ISSN (Print) 0473-453X

Discussion Paper No. 1175

ISSN (Online) 2435-0982

AN EXPERIMENT ON THE NASH PROGRAM:
COMPARING TWOSTRATEGIC MECHANISMS
IMPLEMENTING THE SHAPLEY VALUE
Michela Chessa
Nobuyuki Hanaki
Aymeric Lardon
Takashi Yamada

May 2022

The Institute of Social and Economic Research
Osaka University
6-1 Mihogaoka, Ibaraki, Osaka 567-0047, JapanAn experiment on the Nash program: Comparing two
strategic mechanisms implementing the Shapley value*
Michela Chessa†

Nobuyuki Hanaki‡

Aymeric Lardon§

Takashi Yamada¶May 30, 2022

Abstract
We experimentally compare two well-known mechanisms inducing the Shapley value as an ex ante equilibrium outcome of a noncooperative bargaining procedure: the demand-based Winter’s demand commitment bargaining mechanism andthe offer-based Hart and Mas-Colell procedure. Our results suggest that the offerbased Hart and Mas-Colell mechanism better induces players to cooperate and toagree on an efficient outcome, whereas the demand-based Winter mechanism better implements allocations that reflect players’ effective power.

JEL code: C70, C71, C92
Keywords: Nash program, Bargaining procedures, Shapley value, Experiments* This is a revised version, with an additional experiment, of the Chessa, M., N. Hanaki, A. Lardon, andT. Yamada (2021): “An experiment on the Nash program: Comparing two bargaining implementations of
the Shapley value,” GREDEG Working Paper 2021-07, GREDEG.
†
Université Côte d’Azur, CNRS, GREDEG, France. E-mail: michela.chessa@univ-cotedazur.fr‡
Institute of Social and Economic Research, Osaka University, Japan.
E-mail:nobuyuki.hanaki@iser.osaka-u.ac.jp§
GATE Lyon Saint-Etienne, UMR 5824 CNRS, Université de Lyon.
France.
E-mail:aymeric.lardon@univ-st-etienne.fr¶
Faculty of Global and Science Studies, Yamaguchi University, Japan. E-mail: tyamada@yamaguchiu.ac.jp1

1

Introduction

Whenever a facility is shared by different customers, departments, or other units of anorganization, the problem of how to allocate the costs or the payoffs among players
arises. Relevant examples of this situation include airports, transit systems, water distribution networks, inventory models, and scheduling. These contexts are well known
as cost or payoff allocation (or sharing) problems.1 Similar sharing problems arise in
the context of “co-opetition” (Brandenburger and Nalebuff, 1996) where competitors
cooperate to achieve a common goal. Usually, in such contexts, two approaches basedon game theoretical concepts may be adopted.
One approach is for the players to bargain among themselves to determine how
costs or payoffs should be shared. However, this implies a strategic interaction, which
may result in unnecessary additional costs if it is conducted in an unrestricted fashion(see, e.g., the arguments by Roth and Verrecchia, 1979). Instead, many bargaining
procedures follow the tradition of setting up sequential, perfect information games basedon offers, that is, games in which, at each stage, one of the players becomes a proposerof a cost (payoff) allocation, with a requirement for reaching unanimous agreement.
Such bargaining procedures implement negotiations in the style of the well-known twoplayer bargaining over a pie in Rubinstein’s problem (Rubinstein, 1982), which is thenextended to the n-player case.
Alternatively, one can view the problem as a normative one, in which an external
player, a so-called regulator, designs a pricing (rewarding) scheme that maximizes somemeasures of social welfare or that imposes axioms of equity or stability. Shubik (1962)
was among the original proponents of the Shapley value (Shapley, 1953) as a method
1

Many specific concrete examples of analogous situations come from one of the most typical economic phenomena: consumption and the contribution to public goods.

2

of joint-cost allocation. At present, this value continues to attract the greatest interest
among the allocation schemes predicated on notions of cooperative game theory (see,e.g., Littlechild and Owen, 1973; Schulz and Uhan, 2010; Timmer et al., 2013).
Bridging the gap between the strategic and cooperative approaches is recognized asa fundamental issue of game theory. Attempted resolutions of this issue, well known
as the Nash program (Nash, 1953), have provided many different strategic bargaining
mechanisms that sustain the Shapley value at equilibrium (for example, among others,
Hart and Mas-Colell, 1996; Pérez-Castrillo and Wettstein, 2001). Such mechanisms fit
and unify the two approaches, allowing the players facing an allocation problem (inour specific context) to bargain in a restricted way, and to converge to a stable solution
without the need for an intermediary.
Both the original normative implementation of the Shapley value by a regulator andthe implementation of classical bargaining mechanisms based on offers that lead to the
Shapley value require complete information, either on the part of the regulator, or theplayers. In many contexts, complete, transparent, and accountable information is often
desirable and encouraged. However, in some specific domains, this may represent an
unrealistic assumption, for example, when players are customers of a facility and do
not necessarily know about other customers (Young, 1998), or when computing suchallocations requires data from each player, some of which may be private (McSherry
and Talwar, 2007). We argue that even when such information is publicly available, it isdifficult to guess prices (awards) to charge (give) that are likely to be accepted by every
single player in the bargaining because this requires managing a wide and complex
information set about the structure of the underlying cooperative game and the marginalcontributions of each player.
An alternative but less common approach is to describe a bargaining mechanism
3

based on demands rather than offers. A demand-based mechanism was the basis of theimplementation by Nash (1953) of the cooperative bargaining solution by Nash (1950).
Other examples of mechanisms based on demands, though not common, include Young(1998), who, sharing our concern that complete information may be a difficult assumption in practice, describes a demand revelation mechanism in which potential customers
of a public facility simply bid to be served. Bargaining mechanisms based on demands
resemble oral auctions, in which each player, standing alone, reveals the charges he or
she is ready to pay to be served, or the payoff he or she is ready to accept for offeringcollaboration, and waits for such a request to be met. By allowing the players to bargain,such mechanisms allow them to focus only on their specific role in the organization andon their expectation of how much they should contribute to or obtain from the facility. This approach drastically reduces the information a player is required to possess orprocess to make a proposal. Moreover, in a demand-based mechanism, acceptance of a
proposal by the organization typically depends on objective feasibility conditions rather
than on subjective approval by its members.
In this paper, we aim to contribute to this dispute concerning demand- vs. offerbased bargaining mechanisms. We experimentally compare two well-known mechanisms inducing the Shapley value as an ex ante equilibrium outcome of a noncooperative bargaining procedure. We choose two mechanisms that are based on these opposing
approaches (demand vs. offer) but that remain, in our opinion, similar in terms of theirimplementation and the ease with which they can be understood by the participants in alaboratory experiment.2 The first mechanism is Winter’s demand commitment bargaining mechanism (Winter, 1994, referred to as the Winter mechanism below). The second2

A comparison between offer-based and demand-based mechanisms has been conducted experimentally for voting games by Fréchette et al. (2005a), as well as empirically by, for example, Warwick andDruckman (2001) and Ansolabehere et al. (2005), employing field data.

4

is the Hart and Mas-Colell procedure (Hart and Mas-Colell, 1996, referred to as the
H–MC mechanism below).
Both procedures are described as sequential, perfect information games, where, ateach stage, a player becomes a proposer. In accordance with the theoretical presentationof the two mechanisms, we illustrate and implement the bargaining procedures to define
a sharing of payoffs rather than an allocation of costs.3 In the first mechanism, which isdefined for cooperative games with increasing returns to scale for cooperation (strictly
convex games), the proposer makes a demand for him- or herself concerning the payoffthat he or she is willing to receive from a possible collaboration. In the second mechanism, which is defined for monotonic games (a much weaker assumption), the proposer
makes a proposal to each of the other players concerning the payoff he or she is willingto offer them.
Two main issues arise with most strategic bargaining models, as observed by Fréchette
et al. (2005a) in an experimental work implementing some well-known legislative bargaining processes. First, the theoretical predictions that they propose are very sensitive
to variations in the rules of the game, for example, in our case, whether a demand-basedor an offer-based mechanism is considered. Even if experiments show that actual bargaining behavior is not always as sensitive to the different bargaining rules as the theory
suggests, we expect our analysis to confirm such a statement and the two mechanisms toperform very differently, despite the similar theoretical predictions. Second, the equilibrium solution may require an unrealistic degree of rationality on the part of the players,
such that the experimental evidence is very far from the theoretical prediction. We claimthat the degree of rationality required in a demand-based mechanism is much lower than3

It is straightforward to establish the theoretical implementation of a cost allocation bargaining procedure.

5that for an offer-based mechanism. This is because, as argued above, compared with theoffer-based mechanism, the demand-based mechanism requires a player to know andprocess a smaller amount of information to make a proposal, as he or she can focus only
on his or her own specific role in the organization and his or her marginal contributions,
ignoring the role of all other players. Then, we aim at investigating the consequences ofsuch an issue in the results of our experiment.
It has been argued that the difference between a demand-based vs. an offer-based
mechanism is less relevant when considering two-player games, such as in Rubinstein(1982)’s bargaining-over-a-pie game (see, Fréchette et al., 2005a). However, it becomes
crucial when considering groups with at least three members. In particular, offer-based
mechanisms are comparable with a voting procedure in which all the other players either
accept or reject the proposed utility share put forward by the proposer. As such, they aretheoretically expected to show a high degree of asymmetry between the proposer andall the other players. In our case, both mechanisms are expected to show some form ofproposer advantage. In fact, for both mechanisms the ex post predicted solution strongly
depends on the selected proposer. In the case of the Winter mechanism in particular, iteven depends on the complete ordering.
Our analysis mainly focuses on (i) analyzing whether these mechanisms lead toformation of the grand coalition and (ii) testing the convergence in expected value and,as predicted by the theory, to the Shapley value.
Our results show that the H–MC mechanism results in a higher frequency of grandcoalition formation and a higher efficiency than does the Winter mechanism. Conversely, the Winter mechanism better implements the Shapley value as the average payoff provided that the grand coalition is formed. Therefore, our results suggest that anoffer-based H–MC mechanism better induces players to cooperate and to agree on an
6efficient outcome, whereas a demand-based Winter mechanism better implements allocations that reflect players’ effective power.
The remainder of the paper is organized as follows. Section 2 reviews existingstudies that are most relevant to our work. Section 3 presents the general definition andthe properties of a cooperative transferable utility (TU) game, as well as the Shapley
value. Section 4 presents the two mechanisms that we investigate, namely the Winterand the H–MC mechanisms. Section 5 describes the setting of our experiment. The
results are presented in Section 6, and Section 7 concludes. Additional analyses aimed
at reinforcing our results and at providing new points for reflection are contained in the
Appendix II to V.

2

Related work

Bridging the gap between the noncooperative models, in which the primitives are thesets of possible actions of individual players, and the cooperative models, in which they
are the sets of possible joint actions of groups of players, has been recognized as a fundamental issue of game theory. The very first attempt at this so-called Nash program
dates back almost 70 years to Nash himself (Nash, 1953). His idea was to provide a noncooperative foundation for cooperative solution concepts, and he began implementing it
by designing a noncooperative game that sustained the Nash solution of his two-playerbargaining problem (Nash, 1950) as its equilibrium. Following this first attempt by
Nash, many alternative procedures for implementing solutions of two-player bargaining
problems or n-player pure bargaining problems4 have been implemented. Some mechanisms intended to obtain the Nash solution, exactly or approximately, at equilibrium
4

A pure bargaining problem is a cooperative game in which only the grand coalition N creates a
positive surplus with respect to what each player can achieve if he or she does not cooperate with anyone.

7

(see, among others, Binmore et al., 1986; Trockel, 2002). Others aimed instead to obtain the Kalai–Smorodinsky solution (Kalai and Smorodinsky, 1975), that is, the main
alternative solution to such problems (Moulin, 1984b; Trockel, 1999; Haake, 2000).
Many different theoretical mechanisms have been designed with the aim of implementing other cooperative solution concepts via a strategic interaction of the players formore generic cases, that is, when there are more than two players or when the bargaining problem is not pure. This is the case, for example, in the seminal work of Harsanyi(1974), who reinterpreted the von Neumann–Morgenstern solution as an equilibrium ofa noncooperative bargaining mechanism, and of the many works sustaining the most famous axiomatic solution concept by Shapley (1953), the Shapley value. For a relevant
and extensive review of the theoretical literature on the Nash program, we refer readers
to the surveys by Serrano (2005, 2008, 2014, 2021).
In this section, we focus on the literature devoted to testing cooperative game theory
through experiments. To date, this literature has focused mainly on three different directions. The first direction provides a normative interpretation, as in De Clippel and Rozen(2021), in which subjects designated as decision-makers express their view on what isfair for others by recommending a payoff allocation. De Clippel and Rozen (2021)
show that the decision-maker’s choices can be described as a convex combination of theShapley value and the equal division solution.
The second direction investigates how an unstructured interaction affects the final
agreement. One example is the paper by Kalisch et al. (1954), in which groups ofplayers are asked to freely discuss the formation of coalitions and to reach an agreement on how to split the related values. The authors identify many different factors
influencing the final outcome of such a procedure, including personality differences or
the geometrical arrangement of players around the table. Similarly, but with a greater8

focus on voting games, Montero et al. (2008) propose an unstructured bargaining protocol in which participants propose and vote on how to distribute a fixed budget among
themselves. The paper provides experimental evidence of the so-called paradox of new
members, according to which enlargement of a voting body (i.e., the addition of a newvoter) can increase the voting power of an existing member. Guerci et al. (2014) study
the impact of variations in the experimental protocol of Montero et al. (2008) on theformation of the so-called minimal winning coalitions, that is, coalitions for which eachplayer is crucial.
Most experimental works in the literature follow a third direction, studying the outcome when a more formal (or structured) bargaining protocol is imposed. Our paper
broadens this last direction of research.
Formal bargaining protocols have been implemented to tackle different aspects ofthe cooperative inclination of the players under different settings. For example, Murnighan
and Roth (1977) investigate the effect of various communication/information conditions
on the final outcome in a specific game played by a monopolist and two weaker players. They show how the results over the entire set of conditions closely approximate
the Shapley value, although they often report a clear tendency for an equal split of thepie. Similarly, Murnighan and Roth (1982) introduce bargaining models to investigate
the influence of information shared by subjects about the games (e.g., payoffs) on the
final outcome. They show that the quality of the information has an impact on the final
outcome and that the Nash bargaining solution has a good predictive performance inmany cases. Bolton et al. (2003) investigate how the communication configuration affects coalition negotiation and show how players with weaker alternatives would benefit
from a more constrained structure, especially if they can be the conduit of communication, whereas those endowed with stronger alternatives benefit from working within a9more public communication structure that promotes competitive bidding. Other works
focus more specifically on the coalition formation process, including Nash et al. (2012);
Shinoda and Funaki (2019); Abe et al. (2021). In the first paper, the authors implement
finitely repeated three-person coalition formation games, showing how efficiency requires people’s willingness to accept the agency of others, such as political leaders. The
second paper is then presented as a follow-up, in which the authors maintain the samevalue of the coalitions as in Nash et al. (2012), but implement a different bargaining protocol. They report a rare formation of a grand coalition, which can be induced by someexternal factors, such as the presence of a chat window. The third paper presents a comparison between two mechanisms that invite players to join a meeting simultaneously orsequentially. The authors report that the sequential mechanism induces a higher social
surplus than the simultaneous mechanism. Moreover, players make choices consistent
with the subgame perfect Nash equilibrium (SPNE) in the sequential setting and choose
the dominant strategy in the simultaneous setting, when a dominant strategy exists.
Formal bargaining protocols are mostly based on the implementation of theoretical mechanisms, which are shown to converge to some specific well-known solutions.
This is the case, for example, in Nash (1953) and Harsanyi (1974), which we have referred to above, or in the case of the bargaining mechanism proposed by Raiffa (1953)
to implement the Raiffa solution (as opposed to the Nash solution) to the Nash cooperative bargaining problem. Several experimental implementations have been proposed,with the final goal of testing Nash axioms, or of comparing Nash and Raiffa solutions(see, e.g., Nydegger and Owen, 1975; Rapoport et al., 1977). In addition, there is alarge literature devoted to studying the class of bidding mechanisms. Bidding mechanisms are introduced by Demange (1984) and Moulin (1984a), and Moulin and Jackson(1992) study them in economic environments. They are developed by Pérez-Castrillo
10and Wettstein (2001) and Ju and Wettstein (2009) to implement solution concepts in theframework of cooperative TU games.
In particular, many different theoretical mechanisms have been designed specificallywith the aim of implementing the best-known cooperative solution, the Shapley value(see Shapley, 1953). Because this solution is applied in many economic problems,
supporting it through strategic explanation is considered to be particularly important.
See among others, Harsanyi (1981), Gul (1989), Hart and Moore (1990), Winter (1994),
and Hart and Mas-Colell (1996).5
Despite the large body of existing literature, the Nash program “is not ready for
retirement yet”, but is, on the contrary, “still full of energy” and “waiting for good papers to be written” (Serrano, 2021). In this paper, we aim to contribute to this research
agenda by providing new insights gained from a controlled laboratory experiment. In
particular, we propose an experimental comparison of two mechanisms. The first mechanism is the one-period version developed by Winter (1994) (this simplified version was
also previously used by Bennett and van Damme (1991) to treat Apex games, a type of
weighted majority games). The second mechanism is by Hart and Mas-Colell (1996), in
the specific case in which a proposer whose proposal is rejected leaves the game with aprobability 1. Our work is similar to Fréchette et al. (2005a), who experimentally compare an offer-based model of Baron and Ferejohn (1989) with a demand-based modelof Morelli (1999) in weighted majority voting games. Earlier experimental studies of
the Baron–Ferejohn model include Fréchette et al. (2003, 2005b), and Fréchette et al.
(2005a) provide an experimental study of demand bargaining.6 However, Fréchette
5

Krishna and Serrano (1995) deepen the study of the set of subgame perfect equilibria associated withthe bargaining mechanism proposed by Hart and Mas-Colell (1996).
6
Fiorina and Plott (1978) propose multiple experiments on committee decision-making under majority
rules to test a wide range of solution concepts of noncooperative games.

11et al. (2005a) present the first work to directly compare the two within an experimental
framework. Their results show that proposers have some first-mover advantage in both
the demand and offer games, but their power does not differ nearly as much between thetwo mechanisms as theory predicts.

3

Theoretical model

3.1

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 is thecharacteristic function, which assigns to each coalition S ⊆ N the worth v(S), with
the normalization condition v(∅) = 0. The worth of a coalition represents the valuethat members of S can achieve by agreeing to cooperate. To simplify the notation if noambiguity appears, we consider the set of players N as fixed and we write v instead of(N, v). We use G N to denote the set of all games with player set 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 ,• convex if v(S) + v(T ) ≤ v(S ∪ T ) + v(S ∩ T ), for each S, T ⊆ N , and strictly
convex if the inequality holds strictly.
We observe that convexity ⇒ superadditivity ⇒ monotonicity. In (strictly) convex
games, cooperation becomes increasingly appealing, and a so-called “snowball effect”
12

is expected, leading to the formation of the grand coalition. Another equivalent definition for convexity can be stated as v(S ∪ {i}) − v(S) ≤ v(T ∪ {i}) − v(T ), for each
S ⊆ T ⊆ N \ {i}.
Given a game v ∈ G N , an allocation is an n-dimensional vector (x1 , . . . , xn ) ∈ RN ,assigning to player i the amount xi ∈ R. 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 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 behinda (strong) Nash equilibrium of a noncooperative game, namely an outcome is stable if
no deviation is profitable. For the Nash equilibrium, the possible deviation concerns asingle player, whereas in the core, deviations of groups of players are relevant.
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 best-known solution concept, which is widely applied13

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 to every player his or her expected marginal contribution to
the coalition of players that enter the game before this player, given that every orderof entrance has equal probability. This solution concept has been defined as respectingsome notion of fairness (see Appendix III for more discussion about its properties), but
it is not necessarily stable. However, if the game is superadditive, the Shapley value isan imputation, and if the game is convex, it belongs to the core (in particular, it is its
barycenter).

4

Two mechanisms

In this section, we present the demand-based Winter mechanism (Section 4.1) and theoffer-based H–MC mechanism (Section 4.2) in more detail. Section 4.3 compares theequilibrium predictions of the two mechanisms with a simple example.

4.1

The Winter mechanism

Winter (1994) presented a bargaining model based on sequential demands for strictly
convex cooperative games. As noted, in such games, cooperation becomes increasingly
appealing and a “snowball effect” is expected, leading to the formation of the grandcoalition. Moreover, in convex games, the Shapley value is a central point in the core,
which is always nonempty.
In this model, players announce their demands publicly in turns. That is, the players
14effectively state “I am willing to join any coalition that yields me...” and wait for thesedemands to be met by other players. The bargaining starts with a randomly chosen
player from N , say player i. This player publicly announces his or her demand di andthen points to a second player, who has to state his or her demand. Then, the gameproceeds by having each player introduce a demand then point at a new player to take aturn. If or when, at some point, a compatible demand is introduced, which means that
there exists a coalition S for which the total 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 receive their demands and leave the game, and the bargaining continues
with the rest of the players using the same rule on v restricted on N \ S.
Here, we present the one-period Winter mechanism more formally. This is a simplified version of the more general mechanism in Winter (1994), which allows for more
periods and includes a discount factor. A decision point position at time t of the oneperiod demand commitment game is given by the vector (S1t , S2t , dS2t , j), where:
S1t ⊆ N is the set of players remaining in the game,S2t ⊂ S1t is the set of players who have submitted demands that are not yet met,
dS2t = (di )i∈S2t is the vector of demands submitted by players in S2t , (0 ≤ di ≤maxS⊆N v(S)), andj ∈ S1t \ S2t is the player taking the decision by introducing a demand dj . His
or her demand dj is said to be compatible if there exists some S ⊆ S2t withPv(S ∪ {j}) − i∈S di ≥ dj . Otherwise, dj is not compatible.
With j’s decision, the game proceeds in the following way:15

1) If dj is compatible, then j specifies a compatible coalition S, each player i ∈S ∪ {j} is paid di , and a player k 6= j is randomly chosen from S1t \ S2t . The newposition is now given by (S1t+1 , S2t+1 , dS2t+1 , k), with S1t+1 = S1t \ (S ∪ {j}) andS2t+1 = S2t \ (S ∪ {j}).
2) If dj is noncompatible, then two cases are distinguished:2a ) if S2t = S1t \ {j} (j is the last player to make a demand), then each playeri ∈ S1t (j included) gets his or her individual payoff v({i}), and the game ends;
2b ) if S2t ⊂ S1t \ {j}, then j specifies a new player k 6= j in S1t \ S2t and the newposition is (S1t+1 , S2t+1 , dS2t+1 , k), with S1t+1 = S1t and S2t+1 = S2t ∪ {j}.
The game starts with a randomly chosen 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 S1t ∪ {j} = S2t (see point 2a above).
As shown by Winter for the more generic case, this mechanism has a unique subgame perfect equilibrium, which assigns equal probabilities according to the principle
of indifference. At this equilibrium, the grand coalition forms and the a priori expectedequilibrium payoff coincides with the Shapley value. Moreover, given a specific ordering of the players, the a posteriori equilibrium payoff of each player depends on theorder of players only through the set of the player’s successors but it is not influenced by
the way that these players are ordered, as each player demands a marginal contribution
to the set of successors.

4.2

The Hart and Mas-Colell mechanism

Hart and Mas-Colell (1996) proposed a bargaining procedure for monotonic cooperative
16

games. This is a much weaker assumption compared with the strict convexity requiredby the Winter mechanism. Thus, the H–MC procedure is applicable for a larger set ofcooperative games.
In this mechanism, the bargaining starts with a randomly chosen proposer making
an offer to the other players, with the meaning “If you wish to form a coalition withme, I will give you...”. Then the other players, who act sequentially, may either accept
or reject the proposal. The requirement for agreement is unanimity. The key modelingissue is the specification of what happens if there is no agreement and, as a consequence,
the game moves to the next stage. In our implementation, if the proposal is rejected, theproposer leaves the game with his or her individual value and the bargaining continues
among the rest of the players, with a new player randomly chosen as a new proposer.
We present a more formal description of the H–MC mechanism. A decision point
position at time t is simply given by the vector (S t , j), where:
S t ⊆ N is the set of players remaining in the game,j ∈ S t is the player making an offer to the remaining players (ti )i∈S t \{j} suchP
that i∈S t \{j} ti ≤ v(S t ).
With j’s proposal, the game proceeds now in the following way:
1) If all i ∈ S t \ {j}, who decide sequentially, accept the proposal one after
the other, then players in S t \ {j} are paid (ti )i∈S t \{j} , player j is paid v(S t ) −Pi∈S t \{j} ti , and the game ends;
2) If at least one player i ∈ S t \ {j} refuses the offer, then two cases are distinguished:172a) if |S t | = 2 (only one more player is left, together with j), then they both
receive their individual value v({i}) for each i ∈ S t , and the game ends;2b) if |S t | > 2, then player i is removed from the game, he or she receives his or
her individual payoff v({i}), a new proposer k ∈ S t+1 = S t \ {j} is randomly
selected, and the new position is (S t+1 , k).
The game starts with a randomly chosen player j ∈ N . Then, the initial position is
set to be (N, j). It terminates either when there are no more players in the game (seepoint 2a above), or when the proposal is unanimously accepted (see point 1 above).
Hart and Mas-Colell (1996) show that this game has a unique subgame perfect equilibrium. At this equilibrium, the grand coalition forms and the a priori expected equilibrium payoff coincides with the Shapley value. In contrast to the Winter mechanism,given a specific initial proposer j ∈ N (in the previous mechanism, it was necessary to
specify the order of all the players, whereas in this case only one player, the proposer,
needs to be specified at equilibrium), the a posteriori equilibrium payoff assigns to eachother player his or her Shapley value in the cooperative game, reduced to the set of players N \ {j}, and the proposer is assigned his or her marginal contribution to the grandcoalition v(N ) − v(N \ {j}).

4.3

A comparison between the Winter and the H–MC mechanisms

We illustrate the two mechanisms using the strictly convex three-player game shownin Table 1. Although our experiment is based on four-player games, a three-player
game example is of particular interest because it allows us to graphically represent the
imputation set, the core, and the different solutions, as illustrated in Figure 1.

18

Table 1: A three-player game
S

1

2

3

1,2

1,3

2,3

N

v(S)

20

20

30

45

55

60

100

As we have already observed, the convexity assumption implies the monotonicity. Thus, the game satisfies the assumptions of both the Winter and H–MC mecha
nisms. The Shapley value of this game is given by the vector φ(v) = 170
=
, 185
, 245
6
6
6
(28.33, 30.83, 40.83), which corresponds to the a priori equilibrium payoff for both theWinter and H–MC mechanisms.
We suppose now that player 1 is chosen randomly as the first proposer in both mechanisms. Independently of the order of the following players in the Winter mechanism,
the proposer will receive an a posteriori equilibrium payoff equal to 40 in both mechanisms, which corresponds to his or her marginal contribution to the grand coalitionv(N ) − v(N \ {1}). We can see that both mechanisms lead to a proposer advantage, as, meaning that, as the first proposer, player 1 can obtain more than his or her
40 > 170
6Shapley value.
Suppose now that the total ordering of the players in the Winter mechanism is given
by 1, 2, and 3. The a posteriori equilibrium payoff of the Winter mechanism is given by
the vector SOLW (v) = (40, 30, 30), in which player 2 demands his or her marginal contribution v({2, 3}) − v({3}), and player 3 demands his or her individual value v({3}).
Conversely, in the case of the H–MC mechanism, the proposer offers the Shapley
value of the reduced game to players 2 and 3. Thus, the a posteriori equilibrium payoff
is given by the vector SOLHM C (v) = (40, 25, 35). Even with the disadvantage of not
being the first mover, player 2, as the second mover, manages to obtain more under the19

Figure 1: The core of the three-player game
(100, 0, 0)
Inputation Set
Core

20
x2

x1

45
SOLHM C (v)40
SOLW (v)20φ(v)(0, 100, 0)30(0, 0, 100)55x3Winter mechanism than under the H–MC mechanism even if, in both cases, he or she
obtains less than his or her Shapley value.
Figure 1 shows the imputation set I(v) = co h(20, 50, 30), (50, 20, 30), (20, 20, 60)i,
the core C(v) = co h(40, 30, 30), (40, 20, 40), (25, 20, 55), (20, 25, 55), (20, 45, 35), (25, 45, 30)i,
the Shapley value φ(v), and possible a posteriori solutions SOLW (v) (6 black dots)
and SOLHM C (v) (3 white dots). A point in the simplex corresponds to an allocation(x1 , x2 , x3 ). For example, the height of a point from the edge that is opposite to theapex labeled (100, 0, 0) represents the payoff allocated to player 1. Thus, a point on thebottom edge represents an observed allocation that gives a zero payoff to player 1. Similarly, the height of a point from the edge that is opposite to the apex labeled (0, 0, 100)
represents the payoff allocated to player 3.
We make the following observation to conclude this example and the comparison
between the two mechanisms.
20

Table 2: The games
S

1

2

3

4

1,2

1,3

1,4

2,3

3,4

1,2,3

1,2,4

1,3,4

2,3,4

N

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

0
0

5
20

5
20

10
30

20
20

20
20

25
30

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

50
45

60
55

60
60

60
100

100
100

2,4

Observation 1. The core is always a convex polyhedron. The a posteriori equilibriumof the Winter mechanism always coincides with a vertex of this polyhedron. The aposteriori equilibrium of the H–MC mechanism always provides a vector on a face of
this polyhedron.

5

The experimental setting

5.1

The games

For our analysis, we implement the four four-player games shown in Table 2. These
games are chosen to test the properties of the Shapley value that are presented in Appendix III. Note that:• games 1, 3, and 4 are strictly convex, whereas game 2 is only convex. All fourgames are, by consequence, monotonic. Therefore, all four games respect theassumptions for the implementation of the H–MC mechanism, whereas all except
game 2 respect the assumption for the implementation of the Winter mechanism.
However, with game 2 being only convex, we consider that “strict convexity”
could be relaxed and the mechanism could still be implemented in such a case;• in games 1 and 4, players 2 and 3 are symmetric;
21

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

Game 1
Game 2
Game 3
Game 4

φ1 (v)

φ2 (v)

φ3 (v)

φ4 (v)

22.08
0
22.08
44.16

23.75
28.33
52.08
47.5

23.75
30.83
54.58
47.5

30,42
40.83
71.25
60.83

• in game 2, player 1 is a null player. This is the reason why the game is only
convex, but not strictly convex, as the presence of a null player does not allow, by
definition, the possibility of having a strictly increasing marginal contribution forsuch a player;• game 3 is defined as the sum of games 1 and 2;• game 4 is defined as twice game 1 and it preserves the symmetry of players 2 and3;• the marginal contributions of player 1 are always higher in game 1 than in game
2, and higher in game 4 than in game 3.
The Shapley values of the four games are presented in Table 3. The equal division
payoff vector is simply equal to ED(vk ) = (25, 25, 25, 25) when k = 1, 2, and toED(vk ) = (50, 50, 50, 50) when k = 3, 4.

6

Results

The experiment was conducted at the Institute of Social and Economic Research (ISER),
Osaka University, in January and February 2019 (Winter mechanism) and January and22February 2022 (H–MC mechanism).7 A total of 176 students, who had never participated in similar experiments before, were recruited as subjects of the experiment, 96
playing the Winter mechanism and 80 playing the H–MC mechanism.8 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 four games twice
in one of the following four orderings: 1234, 2143, 3412, and 4321.9 Between eachplay of a game (called a round), players were randomly rematched into groups of four
players, and participants were randomly assigned a new role within the newly createdgroup.10 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 that they earned in these two selected rounds,
with an exchange rate of 20 JPY = 1 point, as well as a 1,500 JPY participation fee.
On average, the experiments lasted for 1 hour 40 minutes for Winter and 1 hour 45
minutes for H–MC, including the instructions (15 minutes for Winter and 11 minutesfor H–MC), a comprehension quiz (5 minutes), and payment.11 The average earnings
7

The experiments were conducted in 2019 and 2022 because the original H–MC experiment conducted in December 2019 (which we refer to as the pseudo-H–MC or H–MCsim in Appendix V) did notreflect the H–MC model precisely, and we have redone the H–MC experiment to correct this. Appendix V
compares the outcomes of the pseudo-H–MC conducted in December 2019 and the (corrected) H–MC
conducted in January–February 2022.
8
The difference in the number of participants between the two mechanisms is a result of variations inthe show-up rate among experimental sessions.
9
We let participants play all four games, instead of just one, in each session. Although this design
choice may have meant participants were slower in learning how to play the game, we consider thathaving within-session variations is desirable because the tests of the axioms involve comparing outcomes
across different games.
10
We implemented random reassignment of the roles across rounds instead of fixing the role. Again,
this may make learning the game slower for players given that their roles change, as Guerci et al. (2014)
suggest. However, given the existence of the null player in one of the four games considered, we chosereassignment of the role to avoid participants feeling the experiment was unfair.
11
Participants received a copy of instruction slides, and a prerecorded instruction video was played.
The quiz was given on the screen after the explanation of the game. The user interface was explained

23were 2,650 JPY for Winter and 2,850 JPY for H–MC.
We first compare the Winter and H–MC mechanisms in terms of the frequency of
grand coalition formation and efficiency. Then, we analyze whether the resulting allocations from the two mechanisms match the Shapley values. We contrast the experimental
results with the allocation predicted under the SPNE as well as under an equal division.
Additional analyses of our experimental results are presented in the Appendix II to IV.

6.1

Grand coalition formation and efficiency

Figure 2 presents the results concerning the grand coalition formation under the H–MC
and Winter mechanisms for the four games.12
We observe that for game 2 and the Winter mechanism, the grand coalition neverforms (because player 1 is a null player and, consequently, the game is only convex andnot strictly convex13 ). Therefore, for game 2, we consider the partition {{1}, {2, 3, 4}}
as a realization of the grand coalition for both the H–MC and Winter mechanisms.
Considering the four games together, the grand coalition (in the case of game 2,either the grand coalition or the {2, 3, 4} coalition) is formed in 61.9% of the cases underthe H–MC mechanism, but only in 40.1% of the cases under the Winter mechanism. In
particular, we observe that the grand coalition is formed more frequently under the H–
MC mechanism than under the Winter mechanism in games 3 and 4 at the 1% and 5%
during the practice rounds, referring to the handout about the computer screen. See Appendix VI for
English translations of the instruction materials and the comprehension quiz.
12
The figure is created based on the estimated coefficients of the following linear regressions: gci =β1 HM Ci + β2 W interi + µi where gci is a dummy variable that takes a value of 1 if the grand coalition
is formed, and zero otherwise, in group i, HM Ci (W interi ) is a dummy variable that takes a value of
1 if the H–MC (Winter) mechanism is used, and zero otherwise. The standard errors are corrected forwithin-session clustering effects. The statistical tests are based on the Wald test for the equality of theestimated coefficients of two treatment dummies.
13
Recall that the Winter mechanism is theoretically defined for strictly convex games. In this game,
Player 1 always has a zero marginal contribution and, as such, can be left out of any coalition at no costfor either him/her or the other players.

24

Figure 2: Proportion of times the grand coalition is formed(a) Game 1(b) Game 2n.s.

(b’) Game 2, allow (2,3,4)
n.s.

***

1
0.8
0.6
0.4
0.2
0

1
0.8
0.6
0.4
0.2
0
H–MC

1
0.8
0.6
0.4
0.2
0

WinterH–MC

Winter(c) Game 3

H–MC

Winter(d) Game 4

***

**

1
0.8
0.6
0.4
0.2
0

1
0.8
0.6
0.4
0.2
0
H–MC

WinterH–MC

WinterNote: The error bars show the one standard error range. The symbols ***, **, and * indicate the proportion of times that theformation of the grand coalition is significantly higher for the H–MC mechanism compared with the Winter mechanism at the 1%,5%, and 10% significance levels (Wald test), respectively.

significance levels, respectively.
As a direct consequence of the grand coalition being formed in less than 100% of
the cases, both mechanisms fail to achieve full efficiency. Efficiency is computed as thesum of the payoffs obtained by the four players as a proportion of the value of the grandcoalition of the considered game (100 for games 1 and 2, and 200 for games 3 and 4).
As Figure 3 shows, efficiency is significantly higher under the H–MC mechanism thanunder the Winter mechanism in games 1 and 4 (both at the 10% level).14
Therefore, we conclude as follows.
14

The figure is created based on the estimated coefficientsof the following linear regressions: Ef fi =Pi πi
β1 HM Ci + β2 W interi + µi , where Ef fi ≡ v(Nistheefficiency measure for group i, HM Ci)(W interi ) is a dummy variable that takes a value of 1 if the H–MC (Winter) mechanism is used, and zerootherwise. 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.

25

Figure 3: Efficiency(a) Game 1(b) Game 2
n.s.

*
1
0.8
0.6
0.4
0.2
0

1
0.8
0.6
0.4
0.2
0
H–MC

H–MC

Winter(c) Game 3

Winter(d) Game 4

n.s.

*
1
0.8
0.6
0.4
0.2
0

1
0.8
0.6
0.4
0.2
0
H–MC

H–MC

Winter

WinterNote: The error bars show the one standard error range. The symbols ***, **, and * indicate that the efficiency for the H–
MC mechanism is significantly higher than for the Winter mechanism, at the 1%, 5%, and 10% significance levels (Wald test),
respectively.

Result 1. Although the grand coalition is not always formed under the two mechanisms,
it is more frequently formed under the H–MC mechanism than under the Winter mechanism. Consequently, efficiency is higher under the H–MC mechanism than under theWinter mechanism.
Note that under the H–MC mechanism, the proposer is forced to offer feasible demands, that is, if S is the set of players remaining in the game, the proposer has topropose a total distribution of payoffs no larger than v(S). Conversely, under the Winter mechanism, the players, speaking one after the other, may make unfeasible demands.
As a result, the formation of a coalition under the H–MC mechanism is simply determined by whether the players choose to accept the proposal or reject it, whereas under
the Winter mechanism, the formation of the coalition can be blocked by unfeasibility

26conditions. Such a difference between the two mechanisms can cause the significantly
higher frequency of the grand coalition formation under the H–MC mechanism compared with the Winter mechanism.15

6.2

Allocations

We use π HM C (vk ) to denote a vector of payoffs obtained by the players in the H–MC
mechanism in game k, with k = 1, 2, 3, 4. Analogously, let π W (vk ) denote a vector of
payoffs obtained by the players under the Winter mechanism. The ex ante theoreticalprediction for both mechanisms states that the mean of such vectors (based on many realizations with different orderings of the players) should converge to the Shapley value.
When players fail to form the grand coalition, the total payoff obtained by the players
is smaller than the value under the grand coalition. As a result, the average realized
payoff vectors are significantly different from the Shapley value, as shown in Figure I.1of Appendix I. Therefore, we focus our analyses on those groups that formed the grandcoalition.16
Our main analyses are based on a set of ordinary least squares (OLS) regressions(using only the data from groups that formed the grand coalition) for the following
15

In Appendix II, we report the frequency of the grand coalition formation and efficiency by separating
the data for the first half (rounds 1–4) and the second half (rounds 5–8) of the experiment. We observe
an increase in both the frequency of the grand coalition formation and efficiency, at least in some of thegames, for both mechanisms. A significantly higher frequency of grand coalition formation and efficiency
is observed under the H–MC mechanism than under the Winter mechanism even in the second half of theexperiment.
16
In Appendix IV, we report the results based on all groups using payoff shares, instead of restricting
our attention to groups that formed the grand coalition.

27

Table 4: Results of linear regression based only on the groups that formed the grandcoalition

g1
g2
g3
g4
R2
Obs.

π1
23.44
(1.23)
13.00(1.98)
45.53(3.82)
47.10(2.46)
0.95
99

H–MC
π2
π3
25.48 25.00(0.50) (0.59)
25.8 28.08(0.63) (0.91)
48.68 50.53(0.65) (2.28)
51.53 50.03(1.02) (1.15)
0.98
0.98
99
99

π4
26.08
(1.29)
33.12(0.80)
55.26(3.19)
51.33(1.27)
0.98
99

g1
g2
g3
g4
R2
Obs.

Winterπ1π2π3π423.09 24.57 22.43 26.65(2.28) (1.11) (1.01) (3.11)
0.0
29.15 31.56 38.81(1.03) (0.54) (0.73)
21.00 52.67 57.33 68.00(3.57) (5.42) (4.11) (5.19)
44.24 48.14 45.57 56.86(3.22) (4.12) (8.58) (4.68)
0.90
0.93
0.91
0.95
77
77
77
77

Note: The standard errors are corrected for within-group clustering effects.

system of equations:
π1 = a1 g1 + a2 g2 + a3 g3 + a4 g4 + u1
π2 = b1 g1 + b2 g2 + b3 g3 + b4 g4 + u2(1)

π3 = c1 g1 + c2 g2 + c3 g3 + c4 g4 + u3
π4 = d1 g1 + d2 g2 + d3 g3 + d4 g4 + u4
where πi is the payoff of player i, gj is a dummy variable that takes a value of 1 if thegame j ∈ {1, 2, 3, 4} is played, and zero otherwise. Because participants play all fourgames twice, we correct the standard errors for within-group clustering effects. Notethat the estimated coefficients aj , bj , cj , and dj are the average payoffs in game j forplayers 1, 2, 3, and 4, respectively. Table 4 reports the results of these regressions,
showing the H–MC (Winter) mechanism in the left (right) panel.
Figure 4 shows the average payoffs obtained by each player in the four games con28

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

Game 260

60

Player 1

Player 2

Player 3

Player 4

Player 1***

Player 2

Game 3Player 3******

Winter***

WinterH–MC

WinterH–MC

0Winter

10

0
H–MC

10
Winter

20

H–MC

30

20

Winter

30

H–MC

40

H–MC

***

40

Winter***

Winter

**

H–MC

50
***

H–MC

50

Player 4

Game 4

120

120

100

100

Player 1

Player 2

Player 3

Player 4

Player 1

Player 2***

Player 3

Winter

**

H–MC

H–MC

WinterH–MC

0Winter

20

0
H–MC

40

20
Winter

40

H–MC

60

Winter

60

H–MC

80

Winter

***

80

H–MC

***

Winter

*

H–MC

***

Winter***Player 4

Note: The horizontal lines indicate the Shapley values. The error bars show the one standard error range. The symbols ***, **, and* indicate the frequency with which the average normalized payoff is significantly different from the Shapley values at the 1%, 5%,
and 10% significance levels (Wald test), respectively.

29

ditional on the grand coalition being formed. The horizontal lines indicate the Shapley
values for each game. It can be observed that for the Winter mechanism, the average
payoffs are not significantly different from the Shapley values for all four players ingames 1, 3, and 4. Conversely, for the H–MC mechanism, they are significantly different from the Shapley values for almost all players in all four games. This indicatesthat provided that the grand coalition is formed, the average payoffs under the Wintermechanism are closer to the Shapley values than those under the H–MC mechanism.
To compare the two mechanisms in terms of how close their average payoffs are to
the Shapley values, we employ the following measure:sX(πi − φi )2Dis2φ =

(2)i

where πi and φi are the average payoff and Shapley value for player i, respectively, inthe given game.
To conduct a statistical test, we employ a bootstrapping technique. For each iteration, we use a sub-sample (with replacement) of our data, and run the system of regressions (Eq. 1). Based on the obtained estimated coefficients (i.e., the average payoffs forthe sub-sample), we compute Dis2φ .
Figure 5 shows the result based on the outcomes of 1,000 repetitions. For all fourgames, Dis2φ is statistically significantly smaller under the Winter mechanism thanunder the H–MC mechanism.17
Result 2. Provided the grand coalition is formed, the average payoffs follow the Shapley
17

Based on two-sample t-test with unequal variance using the sample generated by the bootstrap. The
means Dis2φ (standard errors) for the H–MC mechanism are 5.13 (0.035) in game 1, 15.70 (0.064)
in game 2, 28.99 (0.074) in game 3, and 11.34 (0.047) in game 4. For the Winter mechanism, thecorresponding values are 4.99 (0.047) for game 1, 2.61 (0.030) for game 2, 10.60 (0.178) for game 3, and7.85 (0.094) for game 4.

30

Figure 5: Distance from Shapley
Game 140

Game 2
40

**

20

***

20

0

0
H–MC

H–MC

WinterGame 340

WinterGame 4
40

***

20

***

20

0

0
H–MC

WinterH–MC

WinterNote: The error bars show the one standard error range. The symbols ***, **, and * indicate the significant difference between the
H–MC and the Winter mechanisms at the 1%, 5%, and 10% significance levels (two-sample t-test), respectively.

values more closely under the Winter mechanism than under the H–MC mechanism.

6.3

Realized allocations and a posteriori equilibria

Now, let us analyze the realized payoffs in the light of the a posteriori equilibrium
payoff vectors. We continue to focus only on the groups that formed the grand coalition. We measure the distance between the realized payoff vectors and the allocation
under the SPNE for the four games by their Euclidean distance. Let eqi be the equilibrium payoff for player i for the given game, the realized order of the players (making
a proposal or demand), and the mechanism. The distance of the realized payoff frompP
2 18 We also consider thethe equilibrium is computed as Dis2N E =i (πi − eqi ) .
distance between the realized payoff vectors and equal division payoffs, defined bypP
2
Dis2ED =i (πi − EDi ) where EDi is the equal division payoff for player i for18

For the sake of simplicity, we omit the specifications about the considered mechanism and the game.

31

Figure 6: Mean of the distances of the realized payoff vectors from the SPNE and theequal division(a) Game 1
40

***

***

(b) Game 2

***

n.s.
***

20

***

n.s.

40

***

20

0

0
H–MC

WinterH–MC

Dis2N E

Winter

H–MC

Dis2ED

Winter(c) Game 3
**

ns

H–MC

Winter

Dis2N E

Winter

Dis2ED(d) Game 4

n.s.

100
80
60
40
20
0

H–MC

Dis2N E

100
80
60
40
20
0

***

H–MC

Winter***

H–MC

Dis2ED***

Winter

Dis2N E

***
***

H–MC

Winter

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 equilibrium allocations or from the equal division was significantly different between the H–MC and theWinter implementation, at the 1%, 5%, and 10% significance levels (Wald test), respectively.

the given game.
Figure 6 shows the mean Dis2N E and the mean Dis2ED for the two mechanisms
in the four games.19 We observe that the distance to the equal division is significantly
smaller (at the 1% level) for the H–MC mechanism than for the Winter mechanism in all
four games. This may not be surprising because, as Observation 1 states, the a posteriori equilibrium payoff vectors tend to be less unequal under the H–MC mechanism thanunder the Winter mechanism. In fact, as we can observe, the distance to the equilibrium
allocation is significantly smaller for the H–MC mechanism than for the Winter mecha19

The figure is created based on the estimated coefficients of the following linear regressions: Disi =β1 HM Ci +β2 W interi +µi , where Disi is the relevant distance measure for group i, HM Ci (W interi )is a dummy variable that takes a value of 1 if the H–MC (Winter) mechanism is used, and zero 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.

32

nism in games 1 and 4 (in which the equilibrium payoffs are less unequal than in games
2 and 3) at the 1% level. For games 2 and 3, however, the distance to the equilibrium
allocations is not significantly different between the two mechanisms.
Figure 6 shows that, on the one hand, the payoff vectors realized under the H–MC
mechanism are significantly closer to the equal division than to the equilibrium ones inall but game 2 (in which Dis2N E and Dis2ED are not significantly different). On theother hand, under the Winter mechanism, the realized payoff vectors are significantly
closer to the equal division than to the equilibrium ones only in games 1 and 4, but the
opposite is the case for game 2. In game 3, Dis2N E and Dis2ED are not significantly
different under the Winter mechanism.
Result 3. The H–MC mechanism more often results in payoffs that are closer to theequal division than to the equilibrium payoffs compared with the Winter mechanism.
This indicates that, albeit imperfectly, the Winter mechanism achieves the allocationthat better reflects the power of the players than does the H–MC mechanism.

7

Conclusions

We have experimentally compared two of the best-known bargaining procedures in the
Nash program, the H–MC and the Winter mechanisms. Our main rationale for thischoice is simplicity, which is a key desideratum when considering possible applicability
to the real world. These two mechanisms are simple and similar in their implementation,
making them suitable for a direct comparison. They differ mainly in the way that they
implement bargaining, as the H–MC mechanism is based on offers, and the Winter on
demands.

33

Previous studies have found a certain closeness of the experimental results when
making a similar comparison (see Fréchette et al., 2005a), despite the sharply different
theoretical predictions. Our findings partially contradict these results, showing how twovery similar mechanisms can behave differently, despite the close theoretical predictions. In particular, the H–MC mechanism results in higher frequencies of the grandcoalition formation and, consequently, higher efficiency than the Winter mechanism.
We suggest that the H–MC mechanism is better suited to bargaining over cost or payoff
allocation problems when the main target is efficiency, or when full cooperation represents a crucial goal for society (e.g., full cooperation in the airport problem (Littlechildand Owen, 1973) results in one single airport being built instead of many, and thisis certainly desirable for environmental reasons). Conversely, provided that the grandcoalition is formed, the Winter mechanism results in average payoffs that are closer tothe Shapley values and better satisfy various axioms. We suggest that the Winter mechanism is best suited to allocation problems in which it is important to value players’
effective power (e.g., production games (Owen, 1975), or in which arguments such associal welfare and symmetry are inescapable (e.g., allocation of resources in health orsocial care (Kluge, 2007)).
Our findings suggest that when facing a cost or payoff allocation problem, the choiceof which bargaining procedure to implement, one based on offers or on demands, may
have some unexpected effects, regardless of the theoretical prediction. This should betaken into account when making such a choice in various applications. In fact, different
bargaining mechanisms, even when equivalent from the theoretical point of view, favordifferent properties that are reflected in the resulting allocations. An example of such
effects may be found in the verification of the null player property of the Shapley value
in Appendix III. Theoretically, a player who always has a zero marginal contribution
34

should receive a zero payoff, according to Shapley. According to the theoretical prediction, in a demand-based mechanism, nonnull players should systematically refuse astrictly positive demand by a null player. However, we find that nonnull players seem tobe uncomfortable with making a zero offer to a null player in an offer-based mechanism,and this contributes to a final payoff share that is closer to the equal division solution.
A deeper analysis of how different mechanisms can lead players toward respecting or
violating some properties would be a fruitful direction for future research.
Many potentially important complementary questions can be addressed in future
research. Among others, an analysis of the more complex versions of our proposedmechanisms (e.g., the Winter mechanism with more periods and a discount factor, or
the H–MC mechanism where the proposer whose offer is refused then leaves the gamewith a probability strictly smaller than one) can be compared with our actual results.
Comparing the outcomes of the experiments based on noncooperative mechanisms withthose of unstructured bargaining experiments would be an interesting topic for future
research.

Acknowledgments
The experiments reported in this paper were approved by the Institutional Review Board(IRB) of Yamaguchi University (No. 5, for the experiments conducted in 2019) as
well as by the IRB of ISER, Osaka University (for the experiment conducted in 2022).
We gratefully acknowledge financial support from the Joint Usage/Research Center at
ISER, Osaka University, and Grants-in-Aid for Scientific Research, Japan Society for
the Promotion of Science (15K01180, 15H05728, 18K19954, 20H05631), the Fund for
the Promotion of Joint International Research (Fostering Joint International Research)
35

(15KK0123), and the French government-managed l’Agence Nationale de la Recherche(les Programmes d’Investissements d’Avenir U CAJEDI ) (ANR-15-IDEX-01). In particular, we thank the UCAinACTION project. We thank Guillaume Fréchette and participants of the seminar at the University of Technology Sydney, Monash University,
Osaka University, Tokyo University, and Keio University, as well as participants of ASFEE 2019 (Toulouse, France), Game Theory Workshop 2020 (Tokyo, Japan), SEET
2020 (Naples, Italy), SAET 2021 (Seoul, Korea), and ESA Global 2021 (online) for
comments and suggestions.

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43

Appendix
I

The average payoffs

Figure I.1 shows the mean realized payoffs based on all groups in each of the fourgames, and the horizontal lines indicate the Shapley values for each game.20
Figure I.1: Mean payoffs, all groups

Player 3

Player 4

Player 1

Player 1

Player 2

Player 3

Player 4

Player 1

Player 2***

WinterPlayer 4

WinterPlayer 3******

Winter

**

H–MC

*

H–MC

***

H–MC

***

Winter

***

120
100
80
60
40
20
0
H–MC

***

Winter

***

H–MC

***

Winter***Player 3***

Game 4

H–MC***

Winter

Winter

H–MC

***

H–MC

Game 3
120
100
80
60
40
20
0

***

H–MC

WinterPlayer 2***

Winter

**

H–MC

***

H–MC

Winter***

Winter

***

H–MC

***

H–MC

***

60
50
40
30
20
10
0

WinterPlayer 2***

Winter*Game 2

H–MC

WinterPlayer 1***

Winter

***

H–MC

***

H–MC

Game 1
60
50
40
30
20
10
0

Player 4

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

20

As in Figure 4, the mean and the standard errors are obtained by running the system of linear regressions that take each player’s payoff as the dependent variables and four game dummy variables without
the constant. The standard errors are corrected for session-level clustering effects. The statistical tests are
based on these regressions.

1II

Effect of learning and bargaining dynamics

II.1

Grand coalition formation and efficiency

We have already shown in Section 3 that both mechanisms fail to achive an efficient
outcome. However, H–MC mechanism performs significantly better in this matter. A
possible explanation is because, as we have already observed in Section 3 and with
Result 1, H–MC mechanism forces feasible offers, while Winter mechanism allows for
unfeasible demands which, as a result, lead to inefficiencies. This also naturally leads
to the fact that the grand coalition is formed more often under the H–MC mechanism,
than under the Winter mechanism.
One may hypothesize that this generalized failure (more for Winter, but partially alsofor H-MC) in reaching an efficient outcome can explained by some limited rationality
arguments: even if we chose two mechansims that are in our opinion simple, the games’
optimal dynamics is hard to understand for participants to the experiment especially inthe beginning.
We check this hypothesis by investigating the presence of a learning effect by comparing the outcomes in the first half of four rounds (1-4) and the second half of four
rounds (5-8). Because the number of groups that formed a grand coalition becomes

Table II.1: Number of groups with Grand Coalition
game 1Winter early
10Winter late13
H–MC early
10
H–MC late15

game 210
17
12
13

game 31
5
11
8

Note: game 2 allows {2, 3, 4} to be the grand coalition.

2

game 4912
17
13

Figure II.2: Proportion of times the grand coalition formed in early and late rounds{2, 3, 4} allowed

Game 1
1

n.s.

**

0.8

n.s.

1
n.s.

0.8

0.6

0.6

0.4

0.4

0.2

0.2

0

n.s.

n.s.

n.s.

0
Early

Late

Early

H–MC

Late

Early

Winter

Late

1

***

n.s.

Early

H–MC

Game 3

0.8

n.s.

Late

WinterGame 4

***

1
***

**

n.s.

*
n.s.

0.8

0.6

0.6

0.4

0.4

0.2

0.2

0

0
Early

Late

H–MC

Early

Late

Early

Winter

Late

H–MC

Early

Late

Wintersmall if we separate the data into the first half and second half (see Table II.1), we
do not report the results of the analyses comparing the distance between the averagerealized payoff vectors and the Shapley values, or whether realized payoff vectors satisfy various axioms characterizing the Shapley value. Instead, we investigate only thefrequency of grand coalition formation and efficiency.
Figures II.2 and II.3 show the frequency of the grand coalition formation and theaverage efficiency (i.e., the average total payoff / value of the grand coalition) for the
first half and the second half (i.e., the first four rounds vs. the second four rounds) ofeach game. For H–MC, the frequency of the grand coalition formation and the average

3

Figure II.3: Efficiency in early and late rounds
Game 1
1

*

n.s.

***

Game 2
n.s.

1

0.8

0.8

0.6

0.6

0.4

0.4

0.2

0.2

0

n.s.

n.s.

n.s.

0
Early

Late

Late

Early

H–MC

Early

Winter

Late

n.s.

n.s.

Early

H–MC

Game 3
1

n.s.

Late

WinterGame 4

***
**

1

0.8

0.8

0.6

0.6

0.4

0.4

0.2

0.2

0

*

n.s.

n.s.
n.s.

0
Early

Late

H–MC

Early

Late

Early

Winter

Late

H–MC

Early

Late

Winterefficiency are significantly higher in the later rounds only in game 1; for the remaining
three games, there are no significant differences between the early and late rounds. For
the Winter mechanism, both the frequency of the grand coalition formation and theaverage efficiency are significantly higher in the later rounds only in game 3, with nosignificant differences in other games.
As conclusion, we report no statistically significant learning effect, when implementing either the H–MC or the the Winter mechanism. This does not rule our the
possibility that by implementing a higher number of repetitions, a significant learning
effect could be observed.

4

II.2

Departure from the Shapley payoff share

Here, we investigate some possible dynamics that could lead the experimental resultsof H–MC mechanism to be closer to the equal division solution (Result 3), when compared to the experimental results of the Winter mechanism that are instead closer to the
Shapley value (Result 2).
In Section 6, we have already shown that accepted proposals in the H–MC mechanism go in the sense of equal division. This result is not surprising, as experimental
results of offer-based mechanisms (such as the well-known two-player bargaining over apie of Rubinstein (1982)) often show that, contrary to the theoretical prediction, players
tend to go for an equal split of the pie. We show that, in our experiment, this behavior
becomes more evident after a first rejection of a proposal, as second proposals are closer
to the equal share than the first ones.
In Figure II.2, we show the distance from equal division, devEQL =P

i |aL,i −ED|

where aL,i is the proposed allocation for player i in Lth proposal (for a group) and ED
is the equal division payoff for the game, for the first (L = 1, horizontal axis) and thesecond (vertical axis) proposals (top) and the second (L = 2, horizontal axis) and thethird (vertical axis) proposals (bottom) for each game. Each dot corresponds to a pairof the proposals of a group.
We observe a clear tendency for either the second proposal to be more equal than thefirst one (devEQ1 > devEQ2 ) or the second proposal to be more equal than the third
one (devEQ2 > devEQ3 ) depending on the game.

5

Figure II.4: H–MC mechanism: Distance from equal division for the first and the second
proposals (top) and the second and the third proposals (bottom)
Game 1

Game 2

Game 3

Game 4������

������

������

������

���

���

���

���

���

���

���

���

���

���

���

���

��

��

��

��

��

��

��

��

��

��

��

��

��

��

��

�� �� �� �� ��� ��� ����������� �� �� �� ��� ��� ���

n=15
p=0.003������

���� �� �� �� ��� ��� ���

n=32
p=0.511�������� �� �� �� ��� ��� ���

n=17
p=0.246n=12
p=0.045

������

������

������

������

���

���

���

���

���

���

���

���

���

���

���

���

��

��

��

��

��

��

��

��

��

��

��

��

��

��

��

�� �� �� �� ��� ��� ���

n=7
p=0.798�������� �� �� �� ��� ��� ���������

���� �� �� �� ��� ��� ���

n=19
p=0.015������

n=6
p=0.036�������� �� �� �� ��� ��� ���������

n=3
p=0.181Note: In each panel, only those groups in which the first proposal (top) or the second proposal (bottom) is rejected are plotted.
p-values are based on the Signed-Rank test (two-tailed) with the null hypothesis devEQL = devEQL+1 .

6

III

Testing for the axioms of the Shapley value

We test the axioms that are historically the most relevant to characterizing the Shapley
value. In doing so, we aim to provide greater insight into whether a demand-based
bargaining mechanism is more appropriate than an offer-based bargaining mechanism
for cost or payoff allocation problems when the allocation scheme is constructed on themain axiomatic solution notion of cooperative game theory, that is, the Shapley value.
First, we provide two definitions, which are used in the following.
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 .
In the literature, we find various axiomatic characterizations of cooperative solutions
and, in particular, of the Shapley value. Given a solution ψ : G N → RN , we list someof the most commonly used axioms to provide a characterization.
Axiom 1 (Efficiency): for every v in G N ,Pi∈N ψi (v) = v(N ).

Axiom 2 (Symmetry): if i and j are symmetric players in game v ∈ G N , thenψi (v) = ψj (v).
Axiom 3 (Additivity): for all v, w ∈ G N , ψ(v + w) = ψ(v) + ψ(w).
Axiom 4 (Homogeneity): for all v ∈ G N and a ∈ R, ψ(av) = aψ(v).
Axiom 5 (Null player property): if i is a null player in game v ∈ G N , thenψi (v) = 0.
Axiom 6 (Strong monotonicity): 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).

7

Axiom 7 (Fairness): 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 .
Fairness states that if we add a game w ∈ G N , in which players i and j aresymmetric, to a game v ∈ G N , then the payoffs of players i and j change by thesame amount.
In particular, among many others, the axiomatization of Shapley (1953), which is themost classical one, involves axioms 1, 2, 3, and 5. The axiomatization of Young (1985)
involves axioms 1, 2, and 6, whereas that of van den Brink (2002) involves axioms 1, 5,
and 7. Note that axiom 4, even if not directly involved in any of these axiomatizations,is crucial because, together with axiom 3, it guarantees the linearity of the solution.21
We noted in Section 6 that both mechanisms fail to satisfy efficiency (axiom 1) if we
examine overall data. Here, we examine the remaining six axioms. These axioms aretested based on the estimated coefficients obtained from running the regression of Eq. 1
as follows.
• Symmetry (axiom 2) requires b1 = c1 and b4 = c4 .
• Additivity (axiom 3) and homogeneity (axiom 4) require that x3 = x1 + x2 andx4 = 2x1 for x ∈ {a, b, c, d}, respectively.
• Null player property (axiom 5) requires that a2 = 0.
• Strong monotonicity (axiom 6) requires that a1 > a2 and a4 > a3 .
21

The equal division solution satisfies 1, 2, and 3, but does not satisfy the null player property in 5.
However, it satisfies a similar property when null players are replaced with nullifying players. Player i
is a nullifying player if v(S) = 0 for each S ⊆ N such that i ∈ S. Then, we can state the following
additional axiom that can be called the nullifying player property: if i is a nullifying player in gamev ∈ G N , then ψi (v) = 0. Replacement of the null player property in the axiomatization of the Shapley
value in Shapley (1953) with the nullifying player property characterizes the equal division solution (seevan den Brink, 2006).

8

Table III.2: Results of Wald tests for the verification of the symmetry, additivity, homogeneity, strong monotonicity, and fairness axioms (based only on the groups that formed
a grand coalition)

H–MC
Axiom

H0

a2 = a3d2 = d3
Additivity
c 1 = a1 + b 1
c 2 = a2 + b 2
c 3 = a2 + b 3
c 4 = a4 + b 4
Homogeneityd1 = 2a1
d2 = 2a2
d3 = 2a3
d4 = 2a4
Null player
a2 = 0Strong monotonicity
a1 = b 1(H0 should be rejected)
c1 = d 1
Fairnessb3 − b2 = c 3 − c 2
Symmetry

Winterχ2p-valueχ2p-value

0.35
1.60
6.69
3.23
2.16
1.52
0.00
0.08
0.00
0.06
42.91
10.76
0.16
0.62

0.552
0.206
0.001
0.072
0.142
0.218
0.946
0.772
0.983
0.813
0.000
0.001
0.692
0.433

1.85
0.06
0.13
0.02
0.47
0.78
0.10
0.11
0.00
0.82
.
102.24
26,84
0.74

0.174
0.811
0.721
0.878
0.492
0.376
0.749
0.745
0.947
0.365
.
0.000
0.000
0.391

• Fairness (axiom 7) requires that b3 − b2 = c3 − c2 .
In Table III.2, we present the results of the Wald tests for the verification of these axioms,
together with the null hypothesis (H0 ).
Note that the symmetry (according to which H0 should not be rejected) is confirmed
for the two cases under both the Winter and the H–MC mechanisms. The additivity(according to which H0 should not be rejected) is always confirmed under the Wintermechanism, but is confirmed in only two of four cases under the H–MC mechanism.
The homogeneity (according to which H0 should not be rejected) is always confirmed
for both mechanisms. The null player property (according to which H0 should not be

9Table III.3: Tests of axioms (based only on the groups that formed a grand coalition)
Axiom
Symmetry
Additivity
Homogeneity
Null player property
Strong monotonicity
Fairness

H–MC

Winter

+
+
+

+
+
+
+
+
+

+ indicates that the axiom is considered to be satisfied on average. − indicates the opposite.

rejected) is not confirmed in the H–MC mechanism, but it is confirmed (respected 100%of the time) for the Winter mechanism. The strong monotonicity (according to which
H0 should be rejected) is confirmed for the Winter mechanism but only for half of thetime for the H–MC mechanism. The fairness (according to which H0 should not berejected) is confirmed for both mechanisms.
Let us consider that the axiom is satisfied on average if it is confirmed for strictly
more than half of the cases being tested. Table III.3 summarizes whether each axiom issatisfied on average (+) or not (-) for two mechanisms. We can state the following.
Result 4. Provided the grand coalition is formed, the Winter mechanism better satisfies
axioms that characterize the Shapley value than the H–MC mechanism.

10

IV

Additional results based on payoff shares

In this section, we report the results based on all the groups using the payoff shareinstead of restricting our attention to those that formed the grand coalition. Payoff sharesare defined as πei W (vk ) = P

π H−−M C (vk )πiW (vk )H−−M CP i H−−M C
×v
(N
)
and
π
e
(v
)
=
×
k
i
k
W
(vk )
j∈N πj (vk )j∈N πjvk (N ) for each i = 1, 2, 3, 4.
As in the main text, our analyses are based on running a set of OLS regressions
shown by Eq. 1 but using payoff shares as dependent variables. Table IV.4 shows theresults of the regression.
Table IV.4: Results of linear regression for normalized payoffs

g1
g2
g3
g4
R2
Obs.

πe1
22.98
(0.96)
8.91(1.27)
28.11(4.25)
47.55(1.48)
0.81
160

H–MC
πe2
πe3
24.29 25.01(0.44) (0.67)
26.38 28.51(0.38) (0.82)
50.92 53.45(0.60) (1.42)
48.39 48.78(2.22) (0.92)
0.95
0.95
160
160

πe4
27.72
(0.73)
36.21(0.42)
67.52(3.27)
55.28(2.10)
0.95
160

g1
g2
g3
g4
R2
Obs.

Winter
πe1
πe2
πe3
πe4
21.99 21.91 23.39 32.80(2.78) (3.80) (2.12) (2.09)
0.0
28.70 30.75 40.55(0.61) (0.44) (0.51)
8.95 54.09 61.52 75.55(2.16) (3.86) (2.84) (4.57)
42.18 48.13 45.57 64.13(3.15) (2.34) (5.14) (4.51)
0.73
0.90
0.92
0.93
192
192
192
192

Figure IV.5 shows the mean of the normalized payoffs in the four games, where thehorizontal lines indicate the Shapley values for each game. It can be observed that for
the Winter mechanism, the average normalized payoffs are not significantly different
from the Shapley values for all four players in games 1, 2, and 4. However, for the
H–MC mechanism, the average normalized payoffs for all four players do not respect
the Shapley values in any of the games at the 10% significance level.
11Figure IV.5: Mean of the normalized payoffs
Game 1

Game 260

60
50Player 1

Player 2

Player 3

Player 4

Player 1

Player 2

Game 3Player 3

Winter

***

H–MC

***

WinterH–MC

WinterH–MC

Winter

0
H–MC

10

0
Winter

10
H–MC

20

Winter

30

20

H–MC

30

***

H–MC

***

40

Winter**

Winter

*

40

H–MC

50

Player 4

Game 4

120

120
100

Player 1

Player 2

Player 3

Player 4

Player 1

Player 2

Player 3

Winter

WinterH–MC

WinterH–MC

Winter

0
H–MC

20

0
Winter

20
H–MC

40

Winter

60

40

H–MC

60

***

H–MC

**

80

Winter

**

H–MC

*

Winter

***

80

H–MC

100

Player 4

Note: The horizontal lines indicate the Shapley values. The error bars show the one standard error range. The symbols ***, **, and* indicate the frequency with which the average normalized payoff is significantly different from the Shapley values at the 1%, 5%,
and 10% significance levels, respectively (Wald test).

Figure IV.6 shows the mean Dis2φ (based on the normalized payoff). As in themain text, we use a bootstrapping technique with 1,000 repetitions to create the figure
and conduct the statistical tests. In contrast to the analyses restricted to the groups thatformed a grand coalition, we now observe that Dis2φ is significantly smaller for the
H–MC mechanism than for the Winter mechanism in games 1 and 3. For the other twogames, as before, Dis2φ is significantly smaller for the Winter mechanism than for the12

Figure IV.6: Distance of the normalized payoffs from Shapley
Game 140

Game 2
40

***

20

***

20

0

0
H–MC

H–MC

WinterGame 340

WinterGame 4
40

***

20

***

20

0

0
H–MC

WinterH–MC

WinterNote: The error bars show the one standard error range. The symbols ***, **, and * indicate the significant difference between the
H–MC and Winter mechanisms at the 1%, 5%, and 10% significance levels, respectively (two-sample t-test).

H–MC mechanism.22
Tables IV.5 and IV.6 summarize the results of testing the six axioms. Based on
the normalized payoff, on average, the symmetry, strong monotonicity, and fairness
axioms are now satisfied under the H–MC mechanism. For the Winter mechanism, withnormalized payoffs, the fairness axiom is no longer satisfied.
Thus, if we consider all the groups and normalized payoffs, the Winter and H–
MC mechanisms are comparable in terms of their distance to the Shapley value andsatisfaction of its properties.
Figure IV.7 shows the mean Dis2N E and the mean Dis2ED for the two mechanisms
in the four games computed based on the normalized payoffs using all the groups. The
22

The mean Dis2φ (standard error) values based on the normalized payoff for the H–MC mechanism
are 3.73 (0.034) in game 1, 10.52 (0.056) in game 2, 7.78 (0.120) in game 3, and 7.58 (0.638) in game 4.
For the Winter mechanism, the corresponding values are 4.11 (0.048) in game 1, 0.96 (0.016) in game 2,
16.07 (0.077) in game 3, and 6.91 (0.091) in game 4.

13

Table IV.5: Wald tests for the verification of the symmetry, additivity, homogeneity,strong monotonicity, and fairness axioms for normalized payoffs

H–MC
H0

Axiom

a2 = a3d2 = d3
Additivity
c 1 = a1 + b 1
c 2 = a2 + b 2
c 3 = a2 + b 3
c 4 = a4 + b 4
Homogeneityd1 = 2a1
d2 = 2a2
d3 = 2a3
d4 = 2a4
Null player
a2 = 0Strong monotonicity
a1 = b 1(H0 should be rejected)
c1 = d 1
Fairnessb3 − b2 = c 3 − c 2
Symmetry

Winterχ2p-valueχ2p-value

0.53
0.03
0.99
0.07
0.00
0.92
2.48
0.01
0.31
0.00
49.51
46.26
14.57
0.58

0.466
0.869
0.319
0.790
0.952
0.336
0.115
0.926
0.580
0.963
0.000
0.000
0.001
0.447

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

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

Table IV.6: Tests of axioms for normalized payoffs
Axiom
Symmetry
Additivity
Homogeneity
Null player property
Strong monotonicity
Fairness

H–MC

Winter

+
+
+
+
+

+
+
+
+
+
-

+ indicates that the axiom is considered to be satisfied on average. − indicates the opposite.

normalized payoffs under the H–MC mechanism are significantly closer to the equal
division than those under the Winter mechanism in all four games. Furthermore, thoseunder H–MC are significantly closer to the equilibrium payoffs in games 1 and 4 than

14

Figure IV.7: Mean of the distances of the normalized payoff vectors from the SPNE andthe equal division(a) Game 1
***

*

40

(b) Game 2

***

n.s.
***

20

***

*

40

***

20

0

0
H–MC

WinterH–MC

Dis2N E

Winter

H–MC

Dis2ED

Winter(c) Game 3
100
80
60
40
20
0

n.s.

n.s.

H–MC

Dis2N E

Winter

Dis2ED(d) Game 4

***

100
80
60
40
20
0

***

WinterH–MC

Dis2N E

H–MC

Winter***

H–MC

Dis2ED***

Winter

Dis2N E

***
***

H–MC

Winter

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 equilibrium allocations or from the equal division was significantly different between the H–MC and theWinter mechanism at the 1%, 5%, and 10% significance levels, respectively (Wald test).

those under the Winter mechanism. However, in these games, for both the Winter and
H–MC mechanisms, normalized payoffs are significantly closer to the equal division
than to the equilibrium payoffs. For games 2 and 3, the normalized payoffs under theWinter mechanism are significantly closer to the equilibrium than to the equal division.
Under H–MC, Dis2N E and Dis2ED are not significantly different in games 2 and 3.

15

V

Comparison of a classical H–MC sequential approvalmechanism vs. a pseudo-H–MC simultaneous approvalmechanism

The comparison between sequential mechanisms and simultaneous ones in favoring theformation of efficient coalitions has been the object of recent experimental laboratory
studies (Abe et al., 2021). Experimental evidence shows that subjects may perform very
differently in these two proposed settings. Analogously, we propose a comparison between the performances of the H–MC mechanism and a pseudo-H–MC mechanism (inthe following, denoted as H–MCsim ), whose structure is identical to that of the original
mechanism except that after an offer is proposed, players are asked to either accept orrefuse the proposal simultaneously. Theoretically, the H–MCsim mechanism allows formany more Nash equilibria in which two or more players refuse the proposal. We showthat sometimes, as observed by Fréchette et al. (2005b), bargaining behavior is not as
sensitive to the different bargaining rules as the theory suggests.
The H–MCsim experiment was conducted in December 2019 at ISER at Osaka University. In total, 84 participants, who had never participated in similar experiments
before, were recruited. The experimental procedure was identical to the H–MC experiment reported in the main text. On average, the experiment lasted for 1 hour 30
minutes, including the instructions (11 minutes), a comprehension quiz (5 minutes),and payment. The average earnings were 2,780 JPY.

16

V.1

Grand coalition formation and efficiency

Figures V.8 and V.9 report the results concerning the grand coalition formation andefficiency. The only significant differences reported are for game 2.
Figure V.8: H–MC and H–MCsim mechanisms, proportion of times the grand coalition
is formed(a) Game 1(b) Game 2n.s.

(b’) Game 2, allow (2,3,4)

*

1
0.8
0.6
0.4
0.2
0

***

1
0.8
0.6
0.4
0.2
0
H–MC

H–MCsim

1
0.8
0.6
0.4
0.2
0
H–MC

H–MCsim(c) Game 3

H–MC

H–MCsim(d) Game 4

n.s.

n.s.

1
0.8
0.6
0.4
0.2
0

1
0.8
0.6
0.4
0.2
0
H–MC

H–MCsim

H–MC

H–MCsim

Note: The error bars show the one standard error range. The symbols ***, **, and * indicate the proportion of times the grandcoalition was formed, which was significantly higher for the H–MC implementation compared with the H–MCsim implementation
at the 1%, 5%, and 10% significance levels (Wald test), respectively.

17

Figure V.9: H–MC and H–MCsim mechanisms, efficiency(a) Game 1(b) Game 2

n.s.

***

1
0.8
0.6
0.4
0.2
0

1
0.8
0.6
0.4
0.2
0
H–MC

H–MCsim

H–MC(c) Game 3

Winter(d) Game 4

n.s.

n.s.

1
0.8
0.6
0.4
0.2
0

1
0.8
0.6
0.4
0.2
0
H–MC

WinterH–MC

WinterNote: The error bars show the one standard error range. The symbols ***, **, and * indicate the efficiency for the H–MC mechanism
is significantly higher than for the H–MCsim implementation at the 1%, 5%, and 10% significance levels (Wald test), respectively.

18

V.2

Analyses based only on groups that formed the grand coalition

Table V.7 reports the results of running a set of OLS regressions as in Eq. 1 based ongroups that formed the grand coalition.
Table V.7: H–MCsim mechanism, results of the linear regression based only on thegroups that formed the grand coalition
H–MCsim
π1
π2
π3
π4
g1
23.88 25.63 24.56 25.93(0.70) (0.43) (0.11) (0.42)
g2
11.07 26.07 27.73 35.13(3.31) (1.45) (0.97) (1.95)
g345.88 51.32 49.72 53.08(1.70) (1.19) (0.50) (1.84)
g4
47.83 48.67 50.5 53.00(1.17) (0.89) (0.87) (1.10)
2
R
0.96
0.99
0.99
0.97
Obs.
97
97
97
97
Note: The standard errors are corrected for within-group clustering effects.

Based on the estimated coefficients, Figure V.10 shows the average payoffs obtainedby each player in the four games, conditional on the grand coalition being formed.
The horizontal lines indicate the Shapley values for each game. We observe that thetwo mechanisms perform similarly in that there are players whose average payoff issignificantly different from the Shapley value in all four games under both mechanisms.

19

Figure V.10: H–MC and H–MCsim mechanisms, mean payoffs based only on the groups
that formed the grand coalition
Game 1

Game 260

60

Player 1

Player 2

Player 3

H–MCsim0
H–MC

0
H–MCsim

10
H–MC

20

10
H–MCsim

20

H–MC

30

H–MCsim

30

H–MC

40

***

***

Player 1

Player 4***

Player 2

Game 3******Player 3

***

***

H–MC

***

H–MCsim***

H–MCsim

***

H–MC

**

H–MCsim

***

H–MC

***

H–MCsim

50
***

40

H–MC

50

Player 4

Game 4

120

120
***

***

Player 1

Player 2

Player 3

H–MC

H–MCsim

H–MC

0
H–MCsim

20

0
H–MC

20
H–MCsim

40

H–MC

60

40

H–MCsim

60

H–MC

80

Player 4

Player 1***

Player 2*****

Player 3******

H–MCsim

***

H–MC

***

H–MCsim*

H–MCsim***

H–MCsim

***

H–MC

100
***

80

H–MC

100

Player 4

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

20V.2.1Distance between the ex ante payoffs and the Shapley value

We compute the distance of the average payoffs from the Shapley value, Dis2φ , using
the bootstrapping technique with 1,000 repetitions as we have done in the main text
comparing the H–MC and Winter mechanisms. The result is reported in Figure V.11. It
can be observed that the ex ante payoffs of H–MCsim are closer to the Shapley values
in games 2 and 4, whereas those of H–MC are closer to the Shapley values in games 1
and 3.23
Figure V.11: H–MC and H–MCsim mechanisms, distance from Shapley
Game 140

Game 2
40

***

20

***

20

0

0
H–MC

H–MCsim

H–MC

Game 340

H–MCsimGame 4
40

***

20

***

20

0

0
H–MC

H–MCsim

H–MC

H–MCsim

Note: The error bars show the one standard error range. The symbols ***, **, and * indicate significant differences between the
H–MC and the H–MCsim mechanisms at the 1%, 5%, and 10% significance levels (two-sample t-test), respectively.

23

The mean Dis2φ (standard error) values for the H–MC mechanism are 5.14 (0.035) in game 1, 15.70(0.064) for game 2, 28.99 (0.074) for game 3, and 11.34 (0.047) in game 4. The corresponding valuesfor H–MCsim are 5.28 (0.018), in game 1, 13.09 (0.088) for game 2, 30.55 (0.091) for game 3, and 9.26
(0.057) in game 4.

21

V.2.2

Realized allocations and a posteriori equilibria

In terms of distance from SPNE or equal division, we observe from Figure V.12 that H–
MCsim results in outcomes significantly closer to equal division compared with H–MC
only in game 1, whereas in the other games there is no significant difference.
Figure V.12: H–MC and H–MCsim mechanisms, mean of the distances of the realized
payoff vectors from the SPNE and the equal division(a) Game 1
40

n.s.

***

(b) Game 2

***

n.s.
*

40

20

n.s.

20

0

0
H–MC H–MCsim

H–MC H–MCsim

H–MC H–MCsim

H–MC H–MCsim

Dis2N E

Dis2ED

Dis2N E

Dis2ED(c) Game 3
100
80
60
40
20
0

n.s.

n.s.

(d) Game 4

***
n.s.

**

100
80
60
40
20
0

n.s.

n.s.

***

***
n.s.

H–MC H–MCsim

H–MC H–MCsim

H–MC H–MCsim

H–MC H–MCsim

Dis2N E

Dis2ED

Dis2N E

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 equilibrium allocations or from the equal division was significantly different between the H–MC and the
H–MCsim implementations at the 1%, 5%, and 10% significance levels, respectively (Wald test).

22V.2.3

AxiomsFinally, verification of the axioms (comparing Table V.8 and the left column of Tables III.2 and III.3) indicates that the differences in results between H–MC and H–
MCsim are observed for symmetry and fairness (satisfied in H–MC but not in H–MCsim ).
Table V.8: Results of Wald tests for the verification of the symmetry, additivity, homogeneity, strong monotonicity, and fairness axioms (based only on the groups that formed
the grand coalition)
Axiom

a2 = a3d2 = d3
Additivity
c 1 = a1 + b 1
c2 = a2 + b2
c 3 = a2 + b 3
c 4 = a4 + b 4
Homogeneityd1 = 2a1
d2 = 2a2
d3 = 2a3
d4 = 2a4
Null player
a2 = 0Strong monotonicity
a1 = b 1
c1 = d 1
Fairnessb 3 − b2 = c 3 − c 2
Symmetry

χ2p-value

Test

5.07
1.11
4.84
0.03
14.99
11.10
0.00
13.12
2.25
0.43
9.90
215.83
0.67
3.02

0.024
0.293
0.028
0.861
0.000
0.001
0.983
0.000
0.134
0.513
0.002
0.000
0.411
0.082

-

H0

-

+

-

+ indicates that the axiom is considered to be satisfied on average. − indicates the opposite.

To summarize, there is no systematic difference between the H–MC and the H–
MCsim mechanisms except that the H–MC better satisfies the symmetry and fairness
axioms than does H–MCsim if we focus on the groups that formed grand coalitions.

23

V.3

Analyses based on all the groups but only on normalized payoffs

Below, we compare H–MC and H–MCsim based on the normalized payoffs but using
the data for all groups. Table V.9 reports the results of running a set of OLS regressions
as in Eq. 1 using the normalized payoffs as dependent variables.
Table V.9: Results of linear regression for normalized payoffs
H–MCsim
πe1
πe2
πe3
πe4
g1
21.43 25.38 23.30 29.90(1.88) (1.20) (1.53) (1.60)
g2
5.12 27.52 28.04 39.31(1.63) (0.72) (0.83) (0.92)
g3
38.06 49.84 49.94 62.15(1.75) (1.56) (0.70) (1.96)
g4
44.13 49.03 51.97 54.87(0.56) (2.06) (2.33) (0.81)
2
R
0.83
0.96
0.96
0.96
Obs. 168
168
168
168

Based on the estimated coefficients reported in the left panel of Table IV.4 and Table V.9, Figure V.13 shows the average normalized payoffs obtained by each player inthe four games under H–MC and H–MCsim . The horizontal lines indicate the Shapley
values for each game. We observe that for game 1 under H–MCsim , the average normalized payoffs of each of the four players are not significantly different from the Shapley
values.

24

Figure V.13: Mean of the normalized payoffs
Game 1

Game 260

60

Player 2

Player 3

Player 4

Player 1

Player 2

Game 3***Player 3

***

*

H–MC

***

H–MCsim***

H–MCsim

***

H–MC

***

H–MCsim

H–MCsim

Player 1

H–MCsim

0
H–MC

10

0
H–MCsim

10
H–MC

20

H–MCsim

30

20

H–MC

30

H–MC

40

H–MC

**

H–MCsim

50
*

40

H–MC

50

Player 4

Game 4120

120

Player 1

Player 2

Player 3

Player 4*

Player 1

Player 2

Player 3******

H–MCsim

H–MC

H–MCsim

H–MC

0
H–MCsim

20

0
H–MC

40

20
H–MCsim

40

H–MC

60

H–MCsim

60

H–MC

80

H–MCsim

**

80

H–MC

***

H–MCsim

***

H–MC

*

H–MCsim

***

H–MC

100

100

Player 4

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

V.3.1

Distance between the ex ante payoffs and the Shapley value

We compute the distance of the average normalized payoffs from the Shapley value,
Dis2φ , using the bootstrapping technique with 1,000 repetitions, as we have done above.
The result is reported in Figure V.14. It can be observed that the ex ante normalized
payoffs of H–MCsim are closer to the Shapley values in games 1 and 2, whereas those25

Figure V.14: Distance of the normalized payoffs from Shapley
Game 140

Game 2
40

***

20

***

20

0

0
H–MC

H–MCsim

H–MC

Game 340

H–MCsimGame 4
40

***

***

20

20

0

0
H–MC

H–MC

H–MCsim

H–MCsim

Note: The error bars show the one standard error range. The symbols ***, **, and * indicate the significant difference between the
H–MC and the H–MCsim mechanisms at the 1%, 5%, and 10% significance levels (two-sample t-test), respectively.

of H–MC are closer to Shapley values in games 3 and 4.24

V.3.2

Normalized payoffs and a posteriori equilibria

In terms of distance from SPNE or equal division, we observe from Figure V.15 that
Dis2N E is significantly smaller under H–MC than under H–MCsim only in game 4.
For other games, the values are not significantly different between the two mechanisms.
In terms of Dis2ED , although it is significantly smaller under H–MC in game 2, theopposite is the case for game 3. For games 1 and 4, there is no significant difference
between the two mechanisms. We observe that normalized payoffs are significantly
closer to the equal division than the SPNE for both mechanisms in games 1 and 4. For
24

The mean Dis2φ (standard error) values based on the normalized payoff for H–MC are 3.73 (0.034)
in game 1, 10.52 (0.056) for game 2, 7.78 (0.120) for game 3, and 7.58 (0.638) in game 4. For H–MCsim ,the corresponding values are 2.89 (0.033) in game 1, 6.13 (0.042) for game 2, 19.24 (0.104) for game 3,
and 8.45 (0.058) in game 4.

26

Figure V.15: Mean of the distances of the normalized payoff vectors from the SPNE
and the equal division(a) Game 1
***

n.s.

40

(b) Game 2

***

n.s.
n.s.

40

20

**

n.s.

**

20

0

0
H–MC H–MCsim

H–MC H–MCsim

H–MC H–MCsim

H–MC H–MCsim

Dis2N E

Dis2ED

Dis2N E

Dis2ED(c) Game 3
100
80
60
40
20
0

n.s.

n.s.

(d) Game 4

***

100
80
60
40
20
0

*

**

***

***
n.s.

H–MC H–MCsim

H–MC H–MCsim

H–MC H–MCsim

H–MC H–MCsim

Dis2N E

Dis2ED

Dis2N E

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 equilibrium allocations or from the equal division was significantly different between the H–MC and the
H–MCsim implementations at the 1%, 5%, and 10% significance levels (Wald test), respectively.

H–MCsim , Dis2ED is significantly larger than Dis2N E in game 2, whereas the opposite
is the case for game 3. For H–MC, Dis2N E and Dis2ED are not significantly different
in games 2 and 3.

V.3.3

AxiomsFinally, verification of the axioms (comparing Table V.10 and the left column of Tables IV.5 and IV.6) indicates that the differences in results between H–MC and H–
MCsim are observed for additivity and homogeneity (satisfied in H–MC but not in H–
MCsim ).
To summarize, even comparing the payoff shares using all the groups, there is no
systematic difference between the H–MC and the H–MCsim mechanisms, except that27

Table V.10: H–MCsim normalized payoffs, Wald tests for the verification of the symmetry, additivity, homogeneity, strong monotonicity and fairness axioms
Axiom

H0

a2 = a3d2 = d3
Additivity
c 1 = a1 + b 1
c 2 = a2 + b 2
c 3 = a2 + b 3
c 4 = a4 + b 4
Homogeneityd1 = 2a1
d2 = 2a2
d3 = 2a3
d4 = 2a4
Null player
a2 = 0Strong monotonicity
a1 = b 1(H0 should be rejected)
c1 = d 1
Fairnessb3 − b2 = c 3 − c 2
Symmetry

χ2p-value

Test

1.01
0.47
36.91
1.11
0.53
4.78
0.16
0.28
5.90
3.23
9.90
23.87
11.55
0.15

0.314
0.492
0.000
0.292
0.466
0.0288
0.689
0.598
0.015
0.072
0.002
0.000
0.001
0.694

+
-

-

+
+

+ indicates that the axiom is considered to be satisfied on average. − indicates the opposite.

H–MC better satisfies the additivity and homogeneity axioms than H–MCsim .

28

VI

Translated instruction materials and screenshots ofthe comprehension quiz
• Winter mechanism: https://www.dropbox.com/s/galeo3todbah7iw/
Winter_1_loop_handout.pdf?dl=0• H–MC mechanism: https://www.dropbox.com/s/ctlw85momf96vmx/
HMChandout_seq.pdf?dl=0• Simultaneous voting version of the H–MC mechanism (H–MCsim ): https://www.dropbox.com/s/78lf5bn6qi3qfwp/HMChandout_sim.pdf?dl=
029