ISSN (Print) 0473-453X

Discussion Paper No. 1221

ISSN (Online) 2435-0982

AN EXPERIMENTAL NASH PROGRAM:
A COMPARISON OF NON-COOPERATIVE
V.S. COOPERATIVE BARGAINING
EXPERIMENTS
Michela Chessa
Nobuyuki Hanaki
Aymeric Lardon
Takashi Yamada

November 2023

The Institute of Social and Economic Research
Osaka University
6-1 Mihogaoka, Ibaraki, Osaka 567-0047, JapanAn experimental Nash program:
A comparison of non-cooperative v.s. cooperative
bargaining experiments*
Michela Chessa†

Nobuyuki Hanaki‡

Aymeric Lardon§

Takashi Yamada¶

November 29, 2023
Abstract
This paper aims to contribute to the literature on Nash program by experimentally comparing the results of “structured” (non-cooperative) demand-based andoffer-based mechanisms that implement the Shapley value as an ex-ante equilibrium outcome with the results of corresponding “semi-structured” (cooperative)
bargaining procedures. A significantly higher frequency of the grand coalition formation, the higher efficiency, and the allocation belonging to the bargaining set isobserved in the latter than in the former regardless of whether it is demand-basedor an offer-based. While significant differences in the resulting allocations are observed between the two non-cooperative mechanisms, little difference is observed
between the two cooperative procedures.

JEL code: C70, C71, C92
Keywords: Nash Program, Bargaining procedures, Shapley value* The experiments reported in this paper have been approved by the IRB of Institute of Social andEconomic Research (ISER), Osaka University. 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, 23H00055). Comments andsuggestions from seminar participants at University of Technology Sydney, Monash, UC Davis, Nice,participants of 2021 ESA global Around-the-Clock Virtual Conference (virtual), 2022 annual meetingof French Experimental Economic Association (Lyon), 2022 Conference on Mechanism and Institution
Design (virtual), 2022 Experimental Social Science Conference (Matsumoto), 2022 Summer Workshop
on Economic Theory (Otaru), and workshops at Osaka and Ritsumeikan are gratefully acknowledged.
Yuki Hamada, Hiroko Shibata, and Manami Tsuruta have provided valuable assistance in conducting theexperiments.
†
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. Its main idea “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). Because if different noncooperative approaches yield the same cooperative
solution, it “indicates that the solution is appropriate for a wider variety of situations”(Nash, 1953, p. 136). While many authors have contributed to the development of
the Nash program (see, Serrano, 2005, 2008, 2014, 2021, for surveys), experimental
investigations have been scarce.
To fill this gap in the literature, Chessa et al. (2022, 2023a,b) have conducted a seriesof experiments to compare different non-cooperative mechanisms that are theoretically
expected to implement the Shapley value (Shapley, 1953).
Chessa et al. (2023b) compare Winter’s demand commitment bargaining mechanism(Winter, 1994, referred to as the Winter mechanism below) and the Hart and Mas-Colell
procedure (Hart and Mas-Colell, 1996, referred to as the H–MC mechanism below).
The authors find that while the offer-based H–MC mechanism better induces players toform the grand coalition and thus achieve a higher efficiency, the demand-based Wintermechanism, provided that the grand coalition is formed, better implements allocations
closer to the Shapley value.
Chessa et al. (2022) investigate the effect of choosing a proposer through a bidding
procedure as proposed by Pérez-Castrillo and Wettstein (2001), rather than randomly
selecting one, as in Hart and Mas-Colell (1996). The authors find that choosing the
proposer randomly, instead of via a bidding procedure, not only results in a significantly
greater efficiency, but also provides that the average allocation is closer to the Shapley

2

value for those groups that formed the grand coalition. This is due to the tendency of
proposers who won the bidding to propose an allocation that favors themselves more
often compared to when proposers are selected randomly. Such proposals are more
likely to face rejection.
Finally, Chessa et al. (2023a) investigate the potential impact of the simplification
introduced in the Winter mechanism considered in Chessa et al. (2023b). On one hand,in the original model of Winter (1994), after every player sequentially makes his/herdemand once, those who did not form a coalition, remain in the game, pay a delay costand make another demand. This process continues until there remains only one player
in the game. On the other hand, in Chessa et al. (2023b), instead, those who did notform a coalition after every player has sequentially made his/her demand once, directly
exit the game with their singleton value. Chessa et al. (2023a) compare the simplified
mechanism considered in Chessa et al. (2023b) with some two steps versions of themechanism, in which players who did not form a coalition pay either a small or a highcost, and then move to a second step in which they still do their second demand sequentially. They find that outcomes are not significantly different regardless of whether there
exists a second chance to make a demand or not.1
While Chessa et al. (2022, 2023a,b) experimentally compare the outcomes fromdifferent non-cooperative mechanisms, they have not compared the outcomes of these“structured” (non-cooperative) bargaining experiments with “unstructured” (cooperative) bargaining experiments. In light of the objective of the Nash program, a natural1

The H–MC mechanism considered in Chessa et al. (2023b) is also a simplified version of Hart andMas-Colell (1996). While in Hart and Mas-Colell (1996), if a proposal is rejected, the proposer is removed from the game and receives his/her singleton value with a positive probability (thus, it is also
possible for the proposer to stay in the game and, if chosen, make another proposal), in Chessa et al.
(2023b) the proposer whose proposal is rejected is removed from the game and receives his/her singleton
value with probability one. There is no experimental investigation yet for the effect of this difference.

3and important question is whether these two types of bargaining experiments yield similar outcomes or not. This is the question we aim to answer in this paper.
It is, however, difficult to design an “unstructured” computerized bargaining experiments, because the very same design of a computer interface necessary puts some structure into the bargaining procedure. For example, Shinoda and Funaki (2019) conduct a
computerized “unstructured” three-player bargaining experiment. In their experiment,
participants (acting as players in the bargaining game) can freely propose a coalition
and an associated allocation that is feasible among the members of the proposed coalition. Participants are also free to modify their proposal anytime during the negotiation, and also free to agree on the proposal made by another participant. A coalition isformed if all its members agree. Furthermore, Shinoda and Funaki (2019) also consider
a treatment in which participants can freely send chat messages to others. Similarly, inthree-players games that model negotiable conflicts involving two weak players and onestrong player, Kahan and Rapoport (1980) also consider two communication conditions:
namely a condition where subjects can exchange messages, and a condition where subjects are not allowed to send messages and are unaware that they have been denied thisoption. While these procedures are much less structured compared to those consideredin Chessa et al. (2022, 2023a,b), there remain some constraints in what participants cando and how the coalition is formed.
On the same line than Shinoda and Funaki (2019), it is possible to imagine an alternative procedure where participants, instead of freely proposing a coalition with anassociated allocation among its member, freely make their demand for them to join a
coalition, and a coalition is formed among participants whose demands are compatible (i.e., can be satisfied with the value the coalition generates). And as shown by
Chessa et al. (2023b) for the structured bargaining experiments, such an offer-based and4

a demand-based bargaining procedure may results in quite different outcome even inmuch less structured experiments.
We therefore consider both an offer-based and a demand-based “semi-structured”
bargaining experiment (Duffy et al., 2021). We call our experiment “semi-structured”
because as noted above, while it is much less structured compared to the “structured”
non-cooperative bargaining experiments considered by Chessa et al. (2022, 2023a,b),
there remains some structure in the bargaining procedures.
More specifically, we vary (a) whether or not participants can communicate freely
via online chat during the negotiation, and (b) whether the negotiation is offer-based
or demand-based in our 2×2 between subjects design. The first dimension is motivated by Shinoda and Funaki (2019), who find that the grand coalition is more likely to
be formed with than without a possibility of free-form communication among players
through a chat window. The second dimension is motivated by Chessa et al. (2023b),
who find that demand-based and offer-based mechanism can result in outcomes satisfying much different properties. We also (c) contrast the results of these “semi-structured”
experiments with the results of “structured” experiments of Chessa et al. (2023b), andthis represents the third motivation of our analysis.
We find that “semi-structured” experiments, both offer-based and demand-based,
result in the higher frequency of grand coalition formation and the efficiency than the
structured ones. Unlike Chessa et al. (2023b) who find significant differences in theoutcome of an offer-based and a demand-based procedures, we did not find significant
differences between the two regardless of the possibility of free-form communication.
The latter result is assuring in that, at least for the semi-structured bargaining experiments on the games with non-empty core, exact procedure may have little impact on
the outcomes. As a consequence of the lower efficiency of “structured” experiments, we
5find that the average payoffs deviate significantly from the Shapley values for more players in more games under the “structured” experiments than under the “semi-structured”
ones. While investigating for the possible sources of deviation of the realized allocations
from the Shapley value, applying the approach of Aguiar et al. (2018), we report thatwhich axioms are violated depend on whether the possibility of chatting is allowed, andon whether the experiments are offer-based or demand-based. Finally, in the last partof our analysis, we assess the validity of the prediction of the bargaining set (Aumann
and Maschler, 1964). We report that the frequencies of the belonging to the bargaining
set are significantly higher under the “semi-structured” bargaining experiments, whencompared with the results of the “structured” experiments, and this attests to a greater
stability of the allocations obtained under a “semi-structured” bargaining.
The rest of the paper is organized as follows. Section 2 presents the experimental
design. The results are presented and discussed in Section 3. And Section 4 concludes.

2

The experimental design

We first describe the four-player bargaining games we consider in our experiment. We
then explain our four treatments.

2.1

The games

We consider the four 4-player games shown in Table 1. These games are the same as
those considered in Chessa et al. (2022, 2023a,b). The Shapley values of the four games
is presented in Table 2. The Equal Division payoff vector is simply equal to ED(vk ) =(25, 25, 25, 25) when k = 1, 2, and ED(vk ) = (50, 50, 50, 50) when k = 3, 4.
Following Chessa et al. (2022, 2023a,b), in our experiment, each participants played
6

Table 1: The games
S

1

2

3

4

1,2

1,3

1,4

2,3

3,4

1,2,3

1,2,4

1,3,4

2,3,4

N

v1 (S)
v2 (S)
v3 (S)
v4 (S)

0
0

5
20

5
20

10
30

20
20

20
20

25
30

20 25 25
45 55 60
= v1 (S) + v2 (S)= 2v1 (S)

50
45

60
55

60
60

60
100

100
100

2,4

Table 2: The Shapley value of games 1, 2, 3 and 4

Game 1
Game 2
Game 3
Game 4

φ1 (v)

φ2 (v)

φ3 (v)

φ4 (v)

22.08
0
22.08
44.16

23.75
28.33
52.08
47.5

23.75
30.83
54.58
47.5

30,42
40.83
71.25
60.83

all four games twice. The order of games was counter balanced across sessions. Namely,
participants played these four games in one of the following four orderings: 1234, 2143,
3412, and 4321. At the beginning of a new round (i.e., new play of a game), participants
were randomly rematched into groups of four players, and their roles were randomly
reassigned within a newly created group.

2.2

Treatments

In our 2×2 between subjects design, we vary (a) whether or not participants communicate freely via online chat during the negotiation, and (b) whether the negotiation isoffer-based or demand-based.
In the treatments with free-form communication, participants could freely send chatmessages (except for those messages that can identify oneself and those ones that can

7insult others). Messages could be seen by everyone in the same group, and they can besent anytime during a play of a game (or a negotiation).
In all the treatments, the maximum duration of a negotiation was randomly determined between 300 and 360 seconds. Participants were informed that a negotiation
could continue for at least 300 seconds, but its end would have been terminated at a
randomly chosen moment during the following 60 seconds. The negotiation could end
earlier, when the grand coalition was formed, or when only one player remained without
belonging to any coalition. We now describe the difference between our offer-based anddemand-based negotiation protocol in detail.

2.3

Offer-based protocol

This protocol is similar to the one that Shinoda and Funaki (2019) call “unstructured
bargaining” protocol. Namely, at any time during a negotiation, each player is free topropose or to approve a coalition that include him/herself among players who remainin the game and an associated allocation within the coalition. Below, let proposing or
approving a coalition mean both proposing or approving members of a coalition andthe associated allocation among them. For example, at the beginning of a negotiation
when all the four players remain in the game, player 1 can propose either {1,2}, {1,3},{1,4}, {1,2,3}, {1,2,4}, {1,3,4}, or {1,2,3,4}. Note that single player coalition is notconsidered here as, we explain later, it is the default outcome for the player when s/heends the game without belonging to any coalition. Instead of proposing a coalition, aplayer can also approve a coalition, that include him/herself, proposed by another player.
In our experiment, each player can propose or approve at most one coalition at any
point in time. Thus, if a player has proposed a coalition but would like to approve8

the one proposed by another player, first the player has to withdraw his/her proposal.
Similarly, if a player has approved a coalition proposed by another player but would
like to propose a new one, the player has to first withdraw his/her approval.
If all the members of a proposed coalition approve it, the coalition is formed andits members all exit the negotiation and receive the allocated points. The negotiation
continues with remaining players. If there remains only one player, the negotiation ends.
The players without an agreed coalition at the end of the negotiation (either because of
the time limit or because s/he is the only player left) obtain his/her singleton value.

2.4

Demand-based protocol

In this protocol, at any point during a negotiation, players are free to demand points theywant to obtain. Note that, unlike the offer-based protocol, in doing so, players are notproposing a coalition. Instead, they are expressing the points they want to receive forthem to join a coalition. Each player can make at most one demand at any time during
the negotiation. Players are free to modify their demands anytime during a negotiation.
A coalition can be formed if the sum of the demands made by its members is no
greater than its worth. When there is such a coalition for a player, the player is free toagree to form it. Just as in the offer-based protocol, we exclude a single player coalition
here, because it is the default outcome for the player when s/he ends the game without
belonging to any coalition. Each player can agree to form at most one coalition at any
time during the negotiation. Thus, if players want to form a different coalition thanthe one s/he is currently agreeing to form, they need to withdraw the current agreement
before agreeing to form a new coalition.
A coalition is formed if all its the members agree to form it. Once a coalition is9

Table 3: The number of participants, the mean duration, and the mean payment in four
treatments
Treatment
Demand-based
Without chat (No chat)No. ofParticipants
88(24x2 + 20x2)

Mean
Duration
1h34m

Mean
payment
2810 JPY

When
May-June 2022

Demand-based
With chat (chat)84(24x2,16,20)

1h22m

2860 JPY

May-June 2022

Offer-based
Without chat (No chat)88(24x2 + 20x2)

1h36m

2810 JPY

June-July 2021

Offer-based
With chat (chat)84(20x3 + 24x1)

1h36m

2900 JPY

June-July 2021

formed, its members exit the negotiation and receive the points they have demanded.
Just as in the offer-based protocol, the negotiation continues with remaining players.
If there remains only one player, the negotiation ends. The players without an agreed
coalition at the end of the negotiation (either because of the time limit or because s/heis the only player left) obtain his/her singleton value.

3

Results

The experiment was conducted at the Institute of Social and Economic Research (ISER),
Osaka University, in May and June 2021 (offer-based) and May and June 2022 (demandbased). A total of 344 students, who have never participated in similar experiments
before, were recruited as subjects of the experiment. See Table 3 for the number ofparticipants as well as mean duration and mean payment in each treatment.The experiment was computerized with z-Tree (Fischbacher, 2007) and participants were recruitedusing ORSEE (Greiner, 2015).

10

At the end of the experiment, two rounds (one from the first four rounds and anotherfrom the last four rounds) were randomly selected for payments. Participants received
cash reward based on the point they have earned in these two selected rounds with an
exchange rate of 20 JPY = 1 points in addition to 1500 JPY participation fee. The experiment lasted on average around 90 minutes including the instruction, comprehension
quiz, and payment. Participants received a copy of instruction slides, and a pre-recorded
instruction video were played. Quiz was given on the computer screen after the explanation of the game. The user interface was explained during the practice rounds referringto the handout about the computer screen. See Appendix A for English translations of
the instruction materials and the comprehension quiz.
Table 4 summarizes the duration of a negotiation, the frequency of complete break
down (no coalition being formed), the number of proposals/demands made within anegotiation, and the number of messages sent during a negotiation (in treatments withchat), and the time until the first coalition being formed in the semi-structured experiments.2
The average duration of a negotiation is significantly longer in the offer-based thanin the demand-based protocols.3 The possibility of chat does not significantly affectthe duration of the negotiation.4 Note that while more messages are sent under theoffer-based than the demand-based protocol when chat is possible, the difference is
2

The table is created based on the estimated coefficients of the following linear regressions: yi =β1 ON Ci + β2 OCi + β3 DN Ci + β4 DCi + µi where yi is the statistics of interest in group i, ON Ci ,OCi , DN Ci , and DCi are dummy variables that take value 1 if the treatment is offer-based no chat(ONC), offer-based with chat (OC), demand-based no chat (DNC), and demand-based with chat (DC),respectively, and zero otherwise. The standard errors are corrected for within session clustering effect.
The statistical tests are based on the Wald test for the equality of the estimated coefficients of treatment
dummies.
3
p = 0.0069 and p < 0.0001 for without chat and with chat, respectively. Wald test.
4
p = 0.207 and p = 0.975 for the demand-based and the offer-based protocols, respectively. Wald
test.

11

12

119.27(14.58)96.92(8.63)
177.05(11.33)177.56(11.34)688
0.671

Demand-based
Without chat (No chat)
Demand-based
With chat (chat)Offer-based
Without chat (No chat)Offer-based
With chat (chat)Num. Obs.
R2

0.045(0.029)0.018(0.010)0.131(0.013)0.119(0.023)688
0.265

The frequency ofcomplete failure

The number of
Proposals or
demands1
7.95(0.33)
6.31(0.33)
4.88(0.24)
3.33(0.13)
688
0.773
4.68
(1.33)
336
0.403

2.03
(0.69)The number ofMessages2

3.16(0.62)
336
0.312

1.69
(0.71)The number of
non-greeting
Messages

Note: Standard errors are corrected for session-level clustering effects and shown in parentheses.
1: Include those gruops that did not form any coalition (thus terminated by reaching the maximum duration)
2: Include greeting messages.
3: Do not include those gruops that did not form any coalition.

The
duration1

Treatment

Table 4: Summary statistics of semi-structured bargaining

109.82(9.74)
92.61(9.02)
154.41(11.02)157.89(13.57)
634
0.653

The
duration3

The time until theformation of thefirst coalition3
97.75(6.25)
80.50(11.46)152.55(11.47)156.23(13.97)634
0.657

not statistically significant (p = 0.1200 and p = 0.1619 with and without greetingmessages, respectively. Wald test). The complete failure of the negotiation is more
frequently observed under the offer-based than the demand-based protocols,5 and thepossibilty of chat does not significantly affect the failure rate.6
The longer duration of a negotiation under the offer-based protocol is not just because there are more groups in which negotiation failed completely. Even among thosegroups where a coalition has been formed, the negotiation took longer under the offerbased than the demand-based protocols.7 The same is true for the time it has taken
before the first coalition being formed.8
The number of proposals made under the offer-based protocols is significantly smaller
than the number of demands made under the demand-based protocols.9 The possibilityof chat significantly reduces the number of proposals or the demands made.10 Both thenumber of demands and the proposals made are, however, large compared to what havebeen allowed in Winter and H–MC (maximum is 4) considered in Chessa et al. (2023b).
One may expect that this difference in the number of demands or proposals made between the structured and semi-structured procedures would affect the outcomes. We
now turn to analyzing the outcomes of the negotiation.
56

p = 0.0166 and p = 0.0013 for without chat and with chat, respectively. Wald test.
p = 0.3834 and p = 0.6701 for the demand-based and the offer-based protocols, respectively. Wald

test.
7

p = 0.0084 and p = 0.0011 for without chat and with chat, respectively. Wald test.
p = 0.0008 and p = 0.0008 for without chat and with chat, respectively. Wald test.
9
p < 0.0001 and p < 0.0001 for without chat and with chat, respectively. Wald test.
10
p = 0.0030 and p = 0.0001 for the demand-based and the offer-based protocols, respectively. Wald
test.
8

13

Figure 1: All the games combined
Frequency of Grand Coalition Formation

Efficiency
n.s.

n.s.
n.s.

n.s.
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
Winter

No Chat

Chat

Demand-Based

No Chat**

Chat

Demand-Based

Offer-Based

**
H-MC

Winter

No Chat

Chat

n.s.
***

No Chat

Chat

Offer-Based

Note: Error bars show one standard error range. Horizontal lines marked with Winter or H-MC indicate the outcomes of experiments on these two structured bargaining mechanisms reported in Chessa et al. (2023b). ***, **, and * indicates the outcome issignificantly different from results of the Winter (demand-based) or H-MC (offer-based) at 0.1%, 1%, and 5% significance level,
respectively (what is the test used?). The “n.s.” indicate the absence of significant difference at 5% level between the two treatments
compared.

3.1

Grand coalition formation and efficiency

We first compare the frequency of the grand coalition formation and the efficiency acrossfour treatments. We also compare these outcomes with the results from the experimentson the structured bargaining mechanisms reported in Chessa et al. (2023b). Namely,
the outcomes of a simplified version of the demand-based Winter mechanism (Winter,
1994) and those of a simplified version of the offer-based Hart and MasColell (H-MC)
mechanism (Hart and Mas-Colell, 1996).
Figure 3 presents the frequency of the grand coalition formation (left) and the efficiency (right) of the four treatments. The efficiency is defined as the share of the sumof the points obtained by four players relative to the worth of the grand coalition. We
pool the data from all the sessions and all the games.11 The horizontal line with indi11

The figure is created based on the estimated coefficients of the following linear regressions: yi =β1 ON Ci + β2 OCi + β3 DN Ci + β4 DCi + β5 W interi + β6 H − M Ci + µi where yi is a dummy
variable that takes the value 1 if the grand coalition is formed, and zero otherwise, in group i, for the14

cation of Winter and H-MC are the experimental results of these two non-cooperative
mechanisms reported in Chessa et al. (2023b).
We reject the null hypothesis that the average frequency of the grand coalition formation is the same across four treatments (p = 0.0004, Wald test). We also reject the null
hypothesis that the average efficiency is the same across four treatments (p < 0.0001,
Wald test). The frequency of the grand coalition formation as well as the efficiency areslightly higher in the treatments with chat than those without, although the differenceis significant at 5% only for the frequency of grand coalition formation in the offerbased protocols.12 There is no significant difference between the offer-based and thedemand-based protocols regardless of possibility of chat.13
The frequencies of the grand coalition formation are significantly higher under thesemi-structured bargaining protocols, both with and without chat, than under Winter (forthe demand-based) and H-MC (for the offer-based) mechanisms.14 While the efficiencies are also significantly higher under the semi-structured demand-based protocol thanunder Winter mechanism,15 the difference is significant at 5% level only between thesemi-structured offer-based protocols with chat and H-MC mechanism.16 These resultsPπjj
grand coalition formation and yi = ν(N) for the efficiency, W interi , and H − M Ci are dummy variables that take value 1 if the treatment is winter, and H-MC, respectively, and zero otherwise. Other
treatment dummies are the same as the one explained above. The standard errors are corrected for withinsession clustering effect. The statistical tests are based on the Wald test for the equality of the estimatedcoefficients of treatment dummies.
12
For the frequency of the grand coalition formation, p = 0.2433 for the demand-based, p = 0.0115
for the offer-basedt, Wald test. For the efficiency, p = 0.1723 for the demand-based , p = 0.0608 for theoffer-based, Wald test.
13
For the frequency of the grand coalition formation, p = 0.8228 without chat, p = 0.3686 with chat.
For the efficiency, p = 0.8780 without chat, p = 0.5244 with chat.
14
p = 0.0001, Winter (=0.401, s.e.=0.0339) vs the demand-based without chat, p < 0.0001, Winter vsthe demand-based with chat, p = 0.0082 H-MC (0.619, s.e.=0.0196) vs the offer-based without chat, andp = 0.0016 H-MC vs the offer-based with chat. All are based on Wald tests.
15
p = 0.0075, Winter (=0.779, s.e.=0.0198) vs the demand-based without chat, and p = 0.0001,
Winter vs the demand-based with chat, based on Wald test.
16
p = 0.2235, H-MC (0.859, s.e.=0.0057) vs the offer-based without chat, and p = 0.0142 H-MC vs
the offer-based with chat, based on Wald test.

15

are in line with Kahan and Rapoport (1980) establishing that the type of communication(allowing or not the sending of messages) does not affect significantly the frequency of
coalition structures.

3.2

Allocations

Figure 2 shows the mean payoffs for each player in four games. Treatments are divided
into those demand-based (without chat (No C.), with chat (chat), and Winter) and thoseoffer-based (without chat (No C.), with chat (chat), and H-MC). The horizontal lines
indicate the Shapley values for each player in each game. The mean and the standard
errors are obtained by running a set of ordinary least squares (OLS) regressions for thefollowing system of equations for each treatment separately:

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 a value of 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 within-group clustering effects. Notethat the estimated coefficients aj , bj , cj , and dj are the average payoffs in game j forplayers 1, 2, 3, and 4, respectively.
Because the frequency of grand coalition formation is significantly lower under Winter and H–MC, compared to semi-structured protocols, we observe that the average payoffs tend to be lower under Winter and H–MC than semi-structured ones. As a result,the average payoffs deviate significantly from the Shapley values for more players in
16

Figure 2: Mean payoffs for each player in each treatment
Game 1: Demand Based
60
50
40
30
20
10
0

No C.
Chat
H–MC

*** *** ***

No C.
Chat
H–MC

***

No C.
Chat
H–MC

***

No C.
Chat
H–MC

**

No C.
Chat
Winter

*

No C.
Chat
Winter** *** ***

No C.
Chat
Winter

***

No C.
Chat
Winter**Game 1: Offer Based
60
50
40
30
20
10
0

Player 1

Player 2

Player 3

Player4

Player 1

Player 2

Player 3

Player4

Game 2: Demand Based60
50 *** **
40
30
20
10
0

No C.
Chat
H–MC

*** *** ***

No C.
Chat
H–MC

* *** ***

No C.
Chat
H–MC

** ***

No C.
Chat
H–MC

** *** ***

No C.
Chat
Winter*** *** ***

No C.
Chat
Winter

*

No C.
Chat
Winter

**

No C.
Chat
Winter**Game 2: Offer Based
60
50
40
30
20
10
0

Player 1

Player 2

Player 3

Player4

Player 1

Player 2

Player 3

Player4

Player4*** *** ***

No C.
Chat
H–MC

Player 3*** *** ***

No C.
Chat
H–MC

Player 2

* ***

No C.
Chat
H–MC

*** *** **

No C.
Chat
WinterPlayer 1

*

120
100
80
60
40
20
0

No C.
Chat
H–MC

* ***

No C.
Chat
Winter***Game 3: Offer Based

No C.
Chat
Winter

***

No C.
Chat
WinterGame 3: Demand Based
120
100
80
60
40
20
0

***

Player 1

Player 2

Player 3

Player4

No C.
Chat
WinterNo C.
Chat
WinterPlayer 1

Player 2

Player 3

Player4******* *** ***

No C.
Chat
H–MC

** *** ***

No C.
Chat
H–MC

*

No C.
Chat
H–MC

***

120
100 ***
80
60
40
20
0
No C.
Chat
H–MC

**

No C.
Chat
WinterGame 4: Offer Based

No C.
Chat
WinterGame 4: Demand Based
120
100
80
60
40
20
0

*

Player 1

Player 2

Player 3

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 0.1, 1, and 5 % significance level (Wald test).

17

more games under the structured experiment than under the semi-structured one.
Furthermore, there is a notable difference in terms of the average payoff of the nullplayer, i.e., player 1 in game 2, between the structured and semi-structured experiment.
On one hand, in Winter, the average payoff of the null player was zero and equal to the
Shapley value. On the other hand, in H–MC, the average payoff of the null player ismuch higher. The average payoffs of the null player under the semi-structured experiments are between Winter and H–MC, and is higher when chat is possible. As we will
see next, the null player property tend to fail by a larger extent under the offer-based experiment than under the demand-based experiment regardless of whether it is structured(Winter vs H–MC as shown by Chessa et al., 2023b) or semi-structured, although thedifferece is smaller in the latter. We now investigate the sources of the deviation of therealized allocation of the Shapley value.

3.3

Shapley distance

We apply the approach of Aguiar et al. (2018) to decompose the deviation of the realized
allocation from the Shapley value17 into the failure of efficiency, symmetry, additivity,and null player property.18 The same method of decomposition is used in Chessa et al.
(2022, 2023a,b) in their experiments on non-cooperative mechanisms. Although theprocedure is presented in these papers, in order to be self-contained, we re-present theprocedure.
Let π be the realized allocation (i.e., vector of payoffs) in a game. First, we find an
17

Rapoport (1987) considers alternative metrics to measure the distance between payoff vectors andsolution concepts such as the Bonacich’s error measure (Bonacich, 1979) and the net rate of success(Selten and Krischker, 1983).
18
Aguiar et al. (2018) decompose the deviation into the failure of efficiency, symmetry, and marginality.
While these three components are orthogonal to each other, in our decomposition, the failure due toadditivity and null player property are not orthogonal.

18

allocation, π sym , that satisfies the symmetry and is closest to π. We do so by summing
the payoffs obtained by symmetric players s (players 2 and 3 in games 1 and 4) andP
divide it equally among them. That is, in games 1 and 4, πssym = s∈{2,3} πs /2. For
other players k, πksym = πk .
Second, we find a new allocation, π sym,ef f , that satisfies efficiency and is closest toP
π sym . For each player i = 1, 2, 3, 4, πisym,ef f = πisym + [v(N ) − j∈N πj ]/4.
Third, we find yet another allocation, π sym,ef f,null , that satisfies null player property
and is closest to π sym,ef f . For a null player n (player 1 in game 2), πnsym,ef f,null = 0.
And for other players j in the game, πjsym,ef f,null = πjsym,ef f + πnsym,ef f /3. That is, threeother players in the game equally share πnsym,ef f of the null player. If there is no nullplayer, πisym,ef f,null = πisym,ef f for all i.
f= πisym,ef f − πisym,ef f,null , andLet esym= πi − πisym , eef= πisym − πisym,ef f , enulliii= πisym,ef f,null − φi (v) for all i.
eaddi
Aguiar et al. (2018, Theorem 3) shows that an allocation π from game v can be
decomposed as π = φ(v) + esym + eef f + enull + eadd . Therefore, the Shapley error,eφ = π − φ(v), is eφ = esym + eef f + enull + eadd , and the Shapley distance, ||eφ ||2 , canbe decomposed into||eφ ||2 = ||esym ||2 + ||eef f ||2 + ||enull ||2 + ||eadd ||2 + 2 < eadd , enull >where < ·, · > is the scalar product and for any vector y ∈ Rn , ||y||2 =< y, y >=P2null
and eadd are not orthogonal so thati∈N yi . As noted above, in general, vectors e< eadd , enull > is not equal to zero. Its magnitude, however, is much smaller than theremaining components in our experimental data.
We perform the Shapley distance decomposition of each realized allocation and the19Table 5: Result of Shapley distance decomposition. Based on pooling the data of allgroups and all games||esym ||2
Demand-Based
16.03
No chat(5.29)
Demand-Based
10.53
With chat(4.96)
Offer-Based
7.98
No chat(1.78)
Offer-Based
0.07
With chat(0.07)
Winter
85.18(18.55)H-MC
38.19(12.45)
No. Obs
1040
2
R
0.121
p-value∗
0.0047||eef f ||2 ||enull ||2448.097.03(127.33) (1.37)
241.0819.45(76.67)(6.34)
478.02
13.49(74.28)(2.73)
363.1325.64(119.73) (6.94)
606.81
7.28(99.11)(1.83)
429.9663.97(52.23)(8.08)
1040
1040
0.146
0.079
0.00680.0000||eadd ||2
216.50(20.29)281.07(28.56)164.30(11.71)211.55(13.49)321.49(14.95)270.84(20.25)1040
0.343
0.0000||eφ ||2
687.59(145.27)552.09(98.00)663.74(64.54)600.37(115.54)1020.68(70.63)802.88(61.32)1040
0.300
0.0007

Note: Standard errors are corrected for session-level clustering effects and shownin parentheses. < eadd , enull > are not reported in the table as they are negligible(the mean values are 0.0020, 0.0031, 0.0039, 0.0046, 0.0026, 0.0093 for demandbased no chat, demand-based with chat, offer-based no chat, offer-based with chat,
Winter, and H–MC, respectively.).
*: The null hypothesis is the estimated coefficients of four treatment dummies(excluding Winter and H-MC) are the same (Wald test).

corresponding Shapley value, and compute the average distance, pooling data of allgroups and all games, to compare across four treatments by regressing each of them onto
four treatment dummies (without constant).19 Results of the regressions are presented
in Table 5.
Namely, we run the following regressions ||ei ||2 = β1 ON Ci + β2 OCi + β3 DN Ci + β4 DCi +β5 W interi + β6 H − M Ci + µi where ||ei ||2 is the decomposed distance in group i, ON Ci , OCi ,DN Ci , DCi , W interi , H − M Ci are dummy variables that take value 1 if the treatment is offer based
no chat (ONC), offer based with chat (OC), demand based no chat (DNC), demand based with chat(DC), Winter, and H-MC, respectively, and zero otherwise. The standard errors are corrected for withinsession clustering effect. The statistical tests are based on the Wald test for the equality of the estimatedcoefficients of treatment dummies.
19

20

Table 6: p-values for pair-wise comparisons||esym ||2
Test
Demand, No Chat vs Demand, With Chat
Offer, No Chat vs Offer, With Chat
Demand, No Chat vs Offer, No Chat
Demand, With Chat vs Offer, With Chat
Demand, No Chat vs Winter
Demand, With Chat vs WinterOffer, No Chat vs H–MC
Offer, With Chat vs H–MC||eef f ||2
p-value
0.323
0.867
0.361
0.859
0.220
0.041
0.552
0.550

||enull ||2
Test
Demand, No Chat vs Demand, With Chat
Offer, No Chat vs Offer, With Chat
Demand, No Chat vs Offer, No Chat
Demand, With Chat vs Offer, With Chat
Demand, No Chat vs Winter
Demand, With Chat vs WinterOffer, No Chat vs H–MC
Offer, With Chat vs H–MC

p-value
0.131
0.441
0.859
0.309
0.346
0.014
0.607
0.619

||eadd ||2
p-value
0.135
0.078
0.003
0.637
0.913
0.092
0.00010.0042||eφ ||2
Test
Demand, No Chat vs Demand, With Chat
Offer, No Chat vs Offer, With Chat
Demand, No Chat vs Offer, No Chat
Demand, With Chat vs Offer, With Chat
Demand, No Chat vs Winter
Demand, With Chat vs WinterOffer, No Chat vs H–MC
Offer, With Chat vs H–MC

Test
Demand, No Chat vs Demand, With Chat
Offer, No Chat vs Offer, With Chat
Demand, No Chat vs Offer, No Chat
Demand, With Chat vs Offer, With Chat
Demand, No Chat vs Winter
Demand, With Chat vs WinterOffer, No Chat vs H–MC
Offer, With Chat vs H–MC

p-value
0.364
0.627
0.891
0.699
0.064
0.0026
0.146
0.150

21

Test
Demand, No Chat vs Demand, With Chat
Offer, No Chat vs Offer, With Chat
Demand, No Chat vs Offer, No Chat
Demand, With Chat vs Offer, With Chat
Demand, No Chat vs Winter
Demand, With Chat vs WinterOffer, No Chat vs H–MC
Offer, With Chat vs H–MC

p-value
0.102
0.063
0.035
0.107
0.00220.249
0.00080.0330Column ||eφ ||2 of Table 5 shows that the Shapley distance is lower under the semistructured bargaining than Winter or H–MC. As one can observe from Table 6, however,among these differences between semi-structured and structured protocol, the difference
between the demand-based with chat and Winter is only statistically significant at 5%
level. Tables 5 and 6 further shows that this significant difference of ||eφ ||2 between the
demand-based with chat and Winter is due to the significant difference in ||esym ||2 and||eef f ||2 .
Other significant differences between semi-structured experiments and structured
ones are ||enull ||2 and ||eadd ||2 between offer-based protocol (with and without chat)and H–MC. As we have seen in Figure 2, the average payoffs of the null player wasparticularly high in H–MC, compared to the other treatments. Chessa et al. (2023b)
conjectured this to be caused by proposers trying to avoid their proposal to be rejectedby the null player. Indeed, in the version of H–MC considered in Chessa et al. (2023b), it
was not possible for the proposer to propose a coalition excluding the null player unlessthe null player has been removed from the game already. In the offer-based protocol, it
is possible for non-null players to propose such a coalition without fear of the null player
rejecting it. The possibility of chat makes is more likely for the null player property to
be violated both under the demand-based and the offer-based protocols although thedifferences, from the treatment without chat, are not statistically significant.20

3.4

The bargaining set

Many social scientists have focused on experimentally validating conflict theories using
the cooperative game representation, with a particular emphasis on a set of solutions re20

In the same vein,Kahan and Rapoport (1980) demonstrated that the presence or absence of messages(conditions R and N in their paper, respectively) affects their discrepancy score index from the bargaining
sets defined as the mean absolute deviation of each player’s payoff in a winning coalition from his quota.

22ferred to as bargaining set (Aumann and Maschler, 1964). According to the bargaining
set theory, each resolution to conflict problems specifies a set of payoff vectors, eachelement of which is stable in a psychologically interpretable sense, allowing legitimate
counter-threats to address any threats made by one player against another within thesame coalition. Several experiments have substantiated the significant success of bargaining set theory in accurately predicting payoff vectors in three-person games (see,for example, Kahan and Rapoport (1974); Medlin (1976); Rapoport and Kahan (1976);
Murnighan and Roth (1977); Levinshon and Rapoport (1978), among others). In thecase of games involving more than three players, particularly in the context of fourplayer apex games, Funk et al. (1980) experimentally validated the predictive capacityof bargaining sets, contrasting them with alternative solution concepts like the kernel.
Likewise, Rapoport and Kahan (1984) illustrated the significance of bargaining set predictions in experiments featuring five-player market games.
In this paper, we employ a methodology similar to that of the previously mentioned
studies to assess the validity of the prediction of the bargaining set between semistructured experiments and structured ones applied to the four-player convex games
introduced in Section 2. Notably, in these games, the bargaining set (Aumann andMaschler, 1964) coincides with the kernel (Davis and Maschler, 1965) and the core
(Gillies, 1959).21
Before presenting the analysis, we provide some definitions. Let (N, v) be a cooperP
ative game. Let x ∈ Rn be an imputation, i.e., a payoff vector such that k∈N = v(N ),and for each k ∈ N xk ≥ v({k}). Let i and j be two distinct players in N . An objection of i against j at x is a pair (C, y) satisfying: (i) C ⊂ N , i ∈ C, j ∈/ C; (ii)
21

Similarly, in the context of the three-player games studied by Kahan and Rapoport (1980), when
considering the α-power model, the bargaining set, the kernel and the core all coincide at the point
α = 0.

23Pk∈C yk = v(C); (iii) yk ≥ xk if k ∈ C.

Given a game (N, v) and an imputation x,

let (C, y) be an objection of i against j at x, with i, j distinct players in N . A counterobjection of j against i at y is a pair (D, z) satistying: (i) D ⊂ N , j ∈ D, i ∈/ D; (ii)Pk∈D zk = v(D); (iii) zk ≥ yk is k ∈ C ∩ D; (iv) zk ≥ xk if k ∈ D \ C.
The bargaining set is the set of imputations that have no justified objection, i.e.,
whose objections that always have counter-objections.
Let π be a realized allocation (i.e., vector of payoffs) in one of our games. We firstcompare how frequently these payoff vectors belong to the bargaining set across four
treatments. We also consider a weak bargaining set, in which we relax the hypothesisof efficiency (this to allow to take into account also the payoff vectors when the grandcoalition did not form). We also compare these outcomes with the results from the experiments on the structured bargaining mechanisms reported in Chessa et al. (2023b).
Figure 3 presents the frequency of a payoff vector belonging to the bargaining set, according to the standard definition (on the left) and in its weak version in which we relax
efficiency (on the right) of the four treatments. We pool the data from all the sections
and all the games. The horizontal line with indication of Winter and H-MC are theanalogous experimental results of the two non-cooperative mechanisms implemented in
Chessa et al. (2023b).
We reject the null hypothesis that the average frequency of belonging to the bargaining set is the same across four treatments, both for the standard bargaining set definition,and for its weak version (p < 0.0001, Wald test, in both cases). The frequencies of thebelonging to the bargaining set are higher for the offer-based treatments. We report asignificant difference both when restricting to treatments with chat (p = 0.01394, Waldtest) and to treatments without chat (p = 0.0060, Wald test), for the standard bargaining
set, and a significant difference when restricting to treatments with chat (p < 0.0001,
24

Figure 3: All the games combined
Frequency of Bargaining Set

Frequency of weak Bargaining Set
***

*
n.s.

**
n.s.

1
0.8
0.6
0.4
0.2
0

**

n.s.

n.s.
**

***

**

1
0.8
0.6
0.4
0.2
0

H-MC
Winter

No Chat

Chat

Demand-Based

No Chat**

Chat

Demand-Based

Offer-Based

***

***

H-MC

Winter

No Chat

Chat

n.s.
**

No Chat

Chat

Offer-Based

Note: Error bars show one standard error range. Horizontal lines marked with Winter or H-MC indicate the outcomes of experiments on these two structured bargaining mechanisms reported in Chessa et al. (2023b). ***, **, and * indicates the outcome issignificantly different from results of the Winter (demand-based) or H-MC (offer-based) at 0.1%, 1%, and 5% significance level,
respectively (what is the test used?). The “n.s.” indicate the absence of significant difference at 5% level between the two treatments
compared.

Wald test), for the weak bargaining set. The frequencies of belonging to the bargaining
set, both for the classical and the weak version of the bargaining set are significantly
higher under the semi-structured bargaining protocols, both with and without chat, thanunder Winter (for the demand-based) and H-MC (for the offer-based) mechanisms.22

4

Concluding remarks

This paper aims to contribute to the literature on Nash program (Nash, 1953) by experimentally comparing the results of “structured” (non-cooperative) and “semi-structured”(cooperative) bargaining experiments. In particular, it contrasts the experimental outcomes of two non-cooperative mechanisms that implement the Shapley value (Shapley,22

For the classical bargaining set: Demand No Chat vs. Winter (p = 0.0013), Demand Chat vs. Winter(p = 0.0014), Offer No Chat vs. H-MC (p < 0.0001), Offer Chat vs. H-MC (p = 0.0024). For the weakbargaining set: Demand No Chat vs. Winter (p = 0.0040), Demand Chat vs. Winter (p = 0.0099), Offer
No Chat vs. H-MC (p = 0.0001), Offer Chat vs. H-MC (p = 0.0006). All based on Wald test.

25

1953) as an ex-ante equilibrium outcome considered in Chessa et al. (2023b), simplified
versions of the demand-based mechanism proposed by Winter (1994) and the offerbased mechanism proposed by Hart and Mas-Colell (1996), with those of the corresponding, but much less structured, cooperative bargaining procedure.
We found that semi-structured bargaining procedure resulted in significantly higher
frequency of the grand coalition formation, the higher efficiency, and the higher frequency of allocations belonging to the bargaining set than the structured ones. This ispartly because participants are able to try out much more proposals or demands during anegotiation in the former than the latter. Yet, this also suggests that one should consider
seriously the potential effects of various restrictions, in terms of who can do what andwhen, imposed by various non-cooperative mechanisms on the outcomes if to advance
the Nash program while taking various behavioral biases and cognitive limitations into
account.
We also found significant differences in terms of the duration of the negotiationsor the likelihood of complete failure of the negotiation such that no coalition is being
formed between the offer-based and the demand based semi-structured protocols. However, unlike the sharp differences between the outcomes of the demand-based and theoffer-based structured bargaining reported by Chessa et al. (2023b) in terms of the frequency of the grand coalition formation, the efficiency, and the way realized allocations
deviate from the Shapley value, no significant differences between the demand-based
and the offer-based semi-structured bargaining procedure in these dimensions exceptthat the null player property and additivity are violated by a larger extent under thelatter than the former when there is no chat. In terms of the design of bargaining experiments, this results is encouraging because it suggests that when the participants are lessconstrained in terms of the timing and the number of times they can act, the outcomes26

of the negotiations become similar regardless of whether the protocol is an offer-based
or a demand-based.
In our experiment used only the four games considered by Chessa et al. (2022,
2023a,b). Future studies should consider more varieties of games to better understandthe possible impacts of various behavioral biases, such as fairness consideration andloss aversion, in advancing Nash program while incorporating the fruits of the advancesin the behavioral and experimental economics.

27

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31

A

Instructions

English translation of instructions (of the game and of the experimental software) as wellas comprehension quiz are available at https://osf.io/kqw6n/?view_only=
ea41e1284a1347d09760fba82fda37ea
File names starting with demand-based are for the demand-based treatments, andthose starting with offer-based are for the offer-based treatments.

32