ISSN (Print) 0473-453X

Discussion Paper No. 1152

ISSN (Online) 2435-0982

AN EXPERIMENT ON
DEMAND COMMITMENTBARGAINING

Michela Chessa
Nobuyuki Hanaki
Aymeric Lardon
Takashi Yamada

Revised June 2022
December 2021

The Institute of Social and Economic Research
Osaka University
6-1 Mihogaoka, Ibaraki, Osaka 567-0047, JapanAn experiment on demand commitment
bargaining
Michela Chessa*

Nobuyuki Hanaki†
Takashi Yamada§

Aymeric Lardon‡

June 27, 2022

Abstract
In this experiment, we compare three implementations of the Winter demandcommitment bargaining mechanism: a one-period implementation, a two-period
implementation with low delay costs, and a two-period implementation with highdelay costs. Despite the different theoretical predictions, our results show that thethree different implementations result in similar outcomes in all our investigation
domains: namely, coalition formation, alignment with the Shapley value prediction, and satisfaction of the axioms. Our results suggest that a lighter bargaining
implementation with only one period is often sufficient in providing allocations
that sustain the Shapley value as an appropriate cooperative solution concept, whilesaving unnecessary time and resource costs.

JEL code: C71, C72, C90, D82
Keywords: Nash program, Shapley value, Experiments, Winter mechanism* Université Côte d’Azur, CNRS, GREDEG, France. E-mail: michela.chessa@univ-cotedazur.fr†

Institute of Social and Economic Research, Osaka University,
nobuyuki.hanaki@iser.osaka-u.ac.jp‡
GATE Lyon Saint-Etienne, UMR 5824 CNRS, Université de Lyon.
aymeric.lardon@univ-st-etienne.fr§
Yamaguchi University, Japan. E-mail: tyamada@yamaguchi-u.ac.jp

1

Japan.

E-mail:France.

E-mail:1

Introduction

The aim of the Nash program (Nash, 1953) is to provide a noncooperative foundation of cooperative solution concepts. The Nash program has a long history, having
commenced with Nash, who designed a noncooperative game that sustained the Nashsolution of his cooperative bargaining problem as the equilibrium (Nash, 1950). Since
his seminal paper, the Nash program has continued to grow, with many theoretical andexperimental contributions (For an exhaustive literature review, we refer readers to Serrano, 2005, 2008, 2014, 2021). To cite the first of the aforementioned surveys, and thewords of Nash himself, “The 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 ofview” (Serrano, 2005, p. 220). In fact, “it is rather significant that this different approach yields the same solution. This indicates that the solution is appropriate for awider variety of situations” (Nash, 1953, p. 136).
Most papers that contribute to the Nash program are devoted to sustaining the noncooperative foundation of the Shapley value solution (Shapley, 1953) (see, among others,
Gul, 1989; Harsanyi, 1981; Hart and Moore, 1990; Krishna and Serrano, 1995; Winter,
1994; Hart and Mas-Colell, 1996; Perez-Castrillo and Wettstein, 2001). Because of its
intuitive and desirable properties, the Shapley value has been applied to a variety ofsituations, such as cost or payoff sharing, voting power, fair division, as well as, mostrecently, to many noneconomic contexts, including machine learning, artificial intelligence models, and data analysis (e.g., in one unusual application, the Shapley value
solution was implemented to obtain information about gene expression resulting frommicroarray games (Lucchetti et al., 2010)). As a result, the Shapley value is currently
the most widely used axiomatic cooperative solution concept, the appropriateness of2

which is validated by its many applications and an extensive literature listing its appealing theoretical properties.
In Chessa et al. (2022), we aimed to contribute to the Nash program in sustaining theShapley value by providing a comparison between the experimental results of a demandbased mechanism (a simple one-period version of Winter (1994) model) versus an offerbased (à la Hart and Mas-Colell (1996)) mechanism. Our analysis showed that theWinter mechanism (namely, the Winter demand commitment bargaining mechanism)
better provides allocations that reflect players’ effective bargaining power and satisfy
the axioms that characterize the Shapley value. Conversely, the efficiency and frequencyof grand coalition formation are not very high. This finding suggests that the Shapley
value is indeed an appealing solution in all such situations in which some bargaining
agents interact by expressing their demands about the share that they wish to obtainfrom cooperation. This is a key point when highlighting players’ effective bargaining
power.
However, real-world applications of demand-based bargaining processes may provide the players more time to reach an agreement than one period. Having more time
for agreeing is often costly (obviously in terms of time, but often also in terms of resources), but whether it is more (or less) effective is an open question. Therefore, asimple one-period version of the Winter model may not be suitable to capture all thesenuances of the problem. Thus, in this paper, our main research question is to investigate the robustness of the performances of the Winter mechanism when departing fromits simplified version. In fact, we affirm that for a solution—in our case the Shapley
value—to be relevant, players must agree on it when interacting under different rules.
First, we compare the previously studied one-period (1p) implementation versus a twoperiod (2p) implementation. In the one-period implementation, each player, one after3

another, becomes a proposer and makes a demand concerning the payoff that they arewilling to receive from a possible collaboration. If and when at some point a compatible
demand is introduced, which means that a coalition exists for which the total demands
do not exceed the worth of the coalition, this coalition forms, leaves the game, and thebargaining continues with the rest of the players, until there is at least one player remaining who needs to submit a demand. Players with unsatisfied demands at the end of thefirst period obtain their individual value in the one-period version of the model. In the
two-period implementation, if some players are left with unsatisfied demands after the
first period, they have a second chance to cooperate because the bargaining procedure
repeats for a second time with this set of players by canceling their previous demands
and charging them a fixed delay cost. Second, we compare the performances of the Winter mechanism in its two-period implementation when implementing low (2pL) versus
high (2pH) delay costs.
The theoretical prediction expects all three implementations to provide complete cooperation in the first period and a power share close to the Shapley value (on average,as the ex ante equilibrium). However, the theoretical ex post equilibrium payoff differs
between the different implementations, in particular in terms of the first-mover advantage, which is expected to be smaller in 2p implementations than in 1p implementations.
However, as already observed by Fréchette et al. (2005), experiments often show that
actual bargaining behavior is sometimes not as sensitive to the different bargaining rules
as the theory suggests, and this is what happens in our case.
Our results show that the three different implementations of the Winter mechanism
result in similar outcomes in all our investigation domains: namely, coalition formation,alignment with the theoretical prediction, and satisfaction of the axioms. Moreover, weobserve comparable results when considering the outcome of the first period in a 2p im4

plementation and the outcome of a game with a player who always has a zero marginalcontribution to any coalition, a game that does not fully satisfy the theoretical assumptions of the model. We interpret this finding as a robustness of the Winter mechanism insustaining the Shapley value. Moreover, these results support the implementation of theShapley value as an appealing cooperative solution concept in many real-world applications, both when the decisions must be made rapidly, or when the time for bargaining
is longer. Finally, we suggest that simpler and faster bargaining is often sufficient inproviding allocations that sustain the Shapley value because a second chance to reachan agreement often proves to be ineffective in augmenting the chance of the players
reaching an agreement, or in bringing them closer to the predicted allocation.
The rest of the paper is organized as follows. Section 2 presents the general definition and the properties of a cooperative transferable utility (TU) game, as well as the
Shapley value and its axiomatizations. Section 3 presents the Winter mechanism and its
main theoretical results. Section 4 describes the setting of our experiment and presents
our hypotheses. The results are presented in Section 5. Section 6 concludes.

2

Theoretical model

2.1

Cooperative TU games and solutions

Let N = {1, . . . , n} be a finite set of players. Each subset S ⊆ N is called a coalition
and N is called the grand coalition. A cooperative TU game (from now on, we referto this simply as a cooperative game) consists of a couple (N, v), where N is the set ofplayers and v : 2N → R, with v(∅) = 0, is the characteristic function, which assigns toeach coalition S ⊆ N the worth v(S), that is, the worth that members of S can achieve5

by cooperation. If no ambiguity appears, we consider the set of players N to be fixed
and we write v instead of (N, v). We denote with G N the set of all games with player
set N .
Players i and j are symmetric in v ∈ G N , if v(S ∪ {i}) = v(S ∪ {j}) for all
S ⊆ N \ {i, j}. Player i is a null player in v ∈ G N if v(S) = v(S \ {i}) for all S ⊆ N .
A game v ∈ G N is said to be monotonic if v(S) ≤ v(T ) for each S ⊆ T ⊆ N ,superadditive if v(S) + v(T ) ≤ v(S ∪ T ) whenever S ∩ T = ∅, with S, T ⊆ N andconvex if v(S) + v(T ) ≤ v(S ∪ T ) + v(S ∩ T ), for each S, T ⊆ N (strictly convex if
the inequality holds strictly). Another equivalent definition for convexity can be statedas v(S ∪ {i}) − v(S) ≤ v(T ∪ {i}) − v(T ) for each S ⊆ T ⊆ N \ {i}. In (strictly)
convex games, cooperation becomes increasingly appealing, leading to the formation of
the grand coalition. We may observe that convexity ⇒ superadditivity ⇒ monotonicity.
Given a game v ∈ G N , an allocation is an n-dimensional vector (x1 , . . . , xn ) ∈ RN
that assigns the amount xi ∈ R to player i. For each S ⊆ N , we assume that x(S) =Pi∈S xi . The imputation set is defined by:I(v) = {x ∈ Rn |x(N ) = v(N ) and xi ≥ v({i}) ∀i ∈ N },that is, it contains all the allocations that are efficient (x(N ) = v(N )) and individually
rational (xi ≥ v({i})∀i ∈ N ).
The core is the set of imputations that are also coalitionally rational, that is,C(v) = {x ∈ I(v)|x(S) ≥ v(S) ∀S ⊆ N }.

An element of the core is stable in the sense that if such a vector is proposed as an

6allocation for the grand coalition, no coalition will have an incentive to split off andcooperate on its own. Intuitively, the idea behind the core is analogous to that behind a(strong) Nash equilibrium of a noncooperative game: an outcome is stable if no groupdeviation is profitable. For the Nash equilibrium, the possible deviation is for a single
player, whereas in the core we speak about deviations of groups of players.
A solution is a function ψ : G N → RN that assigns an allocation ψ(v) to every gamev ∈ G N . The Shapley value is the most well-known solution concept, which is widely
applied in economic 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 every player their expected marginal contribution to the coalition of players that entered before them given that every order of entrance has equal
probability. This solution concept was defined as respecting some notion of fairness(see Section 2.2 for more discussion about its properties), but it is not necessarily stable. However, if the game is superadditive, the Shapley value is an imputation and, if
the game is convex, it is the core barycenter (and then, in particular, it belongs to it).
In our analysis, we will consider an easier solution concept, the equal division solution, which distributes the worth v(N ) equally between the players. It is defined as:

EDi (v) =v(N )∀i ∈ N.
nSuch a solution was investigated as an appealing solution for cooperating players
when not considering the worth of the coalitions (see, e.g., de Clippel and Rozen,
forthcoming).

7

2.2

Axiomatizations of the theoretical game solutions

In the literature, we find various axiomatic characterizations of the Shapley value. The
first one is from Shapley (1953) and it involves efficiency, symmetry, additivity, andthe null player. The characterization by Young (1985) involves efficiency, symmetry,and strong monotonicity, and the characterization by van den Brink (2002) involves
efficiency, the null player, and fairness. We list these axioms in Table 1 (for a solution
ψ : G N → RN ).
Table 1: The axioms
Axioms
Efficiency
Symmetry
Additivity
Homogeneity
Null player
Strong monotonicity
FairnessPfor every v in G N , i∈N ψi (v) = v(N )if i and j are symmetric players in game v ∈ G N , then ψi (v) = ψj (v)for all v, w ∈ G N , ψ(v + w) = ψ(v) + ψ(w)for all v ∈ G N and a ∈ R, ψ(av) = aψ(v)if i is a null player in game v ∈ G N , then ψi (v) = 0
if i ∈ N is such that v(S ∪ {i}) − v(S) ≤ w(S ∪ {i}) − w(S)for each S ⊆ N , then ψi (v) ≤ ψi (w)if i, j are symmetric in w ∈ G N , then ψi (v + w) − ψi (v) = ψj (v + w) − ψj (v)for all v ∈ G N

3The Winter mechanism

3.1

Description of the Winter mechanism

In our experiments, we implemented the bargaining model based on sequential demands
for strictly convex cooperative games presented by Winter (1994). In this model, players
in turn announce their demands publicly, indicating “I am willing to join any coalition
yielding me a payoff of ...” and wait for these demands to be met by other players.
8

The bargaining starts with a randomly chosen player from N , say player i. This player
announces their demand di publicly and then points to a second player who must give
their demand. The game proceeds by having each player introduce a demand and thenpoint at a new player to take a turn. If and when at some point a compatible demandis introduced, which means that a coalition S exists for which the total demand forplayers in S does not exceed v(S), then the first player with such a demand selects acompatible coalition S. The players in S receive their demands and leave the game. The
bargaining then continues with the rest of the players using the same rule on v restricted
on N \ S. In a one-period implementation, players with unsatisfied demands at the endof the first period receive their individual value. However, in a T-period implementation
with T > 1, T finite, if some players are left with unsatisfied demands after the first
period, the bargaining procedure repeats for a second time with the set of these players
by canceling their previous demands and charging them a fixed delay cost, and so on,
until the T periods are over.
We present a formal description of the Winter mechanism with T periods (T ≥ 1, T
finite) as defined by Winter (1994). A decision point position m at period t is given by
the vector (S1t,m , S2t,m , dS t,m , j), where:20 ≤ t ≤ T is the current period of the bargaining,
S1t,m ⊆ N is the set of players remaining in the game,
S2t,m ⊂ S1t,m is the set of players who have submitted demands that are not yet
met,
dS t,m = (di )i∈S t,m is the vector of demands submitted by players in S2t,m , (0 ≤22di ≤ maxS⊆N v(S)), and9j ∈ S1t,m \ S2t,m is the player taking the decision by introducing a demand dj .
This player’s demand dj is said to be compatible if there exists some S ⊆ S2t,mP
with v(S ∪ {j}) − i∈S di ≥ dj . Otherwise, dj is not compatible.
With j’s decision, the game proceeds now in the following way:
1) If dj is compatible, then j specifies a compatible coalition S, each player i ∈S ∪ {j} is paid di − tc (where c is the delay cost per period), and nature randomly
chooses a player k 6= j from S1t,m \ S2t,m . The new position is now given by(t, S1t,m+1 , S2t,m+1 , dS t,m+1 , k), with S1t,m+1 = S1t,m \ (S ∪ {j}) and S2t,m+1 =2

S2t,m \ (S ∪ {j}) (we remain at time t and we increment the position from m to
m + 1).
2) If dj is noncompatible, then two cases are distinguished:2a ) if S2t,m = S1t,m \ {j} (j is the last player to give their demand in the current
period), then a new player k is chosen randomly from S1t,m and the new position is given by (S1t+1,1 , ∅, k) (we increment the period from t to t + 1 and werecommence back at period m = 1);
2b ) if S2t,m ⊂ S1t,m \ {j}, then j specifies a new player k 6= j in S1t,m \ S2t,m andthe new position is (S1t,m+1 , S2t,m+1 , dS t,m+1 , k), with S1t,m+1 = S1t,m and S2t,m+1 =2

S2t,m ∪ {j} (we remain at time t and we increment the position from m to m + 1).
The game starts with the randomly chosen player j ∈ N . Then, the initial positionis set to be (N, ∅, d∅ , j). It terminates either when there are no more players in the game(see point 1 above), or when t = T and S1t,m ∪ {j} = S2t,m . In the second case, eachi ∈ S1t,m ∪ {j} is paid v({i}) − T c.
10

3.2

Ex ante and ex post equilibria of the Winter mechanism

The main result of Winter (1994) is stated by the following theorem.
Theorem 1 (Winter (1994)). Let v be a strictly convex game. The demand commitment
game has a unique subgame perfect equilibrium, which assigns equal probabilities at
indifferences. At this equilibrium, the grand coalition forms at the end of the first period.
As c1 approaches zero, the equilibrium payoffs approach the Shapley value of the gamev.
Theorem 1 states that for both the one-period and the T-period implementations, theWinter mechanism has a unique subgame perfect equilibrium in which the grand coalition forms in the first period and the ex ante expected equilibrium payoff coincides with
the Shapley value. The proof of this result is based on a backward induction analysis.
In particular, the expected subgame equilibrium payoff at each period is given by theaverage of the payoffs obtained as ex post equilibria over all the possible orders of theremaining players for the remainder of the game, where different orders are assumed tohave equal probabilities 1/n!2 . Different orders are assumed to have equal probabilities
as, by the definition of the mechanism, the first player at each period is drawn with equal
probabilities. Then, each player chooses the next bidder by assigning equal probabilities
to all potential successors because the player’s equilibrium payoff depends only on theset of the successors and not their actual order.
Regarding the ex post equilibrium payoff, in the one-period implementation, givena specific ordering of the players, each player demands the marginal contribution to the1

In the formulation of this theorem, it is assumed for simplicity that the delay cost c is equal to thesmallest money unit, but the result continues to hold when the delay cost is equal to some integer multipleof the smallest money unit. As stated by Winter (1994), the existence of a smallest money unit is both
motivated by the necessity of reducing players’ sets of actions to finite sets and of making the exact
implementation of the theoretical model possible by experiments.
2
n is then replaced by the number of players still bargaining at that period.

11

set of their successors3 . In the two-period implementation, instead, at equilibrium thefirst player to make a demand asks their Shapley value plus the delay cost times thenumber of remaining players, whereas the other players ask their Shapley value minus
one delay cost. That is, the first player in the first period leaves the other players thepossibility of demanding their expected payoff (received in the second period) during
the first period of the game.4
We observe that the different implementations of the Winter mechanism theoretically present a first-mover advantage. We stress that these theoretical results hold undersome specific assumptions, in addition to the already mentioned strict convexity of thegame. In particular, the delay cost must not be too large5 .

4The experimental setting

4.1

The games

For our analysis, we considered the four four-player games shown in Table 2 together
with corresponding Shapley values. The equal division payoff vector is equal to ED(vk ) =(25, 25, 25, 25) when k = 1, 2, and ED(vk ) = (50, 50, 50, 50) when k = 3, 4.
The games were chosen to be capable of testing the axioms that we presented in
Section 2.2. In Table 3, we detail with which game we aim to test each axiom. Note3

As the mechanism is defined in a discrete version, the first player demands a smallest unit less inorder to leave an extra smallest unit more for the last player and to break his/her indifference between
accepting formation of the grand coalition and receiving their individual value. However, this extra money
unit becomes negligible as it approaches zero.
4
The ex post equilibrium of the T-period implementation with T > 2 resembles the case where T = 2.
5
We refer to the original paper by Winter (1994) for the detailed description of the assumptions underwhich such results hold. As already mentioned, the delay cost must not be too large. Moreover, the results
about the ex ante and the ex post equilibria hold for a discrete version of the mechanism, and when thesmallest money unit approaches zero.

12

13

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

S

0
0

1

5
20

2

5
20

3
10
30

4
20
20

1,2
20
20

1,3
25
30

1,4

2,4

3,4

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

2,3
50
45

1,2,3
60
55

1,2,4
60
60

1,3,4
60
100

2,3,4

Table 2: The games and their Shapley values

100
100

N

22.08
0
22.08
44.16

φ1 (v)

23.75
28.33
52.08
47.5

φ2 (v)

23.75
30.83
54.58
47.5

φ3 (v)

30,42
40.83
71.25
60.83

φ4 (v)

that games 1, 3, and 4 are strictly convex, and game 2 is at least convex. As a consequence, all four games are monotonic. Even if the Winter mechanism is well definedfor any cooperative game, Theorem 1 is not generally true for convex games that are
not strict, such as is the case for game 2. The decision to implement a game that is
not strictly convex was driven by the wish to test the null player axiom. However, thepresence of this null player raises some issues concerning the theoretical prediction of
the equilibrium outcome on such a game. For instance, in 2p implementation with anull player, the delay cost is always too high whatever its size, and the null player who
makes their demand not as the first mover would always prefer to ask zero and leave thegame rather than paying the delay cost and incurring a negative payoff. For this reason,in the following, and when analyzing game 2, we will allow the configuration {1} and{2, 3, 4} to be equivalent to the grand coalition. With game 2 being at least convex, wehave the opportunity to test the robustness of the Winter mechanism when relaxing the
hypothesis of strict convexity, regardless of the theoretical prediction.
Table 3: The axioms and the four games
AxiomsGames

Efficiency
Symmetry

all games
games 1 and 4(symmetry of players 2 and 3)
games 1, 2, and 3(game 3 is defined as the sum of games 1 and 2)
games 1 and 4(game 4 is defined as twice game 1)
game 2(player 2 is a null player)
all games(the marginal contributions of player 1 are always higher in game 1 than in game 2,
and also higher in game 4 than in game 3)
games 1, 2, and 3(game 3 is defined as the sum of games 1 and 2)

Additivity
Homogeneity
Null player
Strong monotonicity

Fairness14

We implemented the Winter mechanism with one period (1p) and two periods andwith low and high delay costs (2pL and 2pH, respectively). Low delay costs are equal
to 0.5 for games 1 and 2, and to 1 for games 3 and 4. High delay costs are equal to 2.5
for games 1 and 2, and to 5 for games 3 and 4.

4.2

The procedure

The experiment was conducted at the Institute of Social and Economic Research (ISER),
Osaka University, between January and August 2019.6 A total of 264 students, who
had never participated in similar experiments before, were recruited as experimental
subjects. There were 96 subjects for 1p treatment, and 84 each for the 2pL and 2pH
treatments7 . The experiment was computerized with z-Tree (Fischbacher, 2007) andparticipants were recruited using ORSEE (Greiner, 2015).
To control for potential ordering effects, each participant played all four games twice
in one of the following four orders : 1234, 2143, 3412, and 4321. Between each gameplay session (called a round), players were randomly rematched into groups of four
players, and participants were randomly assigned a new role within the newly createdgroup. At the end of the experiment, two rounds (one from the first four rounds andanother from the last four rounds) were randomly selected for payments. Participantsreceived cash rewards based on the points they earned in these two selected rounds withan exchange rate of 20 JPY = 1 points in addition to a 1,500 and 1,900 JPY participation
fee for the 1p and 2p implementations, respectively. The experiment lasted on average
100 minutes for the 1p implementation and 130 minutes for the2p implementations,6

All data are available on reasonable request.
The difference in the number of participants between the two mechanisms is a result of variations inthe show-up rate among experimental sessions. The data of the 1p treatment is the same as the one usedin Chessa et al. (2022).
7

15

including the instruction (∼ 15 min), comprehension quiz (∼ 5 min), and payment8 .
The average payments were 2,650 JPY, 3,110 JPY, and 2,960 JPY for the 1p, 2pL, and
2pH implementations, respectively.

5

Results

5.1

Grand coalition formation and efficiency

First, we investigate whether the three implementations succeeded in making the players
reach an agreement and form a grand coalition. In Figure 1, we present the results about
the grand coalition formation for the 1p, 2pL, and 2pH implementations and for the fourgames9 .
As game 2 is not strictly convex with the presence of the null player 1, we allow
the partition {{1}, {2, 3, 4}} as a realization of the grand coalition, as this coalition
structure does not affect the total value that must be shared between the players10 .
We may observe that, at best, the grand coalition formed slightly more (and often,
much less) than 50% of the times for the four games, with no significant difference
between the implementations. We may conclude that all three implementations of theWinter mechanism equally failed in enhancing complete cooperation. We observe that
8

Participants received a copy of instruction slides, and prerecorded instruction movies were played.
See Appendix A for the English translations of the instruction slides and the comprehension quiz.
9
The figure is created based on the estimated coefficients of the following linear regressions: gci =β1 1pi + β2 2pLi + β3 2pHi + µi , where gci is a dummy variable that takes the value of 1 if the grandcoalition is formed, and 0 otherwise, in group i; 1pi , 2pLi , and 2pH are dummy variables that take avalue of 1 if the 1p, 2pL, and 2pH implementations, respectively, are used, and 0 otherwise. The standard
errors are corrected for within-session clustering effects. The statistical tests are based on the Wald test
for the equality of the estimated coefficients of the two treatment dummies.
10
Remember that the Winter mechanism is theoretically defined for strictly convex games. In thisgame, Player 1 always has a zero marginal contribution and, as such, can be left out of any coalition at no
cost for them or the other players.

16Figure 1: Proportion of times the grand coalition formed(a) Game 1(b) Game 2, allow (2,3,4)
n.s.
n.s.

1
0.8
0.6
0.4
0.2
0
1p

n.s.
n.s.

2pL

n.s.

1
0.8
0.6
0.4
0.2
0
2pH

1p

(c) Game 3

2pL

1p

n.s.
n.s.

2pL

2pH

(d) Game 4

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
2pH

1p

n.s.

2pL

2pH

Note: The error bars show the one standard error range. The symbol *** indicates the proportion of times that the grand coalition
formation was significantly different at the 1 % significance level (Wald test).

even when players were given a second chance in a 2p implementation, they did not
manage to perform better in terms of grand coalition formation. In particular, Figure 2
illustrates the number of players whose demands were not met in the first loop. Surprisingly, their number is not higher (and, in some cases, is even significantly smaller) in the
2pL and 2pH implementations when compared with the 1p implementation. We may
recall that the theoretical predictions expect the grand coalition to form in the first loop
for the three implementations. However, one may expect players to demand more in thefirst loop of a 2p implementation because of the second chance of forming a coalition if
their request is not met on the first try.
As a direct consequence of the failure to form the grand coalition, we report a failurein achieving full efficiency, with again no significant difference between the implementations, regardless of the presence of some delay costs in 2pL and 2pH (see Figure 3)11 .
11

The figure is created based on the estimated coefficients of the following linear regressions: Ef fi =17

Figure 2: Average number of players whose demands were not met in the first loop(a) Game 1(b) Game 2

2
1.6
1.2
0.8
0.4
0

*

2
1.6
1.2
0.8
0.4
0
1p

2pL

2pH

1p

(c) Game 3
2
1.6
1.2
0.8
0.4
0

2pL

2pH

(d) Game 4
2
1.6
1.2
0.8
0.4
0

**

1p

*

2pL

2pH

1p

2pL

2pH

Note: The error bars show the one standard error range. The symbols ***, **, and * denote significance at the 1%, 5%, and 10%
significance levels, respectively (Wald test).

Figure 3: Verification of the efficiency axiom(a) Game 1(b) Game 2
n.s.
n.s.

1
0.8
0.6
0.4
0.2
0
1p

n.s.
n.s.

2pL

n.s.

1
0.8
0.6
0.4
0.2
0
2pH

1p

(c) Game 3

2pL

n.s.

1p

n.s.
n.s.

2pL

2pH

(d) Game 4

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
2pH

1p

n.s.

2pL

2pH

Note: Error bars show the one standard error range. The symbols *** and ** indicate the proportion of time that verification of theefficiency axiom was significantly different at the 1% and 5% significance levels, respectively (Wald test).

P

π

i i
β1 1pi + β2 2pLi + β3 2pHi + µi where Ef fi ≡ v(N) is the efficiency measure for group i; and 1pi , 2pL,18
and 2pH are dummy variables that take a value of 1 for the 1p, 2pL, and 2pH treatment, respectively, and0 otherwise. The standard errors are corrected for within-session clustering effects. The statistical tests
are based on the Wald test for the equality of the estimated coefficients of two treatment dummies.

Efficiency is computed as the fraction of the sum of the payoffs obtained by the four
players compared with the worth of the grand coalition of the considered game (100 forgames 1 and 2 and 200 for games 3 and 4).
Therefore, we conclude that:
Result 1. The Winter mechanism in its 1p, 2pL, and 2pH implementations provides
mediocre results in terms of both coalition formation and efficiency.

5.2

Payoff shares: ex ante theoretical prediction

According to the ex ante theoretical prediction, the Winter mechanism in all three implementations is expected to provide approximately the Shapley value on average overthe different orders of the players. We let π 1p (vk ) denote a vector of payoffs obtainedby the players by implementing 1p in game k, with k = 1, 2, 3, 4. Analogously, we use
π 2pL (vk ) to denote a vector of payoffs obtained by the players by implementing 2pL,
and π 2pH (vk ) to denote a vector of payoffs obtained by the players by implementing
2pH. Figure 4 shows the mean of the payoffs in the four games and for the three implementations, with the horizontal lines indicating the Shapley value for each game12 .
As we may observe in Figure 4, as a consequence of the players often failing toform the grand coalition and the resulting lack of efficiency, the average realized payoff vectors are significantly different from the Shapley value. Therefore, we focus on
investigating whether the proportion of the power share, in lieu of the absolute payoffs,
converges to the Shapley value, by considering the normalized (to the worth of the grandcoalition) payoff vectors. Then, in this and in the following sections, we consider the12

The error bars are based on the standard errors that are corrected for within-session clustering effects.
These standard errors are obtained by running the system of linear regressions described in Section 5.4.
The statistical tests are based on these regressions.

19

Figure 4: Mean of the payoffs for the three mechanisms, where the horizontal lines
indicate the Shapley values

Player 1

Player 4

Player 2

Player 1

Player 2

2pH

1p

2pL

2pH

1p

2pL

Player 3*** *** ***Player 4

Player 4

Player 1

Player 2

Player 3

2pH*** *** ***1p** ***

2pL

**

2pH

*** *** ***

1p

2pL

2pH

*** *** ***

1p

2pL

2pH

1p

2pH

2pL

Player 3*** *** ***

120
100
80
60
40
20
0
-20

1p

Game 4*** *** ***

1p

2pL

2pH

*** *** ***

1p

2pH

1p

2pL

*** *** ***** *** ***2pL

Game 3
120
100
80
60
40
20
0
-20

2pH

1p

*** *** ***

2pL

2pL

2pH

***

1p

2pH

1p

2pL

2pL

2pH

Player 3*** *** ***

60
50
40
30
20
10
0
-10

2pL

Player 2*** *** ***

1p

2pL

***

Game 22pH

Player 1

*

2pH

2pH

1p

*** ** ***

2pL

60
50
40
30
20
10
0
-10

1p

Game 1

Player 4

Note: The error bars show the one standard error range. The symbols ***, **, and * indicate the average normalized payoff being
significantly different from the Shapley value at the 1%, 5%, and 10% significance levels, respectively (Wald test).

normalized vectors of payoffs with components πei m (vk ) = P

πim (vk )× vk (N ) for1pj∈N πj (vk )each i = 1, 2, 3, 4 and for each m ∈ {1p, 2pL, 2pH} (remember that the worth of the
grand coalition is equal to 100 for games 1 and 2, and 200 for games 3 and 4).
Figure 5 shows the mean of the normalized payoffs in the four games, with thehorizontal lines indicating the Shapley values for each game.
The three implementations perform well in implementing, on average, the Shapley
value share. Moreover, we do not report any superiority of one of the three implemen-20Figure 5: Mean of the normalized payoffs for the three mechanisms, with the horizontal
lines indicating the Shapley values
Game 1

Player 3

Player 4

Player 1

Player 2

Player 1

Player 2

2pH

1p

2pL

2pH

1p

2pL

Player 3

Player 4

Game 4
120
100
80
60
40
20
0
-20

Player 3

Player 4

Player 1

Player 2

Player 3

2pH

1p

2pL

2pH

1p

2pL

2pL

1p

2pL

2pH

*

1p

2pL

2pH

1p

2pL

**

2pH

*

1p

*

2pL

1p

2pH

1p

2pL

*** *** ***2pH

Game 3
120
100
80
60
40
20
0
-20

*

2pH

*

1p

2pL

2pH

2pH

1p

2pL

2pL

2pH

1p

2pL

2pH

Player 2***

2pH

Player 1

1p

2pH

1p

2pL

*

2pL

60
50
40
30
20
10
0
-10
1p

60
50
40
30
20
10
0
-10

Game 2

Player 4

Note: The error bars show the one standard error range. The symbols ***, **, and * indicate the average normalized payoff being
significantly different from the Shapley value at the 1%, 5%, and 10% significance levels, respectively (Wald test).

tations. Thus, we have the following result.
Result 2. The Winter mechanism in its 1p, 2pL, and 2pH implementations provides goodand comparable results in terms of implementation of the Shapley value power share.

21

Figure 6: Mean of the distances of the normalized payoff vectors from the subgame
perfect Nash equilibriums and the equal division solutions(a) Game 1(b) Game 2
n.s.

80
60
40
20
0
1p

n.s.

n.s.

2pL

2pH

n.s.

1p

Dis2φ

n.s.

n.s.

2pL

***
n.s. ***

80
60
40
20
0

2pH

1p

Dis2ED

2pL

80
60
40
20
0
1p

n.s.

n.s.

2pL

2pH

Dis2φ

2pL

2pH

Dis2ED(d) Game 4
n.s.

1p

1p

Dis2φ

(c) Game 3
n.s.

2pH

n.s.******

n.s.

n.s.

2pL

n.s.

80
60
40
20
0
1p

2pH

n.s.

n.s.

2pL
Dis2φ

Dis2ED

2pH

n.s.

1p

n.s.

n.s.

2pL

2pH

Dis2ED

Note: The error bars show the one standard error range. The symbols ***, **, and * indicate that the distance of the normalized
payoff vectors from the Shapley value or from the equal division solution was significantly different at the 1%, 5%, and 10%
significance levels, respectively (Wald test).

5.3

Payoff shares: Ex post theoretical prediction and first-moveradvantage

The ex post theoretical prediction of the Winter mechanism is dependent on the ordering in which the players make their demands. In particular, a first-mover advantage ispredicted. Furthermore, a higher first-mover advantage is expected in the 1p implementation, followed by the 2pH and then, finally, by the 2pL implementation. As a result, thedistances between the Shapley value and the ex post theoretical predictions are largest
in the 1p implementation, followed by the 2pH, and then the 2pL implementations.
Figure 6 shows the mean of the Euclidean distances of the normalized payoff vectors from the Shapley value, as well as from the equal division solution, for the fourpP
games. Such distances are computed as Dis2mei − φi (v))2 and Dis2ED =
φ =
i (π
22

pP

ei − EDi )2 , where φi (v) denotes the Shapley value for player i in game v 13 .
i (πContrary to the ex post theoretical predictions, we may observe that the distanceof the realized normalized payoffs to the Shapley value in the three implementations
are not significantly different in three out of four games. Only in Game 2 was Dis2φsignificantly larger in 2pH compared with 1p and 2pL. We also observe that the three
implementations perform similarly in terms of distance to the equal division solution
in three out of four games. The only exception is, again, Game 2, in which both 1p
and 2pL are significantly closer to the equal division solution than 2pH. Therefore, we
conclude that:
Result 3. The Winter mechanism in its three implementations provides similar resultsin terms of the distance between the realized normalized payoffs and the Shapley value
as well as the equal division solution.
Figure 7 shows the proportion of times first-mover advantage appears, that is, when
the first mover obtains a higher normalized payoff than their Shapley value. The three
implementations perform in a similar way, and they do not show any first-mover advantage effect (notice that the first mover takes, on average, more than predicted by their
Shapley value half of the time, and, consequently, less than predicted the other half ofthe times). Figure 8 details these results for each single player and Figure 9 presents themean of the amount demanded as first mover. Again, we may observe that the three implementations perform in a similar way, regardless of the differences of the theoreticalprediction. We can now state the following result.
13

The figure is created based on the estimated coefficients of the following linear regressions: Disi =β1 1pi + β2 2pLi + β3 2pHi + µi , where Disi is the relevant distance measure for groups i. 1pi , 2pLi , and
2pH are dummy variables that take a value of 1 if the 1p, 2pL, or 2pH treatments are used, respectively,
and 0 otherwise. The standard errors are corrected for within-session clustering effects. The statisticaltests are based on the Wald test for the equality of the estimated coefficients of the two treatment dummies.

23

Figure 7: Proportion of times first-mover advantage is verified according to the normalized payoff vectors(a) Game 1(b) Game 2

n.s.
n.s.

1
0.8
0.6
0.4
0.2
0
1p

n.s.
n.s.

2pL

2pH

1p

(c) Game 3

2pL

1
0.8
0.6
0.4
0.2
0
1p

2pH

(d) Game 4

n.s.
n.s.

n.s.

**

1
0.8
0.6
0.4
0.2
0

n.s.
n.s.

2pL

n.s.

1
0.8
0.6
0.4
0.2
0
2pH

1p

n.s.

2pL

2pH

Note: The error bars show the one standard error range. The symbols ***, **, and * denote significance at the 1%, 5%, and 10%
significance levels, respectively (Wald test).

Result 4. The Winter mechanism in its 1p, 2pL, and 2pH implementations does notshow any first-mover advantage effect.

5.4

Testing for the axioms

To test for axioms (see Section 2.2), we propose two different approaches. The first
approach investigates verification of the axioms based on the effective average normalized outcomes of each game. The second approach investigates whether different
mechanisms perform differently in terms of decomposition of the Shapley distance à la
Aguiar et al. (2021).

24

Figure 8: Frequency of the first mover demanding more than the Shapley value
Game 2

Player 1

Player 2

Player 3

Player 4

Player 1

Player 2

Game 3Player 3

2pH

1p

2pL

2pH

1p

2pL

**

2pL

1p

2pH

1p

**

2pL

2pH

1p

2pL

0
2pH

0
1p

0.2
2pL

0.4

0.2
2pH

0.4

1p

0.6

2pL

0.6

2pH

0.8

1p

0.8

2pL

1.0

2pH

Game 11.0

Player 4

Game 41.0
0.8

Player 1

Player 2

Player 3

Player 4

Player 1

Player 2

Player 3

2pH

1p

2pL

2pH

1p

2pL

2pL

2pH

*

1p

2pH

1p

2pL

2pH

1p

2pL

0
2pH

0
1p

0.2
2pL

0.2
2pH

0.4

1p

0.4

2pL

0.6

1p

**

0.6

2pL

0.8

2pH

1.0

Player 4

Note: The error bars show the one standard error range. The symbols ** and * indicate significant differences between treatments
at the 5% and 10% significance levels, respectively (Wald test).

5.4.1

Verification of the axioms using the average normalized payoffs

To test symmetry, additivity, homogeneity, strong monotonicity, and fairness, we perform a set of ordinary least squares (OLS) regressions for the following system of equations, with the average normalized payoffs πei as the dependent variables, g1 , g2 , g3 , and25

Figure 9: Mean of the amount demanded as the first moverGame 2

Player 3

Player 1

Player 2

Player 3

Player 1

Player 4

Player 2

2pH

1p

2pL

2pH

2pL

1p

Player 4

Player 3

2pH

1p

2pL

2pH

1p

2pL

1p

2pL

2pH

2pL

2pH

1p

2pL

0
2pH

0
1p

20
2pL

40

20
2pH

40

1p

60

2pL

80

60

2pH

100

80

1p

100

Player 2

Player 3

Game 4120

1p

Game 3120

Player 1

2pH

1p

2pH

1p

2pL

2pH

Player 4**

2pL

Player 2**

2pH

Player 1

1p

0
2pL

0
2pH

10
1p

20

10
2pL

20

2pH

30

1p

40

30

2pL

40

2pH

50

1p

50

2pL

60

2pL

Game 160

Player 4

Note: The error bars show the one standard error range. The symbol ** indicates significant differences between treatments at the
5% significance level (Wald test).

g4 as the independent variables, and without a constant:
πe1 = a1 g1 + a2 g2 + a3 g3 + a4 g4 + u1
πe2 = b1 g1 + b2 g2 + b3 g3 + b4 g4 + u2(1)

πe3 = c1 g1 + c2 g2 + c3 g3 + c4 g4 + u3
πe4 = d1 g1 + d2 g2 + d3 g3 + d4 g4 + u4
where gi is a dummy variable that takes a value of 1 if the game i is played, and 0 otherwise. Various axioms are tested based on the estimated coefficients of these regressions.
26Symmetry requires that b1 = c1 and b4 = c4 . Additivity and homogeneity require thatx3 = x1 + x2 and x4 = 2x1 for x ∈ {a, b, c, d}, respectively. The null player property
requires that a2 = 0, and strong monotonicity requires that a1 > a2 and a4 > a3 . Finally, fairness requires that b3 − b2 = c3 − c2 . In Table 6 in Appendix B, we present theresults of Wald test of the verification of these axioms, together with the null hypothesis(H0 ).
Symmetry (according to which H0 should not be rejected) is fully confirmed by the
1p and the 2pL implementations. However, it is confirmed only for game 4 (half of thetimes) by the 2pH implementation. Additivity and homogeneity (according to which H0should not be rejected) are almost always confirmed by the three implementations. The
null player property (according to which H0 should not be rejected) is confirmed in 1p
and 2pL but not in 2pH. In fact, there is no variation in the normalized payoff for player
1 in game 2 under 1p, and it is zero in all groups. The failure to satisfy the null player
property under 2pH is mainly due to the large delay cost in 2pH. Strong monotonicity(according to which H0 should be rejected) is confirmed by the three implementations.
Fairness (according to which H0 should not be rejected) is rejected by the 1p and the2pH implementations but confirmed by the 2pL implementation. Table 4 summarizes
whether each axiom is satisfied on average (+) or not (-) for the three implementations.
To conclude this first approach for testing for axioms, we can state the following:
Result 5. The Winter mechanism in the 1p, 2pL and 2pH implementations provides
good and comparable results in terms of satisfaction of the axioms.

27

Table 4: Winter mechanisms, axioms

5.4.2

Axiom

1p

2pL

2pH

Efficiency
Symmetry
Additivity
Homogeneity
Null player property
Strong monotonicity
Fairness

–
+
+
+
+
+
–

–
+
+
+
+
+
+

–
–
+
+
–
+
–

Shapley distance (Aguiar et al., 2021)

To strengthen our results on verification of the axioms, we implement the approach
proposed by Aguiar et al. (2021) by computing the Shapley distance, which measuresthe distance of an arbitrary vector of payoffs to the Shapley value, and decomposes it
into the failure of the symmetry, efficiency, and marginality axioms. In the following,
we slightly adapt the original procedure by Aguiar et al. (2021) to test the axiomatization
by Shapley (1953) based on efficiency, symmetry, additivity, and the null player because
our games were selected to test these axioms. Here, we present a formal description of
our procedure.
Given a (nonnormalized) vector of payoffs obtained by the players π = (π1 , π2 , π3 , π4 ):Step 1. Find a vector of payoffs satisfying symmetry that is closest to π.
For symmetric players (players 2 and 3 in games 1 and 4, respectively), we musttake the sum of their payoffs and this sum must be equally shared among them.
For nonsymmetric players, we keep the same payoffs. At the end of this step, we
obtain a new vector of payoffs denoted by π sym = (π1sym , π2sym , π3sym , andπ4sym ).
Step 2. Find a vector of payoffs satisfying efficiency that is closest to π sym .
28

For each player i = 1, 2, 3, 4, compute the new payoff π sym,ef f = π sym + [v(N ) −Pj∈N πj ]/n. At the end of this step, we obtain a new vector of payoffs denotedby π sym,ef f = (π1sym,ef f , π2sym,ef f , π3sym,ef f , andπ4sym,ef f ).
Step 3. Find a vector of payoffs satisfying the null player property that is closest toπ sym,ef f .
If player i is a null player (player 2 in game 2), then their new payoff must be
equal to zero, that is, πisym,ef f,null = 0. Otherwise, for the other players j, thepayoffs of the null players in Step 2 must be equally shared among them, that is,P
πjsym,ef f,null = πjsym,ef f + i∈N f πisym,ef f /(n− | N f |), where N f is the set ofnull players. At the end of this step, we obtain a new vector of payoffs denotedby π sym,ef f,null = (π1sym,ef f,null , π2sym,ef f,null , π3sym,ef f,null , π4sym,ef f,null ).
Step 4. Compute the Shapley distance and its components.
Following Theorem 3 in Aguiar et al. (2021), a vector of payoffs π = (π1 , π2 , π3 , π4 )
obtained when implementing game v can be decomposed as follows: π = φ(v) +esym + eef f + enull + eadd and eφ = esym + eef f + enull + eadd is the Shapley “error”
with:
esym
= πi − πisym for all i,ifeef= πisym − πisym,ef f for all i,ienull= πisym,ef f if player i is a null player and enull=−ij
N f |) for the other players j,
eadd= πisym,ef f,null − φi (v) for all i.
i

29

P

sym,ef f/(n− |i∈N f πiGiven this decomposition, the Shapley distance is given by:||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 >=P
2 14i∈N yi .
To test for differences between the treatments, we run the following OLS regression
by pooling the data from all four games:||ek ||2g = β1 1pg + β2 2pLg + β3 2pHg + U(2)

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 1pg , 2pLg , and 2pHg , which take a valueof 1 if the corresponding treatment is used, and 0 otherwise. The standard errors arecorrected for within-session clustering effects. The results are reported in Table 5.
It can be observed from Table 5 that in all three treatments, the main reasons forthe deviation from the Shapley value are failures of efficiency and additivity. ||eef f ||2
accounts for 59.5%, 62.9%, and 65.1% of the Shapley distance (||eφ ||2 ) in 1p, 2pL,
and 2pH, respectively, and the corresponding values for ||eadd ||2 are 31.5%, 28.8%, and27.8%, respectively. The null player property is largely respected and thus ||enull ||2 isan order of magnitude smaller compared with the other components, accounting for,14

Unlike the original decomposition by Aguiar et al. (2021), which ensures orthogonal components, inour decomposition, in general, vectors enull and eadd are not orthogonal, so that < eadd , enull > is notequal to zero. However, in our data, 2 < eadd , enull > are very small (on average, they are -0.08, -0.08,
and -0.09 in 1p, 2pL, and 2pH, respectively, based on the estimation results reported below) comparedwith other components.

30

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

1p

85.18(18.54)606.81(99.06)

7.28(1.83)

321.49(16.94)1020.68(70.60)2pL

71.48(11.29)613.24(84.04)9.89(2.58)

280.67(8.13)

975.2(82.14)2pH

72.15(9.18)

730.06(72.82)8.26(0.71)

311.69(25.88)1122.07(45.34)

No. ObsR2
p-value∗

528
0.148
0.802

528
0.292
0.491

528
0.102
0.719

528
0.407
0.108

528
0.462
0.252

Standard errors are in parentheses.
* p-values for testing H0 : 1p = 2pL = 2pH (based on the Wald test)

Table 5: Result of Shapley distance decomposition. Based on pooling the data of allgroups and all games
at most, 1% of the Shapley distance. We also observe that there are no statisticallysignificant differences in terms of the size of each of the components across the threetreatments.

6

Conclusions

In this paper, we provide an experimental comparison of three different implementationsof the Winter demand commitment bargaining mechanism: that is, 1p, 2pL, and 2pH
implementations. These three implementations predict the same ex ante outcomes but
differ in terms of ex post outcomes. However, our experiment shows that the three different implementations provide comparable results for both ex ante and ex post outcomes.
No significant difference appeared in any of our investigation domains: that is, coalition
formation, alignment with the theoretical prediction, and satisfaction of axioms.
An example that we borrow from Winter (1994) on bargaining over the formationof a government may help in presenting the implications of our results. Bargaining over31

government formation is a process that naturally resembles the demand commitment
model with more than one period. Parliamentary negotiations are usually based on
demands rather than proposals. These demands are often not compatible in the first
period, and at least a second round of requests is implemented to find an agreement. A
second round may be costly in terms of time and it can make the bargaining process
unnecessarily slow and cumbersome. However, lengthening the bargaining process isoften considered essential and crucial for parties to match and for a coalition to form.
Surprisingly, our experimental results suggest that this may not always be the case,that is, that extending the bargaining process is not always necessary and crucial fora coalition. In fact, our three implementations resulted in similar outcomes in all ourinvestigation domains. The key message of our paper is that a mechanism designer
should implement the lightest possible mechanism for bargaining whenever possible
because refinements may turn out to be not only costly to implement, but ineffective
in terms of quality of the performances. In fact, players converge to similar outcomes(e.g., total or partial cooperation) without taking advantage of any second chances, andregardless of the different theoretical predictions, as the differences are not matchedbehaviorally.

Acknowledgment
The experiments reported in this paper were approved by the Independent Review Boardof 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 from the Japan Society for the Promotion of Science (15K01180, 15H05728,
18K19954, and 20H05631), the Fund for the Promotion of Joint International Research
32

(Fostering Joint International Research) (15KK0123), and the French government-managed
l’Agence Nationale de la Recherche under Investissements d’Avenir U CAJEDI (ANR15-IDEX-01). In particular, we thank the UCAinACTION project.

References
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Behavior, 111, 41–63.
C HESSA , M., N. H ANAKI , A. L ARDON , AND T. YAMADA (2022): “An experiment on
the Nash program: Comparing two strategic mechanisms implementing the Shapley
value,” ISER DP No. 1176, Osaka University.
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35

A

Translated instructions and comprehension quiz

An English translation of the instruction materials as well as the quiz (shown on thescreen) can be downloaded from• https://www.dropbox.com/s/galeo3todbah7iw/Winter_1_loop_
handout.pdf?dl=0 for the Winter one-period implementation,• https://www.dropbox.com/s/kig2i59ncbw9vjw/Winter_2_loop_
handout.pdf?dl=0 for the Winter two-period implementation.

B

Verification of axioms

36

37

H0

Symmetry

b1 = c1
b4 = c4
Additivity
a3 = a1 + a2
b3 = b1 + b2
c3 = c1 + c2
d3 = d1 + d2
Homogeneitya4 = 2a1
b4 = 2b1
c4 = 2c1
d4 = 2d1
Null player
a2 = 0Strong monotonicity
a1 = a2
a4 = a3
Fairnessb3 − b2 = c 3 − c 2

Axiom
0.08
0.14
7.25
0.65
2.54
0.35
0.06
0.37
0.02
0.35
.
62.74
147.12
7.53

χ2
0.781
0.712
0.007
0.422
0.111
0.555
0.805
0.542
0.892
0.552
.
0.000
0.000
0.006

p-value

1p

0.15
0.11
5.51
1.70
0.03
0.50
0.00
0.10
0.02
0.03
0.01
221.37
44.21
1.52

χ2
0.694
0.742
0.019
0.193
0.862
0.479
0.979
0.756
0.892
0.870
0.905
0.007
0.000
0.217

p-value

2pL

12.91
0.30
1.77
0.05
1.37
0.31
1.61
29.04
0.00
0.01
20.27
196.89
25.57
4.09

χ2

0.0003
0.581
0.184
0.815
0.243
0.579
0.204
0.000
0.991
0.922
0.000
0.000
0.000
0.043

p-value

2pH

Table 6: Winter mechanisms’ Wald tests for the verification of the symmetry, additivity, homogeneity, null player, strong
monotonicity, and fairness axioms