ISSN (Print) 0473-453X

Discussion Paper No. 1138

ISSN (Online) 2435-0982

PARTIAL EQUILIBRIUM
MECHANISM AND
INTER-SECTORALCOORDINATION:
AN EXPERIMENT
Nobuyuki Hanaki
Takashi Hayashi
Michele Lombardi
Kazuhito Ogawa

August 2021

The Institute of Social and Economic Research
Osaka University
6-1 Mihogaoka, Ibaraki, Osaka 567-0047, Japan

Partial equilibrium mechanism and inter-sectoral
coordination: an experiment∗
Nobuyuki Hanaki†

Takashi Hayashi‡Michele Lombardi§

Kazuhito Ogawa¶August 4, 2021

Abstract
This study experimentally evaluates the performance of partial equilibrium mechanisms
when different sectors run their mechanisms separately, despite the existence of complementarity between them. In our simple laboratory experiment setting that includes two sectors,
each sector runs the top-trading-cycle mechanism. There is a Pareto-dominant equilibrium,
but it requires coordination across sectors. Our results show that coordination failure occurs more frequently when there is asymmetry between the two sectors compared with the
one-sector benchmark, even without inter-sectoral complementarity. When mechanisms arerun sequentially across the two sectors, such failure is substantially reduced, compared withwhen they are run simultaneously.

Keywords: Partial equilibrium, inter-sectoral coordination, top-trading-cycle mechanism, laboratory experiment
JEL Code: C92, D71, D82∗ This work was supported by Kansai University Expenditures for Support of Training Young Scholars, 20162017, Research fund of Japan Center for Economic Research, 2017, Japan Society for the Promotion of Science,
Grant-in-Aid for Scientific Research, Nos. 15H05728, 18K19954, and 20H05631, and the Joint Usage/ResearchCenter, Institute of Social and Economic Research, Osaka University. The experiments reported in this paper
have been approved by the Institutional Review Board of the Research Institute of Socio-network Strategies,
Kansai University (No. 2017004, No. 2017012, No. 2017043, No. 2018010).
† Institute of Social and Economic Research, Osaka University. E-mail: nobuyuki.hanaki@iser.osaka-u.ac.jp‡ Corresponding
author.
Adam Smith Business School, University of Glasgow.
E-mail:Takashi.Hayashi@glasgow.ac.uk§ Adam Smith Business School, University of Glasgow. E-mail: Michele.Lombardi@glasgow.ac.uk¶ Faculty of Sociology, Research Institute for Socio-network Strategies, Kansai University.
E-mail: kzogawa@kansai-u.ac.jp1

1

Introduction

This study experimentally evaluates the performance of partial equilibrium mechanisms when
they are implemented in an institutionally realistic way. Different sectors run their mechanisms
separately despite the existence of complementarity between them.
The utilization of virtually any good, in which term we include any service, is related tothe utilization of other goods, and vice versa. Hence, apart from the consumption allocation ofother goods, we cannot generically define a preference for the consumption of a single good or
willingness to pay for it.
Therefore, in contrast to the presumption that accompanies operating partial equilibriummechanisms in standard practice, we cannot isolate the issue of allocating a single good from therest of the economy by assuming that other things remain equal. When we apply a partial
equilibrium mechanism to some sectors based on such a false presumption, it will have anunintended effect on the other sectors. In addition, there must be mutual feedback acrosssectors because partial equilibrium mechanisms are adopted in many sectors of our society, andsimultaneity among them is inevitable.
The partial equilibrium approach is justified when handling a sector that is small comparedwith an economy consisting of many sectors. In such a situation, the inter-sectoral dependenceis negligible. This idea dates back to Marshall (1920), and it was formally proven later by Vives(1987). However, economically important problems are typically not minor, and the issue ofinter-sectoral dependence is significant.
A prominent example is school choice and the housing market. Schools and houses are naturalcomplements of each other because they are only valuable when they are in close proximity.
Hence, one cannot have a “preference” over schools independently of one’s house, and viceversa. Thus, if we change something in the school allocation mechanism, it has been empiricallyshown that the housing market is affected (Bonilla-Mejı́a et al., 2020; Chung, 2015; Danielsen
et al., 2015; Reback, 2005). Nevertheless, school choice programs commonly ask each householdto submit only their school preference, while ignoring the complementarity with housing choice.
An ideal solution is for a central planner to solve the allocation problem for the economy.
However, this is typically infeasible. Thus, we have to consider it a given institutional constraintthat the economy is divided into many sectors, and each sector’s authority has a limited viewin which agents only have preferences over objects in that sector. Such a division as the given
institutional constraint results in agents determining how to achieve inter-sectoral coordination.
This typically occurs in the school choice/housing market case, especially when a household2searches for a house and a school simultaneously.
Hayashi and Lombardi (2017) provides a theoretical characterization of the social outcomesthat are viable under this institutional constraint when individuals behave strategically, particularly in the form of Nash equilibria. They consider a society with multiple sectors in which
each sector authority oversees the given sector, and does not communicate with other sector
authorities, and agents send their messages separately to different sector authorities.
This leaves an empirical question about the agents’ behavior regarding inter-sectoral coordination under the institutional constraint. Also, it remains unclear how much allocation efficiency
is indeed achieved. The current study is the first attempt to answer these questions through acontrolled laboratory experiment. We achieve this by constructing a simple setting to extract
the nature of the problem, which arises when several sectors use a partial equilibrium mechanism
separately, based on the mis-specification described above. We focus on whether subjects canoperate in an equilibrium where they report the corresponding marginal rankings induced bytheir true preferences for each sector, and which yield a Pareto-dominant outcome.
Separately running partial equilibrium mechanisms across sectors without participants communicating among themselves is rather ubiquitous. It is unclear which mechanisms we should
begin our investigation with. They may be centralized or decentralized, depending on the natureof the objects that are to be allocated. Although our substantive concerns are ultimately about
real-life problems, especially in the school choice/housing market case, we need to simplify thedetails to focus on the inter-sectoral coordination problem.
In this study, we adopt the top-trading-cycle (TTC) mechanism (Shapley and Scarf, 1974)
for our partial equilibrium exchange mechanism. Although our investigation is motivated by theschool choice/housing market problem, we do not intend to copy the TTC mechanism precisely.
From an analytical perspective, we adopt the TTC mechanism for two reasons. First, it simplifies
the laboratory experiment. We do not exclude other practical partial equilibrium mechanisms,
such as auction mechanisms and decentralized trading mechanisms such as bargaining, matching,and searching. Second, the TTC mechanism has desirable benchmark properties. When it is run
for a single sector, it is strategy-proof (reporting true preference is always a dominant strategy)
and Pareto-efficient. It is also group strategy-proof (no group can gain by joint misreporting)and delivers a unique core allocation.
The simple nature of the TTC mechanism allows us to concentrate on the inter-sectoral
coordination issue because there is no theoretical source of inefficiency or manipulation at thesingle-sector level. In general, other mechanisms are not fully strategy-proof or involve coordi-

3

Table 1: Sector 1 and Sector 2 games

A(a) Sector 1 GameB
eA1  eB1 eB1  eA1eA1  eB1 (eA1 , eB1 ) (eA1 , eB1 )eB1  eA1 (eB1 , eA1 ) (eA1 , eB1 )A(b) Sector 2 GameB
eA2  eB2 eB2  eA2eA2  eB2 (eA2 , eB2 ) (eA2 , eB2 )eB2  eA2 (eB2 , eA2 ) (eA2 , eB2 )nation problems implementing efficient allocations, even when confined to a single sector.
When TTC is run for different sectors separately, however, it is not strategy-proof overall.
This makes efficiency harder to achieve. For example, consider two sectors. In both sectors,
every subject is endowed with one indivisible item, as in Shapley and Scarf (1974). The TTC
mechanism in Sector i ∈ {1, 2} works as each subject submits a ranking for the items in Sector
i. Thus, the allocation of Sector i items is determined by reported rankings for Sector i items.
However, each subject’s reward is determined by the combination of items he or she obtainedin Sectors 1 and 2. We set rewards so that the subjects mutually gain only when they successfully
coordinate their reports across sectors to obtain particular combinations of Sector 1 and Sector
2 items. They are unable to communicate before submitting their rankings.
To demonstrate further, consider the simplest case with two subjects, A and B. A is endowedwith items eA1 and eA2 , while B is endowed with items eB1 and eB2 . If we look at Sector 1 alone,each subject submits a ranking over the two items, and exchange occurs only when they placeeach other’s items on top. The game form is summarized in panel (a) of Table 1. We can see that
if preference over Sector 1 is independent of the allocation in Sector 2, it is always a dominant
strategy to report the true ranking. In other words, one can never gain by misreporting his/herranking.
Likewise, if we look at Sector 2 alone, each subject submits a ranking over the two items, andexchange occurs only when they put each other’s items on top. The game form is summarizedin panel (b) of Table 1. Again, we can see that if the preference over Sector 2 is independent of
the allocation in Sector 1, it is always a dominant strategy to report the true ranking. In otherwords, one can never gain by misreporting his/her ranking.
The entire game form is summarized in Table 2. There is no longer a dominant strategy
when preferences for paired items cannot be separated. Let us assume that while both players
strictly prefer the other’s pair to their own, any mismatched pairs are worthless to them. That
is, A prefers eB1 eB2 over eA1 eA2 , and all the other allocations are equally less ideal than eA1 eA2to him. B prefers eA1 eA2 over eB1 eB2 , and all the other allocations are equally less ideal than4

Table 2: 2-sector gameA

eA1  eB1
eA2  eB2
eA1  eB1
eB2  eA2
eB1  eA1
eA2  eB2
eB1  eA1
eB2  eA2

eA1  eB1
eA2  eB2

B
eA1  eB1
eB2  eA2

eB1  eA1
eA2  eB2

eB1  eA1
eB2  eA2

(eA1 eA2 , eB1 eB2 )(eA1 eA2 , eB1 eB2 )(eA1 eA2 , eB1 eB2 )(eA1 eA2 , eB1 eB2 )(eA1 eB2 , eB1 eA2 )(eA1 eA2 , eB1 eB2 )(eA1 eB2 , eB1 eA2 )(eA1 eA2 , eB1 eB2 )(eB1 eA2 , eA1 eB2 )(eB1 eA2 , eA1 eB2 )(eA1 eA2 , eB1 eB2 )(eA1 eA2 , eB1 eB2 )(eB1 eB2 , eA1 eA2 )(eB1 eA2 , eA1 eB2 )(eA1 eB2 , eB1 eA2 )(eA1 eA2 , eB1 eB2 )eB1 eB2 for her. The efficient allocation (eB1 eB2 , eA1 eA2 ) can be obtained in Nash equilibrium,
but it occurs only when A and B successfully coordinate their reports in both sectors. Note that
it is optimal for A to submit the ranking eB1  eA1 in Sector 1 because he is expecting to get
eB2 instead of eA2 in Sector 2, and it is optimal for him to rank eB2  eA2 in Sector 2 because
he is expecting to get eB1 instead of eA1 in Sector 1. A similar circularity occurs for B as well.
In other words, these “rankings” are equilibrium constructs.
The efficient allocation above is a Pareto-dominant outcome, however, it is somewhat risky
to achieve because a single failure in either sector leads to an allocation with unmatched pairs,
which is the worst case. This is riskier when there are more than two subjects involved in theseexchanges. This leaves the possibility that the subjects send messages so that no trade occurs
in either sector, which is Pareto-inferior but a “safe” second-best. For example, because tradeoccurs only by mutual agreement, no trade is always a trivial equilibrium.
Our research questions are as follows: (i) are subjects able or willing to engage in coordination across sectors to obtain an efficient allocation in the presence of complementarity whenpartial equilibrium mechanisms are run separately? ; (ii) quantitatively, what is the success rateand loss of efficiency? Furthermore, we ask (iii) by how much does sequentially running themechanisms across sectors help the subjects improve their inter-sectoral coordination comparedwith simultaneously running the mechanisms.
We consider five games played by three subjects. Games 1 and 2 have only a single sector,
and they serve as our benchmark. Game 1 is the simplest, in the sense that the TTC mechanism
finalizes allocation in one round.
Game 2 is more complex so that the mechanism finishes in two rounds after one subject
leaves in the first round.
Games 3, 4, and 5 have two sectors. Games 3 and 4 have complementarity between sectors,

5but Game 5 does not have inter-sectoral complementarity. Game 3 is simpler in the sense that thetrading cycles required for the first-best allocation exhibit the same pattern across two sectors,
so that the inter-sectoral coordination is straightforward. Game 4 is more complex because
the first-best allocation requires opposite directions of trading cycles across sectors. Hence, theinter-sectoral coordination is less straightforward in Game 4. There is no need for inter-sectoral
coordination in Game 5 because the two sectors are independent. There are two treatments:
one in which the two-sector games (Games 3,4, and 5) are run simultaneously and the other inwhich they are run sequentially.
We find that when two sectors are lined up in a simpler way (Game 3), the problem of intersectoral coordination is minor. Participants achieve an efficiency level like that in the singlesector problem, despite the need for inter-sectoral coordination. Sequential elicitationdoes not
improve upon the high efficiency of simultaneous elicitation. However, when two sectors are
lined up in more complex ways (Games 4 and 5), efficiency falls compared with the single-sector
case. Surprisingly, this applies even in the absence of inter-sectoral coordination (Game 5), andin such a case, sequential elicitation improves efficiency, compared with simultaneous elicitation.
The remainder of this paper is organized as follows: Section 2 provides a summary of theliterature that is most relevant to our work. The details of the experimental design are presented
in Section 3. Section 4 presents the results, and Section 5 provides concluding remarks.

2

Related Literature

To our knowledge, the current paper is the first experimental approach to the intersectoral
coordination problem, in the context of social choice and implementation. It is motivated by
the theoretical work of Hayashi and Lombardi (2017), which is a natural extension of the Nashimplementation problem (Maskin, 1999) in a multisectoral environment where the sectors do
not communicate with each other. Cabrales et al. (2003) performed an experiment on Maskin’s
canonical mechanism for Nash implementation, in a single-sector environment with three people
and three abstract outcomes, and found that the mechanism successfully implements the desired
outcome in a large majority of instances. It is left unclear, however, whether we can achieve adesired outcome under the institutional constraint in which we must run mechanisms separately
for different sectors despite the existence of complementarity between the sectors.
Our sequential treatment is motivated by the theoretical work of Hayashi and Lombardi(2019), which studies what can be implemented in the multi-period setting in which allocationin the current period must be chosen and finalized, without any commitment to allocations in
6

future periods. This handles more restricted situations than in the existing research on subgameperfect implementation (Moore and Repullo, 1988), which allows that an entire social outcome,once chosen at some stage, can be overturned at any subsequent stage. Some obstacles for
subgame-perfect implementation are reported. Aghion et al. (2018) find that the Moore andRepullo mechanism does not induce truth-telling as theoretically predicted and that individuals
lie because of pessimistic beliefs about subsequent players’ rationality. Fehr et al. (2021) findthat sequential mechanisms induce an individual to retaliate against somebody who chose, in aprevious stage, something socially desirable but unfavorable at a previous stage. In our experiment, on the other hand, sequential treatment improves efficiency over simultaneous treatment.
This is because the allocation of Sector 1 items is finalized at Stage 1, and this makes it morestraightforward to understand how to play in Stage 2, which in turn makes playing in Stage 1
more straightforward.
Another related problem is the auction of multiple items, where the bidders’ preferences arefor complementary bundles, and the question is whether it is better for the seller to bundle theitems or not (see, Avery and Hendershott, 2000; Krishna and Rosenthal, 1996; Levin, 1997; Subramaniam and Venkatesh, 2009). Popkowski Leszczyc and Häubl (2010) experimentally showthat bundling items promotes aggressive bids by the bidders with a preference for complementary bundles when the degrees of complementarity are known. However, the theoretical works
above show that there is a range of environments in which it is beneficial for the seller to run
separate auctions for different items. This is primarily because the necessity of winning all theitems makes certain bidders aggressive in each auction when they are unsure about the degrees
of complementarity for the other bidders. Such situations generate a positive probability of
inefficiency, in the sense that a bidder with a preference for complementarity can win only someitems, which is worthless to the bidder, and thus he has to resell. This point relates to the natureof efficiency loss in our problem.
Although our choice of the TTC mechanism is only meant to be a simplified expression ofgeneral situations in which partial equilibrium mechanisms are being operated, there are actual
instances of allocation problems where this mechanism is applicable, at least at an experimental
level, and some cases are put into practice. In addition, one may have a legitimate question
about whether the TTC mechanism works in the single-sector case.
Abdulkadiroğlu and Sönmez (2003) extends the TTC and Gale–Shapley (GS) mechanisms
to the school choice problem, and shows that they are strategy-proof, in that, it is always
a dominant strategy for each student to submit his/her true preference, in contrast to the7traditional Boston mechanism. They also show that the TTC mechanism achieves efficiency
among students, although, unlike the GS mechanism, it does not eliminate justifiable envy
between students.
Chen and Sönmez (2006) conducts an experimental study of school choice and reports that
the TTC and GS mechanisms improve efficiency over the Boston mechanism. Pais and Pintér(2008) experimentally shows that the TTC mechanism performs better than GS and Boston
mechanisms, which is consistent with the theory, and also that TTC outperforms the other two
in terms of the truthful revelation of preference and is also more robust to deviations. In acomplete information setting, Chen et al. (2016) experimentally shows that the TTC mechanism
outperforms the GS and Boston mechanisms in efficiency. The school choice version of the TTC
mechanism has been put to practice, for example, in the New Orleans Recovery School District(see, Abdulkadiroglu et al., 2017).
Abdulkadiroğlu and Sönmez (1999) extends the TTC mechanism to the house allocation
problem with existing tenants. Chen and Sönmez (2002) experimentally shows that the TTC
mechanism produces significantly more efficient allocations than the random serial dictatorshipmechanism, a mechanism widely used in on-campus university housing allocation.
The TTC mechanism was extended to the kidney exchange program, a rather unexpected
direction, by Roth et al. (2004), so that patients can swap their living donors for a better matchin blood types. The extended TTC mechanism, called top trading cycles and chains, has been
put to practice in the United States (see Roth et al., 2005).
In these applications, preferences for schools and houses and the supply of living donors are
considered exogenous. Our study will serve as an indicator of what we should expect whenthese elements are affected by a potential source of endogeneity due to complementarity withallocations in other sectors.
In the two-sector games in our study, there is a Nash equilibrium that achieves the firstbest although there is generally no dominant strategy. This is reminiscent of the theoreticalobservation that the Boston mechanism is not strategy-proof, but implements efficient outcomesin Nash equilibria (Ergin and Sönmez, 2006). In a school choice experiment, Featherstone andNiederle (2016) shows that the efficiency level achieved by such a non-strategy-proof mechanism
is not high. Our results under simultaneous elicitation when two sectors are asymmetric aligns
with their results.

8

3

Experimental Design and Procedure

3.1

Games

Each experiment consisted of five games, two of which consisted of only one sector (as a benchmark), while the remaining three consisted of two sectors each. Each game was played by threesubjects, so that the coordination problem, as described in the example of the previous section,with two players would be less obvious.
The five games are summarized in Tables 3 and 4. πi (·)s in Table 4, represents the payoffthat player i obtains when he receives none of the three pairs above it. Those shown in bold are
the first-best equilibrium allocations and payoffs.

Games 1 and 2: The one-sector top-trading-cycle mechanism
The purpose of running the two one-sector games was to establish the behavioral benchmark of
the TTC mechanism.
Each of the three subjects received an initial item and a list of monetary evaluations forthe three items. The reward is determined by the value of the item he/she receives as the final
allocation.
Subjects simultaneously submitted their rankings for the three items. See Figure B.1 inOnline Supplementary Material for a screenshot of this. The final allocation of the three items
is determined according to the TTC mechanism based on the submitted rankings.
Given a list of submitted rankings, the TTC mechanism works as follows. The text below
seems to describe a dynamic process. However, this is just an illustration of the algorithm todetermine the outcome. The game itself is static.
Definition 1 Top-trading-cycle (TTC) mechanism
Step 1. Each agent indicates the owner of their favorite item. There is at least one cycle(including a self-cycle) because there are finite agents. Each agent in a cycle is removed
from the market with the assigned item that they indicate. If there is at least one remaining
agent, proceed to the next step.
...
Step k. Each remaining agent points to the owner of their favorite item. There is at least one
cycle (including a self-cycle). Give each agent in a cycle the item they indicate and remove

9

Table 3: one-sector games (Games 1 and 2)A
πA (eC ) = 24
πA (eB ) = 12
πA (eA ) = 6

Game 1B
πB (eA ) = 24
πB (eC ) = 12
πB (eB ) = 6CπC (eB ) = 24
πC (eA ) =12
πC (eC ) = 6

AπA (eA ) = 24
πA (eB ) = 12
πA (eC ) = 6

Game 2B
πB (eA ) = 24
πB (eC ) = 12
πB (eB ) = 6CπC (eA ) = 24
πC (eB ) =12
πC (eC ) = 6

Table 4: two-sector games (Games 3 to 5)

AπA (e1C , e2C ) = 24
πA (e1B , e2B ) = 18
πA (e1A , e2A ) = 12
πA (·) = 6

Game 3B
πB (e1A , e2A ) = 24
πB (e1C , e2C ) =18
πB (e1B , e2B ) = 12
πB (·) = 6CπC (e1B , e2B ) = 24
πC (e1A , e2A ) =18
πC (e1C , e2C ) =12
πC (·) =6A
πA (e1C , e2B ) = 24
πA (e1B , e2C ) = 18
πA (e1A , e2A ) = 12
πA (·) = 6

Game 4B
πB (e1A , e2C ) = 24
πB (e1C , e2A ) =18
πB (e1B , e2B ) = 12
πB (·) = 6CπC (e1B , e2A ) = 24
πC (e1A , e2B ) =18
πC (e1C , e2C ) =12
πC (·) =6A
πA (e1A , e2B ) = 24
πA (e1A , e2C ) = πA (e1B , e2B ) = 18
πA (e1A , e2A ) = πA (e1C , e2B ) = 15
πA (e1B , e2C ) = 12
1
πA (eB , e2A ) = πA (e1C , e2C ) = 9
πA (e1C , e2A ) = 6

AπA (e1A ) = 12
πA (e1B ) = 6
πA (e1C ) = 3

Game 5B
πB (e1A , e2B ) =24
πB (e1A , e2C ) = πB (e1C , e2B ) =18
πB (e1A , e2A ) = πB (e1B , e2B ) =15
πB (e1C , e2C ) = 12
1
πB (eC , e2A ) = πB (e1B , e2C ) =9πB (e1B , e2A ) = 6CπC (e1A , e2B ) = 24
πC (e1A , e2A ) = πC (e1B , e2B ) =18
πC (e1A , e2C ) = πC (e1C , e2B ) =15
πC (e1B , e2A ) =121
πC (eB , e2C ) = πC (e1C , e2A ) =9πC (e1C , e2C ) =6

Two sectors in Game 5 are separable
Sector 1
Sector 2
B
CA
B
πB (e1A ) =12 πC (e1A ) = 12
πA (e2B ) = 12 πB (e2B ) =12
πB (e1C ) =6πC (e1B ) =6
πA (e2C ) = 6
πB (e2C ) =6112πB (eB ) = 3
πC (eC ) =3
πA (eA ) = 3
πB (e2A ) = 3

10CπC (e2B ) = 12
πC (e2A ) = 6
πC (e2C ) =3

them from the market with their assigned item. If there is at least one remaining agent,
proceed with the next step.
...
Because there are a finite number of agents, this process ends after a finite number of rounds.
In the single-sector setting, it is known that reporting the true preference is always a dominant
strategy. There is a unique core allocation, and the mechanism implements the core allocationin the dominant strategy (see, Roth, 1982).
Game 1 is the simplest, and the exchange is completed in one round; hence, only the firstreported item matters. Game 2 is more complicated, and the exchange requires two rounds toprovide the first-best allocation; hence, more than just the item reported first matters. Thus,
we hypothesize that the subjects are more likely to achieve an efficient outcome in Game 1 thanwith Game 2.

Games 3, 4, and 5: two-sector games
We consider three two-sector games, in which there is complementarity between the two sectors
in Games 3 and 4. In the remaining Game 5, the two sectors are independent. In these games,
each of the three subjects receives a pair of items, one for Sector 1 and the other for Sector 2,
and a list of monetary evaluations of the pairs of items. The reward is determined by the valueof the pair of items he or she receives in the final allocation.
In each sector, subjects simultaneously submit rankings for the three items. The timingof the ranking submissions between the two sectors differs depending on the simultaneous andsequential treatments, as we will explain later.
Game 3 is simpler because the directions of trading cycles required for the first-best allocation
are the same across sectors, and thus, coordination is relatively straightforward. Game 4 is more
complex, in that, the first-best allocation requires opposite trading cycle directions across sectors,
and it is more difficult to obtain the first-best allocation. Thus, we expect that subjects are more
likely to achieve an efficient outcome in Game 3 than in Game 4. It should also be noted thatthe preference orderings of the items in Sector 1 for the three best allocations for each subject
in Games 3 and 4 are identical to those of Game 1. Thus, by comparing the rankings submitted
in Game 1 and Sector 1 of Games 3 and 4, we can investigate the effect of having an additional
sector to consider in two-sector games, compared with the one-sector game when the two sectors
are not separable.

11Table 5: Pareto-dominant equilibrium strategies

Game 1
Game 2
Game 3
Game 4
Game 5Player A
Rank eC first
Rank eA first
Rank e1C first
Rank e2C first
Rank e1C first
Rank e2B first
Rank e1A first
Rank e2C above e2A

Player B
Rank eA first
Rank eC above eB
Rank e1A first
Rank e2A first
Rank e1A first
Rank e2C first
Rank e1C above e1B
Rank e2B first

Player C
Rank eB first
Rank eB above eC
Rank e1B first
Rank e2B first
Rank e1B first
Rank e2A first
Rank e1B above e1C
Rank e2A above e2C

In Game 5, the evaluation of pairs is separable across sectors, in the sense that the preference
for Sector 1 items is independent of Sector 2 allocation and vice versa. In fact, each individual’s
total payoff is the sum of those in the two sectors. Thus, in this game, it is a dominant strategy tosubmit the marginal ranking in each sector, as in our two one-sector games. We aim to determine
if this property is preserved under a two-sector setting. It is possible that the subjects fail toplay the dominant strategy because of apparent complications related to facing two sectors thatare not symmetric, even when there is no complementarity between them. Note that, consideredseparately, the preference ordering of Sector 1 of Game 5 for the three players is identical to thatof Game 2, and, although not identical, Sector 2 has a similar structure. Thus, by comparing
the efficiency of the allocation and rankings submitted by subjects between Games 2 and 5, weinvestigate the effect of having two sectors, in the absence of complementarity between the two
sectors, on the behavior and efficiency of the mechanism. Note, however, that the bottom of
Table 4 explaining this separability was not shown to the subjects.
There are many Nash equilibria in the above games because the TTC mechanism leaves
many indifferences for a player when he is not pivotal (see Table 2 for an illustration). This is
typical for mechanisms that are strategy-proof when run for a single sector. Thus, we identify thetheoretical equilibria through certain refinements and compare them with the experimental data.
In particular, we focus on the Pareto-dominant (PD) Nash equilibrium.1 Table 5 summarizes
the characteristics of the PD equilibrium strategies for each player in the five games.
Let us summarize the key comparisons we would like to make among these five games, considering our main research objective of understanding the behavioral consequences of constructing
two separate TTC mechanisms for allocating items over two sectors when, in fact, these decisions
1 We have also considered equilibrium in weakly dominant (WD) strategies in Games 1, 2, and 5. In Game1, truth-telling is a unique WD strategy for each player. In Game 2, there are two WD strategies for Player
A. Player A’s best item is his own endowment, and he can secure his best item by selecting himself without
specifying the ranking between the endowments of Players B and C. Likewise, in Game 5, Player A has two WD
strategies in Sector 1, and Player B has two WD strategies in Sector 2. See Appendix B for frequencies of WD
ranking submission in Games 1, 2, and 5.

12

should not be considered separately.
• Game 1 vs Game 2: the effect of the differences in the TTC mechanism to reach the final
allocation of the subjects’ behavior and efficiency.
• Game 2 vs Game 5: the effect of having two sectors, instead of one, when the two sectors
are separable.
• Game 1 vs Game 3 vs Game 4: the effect of having two sectors, instead of one, when
the two sectors are complementary, and the two sectors are either symmetric (Game 3) orasymmetric (Game 4).

3.2

Two treatments

We consider two treatments, simultaneous and sequential, that differ in the timing of ranking
submissions in the two sectors in Games 3, 4, and 5. This choice is motivated by our interestin mechanism design. The literature of subgame-perfect implementation (following Moore andRepullo, 1988) suggests that the sequential elicitation of information helps the agents achieve
coordination more effectively because it allows for more chances to adjust their informationtransmission across stages. There are, however, two concerns.
First, although most studies in the subgame-perfect implementation literature allow cancelingat any stage, this is not a realistic assumption in the multi-sector setting. Note that whensequential elicitation is applied for the two-sector allocation problem, a realistic specificationis that the allocation for one sector is finalized in the first stage, without any commitment toallocation for the other sector to be made in the second stage, and no cancellation of the first
sector allocation is allowed afterwards (Hayashi and Lombardi, 2019). This is typically the casewhen we must finalize housing allocation before school seats are assigned. The second concern
is that a sequential mechanism may motivate an agent to retaliate against others, depending on
what happened in the previous stage (Fehr et al., 2021).
These concerns motivate an empirical investigation about whether sequential elicitation performs better than simultaneous one in the problem of allocation with multiple sectors, as specified.

Simultaneous treatment
In the simultaneous treatment, in Games 3, 4, and 5, subjects simultaneously submit rankingsfor the three items in each sector separately. See Figure B.3 in Online Supplementary Material

13for a screenshot.2

Sequential treatment
In the sequential treatment, Games 3, 4, and 5 consist of two stages as follows.
Stage 1: The subjects simultaneously submit rankings for the three items in Sector 1. See
Figure B.5 in Online Supplementary Material for a screenshot.3 Then, the TTC algorithm
is run, and the allocation in Sector 1 is determined. The subjects were informed of bothranks and the allocation in their group in Sector 1. See Figure B.6 in Online Supplementary
Material for a screenshot.
Stage 2: The subjects simultaneously rank the three items in Sector 2. See Figure B.7 in
Online Supplementary Material for a screenshot. As one can see from the screenshot, thesubjects are reminded of the allocation in Sector 1 when submitting their rankings for
Sector 2. Then, the TTC algorithm is run and the allocation in Sector 2 is determined.
The subjects were informed of the rankings and the final allocated goods for both sectors
at the end. See Figure B.8 in Online Supplementary Material for a screenshot.
In each game, there is a natural subgame-perfect equilibrium path that yields the samefirst-best allocation, as obtained in the corresponding Nash equilibrium in the simultaneous
treatment.4 We expect that this sequential elicitation of rankings in the two sectors will facilitate
our subjects following equilibrium behavior and improve their welfare.

3.3

Procedure

Our subjects played all five games during the experiment. Some simultaneous treatment sessions
start with one-sector games, while others start with two-sector games. In the sequential treatment, we always started with two-sector games because our focus is on investigating the effectof sequential elicitation in two-sector games in comparison to simultaneous elicitation. The firstgame the subjects played differs across sessions. In half of the sessions starting with a one-sector
game, the subjects played Game 1 first. In the other half, Game 2 was played first. Similarly,
2 Note that in the experiment Sectors 1 and 2 are called Types A and B, respectively.
3 As noted above, in the experiment, Sectors 1 and 2 are called Types A and B, respectively.
4 Sequential implementation leaves a potential problem of manipulation, in that, one may have an incentive
to manipulate the Stage 1 outcome to manipulate the Stage 2 outcome. In general, when an individual claims to
demand a particular item at Stage 1, we cannot determine whether this is because she truly wants this item as
an individual object or if she wants the item that follows in Stage 2 as the consequence of acquiring it. Hayashi
and Lombardi (2019) show that this incentive problem restricts the class of implementable social objectives.
However, this problem is excluded in our payoff setting because the first-best pair of items is PD, and nobody
has an incentive to manipulate the Stage 1 outcome to manipulate that of Stage 2.

14Games 3, 4, and 5 were played first in one-third of the sessions that started with two-sector
games. The order of the remaining two-sector games was counterbalanced across sessions.5
Subjects performed each of the one-sector games thrice, and each of the two-sector games six
times.6 We call one play of a game a round. Thus, all subjects played 24 rounds (3 rounds ×
one-sector games + 6 rounds × two-sector games). At the beginning of each round, subjects wererandomly matched into groups of three. Their roles within their group (A, B, or C)7 are randomly
reassigned once the group is created at the beginning of each round. Subjects were informedthat they would play a total of 24 rounds, although the explanation of the specificity of one- or
two-sector games was given only at the beginning of the relevant games. Thus, those subjects
who started with one-sector games received the instructions regarding the two-sector game only
after they had finished playing six rounds of one-sector games. Similarly, those subjects whostarted with two-sector games received instructions for the one-sector games only after they
had finished playing 18 rounds of two-sector games. See Online Supplementary Material for an
English translation of the instructions for our experiments.
In addition to the 1000 JPY show-up fee, subjects were paid based on the payoff they obtainedin one of the 24 rounds randomly chosen at the end of the experiment. All subjects in the samesession were rewarded, based on the same randomly chosen round. The points obtained in thechosen round were converted to JPY where 1 point = 100 JPY. At the end of the experiment,a set of questionnaires were used to gather the individual characteristics of our subjects.8 The
experiment lasted for an average of 100 minutes for simultaneous treatments and 110 minutes
for sequential treatments.

4

Results

Experiments were conducted between May 2017 and November 2018 at the Center for Experimental Economics, Kansai University, Osaka, Japan.9 Most of the participants were undergraduate students (the others were graduate students). They were recruited using the online billboardof Kansai University. Each participant participated in the experiment only once. There were
5 We have implemented within-subjects design for five games and between-subjects design for two treatments.

The main reason is that sequential treatment experiments followed our simultaneous experiments, as an additional
treatment to respond to the comments we have received while presenting the results from simultaneous treatments.
6 We have opted for more repetitions in two-sector games because we believed that participants needed moretime to familiarize themselves with the mechanism.
7 Called player 1, 2, or 3 in the experiment.
8 The questionnaire included part of the advanced version of Raven’s Progressive Matrix (RPM) test (Raven,1998). We did not provide a monetary incentive for the RPM test. Appendix D emphasizes why we haveconducted an RPM test and summarizes the results of analyses that utilize the RPM test score.
9 Experiments were computerized using z-Tree (Fischbacher, 2007).

15

Table 6: The average total payoffs when all players uniformly randomize their ranking
submission (ΠR ) and the total payoffs under the equilibrium (ΠEq )
game
ΠR
ΠEq

1
32
72

2
39.5
48

3
28.12572

428.125
72

5
39.5
48

a total of 381 subjects, none of whom had participated in similar experiments.10 Participants
earned, on average, JPY 2,590, including the show-up fee.11
We compare (a) the extent to which participants follow equilibrium behavior and (b) the total
welfare of group members. The former is measured by the average frequencies of participants
submitting equilibrium rankings. For the latter, because the deviation from the equilibrium
ranking submission has varying welfare consequences between games, we consider payoffs relative
to the expected payoffs when players completely randomize their ranking submissions.12 We callthis measure the relative efficiency. In particular, the relative efficiency (RE) for group g isdefined as
P
g

RE =

j∈g πj − ΠΠEq − ΠR

R

where πj is the realized payoff for player j in group g, ΠR is the average total payoff (for threeplayers) when all the players are uniformly randomizing their ranking submissions, and ΠEq isthe total payoff for three players under the equilibrium. Table 6 shows ΠR and ΠEq for each
game.

4.1

One-sector games (Games 1 and 2)

We first analyze the results from the two one-sector games. We compared outcomes between
Games 1 and 2 in simultaneous treatment for inexperienced and experienced participants. Inexperienced participants played the game as their first game, and experienced participants were
those who played the game last. We present the first and the last to demonstrate the effect oflearning.13
10 Appendix A summarizes the experimental sessions. The number of participants varied between sessions
because of variation in the attendance rate between sessions.
11 JPY 100 ≈ USD 0.9 at the time of the experiments.
12 We thank an anonymous referee for this suggestion.
13 See Appendix B for the results from each ordering, that is, for participants who played the game as their first,
second, fourth, and fifth game. There is a significant increase in the frequency of equilibrium ranking submission
between participants who played these one-sector games before (the first or the second) or after (the fourth andthe fifth) two-sector games. We do not pool the data of one-sector games from simultaneous and sequential
treatments and use the data only from the simultaneous elicitation treatment. This is because, in the sequential
treatment, one-sector games were always played after the two-sector games. In addition, within one-sector games,Game 1 was always played before Game 2.

16

Figure 1: Game 1 (G1) vs Game 2 (G2)(a) Average frequency ofequilibrium ranking submission***(b) Relative efficiency

***

**

1.0
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0

***

1.0
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
G1

G2

G1

Inexperienced

Experienced

G2

G1

G2

G1

Inexperienced

Experienced

G2

Note. Based on the estimated coefficient of game dummies. Error bars show two standard errors range. Standard errors
are corrected for session-level clustering effect. ***, **, and * stand for statistical significance at the 1%, 5%, and 10%
levels, respectively. n.s., means the difference is not statistically significant at the 10% level based on the Wald test.

Panel (a) Figure 1 shows the average frequency, during three rounds of playing the game,of the subject submitting the equilibrium ranking in Game 1 (G1) and Game 2 (G2).14 We
observe that the frequency of the equilibrium ranking submission is higher for experienced thanfor inexperienced participants. Thus, participants learn, based on the experience of playing
similar games, to submit rankings consistent with equilibrium more frequently.
Figure 1 also shows that a significantly higher frequency of equilibrium ranking submission inG2 than in G1 manifests as the higher average relative efficiency shown in panel (b) of Figure 1.15
Thus, contrary to our expectation that the efficient outcome is more likely to be achieved in G1
than in G2, G2 resulted in higher relative efficiency. This is because, although we did not foresee
this when we constructed the games, the set of PD equilibrium strategies for Players B and C islarger in G2 than in G1.

4.2

Game 2 vs Game 5

Let us now compare G2 and Game 5 (G5). We had mentioned earlier that the two sectors are
separable in G5. Thus, a comparison between G2 and one of the two sectors in G5 captures
the effect of complexity introduced by the mere fact of needing to consider two sectors when
14 Because of the random regrouping of subjects after each round, the observations from subjects who participated in the same experimental sessions are not independent. We adjust for possible within-session correlations
by running linear regressions and using robust standard errors that adjust for the session-clustering effect. For
example, Figure 1 is generated by running the the following linear regression for each panel and each type ofthe subjects: Y = β1 G1 + β2 G2 + µ, where Y is a fraction of three rounds in which a subject has submitted
equilibrium rankings, G1 and G2 are the dummy variables that take the value 1 if the outcome is from subjects
playing the relevant game (Games 1 and 2, respectively). Accordingly, the average frequency for equilibrium
ranking submissions in Games 1 and 2, captured by the estimated coefficients β1 and β2 , respectively. The
results of these regressions are reported in Appendix C.
15 For the average relative efficiency reported in Figure 1, we take the average of groups across three rounds (notethat because of the random rematching, groups are different across rounds within a game). Because of random
regrouping at the beginning of each round, we consider a group-round the unit of analysis and accommodate forthe within-session-clustering effect by running a linear regression like that discussed in footnote 14.

17

Figure 2: Game 2 vs Game 5(a) Equilibrium submission Sector 1
n.s.
***

(b) Equilibrium submission Sector 2
*

***
**

***

*

***

**

1.0
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0

*

***

n.s.

1.0
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
G2

G5sim

G5seq

G2

Inexperienced

G5sim

G5seq

G2

Experienced

G5sim(c) Equilibrium submission both sectors
**
***

***

G2

G5sim

G5seq

Experienced(d) Relative efficiency

***
**

G5seq

Inexperienced

**
n.s.

***

1.0
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0

***
***

***

**

1.0
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
G2

G5sim

G5seq

Inexperienced

G2

G5sim

G5seq

G2

Experienced

G5sim

G5seq

Inexperienced

G2

G5sim

G5seq

Experienced

Note. Based on the estimated coefficient of game dummies. Error bars show two standard errors range. Standard errors
are corrected for session-level clustering effect. ***, **, and * stand for statistical significance at the 1%, 5%, and 10%
levels, respectively. n.s., means the difference is not statistically significant at the 10% level.

deciding the ranking, instead of one.
Figure 2 shows the results of comparing the average frequency of a PD equilibrium ranking
submission in Sector 1 (panel (a)), Sector 2 (panel (b)), and both sectors combined (panel (c))
in G5, both simultaneous (G5sim ) and sequential (G5seq ) treatments, and those in G2. It also
shows, in panel (d) the average relative efficiency for G2, G5sim , and G5seq .16 As we havedone for the comparison of G1 and G2 in each panel above, the outcomes for the inexperienced
participants who played these games as their first game and for the experienced participants
who played these games as their last game are shown.17
For inexperienced participants, the PD equilibrium ranking submission in Sector 1 is significantly less frequent in G5sim than in G2 and G5seq (panel (a)). The same is true for Sector 2(panel (b)), as well as for considering both sectors jointly (panel (c)). When both sectors are
16 The frequency of the equilibrium ranking submission and relative efficiency are computed based on the total
plays of a game, that is, three for G2 and six for G5. The averages reported in the figure are based on theestimated coefficients obtained from running the the following linear regression for each panel and each type ofthe subjects: Y = β1 G2+β2 G5sim +β3 G5seq +µ. The standard errors are corrected for session-clustering effects.
The results of these regressions are reported in Appendix C.
17 Note that the experienced participants considered for G5 here, in both sequential and simultaneous treatments, are those who started with two-sector games and played G5 as their third game before playing one-sector
games. This restriction is required to compare simultaneous and sequential treatments because, in the sequential
treatment, two-sector games were always played before the one-sector games. This remark applies to all theanalyses involving two-sector games.

18considered, the frequency of equilibrium ranking submissions is significantly lower in G5seq thanin G2. This results in significantly lower efficiency in G5sim than in G5seq , which results in significantly lower efficiency than in G2 (panel (d)). A similar tendency is observed for experienced
participants as well, although the frequency of the PD equilibrium ranking submissions in G5sim
and G5seq are no longer significantly different at the 5% level, partly because of a large variancein the outcome of G5sim .
This suggests that the complexity introduced by simultaneously considering two sectors thatare asymmetric, even when the two sectors are independent, makes it difficult for participants
to follow equilibrium behavior. This is especially true when ranking two sectors simultaneously.
This difficulty results in a significantly lower relative efficiency in G5sim than in G5seq , bothof which are significantly lower than G2, not only for inexperienced but also for experienced
participants.

4.3

Comparison between Games 1, 3, and 4

We proceed to compare the outcomes from G1, Game 3 (G3), and Game 4 (G4). Let us firstconsider G3 and G4 under simultaneous treatment, which will be denoted as G3sim and G4sim ,respectively.
Figure 3 shows the average frequencies of subjects submitting equilibrium rankings in Sector 1(panel (a)), Sector 2 (panel (b)), and the two sectors combined (panel (c)), for both inexperienced
and experienced participants. Figure 3 also reports the average relative efficiency (panel (d)).
In each panel, the outcome of G1 is reported as the benchmark.18
Figure 3 clearly demonstrates that, as seen in the case of other games, experienced participants submit equilibrium rankings more frequently, and achieve higher efficiency, than theinexperienced participants in G3sim and G4sim . It also shows that the relative efficiency achievedin G3sim in which two sectors are symmetric is as high as that in G1 for experienced participants(panel (d)), although the frequency of equilibrium ranking submissions is significantly different
between the two (panel (c)). We also note that the asymmetry between the two sectors in G4sim
causes a significant loss in terms of efficiency (panel (d)) compared with G3sim , even though
the frequencies of equilibrium ranking submissions in the two games are similar for experienced
participants (panel (c)).
18 The frequency of the equilibrium ranking submission and relative efficiency are computed based on the totalrounds of playing a game, that is, three for G1 and six for G3 and G4. The averages reported in the figure
are based on the estimated coefficients obtained from running the following linear regression for each panel andeach type of the subjects: Y = β1 G1 + β2 G3sim + β3 G3seq + β4 G4sim + β5 G4seq + µ. The standard errors arecorrected for the session-clustering effect. The results of these regressions are reported in Appendix C.

19

Figure 3: Game 1 vs Game 3 vs Game 4 (Simultaneous elicitation)(a) Equilibrium submission Sector 1
***
n.s.

(b) Equilibrium submission Sector 2

**
n.s.

**

***
n.s.

n.s.

1.0
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0

***
n.s.

***

n.s.

1.0
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
G1

G3sim

G4sim

G1

Inexperienced

G3sim

G4sim

G1

Experienced

G3sim(c) Equilibrium submission both sectors
***
n.s.

***

G1

G3sim

G4sim

Experienced(d) Relative efficiency

***
n.s.

G4sim

Inexperienced

***
n.s.

n.s.

1.0
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0

***
***

n.s.

***

1.0
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
G1

G3sim

G4sim

Inexperienced

G1

G3sim

G4sim

G1

Experienced

G3sim

G4sim

Inexperienced

G1

G3sim

G4sim

Experienced

Note. Based on the estimated coefficient of game dummies. Error bars show two standard errors range. Standard errors
are corrected for session-level clustering effect. ***, **, and * stand for statistical significance at the 1%, 5%, and 10%
levels, respectively. n.s., means the difference is not statistically significant at the 10% level.

The significantly low efficiency in G4sim compared with G3sim is a result of the higher
efficiency loss in G4 compared with G3 when one or two (out of three) individuals in the groupfail to submit a PD equilibrium ranking. To see this, consider the following example: Assume
that Players B and C submit rankings according to the PD equilibrium in both sectors. Thatis, for example, Player B submits esA  esC  esB in both Sectors 1 and 2 (s ∈ 1, 2) in G3, and,in G4, e1A  e1C  e1B in Sector 1 and e2C  e2A  e2B in Sector 2. Imagine also that Player
A deviates from the equilibrium and ranks his endowment first in both Sectors 1 and 2, which
is a natural response under strategic uncertainty because it secures its own initial endowment
pair. As a result, Player A obtains his own endowment and receives a payoff of 12 in G3 andG4. However, the impact of this deviation by Player A on the two remaining players differs
significantly between G3 and G4. In G3, on the one hand, Player B obtains her second-mostpreferred pair with the associated payoff of 18, and Player C obtains the most preferred pairwith an associated payoff of 24. This results in a relative efficiency of 0.590. In G4, on theother hand, Players B and C end up exchanging their endowments in both sectors, and bothobtain a payoff of 6, which is worse than securing their own endowment pairs. This results in a
relative efficiency of -0.094. Indeed, in most of the cases where only one player of the group fails

20Figure 4: Simultaneous vs sequential treatment: Game 3, Game 4(a) Equilibrium ranking submission Sector 1
*

**

***

(b) Equilibrium ranking submission Sector 2

n.s.

n.s.

1.0
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0

n.s.

*

n.s.

1.0
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
G3sim G3seq G4sim G4seq
Inexperienced

G3sim G3seq G4sim G4seq

G3sim G3seq G4sim G4seq

Experienced

G3sim G3seq G4sim G4seq

Inexperienced(c) Conditionally optimal ranking in Sector 2(d) Relative efficiency
n.s.

1.0
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0

Experienced

n.s.

n.s.

n.s.

1.0
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
G3seq
Inexperienced

G4seq

G3seq

G3sim G3seq G4sim G4seq

G4seq

Inexperienced

Experienced

G3sim G3seq G4sim G4seq
Experienced

Note. Based on the estimated coefficient of game dummies. Error bars show two standard errors range. Standard errors
are corrected for session-level clustering effect. ***, **, and * stand for statistical significance at the 1%, 5%, and 10%
levels, respectively. n.s., means the difference is not statistically significant at the 10% level.

to submit the equilibrium ranking in both sectors, the achieved relative efficiency is the sameas that under this example in both games. Furthermore, in 24 out of 44 cases in which only
one player in the group failed to submit the equilibrium ranking among experienced players in
G4, the participant deviating from the equilibrium strategy ranked their own endowment firstin both sectors, as in the above example.
Considering the comparison of G2 and G5 discussed above, as well as the comparison between
G1, G3, and G4, it is suggested that the asymmetry between the two sectors results in a large
efficiency loss regardless of whether complementarity exists between the two sectors. In otherwords, an “escape” to secure one’s initial endowment pair has a more significant impact on
efficiency loss when efficient exchanges are required to occur asymmetrically across sectors.
Does sequential elicitation improve efficiency compared with simultaneous elicitation whencomplementarity exists between the two sectors? Unfortunately, the answer is no for both inexperienced and experienced participants. Note, however, that although the relative efficiency inthe sequential treatment is not statistically significantly different from that in the simultaneous
treatment (see panel (d) of Figure 4), we do observe that inexperienced participants are significantly more likely to submit equilibrium ranking in Sector 1 in the sequential treatment than in21

the simultaneous treatment (see panel (a) of Figure 4).
The low relative efficiency of G4seq compared with G3seq can be understood through the same
example we have discussed above, to understand the low relative efficiency of G4sim comparedwith G3sim . As shown in the above example, even when it is only one player who deviates from
submitting the equilibrium marginal ranking in Sector 1, if the player ranks his own endowment
first, the remaining two players who have submitted the equilibrium marginal ranking cannot
perform much better after the allocation of Sector 1. Indeed, out of 36 cases where only one
player in the group failed to submit the equilibrium ranking in Sector 1, 19 ranked their own
endowment first.

5

Concluding Remarks

This study is the first attempt to construct a simple lab experiment that allows us to test
the performance of partial equilibrium mechanisms when operated under realistic institutional
constraints. Within these constraints, mechanisms are run separately with the presumption thatagents’ preferences are separable across sectors when, in fact, they are not. The experimental
setting allows us to quantify how much we lose by operating partial equilibrium mechanisms
with such a false presumption of separability. We conducted experiments in which the agentsneeded to coordinate their plays across sectors to obtain the first-best allocation overall.
We find that, even in the simple setting considered in our experiment, complexities introduced by a need to consider two sectors simultaneously is indeed a significant burden when thetwo sectors are asymmetric, and the mechanisms are run simultaneously. Surprisingly, this istrue even when the two sectors are independent, and thus, there is no need for inter-sectoral
coordination. Note that such a welfare loss is not observed when the two sectors are symmetric,
even in the presence of complementarity. We also find that running the mechanisms sequentiallycan ease this burden and can reduce welfare loss. This will provide us with a lesson about when
we should be careful using partial equilibrium mechanisms.
We conclude by listing suggestions for future research. In this paper, we considered a simple
case of two sectors and three agents, which turned out to be already difficult to work with. This
results in a question about whether inter-sectoral coordination becomes harder or easier in alarger economy with more sectors and agents. While we adopted the TTC mechanism in thisstudy because it is one of the simplest mechanisms to work with, it is worth considering how
other partial equilibrium mechanisms work. In addition, we have assumed that the degree ofcomplementarity across sectors is homogeneous among subjects. However, there often is hetero22

geneity in the degree of complementarity among agents because it is vital for somebody to winboth items of a particular pair, while it is not so for others. Thus, we will need to consider a moregeneral setting than the current one in which all the subjects use the mechanisms in a symmetricmanner. Finally, note that in the current study, we have restricted our attention to evaluating
the performance of partial equilibrium mechanisms when they are operated under inter-sectoralcomplementarities, rather than identifying a behavioral cause in inter-sectoral coordination andits comparative nature across various parameters, which will naturally be of interest.

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26

A

Summary of experimental sessions
Type of
Experiment
Simultaneous
Simultaneous
Simultaneous
Simultaneous
Simultaneous
Simultaneous
Simultaneous
Simultaneous
Simultaneous
Simultaneous
Simultaneous
Simultaneous
Sequential
Sequential
Sequential
Sequential
Sequential
Sequential

Session ID
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18

Order ofGames
G1, G2, G3, G4, G5
G2, G1, G3, G5, G4
G4, G3, G5, G1, G2
G5, G4, G3, G2, G1
G1, G2, G4, G3, G5
G5, G3, G4, G2, G1
G3, G4, G5, G1, G2
G3, G5, G4, G1, G2
G4, G5, G3, G2, G1
G2, G1, G4, G5, G3
G1, G2, G5, G4, G3
G2, G1, G5, G3, G4
G5, G3, G4, G1, G2
G4, G3, G5, G1, G2
G4, G5, G3, G1, G2
G5, G4, G3, G1, G2
G3, G4, G5, G1, G2
G3, G5, G4, G1, G2

27

Number of
subjects
24
15
18
21
21
24
24
18
21
18
18
21
24
24
27
18
24
21

Average
Earning (JPY)
2,363
2,320
2,567
2,571
2,400
2,525
2,600
2,700
2,414
2,400
2,467
3,413
2,850
2,700
2,600
2,533
2,600
2,600

B

Dynamics

Figure B.1: Average frequency of equilibrium ranking submission across ordering for one-sector
games in simultaneous treatment(a) Game 1 (Weakly dominant)(b) Game 2 (Weakly dominant)

***

n.s.

**

1.0
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0

n.s.

n.s.

1st

1.0
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0

n.s.

2nd

4th

Before two-sector games

n.s.

1st

5th

After two-sector games

**

2nd

4th

Before two-sector games(c) Game 1 (Pareto dominant)

5th

After two-sector games(d) Game 2 (Pareto dominant)
***

***
**

1.0
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0

n.s.

**

1st

2nd

Before two-sector games

4th

***

***

1.0
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0

n.s.

1st

5th

After two-sector games

2nd

Before two-sector games

4th

5th

After two-sector games

Note. Based on the estimated coefficient of order dummies. Error bars show two standard errors range. Standard errors
are corrected for session-level clustering effect. ***, **, and * stand for statistical significance at the 1%, 5%, and 10%
levels, respectively. n.s., means the difference is not statistically significant at the 10% level based on the Wald test.

28

29

1st

2nd

**

***

3rd

1st

2nd

***

n.s.

3rd

Before one-sector games

***

**

3rd

n.s.

4th

**
n.s.

5th

3rd

n.s.

4th

n.s.
n.s.

5th

After one-sector games

***

n.s.

After one-sector games(d) Sector 1

Before one-sector games

n.s.

*

n.s.

1.0
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0

1.0
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
1st
2nd

*
***

3rd

1st
2nd

***
*

3rd

Before one-sector games

***

***

3rd

n.s.

**

4th

**
n.s.

5th

3rd

**

4th

n.s.
n.s.

5th
After one-sector games

***

n.s.

After one-sector gamesn.s.

Game 4(e) Sector 2

Before one-sector games

n.s.

n.s.

Game 3(b) Sector 2

1.0
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0

1.0
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
1st

2nd

**
***

3rd

1st

2nd

***
n.s.

3rd

***

Before one-sector games

***

3rd

n.s.

*

4th

***
n.s.

5th

3rd

*

4th

n.s.
**

5th
After one-sector games

***

n.s.

After one-sector gamesn.s.

(f) Both together

Before one-sector games

n.s.

n.s.

(c) Both together

Note. Based on the estimated coefficient of order dummies. Error bars show two standard errors range. Standard errors are corrected for session-level clustering effect. ***, **, and * standfor statistical significance at the 1%, 5%, and 10% levels, respectively. n.s., means the difference is not statistically significant at the 10% level based on the Wald test.

1.0
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0

1.0
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0

*

(a) Sector 1

Figure B.2: Average frequency of Pareto-dominant equilibrium ranking submission across ordering in simultaneous treatment: Game 3 and Game 4

30

1st

2nd

n.s.

n.s.

3rd

3rd

4th

n.s.
n.s.

1st

2nd

*

n.s.

3rd

Before one-sector games

**

n.s.

5th

3rd

n.s.

4th

n.s.
n.s.

After one-sector games

n.s.

n.s.

5th

After one-sector gamesn.s.

(d) Sector 1 (Pareto dominant)Before one-sector games

n.s.

n.s.

1.0
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0

1.0
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
1st
2nd

n.s.
n.s.

3rd

3rd

n.s.

4th

n.s.
***

1st
2nd

n.s.
n.s.

3rd

Before one-sector games

n.s.

n.s.

5th

3rd

n.s.

4th

n.s.
n.s.

After one-sector games

n.s.

n.s.

5th

After one-sector games*(e) Sector 2 (Pareto dominant)Before one-sector games

n.s.

n.s.

n.s.

(b) Sector 2 (Weakly dominant)

1.0
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0

1.0
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
1st

2nd

n.s.
n.s.

3rd

3rd

n.s.

n.s.

4th

n.s.
n.s.

5th
After one-sector games

n.s.

1st

2nd

n.s.
n.s.

3rd
Before one-sector games

n.s.

n.s.

3rd

n.s.

n.s.

4th

n.s.
n.s.

5th
After one-sector gamesn.s.

(f) Both together (Pareto dominant)Before one-sector games

n.s.

n.s.

(c) Both together (Weakly dominant)Note. Based on the estimated coefficient of order dummies. Error bars show two standard errors range. Standard errors are corrected for session-level clustering effect. ***, **, and * standfor statistical significance at the 1%, 5%, and 10% levels, respectively. n.s., means the difference is not statistically significant at the 10% level based on the Wald test.

1.0
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0

1.0
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0

n.s.

n.s.

(a) Sector 1 (Weakly dominant)Figure B.3: Average frequency of equilibrium ranking submission across ordering in simultaneous treatment: Game 5

31

1st

1st

2nd

n.s.

***

2nd

*

(c) Sector 1

n.s.

n.s.

n.s.

3rd

3rd

1.0
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0

1.0
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0

1st

1st
2nd

n.s.

**

2nd

n.s.

Game 4(d) Sector 2n.sGame 3(b) Sector 2

n.s.

n.s.

3rd

3rd

1.0
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0

1.0
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0

1st

1st

2nd

n.s.
n.s.

**

2nd

*
n.s.

(e) Both togethern.s.

(c) Both together

3rd

3rd

Note. Based on the estimated coefficient of order dummies. Error bars show two standard errors range. Standard errors are corrected for session-level clustering effect. ***, **, and * standfor statistical significance at the 1%, 5%, and 10% levels, respectively. n.s., means the difference is not statistically significant at the 10% level based on the Wald test.

1.0
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0

1.0
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0

(a) Sector 1

Figure B.4: Average frequency of Pareto-dominant equilibrium ranking submission across ordering in sequential treatment: Game 3 and Game 4

32

2nd

*

***

3rd

1st

n.s.

2nd

***

n.s.

3rd

(d) Sector 1 (Pareto dominant)

1st

**

1.0
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0

1.0
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
2nd

n.s.
***

3rd

1st

n.s

2nd

n.s.
n.s.

3rd

(e) Sector 2 (Pareto dominant)

1st

***

(b) Sector 2 (Weakly dominant)

1.0
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0

1.0
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0

2nd

n.s.
**

3rd

1st

n.s.

2nd

***
n.s.

3rd

(f) Both together (Pareto dominant)

1st

***

(c) Both together (Weakly dominant)Note. Based on the estimated coefficient of order dummies. Error bars show two standard errors range. Standard errors are corrected for session-level clustering effect. ***, **, and * standfor statistical significance at the 1%, 5%, and 10% levels, respectively. n.s., means the difference is not statistically significant at the 10% level based on the Wald test.

1.0
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0

1.0
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0

(a) Sector 1 (Weakly dominant)Figure B.5: Average frequency of equilibrium ranking submission across ordering in sequential treatment: Game 5

CRegression results
Table C.1: Regressions corresponding to Figure 1
Pareto dominantInexp.
Exp.
G1
0.656∗∗∗ 0.843∗∗∗(0.017)(0.022)G20.852∗∗∗ 0.972∗∗∗(0.020)(0.011)
N
117
126
R2
0.88
0.94
Hypothesis testing+
G1 = G2
0.00030.0035

Relative efficiency
Inexp.
Exp.
0.440∗∗∗
0.693∗∗∗(0.015)(0.035)0.621∗∗∗
0.941∗∗∗(0.054)(0.042)
117
126
0.52
0.86
0.02420.0060Standard errors (adjusted for session-clustering effect) in parentheses∗ p < 0.10, ∗∗ p < 0.05, ∗∗∗ p < 0.01
+ p-values based on the Wald test are reported

33

34
0.00000.00070.00130.05750.02690.00960.10290.0202

G5seq

0.02600.0111
0.06190.0641

Standard errors (adjusted for session-clustering effect) in parentheses∗ p < 0.10, ∗∗ p < 0.05, ∗∗∗ p < 0.01
+ p-values based on the Wald test are reported

N
R2
Hypothesis testing+G2 = G5sim = G5seq
G2 = G5sim
G2 = G5seq
G5sim = G5seq

0.868∗∗∗(0.00553)
150
0.96

0.813∗∗∗(0.00523)
141
0.93

G5sim
0.794∗∗∗(0.0166)141
0.95

0.704∗∗∗(0.0361)

0.786∗∗∗(0.0307)0.707∗∗∗(0.0334)G2

0.68040.69040.77780.48530.792∗∗∗(0.00553)
150
0.80

0.718∗∗∗(0.0668)

Sector 2
Inexp.
Exp.
0.852∗∗∗ 0.972∗∗∗(0.0193) (0.0117)

Sector 1
Inexp.
Exp.
0.852∗∗∗
0.972∗∗∗(0.0193)(0.0117)0.00020.00090.00010.01990.675∗∗∗(0.000872)
141
0.91

0.504∗∗∗(0.0543)0.00000.00740.00000.27330.694∗∗∗(1.24e-09)
150
0.92

0.583∗∗∗(0.0884)

Both sectors
Inexp.
Exp.
0.852∗∗∗
0.972∗∗∗(0.0193)(0.0117)

Table C.2: Regression corresponding to Figure 2

0.00030.00040.02620.0001

0.462∗∗∗(0.0074)228
0.40

0.173∗∗∗(0.0329)0.00000.00050.00150.04050.592∗∗∗(0.0146)240
0.67

0.273∗∗∗(0.1057)

Relative Efficiency
Inexp.
Exp.
0.621∗∗∗ 0.941∗∗∗(0.0536) (0.0428)

35
0.461∗∗∗(0.0472)240
0.829
0.37250.00510.12100.56120.74330.718∗∗∗(0.0389)
0.763∗∗∗(0.0742)240
0.851
0.00420.00790.62790.00330.61000.479∗∗∗(0.0364)

G3seq

G4sim

0.588∗∗∗(0.00633)
N
240
R2
0.820
Hypothesis testing+
G1 = G3sim
0.3147
G1 = G4sim
0.0012G3sim = G4sim
0.1079G3sim = G3seq
0.0640G4sim = G4seq
0.0141

0.483∗∗∗(0.0457)0.00100.00151.00000.0531
0.23990.596∗∗∗(0.0911)240
0.847

0.718∗∗∗(0.0268)

0.644∗∗∗(0.0316)

Standard errors (adjusted for session-clustering effect) in parentheses∗ p < 0.10, ∗∗ p < 0.05, ∗∗∗ p < 0.01
+ p-values based on the Wald test are reported

G4seq

0.781∗∗∗(0.0079)

0.730∗∗∗(0.0336)G3sim
0.637∗∗∗(0.0151)

0.603∗∗∗(0.0543)0.738∗∗∗(0.0061)

0.595∗∗∗(0.0551)G1
0.718∗∗∗(0.0030)

Sector 2
Inexp.
Exp.
0.656∗∗∗ 0.843∗∗∗(0.0163) (0.0215)

Sector 1
Inexp.
Exp.
0.656∗∗∗ 0.843∗∗∗(0.0163) (0.0215)

0.15040.00020.12970.79100.61800.369∗∗∗(0.0480)240
0.770

0.402∗∗∗(0.0408)

0.563∗∗∗(0.0447)0.540∗∗∗(0.0729)0.00070.00230.67610.00010.24120.578∗∗∗(0.0836)240
0.830

0.694∗∗∗(0.0363)

0.611∗∗∗(0.0113)0.710∗∗∗(0.0030)

Both sectors
Inexp.
Exp.
0.656∗∗∗ 0.843∗∗∗(0.0163) (0.0215)

0.98860.00000.00110.19350.36990.161∗∗(0.0711)417
0.534

0.092∗∗∗(0.0181)0.588∗∗∗(0.0735)

0.442∗∗∗(0.0752)

0.27890.00000.00010.77420.22370.410∗∗∗(0.0621)414
0.641

0.308∗∗∗(0.0449)

0.646∗∗∗(0.0282)

0.655∗∗∗(0.0099)

Relative Efficiency
Inexp.
Exp.
0.440∗∗∗ 0.693∗∗∗(0.0146) (0.0337)

Table C.3: Regression corresponding to Figures 3 and 4

0.03940.00420.729∗∗∗(0.0501)
177
0.894

0.517∗∗∗(0.0075)

0.830∗∗∗(0.0535)

0.643∗∗∗(0.0510)0.00280.00000.911∗∗∗(0.0124)174
0.949

0.702∗∗∗(0.0069)

0.904∗∗∗(0.0305)0.750∗∗∗(0.0151)

Conditional Optimality
Inexp.
Exp.

Table D.1: Distribution of RPM scores
RPM score
No. of Obs.

D

9 and below
103

10
45

11
63

12
55

13 and above
115

Effect of cognitive ability

Let us investigate the effect of varying the level of cognitive ability of participants on their behavior and obtained payoffs. Recent studies have reported that the behavior of less strategicallysophisticated subjects, captured by subjects’ scores on the RPM test, differ (with a negative
payoff consequence) from those with more sophisticated subjects in such games as a beauty contest game (Gill and Prowse, 2016) and repeated cooperation games (Proto et al., 2019), as wellas well-known mechanisms such as strategy-proof differed acceptance mechanism (DA) and immediate acceptance mechanism (IA)(Basteck and Mantovani, 2018).19 Basteck and Mantovani(2016) show that learning from experience in the Boston mechanism is related to participants’cognitive abilities. Hanaki et al. (2016) also report that RPM score is correlated with how subjects respond to the behavioral uncertainty of their opponent in a simple two-player coordination
game. Considering this, it is possible that the effect of complexities introduced by considering
two sectors depends on participants’ cognitive abilities.
In this section, we compare the behavior and resulting payoff to participants with high andlow cognitive ability. The average RPM score of our 381 participants was 10.85, with a standard
deviation of 2.65. The median RPM score was 11. We categorize our participants into three
categories based on their RPM scores. “High” and “low” groups are those with RPM score 13
or above and 9 or below, respectively. There were 115 and 103 participants in the “high” and“low” ability categories, respectively. The remaining 163 participants whose RPM scores areeither 10, 11, or 12 are in the “middle” group. See Table D.1 for a more detailed distribution of
RPM scores.
We measure participants’ behavior based on the average frequency of submitting Paretodominant equilibrium ranking. For simultaneous treatment, Sectors 1 and 2 are jointly considered in two-sector games. For sequential treatment, we consider the equilibrium ranking
submission in Sector 1 and conditionally optimal ranking submission in Sector 2. The payoffs
are normalized to the equilibrium payoff (relative payoff). As before, we consider inexperienced
and experienced participants and take the average across repetitions for both measures (notethat participants’ roles are not fixed across repetitions).
19 Basteck and Mantovani (2018) also report that DA is better than IA in terms of providing a level playing
field to less strategically sophisticated subjects.

36Figure D.1: Cognitive ability, equilibrium behavior, and payoff in simultaneous treatment
Game 1(a) Equilibrium ranking submissionn.s.

(b) Relative payoff

n.s.

**

1.0
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0

n.s.

1.0
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
Low High

Low High

Low High

Low High

Inexperienced

Experienced

Inexperienced

Experienced

Game 2(c) Equilibrium ranking submissionn.s.

(d) Relative payoff

n.s.

1.0
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0

n.s.

n.s.

1.0
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
Low High

Low High

Low High

Low High

Inexperienced

Experienced

Inexperienced

Experienced

Game 3 Simultaneous(e) Equilibrium ranking submission(f) Relative payoff
***

n.s.

**

1.0
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0

n.s.

1.0
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
Low High

Low High

Low High

Low High

Inexperienced

Experienced

Inexperienced

Experienced

Game 4 Simultaneous(g) Equilibrium ranking submission(h) Relative payoff
n.s.

**

n.s.

1.0
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0

***

1.0
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
Low High

Low High

Low High

Low High

Inexperienced

Experienced

Inexperienced

Experienced

Game 5 Simultaneous(i) Equilibrium ranking submission(j) Relative payoff
n.s.

n.s.

1.0
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0

n.s.

n.s.

1.0
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
Low High

Low High

Low High

Low High

Inexperienced

Experienced

Inexperienced

Experienced

Note. Based on the estimated coefficient of game dummies. Error bars show two standard errors range. Standard errors
are corrected for session-level clustering effect. ***, **, and * stand for statistical significance at the 1%, 5%, and 10%
levels, respectively. n.s., means the difference is not statistically significant at the 10% level based on the Wald test.

37

Figure D.2: Cognitive ability, equilibrium behavior, and payoff in sequential treatment(a) Equilibrium Ranking
in Sector 1***

Game 3. Sequential(b) Conditional optimality
in Sector 2

n.s.

1.0
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0

**

(c) Relative payoff

n.s.

1.0
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0

*

n.s.

1.0
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0

Low High

Low High

Low High

Low High

Low High

Low High

Inexperienced

Experienced

Inexperienced

Experienced

Inexperienced

Experienced(d) Equilibrium Ranking
in Sector 1
n.s.

Game 4 Sequential(e) Conditional optimality
in Sector 2

***

1.0
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0

**

(f) Relative payoff

***

1.0
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0

**

n.s.

1.0
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0

Low High

Low High

Low High

Low High

Low High

Low High

Inexperienced

Experienced

Inexperienced

Experienced

Inexperienced

Experienced(g) Equilibrium Ranking
in Sector 1
n.s.

Game 5 Sequential(h) Conditional optimality
in Sector 2

***

1.0
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0

n.s.

(i) Relative payoff

n.s.

1.0
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0

n.s.

**

1.0
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0

Low High

Low High

Low High

Low High

Low High

Low High

Inexperienced

Experienced

Inexperienced

Experienced

Inexperienced

Experienced

Note. Based on the estimated coefficient of game dummies. Error bars show two standard errors range. Standard errors
are corrected for session-level clustering effect. ***, **, and * stand for statistical significance at the 1%, 5%, and 10%
levels, respectively. n.s., means the difference is not statistically significant at the 10% level based on the Wald test.

38

Figures D.1 and D.2 show the results for simultaneous and sequential treatments, respectively.20 We observe that, when inexperienced, the “high” group participants submit the equilibrium ranking or the conditionally optimal ranking more frequently. As a result, these participants obtain higher relative payoffs than “low” group participants. However, the differences
are marginally significant (the 10% level) only for Game 3 in the simultaneous treatment andGames 3 and 4 in the sequential treatment, insignificant for the other games.
The advantage for the “high” group observed among inexperienced participants disappears
among experienced participants except for Games 4 and 5, in which the two sectors are asymmetric. This suggests that, on the one hand, gaining experience helps to reduce the effect ofdifferences in participants’ cognitive ability to achieve equilibrium behavior and obtain a higher
payoff. On the other hand, it also suggests that the required amount of experience to overcome
the impact of cognitive ability increases with the complexity of the allocation problem causedby the existence of multiple sectors that are asymmetric and interrelated.
Table D.2: Regressions corresponding to Figure D.1: Simultaneous treatment.
Equilibrium Ranking submission

Low, Inexp
High, Inexp
Low, Exp
High, Exp
Mid, Inexp
Mid, ExpN
R2
Hypothesis testing+Low = High (Inexp)
Low = High (Exp)Game 10.686∗∗∗(0.060)0.710∗∗∗(0.033)0.750∗∗∗(0.115)0.923∗∗∗(0.011)0.580∗∗∗(0.021)0.810∗∗∗(0.061)129
0.868Game 20.844∗∗∗(0.058)0.881∗∗∗(0.033)0.933∗∗∗(0.055)1.000∗∗∗(0.000)0.840∗∗∗(0.033)0.977∗∗∗(0.009)114
0.963

Game 30.463∗(0.153)0.694∗∗(0.133)0.685∗(0.236)0.652∗∗(0.167)0.484∗∗∗(0.002)0.750∗∗∗(0.026)84
0.775

Game 40.367∗∗(0.062)0.500∗∗∗(0.062)
0.583∗∗∗(0.053)0.719∗∗∗(0.019)0.375∗∗(0.084)0.722∗∗(0.187)81
0.766Game 50.458∗(0.149)0.524∗∗∗(0.002)0.500∗(0.196)
0.681∗∗∗(0.101)0.500∗∗(0.105)0.567∗∗∗(0.017)
87
0.811

0.790
0.188

0.609
0.280

0.001
0.939

0.362
0.028

0.686
0.154

Standard errors (adjusted for session-clustering effect) in parentheses∗ p < 0.10, ∗∗ p < 0.05, ∗∗∗ p < 0.01
+ p-values based on the Wald test are reported

20 The averages reported in the figure are based on the estimated coefficient obtained from running the thefollowing linear regression for each panel and each type of the subjects: Y = β1 LowInexp + β2 HighInexp +β3 LowExp + β4 HighExp + µ where LowInexp is a dummy variable that takes the value 1 if the inexperienced
participant is in the “Low” group and value 0, otherwise. HighInexp, LowExp, and HighExp are dummy
variables that are defined in a similar manner. The standard errors are corrected for the session-clustering effect.
The results of these regressions are reported in Tables D.2 to reg:CAmeanRPseq.

39

Table D.3: Regressions corresponding to Figure D.1: Simultaneous treatment.
Relative Efficiency

Low, Inexp
High, Inexp
Low, Exp
High, Exp
Mid, Inexp
Mid, ExpN
R2
Hypothesis testing+
H0: Low = High (Inexp)
H0: Low = High (Exp)Game 10.642∗∗∗(0.011)0.746∗∗∗(0.026)0.806∗∗∗(0.060)0.827∗∗∗(0.002)0.667∗∗∗(0.045)0.842∗∗∗(0.034)129
0.932

Game 20.967∗∗∗(0.058)0.911∗∗∗(0.030)0.989∗∗∗(0.044)
0.979∗∗∗(0.014)0.933∗∗∗(0.035)0.994∗∗∗(0.005)114
0.978

Game 30.644∗∗∗(0.064)0.719∗∗∗(0.082)0.750∗∗∗(0.101)0.769∗∗∗(0.040)0.633∗∗∗(0.022)0.816∗∗∗(0.013)84
0.960

Game 40.413∗∗∗(0.024)0.440∗∗∗(0.037)0.516∗∗∗(0.010)0.590∗∗∗(0.016)
0.467∗∗∗(0.023)0.597∗∗∗(0.070)81
0.963Game 50.842∗∗∗(0.058)0.860∗∗∗(0.025)0.842∗∗∗(0.060)0.878∗∗∗(0.000)0.814∗∗∗(0.030)0.860∗∗∗(0.004)
87
0.985

0.034
0.730

0.132
0.836

0.024
0.902

0.686
0.001

0.844
0.582

Standard errors (adjusted for session-clustering effect) in parentheses∗ p < 0.10, ∗∗ p < 0.05, ∗∗∗ p < 0.01
+ p-values based on the Wald test are reported

E

English translation of Instruction

Instructions differ slightly between sessions that begin with one-sector games (in Part 1) followedby two-sector games (in Part 2) and sessions that begin with two-sector games (in Part 1) followed
by one-sector game (in Part 2). The parts that differ between the two will be noted as [S1 → S2]
and [S2 → S1].

Instructions for ExperimentThank you very much for participating to our experiment today.
Below, all information needed to participate in the experiment is included. Please read it
carefully. If you have any questions, please raise your hand. We will come to your post to assistyou.
At the beginning of the experiment, you are given 1000 JPY as your initial endowment. You
can earn more money during the experiment by earning “points.” The number of points you canearn will depend on your own and other participants’ decisions.
Talking to other participants and using mobile phones are prohibited. Please only use thefunctions related to the experiment on your computer. We will answer your questions as thoroughly as possible.
The experiment consists of three parts (Parts 1, 2, 3).
40

Table D.4: Regressions corresponding to Figure D.2: Sequential Treatment.
Equilibrium Ranking in Sector 1

Low, Inexp
High, Inexp
Low, Exp
High, Exp
Mid, Inexp
aMid, ExpN
R2
Hypothesis testing+Low = High (Inexp)
Low = High (Exp)Game 30.576∗∗∗(0.058)0.750∗∗∗(0.070)0.819∗∗∗(0.078)0.833∗∗∗(0.100)0.792∗∗∗(0.029)0.719∗∗∗(0.017)90
0.875

Game 40.567∗∗∗(0.034)0.656∗∗∗(0.043)0.692∗∗∗(0.110)0.783∗∗∗(0.098)0.550∗∗∗(0.014)0.844∗∗∗(0.087)96
0.835

Game 50.787∗∗∗(0.002)0.850∗∗∗(0.042)0.806∗∗∗(0.027)0.889∗∗∗(0.016)0.821∗∗∗(0.007)0.889∗∗∗(0.027)
90
0.956

0.001
0.943

0.324
0.005

0.246
0.006

Standard errors (adjusted for session-clustering effect) in parentheses∗ p < 0.10, ∗∗ p < 0.05, ∗∗∗ p < 0.01
+ p-values based on the Wald test are reported• The instruction of Part 3 21 will be given to you after completing Parts 1 and 2.
• In Parts 1 and 2, you and the two other members of your group will decide how to allocate
goods according to a mechanism.
• [S1 → S2] The experiment will be repeated six times in Part 1 and 18 times in Part 2 (atotal of 24 times). Below, we call one play of a game a round. Thus, there are six rounds
in Part 1 and 18 rounds in Part 2 for a total of 24 rounds during the experiment.
• [S2 → S1] The experiment will be repeated 18 times in Part 1 and six times in Part 2 ( atotal of 24 times). Below, we call one play of a game a round. Thus, there are 18 rounds
in Part 1 and six rounds in Part 2 for a total of 24 rounds during the experiment.
• At the beginning of each round, you and two other randomly selected participants will
form a group.
• Within each group, each player will be assigned a role of either Player 1, 2, or 3. Your
role will change randomly for each round, so please pay attention to the indication on yourscreen.
21 In Part 3, we assessed participants’ cognitive abilities based on their responses to the Raven’s advanced

progressive matrices test.

41

Table D.5: Regressions corresponding to Figure D.2: Sequential Treatment.
Conditional Optimality

Low, Inexp
High, Inexp
Low, Exp
High, Exp
Mid, Inexp
Mid, ExpN
R2
Hypothesis testing+Low = High (Inexp)
Low = High (Exp)Game 30.697∗∗∗(0.074)0.883∗∗∗(0.042)0.847∗∗∗(0.086)
0.964∗∗∗(0.021)0.868∗∗∗(0.069)0.895∗∗∗(0.010)90
0.949

Game 4
0.633∗∗∗(0.101)0.854∗∗∗(0.037)0.833∗∗∗(0.000)0.967∗∗∗(0.000)0.700∗∗∗(0.042)0.978∗∗∗(0.012)96
0.934

Game 50.778∗∗∗(0.005)0.783∗∗∗(0.042)0.764∗∗∗(0.045)0.833∗∗∗(0.000)0.821∗∗∗(0.047)0.785∗∗∗(0.013)
90
0.962

0.010
0.166

0.040
0.000

0.914
0.219

Standard errors (adjusted for session-clustering effect) in parentheses∗ p < 0.10, ∗∗ p < 0.05, ∗∗∗ p < 0.01
+ p-values based on the Wald test are reported• The points you can earn during each round will differ depending on the goods you have at
the end of each round. The number of points each player can earn from each good is shownon the screen in each round. Therefore, please pay attention to the information shown on
the screen.
• Once Parts 1 and 2 are completed, one out of the 24 rounds (of Part 1 and Part 2 together)will be chosen randomly. The same round will be chosen for all of you. In addition to yourinitial endowment of 1000 JPY, you will be paid based on the points you earned in thisrandomly chosen round with an exchange rate of 1 point = 100 JPY.
Below, detailed instructions for Part 1 will be provided. There are [S1 → S2] six rounds[S1 → S2]18 rounds in Part 1.
[S1 → S2] The instructions for the one-sector game are given, followed by a quiz for the
one-sector game.
[S2 → S1] The instructions for the two-sector game are given, followed by a quiz for thetwo-sector game.

42

Table D.6: Regressions corresponding to Figure D.2: Sequential Treatment.
Relative Efficiency

Low, Inexp
High, Inexp
Low, Exp
High, Exp
Mid, Inexp
Mid, ExpN
R2
Hypothesis testing+Low = High (Inexp)
Low = High (Exp)Game 30.674∗∗∗(0.044)0.783∗∗∗(0.084)0.767∗∗∗(0.072)0.848∗∗∗(0.024)0.769∗∗∗(0.045)0.748∗∗∗(0.024)90
0.969

Game 40.439∗∗∗(0.029)0.547∗∗∗(0.053)0.569∗∗∗(0.070)0.642∗∗∗(0.028)0.479∗∗∗(0.063)0.736∗∗∗(0.073)96
0.937

Game 50.863∗∗∗(0.006)
0.942∗∗∗(0.033)0.890∗∗∗(0.005)0.932∗∗∗(0.005)0.922∗∗∗(0.025)0.942∗∗∗(0.001)
90
0.991

0.072
0.461

0.020
0.178

0.134
0.023

Standard errors (adjusted for session-clustering effect) in parentheses∗ p < 0.10, ∗∗ p < 0.05, ∗∗∗ p < 0.01
+ p-values based on the Wald test are reported

E.1

Instructions for the one-sector gameAt the beginning of each round, Player 1 will receive Good 1, Player 2 will receive Good 2,
and Player 3 will receive Good 3. In addition, the information regarding how many points eachplayer will receive from each good is displayed.
Based on this piece of information, each player ranks the three goods that they desire mostfrom highest to lowest.
Once all three players submit their rankings, we determine if a trading cycle is possible amongplayers for their most desired goods. If a trading cycle can be established among some players,then these players exchange their goods. Once goods are exchanged, these players receive thepoints according to the good they obtained for this round.
If there is more than one player who did not participate in the exchange, we determine if
a trading cycle is possible among the remaining players. If such a cycle exists, these players
exchange their goods.
If there is still more than one player who did not exchange, we continued the same process.
Here are a few examples:
Example 1: Suppose
Player 1 submits the following ranking of three goods: Goods 1, 2, and 3.
43

Player 2 submits the following ranking of three goods: Goods 2, 3, and 1.
Player 3 submits the following ranking of three goods: Goods 3, 1, and 2.
In this case, all players rank the good they are initially given as the most desired good. Thus,
it is equivalent to the existence of a trading cycle. In this case, all three players received thegood they were initially given, and the round ends.
Example 2: Suppose
Player 1 submits ranking according to: Goods 2, 3, and 1.
Player 2 submits ranking according to: Goods 3, 1, and 2.
Player 3 submits ranking according to: Goods 1, 2, and 3.
In this case, the following trading cycle among the most desired good of each player exists:
Player 1 receives Good 2 from Player 2, Player 2 receives Good 3 from Player 3, and Player
3 receives Good 1 from Player 1. Thus, each player receives their most desired good, and theround ends.
Example 3: Suppose
Player 1 submits ranking according to: Goods 2, 3, and 1.
Player 2 submits ranking according to: Goods 1, 3, and 2.
Player 3 submits ranking according to: Goods 2, 1, and 3.
In this case, only the following trading cycle exists: Player 1 receives Good 2 from Player 2,
and Player 2 receives Good 1 from Player 1. (This is because none of the players rank Good 3that Player 3 has as the most desired good.) Therefore, Players 1 and 2 exchange their goods.
Since only Player 3 remains, no further exchange is possible. Thus, the round ends by Player 1
receiving Good 2, Player 2 receiving Good 1, and Player 3 and receiving Good 3.

E.2

Quiz for the one-sector game[S1 → S2] Practice Quiz

Q1: If each player submits his/her ranking as follows, how will goods be exchanged? Please
complete the empty spaces in the “explanation” section below.
Player 1 submits ranking according to: Goods 1, 2, and 3.
Player 2 submits ranking according to: Goods 2, 3, and 1.
Player 3 submits ranking according to: Goods 3, 1, and 2.
44

“Explanation”
In this case, because all players rank the goods that they initially received as (the most )desired good, an exchange cycle exists. Thus, ( with no ) exchange, all players receive (theirinitial ) good and the round ends. (Note: answers are italics.)
Q2: If each player submits his/her ranking as follows, how will goods be exchanged? Please
complete the empty spaces in the “explanation” section below.
Player 1 submits ranking according to: Goods 2, 3, and 1.
Player 2 submits ranking according to: Goods 3, 1, and 2.
Player 3 submits ranking according to: Goods 1, 2, and 3.
“Explanation”
In this case, an exchange cycle exists in which Player 1 receives Good ( 2 ) from Player 2,
Player 2 receives Good ( 3 ) from Player 3, and Player 3 receives Good ( 1 ) from Player 1 basedon each player’s most desired good. Thus, each player receives the good s/he most desires, andthe round ends.
Q3: If each player submits his/her ranking as follows, how will goods be exchanged? Please
complete the empty spaces in the “explanation” section below.
Player 1 submits ranking according to: Goods 2, 3, and 1.
Player 2 submits ranking according to: Goods 1, 3, and 2.
Player 3 submits ranking according to: Goods 2, 1, and 3.
“Explanation”
In this case, the only possible exchange cycle is that Player 1 receives Good ( 2 ) from Player(2 ), and Player 2 receives Good ( 1 ) from Player ( 1 ). Because none of the players desire Good( 3 ) that Player ( 3 ) has, Player ( 3 ) cannot enter the exchange cycle.) Therefore, an exchange
occurs between Player ( 1 ) and ( ( 2) ). Because only Player ( 3 ) remains, no more exchanges
are possible. Thus, Player 1 receives Good ( 2 ), Player 2 receives Good ( 1 ), Player 3 receives
Good ( 3 ), and the round ends.

E.3

Instruction for the two-sector game (simultaneous running experiment)

At the beginning of each round, each player receives two types of goods, Type A and Type B.
Player 1 will receive Goods A1 and B1 , Player 2 will receive Goods A2 and B2 , and Player 3 will45

receive Goods A3 and B3 . In addition, information regarding how many points each player will
receive from each pair of goods will be displayed.
Based on this information, each player ranks three of each type of good that they desire
most. When submitting the ranking for each type of good, please only enter the number. For
example, when you rank A1 , A2 , and A3 in this order for Type A goods, please enter 1 for themost desired good and 2 for the second most desired good.
Once all the three players submit their rankings for Type A and Type B goods, we will
determine if a trading cycle is possible among players for each type of their most desired goods.
If a trading cycle can be established among some players for Type A goods, then these players
exchange their Type A goods. Similarly, if a trading cycle can be established among some players
for Type B goods, then these players exchange their Type B goods.
Once a player successfully obtained both Type A and Type B goods through exchange cycles,
the player will receive the points according to the pair of each type of goods s/he obtained forthis round.
If there is more than one player who did not participate in the exchange of either type of
good, we determine whether a trading cycle is possible based on these players’ highest ranked
goods for the type of good. If such a cycle exists, these players exchange their goods.
If there is still more than one player who did not exchange either type of good, we continue
the same process.
Let us consider a few examples. Below, we explain only Type A goods, but similar examples
apply for Type B goods.
Example 1: Suppose
Player 1 submits the following ranking of three goods: Good A1 , A2 , and A3 .
Player 2 submits the following ranking of three goods: Good A2 , A3 , and A1 .
Player 3 submits the following ranking of three goods: Good A3 , A1 , and A2 .
In this case, all players rank the good they are initially given as the most desired good. Thus,
it is equivalent to the existence of a trading cycle. In this case, all three players received theType A good they were initially given, and the round ends.
Example 2: Suppose
Player 1 submits ranking according to Good A2 , A3 , and A1 .
Player 2 submits ranking according to Good A3 , A1 , and A2 .

46

Player 3 submits ranking according to Good A1 , A2 , and A3 .
In this case, the following trading cycle among the most desired Type A good of each player
exists: Player 1 receives Good A2 from Player 2, Player 2 receives Good A3 from Player 3, andPlayer 3 receives Good A1 from Player 1. Thus, each player receives their most desired Type A
good, and the round ends.
Example 3: Suppose
Player 1 submits ranking according to Good A2 , A3 , and A1 .
Player 2 submits ranking according to Good A1 , A3 , and A2 .
Player 3 submits ranking according to Good A2 , A1 , and A3 .
In this case, only the following trading cycle exists: Player 1 receives Good A2 from Player
2, and Player 2 receives Good A1 from Player 1. (This is because none of the players rank Good
A3 that Player 3 has as the most desired good.) Therefore, Players 1 and 2 exchange Type A
goods. Since only Player 3 remains, no further exchange is possible. Thus, the round ends byPlayer 1 receiving Good A2 , Player 2 receiving Good A1 , and Player 3 and receiving Good A3 .
Thus far, we have only explained the exchange mechanism of Type A goods. At the beginningof each round, each player has to ranks three goods from the Type A and Type B categories,
separately. Then, the result depends on the type of exchange cycle established for each type of
good.

E.4

Instructions for the two-sector game (sequential running experiment)

At the beginning of each round, each player receives two types of goods, Type A and Type B.
Player 1 will receive Good A1 and B1 , Player 2 will receive Good A2 and B2 , and Player 3 will
receive Good A3 and B3 . In addition, information regarding how many points each player will
receive from each pair of goods will be displayed.
Based on this information, each player submits the ranking for their three most desired Type
A goods. When submitting the ranking, please only enter the number. For example, when you
rank A1 , A2 , and A3 in this order, please enter 1 for the most desired good and 2 for the second
most desired good.
Once all three players submit their rankings for Type A goods, we will determine whether a
trading cycle is possible among players for their most desired goods. If a trading cycle can beestablished among some players, then these players exchange Type A goods.
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If there is more than one player who did not participate in the exchange, we determine
whether a trading cycle is possible based on these players’ highest ranked goods. If such a cycle
exists, these players exchange their goods.
If there is still more than one player who did not exchange a Type A good, we continue thesame process.
Let’s see a few examples.
Example 1: Suppose
Player 1 submits the following ranking of three goods: Good A1 , A2 , and A3 .
Player 2 submits the following ranking of three goods: Good A2 , A3 , and A1 .
Player 3 submits the following ranking of three goods: Good A3 , A1 , and A2 .
In this case, all players rank the good they are initially given as the most desired good. Thus,
it is equivalent to the existence of a trading cycle. In this case, all three players received theType A good they were initially given, and the round ends.
Example 2: Suppose
Player 1 submits ranking according to Good A2 , A3 , and A1 .
Player 2 submits ranking according to Good A3 , A1 , and A2 .
Player 3 submits ranking according to Good A1 , A2 , and A3 .
In this case, the following trading cycle among the most desired Type A good of each player
exists: Player 1 receives Good A2 from Player 2, Player 2 receives Good A3 from Player 3, andPlayer 3 receives Good A1 from Player 1. Thus, each player receives their most desired Type A
good, and the round ends.
Example 3: Suppose
Player 1 submits ranking according to Good A2 , A3 , and A1 .
Player 2 submits ranking according to Good A1 , A3 , and A2 .
Player 3 submits ranking according to Good A2 , A1 , and A3 .
In this case, among the most desired goods of three players, only the following trading cycle
exists: Player 1 receives Good A2 from Player 2, and Player 2 receives Good A1 from Player 1.
(This is because none of the players rank Good A3 that Player 3 has as the most desired good.)
Therefore, Players 1 and 2 exchange Type A goods. Since only Player 3 remains, no further

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exchange is possible. Thus, the round ends by Player 1 receiving Good A2 , Player 2 receiving
Good A1 , and Player 3 and receiving Good A3 .
Once exchanges of Type A goods are completed, each player’s submitted ranking and obtained
good are displayed on the screen. Then, each player submits the ranking for Type B goods
according to their most desired good (the most desired good, the second most desired good, andthe third most desired good for Type B). When submitting the ranking, please only enter thenumber. For example, when you rank B1 , B2 , and B3 in this order, please enter 1 for the mostdesired good and 2 for the second most desired good. When submitting this ranking, information
regarding how many points each player will receive from each pair of goods will be displayed.
Furthermore, which Type A good each player has received through exchange will be shown.
Once all the three players submit their rankings for Type B goods, we will determine whether
a trading cycle is possible among players for their most desired goods. If a trading cycle canbe established among some players, then these players exchange their Type B goods. If there ismore than one player who did not participate in the exchange, we continue the process until it
ends.
Once each player has received their Type A and Type B goods through exchanges, the payoff
they receive is also determined.

E.5

Quiz for the two-sector game (common for simultaneous and sequential running experiment)[S2 → S1] Practice Quiz

Q1: If each player submits his/her ranking as follow, how will goods be exchanged? Please
complete the empty spaces in the “explanation” section below.
Player 1 submits ranking according to Good A1 , A2 , and A3 .
Player 2 submits ranking according to Good A2 , A3 , and A1 .
Player 3 submits ranking according to Good A3 , A1 , and A2 .
“Explanation”
In this case, because all players rank Type A goods initially received as (the most ) desired
Type A good, an exchange cycle exists. Thus, ( with no ) exchange, all players receive (theirinitial ) Type A good and the round ends. (Note: answers are italics.)
Q2: If each player submits his/her ranking as follows, how will goods be exchanged? Please
complete the empty spaces in the “explanation” section below.
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Player 1 submits ranking according to Good A2 , A3 , and A1 .
Player 2 submits ranking according to Good A3 , A1 , and A2 .
Player 3 submits ranking according to Good A1 , A2 , and A3 .
“Explanation”
In this case, an exchange cycle such that Player 1 receives Good ( A2 ) from Player 2, Player
2 receives Good ( A3 ) from Player 3, and Player 3 receives Good ( A1 ) from Player 1 exists
based on each player’s most desired Type A good. Thus, each player receives the Type A good,s/he most desires and the round ends.
Q3: If each player submits his/her ranking as follows, how will goods be exchanged? Please
complete the empty spaces in the “explanation” section below.
Player 1 submits ranking according to Good A2 , A3 , and A1 .
Player 2 submits ranking according to Good A1 , A3 , and A2 .
Player 3 submits ranking according to Good A2 , A1 , and A3 .
“Explanation”
In this case, among the most desired Type A goods of each player, only the exchange cycle
in which Player 1 receives Good ( A2 ) from Player (2 ), Player 2 receives Good ( A1 ) from
Player ( 1 ) exists. Because none of the players desires Good ( A3 ) that Player ( 3 ) has, Player( 3 ) cannot enter the exchange cycle.) Therefore, an exchange takes place between Player ( 1) and ( ( 2) ). Because it is only Player ( 3 ) who remains, no more exchange is possible. Thus,
Player 1 receives Good ( A2 ), Player 2 receives Good ( A1 ), and Player 3 receives Good ( A3), and the round ends.

E.6

Instruction for Part 2
Instruction for Part 2• [S1 → S2] In this part of the experiment, just as in Part 1, you and other members of yourgroup will decide how to allocate goods according to a mechanism. This time, however,there are two types of goods, and you will have to decide how to allocate eachtype of goods.
• [S2 → S1] In this part of the experiment, just as in Part 1, you and other members of yourgroup will decide how to allocate goods according to a mechanism. This time, however,there is only one type of good.
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• [S2 → S1] The experiment will be repeated 18 times. Each repetition is called a round.
• [S1 → S2] The experiment will be repeated six times. Each repetition is called a round.
• At the beginning of each round, you and two other randomly selected participants will
form a group.
• Within each group, each player will be assigned a role of Player 1, 2, or 3. Your role willchange randomly in each round, so please pay attention to the indication on your screen.
Then, the instruction for the relevant game follows. And there was no practice quiz after the
instruction of Part 2.

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F

Screen-shots

Here, we provide screenshots of our experimental software. As the interface was in Japanese,
we have added English translations. In addition to these screenshots, there were screens that
demonstrated the payoff a subject obtained during that round following each round.

Figure F.1: Games 1 and 2. Screen to enter the ranking for the three goods.

Figure F.2: Games 1 and 2. Screen displaying the submitted ranking and resulting allocation.

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Figure F.3: Games 3,4, and 5. Simultaneous treatment. Screen to enter the ranking for thethree goods of Types A and B.

Figure F.4: Games 3,4, and 5. Simultaneous treatment. Screen displaying the submitted
ranking and resulting allocations of the two types of goods.

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Figure F.5: Games 3, 4, and 5. Sequential treatment, stage 1. Screen to enter the ranking forthe three Type A goods.

Figure F.6: Games 3, 4, and 5. Sequential treatment, stage 1. Screen to show the submitted
ranking and resulting allocation of Type A goods.

54Figure F.7: Games 3, 4, and 5. Sequential treatment, stage 2. Screen to enter the ranking
among the three Type B goods knowing the allocation of Type A goods.

Figure F.8: Games 3, 4, and 5. Sequential treatment, stage 2. Screen to show the submitted
ranking of Type B goods and the resulting allocation of Type A and B goods.

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