ISSN (Print) 0473-453X

Discussion Paper No. 1176

ISSN (Online) 2435-0982

THE EFFECT OF CHOOSING A PROPOSER
THROUGH A BIDDING PROCEDURE
IN IMPLEMENTING THE SHAPLEY VALUE

Michela Chessa
Nobuyuki Hanaki
Aymeric Lardon
Takashi Yamada

Revised August 2022
May 2022

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

The effect of choosing a proposer through a bidding
procedure in implementing the Shapley value*
Michela Chessa†

Nobuyuki Hanaki‡

Aymeric Lardon§

Takashi Yamada¶August 13, 2022
Abstract
We experimentally compare a simplified version of two mechanisms that implement the Shapley value as an (ex ante) equilibrium outcome of a noncooperative
bargaining procedure: one proposed by Hart and Mas-Colell (1996, H-MC) andthe other by Perez-Castrillo and Wettstein (2001, PC-W). While H-MC induces
the Shapley value only on average, PC-W does so as a unique equilibrium outcome by introducing an additional bidding stage on top of H-MC. We investigate
the effect of this additional bidding stage on the resulting outcomes such as thefrequency of grand coalition formation, efficiency, and the distance between therealized allocation and the Shapley value. Our experiment shows that H-MC not
only results in significantly greater efficiency than PC-W, but also that the average
allocation is closer to the Shapley value for those groups that formed the grandcoalition. This difference is because those proposers who won the bidding stage in
PC-W tend to offer an allocation that favors themselves more than the randomly
chosen proposers in H-MC, and such offers are more likely to be rejected.

JEL codes: C70, C71, C92
Keywords: Nash program, Bargaining procedures, Shapley value, Experiments* The experiments reported in this paper have been approved by the IRB of Yamaguchi University(No. 5). We gratefully acknowledge financial support from the Joint Usage/Research Center at ISER,
Osaka University, and Grants-in-Aid for Scientific Research, Japan Society for the Promotion of Science(15K01180, 18K19954, 20H05631), Fund for the Promotion of Joint International Research (Fostering
Joint International Research) (15KK0123), and the French government-managed by Agence Nationale
de la Recherche under Investissements d’Avenir UCAJEDI (ANR-15-IDEX-01). In particular, we thankthe UCAinACTION project.
†
Université Côte d’Azur, CNRS, GREDEG, France. E-mail: michela.chessa@univ-cotedazur.fr‡
Institute of Social and Economic Research, Osaka University, Japan.
E-mail:nobuyuki.hanaki@iser.osaka-u.ac.jp§
GATE Lyon Saint-Etienne, UMR 5824 CNRS, Université de Lyon.
France.
E-mail:aymeric.lardon@univ-st-etienne.fr¶
Faculty of Global and Science Studies, Yamaguchi University, Japan. E-mail: tyamada@yamaguchiu.ac.jp1

1

Introduction

The Nash program (Nash, 1953) provides a noncooperative foundation for cooperative
solution concepts. In this seminal work, Nash showed that the solution to the cooperative bargaining problem (Nash, 1950) can be obtained as an equilibrium outcome ofa noncooperative game. The main idea of the Nash program “is both simple and important: the relevance of a concept [...] is enhanced if one arrives at it from different
points of view” (Serrano, 2005, p. 220). Then, noticing that a different noncooperative
approach yields the same solution, which “indicates that the solution is appropriate for awider variety of situations” (Nash, 1953, p. 136), and then that the cooperative solution
is widely applicable (even through a normative approach and without relying on precise
assumptions about the bargaining protocol).
Many authors have contributed to the development of the Nash program (see, Serrano, 2005, 2008, 2014, 2021, for surveys). Since the inception of the Nash program,the cooperative solution that has attracted most attention is the Shapley value (Shapley,1953). The popularity of this solution concept comes from its intuitive definition (anaverage of marginal contributions), its desirable properties (such as efficiency, monotonicity, or symmetry), and consequently, from the many theoretical and real-world
applications (cost and payoff sharing, voting power, fair division, etc.). Indeed, various
papers (for example, Gul, 1989; Harsanyi, 1981; Hart and Moore, 1990; Krishna andSerrano, 1995; Winter, 1994; Hart and Mas-Colell, 1996; Perez-Castrillo and Wettstein,2001) have proposed its implementation.
Chessa et al. (2022a,b) contributed to this literature by providing experimental comparisons between Winter (1994) and Hart and Mas-Colell (1996) in Chessa et al. (2022b)
and among three versions of Winter (1994) in Chessa et al. (2022a). Chessa et al.

2(2022b) found that the demand-based mechanism proposed by Winter (1994) resulted
in an allocation that better reflects players’ effective bargaining power, while the efficiency and the frequency of the grand coalition formation are lower than those from anoffer-based mechanism proposed by Hart and Mas-Colell (1996). However, because thisresult was obtained from a simplified version of the approach of Winter (1994), Chessa
et al. (2022a) investigated the robustness of the outcome by comparing the outcome ofthe simplified one-period version of Winter (1994) considered in Chessa et al. (2022b)with a more complex—but closer to the original theoretical analysis—two-period version of Winter (1994). Chessa et al. (2022a) found that the results of the one-period andtwo-period versions were very similar.
In this paper, we extend the experimental analyses of Chessa et al. (2022a,b) bycomparing the mechanisms proposed by Hart and Mas-Colell (1996) and Perez-Castrillo
and Wettstein (2001). Both mechanisms are offer-based. Namely, in both mechanisms,
there will be a proposer who proposes an allocation, which is voted on by the remaining
players sequentially. The key difference between the two is the way the proposer ischosen. On the one hand, a proposer is chosen randomly among the players in Hart
and Mas-Colell (1996). On the other hand, in Perez-Castrillo and Wettstein (2001),
the proposer is determined through a bidding procedure. Both mechanisms predict at
equilibrium that the grand coalition will form. Moreover, while Hart and Mas-Colell(1996) implement the Shapley value as an ex ante equilibrium payoff (i.e., it is only
achieved as an expected outcome), Perez-Castrillo and Wettstein (2001) implement it asa unique equilibrium outcome of the game.
The bidding procedure introduced by Perez-Castrillo and Wettstein (2001) not only
constitutes an interesting theoretical investigation, but it is also a good representation ofsome real-world scenarios. One main characteristic is that regardless of the random se3

lection of the proposer that many theoretical bidding mechanisms rely on, participants ina negotiation may often be willing to take costly measures to be selected as the proposer
because of the well-known proposer advantage (an individual who makes a proposal
often obtains a greater share than others). Thus, a preliminary bidding stage may be, forexample, a good representation of pre-negotiations in government coalition formation:in Italy, a preliminary round of consultation with the possible prime ministers allows thehead of state to select the “best” prime minister according to her/his bid-like preliminary
proposal.1
The literature on competition to win the proposal for the subsequent bargaining procedure is somewhat vast at present (see, e.g., Yildirim, 2007; Kim and Kim, 2022),
and relies on different approaches such as preliminary lottery contests, ultimatum bargaining games, or bidding procedures. However, these additional preliminary levels inthe bargaining procedures make one wonder whether they would indeed result in thepredicted theoretical solution if the game was played by participants in a laboratory
experiment. For example, Navarro and Veszteg (2011) experimentally investigated theimpact of introducing a bidding stage to determine the proposer in a two-player ultimatum bargaining game. Although they found that participants’ behavior stabilized after
many trials (after 20 out of 30 repetitions) and the payoff gap between the proposer
and the responder becomes smaller as predicted by the theory, the observed behavior,even in these final rounds of the experiment, deviate substantially from the theoreticalprediction.
Furthermore, as demonstrated by the idea of “obviously strategy proofness” (Li,2017), a mechanism needs to be extremely simple for it to result in the outcome in1

Similarly, in France, a similar preliminary round of consultation was recently implemented becauseof the relative majority of prime minister Macron’s parliamentarian group.

4

tended by its designer, and such additional stages certainly make the procedure more
complex. In this paper, we investigate whether the bidding procedure introduced byPerez-Castrillo and Wettstein (2001) improves (as theoretically predicted) or in fact
lowers the performance of the underlying mechanism of Hart and Mas-Colell (1996) in
implementing the Shapley in a laboratory experiment.
Indeed, our experiment shows that the simpler mechanism à la Hart and Mas-Colell(1996) not only results in greater efficiency than that of Perez-Castrillo and Wettstein(2001), but also that the average allocation is closer to the Shapley value for those groups
that formed a grand coalition. Our analyses suggest that this happens because those
proposers who won the bidding stage in Perez-Castrillo and Wettstein (2001) tend to
make offers that favor themselves (thus, they are less equal and deviate more from theShapley value) and are more likely to be rejected than those randomly selected proposers
in Hart and Mas-Colell (1996). We believe that this may be a consequence of the factthat winners at the bidding stage consider themselves to be entitled to a larger share(due to winning at the bidding phase, or because the winners of a bidding phase arethose players who are the most motivated in exploiting the proposer advantage), whereas
randomly selected proposers feel less legitimate in exploiting such an advantage.
The remainder of the paper is organized as follows. We present several theoreticalpreliminaries in cooperative game theory and the two mechanisms considered in our
experiment in Section 2. Section 3 describes the experiment procedure, while the results
are presented in Section 4. Section 5 concludes.

2

Theoretical preliminaries

5

2.1

Cooperative games and values

Let N = {1, . . . , n} be a finite set of players. Each subset S of N is called a coalition while N is called the grand coalition. A cooperative game with transferable utility(hereafter, a cooperative game) on a fixed player set N is a function v : 2N → R suchthat v(∅) = 0. For each coalition S ⊆ N , v(S) describes the worth that membersof S can achieve by agreeing to cooperate. We use G N to denote the set of all gameswith player set N . A game v ∈ G N is said to be monotonic if v(S) ≤ v(T ) for each
S ⊆ T ⊆ N . In monotonic games, the bigger the coalition is, the higher its worth
becomes. A game v ∈ G N is said to be zero-monotonic if v(S) + v({i}) ≤ v(S ∪ {i})for any subset S ⊆ N with i 6∈ S. In zero-monotonic games, there are no negative
externalities when a single player joins a coalition.2

A value is a mapping ψ : G N → RN which uniquely determines, for each v ∈ G N
and each player i ∈ N , a payoff ψi (v) ∈ R for participating to v ∈ G N . The Shapley
value is the best-known solution concept, which is widely applied in economic andpolitical models, and is defined as:φi (v) =(|S| − 1)!(|N | − |S|)!
(v(S) − v(S \ {i})) ∀i ∈ N.
|N|!
S⊆N,i∈SX

The Shapley value assigns to every player his or her expected marginal contribution to
the coalition of players that enter the game before this player, given that every order ofentrance has equal probability. Various characterizations of the Shapley value have been
2

Note that the classes of zero-monotonic and monotonic games are not equivalent. For example,
let N = {1, 2}, v be such that v({1}) = 4, v({2}) = −2 and v({1, 2}) = 3, and w be such thatw({1}) = 3, w({2}) = 2 and w({1, 2}) = 4. We can see that v is zero-monotonic but not monotonic
while w is monotonic but not zero-monotonic. It is worth noting that all the four games we consider inexperiments respect these two assumptions.

6

provided in the literature. One of the most famous characterizations uses efficiency,
symmetry, the null player, and additivity axioms and can be easily deduced from theseminal article by Shapley (1953).

2.2

Two mechanisms

In this subsection, we present the Hart and Mas-Colell (1996, referred to as H-MC
below) mechanism (Section 2.2.1) and the Perez-Castrillo and Wettstein (2001, referred
to as PC-W below) mechanism (Section 2.2.2) in more detail.

2.2.1

H-MC mechanism

The bargaining procedure proposed by Hart and Mas-Colell (1996) implements the
Shapley value in monotonic cooperative games. In this mechanism, the bargaining starts
with a randomly chosen proposer making an offer to the other players (responders). The
responders, sequentially, may either accept or reject the offer. If the offer is acceptedby all responders, it is implemented. If one of the responders rejects the offer, the gamemoves to the next stage. If the offer is rejected, the proposer leaves the game with theworth of her stand-alone coalition with probability p and the bargaining continues withthe remaining players, with a new player randomly chosen as the new proposer. In ourlaboratory implementation, we set p = 1 to make it comparable with the PC-W mechanism. We present here a more formal description of the H-MC mechanism. A decision
point position at time t is given simply by the vector (S t , j), where:
S t ⊆ N is the set of players still in the game, andj ∈ S t is the player making an offer to the remaining players (ti )i∈S t \{j} suchP
that i∈S t \{j} ti ≤ v(S t ).
7

With j’s offer, the game proceeds now in the following way:
1) if all i ∈ S t \ {j} accept the offer one after the other, then players in S t \ {j}P
are paid (ti )i∈S t \{j} , player j is paid v(S t ) − i∈S t \{j} ti , and the game ends;
2) if at least one player i ∈ S t \ {j} refuses the offer, then there are two possibilities:2a) if |S t | = 2 (only one more player is left, together with j), then they bothobtain the worth of their own stand-alone coalition v({i}) for each i ∈ S t , andthe game ends;2b) if |S t | > 2, then player i is removed from the game, she obtains her individual
payoff v({i}), a new proposer k ∈ S t+1 = S t \ {j} is randomly selected, and thenew position is (S t+1 , k).
The game starts with a randomly chosen proposer j ∈ N , so that the initial positionis (N, j). It terminates either when there are no more players in the game (see point 2aabove), or when the offer is unanimously accepted (see point 1 above).
Hart and Mas-Colell (1996) showed that this game has a unique subgame perfectequilibrium that supports the grand coalition and yields the Shapley value payoff vector
in expectation.
Given a specific initial proposer j ∈ N , the a posteriori equilibrium payoff assigns
to each other player her Shapley value in the cooperative game reduced to the set ofplayers N \ {j}, and to the proposer, the marginal contribution to the grand coalitionv(N ) − v(N \ {j}).

82.2.2

PC-W mechanism

Perez-Castrillo and Wettstein (2001) proposed a bidding mechanism to implement the
Shapley value for zero-monotonic cooperative games. In this mechanism, unlike Hart
and Mas-Colell (1996), the bargaining starts with a bidding stage where each of theplayers makes a bid to each of the other players. The proposer is chosen as the player
making the highest net bid, calculated as the difference between the sum of the bids aplayer makes to the others minus the sum of the bids the others make to her. If several
players make the highest net bid, the proposer is then selected randomly from them. At
the end of the bidding stage, the proposer pays the promised bids to the other players.
In the second stage, the proposer makes an offer to the other players (responders). This
offer is sequentially accepted or rejected by the responders. If all the responders accept,
the game ends and the proposer pays the other players according to their offers andreceives what remains of the worth of the coalition. If one of the responders rejects theoffer, the proposer leaves the game and obtains the worth of her stand-alone coalition
minus the bids she has already paid in the first stage. The remaining players keep what
they have received, and they begin a new bargaining round.
We present here a more formal description of the PC-W mechanism. A decision
point position at time t is given simply by the set of players S t ⊆ N remaining in thegame, (when t = 1, S t = N ). The game is described as follows:(1) Each player i ∈ S t makes bids bij to the other players j ∈ S t \{i}. Player i withP
the highest net bid j∈S\{i} (bij − bji ) is chosen to be the proposer and pays bij
to every j ∈ S t \{i}; if several players make the highest net bid, the proposer ischosen randomly among them.
(2) If player i is the proposer, she makes an offer yji to the other players j ∈ S t \{i}.
9(3) With i’s offer, the game proceeds in the following way:(3a) If the offer is sequentially accepted by the other players, each player j ∈P
S t \{i} receives bij + yji , the proposer obtains v(S t ) − j∈S t \{i} (bij + yji ),and the game ends.
(3b) If at least one player j ∈ S t \{i} refuses the offer, then proposer i is removedP
and obtains v({i}) − j∈S t \{i} bij ; the new set of players becomes S t+1 =S t \{i} in which each member j receives bij and a new bidding mechanism
starts among them (if only one player remains in the game, i.e., |S t+1 | = 1,
then that player obtains the worth of her stand-alone coalition and the gameends).
Perez-Castrillo and Wettstein (2001) show that any subgame perfect equilibrium of
this game implements the Shapley value as the a posteriori equilibrium payoff. At
these equilibria, the bid of player i to player j corresponds to j’s Shapley value of theoriginal game minus j’s Shapley value in the cooperative game reduced to the set ofplayers N \ {i}. The balanced contribution property (Myerson, 1980) then ensures that
all net bids are equal to zero, leading to a random selection of the proposer among all
the players. Moreover, player i, if she is randomly selected to be the proposer, offers toplayer j her Shapley value in the reduced cooperative game with player set N \ {i}. It is
worth noting that when the game is strictly zero-monotonic,3 the grand coalition always
forms. Otherwise, rejection of some offers could also constitute a subgame perfectequilibrium (this is the case, for example, in game 2 in Table 1 below when Player 1 ischosen as the proposer).
3

A game is strictly zero-monotonic if v(S) + v({i}) < v(S ∪ {i}) for any nonempty subset S ⊆ Nwith i 6∈ S. Note that only games 1, 3, and 4 that we consider in our experiments are strictly zeromonotonic.

10

3

The experimental setting

We consider the four games that are studied in (Chessa et al., 2022a,b). These games
and their corresponding Shapley values are reported in Table 1.
Table 1: Four games and corresponding Shapley values
Game 1v1 (S)

Game 2v2 (S)

Game 3v3 (S)= v1 (S) + v2 (S)Game 4v4 (S)= 2v1 (S)

0
5
5
10
20
20
25
20
25
25
50
60
60
60
100

0
20
20
30
20
20
30
45
55
60
45
55
60
100
100

0
25
25
40
40
40
55
65
80
85
95
115
120
160
200

0
10
10
20
40
40
50
40
50
50
100
120
120
120
200

S
1
2
3
4
1,2
1,3
1,4
2,3
2,4
3,4
1,2,3
1,2,4
1,3,4
2,3,4
N

Shapley values for each player
φ1 (v)
φ2 (v)
φ3 (v)
φ4 (v)

3.1

22.08
23.75
23.75
30.42

0
28.33
30.83
40.83

22.08
52.08
54.58
71.25

44.16
47.5
47.5
60.84

ProcedureUpon arrival, participants received a copy of the instruction slides. The instructions
were divided into two parts: an explanation of the rules of the game and an explanationof the computer interface. First, a prerecorded video of the first part of the instruc-11

tions (explanation of the rules of the game) was played. Then, a comprehension quiz(computerized) was administered to make sure participants understood the rules of thegame. They needed to answer each question correctly before proceeding to the next one.
Once all the participants finished answering the quiz questions, two practice rounds ofa game in which the Shapley value for all the players was 25 was run to familiarize theparticipants with the software before the real experiment started. During these practice rounds, participants were asked to look at the second part of the instructions about
the computer interface. In the first round of the practice, the experimenter explained
each screen following the instructions. See Appendix D for English translations of the
instruction slides and the comprehension quiz.
In the actual experiment, just as in Chessa et al. (2022a,b), each participant played
all four games twice in one of the following four orderings: 1234, 2143, 3412, and 4321.
Between each play of a game (called a round), players were randomly rematched into
groups of four players, and participants were randomly assigned a new role within the
newly created group.4 At the end of the experiment, two rounds (one from the first four
rounds and another from the last four rounds) were randomly selected for payments.
Participants received cash rewards based on the points they earned in these two selectedrounds with an exchange rate of 20 JPY = 1 point in addition to 1500 JPY and 1900 JPY
participation fees for H-MC and PC-W. The participation fee for PC-W was set larger
4

As Chessa et al. (2022b) note, this design choice may have slowed down participants’ learning how
to play the game, and therefore makes the data noisier compared with a design in which participants play
only one game. Despite this concern, this design is chosen in Chessa et al. (2022b) to have within-session
variations of games because the tests of the axioms involve comparing outcomes across different games.
Furthermore, as suggested by Guerci et al. (2014), this random reassignment of the roles across rounds
instead of fixing the roles may have slowed down participants’ learning how to play the game, and thus
may have made our data noisier. Chessa et al. (2022b) opt for this design, however, to avoid participantsfeeling the experiment is unfair because of the existence of the null player in game 2. Because the dataof H-MC used in this paper are the same as those reported in Appendix V of Chessa et al. (2022b) as HMCsim , we also follow these design choices in PC-W. See Appendix C for analyses as well as discussions
related to subject learning.

12to compensate for the longer experiment, as well as to cover possible losses participants
may have made.

3.2

Simplification of, and an additional difference between, the mechanisms

In our experiment, we simplified both the H-MC and PC-W mechanisms. Namely, after
a proposer made an offer, instead of the remaining players (responders) approving or
rejecting the offer one by one sequentially, all responders decided simultaneously.
Coalition formation processes may perform differently in the two settings (sequential versus simultaneous). For example, recently, Abe et al. (2021) compared experimentally the efficiency of (a) a mechanism in which participants decide to join thecoalition sequentially, as in the original theoretical analysis of Hart and Mas-Colell(1996) and Perez-Castrillo and Wettstein (2001) and (b) an analogous mechanism inwhich participants decide to join the coalition simultaneously, as in our experimental
implementation. The authors found that the former resulted in higher efficiency thanthe latter. However, Chessa et al. (2022b) investigated the effect of this simplification in
H-MC in the same four games we consider in this paper (see their Appendix V). This
simplification gives rise to other Nash equilibria, in which two or more players refuse
the proposal. However, they reported that the simplified H-MC indeed resulted in asignificantly lower frequency of grand coalition formation and efficiency in game 2, but
not in the remaining three games, such that the performances of the two implementations were comparable. Moreover, in our experiment this simplification is done bothfor H-MC and PC-W. Therefore, for the purpose of investigating the effect of adding a
bidding stage to determine the proposer, the impact of this simplification, if any, should13

be analogous on two mechanisms and should not affect their comparison.
In our experiment, in addition to the way the proposer is chosen, there is a differencein the way the simplified version of the H-MC and PC-W is implemented. Namely, on
the one hand, under our H-MC implementation, the responders observed the proposed
allocation to all the players when deciding to accept or reject the proposal. On theother hand, under our PC-W implementation, each responder observed only their own
proposed allocation and not those to other players (see the screenshots entitled “respondent’s input” for H-MC and “STEP 2 (Respondent)” for PC-W included in the English
translation of the instruction material available in Appendix D). The reason for introducing this difference is that in PC-W, responders are presented not only with the offer
in the second stage, but also the allocated amount in the first bidding stage and the sumof the two. Presenting the proposed allocation for all the responders in this manner for
PC-W substantially increases the amount of information that participants must process
when deciding to approve or reject the offer. While this difference in the presentation of
the information does not influence the theoretical analyses (based on the standard set ofassumptions), it may influence the results we report below. We investigate whether andhow this difference in the procedure may have impacted the results in Section 4.2.2.

4

Results

The computerized experiment was conducted at the Institute of Social and Economic
Research (ISER), Osaka University, between June and December 2019. The experiment
was computerized and used z-Tree (Fischbacher, 2007). A total of 164 students among
those registered in the participant database managed by ORSEE (Greiner, 2015), who
had not participated in similar experiments previously, were recruited as subjects of the14

experiment. Among these 164 participants, 84 played H-MC and 80 played PC-W.5 As
noted in footnote 4 above, the data of H-MC used in this paper are the same as thosereported in Appendix V of Chessa et al. (2022b) as H-MCsim .
The experiment lasted on average 1h30m for H-MC and 2h05m for PC-W including
the instructions, comprehension quiz, and payment. The average earning was 2780
JPY for H-MC and 2990 JPY for PC-W (approximately 25.7 and 27.7 USD using an
average exchange rate of 1 USD = 108 JPY for the period over which the experiment
was conducted.)

4.1

Grand coalition formation and efficiency

Figure 1 presents the results relating to the grand coalition formation in the H-MC mechanism and in the PC-W mechanism, for the four games. The figure is created basedon the estimated coefficients of the following linear regression: gci = β1 HM Ci +β2 P CWi + µi where gci is a dummy variable that takes the value 1 if the grand coalition is formed, and zero otherwise, in group i, and HM Ci (P CWi ) is a dummy variable
that takes the value 1 if the H-MC (PC-W) mechanism is used, and zero otherwise.
Standard errors are corrected for within-session clustering effects (there are 8 clusters,
4 clusters within each treatment, in total). The statistical tests are based on the Wald testfor equality of the estimated coefficients of two treatment dummies.
For game 2, we also consider the case in which we allow a coalition without the nullplayer as the grand coalition (panel b’). We observe from Figure 1 that the frequencyof grand coalition formation is higher under H-MC than under PC-W; however, thedifference is not statistically significant.
5

There were four sessions, corresponding to four orderings of the games, for each mechanism. For
H-MC, we had 24 participants in one session (with the order of games being 4321) and 20 participants inthree other sessions. For PC-W, we had 20 participants in all four sessions.

15

Figure 1: H-MC and PC-W mechanisms: proportion of times the grand coalition formed(a) Game 1(b) Game 2n.s.

(b’) Game 2, allow {2,3,4}

n.s.

1
0.8
0.6
0.4
0.2
0

n.s.

1
0.8
0.6
0.4
0.2
0
H-MC

1
0.8
0.6
0.4
0.2
0

PC-W

H-MC

PC-W

(c) Game 3

H-MC

PC-W

(d) Game 4

n.s.

n.s.

1
0.8
0.6
0.4
0.2
0

1
0.8
0.6
0.4
0.2
0
H-MC

PC-W

H-MC

PC-W

Note: Error bars show one standard error range. ***, **, and * indicate the proportion of times that grand coalition formation issignificantly different between the H-MC and the PC-W at the 0.1, 1, and 5% significance levels, respectively (Wald test).

As a direct consequence of the grand coalition not always being formed, both mechanisms fail to achieve full efficiency. Figure 2 compares the efficiency of the two mechanisms for the four games. Efficiency is computed as the fraction of the sum of thepayoffs obtained by the four players compared with the value of the grand coalitionof the considered game (100 for games 1 and 2 and 200 for games 3 and 4). The figure is created based on the estimated coefficients of the following linear regressions:
P

π

i i
is the efficiency measure forEf fi = β1 HM Ci + β2 P CWi + µi where Ef fi ≡ v(N)

group i, HM Ci (P CWi ) is a dummy variable that takes the value 1 if the H-MC (PC-W)
mechanism is used, and zero otherwise. Standard errors are corrected for within-session
clustering effects. The statistical tests are based on the Wald test for the equality of theestimated coefficients of two treatment dummies.
Figure 2 shows that efficiency is higher under the H-MC mechanism than under the16

Figure 2: Efficiency(a) Game 1(b) Game 2

n.s.

n.s.

1
0.8
0.6
0.4
0.2
0

1
0.8
0.6
0.4
0.2
0
H-MC

H-MC

PC-W

(c) Game 3

PC-W(d) Game 4
*

*
1
0.8
0.6
0.4
0.2
0

1
0.8
0.6
0.4
0.2
0
H-MC

H-MC

PC-W

PC-W

Note: Error bars show the one standard error range. ***, **, and * indicate that efficiency is significantly different between H-MCand PC-W at the 0.1, 1, and 5% significance levels, respectively (Wald test).

PC-W mechanism, and that the difference is statistically significant at the 5% level forgames 3 and 4.
The significantly lower efficiency under PC-W compared with H-MC (for game 3,
the mean efficiency is 0.747 and 0.858 for PC-W and H-MC, respectively, and for game
4, it is 0.788 and 0.871) demonstrates that the cost introduced by the bidding procedureof the former is not negligible.

4.2

The first proposals and responses against them

To better understand the reason for the lower efficiency of PC-W compared to H-MC,
we analyze the first proposals and responses against them. We focus on the first proposer
because for the grand coalition to be formed, we require the first proposal to be accepted.
We start by looking at who became the first proposer.
17

Table 2: Frequency of being the first proposer(a) Frequencies in PC-W and H-MC (all games combined)
Player 1 Player 2 Player 3 Player 4
PC-W
21
38
39
62
41
40
45
42
H-MC
(b) Result of multi-nominal logistic regression
12
3
Proposer
∗∗
H-MC
1.059
0.441
0.533∗∗(0.349) (0.318) (0.168)
Constant -1.083∗∗∗ -0.490 -0.464∗∗∗(0.228) (0.264) (0.131)
Note: Proposer 4 is the baseline outcome. Standard errors are corrected for session-level clustering effect. *** and ** are statistically significant at 0.1 and 1% level respectively.

Panel (a) of Table 2 shows the frequency of each player being chosen as the first
proposer in PC-W and H-MC. We have pooled all the games from all the sessions. In
H-MC, by construction, all four players have been selected with similar frequencies tobe a proposer. In PC-W, however, Player 4 is almost three times more likely to become
the proposer than Player 1, and Players 2 and 3 are almost twice more likely to become
the proposer than Player 1.
To test whether there is a significant difference between PC-W and H-MC regarding
the frequency of chosen proposer, we have conducted multi-nominal logistic regression
where the dependent variable is the chosen proposer (Players 1 to 4) and independent
variables are constant and the H-MC dummy that takes value 1 if the treatment is H-MCand zero in case of PC-W. Standard errors are corrected for within-session clustering
effect. The result, with Player 4 used as the base outcome, is shown in panel (b) of
Table 2. As we have observed in panel (a), for PC-W, Players 1 and 3 are significantly(at 1% significance level) less likely to be selected as the proposer than Player 4 (the null18

hypothesis, constant=0, is rejected (p-value < 0.001)). For H-MC, however, the null
hypothesis, constant + H-MC=0, is not rejected (p-value = 0.928, Wald test) suggestingthat all the players are equally likely to become the proposer.

4.2.1

Offer made by the first proposer

We now investigate the offers made by the first proposer in H-MC and PC-W. We havepooled all the games from all the sessions. We consider the following three characteristics regarding the offer made by the first proposers:• Deviation from the equal division: devED =2
j (oj − v(N )/4) where oj is thePoffered amount to player j• Deviation from the Shapley value: devSV =P

j (oj −φj )2where oj is the offeredamount to player j and φj is the Shapley value of player j in the game
• Relative proposer advantage: Adv = (op −φp )/φp where op and φp are the amountthe proposer allocated to him/herself in the proposal and φp is her/his Shapley
value in the gameIn all the measures, for PC-W we include the amount allocated as a result of the firstbidding stage in the first proposal to be able to better compare H-MC and PC-W. Appendix A provides analyses of the bidding stage in PC-W.
Figure 3 shows the empirical cumulative distribution of devED (left), devSV (center), and Adv (right) for H-MC (solid red) and PC-W (dashed blue). In each panel, thedistribution of H-MC lies on the left of that of PC-W, suggesting that first proposals
under H-MC tend to be more equal, closer to SV, and not to give a large amount to the
proposer relative to his/her Shapley value, compared to PC-W.
19

Figure 3: Empirical cumulative distribution of devED, devSV , and Adv
devED

devSV

Adv

ECD
1.0

ECD
1.0

ECD
1.0

0.8

0.8

0.8

0.6

0.6

0.6

0.4

0.4

0.4

0.2

0.2

0.0
0

devED
1000 2000 3000 4000 5000

0.0
0

0.2

devSV
1000 2000 3000 4000 5000

-2

-1

0

1

2

3

4

Adv

Note: Solid Red: H-MC. Dashed Blue: PC-W.

To test whether there are significant differences in terms of these three measuresof the first proposals between H-MC and PC-W, we conduct an ordinary least square(OLS) regression that takes either devED, devSV , or Adv as a dependent variable andconstant and the H-MC dummy as the independent variables (regressions (1), (3), and(5) in Table 3). We also run a regression in which we add the proposer’s ID (1, 2, 3, or
4) as an independent variable in order to control for the difference in the frequency of
proposer between PC-W and H-MC (regressions (2), (4), and (6) in Table 3). Standard
errors are corrected for session-level clustering effect.
The regression results shown in Table 3 confirm our observation from Figure 3.
Namely, the first proposal results in H-MC are significantly more equal, closer to SV,
and the amount allocated to the proposer is significantly closer to his/her Shapley value
than in PC-W.

4.2.2

Responses to the first proposal

We have seen that the first proposals under PC-W tend to allocate a much larger share to
the proposer (relative to her/his Shapley value). This may result in responders rejecting
the first proposal with a higher frequency (which results in a lower frequency of the20Table 3: Results of OLS regression
(1)

H-MC
Proposer(1,2,3,4)
Constant
No. obs
R2

(2)
devED
-888.84∗-765.14(337.28)(330.31)340.12(195.74)
1694.15∗∗∗ 712.04(316.06)(627.46)
328
328
0.012
0.020

(3)

(4)
devSV
-1120.79∗∗ -1096.84∗(307.51)(342.34)65.86(233.75)
2089.07∗∗∗ 1898.91∗(287.99)(803.57)
328
328
0.020
0.020

(5)

(6)
Adv
-0.195∗∗∗ -0.25∗∗∗(0.0383) (0.0214)-0.18∗∗(0.0489)0.38∗∗∗
0.91∗∗∗(0.033)(0.012)
317
317
0.015
0.076

Note: Standard errors are corrected for session-level clustering effect. ***, **, * indicate statistically significant at 0.1%, 1%, and
5% level.

grand coalition formation and also a lower efficiency, as we have observed).
We investigate whether, controlling for the characteristics of the first proposals (using the three measures we have considered above), there are significant differences interms of frequency of the first proposal being accepted or not between H-MC and PCW. In particular, we consider the logit regression6 with the dependent variable being arejection dummy that takes the value 1 if the proposal is rejected and zero otherwise,
and independent variables being constant, the H-MC dummy, one of the three measuresof the proposal (either devED, devSV , and Adv) and its interaction with the H-MC
dummy.
The results reported in Table 4 show that for each of the three measures of the first
proposal (devED, devSV , and Adv), the larger the measure is, the higher is the likelihood of the proposal being rejected. However, except for the regression where we
use Adv as the measure of the proposal, there is no statistically significant difference
between the H-MC and PC-W regarding the likelihood of the proposal being rejected.
6

We obtain qualitatively the same results if we use probit specification

21

Table 4: Results of logit regression

X
X × H-MC
H-MC
Constant
No. Obs
pseudo R2

Dependent variable: Rejection dummy
X = devED X = devSV X = Adv
0.002∗∗
0.0005∗∗
1.8445∗∗∗(0.0008)(0.0002)(0.2681)-0.00070.00001-0.2661(0.0011)(0.00026)(0.4057)-0.214-0.3690-0.481∗∗(0.281)(0.2479)(0.1556)∗-0.425-0.1770-0.034(0.185)(0.2071)(0.1249)
328
328
317
0.15630.084
0.1250Note: Standard errors are corrected for session-level clustering effect. ***, **, * indicate statistically significant at 0.1%, 1%, and
5% level.

Therefore, a lower efficiency observed in PC-W compared to H-MC is a result of thedifferences in the proposals. The results suggest that those participants who won thebidding stage and have actively become the proposer in PC-W, instead of being randomly chosen in H-MC, seem to claim a large amount to themselves, and as a result,
their proposals are more likely to be rejected.
The regression results also suggest that the differences in the way the information
regarding the proposal is presented between the two mechanisms when responders decide to approve or reject it (c.f. Section 3.2) had some, but not too strong, impacts on
the result once the characteristics of the proposal are controlled for.
Let us now analyze the realized allocations.

22

4.3

Allocations

We denote by π HM C (vk ) a vector of payoffs obtained by the players in the H-MC mechanism in game k, with k = 1, 2, 3, 4. Analogously, we denote by π P CW (vk ) a vectorof payoffs obtained by the players in the PC-W mechanism. As noted above, H-MC
implements the Shapley value only on average, whereas PC-W, thanks to the bidding
stage, does so as a unique equilibrium outcome. Therefore, we first compare the averagerealized payoff vectors between the two mechanisms. However, when the players fail toform the grand coalition, the total payoffs obtained by the players are smaller than thevalue of the grand coalition. This results in the average realized payoff vectors being
significantly different from the Shapley value. We therefore focus our analyses on thosegroups that formed the grand coalition.7
Our main analyses are based on a set of OLS regressions (using only the data fromgroups that formed the grand coalition) for the following system of equations.
π1 = a1 g1 + a2 g2 + a3 g3 + a4 g4 + u1
π2 = b1 g1 + b2 g2 + b3 g3 + b4 g4 + u2(1)

π3 = c1 g1 + c2 g2 + c3 g3 + c4 g4 + u3
π4 = d1 g1 + d2 g2 + d3 g3 + d4 g4 + u4 ,where πi is the payoff of player i, gj is a dummy variable that takes the value 1 if thegame j ∈ {1, 2, 3, 4} is played, and zero otherwise. Because participants play all fourgames twice, we correct the standard errors for any within-group clustering effect. Notethat the estimated coefficients aj , bj , cj , and dj are the average payoffs in game j forplayers 1, 2, 3, and 4, respectively.
7

In Appendix B we report the distribution of realized allocation in four games under two mechanisms
for those groups that formed the grand coalition.

23

Table 5: Results of linear regression based only on the groups that formed the grandcoalition

g1
g2
g3
g4
R2
Obs.

π1
23.88
(0.70)
11.07(3.31)
45.88(1.70)
47.83(1.17)
0.96
97

H-MC
π2
π3
25.63 24.56(0.43) (0.11)
26.07 27.73(1.45) (0.97)
51.32 49.72(1.19) (0.50)
48.67 50.5
(0.89) (0.87)
0.99
0.99
97
97

π4
25.93
(0.42)
35.13(1.95)
53.08(1.84)
53.00(1.10)
0.97
97

g1
g2
g3
g4
R2
Obs.

π1
23.93
(0.39)
22.18(4.01)
50.76(0.44)
49.08(1.16)
0.98
67

PC-W
π2
π3
25.00 24.86(1.54) (0.72)
23.45 25.55(1.15) (0.99)
49.18 51.18(1.59) (0.41)
49.64 49.16(0.57) (0.75)
0.98
0.99
67
67

π4
26.21
(1.04)
28.82(3.78)
48.88(2.08)
52.12(1.46)
0.97
67

Note: Standard errors are corrected for any within-group clustering effect.

Table 5 reports the results of these regressions, with H-MC in the left panel and PCW in the right panel. These estimated coefficients and standard errors are also visualized
in Figure 4. In the figure, the horizontal lines indicate the Shapley values for each game.
The stars above each bar indicate that the average payoff is significantly different fromthe corresponding Shapley value at the 0.1, 1, or 5% levels. As one can observe, there
are only a few cases (players 2 and 3 in game 1 as well as Player 2 in game 3 for PCW, and Player 2 in games 2, 3, and 4 for H-MC) where the average payoffs are notsignificantly different from the Shapley value at the 5% level. Thus, neither H-MC nor
PC-W implement the Shapley values.
To better compare the two mechanisms in terms of how close their average payoffs

24Figure 4: Mean payoffs based only on the groups that formed the grand coalition
Game 1

Game 260

60

Player 1

Player 2******

Player 1

Player 4***

Player 2

Game 3*****Player 3

**

**

PC-W

PC-W

Player 3

PC-W

0
H-MC

10

0
H-MC

10
PC-W

20

H-MC

30

20

PC-W

30

H-MC

40

H-MC

***

PC-W

***

H-MC

***

PC-W

***

H-MC

***

PC-W

50
**

40

H-MC

50

Player4

Game 4120

120

Player 1

Player 2

Player 3

Player4*****

Player 1***

Player 2****

Player 3

***

***

PC-W

PC-W

0
H-MC

0
PC-W

20
H-MC

40

20
PC-W

40

H-MC

60

PC-W

60

H-MC

80

H-MC

***

PC-W

***

H-MC

***

PC-W

***

H-MC

***

PC-W

100
***

80

H-MC

100

Player4

Note: The horizontal lines indicate the Shapley values. Error bars show one standard error range. ***, **, and * indicate the average
payoff being significantly different from the Shapley value at the 0.1, 1, and 5% significance levels (χ2 test).

are to the Shapley value, we compute the following measure:s
Disφ =

X(πi − φi )2 ,(2)iwhere πi and φi are the average payoff and Shapley value for player i, respectively, inthe given game. As above, we focus on only those groups that formed a grand coalition.

25

Figure 5: H-MC and PC-W mechanisms: distance from Shapley
Game 140

Game 2
40

n.s.

20

***

20

0

0
H-MC

H-MC

PC-W

Game 3

PC-W

Game 4

60

60
***

***

40

40

20

20

0

0
H-MC

PC-W

H-MC

PC-W

Note: Error bars show the one standard error range. ***, **, and * indicate significant difference between H-MC and PC-W at the
0.1, 1, and 5% significance levels (two-sample t-test).

Furthermore, we employ a bootstrapping technique to conduct statistical tests.8 Namely,in each iteration, we use a subsample (with replacement) of our data, run the system ofregressions presented above (Eq. 1), and compute Disφ based on the obtained estimatedcoefficients (i.e., the average payoffs for the subsample).
Figure 5 shows the results based on the outcomes of 1000 repetitions. As one canobserve, Disφ is significantly higher for games 2, 3, and 4 under PC-W than H-MC.9
Thus, the bidding stage of PC-W not only results in a lower efficiency, but also a larger
deviation from the Shapley value even among those groups that successfully formed the
grand coalition.
8Note that we have one observation of Disφ for each game because Disφ is based on average payoffs.
By using a bootstrapping technique, we generate multiple observations of Disφ using sub-samples.
9
Based on a two-sample t-test with unequal variance using the sample generated by the bootstrap. The
means Disφ (standard errors) for the H–MC mechanism are 5.21 (0.041) in game 1, 13.11 (0.088) in game
2, 30.31 (0.089) in game 3, and 9.21 (0.058) in game 4. For the PC-W mechanism, the corresponding
values are 5.21 (0.041) for game 1, 26.34 (0.098) for game 2, 36.67 (0.053) for game 3, and 10.59 (0.077)
for game 4.

26

4.4

Realized allocations and the a posteriori equilibria

Let us now analyze the realized payoffs in light of the a posteriori equilibrium payoff
vectors. We continue to focus only on the groups that formed the grand coalition. We
measure the distance between the realized payoff vectors and the equilibrium allocation
for the four games by the Euclidean distances between the two. Let eqi be the equilibrium payoff for player i for the given game and the mechanism. Note that eqi depends
also on the realized proposer in the case of H-MC. The distance of the realized paypP2
off from the equilibrium is computed as DisN E =i (πi − eqi ) . We also consider
the distance between the realized payoff vectors and equal division payoffs, defined by
pP
2
DisED =i (πi − EDi ) , where EDi is the equal division payoff for player i forthe given game.
Figure 6 shows the mean DisN E and the mean DisED for the two mechanisms
in the four games.10 Figure 6 shows that, except for game 2, the realized allocations
are closer to the equal division than for the equilibrium under both H-MC and PC-W.
The differences between DisN E and DisED are significant at the 1% significance level,
except for game 2. Because the allocation is closer to the equal division than that forequilibrium, and the equilibrium allocation of PC-W is less extreme than those of HMC, we observe that PC-W results in an allocation closer to that of equilibrium than
H-MC for games 1 and 4 at the 0.1% significance level. For games 2 and 3, DisN E isnot significantly different between the two mechanisms.
10

The figure is created based on the estimated coefficients of the following linear regressions: Disi =β1 HM Ci + β2 P CWi + µi where Disi is the relevant distance measure for group i, HM Ci (P CWi )is a dummy variable that takes the value 1 if the H-MC (P-WC) mechanism is used, and zero otherwise.
Standard errors are corrected for any within-session clustering effect. The statistical tests are based on
the Wald test for the equality of the estimated coefficients of the two treatment dummies.

27

Figure 6: Mean of the distances of the realized payoff vectors from the subgame perfectNash equilbrium and the equal division(a) Game 1
**

***

40

(b) Game 2

***
n.s.

20

n.s.

n.s.

40

n.s.
n.s.

20

0

0
H-MC

PC-W

H-MC

DisN E

PC-W

H-MC

DisED

PC-W

(c) Game 3
***

n.s.

H-MC

PC-W

100
80
60
40
20
0

n.s.

H-MC

DisN E

PC-W

DisED

(d) Game 4

***

100
80
60
40
20
0

H-MC

DisN E

PC-W

H-MC

DisED

***

***

PC-W

DisN E

***
n.s.

H-MC

PC-W

DisED

Note: Error bars show the one standard error range. ***, **, and * indicate that the distance of the payoff vectors from theequilibrium allocations or from the equal division was significantly different between H-MC and PC-W at the 0.1, 1, and 5%
significance levels (Wald test).

4.5

Shapley distance

Finally, following Aguiar et al. (2018), we decompose Shapley distance (the distance of
payoff vectors from the corresponding Shapley value) into the failure of axioms characterizing the Shapley value. While Aguiar et al. (2018) proposes the decomposition
into the failure of the symmetry, the efficiency, and the marginality axioms, we follow
Chessa et al. (2022a) and decompose it into the failure of the axioms considered by
Shapley (1953), namely, the efficiency, the symmetry, the additivity, and the null player.
For this exercise, we consider all the groups, i.e., including those groups that did notform the grand coalition.
Let π = (π1 , π2 , π3 , π4 ) be a vector of payoffs obtained by the four players. Below,28

we summarize the detailed procedure of this decomposition.
(1) First, find a vector of payoffs, π sym , that satisfies symmetry and is closest to π. ThatPis, for symmetric players (j = {2, 3} in games 1 and 4), πjsym = j∈N s πj /|N s |, whereN s is the set of symmetric players in the game. For other players, πksym = πk .
(2) Second, find a vector of payoffs, π sym,ef f , that satisfies efficiency and is closest
to π sym . In particular, for each player i = 1, 2, 3, 4, π sym,ef f = π sym + [v(N ) −Pj∈N πj ]/|N |.
(3) Third, find a vector of payoffs, π sym,ef f,null , that satisfies the null player property
and is closest to π sym,ef f . Namely, if player i is a null player (Player 2 in game 2),P
πisym,ef f,null = 0. For the other players j, πjsym,ef f,null = πjsym,ef f + i∈N n πisym,ef f /(|N |−|N n |), where N n is the set of null players.
(4) Fourth, using these payoff vectors, π, π sym , π sym,ef f , π sym,ef f,null , and the corresponding vector of the Shapley values, φ(v), we can define the following “errors” foreach Player i• eφi = πi − φi (v)• esym= πi − πisym ,
i
f
• eef
= πisym − πisym,ef f ,i= πisym,ef f − πisym,ef f,null ,• enulli• eadd= πisym,ef f,null − φi (v).
iGiven these “errors”, the Shapley distance ||eφ ||2 can be decomposed into the failureof the symmetry, the efficiency, the null player, and the additivity axioms as follows:||eφ ||2 = ||esym ||2 + ||eef f ||2 + ||enull ||2 + ||eadd ||2 + 2 < eadd , enull >,29where < ·, · > is the scalar product.11 As one can observe from the presence of< eadd , enull >, unlike the decomposition of Aguiar et al. (2018) that ensures decomposition into three orthogonal components, in the decomposition of Chessa et al. (2022a),
vectors enull and eadd are not orthogonal, and < eadd , enull > is not equal to zero. However, < eadd , enull > in our data are several orders of magnitude smaller than othercomponents and can be safely ignored (on average, they are 0.014 and 0.007 in PC-W
and H-MC, respectively).
To test for differences between PC-W and H-MC, we run the following OLS regression by pooling the data from all four games:||ek ||2g = β1 HM C + β2 P CW + U.

(3)

The dependent variable is the components of the Shapley distance corresponding to thefour axioms as well as the Shapley distance itself (||ek ||2g with k ∈ {sym, ef f, null, add, φ})for group g and the independent variables are HM C and P CW , which take a value of
1 if the corresponding mechanism is used, and zero otherwise. Standard errors are corrected for within-session clustering effects.
Table 6 shows the results of the regressions. The Shapley distance is significantly
larger for PC-W (1604.27) than for H-MC (769.12, p-value = 0.008, Wald test). In both
PC-W and H-MC, the main components of the Shapley distance are due to the failureof the efficiency and the additivity axioms. They account for 49.2% and 40.5% in PCW and 50.3% and 41.2% in H-MC. Furthermore, as we have observed in Figure 4, thefailure of the null player axiom is significantly larger for PC-W than H-MC (p-value =0.028, Wald test) which accounts for, respectively, 7.3% in PC-W and 5.1% in H-MC
11

For any vector y ∈ Rn , ||y||2 =< y, y >=2i∈N yi .

P30

||esym ||2||eef f ||2||enull ||2||eadd ||2||eφ ||2

PC-W

48.58
(7.67)

789.34(96.88)117.32(26.04)649.15(136.96)1604.27(220.94)

H-MC

25.90
(2.28)

386.64(40.36)

39.55(10.47)317.10(21.81)

769.12(55.17)

No. ObsR2
p-value∗

328
0.103
0.025

328
0.248
0.006

328
0.125
0.028

328
0.265
0.048

328
0.427
0.008

Standard errors are corrected for session-level clustering effects and shown inparentheses.
* p-values for testing H0 : PC-W = H-MC (based on the Wald test)

Table 6: Result of Shapley distance decomposition. Based on pooling the data of allgroups and all gamesof the Shapley distance.

5

Summary and conclusion

In this paper, we aim to experimentally contribute to the Nash program (Nash, 1953) byextending the experimental analyses of Chessa et al. (2022a,b). We do so by comparing
the simplified version of the mechanisms proposed by Hart and Mas-Colell (1996) and
Perez-Castrillo and Wettstein (2001). In the original theoretical investigation, in bothmechanisms, there is a proposer who proposes an allocation, which is voted on sequentially by the remaining players (in our experiment, the voting is done simultaneously).
The proposal is accepted if all the remaining players accept the proposal; if not, withsome probability in H-MC and with probability one in PC-W (in our experiment, bothare set to be with probability 1 to be comparable), the proposer leaves the game withhis/her individual value (minus the sum allocated in the bidding stage in PC-W), andthe game continues with the remaining players. The key difference between the two
mechanisms is the way the proposer is chosen. On the one hand, a proposer is chosen
31randomly from among the players in Hart and Mas-Colell (1996). On the other hand, in
Perez-Castrillo and Wettstein (2001), the proposer is determined through a bidding procedure. While Hart and Mas-Colell (1996) implements the Shapley value as an ex anteequilibrium payoff (i.e., it is only achieved as an expected outcome), Perez-Castrillo
and Wettstein (2001) implements it as a unique equilibrium of the game.
Our experiment shows that the simpler mechanism a la Hart and Mas-Colell (1996)
not only results in higher efficiency than the one by Perez-Castrillo and Wettstein (2001),
but also that the average allocation is closer to the Shapley value for those groups thatformed a grand coalition. Thus, the complexity of the mechanism of Perez-Castrillo
and Wettstein (2001) imposes substantial costs for implementing the Shapley value. In
particular, we find that those proposers who have been selected via the bidding in PC-W
tend to propose an allocation that favors themselves than those selected randomly in HMC. Namely, the allocations proposed in PC-W are significantly less equal and deviate
more from Shapley value than those proposed in H-MC. However, such proposals are
more likely to be rejected by other players, and as a result, the grand coalition are lesslikely to be formed and efficiency is lower under PC-W than under H-MC. This suggests
a possibility that while those who win the bidding stage in PC-W consider themselves
to be entitled to receive a larger share, others do not think in the same way.
Chessa et al. (2022a,b), as well as the present paper, showed that participants havedifficulty forming a grand coalition and achieving full efficiency under various noncooperative mechanisms (namely, Winter, 1994; Hart and Mas-Colell, 1996; Perez-Castrillo
and Wettstein, 2001). A natural question is whether participants can better form a grandcoalition and achieve a higher efficiency if they can negotiate freely in an unstructured
bargaining environment. Establishing such a benchmark in more “cooperative” environments and comparing them with the results obtained under “noncooperative” envi32

ronments would be fruitful future research.
Other possible future work could investigate the performance of modifications of
the mechanism by Perez-Castrillo and Wettstein (2001) when implementing some variations of the Shapley value, such as the egalitarian Shapley value (van den Brink et al.,2013) or when we introduce discounting in the bidding mechanism (van den Brink andFunaki, 20015).

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NASH , J. F. (1950): “The Bargaining Problem,” Econometrica, 18, 155–162.
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P EREZ -C ASTRILLO , D. AND D. W ETTSTEIN (2001): “Bidding for the Surplus : A
Non-cooperative Approach to the Shapley Value,” Journal of Economic Theory, 100,274–294.
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Economicas, 29, 219–258.
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——— (2014): “The Nash Program: A Broader Interpretation,” Ensayos, 33, 105–106.
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VAN DEN B RINK , R. AND Y. F UNAKI (20015): “Implementation and Axiomatizationof Discounted Shapley Values,” Social Choice and Welfare, 45, 329–344.
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A

Bidding stage in PC-W

In this section, we analyze the result of the bidding stage in PC-W. In particular, weinvestigate (1) whether a small number of participants have won the bidding stage mostof the time, and (2) the distribution of the submitted bids.

A.1

Frequency of winning the first bidding stage
Table 7: Frequency of becoming the first proposer

Fr. of winning 1
0
6
1
3
2
4
3
2
4
2
5
3
6
0
7
0
8
0
total
20

Session
2 3 4
3 3 2
5 6 7
7 4 5
2 3 5
1 3 0
1 1 0
1 0 0
0 0 0
0 0 1
20 20 20

Table 7 shows the distribution of the frequency of becoming the first proposer ineach of four sessions. There is one participant in session 4 who won the first bidding
36

Figure 7: Histogram of non-negative total bids and winning total bids

Freq.
300
250
200
150
100
50
0
0

10

20

30

40

∑b

50

Light color: Total bids. Dark color: Total winning bids.

stage and became the first proposer in all the 8 games in which s/he participated. There
are also a few participants who became the first proposer more than half (4) of the times.
Thus, there are some participants who bid aggressively to become a proposer.

A.2

Submitted bids

Figure 7 shows the histogram of the total bids (sum of the bids to three other players
in the group) that are non-negative as well as the winning total bids. Note that there
are 42 (out of 640) cases in which participants submitted a negative total bid to avoid
becoming the proposer (in some cases, the total bid was -300).
Figure 8 shows the distribution of the total bids (that are non-negative). The left
panel shows all the bids, and the right panel shows the only the winning bids. In bothpanels, the sum of the equilibrium bids (equilibrium total bids) is indicated in the x-axis
as well as the straight line in the figure. While there are cases of participants bidding
more than the equilibrium amount, in most of the cases, total bids were less than theequilibrium amounts. The same is true among those who won the bidding stage shown
37

Figure 8: Distribution of total bids vis-a-vis the equilibrium total bids
All non-negative

Winning

∑b

∑b

70
60
50
40
30
20
10

70
60
50
40
30
20
10
10

20

30

∑ eq.b

40

10

20

30

∑ eq.b40

on the right panel.

B

Realized allocations in the grand coalitions

Table 8 shows the distribution of realized allocations in four games for those groups
that formed a grand coalition. For games 1, 3, and 4, there are many groups that opted
for the equal division. There are also groups that decided to allocate points among four
players in the reverse ordering of the Shapley value.

C

Comparison of the first two games and the last twogames

So far, we have considered all the rounds and compared H-MC and PC-W. However,given that PC-W is more complex than H-MC, it is possible that it takes longer for

38

39

π1
50
40
35
20
30
35
38
40
51
70
73

π1
25
20
21
23
25
26
30

π3
25
25
21
23
23
22
24

π4
25
30
21
23
27
26
23
Freq.
16
6
1
1
1
1
1

π2
50
50
55
65
50
65
54
40
51
50
48

π3
50
50
55
45
50
45
54
40
51
50
48

π4
50
60
55
70
70
55
54
80
47
30
31
Freq.
12
3
2
1
1
1
1
1
1
1
1

H-MC, Game 3

π2
25
25
37
31
25
26
23

H-MC, Game 1

π1
50
45
40
47
49
51
52
54
56
60
64

π1
25
15
16
20
25
30
31

π3
25
25
27
20
21
30
25

π4
25
35
32
30
21
30
19
Freq.
8
1
1
1
1
1
1

π2
50
50
45
63
55
51
52
46
40
30
54

π3
50
50
65
45
49
51
52
46
53
55
54

π4
50
55
50
45
47
47
44
54
51
55
28
Freq.
6
2
1
1
1
1
1
1
1
1
1

PC-W, Game 3

π2
25
25
25
30
35
10
25

PC-W, Game 1

π1
50
45
45
25
40
40
44
45
51
55
60

π1
0
0
20
22
0
5
9
10
15
20
23

π3
30
33
30
22
25
30
27
30
26
25
23

π4
40
34
30
34
50
35
37
35
33
30
31
Freq.
2
2
2
2
1
1
1
1
1
1
1

π2
50
50
45
45
45
50
44
50
51
35
50

π3
50
50
65
45
45
50
44
55
51
55
40

π4
50
55
45
85
70
60
68
50
47
55
50

Freq.
17
3
2
1
1
1
1
1
1
1
1

H-MC, Game 4

π2
30
33
20
22
25
30
27
25
26
25
23

H-MC, Game 2

π1
50
44
30
39
40
42
42
51
51
55
55
57
62
65

π1
25
0
16
21
23
23
25
25
26
35

π3
25
30
23
21
24
30
27
25
22
29

π4
25
40
38
33
31
23
24
28
26
24

Freq.
2
1
1
1
1
1
1
1
1
1

π2
50
68
41
44
50
45
56
51
51
30
35
61
46
45

π3
50
44
39
44
50
61
51
51
47
55
55
47
46
45

π4
50
44
90
73
60
52
51
47
51
60
55
35
46
45

Freq.
11
2
1
1
1
1
1
1
1
1
1
1
1
1

PC-W, Game 4

π2
25
30
23
25
22
24
24
22
26
12

PC-W, Game 2

Table 8: Distribution of realized allocation in the grand coalitions.

participants to learn to play better. In this section, therefore, we separately investigate
the outcomes in the early rounds (i.e., the first two games that participants played) andin the late rounds (i.e., the last two games that participants played). The figures below
are generated in a similar way to the corresponding figures presented above.12
Figure 9 shows the frequency of grand coalition formation in four games for the
early and the late rounds. For H-MC, the grand coalition is significantly more frequently
formed in the late rounds than in the early rounds for game 1 (at the 1% significancelevel). For PC-W, the frequency of grand coalition formation is significantly higher inthe late rounds than in the early rounds for games 3 and 4 (at the 0.1 and 1% significancelevels, respectively). A similar tendency can be observed for the efficiency shown in
Figure 10.
Figure 11 shows Disφ for the early and late rounds in H-MC and PC-W for the fourgames. While for H-MC, Disφ becomes smaller in the late rounds than in the earlyrounds for all four games, this is not the case for PC-W.13 For PC-W, Disφ becomes
smaller in the late rounds than in the early rounds only for games 3 and 4, whereas forgames 1 and 2, it becomes larger in the late rounds.14 Thus, gaining experience playing
the game does not necessarily lead to allocation according to the Shapley value under
PC-W.
12

Namely, the figures are created based on the estimated coefficients of the following linear regression:yi = β1 HM Cie + β2 HM Cil + β3 P CWie + β4 P CWil + µi where yi is the outcome variable of interestin group i, and HM Ciτ (P CWiτ ) is a dummy variable that takes the value 1 for τ ∈ {e, l} where e and lstand for early and late rounds of the H-MC (PC-W) mechanism, and zero otherwise. The standard errors
are corrected for any within-session clustering effect. The statistical tests are based on the Wald test for
the equality of the estimated coefficients of two treatment dummies.
13
The mean Disφ (the standard errors) for H-MC in early and late rounds are 6.44 (0.013) and 4.78
(0.027) for game 1, 19.03 (0.09) and 7.38 (0.11) for game 2, 32.08 (0.13) and 29.18 (0.13) for game 3,
and 9.96 (0.09) and 8.78 (0.07) for game 4, respectively.
14
For PC-W, the mean Disφ (the standard errors) in early and late rounds are 4.38 (0.056) and 7.21
(0.057) for game 1, 18.18 (0.17) and 31.70 (0.05) for game 2, 37.16 (0.03) and 36.55 (0.08) for game 3,
and 11.68 (0.11) and 10.16 (0.10) for game 4, respectively.

40

Figure 9: H-MC and PC-W mechanisms: proportion of times the grand coalition wasformed in early and late rounds
Game 2 ({2, 3, 4} allowed)

Game 1
n.s.

**

1

***
n.s.

1

0.8

0.8

0.6

0.6

0.4

0.4

0.2

0.2

0

n.s.

n.s.

n.s.

0
Early

Late

Early

H-MC

Late

Early

PC-W

Late

1

*

n.s.

Late

Early

H-MC

Game 3

0.8

n.s.

PC-W

Game 4

n.s.

1
***

0.8

0.6

0.6

0.4

0.4

0.2

0.2

0

n.s.

n.s.

n.s.
**

0
Early

Late

H-MC

Early

Late

Early

PC-W

Late

H-MC

Early

Late

PC-W

Note: Error bars show the one standard error range. ***, **, and * indicate that the two averages were significantly different at the
0.1, 1, and 5% significance levels (Wald test).

41

Figure 10: H-MC and PC-W mechanisms: efficiency in early and late rounds
Game 1
1

n.s.

***

Game 2

***
n.s.

1

0.8

0.8

0.6

0.6

0.4

0.4

0.2

0.2

0

n.s.

n.s.

n.s.

0
Early

Late

Late

Early

H-MC

Early

PC-W

Late

*

n.s.

Early

H-MC

Game 3
1

n.s.

Late

PC-W

Game 4
*
**

1

0.8

0.8

0.6

0.6

0.4

0.4

0.2

0.2

0

*

n.s.

n.s.
n.s.

0
Early

Late

H-MC

Early

Late

Early

PC-W

Late

H-MC

Early

Late

PC-W

Note: Error bars show the one standard error range. ***, **, and * indicate that the two averages were significantly different at the
0.1, 1, and 5% significance levels (Wald test).

42

Figure 11: Mean Dis2φ in early and late rounds(a) Game 1
***

***

40

(b) Game 2

***
***

20

***

***

40

***
***

20

0

0
Late

Early

Late

Early

H-MC

Late

Early

PC-W

H-MC

(c) Game 3
***

***

40

Late

Early

PC-W

(d) Game 4
***

***
***

20

***

***

40

***

20

0

0
Early

Late
H-MC

Early

Late

Early

PC-W

Late
H-MC

Early

Late

PC-W

Note: Error bars show the one standard error range. ***, **, and * indicate a statistically significant difference between the twomeans, at the 0.1, 1, and 5% significance levels (Wald test).

This point can also be observed for DisN E . Figure 12 shows DisN E (top four
panels) and DisED (bottom four panels) in the early and the late rounds. For game 2, on
the one hand, under H-MC, DisN E becomes significantly smaller (and correspondingly,
DisED becomes significantly larger) in the late rounds compared with the early rounds
at the 1% significance level. On the other hand, under PC-W, the opposite is observed.
Namely, DisN E becomes significantly larger (and correspondingly, DisED becomessignificantly smaller) in the late rounds compared with the early rounds at the 0.1%
significance level. Furthermore, for PC-W in game 4, DisN E in the late rounds issignificantly larger than in the early rounds at the 5% significance level.

43

Figure 12: Mean DisN E and DisED in early and late rounds
DisN E(a) Game 1
***

n.s.

40

(b) Game 2
***

***
n.s.

20

***

**

40

***

20

0

0
Late

Early

Late

Early

H-MC

Late

Early

PC-W

H-MC

(c) Game 3

***

80

n.s.
n.s.

60

PC-W

(d) Game 4

n.s.

80

Late

Early

n.s.

60

40

40

20

20

0

***

n.s.

*

0
Late

Early

Late

Early

H-MC

Late

Early

PC-W

Late

Early

H-MC

PC-W

DisED
(a) Game 1
40

n.s.

n.s.

(b) Game 2

**

40

n.s.

***

***

***
***

20

20

0

0
Late

Early

Late

Early

H-MC

Late

Early

(c) Game 3

PC-W(d) Game 4

n.s.

40

n.s.

***

n.s.

Late

Early

H-MC

PC-W

n.s.

40

n.s.

20

n.s.

n.s.

20

0

0
Early

Late
H-MC

Early

Late

Early

Late
H-MC

PC-W

Early

Late

PC-W

Note: Error bars show the one standard error range. ***, **, and * indicate a statistically significant difference between the twomeans at the 0.1, 1, and 5% significance levels (Wald test).

44

D

English translations of the instruction materials

The instruction materials and screenshots of the quiz for• H-MC can be obtained from https://www.dropbox.com/s/78lf5bn6qi3qfwp/
HMChandout_sim.pdf?dl=0• PC-W can be obtained from https://www.dropbox.com/s/yc584hk4c58ceyr/
PCW_handout.pdf?dl=045