Fairness in Incomplete Information Bargaining: Theory and Widespread Evidence from the Field

Fairness in Incomplete Information Bargaining:

Theory and Widespread Evidence from the Field

∗

Daniel Keniston, Bradley J. Larsen, Shengwu Li, J.J. Prescott,

Bernardo S. Silveira, and Chuan Yu

†

September 21, 2023

Abstract

This paper documents a robust pattern from diverse sequential bargaining settings:

agents favor oﬀers that split the diﬀerence between the previous two oﬀers. Our empirical

settings include used-cars, insurance claims, home sale, trade tariﬀs, a TV game show,

eBay, and auto-rickshaws. These even-split oﬀers are more likely to be accepted, less

likely to spur exit by the opponent, and more likely to be followed by subsequent split-

the-diﬀerence oﬀers if bargaining continues. We propose several theoretical frameworks

to explain this behavior, including an inference argument under which split-the-diﬀerence

oﬀers can be viewed as an equal split of the potential surplus.

JEL Codes: C7, D8, D9

Keywords: Bargaining, negotiation, fairness, split-the-diﬀerence, incomplete informa-

tion, inference, alternating oﬀers

∗

We thank Panle Jia Barwick, Kalyan Chatterjee, Peter Cramton, Jack Fanning, Emin Karagözoğlu, Ted

O’Donoghue, Marek Pycia, Al Roth, Caio Waisman, Alex Wolitzky, and seminar and conference participants

at LSU, UCLA, NYUAD, IACM, and BEET for helpful comments and suggestions. Wendy Yin provided

excellent research assistance. We thank an anonymous real estate brokerage, as well as I

˜n

igo Hernandez-Arenaz

and Nagore Iriberri, for providing data.

†

Keniston: Louisiana State University, Department of Economics and NBER; [email protected]; Larsen:

Washington University in St. Louis, Olin School of Business and NBER; [email protected]; Li: Harvard

University, Department of Economics; [email protected]ard.edu; Prescott: University of Michigan Law

School; jprescott@umich.edu; Silveira: UCLA Department of Economics; [email protected]; Yu: Stanford

University, Department of Economics; chuan[email protected]

1 Introduction

The role of fairness notions in bilateral bargaining has been widely accepted by practitioners

and explored in depth in laboratory experiments. A number of studies document such

inﬂuences, in particular demonstrating a bias toward an equal split of a known pie; see

Camerer (2011) for a review. Little is known, however, about how these norms play out in

the ﬁeld, where assumptions of complete information common in laboratory experiments

are unlikely to hold. In this paper, we document a largely understudied fact: agents in

real-world sequential bargaining settings favor oﬀers that split the diﬀerence between the

two most recent oﬀers on the table.

The term split the diﬀerence appears in previous research (especially experimental work)

on bargaining with diﬀerent meanings. Binmore et al. (1989), for example, use it to describe

bargaining outcomes in which the players divide a known surplus net of their respective

outside options. In the present paper, similar to Bochet et al. (2023) and Backus et al.

(2020), splitting the diﬀerence refers instead to a notion in a sequential-oﬀer game where the

current bargaining oﬀer lies at the midpoint between the two most recent oﬀers (the most

recent oﬀer of the proposer and the most recent oﬀer of the counterparty). As an example,

suppose that in an alternating-oﬀer game, at some point when it is the seller’s turn, the

seller proposes $100, then the buyer proposes $50, and then it is the seller’s turn again. We

refer to the $100 and $50 oﬀers as the two most recent oﬀers, and, if the seller next proposes

$75, we refer to this as a split-the-diﬀerence oﬀer.

Our empirical evidence comes from novel, detailed data on sequential oﬀers from several

vastly diﬀerent bargaining contexts: business-to-business negotiations for used cars in

the U.S., pre-trial settlement bargaining on insurance injury claims in the U.S., street

negotiations from a quirky TV game show in Spain, bargaining for auto rickshaw rides in

India, international trade tariﬀ bargaining, online retail negotiations from eBay.com, and

bargaining over housing through a real estate dealer. In each setting, we ﬁnd strong evidence

of split-the-diﬀerence oﬀers—a clear mode at the 50-50 point between the two most recent

oﬀers. To our knowledge, the widespread nature of this phenomenon—across very diﬀerent

bargaining settings—was previously undocumented.

Our motivation for selecting these particular settings is simple: we use every dataset available to us in

which all back-and-forth oﬀers are recorded, because only with such data can we examine split-the-diﬀerence

behavior. Only recently did researchers start using this type of detailed bargaining data. As such, split-

the-diﬀerence oﬀers have been naturally understudied thus far, and several of the datasets we study are

We ﬁnd that split-the-diﬀerence oﬀers are more likely to be accepted: as an agent

concedes more to the opponent, the agent’s oﬀer is generally more likely to be accepted; this

probability, however, jumps discontinuously at oﬀers that lie halfway between the two most

recent oﬀers, to the extent that a split-the-diﬀerence oﬀer is even more likely to be accepted

than a slightly more generous oﬀer that concedes more than half of the distance between

oﬀers.

Similarly, oﬀers that split the diﬀerence are less likely to result in the opponent

walking away, ending the negotiations.

In the same vein, we show that certain agents are more likely than others to make

split-the-diﬀerence oﬀers, and that a seller is less likely to propose a 50-50 split if one of the

two most recent oﬀers is an extremely low buyer oﬀer—suggesting that the 50-50 norm is

constrained by the disparity between the two previous oﬀers.

We also demonstrate that it is indeed the two most recent oﬀers in a bargaining sequence

to which players gravitate: proposing an oﬀer at a later stage of the game that splits

the diﬀerence between oﬀers other than the previous two is far less common. Similarly,

split-the-diﬀerence oﬀers are substantially more prevalent than oﬀers based on anchor points

that are privately known to the proposer, such as the secret auto-accept and auto-decline

prices set by the seller in the eBay setting or the loss estimate set by the defendant in the

insurance settlement setting. That is, agents systematically split the diﬀerence between the

previous two oﬀers, which are common knowledge, and thus engage in a behavior that is

subject to the scrutiny of all agents involved.

Having demonstrated that split-the-diﬀerence oﬀers are ubiquitous, we ask what model

could explain this behavior. The prevalence of equal splits of a commonly known surplus has

been widely established in lab experiments. Researchers often attribute these patterns to

fairness-related norms.

But it is far from obvious that the patterns that we document are

new to the literature. The housing dataset is completely new. The used-car dataset is largely new; Larsen

(2021) uses some observations from this same setting, but here we take advantage of a larger dataset and

more information on alternating oﬀers that Larsen (2021) does not use. The data on pre-trial settlement

negotiations is also largely new; Prescott et al. (2014) study a small subset of this data to examine high-low

contracts in litigation. Data on auto rickshaw rides comes from Keniston (2011), data on the Spanish TV

show comes from Hernandez-Arenaz and Iriberri (2018), international tariﬀ negotiation data comes from

Bagwell et al. (2020), and the eBay data was ﬁrst analyzed by Backus et al. (2020).

This ﬁnding bears a close resemblance to results from the experimental literature on ultimatum games

showing that (i) respondents are more prone to accept equal divisions of the pie than divisions that are more

favorable to the respondent (Bellemare et al., 2008); and (ii) respondents are discontinuously more likely to

accept 50-50 oﬀers (Lin et al., 2020).

50-50 splits are a common outcome in relatively simple experiments, such as those involving the dictator

and ultimatum games, as well as in less structured bargaining studies allowing the subjects to freely negotiate

over the division of a pie. For excellent reviews of this vast literature, see Roth (1995) and Camerer (2011).

associated with fairness, as the surplus in our setting depends on agents’ private information.

In principle, split-the-diﬀerence oﬀers are “fair” only in the sense that they lie halfway

between the possible range of likely subsequent oﬀers given the current oﬀer history; they

do not necessarily constitute an equitable split of the actual surplus available to the agents.

In fact, unless both parties have made oﬀers that diﬀer by exactly the same amount from

their true valuations, splitting the diﬀerence between oﬀers cannot yield an equal split of

the surplus.

We introduce several theoretical results to explore this question in the context of a

bargaining game with alternating oﬀers and two-sided private information. In this context,

we ﬁrst show that a pattern of split-the-diﬀerence oﬀers may arise in a perfect Bayesian

equilibrium even when the players have no preferences for “fair” outcomes. However, this

equilibrium is far from being unique. Thus, while split-the-diﬀerence oﬀers are consistent

with “standard” game theory, they are by no means predicted by it.

We then propose an explanation for how agents may indeed view an equal split of the

two most recent oﬀers as “fair.” Our argument relies on agents making inferences about the

support of their opponent’s valuation based on the fact that the opponent did not accept

the agent’s previous oﬀer. Consider a case where a seller initially proposes a price of $100.

The buyer rejects this oﬀer and counters at $50. What can the seller infer about the buyer’s

valuation from the fact that the buyer rejected $100? We show that, for all buyer types

(i.e., buyers with some valuation) below $100, rejecting the $100 oﬀer is a sequential best

response (see Battigalli and Siniscalchi 2002) to every belief that the buyer may hold about

the seller’s type. And we show that, for all buyer types above $100, accepting is a sequential

best response to some belief a buyer could hold. In this sense, $100 is the highest buyer

type that the seller cannot rule out (1) based on the buyer’s rejection of $100, and (2)

under the assumption that the seller knows the buyer is behaving rationally according to

some belief. A similar argument applies to the buyer’s inference about the seller’s valuation

if the seller rejects the buyer’s counteroﬀer of $50. Together, these arguments oﬀer an

explanation for how a buyer and a seller might jointly view the gap between the two most

recent oﬀers—[50

100]—as the potential surplus; and hence an oﬀer of $75, halfway between

the two most recent oﬀers, can be viewed as “fair,” an equal split of the potential surplus.

We also discuss a number of other possible explanations for split-the-diﬀerence behavior.

Two existing studies within economics analyze a similar notion of splitting the diﬀerence.

Backus et al. (2020) provide descriptive evidence of several facets of eBay bargaining,

documenting the prevalence of split-the-diﬀerence oﬀers being made and accepted on eBay.

We incorporate their data as one of our seven settings, and we adopt their framework as

a starting point for our analysis.

Bochet et al. (2023) design a lab experiment in which

agents with private information negotiate over multiple issues. The authors ﬁnd that an

alternating-oﬀer regime arises endogenously, with proposals frequently splitting the diﬀerence

between the two most recent oﬀers whether agents negotiate over single items or bundles.

We complement the analyses of these two papers by establishing that split-the-diﬀerence

oﬀers constitute a convention—not an ironclad convention, but a notable convention—among

bargainers in the ﬁeld, and that this behavior is not unique to consumer interactions nor to

U.S. negotiation culture. Similarly, this is not a feature of small-stakes negotiations only; it

also appears in home sales, the largest lifetime purchase for many buyers. Moreover, our

paper establishes a possible grounding for why such behavior could be viewed as “fair” in

some sense—even though the two most recent oﬀers are themselves endogenous equilibrium

objects and thus the halfway point does not actually correspond to an equal split of the

pie. But we stress that we do not view this model as the only possible explanation for the

behavior.

Outside economics, the negotiation literature broadly acknowledges the norm of “even-

split” oﬀers, although it contains diﬀering assessments of the eﬀectiveness of the strategy.

Some authors (Babcock and Laschever, 2008) argue that “split the diﬀerence is a tactic that

often works extremely well” (p. 212), while others (Thompson, 2020) discourage even-split

oﬀers, a perspective exempliﬁed by the best-selling guide to negotiation titled Never Split the

Diﬀerence (Voss and Tahl, 2016). Nevertheless, the negotiation literature universally argues

that split-the-diﬀerence oﬀers operate through an appeal to fairness that may increase the

Many experiments in the lab involve alternating-oﬀer bargaining, but they do not address the questions

we study in this paper; see, for example, Binmore et al. (1985), Ochs and Roth (1989), and Binmore et al.

(1989). These experiments have investigated issues such as time discounting and the relevance of outside

oﬀers in bargaining. Andreoni and Bernheim (2009) focus on bargaining with complete information and

oﬀer a model and experimental results demonstrating how an equal split of a known pie can arise from a

preference to appear equitable (a concern for social image). Other related studies include Roth and Malouf

(1979) and Roth (1985).

Several other studies have also examined this same eBay bargaining data, such as Green and Plunkett

(2022), who analyze how well human agents perform relative to reinforcement learning bots coded to respond

optimally to observed actions in the data, and Freyberger and Larsen (2021), who estimate a structural

model bounding private valuation distributions and bargaining ineﬃciency.

probability of an oﬀer being accepted by one’s bargaining partner.

Popular negotiation

advice echos this motivation: Kwame Christian, lawyer and founder/CEO of the American

Negotiation Institute, a consulting ﬁrm oﬀering training and coaching in negotiation, argues

that the primary beneﬁt of a split-the-diﬀerence oﬀer is that “it can trigger reciprocity and

convey a sense of fairness.”

Overall, we view our primary contribution as documenting a stylized fact that may

discipline the bargaining literature in developing more reasonable models. Theoretical

bargaining models in economics make predictions about a range of things, such as the split

of surplus, delay, and the sequence of oﬀers. Until recently, there have been few (or no)

real-world datasets containing back-and-forth oﬀers with which a researcher could assess

the most important stylized facts regarding the path of play. Our paper documents one

particular set of patterns and demonstrates that they are ecologically robust, in the sense

that they emerge in many, very diﬀerent situations. Given this breadth of applicability, and

given that canonical equilibrium models do not predict these patterns, we suggest that it

would be wise for bargaining theory to attempt to account for them. Likewise, most existing

structural estimation analyses of bargaining models omit systematic behavioral patterns

such as preferences for splitting the diﬀerence. This is the case of workhorse empirical

models that rely on various forms of Nash bargaining (e.g., Crawford and Yurukoglu, 2012,

and subsequent papers), as well as of the majority of studies that bring to the data models

of incomplete information bargaining (e.g., Keniston, 2011; Silveira, 2017; Larsen, 2021).

By accounting for the split-the-diﬀerence behavior that we document, future iterations of

these analyses could yield more realistic estimates and increase the accuracy and reliability

of policy takeaways.

Our study also relates to recent work by Camerer et al. (2019) and Huang et al. (2020),

which examine incomplete-information bargaining in an experimental setting. We elaborate

in Section 5.3 how our results shed light on their ﬁndings, oﬀering insight as to how,

even in incomplete information settings, agents’ behavior is consistent with a notion of

Thompson (2020)’s negotiation textbook captures the tension between even-split oﬀers and an even

division of surplus: “even-split between whatever two oﬀers are currently on the negotiation table ... has an

appealing, almost altruistic ﬂavor to it. So what is the problem with even splits? ... the pattern of oﬀers up

until that point was not ‘equal.”’

https://www.forbes .com/sites/kwamechristian/2023/03/26/splitting-the-difference-in-neg

otiation-a-double-edged-sword/?sh=30ecfa152db5

. Like the academic literature, popular advice diﬀers

when it comes to splitting the diﬀerence. Christian writes, “Although some may say never split the diﬀerence,

there may be situations where the maneuver has validity.”

fairness. Finally, our analysis contributes to a literature that studies behavioral phenomena

in bargaining using real-world data, such as Pope et al. (2015), studying focal points, or

Jiang (2022), studying left-digit bias.

The rest of the paper is organized as follows. Section 2 describes each of our data

settings, and Sections 3–4 contain our empirical results documenting the prevalence of

split-the-diﬀerence oﬀers and related patterns. Section 5 explores several theoretical results

and Section 6 concludes.

2 Description of Field Settings

We now introduce the ﬁeld settings from which we obtain our bargaining data. A beneﬁt

of the question we study in this paper—how the current oﬀer in a sequential bargaining

game relates to the two most recent oﬀers—is that we can address the question in each of

these ﬁeld settings even though the products or outcomes over which agents negotiate diﬀer

drastically from setting to setting. In particular, in each setting, we observe data on many

bargaining sequences (which we also refer to as threads) and, for each sequence, we observe

the full set of sequential oﬀers between negotiating parties. We consider an observation to

be an oﬀer triple: the current oﬀer and two preceding oﬀers.

We drop from every dataset any bargaining sequences in which there are fewer than

three oﬀers. We drop any threads that continue beyond the point where an agent makes

an oﬀer exactly equal to the opponent’s previous oﬀer (which logically should have ended

the game in agreement). We also drop any sequences in which an oﬀer lies outside the two

most recent oﬀers or cases in which a seller’s oﬀer lies below a buyer’s oﬀer.

We describe

additional cleaning steps in Appendix B.

2.1 Business-to-Business Used-Car Bargaining

The ﬁrst dataset comes from the U.S. wholesale used-car industry. In this market, owners of

used-car dealerships buy vehicles from other dealerships as well as from large companies,

such as Hertz (rental cars), Wheels (a ﬂeet company), Bank of America (selling oﬀ-lease

Such behavior likely corresponds to misrecorded data or to cases where some feature of the bargaining

environment changes prior to the current proposed oﬀer, such as the arrival of additional information or a

new outside option for an agent. Dropping such threads eliminates fewer than 2% of observations in most

data settings, but as many as 24% in some data settings where the arrival of new information or misrecorded

oﬀers may be more prevalent. We discuss this in Appendix B.

or repossessed vehicles), or Ford (selling lease buy-back cars). All negotiating agents are

professionals or businesses experienced in these negotiations. This $80 billion industry

underlies the supply side of the used-car market in the U.S., trading 15 million cars annually;

similar platforms exist internationally. For each car, an auction house runs a secret-reserve-

price ascending auction, followed by bilateral bargaining between the seller and the highest

bidder if the auction price falls short of the reserve price. The data we use in our analysis is

generated during this post-auction bargaining stage.

The dataset comes from six auction houses from January 2007 to March 2010. It records

each distinct attempt to sell the vehicle through the mechanism, and, for each attempt,

every alternating oﬀer proposed by either the buyer or the seller, as well as the outcome

of the bargaining. This data overlaps in part with the data used in Larsen (2021), but

it also contains additional bargaining sequences that were dropped in that analysis. This

new inclusion highlights a major beneﬁt of our empirical approach: the structural exercise

of Larsen (2021) required careful data cleaning and controlling for heterogeneity in the

items over which the parties negotiated, whereas our approach only requires looking at

split-the-diﬀerence patterns between oﬀers in a given bargaining sequence, regardless of

heterogeneity across items.

Descriptive statistics for this dataset are shown in the ﬁrst column of Table 1. This

dataset consists of 21,734 total bargaining sequences and 33,356 observations (oﬀer triples).

Bargaining in this market begins if the auction price is below the reserve price, in which case

the auction price becomes the ﬁrst oﬀer in an alternating-oﬀer bargaining game. The seller

can choose to accept the auction price, propose a counteroﬀer, or quit (ending the game).

The bargaining process is typically wrapped up within a day, with an average of several

hours between each oﬀer. The average ﬁrst oﬀer (auction price) is $7,444, followed by an

average counteroﬀer from the seller of $8,918. The average number of oﬀers in a sequence in

the used-car sample is 3.53 and the average last price oﬀered is $8,072.

The negotiation

ends in agreement 59% of the time, at an average accepted price of $7,987. The

objects

reported in Table 1 are deﬁned and discussed in Section 3.

As highlighted at the beginning of Section 2, for every data setting, our analysis conditions on bargaining

sequences that include at least three oﬀers.

Table 1: Descriptive Statistics of Bargaining Settings

Cars Settlement TV Show Rides Housing Trade eBay

# Threads 21,734 74,356 204 2,058 176 44,048 6,976,776

# Oﬀer Triples 33,356 208,463 714 2,986 176 46,985 9,789,903

Rounds 3.53 4.80 5.50 4.32 3.00 3.07 3.40

Pr(Agree) 0.59 0.94 0.91 0.39 0.73 0.08 0.29

First Oﬀer $7,444 $36,391 e19.50 |51.69 $460,849 0.00 $151.06

Second Oﬀer $8,918 $64,042 e124.27 |36.73 $430,071 69.60 $88.71

Final Oﬀer $8,072 $24,728 e56.67 |39.50 $451,248 38.30 $122.70

Accept Price $7,897 $21,838 e55.37 |42.54 $469,658 24.29 $91.40

0.39 0.29 0.32 0.28 0.71 0.44 0.42

0.18 0.43 0.31 0.24 0.26 0.38

0.38 0.29 0.25 0.18 0.33 0.23

0.14 0.39 0.28 0.16 0.31

0.33 0.30 0.19 0.17 0.19

0.12 0.38 0.34 0.27

0.45 0.32 0.09

0.00 0.38 0.38

Notes: Table shows the number of sequences/threads and number of oﬀer triples in each data setting, as well as

averages for several variables in the data. For t ≥ 3, the variable γ

denotes the average concession weight in

bargaining round t, as deﬁned in Section 3. The units for the average ﬁrst, second, ﬁnal, and accept prices are euros

for the TV Show and Indian rupees for the Rides setting. The trade setting combines oﬀer sequences in which the

units are percentages and sequences where the units are a currency. Units are unimportant for our analysis, which

relies on the unit-less γ

weights.

2.2 Pre-trial Settlement Bargaining from Insurance Claims

The second dataset comes from pre-trial settlement bargaining over injury claims made

under U.S. auto and general liability insurance policies.

This dataset consists of extensive

proprietary information about all claims made to a large national auto and general liability

insurer that closed between January 1, 2004, and March 31, 2009. The data contains details

about the underlying accident, the alleged injury, the involved parties, the insurance contract,

and all attempts by the parties to resolve the associated dispute.

In these insurance cases, an injured person alleges that an injurer caused harm covered

by the insurer’s policies. If a claimant asserts damages within policy limits or declines to

pursue the insured individually for any excess—or if the insurer agrees to cover any damages

in excess of the insurance contract’s policy limit, which eﬀectively it must do to act in “good

faith” if it turns down a demand by the claimant for the policy limit or less—the insurer

eﬀectively replaces the injurer as the claimant’s counterparty in any dispute, which occurs

Prescott et al. (2014) study a small subset of this information, but the data we use in this paper—

particularly the bargaining threads—remains largely unexplored.

in virtually all cases. Under these circumstances, the claimant and the insurer bargain over

the amount that should be paid by the insurer to the claimant. This process can take many

months to reach a conclusion. If the two parties cannot reach an agreement, the claimant

will often pursue litigation, with the claimant (now plaintiﬀ) ﬁling a complaint in court

demanding damages against the insurer (now defendant). The parties may also negotiate

and settle the claim during the litigation stage. Importantly, the insurer records the amount

of each back-and-forth proposal made by either the insurer or the claimant and whether the

parties reach an agreement in these negotiations.

After cleaning, our sample contains 74,356 bargaining threads and 208,463 oﬀer triples.

Table 1 shows that the average ﬁrst oﬀer is $36,391, followed by a counteroﬀer of $64,042

from the opposing party, and an average ﬁnal oﬀer of $24,728. The negotiations include 4.8

oﬀers on average and end in agreement 94% of the time, at an average accepted price of

$21,838. This number is not between the average ﬁrst and second oﬀers because of a feature

of negotiations in this setting: the ﬁrst oﬀer may come from the claimant or the insurer,

and, in computing the average ﬁrst and second oﬀers, Table 1 pools over these two cases.

Appendix B discusses this point in more detail and describes how we clean the data to form

alternating-oﬀer sequences.

2.3 Street Bargaining from a TV Game Show in Spain

The third dataset comes from a TV game show in Spain titled Negocia Como Puedas

(roughly, “Bargain However You Can”) analyzed in Hernandez-Arenaz and Iriberri (2018).

This data was generated in the streets of several major Spanish cities in summer 2013. In a

typical episode of the show, the host approaches individuals in the street and invites them

to participate in the game. Upon acceptance, an individual (the proposer) is endowed with

a potential pie of 100 euros and is asked an easy question. The proposer is not allowed

to answer the question herself. Instead, she must, within a three-minute limit, (i) ﬁnd a

passer-by (the responder) able to provide an answer to the question that the proposer ﬁnds

satisfactory, and (ii) negotiate a price that the proposer will pay the responder to be able to

use that answer.

If the negotiations succeed and the responder’s answer to the original question is correct

Examining these two cases separately, the average accepted price is indeed between the average ﬁrst and

second oﬀers.

(as determined by the host), the proposer pays the responder the agreed amount. The

proposer then moves on to a new question, referred to as a new stage of the game, where

the process is repeated. In the new stage, the size of the pie increases (by 200 euros in the

second stage, 300 in the third, and 1,000 in the fourth). In any stage of the game, if the

proposer does not reach an agreement within the three-minute time limit, the game ends,

and the proposer gets nothing. Throughout the game, the size of the pie is only known to the

proposer, not the responder. In any given stage of the game, the bargaining is unstructured,

but the negotiations typically follow an alternating-oﬀer structure, with the proposer making

the ﬁrst oﬀer. We only keep those threads in which oﬀers clearly alternate between parties.

Additional details on the cleaning of the data are found in Appendix B.

In the data, we have 204 sequences and 714 oﬀer triples. Table 1 shows that the average

ﬁrst oﬀer is 19.5 euros, followed by an average second oﬀer of 124.3 euros and an average

ﬁnal oﬀer of 56.7 euros. The game ends in agreement most of the time (91%), after an

average of 5.5 rounds and at an average accepted price of 55.4 euros.

2.4 Auto Rickshaw Rides Bargaining in India

The fourth dataset comes from the local transportation market by auto rickshaw in Jaipur,

India. An auto rickshaw is a form of three-wheeled mini-taxi, oﬃcially capable of carrying

three passengers in a semi-enclosed back seat. Auto rickshaws are the primary means of

rented transportation in Jaipur. During the period in which Keniston (2011) collected the

data (January 2008 to January 2009), all prices were set by negotiation.

The data was collected by surveyors (whom we also refer to as buyers) who followed one

of two possible protocols. In real bargaining, buyers were assigned to travel by auto rickshaw

along ﬁxed routes through the city, bargaining for the price of each ride. At the beginning of

each route, buyers were paid a lump sum slightly higher than the expected cost of the route

and were allowed to keep any money not spent on auto rickshaw fares. At the end of their

assigned trip, they were free to return to their homes or alternate employment. Thus, their

ﬁnancial incentives and cost of time were similar to real trips taken for personal purposes. In

scripted bargaining, buyers negotiated with sellers according to a written bargaining script

(prepared by Keniston) consisting of a sequence of pre-determined counteroﬀers. Scripted

surveyors were instructed to act as if they were bargaining in a realistic manner so that

drivers (whom we also refer to as sellers) would respond as naturally as possible. After the

conclusion of the bargaining, surveyors wrote down the series of oﬀers made by the drivers

and themselves. Drivers were not aware that they were part of a ﬁeld experiment. The

average negotiation took 55 seconds to complete.

In our analysis, we exclude any oﬀers or responses that come from scripted surveyors, as

they do not represent actual reactions. Additional details on data cleaning are in Appendix

B. Our main sample consists of 2,058 bargaining threads and 2,986 oﬀer triples. Table 1

demonstrates that the average ﬁrst oﬀer is 52 rupees, followed by an average counteroﬀer

of 37 rupees and a ﬁnal oﬀer of 40 rupees. Negotiation concludes after an average of 4.32

oﬀers, ending in agreement 39% of the time at an average accepted price of 43 rupees.

2.5 Bargaining in Residential Real Estate

The ﬁfth dataset we use is new to the literature and comes from a growing residential

real estate brokerage company that oﬀers discounted agency fees of 2–3% rather than the

traditional 6%. We collected this dataset in collaboration with the company, covering a

number of houses on the market from 2015 to 2019 in Colorado. This dataset diﬀers from

the others we analyze in that we only observe oﬀers placed by potential buyers, not by the

seller. The company informs us that seller counteroﬀers are indeed quite rare in this market.

Rather, the typical negotiation proceeds with a seller list price (which we treat as the ﬁrst

oﬀer) followed by sequential oﬀers from the buyer. For a given home, we observe the seller’s

list price and each oﬀer placed by potential buyers. The time on the market for a given

home can be several weeks or several months. In this setting, we consider split-the-diﬀerence

behavior to be cases in which a buyer makes an oﬀer and then, if that oﬀer is rejected,

subsequently makes an oﬀer that splits the diﬀerence between the list price and the buyer’s

initial oﬀer.

Table 1 shows that our sample has 176 threads and the same number of oﬀer triples.

This feature is by construction: our main object introduced in Section 3 cannot be deﬁned in

cases where one party makes three consecutive oﬀers; thus, we only keep the seller’s list price

and the ﬁrst two oﬀers from the buyer, i.e., one oﬀer triple for each bargaining sequence.

These bargaining sequences end in agreement 73% of the time. Bargaining begins with an

average list price of $460,849, followed by an average second oﬀer of $430,071, and a ﬁnal

oﬀer of $450,248. When parties agree, they end at an average price of $469,658. Additional

details on data cleaning are found in Appendix B.

2.6 International Trade Tariﬀ Bargaining

The sixth dataset contains detailed information on international trade negotiations recently

declassiﬁed by the World Trade Organization (WTO). In these negotiations, countries

bargain over commitments on their respective import tariﬀs. Despite the multilateral nature

of both the WTO and its predecessor, the General Agreement on Tariﬀs and Trade (GATT),

the negotiations are mostly bilateral, with individual country pairs making requests and

oﬀers over the tariﬀ for a speciﬁc tariﬀ-line (a product code).

We use the dataset of Bagwell et al. (2020), which comprises the Torquay Round (1950–

1951). This data includes 298 bilateral bargaining pairs from 37 countries, negotiating

tariﬀs over thousands of tariﬀ-line products. A bargaining thread is deﬁned as two countries

(proposer and target) negotiating a tariﬀ over a tariﬀ-line product. In each thread, we observe

all requests from the proposer and oﬀers from the target. As documented in Bagwell et al.

(2020), relatively few back-and-forth oﬀers and counteroﬀers take place in any given thread.

For our analysis below, we consider whether a proposer requests a tariﬀ that splits the

diﬀerence between a zero tariﬀ and the existing tariﬀ (the status quo before the negotiations).

To map the possibilities of a zero tariﬀ and existing tariﬀ into the same framework as the

other data settings, Table 1 considers a zero tariﬀ as the de facto initial request from the

proposer (and, indeed, a zero tariﬀ is frequently an actual request made in this setting).

Similarly, Table 1 considers the existing tariﬀ as the initial oﬀer from the target (the row

of “second oﬀer”). We discuss this point further in Section D. All subsequent requests and

oﬀers are also recorded in the data.

Table 1 shows that the sample consists of 44,048 bargaining sequences and 46,985 oﬀer

triple observations. The game proceeds for 3.07 oﬀers on average, ending in agreement 8%

of the time. Note that here we combine sequences in which the units are percentages and

sequences in which the units are a currency; thus, we provide no units on these averages of

oﬀers in Table 1 and their values are hard to interpret. However, units are unimportant for

our analysis, which relies on the unit-less

weights (described in Section 3). Additional

details on data cleaning are found in Appendix B.

2.7 Bargaining on eBay’s Best Oﬀer Platform

Our ﬁnal ﬁeld setting comes from eBay’s Best Oﬀer negotiation platform. eBay is well known

as a platform for buying and selling via auctions or ﬁxed prices. Less well known is the

bargaining mechanism on eBay, through which a buyer and seller negotiate via alternating

oﬀers (limited to three oﬀers by each party). The game begins with the seller posting a

list price. An interested buyer can pay this price or make an oﬀer. Oﬀers are sent through

the eBay platform, and the receiving party has 48 hours to respond by either accepting,

declining, or proposing a counteroﬀer. Our data comes from internal data collected by Larsen

and coauthors for a separate project (Backus et al. 2020), and it consists of all bargaining

sequences placed by buyers and sellers on eBay (regardless of the product) from June 2012

through May 2013.

The sample consists of 6,976,776 bargaining sequences, comprising 9,789,903 oﬀer triple

observations. These sequences contain an average of 3.4 oﬀers. The average list price is

$151, the average second oﬀer is $89, and the average ﬁnal oﬀer is $123. When trade occurs

(which happens 29% of the time), the ﬁnal agreed-upon price is $91.

3 Split-the-Diﬀerence Oﬀers are Widespread

We now demonstrate a key empirical pattern: agents favor oﬀers that split the diﬀerence

between the two most recent oﬀers. To show this, we ﬁrst introduce some useful notations.

For each data setting, we organize each sequential bargaining thread in the following way, as

in Backus et al. (2020). For each round

= 1

, ...

in the bargaining thread

, we observe

the proposed amount,

j,t

. This proposed amount is an oﬀer made by the seller/buyer,

insurer/claimant, proposer/respondent, driver/surveyor, proposer/target, etc. If

j,t

comes

from one player,

j,t+1

must come from the opponent—with the exception of threads in the

housing dataset, as we note above and explain further later in this section.

We can write the proposed amount in round

t ≥

3 as a weighted average of the proposed

amount in the previous two rounds:

j,t

j,t−1

+ (1

− γ

j,t

)

j,t−2

. Therefore,

j,t

represents, for bargaining thread

, the weight that the player in round

places on the

opponent’s previous oﬀer, and 1

− γ

j,t

represents the weight the player places on her own

previous oﬀer. We can think of

j,t

as how much a player concedes to her opponent when

she is making a counteroﬀer, or her concession weight. Rearranging to solve for γ

j,t

yields

j,t

− p

j,t−2

j,t−1

− p

j,t−2

. (1)

In particular, a split-the-diﬀerence oﬀer corresponds to γ

j,t

= 0.5.

As highlighted in Section 2.5, in the housing dataset, we observe a seller’s list price and

a given buyer’s oﬀers, and in no bargaining thread does the number of oﬀers from the same

buyer exceed two. In this setting, we only deﬁne the concession rate for the second buyer’s

oﬀer. Speciﬁcally, we let

j,3

− p

j,2

j,1

− p

j,2

, (2)

where

j,1

is the list price,

j,2

is the buyer’s ﬁrst oﬀer, and

j,3

is the buyer’s second oﬀer.

Thus, in the housing dataset,

j,3

represents the weight a buyer places on the list price, and

1 − γ

j,3

represents the weight she places on her own initial oﬀer.

The advantage of focusing on concession weights is that they are unit-less and do not

require considering any heterogeneity across negotiation threads. Table 1 shows the average

j,t

for each round of the game, beginning with

= 3. We observe as many as ten

j,t

’s in

some settings,

and, as highlighted above, only the round-3

j,t

in the housing setting. The

average

j,t

in each round is below 0.5 (other than for housing, where it is 0.71), suggesting

that most agents make oﬀers closer to their own previous oﬀers than to their opponent’s. In

the settlement data, the average

j,t

is roughly 0.3–0.4 in every round of the game, whereas

in the cars setting,

j,t

tends to be much smaller in even rounds, which correspond to the

seller’s turns, suggesting that sellers generally concede less than buyers in this context.

The main pattern we wish to examine is whether agents in each of these settings exhibit

a tendency to make oﬀers that lie halfway between the two most recent oﬀers on the table

(i.e.,

j,t

close to 0.5). Figure 1 plots a histogram of these concession weights, with each

panel corresponding to one data setting. We pool together

j,t

for all

t ≥

3; our results are

similar if we analyze

j,t

from each round

separately.

An interesting pattern stands out

in all datasets: there is a mass point at 0.5—counteroﬀers that are halfway between the

Some settings have bargaining threads exceeding ten rounds, and we truncate them to the ﬁrst ten

rounds in Table 1.

For the eBay setting, Figure 1.G is not new to the literature, capturing a result that is illustrated in

several panels in Figure 8 of Backus et al. (2020), but here the result is pooled across rounds. Similarly, our

Figure A4.D and column 6 of Table 2 capture the same information as Figure 9 and Table 8 of Backus et al.

(2020). All other results in this study are new.

previous two oﬀers, or split-the-diﬀerence oﬀers.

There are a number of diﬀerences across our data settings. The trade tariﬀ setting, for

example, likely involves agents bargaining over multiple issues simultaneously, unlike, say,

the rides setting. In practice, agents in a multi-issue setting might be expected to fully

concede on one issue in return for the other party’s concession on a separate issue. These

possibility makes it all the more striking that split-the-diﬀerence oﬀers within a given thread

appear even in the trade setting.

The various settings certainly diﬀer in the relative prevalence of split-the-diﬀerence

behavior. Figure 1 shows that housing and settlement appear to have lower frequencies of

split oﬀers relative to non-split oﬀers.

We note that both settings often involve principals

and agents on each side, with the agent formally bargaining with the counterparty’s agent

while the principals, in theory, make decisions about what to oﬀer and when to accept on the

basis of communications from their agents. This insulated arrangement may dampen some

eﬀects of principals’ preferences, such as a preference for fairness. Also, parties bargaining

over the resolution of an insurance claim may anticipate bargaining over more rounds on

average. Early oﬀers and counter-oﬀers may serve some other purpose, whereas split-the-

diﬀerence oﬀers may represent a genuine attempt to resolve the dispute without (or with

very few) further rounds. There are other candidate explanations for this variation as well.

In the housing setting, for instance, the split-the-diﬀerence bargaining may be less likely

when sellers have a particularly strong position in the market (with few sellers and many

buyers), such that a split-the-diﬀerence oﬀer corresponds to too much concession from the

seller’s standpoint.

Despite the drastic diﬀerences in the environments, products, outcomes, or agents, the

presence of a split-the-diﬀerence pattern is clear in all seven settings. To our knowledge,

the widespread nature of this split-the-diﬀerence pattern in negotiations in the ﬁeld has not

previously been documented. Unlike laboratory experiments, where prior research shows

that players split a commonly known pie, in our settings, no player knows the true valuation

of her opponent. A challenge is then to explain why, in real-world negotiations with private

information, players propose oﬀers that split the diﬀerence between endogenous oﬀers.

Another common mass point in these histograms is at zero, representing cases where a player does not

budge at all. In the housing dataset, a mass point at 1 is also common, representing that an agent fully

concedes to the seller’s list price.

This can also be seen in the “split rate” row in Table 2, discussed in the following section.

Figure 1: Distribution of Concession Weights γ

j,t

Panel A: Used Car Bargaining Panel B: Pre-trial Settlement Bargaining

0 .25 .5 .75 1

j,t

33,356 observations

.02

.04

.06

.08

0 .25 .5 .75 1

j,t

208,463 observations

Panel C: Street Bargaining in a TV Show Panel D: Auto Rickshaw Rides Bargaining

0 .25 .5 .75 1

j,t

714 observations

.05

.15

.25

0 .25 .5 .75 1

j,t

2,986 observations

Panel E: Bargaining Over Housing Panel F: Trade Tariﬀ Bargaining

0 .25 .5 .75 1

j,t

176 observations

.05

.15

0 .25 .5 .75 1

j,t

46,985 observations

Panel G: eBay Best Oﬀer Bargaining

.05

.15

0 .25 .5 .75 1

j,t

9,789,903 observations

Notes: Each panel shows a histogram of γ

j,t

in a given data setting, as deﬁned in Section 3.

4 Split-the-Diﬀerence Oﬀers as a Norm

We now present evidence that the split-the-diﬀerence pattern that we document in our

empirical settings constitutes a social norm. Following Fehr and Gächter (2000), we deﬁne

a social norm as a behavioral pattern that (i) relies on a socially shared belief about

what constitutes appropriate behavior; and (ii) triggers the enforcement of the prescribed

behavior by informal social sanctions. This characterization of split-the-diﬀerence as a social

norm is echoed by veteran business and hostage negotiator Chris Voss, who writes, “The

traditional negotiating logic that’s drilled into us from an early age, the kind that exalts

compromise, says ‘Let’s just split the diﬀerence...Then everybody’s happy”’ (Voss and Tahl,

2016, p.115). Below we show results suggesting that agents that make split-the-diﬀerence

oﬀers are systematically rewarded by the opposing party (or, conversely, that the failure

to make a split-the-diﬀerence oﬀer does not yield this reward), conﬁrming the view that

these oﬀers, as part of a social norm, are enforced by informal rewards. These rewards

come in the form of higher probabilities of acceptance and higher probability of subsequent

split-the-diﬀerence oﬀers within the same bargaining sequence. Buyers who do not comply

with the norm experience a higher probability of their opponent exiting the negotiations,

which could impose ﬁnancial or time costs, depending on the setting.

We also show that the two most recent oﬀers are special: oﬀers that split the diﬀerence

between other anchor points as less prevalent. These other anchor points include earlier

oﬀers in the game other than the two most recent oﬀers or privately known reference points

(observable only to one bargaining party and not the other), which we observe in some of

our settings. This latter results suggests that agents gravitate to a behavior that is easily

veriﬁable by all parties, consistent with the idea that split-the-diﬀerence oﬀers are subject

to social enforcement. Additionally, we show that an agent’s oﬀer is more likely to split the

diﬀerence if either the previous oﬀer by the opponent or that agent’s own previous oﬀer

was a split oﬀer. Split-the-diﬀerence oﬀers are also more likely to occur if the opponent’s

previous oﬀer was arguably reasonable. For example, a seller is more likely to propose a

split-the-diﬀerence oﬀer at the third round of the game if the buyer’s oﬀer at round 2 was

not well below the seller’s initial oﬀer (i.e., the buyer was not low-balling the seller).

4.1

Split-the-Diﬀerence Oﬀers Are More Likely to be Accepted, Less

Likely to Cause Exit

In this section, we explore how agents respond when they receive split-the-diﬀerence oﬀers

compared to when they do not. We show that split oﬀers are discontinuously more likely

to be accepted than non-split oﬀers, suggesting that splitting the diﬀerence is not merely

a heuristic used by the oﬀering party, but rather that it is viewed as preferable (arguably

fair) behavior by both the oﬀerer and the receiver. Consistent with this interpretation,

after agents make split oﬀers, their opponents are less likely to abandon negotiations. This

suggests that the choice to exit a bargaining interaction may be motivated not just by the

lack of expected surplus, but also by the perception that the opponent is not being “fair.”

Speciﬁcally, we examine how a player’s choice of oﬀer, as measured by the concession

weight,

j,t

, relates to the probability that the oﬀer is accepted or leads to a breakdown

of negotiations. We create a measure for whether the oﬀer is a “split” oﬀer by creating

an indicator

Split

j,t

that is equal to one if

j,t

is equal to 0.5 (after being rounded to the

nearest hundredth, or

j,t

∈

495

505]) for each

t ≥

3. We then estimate the following

linear probability regressions:

j,t

= βSplit

j,t

+ f (γ

j,t

) + τ

+ ϵ

j,t

(3)

Exit

j,t

= βSplit

j,t

+ f (γ

j,t

) + τ

+ ϵ

j,t

, (4)

where

j,t

is an indicator for whether the oﬀer is accepted,

Exit

j,t

is an indicator that

the opponent exits before period

+ 1,

is a round ﬁxed eﬀect, and

(

j,t

) is a ﬂexible

function of

j,t

. We specify

(

j,t

) as a third-order polynomial of

j,t

The results are

reported in Table 2, where each column corresponds to one dataset. We also report the

frequency of acceptance, exit, and split oﬀers. The acceptance rate varies across settings

from 7% to 73%, the exit rates are between 3% and 86%, and the fraction of split oﬀers

ranges from 4% to 19%.

In Panel A, we see a positive coeﬃcient before the “split” oﬀer indicator in all of our

datasets, and this is statistically signiﬁcant in all columns except the cases of auto rickshaw

rides and housing (columns 4 and 5). This means that an oﬀer in bargaining is more likely to

The results are not sensitive to this choice; we ﬁnd similar results with second-, fourth-, or ﬁfth-order

polynomial approximations. We also ﬁnd similar results deﬁning “split” oﬀers using other bandwidths,

including 0.01 (i.e., γ

j,t

∈ [0.49, 0.51]) and 0.05 (i.e., γ

j,t

∈ [0.45, 0.55]).

Table 2: Acceptance and Exit Following Split Oﬀers

(1) (2) (3) (4) (5) (6) (7)

Cars Settlement TV Show Rides Housing Trade eBay

Panel A: Acceptance

Split 0.120

∗∗∗

0.219

∗∗∗

0.151

∗∗

0.0569 0.158 0.0676

∗∗∗

0.0849

∗∗∗

(0.00830) (0.00555) (0.0615) (0.0354) (0.119) (0.00341) (0.000484)

Panel B: Opponent Exit

Split -0.0402

∗∗∗

-0.00977 -0.103

∗∗∗

-0.0511

∗∗∗

-0.0198

∗∗∗

(0.00577) (0.0122) (0.0392) (0.00453) (0.000504)

N 33356 208463 714 3010 176 46985 9789903

Order of γ

j,t

3 3 3 3 3 3 3

Round FE Yes Yes Yes Yes Yes Yes Yes

Accept rate 0.38 0.34 0.26 0.20 0.73 0.07 0.20

Exit rate 0.27 0.03 0.41 0.86 0.51

Split rate 0.18 0.04 0.14 0.19 0.08 0.17 0.12

0.146 0.462 0.0295 0.160 0.0336 0.110 0.113

Notes: Table shows the estimated coeﬃcient on the split indicator from the regressions described by equations (3) and

(4). Each column corresponds to a separate data setting. The accept and exit rates are the means of the dependent

variables, and the split rate is the mean of the split indicator. Standard errors are shown in parentheses. The number

of observations for the rides setting diﬀers from that in Table 1 because some scripted bargaining acceptance must be

dropped and some scripted bargaining oﬀers can be included here. The number of observations for the settlement

setting diﬀers from that in Table 1 because some sequences end at round 10, and we have no data on whether round

10 oﬀers were accepted. See Appendix B for details. ∗ : p < 0.10, **: p < 0.05, and ***: p < 0.01

be accepted if it is a split oﬀer than if it is not. This eﬀect is surprisingly large in magnitude,

varying from 5.7% to 21.9%. Conversely, in Panel B, “split” oﬀers are negatively associated

with the opponent exiting the bargaining interaction. While sizes of the eﬀect on exit are

usually smaller than those on acceptance, they are signiﬁcant in all cases except for the

TV show, where exit is rare. The structure of the data on legal settlements and housing

(columns 2 and 5) do not allow us to examine exit as an outcome in these contexts.

Our result in the housing setting is likely insigniﬁcant due to the small sample size (176

oﬀer triples). One possible explanation for the lack of a signiﬁcant eﬀect on acceptance in

the setting of auto rickshaw rides is, this speciﬁcation compares the acceptance rate of a

split oﬀer with nearby oﬀers and, as shown Figure 1, the nearby oﬀers in this dataset are

quite sparse. This is also reﬂected in the high rate of split oﬀers in the fourth column (19%).

Therefore, we likely lack the power to detect a signiﬁcant eﬀect in this dataset. This dataset,

however, has a unique subset—the scripted bargaining oﬀers that Keniston (2011) randomly

assigned—in which we can obtain estimates that are closer to a causal eﬀect of split oﬀers

on acceptance rates. As highlighted in Section 2.4, buyers in these scripted sequences make

oﬀers that are assigned by the experiment designer rather than arising endogenously. When

we estimate equation (3) in this subset of the data, shown in columns 4–5 of Appendix

Table A2, we ﬁnd positive point estimates, and a particularly large, positive eﬀect in those

sequences that begin with a buyer oﬀer. Appendix Table A3 presents the results of a similar

subsample analysis for equation (4).

In Figure 2, we oﬀer an even more ﬂexible approach to this question, plotting a weighted

local linear ﬁt of acceptance and

j,t

, using observations where

j,t

is not a split oﬀer. We

also plot in Figure 2 the average acceptance probability for observations that are split

oﬀers, along with the 95% conﬁdence bound around this mean. We ﬁnd that the underlying

relationship between the acceptance and

j,t

is monotonic in most regions, and split oﬀers

are substantially more likely to be accepted than nearby oﬀers with similar

j,t

in all datasets

except the settings of auto rickshaw rides (where we again lack power locally around 0.5)

and housing (where the number of observations is small). In the latter two datasets, the

point estimates are still higher at split oﬀers.

The striking implication overall is that, even across these widely varying ﬁeld settings,

split-the-diﬀerence oﬀers are more likely to be accepted than even a slightly more favorable

oﬀer. This suggests that a preference toward splitting the diﬀerence between the two most

recent oﬀers is a norm followed not only by the proposer of these oﬀers but also by the

receiver. These results are reminiscent of ﬁndings in a wide range of laboratory ultimatum

games (e.g., Roth et al. 1991), which show receivers frequently rejecting oﬀers of less than

half of the surplus, but accepting “fair” oﬀers of a 50-50 split of the surplus nearly 100% of

the time.

4.2 Split Oﬀers Are Frequently Followed by Split Oﬀers

In this section, we address two empirical questions. First, do split oﬀers in period

by one

party tend to be followed by the opponent proposing a split oﬀer in period

+ 1? Second, do

split oﬀers in period

tend to be followed by split oﬀers by the same party in period

+ 2?

Both of these points speak to the question of whether splitting the diﬀerence between the

two most recent oﬀers is a norm that is perhaps followed more consistently by some agents

than others and that, when invoked by one agent, tends to be adopted by the opponent.

To examine this question, we analyze the following linear probability model:

Split

j,t

= βSplit

j,t−1

+ ϵ

j,t

, (5)

Figure 2: Probability of an Oﬀer Being Accepted

Panel A: Used Car Bargaining Panel B: Pre-trial Settlement Bargaining

Acceptance Probability

0 .25 .5 .75 1

j,t

Acceptance Probability

0 .25 .5 .75 1

j,t

Panel C: Street Bargaining in a TV Show Panel D: Auto Rickshaw Rides Bargaining

Acceptance Probability

0 .25 .5 .75 1

j,t

Acceptance Probability

0 .25 .5 .75 1

j,t

Panel E: Bargaining Over Housing Panel F: Trade Tariﬀ Bargaining

Acceptance Probability

0 .25 .5 .75 1

j,t

Acceptance Probability

0 .25 .5 .75 1

j,t

Panel G: eBay Best Oﬀer Bargaining

Acceptance Probability

0 .25 .5 .75 1

j,t

Notes: Each panel shows a local linear ﬁt of the acceptance probability as a function of

j,t

along with the acceptance

probability of split oﬀers (plus the 95% conﬁdence interval around this point). The ﬁtted values are estimated using

locally weighted least squares with a tricube weighting function. To facilitate computation of the local linear estimator

in the eBay setting, we use a random sample of 100,000 threads.

where we regress the indicator of whether, in period

, an agent proposes a split oﬀer,

Split

j,t

on the indicator of whether the most recent oﬀer,

Split

j,t−1

(which naturally comes from

the opponent), also corresponds to a split oﬀer. Panel A in Table 3 shows the results.

observe positive point estimates in each setting, and these estimates are signiﬁcant in most

columns, suggesting that agents are more likely to propose a split oﬀer when the opponent

has just done the same.

In panel B, we consider a version of equation (5) in which we use

t −

2 actions on

the right-hand side rather than

t −

1, allowing us to examine whether agents who make

split-the-diﬀerence oﬀers earlier in the game are more likely to do so again. We ﬁnd evidence

of this eﬀect for the case of settlement and eBay negotiations and a marginally signiﬁcant

eﬀect in the case of auto rickshaw rides.

Table 3: Repeat Split-the-Diﬀerence Behavior

(1) (2) (3) (4) (5) (6)

Cars Settlement TV Show Rides Trade eBay

A: Eﬀect of Opponent’s Split Oﬀer

Split

j,t−1

0.00304 0.144

∗∗∗

0.117

∗∗

0.00358 0.0928

∗∗∗

0.125

∗∗∗

(0.00763) (0.00672) (0.0591) (0.0193) (0.0188) (0.000840)

N 11622 134107 510 2714 2937 2813127

B: Eﬀect of Agent’s Own Split Oﬀer

Split

j,t−2

0.0440 0.0807

∗∗∗

0.0203 0.0708

∗

-0.000644 0.0699

∗∗∗

(0.0276) (0.00991) (0.0724) (0.0382) (0.0519) (0.00137)

N 2155 83296 343 1121 250 1100490

Notes: Panel A shows the estimated coeﬃcient on the one-period-lagged split indicator from the regression described

by equation (5). Panel B shows results instead using the two-period-lagged split indicator. Each column corresponds

to a separate data setting. The housing data setting is omitted because no sequence contains more than three oﬀers.

∗ : p < 0.10, **: p < 0.05, and ***: p < 0.01

We examine this latter eﬀect further by exploring whether splitting the diﬀerence is a

norm followed by certain agents more than by others. To do so, we take advantage of agent

identiﬁers, which we observe in the used car, trade, and eBay bargaining settings.

In a

given dataset, we sort agents by the fraction of split oﬀers among all oﬀers they make. We

then plot, on the horizontal axis in each panel of Figure 3, the cumulative share of oﬀers

made by agents. The solid line corresponds to the cumulative share of oﬀers by these agents

We cannot examine this question in the housing dataset as we only observe a single oﬀer triple in each

sequence in that setting.

In our other data settings, we have no consistent means of tracking an agent across diﬀerent bargaining

sequences.

that are split oﬀers. If the propensity to propose split oﬀers is roughly equal across all

agents, and if each agent makes a large number of oﬀers, the solid line should be close to the

45-degree line (the dotted one). This comparison can thus be considered a modiﬁed Lorenz

curve that measures the “inequality” of split oﬀers among agents.

Figure 3: Some Agents More Likely to Make Split Oﬀers

Panel A: Used Car Bargaining Panel B: Trade Tariﬀ Bargaining

0 .2 .4 .6 .8 1

Cumulative Share of Split Offers

0 .2 .4 .6 .8 1

Cumulative Share of Offers

0 .2 .4 .6 .8 1

Cumulative Share of Split Offers

0 .2 .4 .6 .8 1

Cumulative Share of Offers

Panel C: eBay Best Oﬀer Bargaining

0 .2 .4 .6 .8 1

Cumulative Share of Split Offers

0 .2 .4 .6 .8 1

Cumulative Share of Offers

Notes: Each panel ranks agents by the fraction of split oﬀers among all oﬀers they make and plots their cumulative

share of total oﬀers on the

-axis and the cumulative share split oﬀers on the

-axis. The solid lines use the real data.

The dashed lines use simulated split indicators assuming every agent has the same propensity to propose split oﬀers.

The dotted line indicates the 45-degree line.

Figure 3 indeed shows a gap between the solid and dotted lines in each panel. In reality,

however, we only observe a few oﬀers made by a given agent, and hence a large part of the

area between the solid line and the 45-degree line is due to sampling noise.

To construct a

more meaningful benchmark, we consider a case where each agent has the same probability

to propose a split oﬀer, with this probability given by the split rate reported in Table 2.

For example, suppose every agent has the same propensity

to propose a split oﬀer, but each agent only

ever proposes one oﬀer in the data. We would observe roughly a fraction of

players proposing one split

oﬀer and a fraction of 1

− q

players proposing no split oﬀers, and the curve will be far oﬀ the 45-degree line.

Using this split rate, we simulate a fake split indicator for each observation following a

Bernoulli distribution. We plot our cumulative share of split oﬀers based on these fake

indicators using the dashed line in Figure 3, which should lie between the 45-degree line and

the curve plotted using the real data.

Comparing the dashed and solid lines in Figure 3, we ﬁnd evidence that some agents

have a stronger proclivity toward split-the-diﬀerence than others. This is particularly the

case in the eBay and trade settings, where the dashed line is farther from the solid line. In

the used-car setting, the solid and dashed lines are close, indicating that the propensity to

make split-the-diﬀerence oﬀers is roughly uniform across agents.

4.3 The Two Most Recent Oﬀers Are Special

Our analysis in Section 3 demonstrates robust evidence across a wide spectrum of settings

that a modal strategy in real-world bargaining is to make oﬀers that split the diﬀerence

between the two most recent oﬀers. Here, we explore whether it is indeed the two most

recent oﬀers that serve as the most prominent anchor points that players tend to split in

half, or whether other anchor points are equally common. For example, it is possible that

oﬀers splitting the diﬀerence between earlier oﬀers in the game (prior to the two most recent

oﬀers) are also common. For example, in a sequence

{

100

}

, a subsequent oﬀer

splitting the diﬀerence between the two most recent oﬀers would be 80, but it may be that

75 (which splits the diﬀerence between the ﬁrst two oﬀers) is also a focal point for players in

this game.

To examine this possibility, we deﬁne placebo concession by treating the proposed amount

in round

as a convex combination of oﬀers from earlier rounds of the game. For example,

for

= 4, we can treat the proposed amount as a convex combination of oﬀers from the ﬁrst

and second rounds, which we deﬁne as

j,4

− p

j,1

j,2

− p

j,1

. In general, for

t ≥

4 and

s < t

, the

placebo concession is deﬁned as follows:

j,t

− p

j,s−2

j,s−1

− p

j,s−2

, t ≥ 4, 3 ≤ s < t, (6)

where

is the round in which the current oﬀer is evaluated, and

s < t

is the round in which

oﬀers from rounds

s −

1 and

s −

2 actually were the two most recent oﬀers. If negotiating

agents care most about fairness as deﬁned by an equal split of the two most recent oﬀers, we

expect less mass at 0.5 for the placebo concession than in our main results from Section 3.

For this analysis, we focus on the three datasets for which we have the longest sequences,

as they allow us to construct the placebo concession metric: used car bargaining, pre-

trial settlement bargaining, and eBay bargaining. Figure 4 plots the distribution of true

concession and placebo concession in these three datasets. In the left column, we replicate

the histograms of true concession weights from Figure 1. In the middle column, we plot

histograms of the placebo concession weights based on bargaining oﬀers in earlier rounds,

j,t

, t ≥ 4, 3 ≤ s < t, as deﬁned in equation (6).

Figure 4: Distribution of True Concession and Placebo Concession

True Placebo Placebo, Restricted

Panel A: Used Car Bargaining

0 .25 .5 .75 1

j,t

0 .25 .5 .75 1

j,t

0 .25 .5 .75 1

j,t

Panel B: Pre-trial Settlement Bargaining

.03

.06

.09

0 .25 .5 .75 1

j,t

.03

.06

.09

0 .25 .5 .75 1

j,t

.03

.06

.09

0 .25 .5 .75 1

j,t

Panel C: eBay Best Oﬀer Bargaining

.05

.15

0 .25 .5 .75 1

j,t

.05

.15

0 .25 .5 .75 1

j,t

.05

.15

0 .25 .5 .75 1

j,t

Notes: Figure shows, in left plots, histograms of true concession weights (γ

j,t

) as in Figure 1 for the cars (panel A),

settlement (panel B), and eBay (panel C) settings. In the middle plots, we show histograms of the placebo concession

weights. In the right plots, we show histograms of the placebo concession weights restricting the sample to exclude

observations that are mechanically equal to 0.5.

In the third column of Figure 4, we focus on a restricted sample in which we drop cases

that can mechanically lead to mass points at 0.5 even in the placebo concession weights. As

an example, consider an oﬀer sequence with the ﬁrst four oﬀers being

{

100

}

. A

split-the-diﬀerence oﬀer at the ﬁfth round would be 80, but this oﬀer would also represent

splitting the diﬀerence between the earlier oﬀers of 100 and 60. Our restricted sample

excludes placebo concession weights that are exactly equal to the true concession weights

for a given round.

Panels A and C of Figure 4 demonstrate that counteroﬀers occur halfway between earlier

oﬀers of the game (the middle and right columns) less frequently than between the two

most recent oﬀers (the left column). In panel B, the pre-trial settlement bargaining, placebo

split-the-diﬀerence oﬀers are also frequent. However, in both the middle and right ﬁgures in

panel B, the mass is more uniformly distributed across concession weights in the placebo

cases than in the left column, suggesting that the relative likelihood of split-the-diﬀerence

oﬀers is the highest when considering the two most recent oﬀers.

To formally test whether the masses at 0.5 are diﬀerent in the true vs. placebo concession

weights, we calculate the fraction of split-the-diﬀerence oﬀers (as in Section 4.1) for the true

concession weights, placebo concession weights, and their diﬀerences, as well as standard

errors on each of these.

We also compute the fraction of split oﬀers among placebo

concession weights in the restricted sample. Table 4 presents the results. In each dataset,

we detect a (statistically signiﬁcantly) larger fraction of split oﬀers using the true concession

weights than the placebo concession weights, suggesting that players indeed rely more

strongly on the two most recent oﬀers than on earlier oﬀers in determining a 50-50 split.

4.4 Splitting Based on Private Information

We also analyze whether agents tend to propose oﬀers that split the diﬀerence between a

publicly known threshold (a previous oﬀer) and a privately known quantity that relates to

agents’ values. For this analysis, we exploit a number of privately known variables speciﬁc

to several of our settings, including the secret reserve price known only to the seller in the

used-car setting; the reserve/loss estimate known only to the “buyer” (the insurer) in the

settlement negotiation setting; and the auto-accept or auto-decline prices that are known

only to the seller in the eBay setting. The importance of these variables is that they are

Standard errors on the diﬀerence are constructed from 100 nonparametric bootstrap-sample estimates of

the true and placebo rates, sampling at the thread level.

Table 4: Fraction of Split Oﬀers in True vs. Placebo Concession Weights

True Placebo Diﬀerence Placebo Diﬀerence

Restricted

Panel A: Used Car Bargaining

Split 0.1823 0.0503 0.1319 0.0614 0.1208

(0.0019) (0.0020) (0.0023) (0.0033) (0.0034)

N 33,356 14,721 6,674

Panel B: Pre-trial Settlement Bargaining

Split 0.0435 0.0218 0.0217 0.0190 0.0245

(0.0004) (0.0003) (0.0005) (0.0003) (0.0005)

N 208,463 313,216 292,470

Panel C: eBay Best Oﬀer Bargaining

Split 0.1176 0.0437 0.0739 0.0424 0.0753

(0.0001) (0.0001) (0.0001) (0.0001) (0.0001)

N 9,789,903 4,353,489 3,880,706

Notes: Table shows the fraction of split oﬀers among the true concession weights (ﬁrst column) vs. the placebo

concession weights (second column), as well as the diﬀerence of these means (third column). The fourth column shows

the fraction of split oﬀers among the placebo concession weights in the restricted sample, and the ﬁfth column shows

the diﬀerence between this mean and the true concession weight fraction. Standard errors are shown in parentheses.

Panel A shows this analysis for the cars sample, panel B for the pre-trial settlement sample, and panel C for the eBay

sample.

each privately known to only one side; thus, these variables allow us to examine whether

agents’ behavior is more consistent with equally splitting some quantity that is based on the

public information contained in agents’ previous oﬀers or whether instead agents appear to

oﬀer prices that constitute an equal split relative to their own private information.

For this analysis, we follow our construction of placebo concession weights from Section

4.3, replacing some oﬀer information with these privately known quantities. We describe

this analysis and results in detail in Appendix C. The bulk of the evidence from this exercise

suggests that split-the-diﬀerence behavior is less related to such privately known values and

more related to the two most recent oﬀers.

4.5 When are Split-the-Diﬀerence Oﬀers Made?

It is possible that the set of cases in which split-the-diﬀerence oﬀers might occur is constrained,

either because the mid-point between the two previous oﬀers might be lower than the seller’s

true willingness to sell or higher than the buyer’s willingness to pay, or because a very

demanding oﬀer from the opponent might be seen as “unfair” (a buyer low-balling the seller,

for example) and thus not deserving of a “good faith” split-the-diﬀerence counteroﬀer. Both

of these arguments suggest that when a buyer’s oﬀer is only a small fraction of the seller’s

initial oﬀer, we are unlikely to observe a subsequent split-the-diﬀerence oﬀer, with the same

being true for relatively high seller oﬀers. Conversely, when the diﬀerence between previous

oﬀers is very small—for instance, within a few percentage points—the size of potential

surplus may be small, and fairness concerns might be less relevant.

Figure 5 examines these patterns, showing the probability that an agent makes a split

oﬀer in round

= 3 as a function of the ratio of the ﬁrst two oﬀers. Regardless of the order

of the moves (that is, whether the seller or the buyer make the ﬁrst oﬀer in the thread), this

is the ratio of the buyer’s ﬁrst oﬀer to the seller’s ﬁrst oﬀer; thus, the ratio is always between

zero and one. We cannot examine this in the trade setting, where the ﬁrst oﬀer is zero.

In all six datasets in Figure 5, we ﬁnd that the probability of a split oﬀer is lower when

the buyer’s oﬀer is a very small fraction of the seller’s (that is, when the gap is relatively

large), and then increases initially. When the buyer’s and seller’s previous oﬀers are relatively

close, the chance of a split counteroﬀer decreases again in the TV, rides, and housing cases.

In these cases, split oﬀers are most likely when the ratio of previous oﬀers lies in the 60%-90%

range. For these values, where the opponent’s oﬀer is not too “unreasonable,” but the gains

from bargaining may still be substantial, agents are most likely to adopt the fairness norm.

For the other three settings—cars, settlement, and eBay, which also correspond to the cases

where we have the most data—we see the frequency of split oﬀers increase globally as the

ratio increases. Interestingly, these results are related to the ﬁndings by Karagözoğlu and

Riedl (2015) that, in the context of unstructured bargaining lab experiments, the diﬀerence

between the ﬁrst two oﬀers in a negotiation (i.e., tension or conﬂict in opening oﬀers) is a

very good predictor of bargaining duration and the probability of disagreement.