ISSN (Print) 0473-453X

Discussion Paper No. 1152

ISSN (Online) 2435-0982

AN EXPERIMENTON THE WINTER DEMAND
COMMITMENT BARGAINING MECHANISM

Michela Chessa
Nobuyuki Hanaki
Aymeric Lardon
Takashi Yamada

December 2021

The Institute of Social and Economic Research
Osaka University
6-1 Mihogaoka, Ibaraki, Osaka 567-0047, JapanAn experiment on the Winter demandcommitment bargaining mechanism
Michela Chessa∗

Nobuyuki Hanaki†
Takashi Yamada§

Aymeric Lardon‡

November 30, 2021

Abstract
In this paper we experimentally compare three implementations of Winter demand commitment bargaining mechanism: a one-period implementation, a twoperiod implementation with low and with high delay costs. Despite the different
theoretical predictions, our results show that the three different implementations
result in similar outcomes in all our domains of investigation, namely: coalition
formation, alignment with the Shapley value prediction and axioms satisfaction.
Our results suggest that a lighter bargaining implementation with only one period
is often sufficient in providing allocations that sustain the Shapley value as appropriate cooperative solution concept, while saving unnecessary costs in terms oftime and resources.

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, 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§
Yamaguchi University, Japan. E-mail: tyamada@yamaguchi-u.ac.jp
†

1

1

Introduction

The aim of the Nash program (Nash, 1953) is to provide a noncooperative foundation ofcooperative solution concepts. Nash started such program designing a noncooperative
game which sustained as equilibrium the Nash solution of his cooperative bargaining
problem (Nash, 1950). Since then, the Nash program had a very long history and it kept
on growing thanks to many theoretical and experimental contributions (the reader isreferred to Serrano, 2005, 2008, 2014, 2020, for an exhaustive literature review). Citing
the first of the aforementioned surveys, and the words of Nash himself, “The idea” of
the Nash program “is both simple and important: the relevance of a concept [...] isenhanced if one arrives at it from different points of view” (Serrano, 2005, pag. 220). In
fact, “it is rather significant that this different approach yields the same solution. This
indicates that the solution is appropriate for a wider variety of situations” (Nash, 1953,
pag. 136).
Between the many, most of the existing papers contributing to the Nash program are
devoted to sustain 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 andWettstein, 2001). In fact, thanks to its intuitive and desirable properties, the Shapley
value has seen many applications on a variety of situations, such as cost or payoff sharing, voting power, fair division, and, most recently, on many non-economics focused
contexts such as machine learning, AI models or data analysis (we can cite, between
the less standard applications, the Shapley value implemented to get information about
gene expression thanks to microarray games (Lucchetti et al., 2010)). As a result, theShapley value is nowadays and undoubtedly the “most famous” axiomatic cooperative2

solution concept, whose appropriateness is validated by an extensive literature listing its
appealing theoretical properties, and by its many applications.
In Chessa et al. (2021), we aimed to contribute to the Nash program in sustaining theShapley value, by providing a comparison between the experimental results of a demand
based (Winter, 1994) versus an offer based (à la Hart and Mas-Colell (1996)) mechanism. Our analysis showed that the Winter mechanism (namely, the Winter demandcommitment bargaining mechanism) better provides allocations that reflect players’ effective bargaining power, and that satisfy the axioms that characterize the Shapley value.
The efficiency and frequency of grand coalition formation, instead, are not very high.
This finding suggests that the Shapley value is indeed an appealing solution in all suchsituations in which some bargaining agents interact by expressing their demands about
the share they wish to obtain from cooperation, and when highlighting players’ effective bargaining power is a key point. However, in this previous work such results wereobtained under an important simplification in the proposed mechanism as compared toits more generic theoretical definition, i.e., implementing a one-period version of themodel. In the Winter mechanism, each player one after another becomes a proposer andmakes a demand for herself, of the payoff she is willing to get from a possible collaboration. If and when at some point a compatible demand is introduced, which meansthat there exists a coalition for which the total demands do not exceed the value of thecoalition, such coalition forms, leaves the game, and the bargaining continues with therest of the players, till there is at least one to still have to submit a demand. Playerswith unsatisfied demands at the end of the first period get their individual value in the
one-period version of the model.
However, real world applications of demand-based bargaining processes may provide the players more time to get an agreement. Having more time for agreeing is often
3

costly (surely in terms of time and often as well in terms of resources), but it may be
more (or less) effective. Thus, in this paper, our primarily research question is to investigate the performances of the Winter mechanism when modifying the way the playersinteract. In fact, we affirm that for a solution -in our case the Shapley value- to be relevant, players need to agree on it when interacting under different rules. Then, firstly, we
compare a one-period implementation versus a two-period implementation. In the twoperiod implementation, if some players are left with unsatisfied demands after the first
period, they have a second chance for cooperating, as the bargaining procedure repeats
for a second time on the set of these players, by canceling their previous demands andcharging them a fixed delay cost. Secondly, we compare the performances of the Wintermechanism in its two-period implementation when implementing low versus high delay
costs.
The theoretical prediction expects all the three implementations to provide complete
cooperation already in the first period and a power share close to the Shapley value(in average, as ex ante equilibrium). However, the theoretical ex post equilibrium payoff differs between the different implementations, in particular in terms of first moveradvantage. The first mover advantage is expected to be smaller in two-period implementations than in one-period implementation. But 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 also in our case.
Our results show that the three different implementations of the Winter mechanism
result in similar outcomes in all our domains of investigation, namely: coalition formation, alignment with the theoretical prediction and axioms satisfaction. Moreover,
we observe similar results also when observing the outcome of the first period in a4

two-period implementation. We interpret this finding as a robustness of the Wintermechanism in sustaining the Shapley value. Moreover, such results support the implementation of the Shapley value as an appealing cooperative solution concept in many
real-world applications, both when the decisions have to be taken rapidly, or when thetime for bargaining is longer. Finally, we suggest that a simpler and faster bargaining
is often sufficient in providing allocations that sustain the Shapley value, as a second
chance for reaching an agreement is proved to be ineffective in augmenting the chanceof the players of finding an agreement, or to get 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,
describes the setting of our experiment, and presents our hypotheses. The results arepresented in Section 4. Section 5 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, cooperativegame) consists of a couple (N, v), where N is the set of players and v : 2N → R,with v(∅) = 0, is the characteristic function, which assigns to each coalition S ⊆ N
the worth v(S), i.e., the value that members of S can achieve by cooperation. If noambiguity appears, we consider the set of players N fixed and we write v instead of(N, v). We denote with G N the set of all games with player set N .

5

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
P
assigning to player i the amount xi ∈ R. For each S ⊆ N , we denote x(S) = i∈S xi .
The imputation set is defined by

I(v) = {x ∈ Rn |x(N ) = v(N ) and xi ≥ v({i}) ∀i ∈ N },i.e., 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, i.e.,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 anallocation 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 group6

deviation is profitable. For the Nash equilibrium the possible deviation is for a single
player, while 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 asφi (v) =(|S| − 1)!(|N | − |S|)!
(v(S) − v(S \ {i})) ∀i ∈ N.
|N |!
S⊆N,i∈SX

The Shapley value assigns every player its expected marginal contribution to the coalition of players that enter before him, given that every order of entrance has equal probability. This solution concept has been defined as respecting some notion of fairness (seeSection 2.2 for more discussion about its properties), but it is not, on the contrary, necessary stable. However, if the game is superadditive, the Shapley value is an imputation,and if the game is convex, is the core barycenter (and then, in particular, it belongs toit).
In our analysis, we will also consider an easier solution concept, the equal division solution, which distributes the worth v(N ) equally between the players. It is thendefined as
EDi (v) =

2.2

v(N )
∀i ∈ N.
n

Axiomatizations of the game theoretical solutions

In the literature, we find various axiomatic characterizations of the Shapley value. The
most classical one is from Shapley (1953), and it involves Efficiency, Symmetry, Additivity and Null player. The one by Young (1985) involves Efficiency, Symmetry andStrong monotonicity. The characterization by van den Brink (2002), instead, involves
7Efficiency, Null player and Fairness. We list in Table 1 these axioms (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 experimental setting

3.1

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 turns announce publicly their demands, meaning “I am willing to join any coalition
yielding me a payoff of ...” and wait for these demands to be met by other players.
The bargaining starts with a randomly chosen player from N , say player i. This player
announces publicly her demand di and then points a second player who has to give her
demand. Then, the game proceeds by having each player pointing at a new player totake her turn after introducing a demand herself. If and when at some point a compatible
demand is introduced, which means that there exists a coalition S for which the total

8

demand for players in S does not exceed v(S), then the first player with such a demand
selects a compatible coalition S. The players in S get their demands, leave the game, andthe bargaining 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 end of
the first period get their individual value. In a T-period implementation with |T | > 1, T
finite, instead, if some players are left with unsatisfied demands after the first period, thebargaining procedure repeats for a second time on the set of these players, by cancelingtheir previous demands and charging them a fixed delay cost, and so on till periods areover. A more formal description of the Winter mechanism with T periods (|T | ≥ 1, T
finite) is presented in Appendix A.
For both the one-period and the T-period implementations, this mechanism has aunique subgame perfect equilibrium which assigns equal probabilities at indifference.
At this equilibrium, the grand coalition forms in the first period and the ex ante expectedequilibrium payoff coincides with the Shapley value. Regarding the ex post equilibrium
payoff, in the one-period implementation, given a specific ordering of the players, eachplayer’s demand depends on the ordering, but only through the set of her successors andnot through the way these players are ordered. In fact, each player demands the marginalcontribution to the set of her successors. In the two-period implementation (which is our
second case of interest), instead, at equilibrium the first player to make a demand asks
her Shapley value plus the delay cost times the number of remaining players, while theother players ask their Shapley value minus once the delay cost1 . More generally, alldifferent implementations of the Winter mechanism present, theoretically, a first moveradvantage. We stress that these theoretical results hold under some specific assumptions,1

The ex-post equilibrium of the T-period implementation with |T | > 2 much resembles the case|T | = 2.

9in addition to the already mentioned strict convexity of the game. In particular, delay
cost must not be too large2 .

3.2

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 choice of the games has been done in order to be capable of testing the axioms
we presented in Section 2.2. In Table 3 we detail for each axiom which game andthanks to which characteristics is introduced in our analysis to test it. Notice that games
1, 3 and 4 are strictly convex, while game 2 is only convex. All the four games are,by consequence, monotonic. Therefore, all but game 2 respect the assumption for the
implementation of Winter mechanism. With game 2 being at least convex, however, we
guessed that “strict convexity” could be relaxed and the mechanism could still provide
convincing results3 .
We implemented the Winter mechanism with one period, from now on referred to as
one-period (1p) implementation, and the mechanism with two periods and with low andhigh delay costs, from now on referred to as two-period low cost (2pL) implementationand two-period high cost (2pH) implementation respectively. Low delay costs are equal2

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. Moreovoer, theresults about the ex ante and the ex post equilibria hold for a discrete version of the mechanism, and when
the smallest money unit approaches zero.
3
The choice of implementing a game that is not strictly convex, is given by the wish to test the nullplayer axiom. However, the presence of this null player raises some issues concerning the theoreticalprediction of the equilibrium outcome on such game. For instance in a two-period implementation witha null player, the delay cost is always too high whatever its size, and the null player making her demand
not as first mover would always prefer to ask zero and leave the game, better than paying the delay costand incurring a negative payoff. Then, in the following of the paper and when analyzing game 2, we willallow the configuration {1} and {2, 3, 4} as equivalent to the grand coalition.

10

11

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)

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

Fairness

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.

3.3

The procedure

The experiment was conducted at the Institute of Social and Economic Research (ISER),
Osaka University, between January 2019 and August 2019. A total of 264 students, who
had never participated in similar experiments before, were recruited as subjects of theexperiment. 96 for the one-period treatment, and 84 each for the two-period low andhigh cost treatments.4 The experiment was computerized with z-Tree (Fischbacher,2007) and participants were recruited using ORSEE (Greiner, 2015).
To control for potential ordering effects, each participant played all the four games
twice in one of the following four orderings: 1234, 2143, 3412 and 4321. Between
4

The difference in the number of participants between the two mechanisms is a result of variations inthe show-up rate among experimental sessions.

12each play of a game (called a round), players were randomly re-matched into groupsof four players, and participants were randomly assigned a new role within the newly
created group. At the end of the experiment, two rounds (one from the first four rounds
and another from the last four rounds) were randomly selected for payments. Participants received cash reward based on the point they have earned in these two selectedrounds with an exchange rate of 20 JPY = 1 points in addition to 1500 and 1900 JPY
participation fee for the one-period and the two-period implementations, respectively.
The experiment lasted on average 100 min for the one-period implementation and 130
min for the two-period implementations including the instruction (∼ 15 min), comprehension quiz (∼ 5 min), and payment.5 The average earning was 2650 JPY, 3110
JPY, and 2960 JPY for the one-period, 2-period low costs, and two-period high costs
implementations, respectively.

4

Results

4.1

Grand coalition formation and efficiency

At first, we investigate whether the three implementations succeeded in making the players find an agreement and form the grand coalition. In Figure 1, we present the results
about the grand coalition formation for the 1p, the 2pL and the 2pH implementations
and for the four games.6
5

Participants received a copy of instruction slides, and pre-recorded instruction movies were played.
See Appendix B for English translations of the instruction slides and the comprehension quiz.
6
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 1 if the grand coalition
is formed, and zero otherwise, in group i, 1pi (2pLi or 2pH) is a dummy variable that takes value 1 if
the 1p (2pL and 2pH) implementation is used, and zero otherwise. The standard errors are corrected forwithin session clustering effect. The statistical tests are based on the Wald test for the equality of theestimated coefficients of two treatment dummies.

13

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

As game 2 is not strictly convex with the presence of the null player 1, we allow thepartition {1}, {2, 3, 4} as a realization of the grand coalition, as this coalition structure
does not affect the total value which has to be shared between the players7
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 the three implementations of
the Winter mechanism equally failed in enhancing complete cooperation. Observe that,even when players were given a second chance in a two-period 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 same cases, is even significantly
7

Remember that the Winter mechanism is theoretically defined for strictly convex games. Player 1, in
this game, has always a zero marginal contribution and, as such, can be left out from any coalition at no
cost for either him/her or the other players.

14

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: Error bars show one standard error range.***, **, * significant at 1, 5, and 10% significance level, respectively (Wald test).

smaller) in the 2pL and 2pH implementations when compared to the 1p implementation.
We may recall that the theoretical predictions expects the grand coalition to form in thefirst loop for the three implementations. However, one may expect players to demand
more in the first loop of a two-period implementation, because of the second chance offorming a coalition if their requests is not met on the first try.
As a direct consequence of the failure in forming the grand coalition, we report alsoa failure in achieving full efficiency, with again no significant difference between theimplementations, regardless of the presence of some delay costs in 2pL and 2pH (seeFigure 3)8 . Efficiency is computed as the fraction of the sum of the payoffs obtainedby the four players compared to the value of the grand coalition of the considered game8

The figure is created based on the estimated coefficientsof the following linear regressions: Ef fi =Pi πi
β1 1pi + β2 2pLi + β3 2pHi + µi where Ef fi ≡ v(N ) is the efficiency measure for group i, 1pi (2pL
and 2pH) is a dummy variable that takes value 1 for the 1p (2pL or 2pH) treatment, and zero otherwise.
The standard errors are corrected for within session clustering effect. The statistical tests are based on theWald test for the equality of the estimated coefficients of two treatment dummies.

15

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

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: Error bars show one standard error range. *** and ** indicate the proportion of time that verification of the efficiency axiom
was significantly different at the 1 and 5% significance level (Wald test).

(100 for games 1 and 2 and 200 for games 3 and 4).
Therefore, we conclude that
Result 1. The Winter mechanism in its one-period, two-period low cost and two-periodhigh cost implementations provides mediocre and comparable results in terms of coalition formation and efficiency.

4.2

Payoff shares: ex ante theoretical prediction

According to the ex ante theoretical prediction, the Winter mechanism in all the three
implementations is expected to provide approximately the Shapley value, in averageover the different orderings of the players. We denote by π 1p (vk ) a vector of payoffs
obtained by the players and by implementing 1p and on game k, with k = 1, 2, 3, 4.
Analogously, we denote by π 2pL (vk ) a vector of payoffs obtained by the players by
16

Figure 4: Mean of the payoffs for the three mechanisms, the horizontal lines indicating
the Shapley values.

Player 3

Player 2

Player 2

2pH

1p

2pL

2pH

1p

Player 3*** *** ***Player4

Player4

Player 1

Player 2

Player 3

2pH

1p*** *** ***2pL** ***

2pH

**

1p

2pL

2pH

*** *** ***

1p

2pL

2pH

*** *** ***

1p

2pL

2pH

1p

2pL

2pH

Player 3*** *** ***

120
100
80
60
40
20
0
-20

2pL

Game 4*** *** ***

1p

2pL

1p

Player 1

2pH*** *** ***2pH*** *** ***

2pL

120
100
80
60
40
20
0
-20

1p

Game 3** *** ***

2pL

1p

Player 1

Player4

2pH*** *** ***

2pL

2pL

1p

***

2pH

60
50
40
30
20
10
0
-10

2pH

1p

*** *** ***

2pL

2pL

2pH

Player 22pH

1p

Player 1

Game 2*** *** ***

1p

***

2pL

*

1p

*** ** ***

2pL

60
50
40
30
20
10
0
-10

2pH

Game 1

Player4

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

implementing 2pL and by π 2pH (vk ) 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 thethree implementations, the horizontal lines indicating the Shapley value for each game9 .
As we may observe in Figure 4, as a consequence of the players often failing toform the grand coalition and, consequently, because of a lack of efficiency, the average
9

The error bars are based on the standard errors that are corrected for withing session clustering effect.
These standard errors are obtained running the system of linear regressions described in Section 4.4. The
statistical tests are based on these regressions.

17

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 theabsolute payoffs, converges to the Shapley value, by considering the normalized (tothe value of the grand coalition) payoff vectors. Then, in this and in the following
sections we consider the normalized vectors of payoffs with components πei m (vk ) =P

πim (vk )×vk (N ) for each i = 1, 2, 3, 4 and for each m ∈ {1p, 2pL, 2pH} (remember
1pj∈N πj (vk )that the value of the grand coalition is equal to 100 for games 1 and 2 and equal to 200
for games 3 and 4).
Figure 5 shows the mean of the normalized payoffs in the four games, the horizontal
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 implementations. Thus, the following result follows.
Result 2. The Winter mechanism in its one-period, two-period low cost and two-periodhigh cost implementations provides good and comparable results in terms of implementation of the Shapley value power share.

4.3

Payoff shares: ex post theoretical prediction and first mover advantage

The ex post theoretical prediction of the Winter mechanism is dependent on the ordering in which the players make their demand. In particular, a first mover advantage ispredicted. Furthermore, a higher first mover advantage is expected in the 1p implementation, followed by the 2pH and, finally by the 2pL implementation. As a result, thedistances between the Shapley value and the ex post theoretical predictions are largest
18

Figure 5: Mean of the normalized payoffs for the three mechanisms, the horizontal lines
indicating the Shapley values.

Player 3

Player4

Player 1

Player 2

Player 1

Player 2

1p

2pL

2pH

1p

2pL

Player 3

Player4

Game 4
120
100
80
60
40
20
0
-20

Player 3

Player4

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

2pH

Game 1

Player4

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

in 1p implementation, followed by 2pH and then 2pL.
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 Dis2m=ei − φi (v))2 and Dis2ED =
φ
i (π
pP
ei − EDi )2 where φi (v) denotes the Shapley value for player i in game v.10i (π10The 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 group i, 1pi (2pLi and2pH) is a dummy variable that takes value 1 if 1p (2pL or 2pH) treatment is used, and zero otherwise.
The standard errors are corrected for within session clustering effect. The statistical tests are based on the19Figure 6: Mean of the distances of the normalized payoff vectors from the SPNEs andthe 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
Dis2φ

2pH

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

2pH

Dis2φ

Dis2ED

n.s.

1p

n.s.

n.s.

2pL

2pH

Dis2ED

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

We may observe that, contrary to the ex post theoretical predictions, 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, Dis2φ wassignificantly larger in 2pH compared to 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 provide similar results interms of the distance between the realized normalized payoffs and the Shapley value aswell as the equal division solution.
Wald test for the equality of the estimated coefficients of two treatment dummies.

20

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

n.s.
n.s.

2pL

2pH

(d) Game 4

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
2pH

1p

n.s.

2pL

2pH

Note: Error bars show one standard error range.***, **, * significant at 1, 5, and 10% significance level, respectively (Wald test)

Figure 7 shows the proportion of times first mover advantage appears, i.e., when thefirst mover obtains a higher normalized payoff than her 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 in average more than predicted by her Shapley
value half of the time, and, consequently, less than predicted the other half of the times).
Figure 8 details these results for each single player and Figure 9 presents the mean 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 theoretical prediction.
We can now state the following result.
Result 4. The Winter mechanism in its one-period, two-period low cost and two-periodhigh cost implementations do not show any first mover advantage effect.

21

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

Player 1

Player 2

Player 3

Player4

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

Player4

Game 41.0
0.8

Player 1

Player 2

Player 3

Player4

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

Player4

Note: Error bars show one standard error range. ** and * indicate significant difference between treatments at 5% and 10%
significance level (Wald test).

4.4

Testing for the axioms

From Figure 5 we may notice that both the one-period and the two-period low cost
implementations verify the null player property. In particular, the one-period implementation satisfies it fully (100% of the times), always assigning a payoff equal to 0 to
the null player 1, in game 2. Instead, the other two implementations assign in averagea negative payoff to the null player, as a consequence of the delay costs. As expected,
small delay costs, as in 2pL, let the mean be close to 0, while on the other side, large
delay costs, as in 2pH, let the final average payoff for player 1 in game 2 be significantly

22

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

Player 3

Player 1

Player 2

Player 3

Player 1

Player4

Player 2

2pH

1p

2pL

2pH

2pL

1p

Player4

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

Player4**

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

Player4

Note: Error bars show one standard error range. ** indicates significant difference between treatments 5% significance level (Waldtest).

different from 0.
In order to test symmetry, additivity, homogeneity, strong monotonicity and fairness, we perform a set of OLS regressions for the following system of equations, withdependent variables the average normalized payoffs πei , and as independent variables g1 ,23

g2 , g3 , g4 and without constant:
πe1 = a1 g1 + a2 g2 + a3 g3 + a4 g4 + u1
πe2 = b1 g1 + b2 g2 + b3 g3 + b4 g4 + u2
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 value 1 if the game i is played, and zero otherwise. Various axioms are tested based on the estimated coefficients of these regressions. Symmetry requires b1 = c1 and b4 = c4 . Additivity and Homogeneity requirex3 = x1 + x2 and x4 = 2x1 for x ∈ {a, b, c, d}, respectively. Strong monotonicity
requires a1 > a2 and a4 > a3 . Finally, fairness requires, b3 − b2 = c3 − c2 . In Table 5in Appendix C, we present the results 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, and only for game 4 (then, half of the times) by the
2pH implementation. The additivity and homogeneity (according to which H0 should
not be rejected) are almost always confirmed by the three implementations. The strong
monotonicity (according to which H0 should be rejected) is confirmed by the three implementations. The fairness (according to which H0 should not be rejected) is rejectedby the 1p and the 2pH implementations, but confirmed by the 2pL implementation. Table 4 summarizes whether each axiom is satisfied in average (+) or not (–) for the three
implementations. To conclude this section, we can state as follows:
Result 5. The Winter mechanism in its one-period, two-period with low costs and twoperiod with highs cost implementations provides good and comparable results in terms

24

Table 4: Winter mechanisms, axioms
Axiom

1p

2pL

2pH

Efficiency
Symmetry
Additivity
Homogeneity
Null player property
Strong monotonicity
Fairness

–
+
+
+
+
+
–

–
+
+
+
+
+
+

–
–
+
+
–
+
–

of satisfaction of the axioms.

5

Conclusions

In this paper we provide an experimental comparison of three different implementationsof the Winter demand commitment bargaining mechanism: one-period, two-period withlow costs and two-period with high costs. These three implementations predict thesame ex-ante outcomes, while they differ in terms of ex-post outcomes. Our experimentshows that, however, these three different implementations provide comparable results
for both ex-ante and ex-post outcomes. No significant difference appeared in any of ourdomains of investigation: coalition formation, alignment with the theoretical prediction
and axioms satisfaction.
An example we borrow by Winter (1994) on the bargaining over government formation may help presenting the implications of our results. Bargaining over government
formation is a process that naturally resembles the demand commitment model withmore than one period. Parliamentarian negotiations are usually based on demands rather
than proposals, and often these demands are not compatible in the first period, and at25

least the 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 is often considered
as essential and crucial for parties to match and a coalition to form.
Surprisingly, our experimental results suggest that this could not be always the case.
In fact, our three implementations resulted in similar outcomes in all our domains ofinvestigation. The take-off message of our paper suggests a mechanism designer toimplement, whenever possible, the lightest possible mechanism for bargaining, as refinements may turn out to be costly to implement, but ineffective in terms of qualityof the performances. In fact, players converge to similar outcomes (e.g., total or partial cooperation) without taking advantage of any second chance, and regardless of thedifferent theoretical predictions, differences that are not matched behaviorally.

Acknowledgement
The experiments reported in this paper has been approved by IRB of Yamaguchi University (No. 5). We gratefully acknowledge financial support from the Joint Usage/ResearchCenter at ISER, Osaka University, and Grant-in-aid for Scientific Research, Japan Society for the Promotion of Science (15K01180, 15H05728, 18K19954, 20H05631), Fundfor the Promotion of Joint International Research (Fostering Joint International Research) (15KK0123), the French government-managed l’Agence Nationale de la Recherche
under Investissements d’Avenir U CAJEDI (ANR-15-IDEX-01). In particular, we thankthe UCAinACTION project.

26

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29

A

The Winter Mechanism

We present here more formally the T period Winter mechanism 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),2where:
0 ≤ 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 which 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)), andj ∈ S1t,m \ S2t,m is the player taking the decision by introducing a demand dj .
Her demand dj is said to be compatible if there exists some S ⊆ S2t,m with v(S ∪P{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 playeri ∈ S ∪ {j} is paid di − tc (c is the delay cost per period), and nature chooses
randomly 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 non-compatible, then two cases are distinguished:302a ) if S2t,m = S1t,m \ {j} (j is the last player to demand in the current period),
then a new player k is chosen randomly from S1t,m and the new position is givenby (S1t+1,1 , ∅, k) (we increment the period from t to t + 1 and we start 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 randomly chosen a player j ∈ N . Then the initial position is
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.

B

Translated instructions and comprehension quiz

An English translation of the instruction handout can be downloaded from• https://bit.ly/33IzgMM for Winter 1 period implementation• http://bit.ly/3avJPoq for Winter 2 period implementation
An English translation of the comprehension quiz can be downloaded fromhttp://bit.ly/3oOMVsLCVerification of Axioms

31

32

H0

Symmetry

b1 = c 1
b4 = c 4
Additivity
a3 = a1 + a2
b3 = b1 + b2
c3 = c1 + c2
d3 = d1 + d2
Homogeneitya4 = 2a1
b4 = 2b1
c4 = 2c1
d4 = 2d1
Strong 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
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
0.007
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
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
0.043

p-value

2pH

Table 5: Winter mechanisms, Wald tests for the verification of the symmetry, additivity, homogeneity, strong monotonicity
and fairness axioms