013). How sensitive are bargaining outcomes to changes indisagreement payoffs? Experimental Economics, 16(4):560–596.
Anbarci, N. and Feltovich, N. (2018). How fully do people exploit their bargaining position?
the effects of bargaining institution and the 50–50 norm. Journal of Economic Behavior &Organization, 145:320–334.
Baranski, A. (2016). Voluntary contributions and collective redistribution. American Economic
Journal: Microeconomics, 8(4):149–73.

14

Baranski, A. (2019). Endogenous claims and collective production: An experimental study on thetiming of profit-sharing negotiations and production. Experimental Economics, 22(4):857–884.
Bastianello, L. and LiCalzi, M. (2019). The probability to reach an agreement as a foundation
for axiomatic bargaining. Econometrica, 87(3):837–865.
Binmore, K., Rubinstein, A., and Wolinsky, A. (1986). The nash bargaining solution in economic
modelling. The RAND Journal of Economics, pages 176–188.
Binmore, K., Samuelson, L., and Young, P. (2003). Equilibrium selection in bargaining models.
Games and Economic Behavior, 45(2):296–328.
Birkeland, S. and Tungodden, B. (2014). Fairness motivation in bargaining: A matter of principle.
Theory and Decision, 77(1):125–151.
Blurton-Jones, N. (1984). A selfish origin for human food sharing: tolerated theft. Ethology andsociobiology, 5(1):1–3.
Cappelen, A. W., Hole, A. D., Sørensen, E. Ø., and Tungodden, B. (2007). The pluralism offairness ideals: An experimental approach. The American Economic Review, 97(3):818–827.
Compte, O. and Jehiel, P. (2003). Bargaining with reference dependent preferences. Unpublished.
Dawid, H. and Dermietzel, J. (2006). How robust is the equal split norm? responsive strategies,
selection mechanisms and the need for economic interpretation of simulation parameters.
Computational Economics, 28(4):371–397.
Ellingsen, T. (1997). The evolution of bargaining behavior. The Quarterly Journal of Economics,
112(2):581–602.
Güth, W. and Kocher, M. G. (2014). More than thirty years of ultimatum bargaining experiments:
Motives, variations, and a survey of the recent literature. Journal of Economic Behavior &Organization, 108:396–409.
Hoffman, E. and Spitzer, M. L. (1982). The coase theorem: Some experimental tests. The
Journal of Law and Economics, 25(1):73–98.
Kahneman, D. and Tversky, A. (1996). On the reality of cognitive illusions.
Kamijo, Y. (2023a). A comparative statics on the nash solution. WINPEC Working Paper
Series.
Kamijo, Y. (2023b). A note on the risk dominance of the nash demand game. WINPEC Working
Paper Series, E2210.
Kamijo, Y. and Yokote, K. (2022). Behavioral bargaining theory: Equality bias, risk attitude,
and reference-dependent utility. WINPEC Working Paper Series, E2208.
Kaplan, H., Hill, K., Cadelina, R. V., Hayden, B., Hyndman, D. C., Preston, R. J., Smith,
E. A., Stuart, D. E., and Yesner, D. R. (1985). Food sharing among ache foragers: Tests of
explanatory hypotheses [and comments and reply]. Current anthropology, 26(2):223–246.
Luhan, W. J., Poulsen, O., and Roos, M. W. (2019). Money or morality: Fairness ideals in
unstructured bargaining. Social Choice and Welfare, 53(4):655–675.
Malouf, M. W. and Roth, A. E. (1981). Disagreement in bargaining: An experimental study.
Journal of Conflict Resolution, 25(2):329–348.
Murnighan, J. K., Roth, A. E., and Schoumaker, F. (1988). Risk aversion in bargaining: An
experimental study. Journal of Risk and Uncertainty, 1(1):101–124.

15

Nash, J. F. (1950). The bargaining problem. Econometrica, 18(2):155–162.
Nash, J. F. (1953). Two-person cooperative games. Econometrica, 21(1):128–140.
Nydegger, R. V. and Owen, G. (1974). Two-person bargaining: An experimental test of the nash
axioms. International Journal of Game Theory, 3(4):239–249.
Raiffa, H. (1982). The art and science of negotiation. Harvard University Press.
Roth, A. E. (1979). Axiomatic models of bargaining,. In Lecture Notes in Economics andMathematical Systems #170, pages 1–121. Princeton University Press.
Roth, A. E. (1995). Bargaining experiments. In The Handbook of Experimental Economics,
pages 253–348. Princeton University Press.
Roth, A. E. and Malouf, M. W. (1979). Game-theoretic models and the role of information in
bargaining. Psychological review, 86(6):574.
Roth, A. E. and Murnighan, J. K. (1982). The role of information in bargaining: An experimental
study. Econometrica, 50(5):1123–1142.
Rubinstein, A., Safra, Z., and Thomson, W. (1992). On the interpretation of the nash bargaining
solution and its extension to non-expected utility preferences. Econometrica, 60(5):1171–1186.
Shalev, J. (2002). Loss aversion and bargaining. Theory and Decision, 52(3):201–232.
Takeuchi, A., Vesteg, R., Kamijo, Y., and Funaki, Y. (2022). Bargaining over a jointly produced
pie: The effect of the production function on bargaining outcomes. Games and Economic
Behavior, 134:169–198.
Thomson, W. (1994). Cooperative models of bargaining. In Aumann, R. and Hart, S., editors,
Handbook of Game Theory with Economic Applications, chapter 35, pages 1237–128. Elsevier.
Tversky, A. and Kahneman, D. (1979). Prospect theory: An analysis of decisions under risk.
Econometrica, 47(2):263–292.
von Neumann, J. and Morgenstern, O. (1944). Theory of games and economic behavior (sixtiethanniversary edition published in 2007). Princeton University Press.
Young, H. P. (1993). An evolutionary model of bargaining. Journal of Economic Theory,59(1):145–168.

16