qjaa003

SEQUENTIAL BARGAINING IN THE FIELD: EVIDENCE

FROM MILLIONS OF ONLINE BARGAINING

INTERACTIONS

∗

MATTHEW BACKUS

THOMAS BLAKE

BRAD LARSEN

STEVEN TADELIS

We study patterns of behavior in bilateral bargaining situations using a rich

new data set describing back-and-forth sequential bargaining occurring in over

25 million listings from eBay’s Best Offer platform. We compare observed behav-

ior to predictions from the large theoretical bargaining literature. One-third of

bargaining interactions end in immediate agreement, as predicted by complete-

information models. The majority of sequences play out differently, ending in dis-

agreement or delayed agreement, which have been rationalized by incomplete

information models. We ﬁnd that stronger bargaining power and better outside

options improve agents’ outcomes. Robust empirical ﬁndings that existing mod-

els cannot rationalize include reciprocal (and gradual) concession behavior and

delayed disagreement. Another robust pattern at odds with existing theory is

that players exhibit a preference for making and accepting offers that split the

difference between the two most recent offers. These observations suggest that

behavioral norms, which are neither incorporated nor explained by existing the-

ories, play an important role in the success of bargaining outcomes. JEL Codes:

C78, D82, D83, M21.

I. INTRODUCTION

Bilateral bargaining is one of the oldest and most common

forms of trade. Nations negotiate trade deals, arms control, and

climate change mitigation; legislators engage in horse-trading

to build coalitions and pass legislation; business people haggle

over contracts from corporate acquisitions to labor agreements;

lawyers wrangle settlements both civil and criminal, and private

individuals bargain over wages, real estate, and the allocation of

∗

We are grateful to eBay for allowing us to make the data used herein acces-

sible and publicly available. We thank Luis Cabral, Peter Cramton, John Horton,

Emin Kara

gozo

glu, Saul Lach, Axel Ockenfels, and Jean Tirole for thoughtful

comments. We thank Brenden Eum, Yuyang Jiang, Ziao Ju, Rebecca Li, Carol Lu,

Sharon Shiao, Caio Waisman, and Chuan Yu for outstanding research assistance.

Part of this research was supported by NSF Grant SES-1629060.

⃝

The Author(s) 2020. Published by Oxford University Press on behalf of the Presi-

journals.permissions@oup.com

The Quarterly Journal of Economics (2020), 1–43. doi:10.1093/qje/qjaa003.

2 THE QUARTERLY JOURNAL OF ECONOMICS

household chores. Bargaining determines the allocation of surplus

in these settings, as well as the likelihood of breakdown—the lat-

ter with real economic and human costs. Therefore, understand-

ing how people bargain, and the institutions, norms, and practices

that affect bargaining outcomes, is of ﬁrst-order importance.

Over the past 60 years, a large body of literature in economics

has examined various aspects of bargaining in theory and in

laboratory experiments. The theoretical literature typically

assumes a particular information structure and extensive form of

the game, which can sometimes be implemented in a controlled

laboratory setting. Bargaining in real-world settings, however,

tends to be less structured, and as a consequence, little evidence

has been presented about how people bargain in the ﬁeld and

how prices actually form in real-world negotiations. Indeed,

Fudenberg, Levine, and Tirole (1985) explained that the “thorny

issue” arising in much of the bargaining literature is that the

researcher does not actually know the extensive form of real-

world, often unstructured bargaining scenarios. For example, a

street vendor bargaining over price might state an offer, watch

the reaction of the buyer, and immediately state a lower price

without waiting for a spoken response. It is unclear whether this

situation should be modeled with alternating offers, one-sided

offers, a concession game, or some other structure.

The advent of online marketplaces provides new opportuni-

ties to study negotiations in a real-world setting, where the exten-

sive form of the game is similar to those studied in the theoretical

and experimental literature, and where the data collected are on

a massive scale. In this article, we use data from over 88 million

listings on the eBay.com Best Offer platform, where sellers offer

items at a listed price and invite buyers to engage in alternating,

sequential-offer bargaining, very much in the spirit of Rubinstein

(1982).Inalargefractionoftheselistings,buyerschosetomake

an offer, initiating the alternating-offer game, resulting in over

25 million bargaining sequences. In this s etting, we document a

variety of facts on how bargaining proceeds and how prices form,

and we ﬁnd evidence consistent with the most salient predictions

of economic theory. At the same time, we document robust pat-

terns that suggest that behavioral factors based on reciprocal and

equitable norms play a signiﬁcant role in bargaining outcomes.

Although more widely known for its auctions and ﬁxed-price

listings, eBay has offered sales through alternating-offer bargain-

ing for over a decade. Our data come from eBay’s Best Offer

SEQUENTIAL BARGAINING IN THE FIELD 3

platform, through which almost 10% of eBay transaction volume

occurs as buyer-seller pairs engage in alternating-offer bargain-

ing. Given the sheer volume of trade on eBay and the simple

extensive form of the game, the Best Offer platform provides a

useful setting for studying the determination of agreed-on prices

in sequential bargaining situations. The bargaining in this setting

is only over a single dimension (price), making it more straight-

forward to analyze than many other bargaining settings (such

as procurement contracts; Bajari, McMillan, and Tadelis 2009),

while still yielding the beneﬁt of being a real-world setting. Fur-

thermore, the data allow us to link buyers and sellers over time.

Our data set is, to our knowledge, the largest offer-level negotia-

tions data set to be analyzed in the literature.

Section II describes background on the Best Offer platform

and introduces our data set. Section III then documents how

patterns observed in the data relate to a variety of game-theoretic

models of bargaining. We provide a breakdown of how bargaining

sequences unfold in practice and the frequency with which

different responses and outcomes occur. We ﬁnd that there are

often few back-and-forth offers in a given bargaining pair, which

is predicted by complete-information, common-priors models of

bargaining, such as the classical Rubinstein (1982) model. Bar-

gaining also frequently ends in disagreement early on, consistent

with the incomplete information model of Perry (1986).Some

interactions involve a delay and end in agreement, consistent

with other models of incomplete information (Rubinstein 1985;

Grossman and Perry 1986; Gul and Sonnenschein 1988). How-

ever, a number of sequences end in disagreement after a delay,

afeaturethatisabsentinnearlyallexistingbargainingmodels,

with Cramton (1992) being the only exception we are aware of.

In Section IV we examine several conventional drivers of

bargaining outcomes. Bargaining differs when players bargain

over expensive versus inexpensive products in a way that is

consistent with ﬁxed costs of bargaining playing a role, as op-

posed to the more commonly modeled discount-factor approach.

We examine several forms of bargaining strength. We ﬁnd that

1. In cooperation with eBay, we anonymized the data set and made

it publicly available for research purposes. The data can be accessed at

http://www.nber.org/data/bargaining.html or by contacting the authors. We hope

that it will further fuel the recent surge of empirical work studying bargain-

ing in economics and stimulate additional work in the area, both empirical and

theoretical.

4 THE QUARTERLY JOURNAL OF ECONOMICS

buyers who are more experienced in bargaining on this platform

(as measured by the number of previous Best Offer negotiations

the buyer has participated in) also tend to achieve lower ﬁnal

prices, and experienced sellers achieve higher ﬁnal prices. These

results are consistent with common models of bargaining in which

aplayer’sbargainingpoweraffectsoutcomes(Rubinstein 1982,

1985; Watson 1998); they are also consistent with laboratory ev-

idence (Rapoport, Erev, and Zwick 1995)andsurveydata(Scott

Morton, Silva-Risso, and Zettelmeyer 2011), but to our knowledge

have not been previously conﬁrmed with data from actual bargain-

ing outcomes.

We also ﬁnd evidence that bargaining games in

which multiple buyers negotiate with the same seller—improving

the seller’s outside option with each buyer—yield higher prices for

the seller, and vice versa. Furthermore, we document that listings

containing more photos (a measure of reduced severity of adverse

selection) tend to more quickly receive buyer offers and tend to

yield higher negotiated prices.

In Section V we document some patterns unexplained by ex-

isting bargaining theories, which exhibit ﬂavors of reciprocity.

First, we ﬁnd that bargaining offers tend to change gradually over

the course of the bargaining interaction. This is in contrast to a

number of existing bargaining models, in which any delay in bar-

gaining is war-of-attrition-like delay: no information is revealed

until one party concedes everything, completely revealing her val-

uation to the opponent. In contrast, the behavior we observe is

consistent with a gradual revelation of information and a gradual

concession of bargaining positions. We demonstrate furthermore

that this gradualism is reciprocal: the opponent responds to stub-

bornness by not conceding and to concession by reciprocating with

more concession.

We show that players often make offers lying halfway

between their previous offer and their opponent’s current offer.

We further demonstrate that such “split-the-difference” offers

have a higher likelihood of being accepted—higher even than

some offers that would be even more favorable in money terms

for the accepting party. Such behavior is not consistent with any

existing theory of rational behavior, but it may be consistent with

2. One study documenting causal evidence on delay in bargaining from ﬁeld

data (but not actual offer data) is Ambrus, Chaney, and Salitsky (2018), studying

delay induced by travel times of Spanish ransom teams negotiating with North

African pirates in the 1600s.

SEQUENTIAL BARGAINING IN THE FIELD 5

behavioral norms; similar split-the-difference behavior is dis-

cussed throughout the experimental and theoretical behavioral

bargaining literature, in which market participants may care

about fairness and often favor a s plit-the-difference strategy (Roth

and Malouf 1979; Roth 1985; Binmore, Shaked, and Sutton 1985;

Bolton 1991; Bolton and Ockenfels 2000; Charness and Rabin

2002; Andreoni and Bernheim 2009). A distinction in the setting

we study is that here players are splitting the difference between

two previous price offers, not necessarily a known surplus.

Our article is related to a growing literature studying ne-

gotiated price settings. Many such papers have only data on ﬁ-

nal negotiated prices and only for cases in which trade occurs,

such as in Crawford and Yurukoglu (2012).Incontrast,weob-

serve all back-and-forth offers, even for bargaining interactions

that failed to reach an agreement. Similar data sets, although

smaller than ours in size and scope, are analyzed in Keniston

(2011), Bagwell, Staiger, and Yurukoglu (2017), Hernandez-

Arenaz and Iriberri (2018), Larsen and Zhang (2018), Larsen

(2019),andBackus, Blake, and Tadelis (2019a, 2019b).Several

papers provide tests of implications of bargaining theory, as ours

does, such as Scott Morton, Silva-Risso, and Zettelmeyer (2011)

and Grennan and Swanson (2019).

The large scale of our data and the variation across several

measures of heterogeneity help paint a useful and comprehen-

sive picture of sequential bargaining in the real world that adds

great detail to the existing literature. The patterns we uncover

conﬁrm some of the most basic insights of bargaining theory, yet

reveal behaviors that are not explained by conventional theoret-

ical approaches. As such, the patterns we uncover suggest that

further developments in the theory of bargaining are warranted,

especially those that more seriously consider aspects of reciprocity

and fairness. In turn, such new theoretical insights can shed light

on how to better design platforms that increase gains from trade.

II.

EBAY’S BEST OFFER MECHANISM:FACTS AND DATA

eBay is one of the world’s largest online marketplaces

for consumer-to-consumer transactions. It began in 1995 using

second-price-like auctions as the sole format for transacting on

its platform. The site eventually allowed users the option of

selling goods through a single posted ﬁxed price. In 2005, the

site began to allow sellers to sell through an alternating-offer

6 THE QUARTERLY JOURNAL OF ECONOMICS

FIGURE I

Growth of Best Offer

This ﬁgure depicts the percentage of gross market value made up by three

mechanisms on the eBay platform—Best Offer, auctions, and ﬁxed-price listings—

from 2005 to 2016, computed from internal eBay data. The calculation for ﬁxed-

price listings includes Best Offer sales. The calculation for Best Offer includes only

listings that were bargained; it does not include Best-Offer-enabled listings that

sold at the listing price.

protocol referred to as Best Offer. This feature can be enabled (at

no cost) by the seller at the creation of the listing and is only avail-

able for ﬁxed-price listings—there is no equivalent mechanism for

auctions.

The Best Offer platform is currently a fast-growing sales for-

mat on eBay. Figure I shows the growth of this format relative to

auctions and ﬁxed-price listings over the past 10 years. In 2005,

when the format was ﬁrst rolled out, only a tiny fraction of list-

ings were Best Offer listings, less than 1% of all eBay transactions

occurred through a buyer actually placing an offer (rather than

accepting the Buy It Now price). By 2012, that fraction had grown

to just under 9%.

Goods offered for sale under the Best Offer format are listed

as “accepts Best Offer” in eBay search results.

Throughout, we

refer to these postings as Best Offer listings. A buyer viewing a

Best Offer listing sees similar information to a buyer viewing a

ﬁxed-price listing (referred to as a Buy-It-Now listing), including

3. Potential buyers may ﬁlter search results to display only those listings that

accept offers.

SEQUENTIAL BARGAINING IN THE FIELD 7

FIGURE II

Best Offer User Interface

This ﬁgure depicts the “view item” page for a listing with Best Offer enabled.

The potential buyer may click on “Buy It Now” to purchase the painting at the

listed price of $746.40—or they may click on “Make Offer” and be prompted to

propose a price.

the listing title, seller ID and feedback score, at least one picture

of the item, and any other information about the item that the

seller decides to display.

The buyer sees the Buy-It-Now price,

as in a standard ﬁxed-price listing, but also sees an additional

option, a button labeled “Make Offer,” as illustrated in Figure II.

Selecting the Make Offer button allows the buyer to send an offer

to the seller. As such, we treat the Buy-It-Now price (equivalently,

“listing price”) as the seller’s ﬁrst offer to any buyer who wished

to bargain.

Upon receiving this offer, the seller may accept the offer,

make a counteroffer, or decline the offer (without making a

4. Throughout, we use the term “buyer” to refer to the user interested in

potentially buying the item whether or not the transaction actually occurs.

5. The observant reader will notice the “Add a message to seller” option in

Figure II.Thedatasetaccompanyingthisarticleincludesadummyvariablefor

whether a message was sent along with an offer, but does not include the text

of that message for risk of deanonymizing users. In Backus, Blake, and Tadelis

(2019a),theauthorsidentifyanaturalexperimentintheavailabilityofthistext

communication (on the German version of eBay’s Best Offer mechanism). They

ﬁnd that communication substantially improves the probability that a bargaining

interaction leads to a transaction.

8 THE QUARTERLY JOURNAL OF ECONOMICS

counteroffer).

If the seller makes a counteroffer, the buyer can

accept, decline, or counter in response. Play continues until either

party accepts or until the buyer declines. If the seller declines,

the buyer may still respond with a counteroffer or can, at any

time, purchase at the Buy-It-Now price. Each party is limited to

three offers (not including the listing price), and each offer expires

48 hours after being placed.

We refer to a sequence of back-and-

forth offers—a given buyer and s eller pair bargaining over a given

item—as a thread.

To form our primary data set, we obtain internal eBay data

from all Best-Offer-enabled, single-unit listings created in May 31,

2012–June 1, 2013 from the US eBay site. This consists of over 90

million Best-Offer-enabled listings. This data set, anonymized to

remove all identiﬁable information, constitutes the data set that

we have arranged to have released publicly for research purposes.

For the primary analysis in this article, we restrict attention to

listings with Buy-It-Now prices between $0.99 and $1,000.00, and

eliminate listings with apparent data errors (e.g., cases where we

could not locate the original offer corresponding to a counterof-

fer). Details on our sample construction criteria appear in Online

Appendix B.

Our ﬁnal data set analyzed in this article contains

approximately 88.4 million listings. Of these, 18 million receive

an offer, in some cases multiple offers, generating 25.5 million

bargaining threads (deﬁned as a listing-buyer pair). These bar-

gaining threads involve 1.2 million sellers and 4.7 million buyers.

Table I presents descriptive statistics for our sample. The top

panel contains statistics at the level of the listing. The average list

price (Buy-It-Now) is $95, and the average sale price is 83% of the

list price. Sales include Buy-It-Now choices as well—conditional

on bargaining occurring, the average sale price comes down to 73%

of the list price. We note that almost 80% of listings never receive

an offer. Of the listings, 54.8% are for used goods, and 26.3% have

6. As a time-saving device for sellers, the platform offers sellers the option

to specify an “auto-accept” price, which is unobserved to buyers and which, if

exceeded by the buyer’s offer, will result in the platform accepting the offer on

behalf of the seller. Sellers can similarly specify an “auto-decline” price.

7. In 2017 this maximum limit was changed to ﬁve offers. In our sample,

approximately 1.1 percent of interactions reach the binding limit, and most of

these fail; see Figure III later. The limit was extended in an attempt to encourage

these negotiators to succeed. The time frame of our data does not permit us to

evaluate this policy change.

8. This and all other appendix material is found in the Online Appendix.

SEQUENTIAL BARGAINING IN THE FIELD 9

TABLE I

DESCRIPTIVE STATISTICS FOR THE MAIN SAMPLE

Mean Std. dev. Min Median Max

Listing-level data

Listing price 94.6 164 0.01 30 1,000

Used 0.548 0.498 0 1 1

Revised 0.263 0.44 0 0 1

Sold 0.215 0.411 0 0 1

Sold by Best Offer 0.132 0.338 0 0 1

Received an offer 0.206 0.404 0 0 1

No. photos 2.69 2.68 0 1 12

Sale price 69.7 119 0.01 25 1,000

Sale price / list price 0.832 0.175 0.00099 0.857 1

Bargained price 74.1 121 0.99 28 1,000

Bargained price / list price 0.727 0.146 0.00099 0.75 1

No. listings 88,386,471

Seller-level data

Feedback positive percent 99.4 5.3 0 100 100

No. listings 73.8 1,941 1 3 1,084,709

No. sales 15.9 158 0 1 66,977

No. sales by Best Offer 9.71 101 0 1 56,473

No. sellers 1,197,397

Buyer-level data

No. bargaining threads 5.12 17.9 1 2 5,697

No. offers 8.48 30 1 3 7,823

No. purchases 3.21 9.27 1 1 4,095

No. bargained purchases 2.47 7.39 0 1 3,329

No. buyers 4,701,301

Thread-level data

No. offers 1.66 0.942 1 1 6

No. offers if sold 1.48 0.891 1 1 6

Agreement reached 0.454 0.498 0 0 1

Seller experience 3,883 18,350 1 450 385,419

Buyer experience 129 598 1 26 32,909

First buyer offer 86.6 126 0 35 1,000

First buyer offer / list price 0.608 0.193 0 0.626 1

No. threads 25,453,072

Notes. This table presents summary statistics for the main data set. Note that indicator “Used” (for used

versus new status of item) is missing for 27,678,157 listings, and the feedback variable in the seller panel is

missing for 51,992 sellers. See Online Appendix A for an extensive discussion of exclusion criteria.

the Buy-It-Now price revised at some point by the seller during

the listing life.

Table I also includes detailed information on market

participants. On average, s ellers have a 99.4% positive feedback

score. Although there are many one-time sellers, the market is

10 THE QUARTERLY JOURNAL OF ECONOMICS

skewed toward experienced sellers: the average number of listings

per seller is 74, 16 of which sell through Best Offer or Buy-It-Now

and 10 of which sell speciﬁcally through Best Offer. Most of the

sales in our data set are made by a relatively small fraction of the

sellers. The population of buyers is skewed, but less so: on aver-

age, buyers in our sample are observed in 5 bargaining threads,

make 8 offers, and purchase 3 items (2.5 of these coming through

bargaining). Finally, at the thread level, Table I shows that most

bargaining threads are short (only 1.66 offers, on average, where

the ﬁrst offer is always made by the buyer). Seller and buyer expe-

rience in this table are counts, at the t ime of the current thread, of

the number of bargaining threads including the current thread in

which the player has ever participated since the inception of the

Best Offer mechanism. The average thread includes a seller hav-

ing several thousand previous negotiations and buyer with over

100, again reﬂected a skew toward experienced players. On aver-

age, buyers offer $86.60, which represents 61% of the list price.

Bargaining is ultimately successful 45% of the time.

Table II contains means of several variables at the listing and

thread level for the main sample and for broad category subsam-

ples: collectibles, electronics, fashion, media, toys, business and

industrial, and other. Table II demonstrates that the majority of

listings are for items that could reasonably be characterized as id-

iosyncratic or one-of-a-kind inventory, such as collectibles or fash-

ion. Categories with more well-deﬁned, frequently sold products,

such as media products or electronics, make up a smaller fraction

of the data. Interestingly, Table II reveals that collectibles are less

likely to receive offers or sell through bargaining, than are elec-

tronics, but when they do sell through bargaining they do so at a

smaller fraction of the list price (the ratio of the bargained price to

the list price is 0.70 in the former and 0.79 in the latter). The prob-

ability of agreement is highest in bargaining threads over media

listings (0.596) and least likely for electronics (0.317). Sellers are

most experienced at bargaining in fashion, and buyers are most

experienced in the business category. A number of statistics are

remarkably stable across categories, such as the number of offers

per thread (1.54–1.7) and the ratio of the ﬁrst buyer offer to the

list price (0.575–0.660).

9. It would be tempting to use cross-category comparisons to learn about

bargaining on the assumption that some will intuitively manifest more asym-

metric information than others (e.g., we intuit that collectibles might have more

SEQUENTIAL BARGAINING IN THE FIELD 11

TABLE II

EANS BY CATEGORY

Full sample Collectibles Electronics Fashion Media Toys Business Others

Listing-level data

Listing price 94.6 75.7 146 122 31.6 73.7 200 126

Used 0.548 0.56 0.639 0.466 0.724 0.503 0.613 0.413

Revised 0.263 0.236 0.242 0.31 0.296 0.238 0.208 0.269

Sold 0.215 0.177 0.372 0.222 0.148 0.287 0.213 0.273

Sold by Best Offer 0.132 0.115 0.194 0.15 0.0714 0.164 0.128 0.151

Received an offer 0.206 0.176 0.345 0.229 0.102 0.268 0.186 0.24

No. photos 2.69 2.47 2.67 3.36 1.69 2.9 2.72 2.57

Sale price 69.7 54.8 124 65.9 22.6 62.5 143 77.4

Sale price / list price 0.832 0.805 0.89 0.814 0.855 0.858 0.827 0.858

Bargained price 74.1 60.7 135 68.9 25.8 68.2 145 81.7

Bargained price / list price 0.727 0.701 0.791 0.728 0.703 0.753 0.714 0.747

No. listings 88,386,471 34,809,404 6,796,662 23,416,322 9,350,356 7,760,006 3,060,080 3,193,641

Thread-level data

No. offers 1.66 1.64 1.7 1.66 1.54 1.68 1.61 1.66

No. offers if sold 1.48 1.46 1.58 1.46 1.37 1.53 1.45 1.48

Agreement reached 0.454 0.491 0.317 0.474 0.596 0.424 0.543 0.451

Seller experience 3,883 3,248 2,227 6,671 2,177 2,650 2,329 2,174

Buyer experience 129 200 121 75.2 79.6 58.7 377 65.1

First buyer offer 86.6 67.6 150 79 26.9 77.1 138 92.2

First buyer offer / list price 0.608 0.575 0.66 0.607 0.602 0.624 0.596 0.625

No. threads 25,453,072 8,106,675 4,126,466 7,348,818 1,113,603 2,976,443 719,044 1,062,023

Notes. This table presents means for variables in the main data set in the ﬁrst column and in a mutually exclusive and exhaustive set of broad category subsamples in the remaining

columns.

12 THE QUARTERLY JOURNAL OF ECONOMICS

TABLE III

ISTINGS RECEIVING AN OFFER VERSUS NOT

Full sample Received at least Never received

one offer any offer

Listing-level data

Listing price 94.6 119 88.3

Used 0.548 0.558 0.545

Revised 0.263 0.263 0.263

Sold 0.215 0.739 0.0789

No. photos 2.69 3.26 2.54

Sale price 69.6 78.7 47.6

Sale price / list price 0.832 0.765 0.996

No. listings 88,350,383 18,214,827 70,135,556

Seller-level data

Feedback postitive percent 99.4 99.4 99.4

No. listings 73.8 21.1 71.9

No. sales 15.9 15.6 5.67

No. sellers 1,196,955 861,757 975,772

Buyer-level data

No. purchases 3.21 2.82 2.19

No. buyers 4,698,748 4,481,259 1,116,258

Notes. This table presents summary statistics for the listings in our data that ever receive an offer compared

to those listings that never receive an offer.

Much of our analysis in the remainder of the article focuses on

the subsample of listings in which bargaining takes place. These

listings, and the buyers and sellers involved in them, may differ

from those in which no bargaining offer is ever observed. In the

top panel of Table III,wecomparethefullsampleandthesetwo

subsamples of listings. We ﬁnd that listings that received at least

one offer were much more likely to sell, and at a higher price.

(Note that this ﬁnding would not have been a foregone conclusion:

listings that receive no offers can still sell through the Buy-It-

Now option.) We ﬁnd that listings bargained over and those not

bargained over are equally likely to be used items and to have the

Buy-It-Now price revised at some point. In the bottom two panels

of Table III,wecomparethebuyersandsellerinvolvedinlistings

idiosyncratic inventory than electronics). This is undermined by the stability of

the statistics. We contend, and our casual browsing conﬁrms, that there is sub-

stantial heterogeneity in all categories, and that conditional on using the Best

Offer mechanism, which is an endogenous decision of the seller, these intuitive

preconceptions often do not hold.

SEQUENTIAL BARGAINING IN THE FIELD 13

with no bargaining and those with bargaining. In the sample of

listings that receive offers, sellers post fewer listings on average

and sell more listings but have similar feedback ratings. Buyers

in this sample tend to make more purchases than those who are

not.

III. BARGAINING THEORIES AND EMPIRICAL EVIDENCE

In this section, we walk through a number of existing theoret-

ical models of bargaining, examine their empirical implications,

and study how much of the observed data each model might be

able to explain. Models are abstractions from reality, and as such

cannot be expected to explain all features of real-world negotia-

tions. We use these models simply as a framework for highlighting

features of real-world bargaining that existing theory can or can-

not explain well. We focus our discussion in this section on three

features that have played a prominent role in motivating existing

theory: (i) whether agreement or disagreement occurs, (ii) when

agreement or disagreement occurs (immediately or after a delay),

and (iii) how the ﬁnal negotiated price is reached (suddenly or

gradually).

Points (i) and (ii) relate to how bargaining ends. To visual-

ize how bargaining ends in the actual data, we display a tree in

Figure III representing the extensive form of the bargaining game.

Square boxes represent the identity of the player (B for buyer and

S for seller). At the right of each box, we display the number of

observations that reach the node. Below each node are edges rep-

resenting the player’s decision to make an initial offer (O), accept

(A), decline (D), or counter (C). Each edge shows the percent of

observations passing through that edge corresponding to a given

action being chosen.

We refer to these percentages as we explore

10. Note that unlike the top panel of Table III, the comparison of sellers

involved in listings that never receive an offer and sellers involved in listings that

do receive an offer is not a comparison of mutually exclusive samples. This is also

true for the buyer comparison in the bottom panel. Note also that the sale/list price

ratio in the “never received an offer” column is not exactly equal to 1; this appears

to be due to a small fraction of listings in which the Buy-It-Now price was revised

by the seller and the sale price variable was not updated in eBay’s database to

reﬂect this change.

11. For the sake of visual clarity, Figure III does not display the buyer’s option

to buy at the Buy-It-Now price later in the bargaining sequence, which is always

an option buyers have available.

14 THE QUARTERLY JOURNAL OF ECONOMICS

FIGURE III

Bargaining Sequence Patterns

This ﬁgure summarizes the offer-level data in terms of the game tree of bargain-

ing. See text for detailed discussion.

what fraction of the data existing models could potentially explain.

Throughout the discussion that follows, we attempt to be gener-

ous in attributing what fraction of observations can be explained

by a given model. In particular, each observation in the data is

asequenceofofferscorrespondingtosomegame-tree-pathfrom

Figure III;ifagivenmodelcangeneratethispath—andinsome

cases a particular rate of concession of players’ counteroffers—we

state that this model could have plausibly generated this obser-

vation.

We emphasize that the discussion is not meant to suggest that

any of these theoretical models are “wrong” in any sense—each is

SEQUENTIAL BARGAINING IN THE FIELD 15

entirely accurate if its corresponding assumptions are satisﬁed.

Rather, by pointing out empirical regularities that some models

fail to generate, this discussion highlights that the assumptions of

these models (such as complete information of one or both parties)

may not be satisﬁed in real-life settings.

We also note that existing game theoretic models of bargain-

ing focus on particular, interesting equilibria, that satisfy some

reasonable equilibrium selection restrictions. Indeed, the patterns

we observe that do not arise in existing equilibrium models can be

trivially supported as equilibria in some cases, but these trivial

equilibria are neither meaningful nor interesting. For example,

an equilibrium in which players’ counteroffers gradually concede

at a given rate can be sustained by a perfect Bayes equilibrium

in which any off-equilibrium deviation from this prespeciﬁed con-

cession rate leads the opposing player to believe she is facing the

weakest bargaining type. In this sense, it isn’t necessarily that

existing theory cannot explain patterns we document below, but

that it has very little predictive power, in that it can trivially

generate nearly any type of behavior in a sequential bargaining

game.

III.A. Immediate Agreement: Rubinstein (1982)

Perhaps the two most inﬂuential bargaining models that

come to an economist’s mind are those of Nash (1950) and

Rubinstein (1982).ThemodelofRubinstein (1982) consists of two

players who s equentially alternate their offers. Each player has

some cost of bargaining, such as a per period discount factor or a

per offer cost applied to each time period of the game, and players

have complete information about their opponents’ valuations.

The unique subgame perfect Nash equilibrium (SPNE) of this

game is related to the axiomatic, cooperative solution proposed

in Nash (1950),inwhichplayerschooseadivisionofsurplusthat

maximizes a weighted, joint payoff of the two bargaining parties,

with each party’s weight depending on a “bargaining power” pa-

rameter. When this bargaining power is given by players’ discount

factors in Rubinstein’s model, the Nash solution corresponds

precisely to the unique SPNE of the Rubinstein game.

This model has several immediate empirical restrictions.

First, all bargaining interactions should end in agreement. Sec-

ond, the player who moves ﬁrst makes an offer that is immediately

16 THE QUARTERLY JOURNAL OF ECONOMICS

accepted by the second mover.

These two restrictions imply that

for an observed interaction to be rationalized by the Rubinstein

model, it must be the case t hat the game ends with the ﬁrst of-

fer being accepted immediately by the opponent. As shown in

Figure III,immediateagreementoccursin32%ofourobserva-

tions. The remaining 68% of observations are inconsistent with

Rubinstein’s strategic model or Nash’s axiomatic model.

Below

we discuss a number of theoretical approaches that could explain

some of these outcomes.

III.B. Immediate Disagreement: Perry (1986)

Asalientfeatureofreal-worldbargainingisthatsomene-

gotiations end in impasse, and this cannot be explained by

the complete-information models of Nash (1950) and Rubinstein

(1982).Anumberofstudiesexplainsuchimpassebyrelaxing

the complete information assumption and incorporating incom-

plete information into a sequential bargaining game or a mecha-

nism design framework. For example, incomplete information is

the key to the seminal theorem of Myerson and Satterthwaite

(1983).Animportantcontributiontomodelinganextensive-form,

alternating-offer bargaining game with incomplete information

is that of Perry (1986),inwhichboththebuyerandsellerhave

private valuations that are not commonly known to the parties.

Buyers face a cost (common to all buyers) of making each offer and

the seller faces a per offer cost common to all sellers. The unique

sequential equilibrium of this game is that the side with the lowest

bargaining cost makes an offer and the other party accepts or re-

jects, but never makes a counteroffer. Figure III implies that this

model can rationalize 57% of the observed bargaining sequences;

this includes a large percentage that cannot be rationalized by

the Rubinstein (1982) or Nash (1950) models.

The Perry (1986)

12. A third implication of the Rubinstein and Nash bargaining models is that

a player who has more “bargaining power” (or lower bargaining costs) should

get a better deal. Testing this implication is not simple, as bargaining power

and bargaining costs are not tangible or well-deﬁned objects and are difﬁcult to

measure in data. We provide one approach to such an analysis in Section IV.

13. Note that for the purposes of these calculations we consider the buyer’s

ﬁrst offer, rather than the Buy-It-Now price, to be the ﬁrst bargaining offer.

14. This 57% includes the 32% of immediate-agreement observations and an

additional 25% of observations—cases where the seller immediately declines (40%)

followed by the buyer declining (62%): that is, 40% × 62% = 25%.

SEQUENTIAL BARGAINING IN THE FIELD 17

model cannot rationalize cases in which multiple offers or delays

occur in equilibrium. A number of other bargaining models of in-

complete information share this feature, such as the “k-double

auction” of Chatterjee and Samuelson (1983):bargainingcanend

in agreement or in disagreement, but it always ends immediately

(in this case because the game is assumed to be static).

III.C. Delayed Agreement: Gul and Sonnenschein (1988) and

Others

Alargebranchofthetheoreticalbargainingliteraturein-

cludes models in which parties may delay reaching an agreement,

but nonetheless always agree. Models that generate delayed

agreement differ widely in how delay is generated. Rubinstein

(1985) studies an alternating-offer game in which there may be

two offers in equilibrium before agreement occurs: if the offer of

the player moving ﬁrst is not accepted, the player who rejected

that offer then makes an offer that is accepted immediately.

Grossman and Perry (1986) and Gul and Sonnenschein (1988)

provide a model where an informed buyer (with a private valua-

tion) alternates offers with an uninformed seller (with a known

valuation). In the equilibria they study, parties always agree, and

that agreement may occur immediately or after several back-

and-forth offers. Interpreted generously, these last two models

could explain any bargaining sequences ending in agreement,

which occurs 45.4% of the time (Table I). However, Gul and

Sonnenschein (1988) demonstrate that a Coase conjecture result

(Gul, Sonnenschein, and Wilson 1986)holds:asthetimebetween

offers decreases, agreement takes place immediately.

Two separate models that can also generate delayed agree-

ment in alternating-offer bargaining are Cramton (1992) and

Abreu and Gul (2000).Wediscussthesebelow,astheyalsopro-

vide testable predictions of how offers themselves change over the

course of the bargaining.

15. The authors also demonstrate that the equilibria they study have the

property that all unaccepted offers made by buyers in a given period of the game

must be the same for all buyers. The class of equilibria they study nests that

of Grossman and Perry (1986). There are a number of other theories of delayed

agreement—too many to treat here. For example, Feinberg and Skrzypacz (2005)

obtain delayed agreement in a model in which one party has private information

about her v aluation and the other has private information about his beliefs about

the other party’s valuation.

18 THE QUARTERLY JOURNAL OF ECONOMICS

III.D. Delayed Disagreement: Cramton (1992)

The models discussed above cannot generate delayed

disagreement—cases where a buyer and seller exchange multi-

ple offers and then walk away without trading. This is another

salient feature of the data shown in Figure III.Theonemodelof

alternating offers of which we are aware that allows for delayed

disagreement is Cramton (1992),inwhichbothpartieshavepri-

vate information about their valuations and may discover at some

point that there are no gains from trade and will then walk away.

In the model, parties effectively play a war-of-attrition game: once

apartymakesanoffer,theofferitselfandthetimingoftheoffer

completely reveal the offering party’s valuation. The model can

be augmented to allow for multiple back-and-forth offers as long

as any early offers (offers other than the last two) are nonserious

(that is, offers that would not be accepted by any opponent type

and that are only intended to inform the opposing party of the

offerer’s unwillingness to budge yet). Assuming that early offers

are nonserious, the Cramton (1992) model could potentially ex-

plain all sequences in the data. We provide evidence that offers

tend to change gradually, suggesting that they are indeed serious.

Assuming offers are serious, the model can potentially rationalize

any cases in which, after an offer is made, the opposing party (i)

immediately accepts or declines it or (ii) makes a counteroffer that

is immediately accepted.

The empirical prevalence of these Cramton (1992) cases can

be computed in a number of ways. First, if the Buy-It-Now price

is viewed as not fully revealing of the seller’s valuation, and the

buyer’s ﬁrst offer (and its timing) is viewed as arising from the

Cramton separating equilibrium, then the seller should respond to

this ﬁrst buyer offer by immediately accepting, declining, or mak-

ing an offer that i s guaranteed to be accepted. Such cases make

up 62% of bargaining sequences in Figure III.

If instead the ﬁrst

bargaining offer is viewed as uninformative to the seller (and sim-

ply as a signal that the buyer wishes to enter negotiations), then

the ﬁrst offer with the potential to fully reveal a player’s valuation

is the seller’s ﬁrst counteroffer. The buyer should respond to this

16. The model of Cramton (1992) is a two-sided-offer, continuous-types exten-

sion of the one-sided offer game of Admati and Perry (1987).

17. This quantity comes from 32% immediate agreement, 40% × 62% imme-

diate disagreement, as in the Perry (1986) model, plus an additional 28% × 17%

of cases where the seller counters at a price that the buyer then accepts.

SEQUENTIAL BARGAINING IN THE FIELD 19

by accepting or rejecting or by countering at a price acceptable

to the seller. This behavior is consistent with 83% of the observa-

tions in which the seller makes a counteroffer in response to the

buyer’s ﬁrst offer (which occurs 28% of the time).

In either sce-

nario, a signiﬁcant fraction of bargaining sequences in the data

are too long to be explained fully by the Cramton (1992) equilib-

rium. Moreover, the war-of-attrition nature of the Cramton model

does not match the actual protocol used in the data; the proto-

col dictates whose turn is next, which in and of itself poses some

difﬁculty for this model in explaining patterns of behavior in our

game.

III.E. What Theory Struggles to Explain: Gradual Offers

Asalientfeatureofthedatathatexistingtheoriesfailtorepli-

cate is that counteroffers on both sides tend to gradually be more

and more favorable to the opposing party with each subsequent

offer. This is illustrated in Figure IV.Ineachpanel,thevertical

axis shows the average amount of the offer and, on the horizon-

tal axis, the period of the game in which the offer is made, with

the t = 0offerrepresentingthelist(Buy-It-Now)price.Panels

on the left include sequences ending in agreement, and panels

on the right include those ending in disagreement. We analyze

separately those sequences that end in period 6, where the seller

declines (and the buyer takes no further action) or the seller ac-

cepts, and those t hat end in period 7, with the buyer accepting

or declining. Each panel also displays the average change in offer

price the seller makes from one offer to the next, averaged across

all periods of the game, and similarly for the buyer.

This gradualism of offers seems quite intuitive to anyone

who has engaged in bargaining in practice—it is exactly how one

would expect bargaining offers/counteroffers to evolve over the

course of negotiations. However, theoretical models generating

this type of pattern are almost nonexistent. As highlighted al-

ready, most models can rationalize only immediate disagreement

or immediate agreement. The equilibria of Grossman and Perry

(1986) or Gul and Sonnenschein (1988) can generate gradually

18. This 83% ﬁgure can be seen in Figure III, following the seller’s ﬁrst coun-

teroffer, as the sum of 17% + 58% + 25% × 31%.

19. In creating these ﬁgures, we include only sequences from the left side of

the game tree (Figure III); this excludes sequences that involve a seller’s decline

followed by additional action on the buyer’s part.

20 THE QUARTERLY JOURNAL OF ECONOMICS

(A) End at t = 6, seller accepts (B) End at t = 6, seller declines

FIGURE IV

Average Offers over the Duration of Bargaining (t = 6, 7)

Figure displays the average ﬁrst offer (Buy-It-Now price), average second offer,

and so on for bargaining sequences that ended in six (Panels A and B) or seven

(Panels C and D) periods. Panels on the left ended in acceptance, and panels on

the right ended in decline. Units are U.S. dollars.

changing offers by one party (the uninformed party) but not

by both parties. As highlighted, the model of Cramton (1992)

can accommodate many offers by each party i n equilibrium, but

only if all offers (other than the last offer made by each party)

are completely uninformative, nonserious offers; offers cannot

gradually become more favorable to the opposing party.

Indeed,

in the model of Cramton (1992),whenaplayerconcedestothe

opposing player, the concession must be a precipitous jump.

20. In the equilibrium of Cramton (1992), the timing of offers is endogenous.

This feature can be viewed as literally allowing a player to signal her valua-

tion when she ﬁrst makes an offer—and that is the interpretation we adopt in

Section III.D—or instead allowing the player to delay through making nonserious

offers whenever it is her turn, and then ﬁnally making a serious offer and signal-

ing her valuation through the cumulative total time passed since the game began

until the serious offer is made.

SEQUENTIAL BARGAINING IN THE FIELD 21

Other models have a similar war-of-attrition ﬂavor to the

Cramton (1992) model and can thus explain sudden, but not grad-

ual, offer changes. One example includes Abreu and Gul (2000).

Their model applies reputational game theory concepts (Kreps

et al. 1982)tobargaining,assuggestedbyMyerson (1991),where

with some probability an opponent is a “crazy” or “obstinate” type

who will never budge from her initial offer. Rational types ﬁnd it

proﬁtable to mimic obstinate types until the game ends, at which

point they concede. This model yields sudden offer changes when

arationalplayerdoesconcede,butnogradualchangesalongthe

way.

We are aware of only two previous models generating grad-

ually changing offers on both sides: Abreu and Pearce (2003) and

Compte and Jehiel (2004).Intheﬁrstpaper,theauthorsmodel

behavioral types who, for exogenous reasons, concede differently

than rational agents. The model shares an unrealistic feature of

many of the above models in that it only allows for equilibrium

agreement, not disagreement. In the second paper, the authors

model a setting in which both parties have complete information,

but by making offers, parties can inﬂuence the outside-option pay-

off to the opposing party. Disagreement can occur in a special case

of the Compte and Jehiel (2004) model, but that special case in-

volves no gradual offers. We explore the gradualism behavior that

we observe in more detail in Section V.A.

IV. C

ONVENTIONAL DRIVERS OF BARGAINING OUTCOMES

In this section we analyze three features that play impor-

tant roles and drive outcomes in a number of bargaining models:

costs of bargaining, bargaining power, and players’ outside op-

tions. Each feature can be related to the others and are not nec-

essarily distinct in their empirical implications. We also explore

21. A gradual change in offers is the equilibrium outcome of several models

of one-sided bargaining, where only one party gets to make offers to the opposing

party, and the opposing party only gets to accept or reject these offers. For example,

Fudenberg, Levine, and Tirole (1985) and Gul, Sonnenschein, and Wilson (1986)

allow for a seller (with no private valuation for the good) to screen a buyer (who

does have a privately known valuation) by making successively decreasing offers.

This generates gradual offers, but only through the constraint that only one party

is allowed to make offers, which is perhaps not a realistic constraint for many bar-

gaining settings (and clearly not for ours). Other papers with similar assumptions

and gradualism include Deneckere and Liang (2006), Fuchs and Skrzypacz (2013),

and Gerardi, H

orner, and Maestric (2014).

22 THE QUARTERLY JOURNAL OF ECONOMICS

afourthdimensionofanalysis:examiningtheeffectofreducing

adverse selection on bargaining outcomes. Online Appendix C con-

tains additional analysis of bargaining outcomes, demonstrating

that variation in bargaining outcomes is explained more by vari-

ation in who is bargaining than what is being bargained over.

IV.A. Bargaining Costs

Rubinstein (1982) proposed two models of bargaining costs:

one in which the surplus at stake is discounted exponentially, as

if the primary cost of bargaining were delayed consumption, and

asecondinwhichthereisaﬁxedcostofmakingoffers.Intheﬁrst

case, bargaining costs scale up with the value of the transaction,

whereas in the latter they are ﬁxed. Many subsequent bargaining

models have also adopted one or the other (or both) of these types

of costs (e.g., Cramton 1991). Because these types of bargaining

costs differ in how they relate to the value of the transaction, in

this section we search for evidence of s uch costs by examining how

bargaining outcomes differ at different levels of the listing price.

Figures V and VI present smoothed (weighted local linear re-

gressions, LOWESS) plots of expected outcomes against t he listing

price for our sample. To construct these plots, we used a strati-

ﬁed subsampling approach, sampling 10,000 listings each from

20 bins of $50 in length (inclusive on the upper extreme).

The

distribution of listing prices is presented in Figure V,PanelA,

where we see that the vast majority of listings fall in the $0.99

to $100 range. While average ﬁrst offers relative to list price are

decreasing throughout the range (Panel B), bargained prices are

initially rising and then fall (Panel C), and the slope of the ex-

pected sale prices ﬂips from negative to positive and back again

(Panel D). Figure VI provides some insight into this pattern. For

very cheap items, more buyers exercise the Buy-It-Now option

and forgo bargaining (Panels B and C). Moreover, sellers who do

receive offers on cheaper items tend to accept them immediately

(Panel D).

We interpret these outcomes as informative about the costs of

bargaining. Assuming higher listing prices correspond to settings

22. For the LOWESS plots in Figures V and VI, as well as those in

Figure IX later, we use default Stata options for LOWESS plots: tricube weighting,

and centered subsets of size 0.8

∗

N (where N is the number of observations in a

given plot) to construct a smoothed prediction about each point. For endpoints,

smaller, uncentered subsets are used as described in Stata documentation.

SEQUENTIAL BARGAINING IN THE FIELD 23

FIGURE V

Bargaining Outcomes by Listing Price

Panel A depicts a histogram of the listing prices for the full sample of listings.

The remaining panels depict LOWESS plots of the outcome variables in terms

of the listing price. In Panel B the variable of interest is the mean ﬁrst offer of

bargaining threads; in Panel C it is the bargained price, conditional on sale and

the buyer not executing the Buy-It-Now option; and in Panel D we are interested

in the sale price, conditional on sale.

with a larger surplus on the table, our data are consistent with

the existence of ﬁxed costs of bargaining: when the listing price

is greater and the amount of surplus to be negotiated is large,

parties are more willing to engage in the back and forth of ne-

gotiation; when the listing price is low and t here is little surplus

on the table, bargaining power tends to sit with whoever makes

the current offer. This model of costs is consistent with the ﬁnd-

ings of Jindal and Newberry (2018) in the setting of negotiation

over retail appliances; it is also consistent casual empiricism: bar-

gaining in street markets is less frequent in developed economies

with higher incomes—it is in some sense an inferior good—but

bargaining remains prevalent among high-value transactions, for

example, salary negotiations, plea bargaining, terms of a merger,

and trade deals; or even big-ticket consumer transactions, such as

24 THE QUARTERLY JOURNAL OF ECONOMICS

(A) Probability of Sale (B) Probability of Bargained Sale Conditional on Sale

(E) Number of Threads (F) Number of Offers per Thread

FIGURE VI

More Bargaining Outcomes by Listing Price

These panels depict LOWESS plots of bargaining outcomes in terms of the listing

price. Panel A concerns the probability of sale for all listings; Panel B restricts at-

tention to successful listings and plots the likelihood that the price was bargained

(as opposed to a buyer executing the Buy-it-Now option); Panel C concerns the

empirical likelihood of receiving any offer; Panel D concerns the likelihood that,

conditional on such an offer arriving, it is immediately accepted; Panel E concerns

the number of bargaining threads per listing; and Panel F measures the number

of offers associated with each thread, not including the listing price as an offer.

cars, large appliances, or homes. Fixed costs of bargaining are not

the only possible explanation for the ﬁndings; another possible

explanation would be that the types of players or the equilibrium

of the game differs markedly between high- and low-listing price

sequences.

SEQUENTIAL BARGAINING IN THE FIELD 25

These ﬁndings are especially important insofar as most test-

ing of theoretical models of bargaining has been primarily done

in the lab. For reasons of feabsility, experimental work focuses

on low-stakes bargaining. If players behave differently when the

stakes are high—because there are ﬁxed costs of bargaining or

because of some other reason—then this implies an important

caveat to the external validity of those ﬁndings. It also highlights

the importance of complementing this experimental testing in the

lab with evidence from the ﬁeld.

IV.B. Bargaining Power

An additional feature of many theoretical models of bargain-

ing is that players with more bargaining power obtain a greater

share of the surplus. This “bargaining power” is captured differ-

ently in different contexts. In some models, such as Rubinstein

(1982) and Rubinstein (1985),bargainingpowerisexplicitlyrep-

resented by a player’s patience (discount factor). In other bargain-

ing models, in particular many recent models applied in empirical

research in bargaining settings (e.g., Crawford and Yurukoglu

2012; Grennan 2013), bargaining power is instead a reduced-form

feature of the model rather than an underlying primitive, with a

direct correspondence to the share of the surplus the player would

receive in a static Nash bargaining game, where both players

agree to maximize the total surplus weighted by the bargaining

power weights (see Binmore, Rubinstein, and Wolinsky 1986). In

these models, bargaining power can represent concepts such as a

bargaining party’s negotiation skill or experience.

Here we use a simple approach to identify buyers who may

have a greater degree of patience than others. In particular, we

identify patient buyers as those who, ex post (after the bargaining

ends), choose the slowest shipping option when multiple options

are available. Namely, at checkout, a buyer can sometimes choose

between several shipping options, where faster shipping is more

expensive than slower shipping. Hence, by revealed preference,

buyers who choose a slower shipping method reveal that they are

willing to wait rather than spend more money, and are thus more

patient than buyers who opt for faster shipping at a higher price.

We also construct a measure of experience for buyers and sellers

using their accumulated number of previous bargaining threads

participated in (summary statistics for this measure are shown in

Table I).

26 THE QUARTERLY JOURNAL OF ECONOMICS

For this analysis, we rely on a subsample of the data for

which we can compute a reference price for each good. We con-

struct these reference prices by limiting to listings of products

that can be linked to third-party catalogs. For each such product

and item condition pair (where condition is used versus new), we

construct a reference price by taking the average Buy-It-Now price

for all listings of this product and condition sold during our sam-

ple through Buy-It-Now listings that did not have the bargaining

mechanism enabled; thus, reference prices are constructed using

listings outside of our data but during the same sample period.

See Online Appendix B for a discussion of and s ummary statistics

for this sample. In Table IV,thedependentvariableineachre-

gression is the ﬁnal price from a bargaining transaction in which

agreement occurred, divided by the reference price for that item.

All regressions include ﬁxed effects at the ﬁnest category level

available in the eBay data (referred to as leaf category).

Table IV shows the results of regressing the normalized

price on our measures of buyer patience (controlling for whether

multiple shipping options are available) and on both parties’ expe-

rience. We ﬁnd negative point estimates for the slowest shipping

variable in each column. This sign is consistent with more patient

buyers receiving lower ﬁnal prices in bargaining, as in many the-

oretical models; however, none of these estimates are statistically

signiﬁcant.

We ﬁnd stronger evidence for bargaining power be-

ing captured by players’ experience. Table IV demonstrates that

more experienced sellers tend to obtain higher prices (statistically

signiﬁcantly so for new goods) and more experienced buyers tend

to obtain lower prices (statistically signiﬁcantly so for used

goods). For both buyers and sellers, the results suggest a relative

23. This insigniﬁcance is likely due to the fact that our measure of patience

is imperfect and to the speciﬁcation we adopt here by including leaf category

ﬁxed effects. When we remove leaf category ﬁxed effects, we ﬁnd negative and

signiﬁcant estimates for used goods of approximately −0.07, suggesting that more

patient buyers obtain prices that are lower by 7 percentage points of the reference

price. It is important to note that while this evidence is consistent with a role for

patience, these regressions may also simply be capturing an effect of willingness

or ability to pay: buyers who are willing/able to pay less for the item may also

be willing/able to pay less for fast shipping, and hence the better deal obtained

by buyers whom we label as “patient” may actually be due to those buyers’ low

willingness/ability to pay. This highlights an important point: a player’s patience,

willingness/ability to pay, bargaining costs (from the previous section), or outside

options (discussed in the following section) can all affect the strength of a player’s

bargaining position and the surplus the player obtains.

SEQUENTIAL BARGAINING IN THE FIELD 27

TABLE IV

NEGOTIATED PRICES REGRESSED ON BARGAINING POWER MEASURES

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

Slowest shipping −0.0188 −0.0164 −0.0205 −0.0213

(0.0314) (0.0314) (0.0124) (0.0124)

Multiple shipping options 0.214

∗∗∗

0.202

∗∗∗

0.102

∗∗∗

0.100

∗∗∗

(0.0220) (0.0217) (0.00977) (0.00984)

Log(seller experience) 0.0192

∗∗∗

0.0158

∗∗∗

0.00262

∗∗

0.000119

(0.00174) (0.00165) (0.000926) (0.000952)

Log(buyer experience) −0.00401 −0.00289 −0.0220

∗∗∗

−0.0216

∗∗∗

(0.00251) (0.00251) (0.00134) (0.00134)

Constant 0.989

∗∗∗

0.942

∗∗∗

0.924

∗∗∗

1.115

∗∗∗

1.179

∗∗∗

1.175

∗∗∗

(0.00362) (0.0110) (0.0112) (0.00247) (0.00667) (0.00667)

Condition New New New Used Used Used

0.0300 0.0282 0.0306 0.0452 0.0453 0.0459

No. leaf FE 404 404 404 398 398 398

N 125,418 125,418 125,418 328,029 328,029 328,029

Notes. This table presents results from regressions where the dependent variable is the normalized price (see text for a discussion of the construction of reference prices) and the

regressors are buyer and seller attributes. All regressions include leaf category ﬁxed effects. Robust standard errors are presented in parentheses.

∗

: α = 0.10,

∗∗

: α = 0.05, and

∗∗∗

: α = 0.01.

28 THE QUARTERLY JOURNAL OF ECONOMICS

improvement in prices of about 2 percentage points given a

1logpointincreaseinexperience.Weshowadditionalevidencein

Online Appendix D (Table A-4) that buyers’ and sellers’ counterof-

fer behavior follows a similar pattern: players concede more in

nearly every round of the game when facing a more experienced

opponent and concede less when they are more experienced

themselves.

IV.C. Outside Options

Another feature that can affect outcomes is the outside option

of a negotiator. In our setting, the outside option of a seller is to

exit and search for another player with whom to negotiate (or to

leave the platform). In some cases, a player may even be engaged

in bargaining simultaneously with multiple parties at the same

time. Of the listings in our data, 7.8% have at least two buyers

whose bargaining threads overlap in time with the s ame seller,

and these threads make up 14.2% of all bargaining threads. In

2% of threads, a seller’s decision to accept a buyer offer results in

the seller declining an outstanding offer from a competing buyer.

On the buyer side, in the subset of our data containing cataloged

product identiﬁers, we see that 2.1% of listings involve a given

buyer bargaining with more than one seller of the same cataloged

product at the same time, and these threads make up 4.8% of all

cataloged product threads. When a buyer fails to reach agreement

with a seller of a given cataloged product, in 2.9% of threads the

buyer trades with another seller of the same product within a

day’s time.

The option of a player to end negotiations with one player

and engage with another can affect prices and other bargaining

outcomes. To examine this, we run regressions of the ﬁnal price

(normalized by the reference price, as in Table IV)onseveral

measures of competition and outside options, including the log of

the number of buyers with whom the current seller is bargaining

simultaneously for this item (labeled “competing buyers” in

Table V), the log of the number of sellers (selling the same

24. A related interesting statistic is the fraction of repeat interactions by

the same buyer and seller pair. We ﬁnd that 9% of buyer-seller pairs meet in at

least two separate bargaining threads. Together, these repeat buyer-seller pairs

constitute 23.5% of the interactions in the data. This feature is another interesting

aspect that could be exploited in future research with our public data set, studying,

for example, reputation-building and learning in bargaining.

SEQUENTIAL BARGAINING IN THE FIELD 29

TABLE V

EGOTIATED PRICES REGRESSED ON COMPETITION MEASURES

(1) (2) (3) (4) (5) (6) (7) (8)

Log(competing buyers + 1) −0.00175 −0.00631 0.0193

∗∗∗

0.0208

∗∗∗

(0.00767) (0.00802) (0.00473) (0.00481)

Log(competing sellers + 1) 0.0202 −0.0123 −0.0619

∗∗∗

−0.0566

∗∗∗

(0.0610) (0.0589) (0.00603) (0.00626)

Log(competing listings + 1) 0.0369

∗∗∗

0.0374

∗∗∗

−0.00734

∗∗∗

−0.00577

∗∗

(0.00735) (0.00671) (0.00173) (0.00182)

Constant 1.023

∗∗∗

1.022

∗∗∗

0.991

∗∗∗

0.992

∗∗∗

1.128

∗∗∗

1.134

∗∗∗

1.141

∗∗∗

1.137

∗∗∗

(0.00433) (0.00389) (0.00616) (0.00585) (0.00233) (0.00237) (0.00319) (0.00312)

Condition New New New New Used Used Used Used

0.0272 0.0272 0.0276 0.0276 0.0446 0.0446 0.0446 0.0447

No. leaf FE 404 404 404 404 398 398 398 398

N 125,418 125,418 125,418 125,418 328,029 328,029 328,029 328,029

Notes. This table presents results from regressions where the dependent variable is the normalized price (see text for a discussion of the construction of reference prices) and the

regressors are the (log of) the number of overlapping buyers competing on the same thread for a given product, the number of sellers offering a given product, and the number of

listings of the same product live on the site at the same time as a given listing. All regressions include leaf category ﬁxed effects. Robust standard errors are presented in parentheses.

∗

: α = 0.10,

∗∗

: α = 0.05, and

∗∗∗

: α = 0.01.

30 THE QUARTERLY JOURNAL OF ECONOMICS

cataloged product) with whom the current buyer is bargaining

simultaneously (labeled “competing sellers”), and the log of the

number of other listings available at the same time as the current

listing offering the same product as the current listings (labeled

“competing listings”). In these regressions, we also include

leaf-category ﬁxed effects.

The results are displayed in Table V.Theﬁrstfourcolumns

display results for new items, and the last four columns display re-

sults for used items. For new goods we do not detect any signiﬁcant

effects of competition and outside options on the ﬁnal price except

for a signiﬁcant and positive coefﬁcient on the number of compet-

ing listings, suggesting that prices are actually higher when more

listings of a particular product are available; this likely reﬂects an

equilibrium response of supply. Among used items, we see a pos-

itive and signiﬁcant increase i n price (2% of the reference price)

when the number of competing buyers on a given listing increases

by 1 log point. Conversely, we see a negative and signiﬁcant drop

of 6% of the reference price when the number of competing sellers

increases by 1 log point, and a drop of 0.6% when the number

of competing products increases by 1 log point. These results for

used goods are consistent with the role one would expect for mar-

ket thickness and outside options of one side of the market or the

other.

This simple analysis only touches the s urface of possible ef-

fects of outside options on bargaining outcomes in dynamic mar-

ketplaces. For example, eBay is a marketplace that is constantly

changing, with goods continuously arriving and exiting from in-

ventory. Indeed, there is no reason to expect that the outside

options of two bargaining parties will stay constant even dur-

ing the short window of negotiation over which we observe them

(and outside options of negotiators off the eBay platform may be

changing as well). In this light, dynamic outside options may be

an additional explanation for the delayed agreement and delayed

disagreement documented in Section III.

IV.D. Adverse Selection

Another area of interest in the theory literature on bargain-

ing is adverse selection (Evans 1989 ; Vincent 1989; Deneckere and

Liang 2006). We do not attempt a full survey of the implications of

adverse selection for bargaining here, but we do present several

salient facts that may help guide future theoretical and empirical

SEQUENTIAL BARGAINING IN THE FIELD 31

FIGURE VII

Time to First Offer by Number of Photos

Figure displays the average time (in days) from when the listing is posted until

the ﬁrst offer arrives conditional on the number of photos.

work, in particular related to information disclosure by sellers. In

previous work studying online ﬁxed-price and auction markets on

eBay Motors (a marketplace for vehicles), Lewis (2011) demon-

strated that the number of photos included in a listing serves as

ausefulmeasureoftheamountofinformationrevealedbysell-

ers about an item’s quality.

Here we examine how bargaining

outcomes differ with the number of photos.

In Figure VII,weexaminehowmanydaysittakesfortheﬁrst

bargaining offer to arrive as a function of the number of photos

in a listing. We ﬁnd that conditional on having at least one photo,

the larger the photo count, the quicker the ﬁrst offer arrives. This

suggests that buyers may be more willing to engage in bargaining

25. Note that our data does not include data from eBay Motors, which is a

platform primarily used by sellers as a classiﬁed advertising mechanism and not

as a setting for negotiating over or selling cars.

26. Note an important distinction between the type of information revela-

tion that can be captured in the number of photos and the private informa-

tion modeled in Myerson and Satterthwaite (1983), Perry (1986),andothers

(see Section III.B): in those models, each bargaining party has private information

about her own valuation, but no private information about factors affecting her

opponent’s valuation.

32 THE QUARTERLY JOURNAL OF ECONOMICS

TABLE VI

ARGAINING OUTCOMES BY NUMBER OF PHOTOS

Received Received Sold by Sold by Norm. Norm.

offer offer Best Offer Best Offer price price

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

Log(photos + 1) 0.0611

∗∗∗

0.0353

∗∗∗

0.0344

∗∗∗

0.0259

∗∗∗

0.193

∗∗∗

0.225

∗∗∗

(0.000610) (0.00106) (0.000574) (0.000971) (0.00494) (0.00975)

Constant 0.368

∗∗∗

0.414

∗∗∗

0.194

∗∗∗

0.204

∗∗∗

0.908

∗∗∗

0.860

∗∗∗

(0.000662) (0.000899) (0.000603) (0.000793) (0.00470) (0.00516)

Condition Used New Used New Used New

0.214 0.124 0.0735 0.0264 0.0530 0.0367

No. leaf FE 494 492 494 492 398 404

N 1,469,560 574,560 1,469,560 574,560 328,029 125,418

Notes. This table presents results from regressions where the dependent variable is the normalized price

(see text for a discussion of the construction of reference prices) and the right-hand-side variable of interest is

the log of 1 plus the number of photos of the listing. All regressions include leaf category ﬁxed effects. Robust

standard errors are presented in parentheses.

∗

: α = 0.10,

∗∗

: α = 0.05, and

∗∗∗

: α = 0.01.

when there is less asymmetric information; however, the choice to

disclose that information may itself be endogenous to t he content.

Table VI presents regressions of bargaining outcomes on the log of

the number of photos included with the listing. These regressions

use our reference price sample, as in Tables IV and V.Weﬁndthat

the more photos a listing has, the more likely it is to receive an

offer and to sell through bargaining (as opposed to selling through

the Buy-It-Now option or not selling at all). We also see that the

normalized price is higher by 19 percentage points for used goods

and 22.5 percentage points for new goods when the number of

photos increases by 1 log point.

V. U NEXPLAINED BEHAVIORAL PATTERNS

We now turn to players’ choices of counteroffers to further

explore the gradual concessions we discussed in Section III.E and,

in the process, uncover another interesting behavioral pattern

that conventional bargaining theories cannot explain. In general,

we are interested in exploring further patterns about how the

offer in period t relates to the offers in periods s < t.

Let γ

,and,fort = 2, 3, ..., 6, let γ

∈ [0, 1] be the weight

such that p

= γ

t−1

+ (1 − γ

t − 2

.Therefore,γ

represents the

27. Another beneﬁt of disclosing information with photos is to better match

buyers with different quality levels of goods based on their preference for quality

levels, as in Tadelis and Zettelmeyer (2015).

SEQUENTIAL BARGAINING IN THE FIELD 33

weight that player t places on the opponent’s previous offer, and

1 − γ

represents the weight the player places on their own pre-

vious offer. Note that by deﬁnition, p

= γ

+ (1 − γ

)0, so we

can think of the buyer’s “previous” offer when he makes his ﬁrst

offer as his bliss point of paying nothing for the good. It is useful

to think of γ

as how much a player concedes to her opponent

when she is making a counteroffer, or her concession weight. This

notion will prove useful for exploring concession behavior in more

detail as we show below. In the remainder of this section, we limit

the sample to threads with back-and-forth sequences correspond-

ing to the left side of the game tree displayed in Figure III,asin

Section III.E.Wealsorestricttothreadswhereγ

∈ [0, 1] for all

offers.

V.A. Recipro cal Gradualism

As we explained in Section III.E,existingtheorieshavetrou-

ble explaining the appearance of gradual changes in counterof-

fers, and as Figure IV showed, the path of average counteroffers

exhibits very strong gradual concessions. In fact, gradual changes

in offers are vastly more common in the data than are the sud-

den changes predicted by most theoretical analyses. In the data,

among observations in which the seller makes at least two offers

(beyond the Buy-it-Now price), we observe that only 0.77% of ob-

servations involve a seller standing ﬁrm at the Buy-It-Now price

for several periods and then conceding. In contrast, in 98.8% of

the observed sequences the seller’s ﬁrst counteroffer already con-

cedes a bit relative to the Buy-It-Now price. Examining analogous

numbers for buyers, we ﬁnd that only 0.42% involve concession

only suddenly after holding ﬁrm at the previous offer for at least

one period, whereas 95.9% involve a buyer conceding gradually.

To explore concession behavior within individual bargaining

threads (rather than average behavior across bargaining threads),

Table VII presents results from regressing a player’s concession

weight, γ

,ontheopponent’spreviousconcessionweight,γ

t−1

where the unit of observation is one movement along each bargain-

ing thread starting from the buyer’s ﬁrst counteroffer in period

t = 3.

Alargerγ

t−1

means that the player’s opponent conceded

28. For these regressions, to guarantee that the offers constitute all relevant

information that passes between parties, we eliminate threads in which prices

were auto-accepted or auto-declined and threads in which the buyer or seller

communicated via a message.

34 THE QUARTERLY JOURNAL OF ECONOMICS

TABLE VII

CONCESSIONS REGRESSED ON PREVIOUS CONCESSIONS BY ONE’S OPPONENT

(1) (2) (3) (4) (5) (6) (7) (8)

t−1

0.125

∗∗∗

0.120

∗∗∗

0.226

∗∗∗

0.223

∗∗∗

0.156

∗∗∗

0.149

∗∗∗

0.124

∗∗∗

0.166

∗∗∗

(0.00185) (0.00145) (0.00362) (0.00281) (0.00712) (0.00536) (0.0113) (0.00793)

Constant 0.347

∗∗∗

0.345

∗∗∗

0.166

∗∗∗

0.161

∗∗∗

0.281

∗∗∗

0.289

∗∗∗

0.174

∗∗∗

0.160

∗∗∗

(0.000820) (0.000654) (0.00127) (0.000990) (0.00197) (0.00148) (0.00323) (0.00228)

Condition Used New Used New Used New Used New

0.0605 0.0550 0.110 0.116 0.140 0.133 0.198 0.197

No. leaf FE 8,833 12,320 6,524 9,386 3,794 5,822 2,442 4,016

N 345,961 529,193 128,597 210,445 35,447 60,786 15,953 29,098

Notes. This table presents results from regressions of γ

on γ

t−1

for t = 3, 4, 5, and 6. All regressions include leaf category ﬁxed effects. Robust standard errors are presented in

parentheses.

∗

: α = 0.10,

∗∗

: α = 0.05, and

∗∗∗

: α = 0.01.

SEQUENTIAL BARGAINING IN THE FIELD 35

more in his previous round, and a positive coefﬁcient means that

the player in period t is herself conceding more in response to a

larger concession. This is the pattern we see for all periods t = 3,

4, 5, and 6, for new and used goods: the more a player concedes

in period t − 1, the more will his opponent concede in period t.

Hence, gradual behavior is reciprocal, in that deeper concessions

from one player are “rewarded” by deeper concessions from the

following player.

Together with the discussion in Section III.E,thedata

strongly suggest that a major gap in the bargaining literature

is a theory that can generate—from both parties in equilibrium—

agreement and disagreement, delay and immediate termination,

and the very robust pattern of reciprocal gradualism.

V.B. “Split-the-Difference” Offers

In deﬁning the concession weight γ

,itisinsightfultoobserve

the distributions of γ

in different rounds of the game. Figure VIII

displays histograms of these concession weights for the bargaining

threads observed in the data. Panel A plots a histogram of γ

Panel B plots a histogram of γ

, limiting to those threads in the

data in which a period-2 offer was made, and so on.

Several interesting patterns are evident in this analysis. We

note ﬁrst that offers typically make nonzero concessions that are

closer to the sender’s own prior offer than the other party’s (conces-

sion weights below 0.5), with the exception of the ﬁrst offer, which

is often close to the Buy-It-Now price. Second, some common mass

points emerge, and of particular notice are counteroffers that are

halfway between the previous two offers, or “split-the-difference”

counteroffers. The pattern even holds for buyers’ ﬁrst offers (γ

where the modal initial offer is half of the Buy-It-Now. The mid-

point offer is also the modal offer for the ﬁrst seller counter (γ

the ﬁrst buyer counter (γ

)andthesecondbuyercounter(γ

). For

other counters, the modal offer gives zero or nearly zero weight

to the buyer’s most recent offer, and second only to this choice is

again the split-the-difference point.

This pattern is consistent with previously documented labo-

ratory evidence and behavioral economic theory (Roth and Malouf

29. We ﬁnd the same pattern of reciprocal gradualism if instead of regressing

on γ

t−1

we regress the percent change in a player’s offer on the percent change

in the opponent’s offer (regressing, for example,

−P

). See Online

Appendix Table A-5.

36 THE QUARTERLY JOURNAL OF ECONOMICS

FIGURE VIII

Where Current Offer Lies Relative to Previous Offers

Each panel displays a histogram of offer weights deﬁning how the current offer

relates to the previous offers, where γ

,and,fort = 2, 3, ..., 6, γ

is such that

= γ

t−1

+ (1 − γ

t − 2

1979; Roth 1985; Binmore, Shaked, and Sutton 1985; Bolton 1991;

Bolton and Ockenfels 2000; Charness and Rabin 2002; Andreoni

and Bernheim 2009), in which market participants may care about

notions of fairness and may favor a split-the-difference strategy

in negotiations. Interestingly, however, the split-the-difference

pattern we observe is not necessarily a pattern of splitting sur-

plus between the two parties, as the surplus is not necessarily

SEQUENTIAL BARGAINING IN THE FIELD 37

TABLE VIII

PROBABILITY OF SPLIT OFFER ACCEPTED

t = 1 t = 2 t = 3 t = 4 t = 5 t = 6

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

Split 0.0381

∗∗∗

0.0628

∗∗∗

0.0848

∗∗∗

0.0978

∗∗∗

0.0951

∗∗∗

0.0998

∗∗∗

(0.000343) (0.000556) (0.00110) (0.00219) (0.00344) (0.00691)

0.932

∗∗∗

0.617

∗∗∗

1.191

∗∗∗

0.811

∗∗∗

1.019

∗∗∗

0.771

∗∗∗

(0.00167) (0.00266) (0.00473) (0.00635) (0.0106) (0.0125)

Constant 0.188

∗∗∗

0.182

∗∗∗

0.411

∗∗∗

0.342

∗∗∗

0.458

∗∗∗

0.488

∗∗∗

(0.000131) (0.000247) (0.000581) (0.00117) (0.00178) (0.00347)

0.114 0.0479 0.108 0.118 0.167 0.204

No. leaf FE 17,051 16,190 14,486 11,777 8,748 6,154

N 9,638,253 3,185,044 1,010,604 362,583 138,571 62,953

Notes. This table displays the results from a linear regression where the dependent variable is an indicator

for whether the offer in period t of the game is accepted and this is regressed on the offer weight, γ

,andon

an indicator for whether γ

is approximately equal to 0.5. Columns (1)–(6) display results for γ

for t = 1,

..., 6. All regressions include leaf category ﬁxed effects. Robust standard errors are presented in parentheses.

∗

: α = 0.10,

∗∗

: α = 0.05, and

∗∗∗

: α = 0.01.

known to the players given the potential presence of incomplete

information about opponent valuations. Rather, the split-the-

difference phenomenon we observe in our data refers to splitting

the two most recent offers, regardless of how those offers relate to

surplus.

We now explore how a player’s choice of offer , as measured

by the weight, γ

,relatestolateroutcomesinthebargaining

game. We create a measure for whether the offer is a “split” offer

by creating an i ndicator that is equal to 1 if γ

is equal to 0.5

(after being rounded to the nearest hundredth) for each t ∈ 1, 2,

3, 4, 5, 6. We ﬁnd that about 8% of offers are split offers by this

deﬁnition.

We estimate a local linear regression of an indicator

for whether each offer is accepted on both this split indicator and

the underlying γ

Results are shown in Table VIII.

Table VIII demonstrates that, as would be expected, the co-

efﬁcient on the concession rate (γ )ispositive:themoreaplayer

concedes relative to previous offers, the more likely it is that the

opposing player accepts the offer. The key result of Table VIII,

however, is that an offer in bargaining is more likely to be ac-

cepted if it is a split offer than if it is not, and this effect is both

30. Broader deﬁnitions of split, by rounding γ

to the nearest ﬁve hundredths

or nearest tenth yield 11% and 16% split rates, respectively.

31. We follow Fan and Gi

jbels (1992) in the construction of the optimal variable

bandwidth for estimation of the effect at 0.5 using a rectangular kernel.

38 THE QUARTERLY JOURNAL OF ECONOMICS

FIGURE IX

Probability of Split Offer Accepted

This ﬁgure displays a LOWESS ﬁt of the probability of an offer being accepted

regressed on the offer weight, γ

, and on an indicator for whether γ

is approxi-

mately equal to 0.5. From left to right, top to bottom, the panels display results

for γ

,wheret ranges from 1 to 6.

statistically signiﬁcant and surprisingly large in magnitude, as

well as being curiously stable across periods of the bargaining,

lying in a range of 5–10% independent of what point in the bar-

gaining game the split offer occurs. We supplement this approach

with a more ﬂexible ﬁt of γ

by plotting LOWESS ﬁts of accep-

tance and γ

in Figure IX using observations where γ

is not a

split offer. We then also plot in Figure IX the average acceptance

SEQUENTIAL BARGAINING IN THE FIELD 39

probability for observations that are split offers, along with 95%

conﬁdence bound about this mean. As can be seen, the underlying

relationship between γ

and acceptance is positive and split offers

are substantially more likely to be accepted.

This pattern of behavior is particularly surprising because,

taken seriously, it implies a nonmonotonicity in the likelihood

of acceptance—that is, a player is more likely to accept a split-

the-difference offer than an even slightly more favorable offer.

This kind of discrete and nonmonotonic behavior is not easy to

rationalize with existing theories, even those that incorporate

other-regarding preferences, such as altruism or inequity aver-

sion. Other-regarding preferences can explain why the well-being

of the player making the offer would be accounted for by the

responder, but this cannot rationalize the observed behavior.

Namely, a discontinuous drop in the probability of accepting an

offer that is slightly more generous for the respondent hurts

both players, and as such, it seems suggestive of some norm that

players are expected to follow. For that matter, any standard

preferences for which bargaining can be modeled as a Bayesian

game would be unable to explain the observed behavior. Namely, if

one considers the probability of trade as an equilibrium outcome,

then a general mechanism design approach as in Myerson and

Satterthwaite (1983) implies that the probability of trade should

be monotonic along some notion of private types if standard no-

tions of interim incentive compatibility hold. Hence, the observed

behavior presents a challenge to standard equilibrium theory

and suggests that there is some kind of “numerosity” effect at the

split-the-difference points that seems to go beyond preferences,

possibly some kind of norm, or some notion of salience.

VI. C

ONCLUSION

In this article we analyzed a novel data set of bilateral

bargaining used by millions of users in a live ecosystem. We

documented a number of facts consistent with rational theories

of bargaining behavior. In particular, we found that existing

theories can explain a nontrivial fraction of the data in terms of

how the bargaining ends (immediately versus with delay, and in

agreement vs. disagreement). We also found evidence favoring

ﬁxed over proportional costs of bargaining. This is important

for understanding mechanism selection but also for thinking

about the external validity of experimental studies of low-stakes

bargaining: if our conclusion is correct, behavioral patterns in

40 THE QUARTERLY JOURNAL OF ECONOMICS

high- and low-stakes bargaining will be very different. We also

found that more patient or more experienced players obtain

better deals, as do players facing less competition (or having a

better outside option). We ﬁnd that listings with more photos—a

potential measure of reduced adverse selection—tend to receive

offers more quickly and sell for higher negotiated prices.

We then documented several features of the data that are

clearly not consistent with any existing models of bargaining.

First, we showed that bargaining often ends in disagreement af-

ter several back-and-forth offers. Second, we showed that buyer

and seller offers tend to change gradually, not suddenly, over the

course of the bargaining sequence, and this gradualism is not one-

sided but is instead bilateral and gradual: more concession by one

player is associated with more concession by her opponent as well.

These two data patterns stand in stark contrast with existing bar-

gaining theory models.

Finally, we offered stark evidence of “splitting-the-difference”

behavior, a result that supports the incorporation of behavioral

elements to understanding bargaining dynamics. We documented

the surprising fact that counteroffers lying halfway between the

two preceding offers are signiﬁcantly more likely to be accepted

by the opposing party than are offers that are even slightly more

favorable to the opposing party.

We believe that the rich data we used herein, which we have

made publicly available, offers opportunities to explore how people

bargain and can help shed light on what determines bargaining

outcomes in the real world.

OLUMBIA UNIVERSITY,NATIONAL BUREAU OF ECONOMIC RESEARCH,

AND CENTER FOR ECONOMIC POLICY AND RESEARCH

BAY RESEARCH

STANFORD UNIVERSITY AND NATIONAL BUREAU OF ECONOMIC

RESEARCH

UNIVERSITY OF CALIFORNIA AT BERKELEY,NATIONAL BUREAU O F

ECONOMIC RESEARCH,CENTER FOR ECONOMIC POLICY AND RESEARCH,

AND CESIFO

SUPPLEMENTARY MATERIAL

An Online Appendix for t his article can be found at The

Quarterly Journal of Economics online. Data and code replicat-

ing tables and ﬁgures in this article can be found in Backus et al.

(2020),intheHarvardDataverse,doi:10.7910/DVN/ZOWWO7.

SEQUENTIAL BARGAINING IN THE FIELD 41

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