Paying to Match: Decentralized Markets with Information Frictions*

Paying to Match: Decentralized Markets with

Information Frictions

Marina Agranov

†

Ahrash Dianat

‡

Larry Samuelson

Leeat Yariv

January 24, 2023

Abstract

We experimentally study decentralized one-to-one matching markets with transfers. We vary the informa-

tion available to participants, complete or incomplete, and the surplus structure, supermodular or submod-

ular. Several insights emerge. First, while markets often culminate in eﬃcient matchings, stability is more

elusive, reﬂecting the diﬃculty of arranging attendant transfers. Second, incomplete information and sub-

modularity present hurdles to eﬃciency and especially stability; their combination drastically diminishes

stability’s likelihood. Third, matchings form “from the top down” in complete-information supermodular

markets, but exhibit many more and less-obviously ordered oﬀers otherwise. Last, participants’ market

positions matter far more than their dynamic bargaining styles for outcomes.

Keywords: Matching Markets, Transfers, Incomplete Information, Experiments.

JEL: C78, C92, D02, D47.

We thank Jeongbin Kim and Pellumb Reshidi for superb research assistance. We gratefully acknowledge ﬁnancial

support from the International Foundation for Research in Experimental Economics and from the NSF, through grant

SES 1629613.

†

Division of the Humanities and Social Sciences, California Institute of Technology, Email: [email protected]

‡

Department of Economics, University of Essex, Email: [email protected]

Department of Economics, Yale University, Email: [email protected]

Department of Economics, Princeton University, Email: [email protected]

1 Introduction

1.1 Overview

A large and successful literature studies matching markets with complete information. It as-

sumes individuals know not only their own preferences over possible allocations, but also all

others’ preferences. This assumption is often made for tractability and is reasonable for some

applications. In large school choice systems, for example, historical data on admission patterns

might be persistent and therefore predictable. Nevertheless, there are numerous applications in

which incomplete information regarding others’ preferences, at times even one’s own, might be

present. Newly-minted doctors seeking a residency in a speciﬁc specialty may not be privy to

the idiosyncratic preferences of both their peers and the hospitals they interview with. Similarly,

fresh academic Ph.D.’s may not be fully informed of other applicants’ preferences or of universi-

ties’ particular research and teaching needs. Outside the labor market, potential adoptive parents

are unlikely to know the preferences over child characteristics of other potential adoptive parents

seeking children, or of the birth mothers relinquishing them. In marriage markets, an entire con-

sulting industry builds on the premise that individuals frequently misread others’ preferences,

even after lengthy interactions. While incomplete information is prevalent across these applica-

tions, there is a paucity of empirical data to elucidate its impacts.

We report results from an array of lab experiments in which participants match through de-

centralized interactions, as in the examples above. Our treatments vary the surplus structure and

the initial information available to market participants about others’ preferences. We illustrate

the substantial eﬀect limited information has on outcomes and the regularities it produces.

Since many markets operate through transfers—labor markets often entail targeted wages,

housing markets generate exchanges via prices, etc.—we consider decentralized matching mar-

kets allowing for transfers. While common wisdom suggests that transfers assist in equilibrating

economic systems, they also allow for various bargaining frictions to emerge when negotiations

are unconstrained. Diﬀerences in bargaining styles, failures to follow up with appropriate oﬀers,

and the like could, in principle, aﬀect the generated match surpluses and their split between par-

ties. Furthermore, in practice, implementing the “right” transfers might be challenging: agents

often interact with a small set of potential partners and have limited information on what others

should reasonably expect to receive. The challenge of ﬁnding the “right” transfers is compounded

by agents’ incomplete information about others’ preferences. Our design allows us to discern how

transfers evolve and get determined.

Experimental Design. While our study speaks to many applications, ﬁeld data are diﬃcult to

gather. Rarely do we know market participants’ true preferences, their detailed interactions, or

the information available to them. The lab setting allows us to control for diﬀerent market features

and observe how individuals respond.

We consider two-sided one-to-one markets in which agents can make match oﬀers to one an-

other. Agents in the market are characterized by their “types,” which can stand for education level

or expertise in labor markets, age in child-adoption processes, or pizzazz in the marriage setting.

All agents prefer to be matched with higher-type agents: they generate a greater match surplus.

Going back to Becker (1973), the matching literature has often assumed match surpluses

exhibit either supermodularity or submodularity; see Chade, Eeckhout, and Smith (2017) and

Chiappori (2017) for literature surveys. We accordingly consider two surplus structures: super-

modular and submodular. In supermodular markets, higher types experience a greater marginal

increase in generated surplus from matching with higher types. Consequently, the utilitarian

eﬃcient matching is positively assortative. In submodular markets, the reverse holds: it is the

lower-type agents who experience the greatest marginal beneﬁt from increasing their partner’s

quality. In such markets, utilitarian eﬃciency implies negatively assortative matchings.

In our experimental markets, interactions are decentralized: individuals can make oﬀers to

one another in an unscripted fashion, with the sole restriction that only one oﬀer can be made or

accepted at any time. An oﬀer indicates a split of the match surplus. Accepted oﬀers generate ten-

tative matches—they can be broken by making a diﬀerent oﬀer that gets accepted, or by accepting

an alternative oﬀer. A market terminates after 30 seconds of inactivity. We use this matching pro-

tocol for three reasons. First, it echoes the narrative underlying the cooperative underpinnings of

stable matchings with complete information: any coalition can be formed throughout a market’s

duration. Second, it provides participants ample opportunities to learn about others’ match qual-

ities, which is important for some of our analysis. Last, it captures features of many real-world

matching settings, where pairs negotiate with a variety of partners before settling on a permanent

match. For example, employees can negotiate with one ﬁrm and switch to negotiations with an-

other, home buyers inspecting a property can decide to forgo that property and inspect others,

and individuals seeking a life partner often date multiple individuals sequentially.

In our complete-information treatments, participants know the surplus generated by each pair

of participants in the market. In our incomplete-information treatments, participants enter the

market knowing the proﬁle of match surpluses they can generate. That is, they know their own

type. They do not, however, know the types of other participants. In particular, at the start of the

market, they do not know who will help them generate which surplus. Learning occurs through

market interactions. Whenever an oﬀer is accepted, both partners learn the match surplus they

jointly generate and the resulting payoﬀs they receive.

The full market observes the creation of

matches, as well as the surplus that recipients of accepted oﬀers receive, but not the total surplus

such matches generate.

Empirical Regularities. Several insights emerge from our analysis. As we describe in our liter-

ature review below, prior experimental work supports the impression that, at least without trans-

fers, decentralized interaction yields stable outcomes. Our results show that matching markets

with transfers often produce utilitarian eﬃcient matchings, but stable outcomes are less com-

mon.

Thus, there is a sense in which the common wisdom on the beneﬁts of transfers holds

water in our data, albeit at the cost of stability. This eﬀect is particularly salient when the surplus

is submodular. Speciﬁcally, with complete information, 94% of our supermodular markets cul-

minate in eﬃcient, positively assortative outcomes and 78% of market results are stable. When

preferences are submodular, only 73% of our markets yield eﬃcient, negatively assortative out-

comes and a mere 14% of ultimate outcomes are stable.

When information is incomplete, the theoretical literature oﬀers little guidance as to the dy-

namics and outcomes one might expect; see our review of the literature. In particular, there is

little work inspecting the information revealed through the operations of a market that starts

While markets’ termination depends on all market participants ceasing activity, any particular pair can stop search-

ing for alternatives sooner by maintaining their pairing till the market ends.

Oﬀers still entail the speciﬁcation of a transfer. Absent information on match surpluses, certain oﬀers can lead

to negative payoﬀs. We allow participants to cancel such oﬀers ex-post. We do not see participants “gaming” the

experiments through this option.

Utilitarian eﬃcient matchings maximize the sum of match surpluses. Stability with transfers corresponds to a

matching and a proﬁle of transfers such that no pair of agents can ﬁnd a transfer that, if implemented, would make

their match desirable relative to what they already receive. It is well known that stable matchings are utilitarian

eﬃcient (e.g. Roth and Sotomayor, 1990).

with incomplete information. In simple incomplete-information markets such as ours, where par-

ticipants enter with considerable information and every accepted or rejected oﬀer yields more

information, interactions could plausibly lead to complete information. Therefore, throughout

much of our analysis, we use complete-information stability as a benchmark, returning to this

issue in Section 5.1.

In our data, incomplete information and submodularity constitute substantial obstacles to

both eﬃciency and stability. Indeed, with incomplete information, in markets with a supermod-

ular surplus structure, 84% of outcomes are eﬃcient, while 54% are stable. With submodular

surpluses, 39% of outcomes are eﬃcient, and a mere 3% are stable. In other words, the combina-

tion of incomplete information with submodularity nearly eliminates the likelihood of achieving

stable outcomes. This observation opens the door for richer theories of matching with incom-

plete information: ones that allow for some market learning, but still yield outcomes that are not

complete-information stable.

The analysis of oﬀer dynamics provides a lens into how market outcomes are generated. When

information is complete, markets governed by supermodular surpluses “match from the top,”

in the sense that high-type agents (who generate the greatest surplus) match ﬁrst, followed by

medium-type agents, and ﬁnally low-type agents.

In contrast, the sequenceing of matches is

more haphazard in submodular markets. Many more oﬀers are made throughout those market

operations. Early matches that “stick” are often between high-type agents, and generate a barrier

to achieving market-wide eﬃciency. Medium-type agents follow suit, while low-type agents tend

to match late, regardless of their partner’s type.

When information is incomplete, regardless of the underlying surplus structure, initial matches

are often transient and broken later in the market’s operations as participants learn. This leads to

far more market activity. Matches between incongruent types, low and high, occur earlier. Inter-

estingly, the enhanced market activity we observe implies that, by the end of our markets’ oper-

ations, most participants learn with whom they can generate which surplus, consistent with our

focus on complete-information stability. However, even matches formed later in our incomplete-

information markets do not echo the patterns seen in their complete-information counterparts.

Why do outcomes not mimic those in our complete-information markets? We show that our data

This sequence resembles the Marshallian path studied by Plott, Roy, and Tong (2013).

are consistent with simple satisﬁcing heuristics of participants. Based on their type, participants

aim at a certain payoﬀ and actively search for partners only below their satisﬁcing threshold. This

observation echoes evidence documented in individual decision-making; see Caplin, Dean, and

Martin (2011). In our setting, satisﬁcing emerges through dynamic market interactions.

A large literature, scientiﬁc and popular, suggests the importance of bargaining strategies

in determining outcomes—for reviews, see Bertrand (2011) and Shell (2006). One may wonder

whether our participants’ individual tendencies help explain their obtained payoﬀs. We indeed

see large variation in individuals’ bargaining styles, in terms of oﬀer volume, number of matches

created throughout a market, how demanding oﬀers are, and the timing of interactions. Nonethe-

less, none of these has any substantial explanatory power for outcomes. Ultimately, market fea-

tures and one’s position in the market—namely, the type—appear to explain the lion’s share of the

payoﬀs participants receive. This result may be comforting to some: while individual characteris-

tics and access to opportunities for self-improvement may greatly aﬀect one’s consequences, they

do so only insofar as they alter one’s position, or “quality,” in the market. Once in the market,

outcomes appear resilient to individual variations that go beyond participants’ contributions to

the surpluses they can generate.

Implications. Taken together, our results carry several important implications for market de-

sign. First, when transfers are available, stability through decentralized interaction is not guaran-

teed and depends crucially on the underlying preferences in the market. In particular, centralized

clearinghouses may be especially beneﬁcial when preferences are submodular. Second, incom-

plete information presents a substantial hurdle for achieving stability. Interventions directed at

alleviating information frictions may therefore prove useful. Such interventions could entail al-

lowing for more communication between participants, introducing public releases of information

about market attributes, etc. Alternatively, centralization of the matching process may be par-

ticularly important when participants are not perfectly informed. The existing literature rarely

considers clearinghouses with imperfectly-informed participants, and the consideration of those

may be necessary. Last, educating participants to achieve their maximal potential “quality” as

they enter a matching market may be of far greater importance than educating participants on

how to negotiate their interactions once in the market.

1.2 Related Literature

A large theoretical literature considers matching markets with complete information, with or

without transfers; see Roth and Sotomayor (1990). Incomplete information poses many challenges

to the analysis of matching markets. Even the basic stability notion is diﬃcult to translate when

agents are not fully informed of everyone’s preferences. Pairs that would block a matching with

complete information—pairs of agents that prefer one another over their allocated match—may

not form if agents cannot discern who would prefer a match with them. Furthermore, any at-

tempt to form a blocking pair allows other participants to partially infer the preferences of agents

involved, possibly others’ when those are correlated.

Starting from Liu, Mailath, Postlewaite, and Samuelson (2014), several recent papers suggest

stability notions for incomplete information markets, mostly restricting attention to incomplete

information only on one market side. Bikhchandani (2017) considers matching markets without

transfers. Liu et al. (2014), and relatedly Alston (2020) and Liu (2022), consider matching with

transfers as we do. At the heart of the stability notions in these papers is the requirement that any

agent’s “deviation” from a matching, with its attendant transfers, ultimately leaves her (weakly)

worse oﬀ. The stability notions in these papers are deﬁned with respect to a given matching and

a ﬁxed speciﬁcation of the uncertainty agents have about others’ types. They do not consider the

possibility that the match formation process may reveal information, rendering the speciﬁcation

of uncertainty endogenous and possibly leading to eﬀectively complete information.

Stability notions are interesting due to the presumption that, in frictionless decentralized mar-

kets, agents would ultimately converge to a stable outcome. Nonetheless, only a few papers explic-

itly model the decentralized process that generates such stable outcomes. Roth and Vate (1990)

oﬀer non-strategic dynamics that yield stable outcomes, whereby at each stage, a random block-

ing pair is implemented. Complete information is implicitly assumed, as all blocking pairs are

considered at every step. Ferdowsian, Niederle, and Yariv (2022) consider a decentralized mar-

ket game in which ﬁrms and workers interact in a dynamic fashion. They illustrate the hurdles

incomplete information or time frictions generate in establishing stability as a property of equi-

librium outcomes. Chen and Hu (2020) focus on incomplete-information settings and construct

an adaptive learning process leading to stable outcomes as they deﬁne them.

Hakimov and Kübler (2021) present a general overview of the experimental matching liter-

ature. Several papers in that literature share features with our design.

Concurrent work by

He, Wu, and Zhang (2022) considers decentralized matching with complete information, allow-

ing for transfers between participants. He et al. (2022)’s design resembles that of our complete-

information treatment, although their experimental markets have a pre-speciﬁed time of opera-

tion.

In line with our results, they observe far more stable outcomes in assortative than in anti-

assortative markets. In contrast with our results, they observe a substantial fraction of unmatched

individuals, potentially due to the limited-market horizon. Outside the matching context, Agra-

nov and Elliott (2021) identify the important impacts of transfers on bargaining outcomes.

Nalbantian and Schotter (1995) report on experiments emulating the market for professional

baseball players in the free-agent years. They analyze several allocation procedures with trans-

fers, including a decentralized procedure in which agents are informed of their own bargaining

outcomes, but not of others’. Pais, Pintér, and Veszteg (2020) study decentralized markets with

multiple stable matchings. There are no transfers and markets operate over a ﬁxed duration. In

some treatments, participants are privately informed of their preferences. Incomplete informa-

tion in their setting does not aﬀect the stability or the eﬃciency of the ﬁnal outcome. However,

similar to what we observe, it boosts market activity. Niederle and Roth (2009) study unrav-

eling by considering an incomplete-information setting in which one side of the market makes

proposals to the other side over three experimental periods and information is exogenously and

incrementally released. They illustrate the impact of proposal structure on matching outcomes

and eﬃciency. Incomplete information has also been considered in centralized markets, mostly

focusing on school-choice mechanisms and, therefore, absent transfers. See, for example, Chen

and Sönmez (2006), Pais and Pintér (2008), and work that followed.

2 Experimental Protocol

In our experiments, participants engage in a sequence of decentralized markets varying in two

dimensions: the surplus structure, which is either supermodular or submodular, and the infor-

mation available. In some matching markets, participants are fully informed of everyone’s pref-

There are also several papers studying decentralized matching with complete information and no transfers, see

Echenique and Yariv (2013) and references therein.

In their design, participants whose match is broken in the last 15 seconds can make oﬀers within 15 seconds.

There is also recent empirical work that highlights the consequences of incomplete information in school choice,

see Kapor, Neilson, and Zimmerman (2020).

Table 1: Match Surpluses (Eﬃcient Allocation in Bold)

Kiwi Mango Plum

Blue 8 16 24

Crimson 16 32 48

White 24 48 72

(a) Supermodular

Kiwi Mango Plum

Blue 8 32 56

Crimson 32 48 64

White 56 64 72

(b) Submodular

erences, in others they receive only partial information. We describe each of these dimensions in

turn. Sample experimental instructions are presented in Online Appendix C.

Our two-sided one-to-one matching markets are comprised of six participants, three on each

side. Each participant has a role: a color or a food. These roles capture the market sides and can

be metaphors for ﬁrms and workers in labor markets, men and women in the marriage market,

etc. There are three payoﬀ-relevant types for each role. Speciﬁcally, the three color types are Blue,

Crimson, and White. The three food types are Kiwi, Mango, and Plum. A matching speciﬁes

which colors and foods are unmatched, as well as a mapping between the remaining colors and

foods that is one-to-one: if a color type, say Blue, is assigned to a food type, say Kiwi, then the

reverse holds as well: Kiwi is assigned to Blue.

Table 1 depicts the match surpluses governing preferences in our markets. Panel (a) corre-

sponds to our supermodular surpluses, while panel (b) captures our submodular surpluses. In

each, the entry corresponding to each color and food pair is the match surplus generated by that

pair. Unmatched agents receive a payoﬀ of 0.

We consider two information structures. In our complete-information treatments, partici-

pants are fully informed of surpluses generated by all pairs in the market. In particular, they

can readily rank their potential partners in terms of the surpluses they would jointly generate. In

our incomplete-information treatments, participants know their own type: at the market’s outset,

they are informed of the three possible surpluses they can generate with the three potential part-

ners on the other side of the market. However, they do not know which partner generates which

surplus, nor which of the remaining two types have been assigned to which other agent on their

side of the market. Formally, we permute the type labels in markets and inform participants only

of their own type. The structure of payoﬀs is transparent, as well as the randomization procedure.

The general structure of each session is as follows. The governing match surpluses, either su-

In our experimental results, the incidence of unmatched participants is only 0.5% across all sessions.

permodular or submodular, as well as the information structure, are ﬁxed throughout the session.

Participants engage in one practice round, followed by 10 real rounds, each corresponding to a

new market (of the same sort in terms of both preferences and information), with a freshly drawn

random set of participants within the session. Participants maintain their role, a color or a food,

throughout the session. However, in each round, they are randomly assigned one of the three

types corresponding to their role.

In our complete-information treatments, each participant observes the full surplus matrix

throughout every round.

In our incomplete-information treatments, each participant only ob-

serves her own possible match surpluses at the start of each round.

As we soon explain, inter-

actions in the market can reveal some information on who generates which surplus.

We now describe the rules of the matching protocol. In each round, participants start oﬀ un-

matched. Each participant is free to make at most one match proposal to any individual of the

opposite role at any given time. In the complete-information treatments, a match proposal speci-

ﬁes how the match surplus will be split among the two individuals—i.e., the proposer’s payoﬀ and

the responder’s payoﬀ, summing up to the match surplus. In the incomplete-information treat-

ments, a match proposal only speciﬁes the responder’s payoﬀ, with the attendant proposer payoﬀ

revealed if the responder accepts. Thus, the proposer bears the payoﬀ risk under incomplete infor-

mation. This design choice was made to echo many applications in which the proposing side has

limited information on the returns to her proposal. For instance, ﬁrms oﬀering employment often

cannot assess workers’ abilities. Nonetheless, we limited the scope of risk by allowing proposers

to immediately retract an oﬀer if it turned out to generate a strict loss. For example, suppose Blue

and Kiwi generate a match surplus of 8. If Blue oﬀers Kiwi a payoﬀ of 16, and Kiwi accepts, then

Blue earns a payoﬀ of -8. Since Blue earns a negative payoﬀ, we would allow her to unilaterally

cancel the match.

This design choice was made both for practical reasons, in order to limit the

liability participants face, and also to mimic applications in which catastrophic relationships can

be severed promptly. For instance, a worker who does not have their presumed credentials can be

To prevent participants from reacting to cosmetic features of the surplus matrix, we shuﬄed the rows and columns

of the surplus matrix between rounds. In particular, speciﬁc colors and foods correspond to diﬀerent match-surplus

proﬁles in each round. As a result, the eﬃcient matchings do not always coincide with the diagonal (in the positively

assortative case) or anti-diagonal (in the negatively assortative case) of the surplus matrix.

For example, Blue in panel (a) of Table 1 would know that her possible match surpluses are 8, 16, and 24. However,

she would not know which surplus corresponds to which match.

In our data, 88% of negative oﬀers were canceled by the proposers immediately. Other cases were associated with

immediate new oﬀers by the proposers. Either way, we never see negative payoﬀs for any participant at any round.

ﬁred quickly by the ﬁrm that hired them in many labor-market settings. Whenever an oﬀer that

gives a positive payoﬀ for both parties is accepted, the match can be dissolved only upon one of

its participants making or receiving an alternative oﬀer that is accepted.

A match generating a loss certainly conveys some information to the proposer. In general,

whenever a match proposal is made and accepted, both the proposer and the responder observe

their realized payoﬀs. Other participants observe the formation of the match and the responder’s

payoﬀ. This is the main channel through which learning occurs under incomplete information.

In terms of market activity, participants are free to accept or reject match proposals they re-

ceive from other individuals. Any match proposal that is not accepted or rejected within 10 sec-

onds automatically disappears and is eﬀectively interpreted as a rejection. An acceptance forms a

match. Participants who are currently matched can still make and receive new match proposals,

with their current match persisting until such a new match proposal is accepted, at which point

(only) the current match is dissolved. A round ends after 30 seconds of inactivity. In incomplete-

information treatments, in order to allow participants to avoid excessive waiting, we introduced a

“move on to next round” button. Whenever at least 5 participants pressed the button, the market

for that group and round terminated.

A participant’s payoﬀ in their current match at the end

of a round is their ﬁnal payoﬀ for that round. If a participant is unmatched at the end of a round,

their ﬁnal payoﬀ is 0 for that round.

At the end of each session, participants also complete two risk-elicitation tasks and two al-

truism tasks. We use a version of the Gneezy and Potters (1997) investment task to measure

risk preferences and a dictator game to measure altruism.

All payoﬀs in the experiment are

expressed in tokens. For the main part of the experiment, 10 tokens correspond to $1. For the

elicitation tasks, one token corresponds to $0.01. Participants’ total earnings in the experiment

include a $10 show-up payment, the sum of their ﬁnal payoﬀs across the 10 rounds, and their

payoﬀs from the auxiliary elicitation tasks.

We were particularly concerned about the sensitivity of our incomplete-information treatments to speciﬁc indi-

viduals prolonging rounds after long waits. Whenever 5 of the 6 group members are ready to halt market activity,

we suspected the choice of the sixth individual to delay the end time would not aﬀect outcomes. As will be seen, our

incomplete-information treatments indeed took much longer in terms of both time and market activity.

Speciﬁcally, in each of the two risk-elicitation tasks, participants were provided with 200 tokens that they had to

allocate between a safe project, returning token for token, and a risky project. In the ﬁrst task, each token invested in

the risky project returned 2.5 tokens with 50% probability, and 0 otherwise. In the second task, each token invested

returned 3 tokens with 40% probability, and 0 otherwise. Altruism was elicited using two dictator-game elicitations,

one with an endowment of 100 tokens and another with an endowment of 200 tokens.

The experimental sessions were run at the Experimental and Behavioral Economics Laboratory

(EBEL) at UC Santa Barbara between July 2018 and October 2018. Each treatment, characterized

by a combination of underlying preferences and information structure, was run in 4 sessions, each

consisting of 12 participants. This generated 16 sessions overall and a total of 192 participants.

Each session lasted approximately 90 minutes, and paid an average of $24.75, in addition to a $10

show-up fee. The experiment was programmed and conducted with the oTree software (Chen,

Schonger, and Wickens, 2016).

3 Theoretical Preliminaries

In both our supermodular and submoduar markets, types can be ordered. In Table 1, White

generates higher surplus than Crimson, who generates higher surplus than Blue, for any food.

Similarly, Plum generates higher surplus than Mango, who generates a higher surplus than Kiwi,

for any color. Since labels of diﬀerent types are shuﬄed across rounds, and the two sides of

our markets are fully symmetric, we refer to the types generating the highest surplus as high

types, those generating the intermediate surplus as medium types, and those generating the lowest

surplus as low types.

Formally, let C be the set of colors and F the set of foods, with generic elements f and c.

Let π : C × F → R identify the surplus associated with each pair of agents. A matching (with no

unmatched agents) is a bijection m : C ∪ F → C ∪ F, where m(C) ⊆ F, m(F) ⊆ C, satisfying the

restriction that if m(c) = f , then m(f ) = c for all c ∈ C and f ∈ F. An outcome is a matching m and

a pair of transfer schemes τ

: C → R and τ

: F → R.

Under complete information, an outcome (m,τ

,τ

) is stable if it satisﬁes

(c), τ

(f ) ≥ 0 ∀(c, f ) ∈ C × F

(m(c)) = π(c,m(c)) ∀c ∈ C

(f ) ≥ π(c, f ) ∀(c,f ) ∈ C × F.

The ﬁrst condition is an individual rationality requirement guaranteeing that no agent prefers to

be unmatched. The second condition is a feasibility condition guaranteeing that transfers in a

matched pair do not exceed the generated surplus. The third condition is a no-blocking condition

that guarantees there is no pair of agents who can increase their transfers by matching with one

another and appropriately dividing the attendant surplus. We say that a matching m is stable if it

is consistent with some stable outcome.

While all our treatments allow participants to make transfers to one another, it is useful to

ﬁrst contemplate a no-transfer benchmark, in which surpluses are divided at some ﬁxed ratios

between the two sides of the market.

In this case, there is a unique stable matching in both the

supermodular and submodular markets: the positive assortative matching (PAM) that matches

individuals of the same type. In the markets depicted in Table 1, the unique stable matching

would match the highest types, White and Plum, then the medium types, Crimson and Mango,

and ﬁnally the low types, Blue and Kiwi.

When transfers are available, stable outcomes induce matchings that maximize the sum of

surpluses: the utilitarian eﬃcient matchings, hereafter referred to as simply eﬃcient. For the

two markets we consider, there may be many transfer proﬁles that sustain stable outcomes, see

Appendix B for a formal characterization, but the matching itself is unique. In our supermod-

ular markets, the matching maximizing utilitarian eﬃciency is the positive assortative matching

(PAM). Furthermore, PAM together with transfers that split each surplus equally constitute a sta-

ble outcome. In our submodular markets, the matching maximizing utilitarian eﬃciency is the

negative assortative matching (NAM). Importantly, there is no stable outcome in which each sur-

plus is split equally. Instead, in the matches between low and high agents (Blue with Plum and

White with Kiwi), the high agents (White and Plum) must receive more than half the surplus.

The existing theoretical literature oﬀers less deﬁnitive guidance as to the behavior and out-

comes one might expect under incomplete information. We adopt a permissive analog of complete-

information stability, deeming an outcome as unstable if there is a pair of agents with the property

that (i) one of the agents can propose matching under terms that will make both agents better oﬀ,

and (ii) the proposing agent knows there is no chance the proposal could make her worse oﬀ.

To make this precise, for each color c, let the set E

⊂ R

, with generic element (π(c,·)), con-

tain vectors specifying the surpluses that c can generate with each food f . Let E

be similarly

deﬁned for each food f . We say that the set E

(or E

) is consistent with color c’s (or food f ’s)

experience if it excludes those values that the experience has revealed to be impossible. Then

That is, we can restrict τ(c) = απ(c,m(c)) and τ(f ) = (1 − α)π(m(f ),f ) for all c and f , where α ∈ (0,1).

(m,τ

,τ

,(E

)

c∈C

,(E

)

f ∈F

) is incomplete-information stable if the sets (E

)

c∈C

,(E

)

f ∈F

are consis-

tent and

(c), τ

(f ) ≥ 0 ∀(c, f ) ∈ C × F

(m(c)) = π(c,m(c)) ∀c ∈ C

(f ) ≥ min

(

π(c,·))∈E

π(c,f ) ∀c ∈ C

(f ) ≥ min

(

π(·,f ))∈E

π(c,f ) ∀f ∈ F.

The ﬁrst two conditions are the individual rationality and feasibility conditions. The latter two

conditions are the no-blocking conditions, stated separately for colors and foods. For example, the

colors’ no-blocking condition asks each color c to identify the minimum surplus generated with

each food f , given what c has learned. The condition guarantees that no split of that (pessimistic)

surplus can leave both c and f better oﬀ. The last condition is the analog for foods.

If at least one of the participants in each possible match knows the surplus that would be gen-

erated by the match, then this notion of incomplete-information stability reduces to complete-

information stability. This suﬃcient condition will be met in our case if at least 5 market par-

ticipants know the surplus they can generate with each opponent. Our results (Table 5 below)

show that the average number of participants who learn from their direct experience—i.e., from

matches in which they have at least temporarily participated—the surpluses they can generate

with each opponent exceeds 5. The matching process provides yet more information. We accord-

ingly take complete-information stability as our benchmark throughout.

Despite the abundance of information participants accumulate through market interactions,

we show that our incomplete-information markets do not routinely produce stable outcomes. In

latter parts of the paper, we consider two explanations. First, market participants may make

use of even less information than that which they have gained through their direct experience.

Second, incomplete information may introduce glitches into the matching process beyond the lack

of information—some oﬀers a potential proposer knows impose no risk of a reduced payoﬀ and

beneﬁt both parties involved, may not be made. In particular, participants may pursue decision

criteria diﬀerent from payoﬀ maximization. Our analysis indicates the room for a richer theory of

matching with incomplete information.

4 Aggregate Outcomes

We start by describing the aggregate outcomes in our experimental markets. We use all 10 rounds

played in each session and omit the practice round.

4.1 Eﬃciency and Stability

Table 2 depicts the fraction of eﬃcient outcomes, those corresponding to a matching that maxi-

mizes the sum of match surpluses, i.e. the fraction of PAM for supermodular markets and NAM

for submodular markets. The table also depicts the fraction of stable outcomes, consisting of both

an eﬃcient matching and a stable transfer proﬁle. For the latter, we also display the fraction of

outcomes that are “almost stable” in that the matching is eﬃcient, and no pair can generate more

than a small additional amount, 2 or 5 tokens, by blocking the observed outcome.

Table 2: Final Outcomes (allocations)

Complete Information Incomplete Information

Supermodular Submodular Supermodular Submodular

mean (se) mean (se) mean (se) mean (se)

Eﬃciency 0.94 (0.04) 0.73 (0.07) 0.84 (0.04) 0.39 (0.10)

Matching

PAM 0.94 (0.04) 0.01 (0.01) 0.84 (0.04) 0.08 (0.06)

NAM 0.00 (0.00) 0.73 (0.07) 0.00 (0.00) 0.39 (0.10)

Stability

exact 0.78 (0.10) 0.14 (0.09) 0.54 (0.07) 0.03 (0.03)

- 2 tokens 0.89 (0.06) 0.33 (0.11) 0.68 (0.06) 0.06 (0.05)

- 5 tokens 0.93 (0.05) 0.58 (0.12) 0.78 (0.06) 0.19 (0.11)

Notes: Robust standard errors (reported in parentheses) are obtained by clustering at the session level to account for

interdependencies of observations that come from the same session. An eﬃcient outcome is one that maximizes the sum

of match surpluses. PAM indicates positive-assortative matching, while NAM indicates negative-assortative matching.

An allocation is stable (or stable −x tokens) if the matching is utilitarian eﬃcient and no pair can generate any gain (of

more than x tokens) by blocking the observed outcome.

Table 2 suggests several insights. First, a substantial fraction of all of our markets culminates

in eﬃcient matchings. For the most part, matchings are positively assortative when PAM is eﬃ-

cient and negatively assortative when NAM is eﬃcient.

Second, incomplete information poses an important obstacle for eﬃciency, particularly for

submodular markets. For supermodular markets, moving from complete to incomplete infor-

mation generates a reduction of 10 percentage points in the fraction of eﬃcient matchings. In

contrast, for our submodular markets, the fraction of eﬃcient matchings almost halves, dropping

by 34 percentage points.

Third, there are fewer stable than eﬃcient outcomes. Much of the instability we observe is due

to unstable transfer proﬁles, as captured by the diﬀerence between the fraction of outcomes that

entail an eﬃcient matching and those that are stable. For example, with complete information, 6%

of supermodular markets do not culminate in an eﬃcient matching, while 16% of supermodular

markets yield an eﬃcient matching but with unstable transfers. This wedge is even more pro-

nounced for submodular markets. With complete information, 27% of markets do not culminate

in eﬃcient matchings, while 59% produce an eﬃcient matching with unstable transfers.

Fourth, the surplus structure has important impacts on whether emergent outcomes are sta-

ble. Submodular markets generate signiﬁcantly and substantially fewer stable outcomes than do

supermodular markets, especially under incomplete information, with diﬀerences that are more

pronounced than the comparisons of eﬃcient outcomes. With complete information, 78% of su-

permodular markets culminate in a stable outcome, while for submodular markets, the corre-

sponding ﬁgure is 14%. This is the case even when allowing for minimal “mistakes” in terms of

the transfer proﬁle. With incomplete information, the fraction of stable outcomes in supermod-

ular markets falls to 54%, while the fraction of stable outcomes in submodular markets nearly

vanishes, standing at 3%.

These comparisons are mirrored by the pattern of blocking pairs. For supermodular markets,

unstable outcomes are associated with an average of 1.56 blocking pairs with complete informa-

tion and an average of 1.65 blocking pairs with incomplete information. The corresponding ﬁg-

ures for our submodular markets are 1.67 and 2.86. In particular, with incomplete information,

submodular preferences yield signiﬁcantly more blocking pairs than supermodular preferences

(at the 1% signiﬁcance level). The patterns are more pronounced when considering eﬃcient, but

unstable, outcomes. In those, we see an average of one blocking pair with complete information

and an average of 1.25 blocking pairs with incomplete information when surpluses are super-

modular. The ﬁgures for our submodular markets are 1.32 and 1.93, both signiﬁcantly higher (at

the 1% signiﬁcance levels). A similar pattern emerges when considering the average surplus gen-

erated by blocking pairs of unstable outcomes, with incomplete information and submodularity

leading to greater average surplus available across blocking pairs.

These results suggest that,

even with fairly unrestricted and decentralized interactions, market participants facing incom-

plete information are not gleaning enough information to attain complete-information stability.

4.2 Payoﬀs

Table 3 provides aggregate statistics of participants’ payoﬀs and transfers. The top panel of the

table depicts the average payoﬀs by type. Theoretically, all stable outcomes of our supermodular

markets entail higher types gaining greater payoﬀs. In our submodular markets, however, a uni-

form distribution over stable outcomes would yield only 50% of stable transfer proﬁles exhibiting

this monotonic pattern (see our Appendix B for details on the constraints stability with transfers

imposes). Interestingly, when considering average payments, in all of our treatments we see high

types garnering signiﬁcantly greater payoﬀs than medium types, who end up with signiﬁcantly

greater payoﬀs than low types. Nonetheless, when we consider these comparisons market by mar-

ket, we see some diﬀerences, echoing our theoretical predictions. When information is complete,

all of our supermodular markets, but only 83% of our submodular markets, end up with higher

types gaining greater payoﬀs. When information is incomplete, 93% of our supermodular markets

and 61% of our submodular markets generate greater payoﬀs for higher types.

Equal splits are common in experimental work (e.g. Andreoni and Bernheim, 2009). While

equal splits are consistent with stability for supermodular markets, that is not the case for sub-

modular markets. In fact, were participants simply following an egalitarian norm and always

proposing an equal split of match surpluses, we would expect to observe PAM outcomes in all

our markets, as described in Section 3. Table 2 suggests that egalitarian norms cannot be medi-

ating all of our results. Indeed, in our submodular markets, we see PAM outcomes quite rarely.

In fact, while the transfers required for stable outcomes seem elusive, particularly in submodular

markets, the overall eﬃciency in those markets is far greater than under PAM. While PAM would

generate a total surplus of 128 tokens, the top panel of Table 3 indicates an average total surplus

of 157.5 tokens under complete information and an average total surplus of 150.5 tokens under

incomplete information. In other words, transfers do allow for greater eﬃciency, even in cases in

In supermodular markets that culminated in an unstable outcome, the average surplus available to blocking pairs

was 7.06 tokens with complete information and 7.19 tokens with incomplete information. For submodular markets,

the corresponding ﬁgures were 5.33 and 7.90 tokens. In particular, the greatest number of blocking pairs and available

surplus per blocking pair were observed for submodular markets with incomplete information.

Table 3: Players’ Final Payoﬀs

Complete Information Incomplete Information

Supermodular Submodular Supermodular Submodular

mean (se) mean (se) mean (se) mean (se)

Payoﬀs by Type

Low Type 4.11 (0.06) 16.07 (0.37) 4.41 (0.15) 14.72 (0.60)

Medium Type 15.83 (0.12) 23.81 (0.25) 15.60 (0.51) 24.19 (0.14)

High Type 35.86 (0.10) 38.87 (0.45) 35.29 (0.27) 36.33 (0.54)

Equal Splits

All allocations 0.79 (0.04) 0.32 (0.01) 0.67 (0.06) 0.39 (0.06)

First two deals 0.64 (0.03) 0.60 (0.03) 0.15 (0.01) 0.23 (0.02)

Eﬃcient allocations 0.81 (0.03) 0.31 (0.01) 0.69 (0.05) 0.29 (0.02)

Stable allocations 0.87 (0.02) 0.30 (0.01) 0.80 (0.06) 0.33 (0.21)

Notes: Average payoﬀs by players’ types are reported in the ﬁrst three rows. Robust standard errors are clustered at

the session level and reported in the parentheses. Equal splits correspond to ratios of payoﬀs between two matched

players falling between 45% and 55%. For the ﬁrst two deals in the incomplete-information markets, we focus on deals

in which the proposer receives non-negative payoﬀs.

which stability appears the most challenging to achieve.

The bottom panel of Table 3 displays explicitly the frequency of (roughly) 50 − 50 splits of

the match surplus in the ﬁnal outcomes observed across markets. In order to allow for some

margin of error, we consider oﬀers that range from 45% to 55% of the surplus. Since there are

many opportunities for participants to learn their match surpluses with others through markets’

operations—see Section 5.2 below—we consider the possibility of equal splits for incomplete-

information markets as well. Such roughly equal splits are prevalent, albeit not universal, in our

supermodular markets: hovering around 80% with complete information and around 70% with

incomplete information. They are far less common in our submodular markets, comprising fewer

than 40% of the outcomes regardless of the information available.

Figure 1 describes the cumulative distribution functions corresponding to payoﬀs across our

treatments. While complete-information treatments generally yield more pronounced “jumps”

the underlying preferences have a clear impact. With supermodular preferences, many of the

payoﬀs take one of three values. Indeed, when superimposing the distribution of payoﬀs that

would be generated by equally splitting the surplus between agents in the eﬃcient matching (the

Interestingly, when looking at initial accepted oﬀers in our markets, we see far more equal shares of match sur-

pluses for matches between congruent types in our submodular markets. In the ﬁrst two accepted oﬀers, equal sharing

occurs in over 89% of deals between participants of the same type. Nonetheless, even in these additional deals, partic-

ipants of diﬀerent types do not split the surpluses equally. In particular, low types matching with high types get only

20% of the surplus. We provide more details on behavior at the start of our markets’ operations in Appendix A.

Figure 1: Cumulative Distributions of Players’ Payoﬀs

Cumulative Probability

0 10 20 30 40 50

Players' Payoffs

Complete Supermodular Complete Submodular

Incomplete Supermodular Incomplete Submodular

orange solid line, corresponding to equal masses receiving payoﬀs of 4, 16, and 36), we see sub-

stantial overlap. Submodular preferences, however, yield a nearly uniform distribution of payoﬀs,

particularly when information is incomplete. With complete information, we do see a large frac-

tion of individuals receiving a payoﬀ of 24, an observation we soon elaborate on. Furthermore, the

payoﬀ distributions corresponding to submodular preferences ﬁrst order stochastically dominate

those corresponding to supermodular preferences. This, again, suggests that participants are not

simply implementing the eﬃcient matching and splitting the surplus in those markets. Indeed,

the resulting distribution (depicted in the blue solid line, corresponding to one third of the popu-

lation receiving a payoﬀ of 24 and two thirds receiving a payoﬀ of 28) would second order, but not

ﬁrst order, stochastically dominate the corresponding distribution for our supermodular markets.

Figure 1 presents payoﬀ distributions aggregated across all market participants. At ﬁrst blush,

it indicates similarity in overall payoﬀ distributions between complete- and incomplete-information

markets of the same surplus variety. Figure 2 presents the distribution of payoﬀs by types of

agents in markets that culminated in “nearly stable” outcomes, ones in which the matching is eﬃ-

cient and there is a stable allocation of the surpluses such that each participant’s payoﬀ is at most

2 tokens away from what they would receive in that allocation.

The ﬁgure depicts how observed

Allowing for a small margin of error ensures that the samples we discuss are of suﬃcient size.

Figure 2: Distribution of Payoﬀs by Types in Stable (-$2) Allocations

0 .2 .4 .6 .8

Fraction

0 4 8 12 16 20 24 28 32

Submodular: Low Types

0 .2 .4 .6 .8

Fraction

0 4 8 12 16 20 24 28 32

Supermodular: Low Types

0 .2 .4 .6 .8

Fraction

0 4 8 12 16 20 24 28 32 36 40

Supermodular: Medium Types

0 .2 .4 .6 .8

Fraction

0 4 8 12 16 20 24 28 32 36 40

Submodular: Medium Types

0 .2 .4 .6 .8

Fraction

20 24 28 32 36 40 44 48 52 56

Supermodular: High Types

0 .2 .4 .6 .8

Fraction

20 24 28 32 36 40 44 48 52 56

Submodular: High Types

PayoffsHighStableT2Sub 5/31/20, 11:38 AM

0 .2 .4 .6 .8

Fraction

20 24 28 32 36 40 44 48 52 56

Complete Incomplete

Submodular: High Types

Notes: The red line on the horizontal axis of each graph depicts the range of payoﬀs corresponding to stable outcomes

predicted by theory.

payoﬀs relate to the range of payoﬀs corresponding to stable outcomes, captured by the red solid

line in each panel (see Appendix B for computational details). In line with our discussion above,

the distribution of payoﬀs for all agents and across our markets exhibits a spike at the midpoint

of stable allocations. However, the ﬁgure conveys several additional messages. Considering our

complete-information treatments, payoﬀs corresponding to the midpoint of the interval of stable

transfers are far more prevalent in our supermodular markets. In fact, in our submodular mar-

kets, while over 70% of the medium-type agents receive the mid-point of their stable payoﬀs, low-

and high-type agents have far more disperse payoﬀ distributions. With incomplete information,

payoﬀs are noticeably more diﬀuse, particularly when preferences are submodular. In fact, for

submodular preferences, for all types, no allocation carries more than 30% of our data.

4.3 Market Activity

Table 4 summarizes the aggregate features of activity in our markets. As can be seen, market

activity took, on average, between 3 and 5 minutes. Our incomplete-information markets corre-

Table 4: Market Characteristics

Complete Information Incomplete Information

Supermodular Submodular Supermodular Submodular

mean (se) mean (se) mean (se) mean (se)

Market duration (in sec) 158.6 (14.0) 171.5 (16.1) 191.6 (16.5) 292.9 (10.6)

Nb of oﬀers extended

overall per market 24.8 (1.5) 24.1 (3.0) 42.4 (4.3) 45.9 (5.3)

by Low type 9.6 (1.4) 10.7 (2.0) 19.9 (1.5) 21.0 (2.7)

by Medium type 8.0 (0.8) 8.4 (0.7) 14.2 (2.0) 15.8 (2.2)

by High type 7.2 (0.8) 5.2 (0.3) 8.3 (0.9) 9.1 (1.4)

Nb of oﬀers accepted

overall per market 5.6 (0.3) 8.8 (1.3) 15.9 (1.0) 17.3 (1.1)

by Low type 1.7 (0.04) 2.1 (0.26) 5.9 (0.36) 5.7 (0.38)

by Medium type 2.1 (0.16) 3.1 (0.48) 5.9 (0.48) 5.7 (0.35)

by High type 1.8 (0.12) 3.5 (0.55) 4.2 (0.43) 6.0 (0.37)

Notes: Robust standard errors are clustered at the session level and reported in parentheses.

spond to signiﬁcantly longer durations and a greater volume of oﬀers made and accepted. Our

submodular markets correspond to somewhat longer times to reach an outcome. However, the

volume of oﬀers made or accepted does not seem to hinge on the underlying preference structure.

At the aggregate level, there are no great diﬀerences across agent types in terms of overall oﬀers

made or accepted. One exception corresponds to the low-type agents, who make signiﬁcantly

more oﬀers than the high-type agents, particularly when information is incomplete.

4.4 Summary of Aggregate Outcomes

Overall, participants often establish eﬃcient matchings. Stability is more elusive, indicating that

it is easier to settle on an eﬃcient matching than to arrange the attendant transfers. Incomplete

information and submodular surplus structures present hurdles to eﬃciency and especially sta-

bility, with the combination of incomplete information and submodularity reducing stability to a

tiny percentage of outcomes. Incomplete information appears to introduce “complexity” in that

markets with incomplete information take longer to converge and involve substantially more par-

ticipant interactions to reach an outcome.

5 Market Dynamics

5.1 Match Evolution

Figure 3: Transient and Permanent Matches, by Quintiles

Complete

Supermodular

Quintile 1 Quintile 2

Quintile 3 Quintile 4 Quintile 5

PayoffsHighStableT2Sub 5/31/20, 11:38 AM

0 .2 .4 .6 .8

Fraction

20 24 28 32 36 40 44 48 52 56

Complete Incomplete

Submodular: High Types

TransientPermanent

Complete

Submodular

Incomplete

Supermodular

Incomplete

Submodular

0. 2

0. 4

0. 6

0. 8

LL MM HH LM LH MH

0. 2

0. 4

0. 6

0. 8

LL MM HH LM LH MH

0. 2

0. 4

0. 6

0. 8

LL MM HH LM LH MH

0. 2

0. 4

0. 6

0. 8

LL MM HH LM LH MH

0. 2

0. 4

0. 6

0. 8

LL MM HH LM LH MH

0. 2

0. 4

0. 6

0. 8

LL MM HH LM LH MH

0. 2

0. 4

0. 6

0. 8

LL MM HH LM LH MH

0. 2

0. 4

0. 6

0. 8

LL MM HH LM LH MH

0. 2

0. 4

0. 6

0. 8

LL MM HH LM LH MH

0. 2

0. 4

0. 6

0. 8

LL MM HH LM LH MH

0. 2

0. 4

0. 6

0. 8

1. 2

1. 4

LL MM HH LM LH MH

0. 2

0. 4

0. 6

0. 8

1. 2

1. 4

LL MM HH LM LH MH

0. 2

0. 4

0. 6

0. 8

1. 2

1. 4

LL MM HH LM LH MH

0. 2

0. 4

0. 6

0. 8

1. 2

1. 4

LL MM HH LM LH MH

0. 2

0. 4

0. 6

0. 8

1. 2

1. 4

LL MM HH LM LH MH

0. 2

0. 4

0. 6

0. 8

1. 2

1. 4

1. 6

LL MM HH LM LH MH

0. 2

0. 4

0. 6

0. 8

1. 2

1. 4

1. 6

LL MM HH LM LH MH

0. 2

0. 4

0. 6

0. 8

1. 2

1. 4

1. 6

LL MM HH LM LH MH

0. 2

0. 4

0. 6

0. 8

1. 2

1. 4

1. 6

LL MM HH LM LH MH

0. 2

0. 4

0. 6

0. 8

1. 2

1. 4

1. 6

LL MM HH LM LH MH

Notes: Quintiles are deﬁned relative to the total time the average market takes to reach a ﬁnal outcome, where average

market duration is calculated as the mean time diﬀerence between the last and ﬁrst activity in that market. The white

parts of the bars indicate the transient matches, i.e. those that are broken later on. Grey bars depict the ﬁnal matches

that remain intact until the market’s termination.

We start by considering the progression of matches over the course of a market’s operation. In

order to depict our markets’ evolution, we inspect quintiles of activity in each market’s operation.

Namely, for each round in our sessions, we consider the time between the ﬁrst and last moments

of activity. We split that time span into ﬁve quintiles, thereby eﬀectively normalizing the length

of activity in each of our experimental markets.

Figure 3 reports the mean number of transient and permanent matches formed at each quin-

tile of market duration across treatments, for each combination of matched types—LL denotes

two low-type agents, LM denotes one low- and one medium-type agent, and so on. Several obser-

vations emerge. First, there are substantial diﬀerences in match dynamics across supermodular

and submodular markets. The complete-information treatments are particularly telling. With

supermodular preferences, both transient and permanent matches are ﬁrst formed between the

high-type agents. They are followed by medium-type agents, and last by low-type agents. In

contrast, with submodular preferences, extremal types form matches ﬁrst: we see the most early

transient matches between medium- and high-type agents and the most early permanent matches

between low- and high-type agents. These are followed by matches between congruent medium-

type agents. The early permanent matches between low- and high-type agents occur at substan-

tially lower rates than the corresponding congruent and permanent early matches in our super-

modular markets, particularly when accounting for the fact that each market contains two incon-

gruent low- and high-type pairs. Last, we see a substantial fraction of matches between low- and

medium-type agents at later market phases. These late matches are not part of stable outcomes.

The second observation from Figure 3 is that incomplete-information treatments are associ-

ated with far more oﬀers, in every quintile and particularly early on. Some of these oﬀers act to re-

veal information about the types of agents on the other side of the market—indeed, early-quintile

oﬀers are frequently transient. This presumably explains why some features of the complete-

information markets remain. Speciﬁcally, we see more transient and permanent early matches

involving high-type agents. In contrast, the lower-type agents take more time to ﬁnd partners.

5.2 Learning through Market Operations

As incomplete-information markets evolve, participants have opportunities to learn others’ types.

Indeed, whenever an oﬀer is accepted, individuals learn the resulting payoﬀs and can thus identify

the type of individual on the other side of the market. Table 5 documents this type of learning

throughout quintiles of our sessions. In both supermodular and submodular markets, by the

Repeat oﬀers occur frequently, even in later parts of markets’ operations. Consider the average fraction of repeat

oﬀers in the second half of each round. In complete-information supermodular markets, these range from 0.27 for

low-type agents to 0.34 for medium-type agents to 0.13 for high-type agents. In incomplete-information supermodular

markets, these range from 0.34 for low-type agents to 0.29 for medium-type agents to 0.25 for high-type agents. Our

submodular markets yield nearly identical frequencies of repeat oﬀers.

Table 5: Discovering Partners in Incomplete Information Markets

Quintile 1 Quintile 2 Quintile 3 Quintile 4 Quintile 5

Supermodular

all players know partners (frac of markets) 0.14 0.38 0.51 0.56 0.60

av # of players who know their partners 3.38 4.84 5.18 5.29 5.41

Submodular

all players know partners (frac of markets) 0.21 0.41 0.63 0.65 0.80

av # of players who know their partners 3.44 4.75 5.35 5.40 5.66

Notes: For each type of market, supermodular or submodular, we report the fraction of markets in which all players

discover their partners’ surpluses in the corresponding quintile (the ﬁrst row) and the average number of players (out

of six) who discover their partners’ surpluses in each quintile (the second row).

third quintile, all participants are fully informed about the surpluses they can generate with all

those on the other side of the market in over 50% of markets, with an average of over 5 out of 6

participants being informed. By the termination of each market, these numbers are even higher.

The experimental protocol also ensures that whenever a match is formed, everyone learns the

payoﬀ of the participant accepting the corresponding oﬀer.

Having 5 out of 6 participants informed is suﬃcient for our stability notion to coincide with

complete-information stability, see Section 3. The dearth of stable outcomes in our incomplete-

information treatments is thus puzzling. The following subsection suggests explanations.

5.3 Patterns of Market Interactions

Matching theory provides limited guidance on the equilibrating dynamics one might expect in

decentralized markets with transfers. We now discuss how certain beliefs might be consistent with

a modiﬁed version of stability explaining our markets’ outcomes. We also illustrate behaviors in

our data consistent with participants following a satisﬁcing heuristic, reducing substantially their

eﬀorts once hitting a (type-dependent) payoﬀ target.

5.3.1 Stability with Pessimistic Beliefs

Suppose that instead of using all the information participants learn from their market interac-

tions, they assemble only the minimal information consistent with the observed ﬁnal matching.

Learning in submodular markets appears slightly faster than in our supermodular markets. However, this diﬀer-

ence is signiﬁcant only in the last quintile. Indeed, regression analysis clustering errors at the session level suggests a

corresponding p-value of p = 0.026 when looking at the last quintile, whereas p > 0.10 for earlier quintiles.

Suppose, further, that participants form pessimistic expectations (from the point of view of block-

ing), recognizing the possible match surpluses in the market. Namely, when observing a match

in which the receiver realizes a payoﬀ of x, they assume the matched partner receives the max-

imal feasible payoﬀ of 72 − x. Then, in those incomplete-information outcomes that fail to be

complete-information stable, there is an average of only 0.59 blocking pairs in the supermodular

markets and 1.04 blocking pairs in the submodular markets. These numbers are considerably

smaller than those reported above for our complete-information markets (1.56 and 1.67, respec-

tively). This indicates that there are beliefs consistent with the information revealed under which

participants come as close to stable outcomes as they do under complete information. In prac-

tice, we might expect participants to be less pessimistic and to glean yet more information from

the matching process (though perhaps not as much as do the hyper-rational agents in theoret-

ical treatments of incomplete-information stability such as Liu et al., 2014). We conclude that

the matching process in incomplete-information markets does not convey enough information to

bring the agents to complete-information stability. We leave as an important topic for future work

the more reﬁned investigation of whether the resulting behavior reﬂects stability on the part of

timid, under-inferring agents, or failures of less timid and more clever agents to achieve stability.

While pessimistic beliefs provide an eﬀective way to organize the market outcomes we observe,

they do not, on their own, explain the dynamics markets follow.

In what follows, we illustrate a

simple heuristic consistent with the market activity we observe.

5.3.2 Satisﬁcing Patterns

We now consider a satisﬁcing heuristic, whereby participants target a particular payoﬀ, possibly

dependent on their market type. Caplin et al. (2011) document the use and prevalence of sat-

isﬁcing heuristics in individual choices. Our design allows for satisﬁcing behavior to emerge in

a market setting, where the satisﬁcing threshold depends on global market features. Satisﬁcing

would suggest that participants make oﬀers before hitting their target payoﬀ, and cease making

oﬀers once that target is achieved.

For each market, we compute the number of oﬀers made by players of each type—low, medium,

For instance, pessimistic beliefs at the start of each market’s activity could correspond to each participant sus-

pecting the minimal match surplus available to them. However, initial oﬀers in our incomplete-information markets

signiﬁcantly exceed half that surplus for each of the three types. Furthermore, initial oﬀers do not reﬂect a persistent

fraction other than 50% of that “pessimistic” surplus.

or high—conditional on their current payoﬀ, excluding unmatched individuals. We then group

oﬀers in bins corresponding to participants’ current payoﬀ. We normalize the number of oﬀers

by the volume of oﬀers made by the corresponding type across all rounds. Table 6 presents the

resulting fraction of oﬀers by each type in each of our treatments, conditioning on the current pay

bin. Horizontal lines in the table correspond to a satisﬁcing rule, allowing for a 20% mistake rate.

That is, payoﬀs exceeding those the line represents yield fewer than 20% of oﬀers. As can be seen,

similar qualitative conclusions result from using lower mistake thresholds.

Table 6: Frequency of Oﬀers Made Conditional on the Current Pay

Supermodular

Low Type Medium Type High Type

Current Pay Complete Incomplete Complete Incomplete Complete Incomplete

1 to 4 0.87 0.64 0.03 0.08 0.00 0.04

5 to 8 0.13 0.28 0.29 0.25 0.00 0.05

9 to 12 0.00 0.05 0.28 0.23 0.03 0.16

13 to 16 0.00 0.01 0.35 0.33 0.06 0.09

17 to 24 0.00 0.01 0.04 0.09 0.39 0.30

25 to 32 0.00 0.00 0.01 0.01 0.33 0.23

33 to 40 0.00 0.00 0.00 0.00 0.18 0.12

41 and above 0.00 0.00 0.00 0.01 0.00 0.01

Submodular

Low Type Medium Type High Type

Current Pay Complete Incomplete Complete Incomplete Complete Incomplete

1 to 4 0.33 0.26 0.00 0.06 0.00 0.00

5 to 8 0.17 0.24 0.00 0.12 0.00 0.00

9 to 12 0.13 0.11 0.03 0.10 0.00 0.01

13 to 16 0.27 0.24 0.27 0.17 0.01 0.01

17 to 24 0.11 0.11 0.60 0.37 0.05 0.18

25 to 32 0.00 0.04 0.11 0.17 0.37 0.53

33 to 40 0.00 0.00 0.00 0.01 0.51 0.23

41 and above 0.00 0.00 0.00 0.00 0.05 0.04

Notes: Fractions represent the frequency of oﬀers of a particular type in a particular market conditional on current-

payoﬀ bin. The horizontal line corresponds to the satisﬁcing thresholds with a 20% mistake threshold: payoﬀs beyond

the threshold lead to fewer than 20% of oﬀers.

Estimated satisﬁcing thresholds are remarkably consistent across complete- and incomplete-

information markets across types, with only one exception corresponding to low types in super-

modular markets. This consistency is reassuring and suggests that market positions are among the

strongest forces shaping our participants’ behavior. The next section explores this idea further.

Similar levels are derived when considering the ﬁrst and second half of each market’s operations separately. In

addition, we do not see substantial diﬀerences across early and later rounds. Oﬀer rates documented in Table 6 nat-

urally reﬂect a combination of how active people are conditional on their current pay and how frequently they ﬁnd

themselves with that pay.

Table 7: Impact of Number of Oﬀers on Final Payoﬀs

Low Medium High

Complete

Supermodular 0.19 (0.09) 0.16 (0.08) 0.14 (0.09)

Submodular 0.21 (0.07) 0.18 (0.06) 0.11 (0.11)

Incomplete

Supermodular 0.25 (0.10) 0.17 (0.08) 0.10 (0.05)

Submodular 0.22 (0.08) 0.17 (0.09) 0.11 (0.07)

(a) Fraction of Oﬀers by Type

low

med

high

low

= β

med

low

= β

high

med

= β

high

(n)

Complete -0.17 -1.77 13.33 0.35 0.12 0.11 0.94 (480)

Supermodular (0.61) (1.74) (6.60)

Complete 1.05 1.18 -2.05 0.98 0.69 0.39 0.84 (480)

Submodular (4.70) (2.53) (5.04)

Incomplete -1.44 -10.64

∗

-5.72 0.04 0.45 0.44 0.93 (480)

Supermodular (1.05) (3.76) (5.37)

Incomplete -17.72 -14.30

∗

-9.96 0.79 0.41 0.63 0.74 (480)

Submodular (11.11) (4.81) (4.43)

(b) Payoﬀ Regressions

Notes: Panel (a) reports the average number of oﬀers made by each participant type as a fraction of all oﬀers made

in that speciﬁc market, with standard deviations in parentheses. Rows add up to 50%, modulo rounding. Panel (b)

reports the estimated coeﬃcients from linear regressions in which the dependent variable is the participants’ payoﬀ

and the independent variables include the constant, the dummies for medium and high types (leaving the low types

as the base group), the round and the round squared, two measures of altruism per participant, two measures of risk

attitudes per participant, and the fractions of oﬀers extended by low, medium, and high types. The ﬁrst three columns

in panel (b) report the estimated coeﬃcients, and robust standard errors in parentheses, on the fraction of oﬀers made

by low, medium, and high types, respectively. Standard errors are clustered at the session level. The next three columns

report p-value tests of coeﬃcients based on these regressions. The ﬁnal column reports the number of observations and

the R-squared.

5.3.3 Bargaining Styles

We now turn to the various bargaining styles participants used and how they aﬀect outcomes in

our markets. The main message is that, while individuals vary widely in how they approach our

markets, in terms of the number of oﬀers they make, their insistence on higher shares, etc., their

payoﬀs are not strongly aﬀected by these diﬀering interaction styles. By and large, payoﬀs are

determined by the governing match surpluses: supermodular or submodular, and the position

one takes in the market in terms of types.

We focus our discussion on one aspect of behavior: the number of oﬀers made in a market. In

the Appendix, we include analogous analyses relating to other aspects: number of matches made

throughout the market, the share of surpluses demanded in oﬀers, and which quintile the ﬁrst

and last matches were created. The messages from these are qualitatively identical.

Panel (a) of Table 7 illustrates, for each of our treatments, the fraction of oﬀers made by type,

as well as their standard deviation.

There is substantial variation in the number of oﬀers partic-

ipants make, reﬂected in the relatively large standard deviations. While there are some notable

diﬀerences across types, with lower types generally making more oﬀers than higher types, diﬀer-

ences across information or preference treatments are insigniﬁcant.

Panel (b) of Table 7 illustrates the impacts of the number of oﬀers on payoﬀs. In the reported

regressions, we explain the ﬁnal payoﬀ of a participant by their position, market characteristics,

and the number of oﬀers they make, allowing for ﬁxed-eﬀects by participant and clustering errors

at the session level. The association is insigniﬁcant in all but one case, pertaining to medium types

in the incomplete-information supermodular markets, where we observe a moderately positive

association between fraction of oﬀers made and ﬁnal payoﬀ.

A similar picture emerges when considering variations at the individual level. We consider

participants who are “high earners,” receiving a ﬁnal payoﬀ that is above the average payoﬀ for

the role they take in at least 7 of the 10 rounds played. Analogously, we consider “low earners”

as those receiving a ﬁnal payoﬀ below the average for their role in at least 7 of the 10 rounds.

“Medium earners” are all those that remain. The distribution of the various types of earners is

similar to that which would be generated by random choice, with somewhat more “high earn-

ers.”

We see minute diﬀerences between the three categories of earners in terms of fraction

of oﬀers made, when controlling for market features, position in the market, session round, and

overall number of oﬀers made in the market (see the Appendix for detailed regression results).

Figure 4 summarizes this discussion. For each of our treatments, and each market position,

or type, we classify participants according to the fraction of oﬀers they made, divided into 9

categories. We see a substantial number of observations in each of these categories, particularly

Since there are 6 participants in each market, if all made oﬀers at equal volumes, we would expect each to make

16.66% of the oﬀers observed.

For example, were payoﬀs determined at random, we would expect 17.19% of participants to be classiﬁed as

“high earners.” In our data, we see 27.08%, 35.42%, 31.25%, and 22.92% high earners in the complete supermodular,

incomplete supermodular, complete submodular, and incomplete submodular treatments, respectively.

Figure 4: Payoﬀs by Type as a Function of Fraction of Oﬀers Made

< 0.05 0.05 -

0.10

0.10 -

0.15

0.15 -

0.20

0.20 -

0.25

0.25 -

0.30

0.30 -

0.35

0.35 -

0.40

>0.40

Incomplete Submodular

< 0.05 0.05 -

0.10

0.10 -

0.15

0.15 -

0.20

0.20 -

0.25

0.25 -

0.30

0.30 -

0.35

0.35 -

0.40

>0.40

Incomplete Supermodular

< 0.05 0.05 -

0.10

0.10 -

0.15

0.15 -

0.20

0.20 -

0.25

0.25 -

0.30

0.30 -

0.35

0.35 -

0.40

>0.40

Complete Supermodular

< 0.05 0.05 -

0.10

0.10 -

0.15

0.15 -

0.20

0.20 -

0.25

0.25 -

0.30

0.30 -

0.35

0.35 -

0.40

>0.40

Complete Submodular

High type Medium type

Low type

Notes: The height of white bars represents the average payoﬀ of low-type participants in a particular market for a

given fraction of market oﬀers they make. Similarly, the heights of the orange and grey bars represent average payoﬀs

of medium- and high-type players, respectively, conditional on the fraction of oﬀers they make in a market.

for those pertaining to fewer than 0.25 of market oﬀers (see Table A.3 in Appendix A). However,

as the ﬁgure illustrates, we see little impact of this volume of oﬀers on average payoﬀs for any of

the types.

These results are interesting in view of the vast literature suggesting that bargaining strategies

may be important in various contexts, most notably in labor markets; see Bertrand (2011) and

Shell (2006). While our settings are certainly simplistic in many ways, they suggest a very diﬀerent

story. Market position appears to have a dominant impact on outcomes. Who one matches with,

and with what transfer, depends importantly on the role one occupies in the market, while being

relatively insensitive to how one bargains.

6 Conclusions

We report an array of results examining outcomes and the dynamics of interactions in decentral-

ized matching markets. We allow for transfers and consider markets with complete and incom-

plete information, and with supermodular or submodular surplus structures.

Without transfers, prior literature has suggested that decentralized interactions often culmi-

nate in stable, and hence eﬃcient, outcomes. Our ﬁrst main message is that, in the presence

of transfers, a substantial fraction of outcomes is still eﬃcient, while fewer outcomes are stable.

Participants often establish the matches required for eﬃciency, but ﬁnd it more diﬃcult to settle

on transfers that induce stable outcomes. Our second message is that limited information and

the shift from a supermodular to a submodular surplus structure have important impacts. Lim-

ited information constitutes an important hurdle for stability, even when participants have ample

opportunities to learn about others through market interactions. Stability is similarly more chal-

lenging to achieve with submodular surpluses. Together, incomplete information and submodu-

larity are deadly for stability—in those markets, less than 5% of ultimate outcomes are stable. Our

third message pertains to the dynamics of interactions. When surpluses are supermodular, high

types tend to match ﬁrst, followed by medium types, and then by low types. When surpluses are

submodular, there is substantial variation in the order of match formation and resulting payoﬀs.

Indeed, the ultimate payoﬀ distribution is nearly uniform. Participants’ behavior is consistent

with a satisﬁcing heuristic in which payoﬀ targets are type dependent. The ﬁnal message emerg-

ing from our data regards bargaining styles. While we see variation in bargaining patterns across

individuals, those seem to have little impact on ﬁnal payoﬀs. The main determinants of partici-

pants’ payoﬀs in our markets are the governing surplus structure and their positions, or types.

We believe our study opens the door for several further investigations, both theoretical and

experimental. As described in our literature review, several recent papers have suggested some

natural stability notions allowing for incomplete information on one side. The results from those

papers cannot be easily reconciled with our data. In part, this may be due to the fact that the

suggested notions are agnostic about the decentralized process that takes place in the market.

In particular, these solution concepts typically take as ﬁxed a speciﬁcation of uncertainty and

a candidate stable outcome, ignoring the fact that the residual uncertainty is likely to depend

on the process by which the candidate outcome is attained. We hope our experimental results

might be useful in reﬁning those stability notions to richer settings. Our results point to the need

for a model allowing agents to glean complete information about the other side of the market

through the matching process, while settling on an outcome that is not complete-information

stable. Our experiments also suggest that, when information is incomplete, there might be great

value for centralization if (complete-information) stability is an objective. Nonetheless, the extant

literature oﬀers little insights on the design of matching markets in the presence of incomplete

information, for any objective. We believe this would be a fruitful direction for future research.

Our experiments examine bargaining frictions due to transfers and incomplete information.

There are certainly other frictions that would be interesting to inspect. In applications, oﬀers are

often costly—in labor markets, employers frequently have limited interview slots or limited oﬀers

to make, regardless of their yield; in marriage markets, dating is costly in terms of both time and

money. Naturally, frictions pertaining to oﬀer costs may dampen market activity. That, in turn,

would limit learning even further, as participants “try out” fewer matches. The ineﬃciencies we

identify may then oﬀer lower bounds on the ineﬃciencies one might expect in practice. There are

other details that our basic design does not incorporate: whether oﬀers are exploding, commu-

nication channels between market participants, etc. We hope our study sets the stage for further

investigations of such additional frictions.

References

Agranov, M. and M. Elliott (2021): “Commitment and (in)Eﬃciency: A Bargaining Experi-

ment,” Journal of the European Economic Association, 19, 790–838.

Alston, M. (2020): “On the Non-existence of Stable Matches with Incomplete Information,”

Games and Economic Behavior, 120, 336–344.

Andreoni, J. and B. D. Bernheim (2009): “Social Image and the 50–50 Norm: A Theoretical and

Experimental Analysis of Audience Eﬀects,” Econometrica, 77, 1607–1636.

Becker, G. S. (1973): “A Theory of Marriage: Part I,” Journal of Political Economy, 81, 813–846.

Bertrand, M. (2011): “New Perspectives on Gender,” in Handbook of Labor Economics, vol 4b, ed.

by O. Ashenfelter and D. Card, Amsterdam: Elsevier, 1543–1590.

Bikhchandani, S. (2017): “Stability with One-sided Incomplete Information,” Journal of Economic

Theory, 168, 372–399.

Caplin, A., M. Dean, and D. Martin (2011): “Search and Satisﬁcing,” American Economic Review,

101, 2899–2922.

Chade, H., J. Eeckhout, and L. Smith (2017): “Sorting through search and matching models in

economics,” Journal of Economic Literature, 55, 493–544.

Chen, D. L., M. Schonger, and C. Wickens (2016): “oTree—An Open-source Platform for Labo-

ratory, Online, and Field Experiments,” Journal of Behavioral and Experimental Finance, 9, 88–97.

Chen, Y. and T. Sönmez (2006): “School Choice: An Experimental Study,” Journal of Economic

theory, 127, 202–231.

Chen, Y.-C. and G. Hu (2020): “Learning by Matching,” Theoretical Economics, 15, 29–56.

Chiappori, P.-A. (2017): Matching with Transfers: The Economics of Love and Marriage, Princeton

University Press.

Echenique, F. and L. Yariv (2013): “An Experimental Study of Decentralized Matching,” .

Ferdowsian, A., M. Niederle, and L. Yariv (2022): “Decentralized Matching with Aligned Pref-

erences,” .

Gneezy, U. and J. Potters (1997): “An Experiment on Risk Taking and Evaluation Periods,” The

Quarterly Journal of Economics, 112, 631–645.

Hakimov, R. and D. Kübler (2021): “Experiments on centralized school choice and college admis-

sions: A Survey,” Experimental Economics, 24, 434–488.

He, S., J. Wu, and H. Zhang (2022): “Decentralized Matching with Transfers: Experimental and

Noncooperative Analyses,” .

Kapor, A. J., C. A. Neilson, and S. D. Zimmerman (2020): “Heterogeneous Beliefs and School

Choice Mechanisms,” American Economic Review, 110, 1274–1315.

Liu, Q. (2022): “Stability and Bayesian Consistency in Two-Sided Markets,” American Economic

Review, 110, 2625–2666.

Liu, Q., G. J. Mailath, A. Postlewaite, and L. Samuelson (2014): “Stable Matching with Incom-

plete Information,” Econometrica, 82, 541–587.

Nalbantian, H. R. and A. Schotter (1995): “Matching and Eﬃciency in the Baseball Free-Agent

System: An Experimental Examination,” Journal of Labor Economics, 13, 1–31.

Niederle, M. and A. E. Roth (2009): “Market Culture: How Rules Governing Exploding Oﬀers

Aﬀect Market Performance,” American Economic Journal: Microeconomics, 1, 199–219.

Pais, J. and Á. Pintér (2008): “School Choice and Information: An Experimental Study on Match-

ing Mechanisms,” Games and Economic Behavior, 64, 303–328.

Pais, J., Á. Pintér, and R. F. Veszteg (2020): “Decentralized Matching Markets with(out) Frictions:

A Laboratory Experiment,” Experimental Economics, 23, 212–239.

Plott, C., N. Roy, and B. Tong (2013): “Marshall and Walras, Disequilibrium Trades and the

Dynamics of Equilibration in the Continuous Double Auction Market,” Journal of Economic Be-

havior and Organization, 94, 109–205.

Roth, A. E. and M. Sotomayor (1990): “Two-Sided Matching: A Study in Game-theoretic Model-

ing and Analysis,” Econometric Society Monographs, 18.

Roth, A. E. and J. H. V. Vate (1990): “Random paths to stability in two-sided matching,” Econo-

metrica: Journal of the Econometric Society, 1475–1480.

Shell, G. R. (2006): Bargaining for Advantage: Negotiation Strategies for Reasonable People, Penguin.

A Additional Analysis

A.1 Do Markets Start the Same Way?

Since most of our results consider the entirety of participants’ market interactions, one may won-

der whether the impact of the surplus variety occurs through market dynamics alone. In this

section, we show that even initial activity varies by the sort of surplus that governs preferences.

We focus on complete-information markets, since initial interactions in incomplete-information

markets are naturally noisy due to the limited knowledge participants have regarding their own

and others’ preferences.

Figure A.1 illustrates the frequency of matches formed and oﬀers made at the start of our

markets. As can be seen, diﬀerent patterns emerge across our markets. For example, in our

supermodular markets, initial matches are mostly formed between stable partners, with a pre-

dominance of matches and oﬀers between high types. In contrast, in our submodular markets,

while early matches between high types are still prevalent, most oﬀers occur between low and

high types, or between medium and high types. Furthermore, there is roughly an equal share of

matches between low and high, medium and high, or two medium types.

Table A.2 suggests that the proﬁle of transfers also diﬀers across market classes from the start.

Interestingly, submodular markets are characterized by more equal splits at the start of interac-

tions, even though, as we showed in the paper, this pattern reverses as markets evolve.

Table A.2: Frequency of Equal Splits in First Deals/Oﬀers

First two deals First two oﬀers

Supermodular Submodular Supermodular Submodular

LL 0.78 1.00 0.00

LM 0.71 1.00 0.64 0.50

LH 0.29 0.20 0.55 0.42

MM 0.83 0.95 0.77 0.75

MH 0.47 0.60 0.59 0.70

HH 0.75 0.89 0.67 0.79

Notes: Equal split corresponds to between 45% and 55% of the total surplus.

A.2 Bargaining Styles

Table A.3 below is the companion table for Figure 4 in the main text of the paper. It depicts the

number of observations we have for each of the categories representing the fraction of oﬀers made

Figure A.1: First Two Oﬀers and Deals with Complete Information

10%

15%

20%

25%

30%

35%

Supermodular Submodular Supermodular Submodular

First two deals First two offers

How do Complete Info markets start?

LL LM LH MM MH HH

Note: Numbers for LM, LH, and ML are divided by two because there are twice as many opportunities to

make these deals in the markets than the congruent one like LL, MM, and HH.

Notes: The frequencies for non-congruent matches (LM, LH, MH) are divided by two, since there are twice as many

opportunities to make such oﬀers/deals than congruent ones (LL, MM, and HH).

in a market by a particular player type in a particular market variety. There are 160 observations

for each player type in each market (8 groups × 10 markets × 2 players of each type).

Next, we replicate the analysis presented in the main body of the paper regarding bargain-

ing styles of market participants with respect to other characteristics of market behavior such as

number of matches made throughout the market, the share of surpluses demanded when making

oﬀers, and the timing in which the ﬁrst and last matches were created. For each of these four

characteristics, we present the variation across diﬀerent types of players and the eﬀects of this

variation on ﬁnal payoﬀs (using regression analysis similar to that reported in Table 6).

In Table A.6 we report tests pertaining to comparisons of coeﬃcients based on regression anal-

ysis of the bargaining style of “high earners.” Speciﬁcally, for each market structure, correspond-

ing to the governing surplus and information available, we run a linear regression in which the

dependent variable is one of the market behavior characteristics—number of oﬀers, number of

deals, share of surplus requested in oﬀers, and timing of the ﬁrst and last match—on a constant,

indicators for medium- and high-type participants, interaction terms corresponding to high earn-

ings and type, and control variables. The controls include the round number, the ﬁxed eﬀects

Table A.3: Number of observations for each category in Figure 4

Complete Supermodular Complete Submodular

low medium high low medium high

less than 0.05 21 28 39 6 13 41

0.05 - 0.10 30 33 26 15 20 45

0.10 - 0.15 21 23 28 35 33 31

0.15 - 0.20 19 24 22 23 25 15

0.20 - 0.25 18 11 19 19 33 12

0.25 - 0.30 15 16 8 30 12 9

0.30 - 0.35 16 12 7 14 12 4

0.35 - 0.40 7 6 5 7 7 1

more than 0.40 13 7 6 11 5 2

Incomplete Supermodular Incomplete Submodular

low medium high low medium high

less than 0.05 4 14 46 2 17 51

0.05 - 0.10 9 30 50 15 33 47

0.10 - 0.15 23 37 33 32 30 22

0.15 - 0.20 24 23 19 23 21 18

0.20 - 0.25 30 22 2 26 23 9

0.25 - 0.30 26 15 17 26 19 8

0.30 - 0.35 23 6 2 15 7 2

0.35 - 0.40 7 6 1 8 5 1

more than 0.40 14 7 0 13 5 2

Notes: We present the number of observations per category; there are 160 observations per market and type.

for individual participants, the total number of oﬀers made in the market in the ﬁrst set of re-

gressions, and the total number of deals made in the market in the second set of regressions. We

cluster standard errors at the session level to account for interdependencies of observations that

come from the same session. We report the estimated coeﬃcients and the robust standard errors

in parentheses for the interaction terms of high earners with their type and test whether those in-

teraction coeﬃcients are signiﬁcantly diﬀerent from zero. If they are, this means that high earners

of the speciﬁed type behave in a systematically diﬀerent way than low earners.

Table A.4: Variation in Market Behavior

Number of Matches by Type (frac) Share of Surplus in Oﬀers (out of 100%)

Low Medium High Low Medium High

Complete

Supermodular 0.31 (0.06) 0.36 (0.06) 0.33 (0.06) 49.15 (6.58) 49.31 (6.36) 55.20 (4.79)

Submodular 0.24 (0.06) 0.34 (0.06) 0.40 (0.08) 40.53 (6.56) 48.85 (4.22) 58.67 (8.07)

Incomplete

Supermodular 0.38 (0.08) 0.37 (0.06) 0.25 (0.08) 47.45 (7.31) 49.68 (7.00) 52.63 (7.86)

Submodular 0.33 (0.06) 0.32 (0.07) 0.34 (0.09) 43.59 (7.49) 50.85 (6.69) 55.36 (5.95)

First Match (quintile) Last Match (quintile)

Low Medium High Low Medium High

Complete

Supermodular 3.02 (1.09) 1.98 (0.71) 1.88 (0.77) 4.00 (0.89) 3.58 (0.98) 3.15 (1.02)

Submodular 2.40 (0.99) 1.82 (0.77) 1.39 (0.48) 3.93 (0.86) 4.06 (0.73) 3.72 (0.70)

Incomplete

Supermodular 1.09 (0.15) 1.08 (0.14) 1.05 (0.11) 4.47 (0.70) 4.00 (0.91) 2.57 (0.88)

Submodular 1.13 (0.24) 1.08 (0.19) 1.05 (0.12) 4.24 (0.70) 3.95 (0.85) 3.65 (0.97)

Table A.5: Does Market Behavior Aﬀects Final Payoﬀs? Regression Analysis

low

med

high

low

= β

med

low

= β

high

med

= β

high

(n)

Complete

Supermodular 0.82 (1.22) -2.53 (1.31) 20.10 (17.51) 0.18 0.32 0.30 0.94 (480)

Submodular -3.67 (3.04) -6.04

∗

(2.44) 2.70 (1.79) 0.35 0.26 0.08 0.84 (480)

Incomplete

Supermodular 2.20 (2.19) -9.13

∗

(3.28) -9.30

∗∗

(2.34) 0.08 0.03 0.97 0.91 (480)

Submodular -11.63 (6.59) -8.66

∗∗

(2.12) 1.44 (5.22) 0.73 0.01 0.23 0.68 (480)

(a) Number of Matches by Type

low

med

high

low

= β

med

low

= β

high

med

= β

high

(n)

Complete

Supermodular 0.01 (0.01) 0.01 (0.02) 0.02 (0.02) 0.83 0.57 0.61 0.92 (428)

Submodular -0.02 (0.07) -0.04 (0.02) -0.01 (0.05) 0.81 0.81 0.37 0.83 (448)

Incomplete

Supermodular 0.003 (0.01) 0.10 (0.05) 0.04 (0.04) 0.19 0.48 0.22 0.91 (397)

Submodular 0.05 (0.04) -0.04 (0.09) 0.14 (0.08) 0.46 0.52 0.05 0.69 (427)

(b) Share of Surplus in Oﬀers

low

med

high

low

= β

med

low

= β

high

med

= β

high

(n)

Complete

Supermodular -0.04 (0.03) 0.17 (0.11) -0.03 (0.08) 0.23 0.86 0.14 0.96 (478)

Submodular -0.29 (0.15) 0.21 (0.14) 0.43 (0.51) 0.02 0.29 0.71 0.84 (480)

Incomplete

Supermodular -0.38 (0.55) 0.84

∗∗

(0.16) 0.54

∗

(0.16) 0.12 0.23 0.26 0.91 (480)

Submodular -0.26 (0.97) 0.99 (1.91) -0.65 (1.28) 0.30 0.50 0.15 0.68 (480)

low

med

high

low

= β

med

low

= β

high

med

= β

high

(n)

Complete

Supermodular 0.006 (0.03) -0.01 (0.13) 0.30 (0.17) 0.92 0.21 0.09 0.96 (466)

Submodular -0.37 (0.28) -0.19 (0.23) 0.11 (0.36) 0.42 0.51 0.61 0.84 (476)

Incomplete

Supermodular 0.16 (0.17) -0.26 (0.24) -0.47 (0.16) 0.38 0.00 0.61 0.91 (480)

Submodular -0.85 (0.74) -0.83 (0.30) 0.43 (0.28) 0.98 0.19 0.10 0.68 (480)

(d) Timing of Last Match (quintile)

Notes: Each row corresponds to one regression. The dependent variable is the ﬁnal payoﬀ of a player. The independent

variables contain a constant, the dummies for medium- and high-type players, interaction between the relevant charac-

teristics of market behavior (number of deals, share of surplus, timing of the ﬁrst and the last match) and player type,

as well as individual controls for risk attitudes and degree of altruism as captured by the auxiliary tasks conducted at

the end of the experiment. We cluster the standard errors by session. The number of observations in the panel (b) is

smaller than in other panels because some participants never made an oﬀer in a particular market.

Table A.6: Individual Analysis of Bargaining Styles and Its Eﬀect on Payoﬀs

Number of Oﬀers Number of Deals

Low Medium High Low Medium High

Complete

Supermodular 0.05

∗∗

(0.01) 0.003 (0.01) 0.0001 (0.01) -0.003 (0.02) -0.02

∗∗

(0.006) -0.01 (0.02)

Submodular 0.07

∗

(0.03) 0.02

∗∗

(0.006) 0.02 (0.02) 0.02 (0.02) 0.01 (0.02) 0.03 (0.02)

Incomplete

Supermodular 0.11

∗∗

(0.02) 0.10

∗∗

(0.01) 0.11

∗∗

(0.01) -0.10

∗∗

(0.03) -0.08

∗∗

(0.009) -0.04 (0.02)

Submodular 0.18

∗∗

(0.02) 0.14

∗∗

(0.03) 0.19

∗∗

(0.03) 0.01 (0.03) -0.02 (0.04) -0.00004 (0.0002)

Share of Surplus in Oﬀers

Low Medium High

Complete

Supermodular 3.77

∗∗

(0.66) 4.47

∗∗

(1.18) 1.97

∗

(0.75)

Submodular 3.84 (2.45) 5.74

∗∗

(1.57) 5.11

∗∗

(1.17)

Incomplete

Supermodular 1.29 (1.97) 1.31 (0.61) 0.48 (1.32)

Submodular -7.17

∗∗

(1.64) -7.80

∗∗

(1.89) -3.22

∗∗

(0.76)

Timing of First Deal (in quintiles) Timing of Last Deal (in quintiles)

Low Medium High Low Medium High

Complete

Supermodular 0.31 (0.25) -0.09 (0.11) -0.30 (0.22) 0.58

∗∗

(0.12) -0.12 (0.09) -0.08 (0.12)

Submodular -1.34

∗∗

(0.12) -1.28

∗∗

(0.20) -1.37

∗∗

(0.18) -0.04 (0.06) -0.39 (0.18) -0.20 (0.14)

Incomplete

Supermodular -0.11 (0.06) -0.10

∗∗

(0.01) -0.09

∗

(0.04) -0.48 (0.36) -0.69

∗∗

(0.16) -0.19 (0.11)

Submodular 0.05 (0.03) 0.12

∗∗

(0.02) 0.11

∗∗

(0.02) 0.42 (0.34) 0.27 (0.20) 1.00

∗∗

(0.16)

Notes: see description above

B Predictions from Stability with Transfers

As described in Section 3, a stable outcome is an individually rational outcome that has no block-

ing pairs. When transfers are available, there exist many transfer proﬁles that sustain stable

outcomes in both our supermodular and submodular markets. The associated stable matching,

however, is unique for both market types. The table below shows the system of inequalities for

each market type that identiﬁes the set of stable transfer proﬁles. We use x

to denote the payoﬀ

of market participant i, using the labels in Table 1 in the main text. The ﬁrst column corresponds

to our supermodular market, the second to our submodular market.

Supermodular market Submodular market

Kiwi

≥ 0 x

Kiwi

≥ 0

Mango

≥ 0 x

Mango

≥ 0

Plum

≥ 0 x

Plum

≥ 0

Blue

≥ 0 x

Blue

≥ 0

Crimson

≥ 0 x

Crimson

≥ 0

White

≥ 0 x

White

≥ 0

Kiwi

+ x

Blue

≥ 8 x

Kiwi

+ x

Blue

≥ 8

Kiwi

+ x

Crimson

≥ 16 x

Kiwi

+ x

Crimson

≥ 32

Kiwi

+ x

White

≥ 24 x

Kiwi

+ x

White

≥ 56

Mango

+ x

Blue

≥ 16 x

Mango

+ x

Blue

≥ 32

Mango

+ x

Crimson

≥ 32 x

Mango

+ x

Crimson

≥ 48

Mango

+ x

White

≥ 48 x

Mango

+ x

White

≥ 64

Plum

+ x

Blue

≥ 24 x

Plum

+ x

Blue

≥ 56

Plum

+ x

Crimson

≥ 48 x

Plum

+ x

Crimson

≥ 64

Plum

+ x

White

≥ 72 x

Plum

+ x

White

≥ 72

Kiwi

+ x

Mango

+ x

Plum

+ x

Kiwi

+ x

Mango

+ x

Plum

Blue

+ x

Crimson

+ x

White

≥ 112 +x

Blue

+ x

Crimson

+ x

White

≥ 160

C ONLINE APPENDIX: Sample Instructions

Welcome. Welcome to EBEL and thank you for participating in today’s experiment. Please place

all of your personal belongings away so that we can have your complete attention. Please use the

laptops as instructed. In particular, please do not attempt to browse the web or use programs

unrelated to the experiment.

Guidelines. You will be paid in private and in cash at the end of the experiment. The amount

that you ultimately earn in the experiment depends on your decisions, the decisions of others, and

random chance. You have each earned a $10 payment for showing up on time. You will be using

laptops for the entire experiment, and all interactions between yourself and others will take place

via the laptop?s terminal. Please DO NOT socialize or talk.

Overview. Today’s experiment is about matching. The main part of the experiment consists

of 11 rounds: one practice round and 10 actual rounds. We will also ask you to complete several

simple tasks at the end.

At the beginning of the experiment, you will be randomly assigned a role: either a color or a

food. Your role will remain ﬁxed across all rounds of the experiment. At the beginning of each

round, you will be randomly assigned a type:

• If you are a color, you can be one of three types: Blue, Crimson, or White.

• If you are a food, you can be one of three types: Kiwi, Mango, or Plum.

Your type (but not your role) may change across each round of the experiment.

In each round, you will be randomly assigned to a group of six: three colors (one Blue, one

Crimson, and one White) and three foods (one Kiwi, one Mango, and one Plum). Your goal in

each round is to match with an individual from the opposite role. If you are a color, your goal is

to match with a food. If you are a food, your goal is to match with a color.

Match Payoﬀ.

KIWI MANGO PLUM

BLUE 8 16 24

CRIMSON 16 32 48

WHITE 24 48 72

A color and food who match together earn a match payoﬀ (in tokens). You will earn a diﬀerent

payoﬀ depending on whom you match with and how you choose to split the match payoﬀ.

Example 1:

KIWI MANGO PLUM

BLUE 8 16 24

CRIMSON 16 32 48

WHITE 24 48 72

Blue and Mango can match together, with the sum of payoﬀs for Blue and Mango yielding 16

tokens. For example, Blue could receive a payoﬀ of 10 tokens and Mango would then receive a

payoﬀ of 6 tokens. Another example: Blue could receive a payoﬀ of 7 tokens and Mango would

then receive a payoﬀ of 9 tokens.

Example 2:

KIWI MANGO PLUM

BLUE 8 16 24

CRIMSON 16 32 48

WHITE 24 48 72

Crimson and Plum can match together, with the sum of payoﬀs for Crimson and Plum yield-

ing 48 tokens. For example, Crimson could receive a payoﬀ of 30 tokens and Plum would then

receive a payoﬀ of 18 tokens. Another example: Crimson could receive a payoﬀ of 23 tokens and

Plum would then receive a payoﬀ of 25 tokens.

Example 3:

KIWI MANGO PLUM

BLUE 8 16 24

CRIMSON 16 32 48

WHITE 24 48 72

White and Kiwi can match together, with the sum of payoﬀs for White and Kiwi yielding 24

tokens. For example, White could receive a payoﬀ of 18 tokens and Kiwi would then receive a

payoﬀ of 6 tokens. Another example: White could receive a payoﬀ of 9 tokens and Kiwi would

then receive a payoﬀ of 15 tokens.

Information about the payoﬀ matrix. At the beginning of each round, we will reshuﬄe the

rows and columns of the matrix.

KIWI MANGO PLUM

BLUE 8 16 24

CRIMSON 16 32 48

WHITE 24 48 72

Examples:

KIWI MANGO PLUM

BLUE 8 16 24

CRIMSON 16 32 48

WHITE 24 48 72

KIWI MANGO PLUM

BLUE 16 32 48

CRIMSON 24 48 72

WHITE 8 16 24

KIWI MANGO PLUM

BLUE 24 16 8

CRIMSON 48 32 16

WHITE 72 48 24

KIWI MANGO PLUM

BLUE 48 32 16

CRIMSON 72 48 24

WHITE 24 16 8

All possible reshuﬄings are equally likely. At the beginning of each round, you will not know

how the matrix was reshuﬄed, but you will know which payoﬀs you might receive.

Examples. Suppose you are Blue, and you learn that your payoﬀs might be 16, 32, 48. It might

be that a match with Kiwi generates 16, a match with Mango generates 32, and a match with Plum

generates 48.

KIWI MANGO PLUM

BLUE 16 32 48

CRIMSON 24 48 72

WHITE 8 16 24

It might be that a match with Kiwi generates 48, a match with Mango generates 32, and a

match with Plum generates 16.

KIWI MANGO PLUM

BLUE 48 32 16

CRIMSON 72 48 24

WHITE 24 16 8

All possible reshuﬄings are equally likely. While you will know your possible payoﬀs, you

will not know which payoﬀ corresponds to which matching. Your payoﬀs will appear on the top

part of your interface. Note that no other subject in your group observes your possible match

payoﬀs. Each person only sees his or her own possible match payoﬀs.

Rules of the Experiment. In each round, you will start oﬀ unmatched. You are free to make

at most one match proposal to any individual of the opposite role at any given time.

A match proposal must specify what your partner will get. If your proposal is accepted, then

you will get the match payoﬀ minus what your partner will get. You are also free to accept or

reject match proposals that you receive from other individuals. Any match proposal that is not

accepted or rejected within 10 seconds will automatically disappear. When a match is accepted,

both the proposer and the receiver see the payoﬀs they get from the match.

Example 1: Suppose you are Blue, and you know your possible match payoﬀs are 16, 32, and

48. If you make an oﬀer of 10 to Kiwi, which gets accepted, you might see that you get 6 and Kiwi

gets 10. This implies that you and Kiwi have a match payoﬀ of 6+10 = 16.

Example 2: Suppose you are Blue, and you know your possible match payoﬀs are 16, 32, and

48. If you make an oﬀer of 10 to Kiwi, which gets accepted, you might see that you get 38 and

Kiwi gets 10. This implies that you and Kiwi have a match payoﬀ of 38+10 = 48.

Click on the square to

make an offer

Move slider to suggest

the responder’s payoff

Your possible match

payoffs

Click on the square to

make an offer

Move slider to suggest

the responder’s payoff

Your possible match

payoffs

Confirmation of offer

Decision to accept

or reject

Timer

When your offer

is accepted, you

see the payoffs

When you accept

an offer, you

see the payoffs

Current match

entries are

shaded in orange

Rules of the Experiment, cont... If you proposed a match which was accepted and in which

your own earnings are negative, then you will have an opportunity to unilaterally break this match

by clicking a “cancel” button.

Negative payoff

Click Cancel button to

cancel this match

If you are currently matched, you can still make a match proposal to another individual. If the

match proposal is accepted, then your current match will be broken. If you are currently matched,

you can still receive match proposals from other individuals. If you accept a match proposal, then

your current match will be broken. When you accept a match, or someone else accepts a match

with you, any proposal you had made or received before will be automatically rejected.

You can make an

offer while matched

Confirmation of

offer while matched

Confirmation of

offer while matched

A round ends after 30 seconds of inactivity (i.e., after 30 seconds in which no new match

proposals have been made). Whenever you are ready to proceed, you can also press the “Move on

to Next Round” button. When at least 5 in your group press the button, you will automatically

move to the next round (without waiting the 30 seconds). If any activity occurs after you press

the button, the button will get released.

Press whenever

ready to proceed

to next round

Payoﬀs for a Round. Your payoﬀ when the round ends is your ﬁnal payoﬀ for that round. If

you are not matched at the end of the round, your ﬁnal payoﬀ for that round is 0. Your payoﬀ

from temporary matches that are made and then broken during a round do not count for your

ﬁnal payoﬀ.

Timer for the round

(round ends after

30 seconds of

inactivity)

Post-Experiment. At the end of the experiment, you will be paid the sum of your payoﬀs

across rounds (excluding the practice round). You will also be asked to complete several simple

tasks at the end. You can earn additional money based on your decisions in these tasks.

Your Earnings. Your total earnings in the experiment are the sum of the following amounts:

• $10 show-up payment

• payoﬀ from 10 main rounds: 10 tokens = 1 dollar

• payoﬀ from the simple tasks:

You need not tell any other participant how much you earned. If there are no questions, we

will now begin the actual experiment.