ISSN (Print) 0473-453X

Discussion Paper No. 1175

ISSN (Online) 2435-0982

AN EXPERIMENT ON THE NASH PROGRAM:
A COMPARISON OFTWO STRATEGIC MECHANISMS
IMPLEMENTING THE SHAPLEY VALUE
Michela Chessa
Nobuyuki Hanaki
Aymeric Lardon
Takashi Yamada

Revised February 2023
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: A comparison of two
strategic mechanisms implementing the Shapley value
Michela Chessa*

Nobuyuki Hanaki†

Aymeric Lardon‡

Takashi Yamada§February 21, 2023

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, provided that the
grand coalition is formed.

JEL code: C70, C71, C92
Keywords: Nash program, Bargaining procedures, Shapley value, Experiments* 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. 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 represent 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 methodof joint-cost allocation. At present, this value continues to attract the greatest interest

2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 who face 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 playing of classical bargaining mechanisms based on offers that lead to the Shapley
value require one single agent, either the regulator or one of the players, to propose acomplete allocation. In many contexts, the centralization in the hands of a single player
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 anddo 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).
An alternative but less common approach is to describe a bargaining mechanism
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 describes a demand revelation mechanism in which potential customers ofa public facility simply bid to be served. Bargaining mechanisms based on demands
3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. 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 investigate the differences between demand- and offer-based
bargaining mechanisms by experimentally comparing the 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 the easewith which they can be understood by the participants in a laboratory experiment.1 The
first mechanism is Winter’s demand commitment bargaining mechanism (Winter, 1994,
referred to as the Winter mechanism below). The second 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 the bargaining procedures to define a sharing of
payoffs rather than an allocation of costs.2 In the first mechanism, which is definedfor cooperative games with increasing returns to scale for cooperation (strictly convex
games), the proposer makes a demand for him- or herself concerning the payoff thathe 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
1

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.
2
It is straightforward to theoretically establish the equivalence between these two.

4

a proposal to each of the other players concerning the payoff he or she is willing to offerthem.
Two main issues arise with most strategic bargaining models, as observed by Fréchette
et al. (2005a), in experimental analyses of some well-known legislative bargaining processes. First, 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. Second, partly related to the first point, while the theoretical predictions
are very sensitive to variations in the rules of the game, the observed bargaining behaviors in the experiment are not always so. In our case, although two mechanisms havethe same ex ante prediction (the Shapley value as expected payoffs), it is possible thatthe degree to which the observed behavior deviates from the theoretical prediction, andthe reason for doing so, may differ greatly between the two.
Note that, it has been argued that the difference between a demand-based vs. anoffer-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 may become crucial when considering groups with at least three members. Notice
that, on one hand, in an offer-based mechanism, because proposers propose an allocation
by dividing the worth of the coalition, it is not difficult for a proposal to satisfy boththe feasibility and efficiency conditions. In demand-based mechanisms, on the other
hand, because each player separately makes his/her demand, coordination among themto make the submitted set of demands as a whole to satisfy these two conditions becomes
more difficult as the group becomes larger.
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.
5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
efficient outcome, whereas a demand-based Winter mechanism better implements allocations that reflect players’ effective power provided the grand coalition is formed.
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 that
supplement our results and provide new points for reflection are contained in the Online
Appendices I to VI.

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 anoncooperative foundation for cooperative solution concepts, and he began doing so
by proposing a noncooperative game that sustained the Nash solution of his two-player
6bargaining 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 problems3 have been proposed. Some mechanisms intended to obtain the Nash solution, exactly or approximately, at equilibrium(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(2022), in which subjects designated as decision-makers express their view on what isfair for others by recommending a payoff allocation. De Clippel and Rozen (2022)show that the decision-maker’s choices can be described as a convex combination of theShapley value and the equal division solution.
3

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

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 that
influence the final outcome of such a procedure, including personality differences or
the geometrical arrangement of players around the table. Similarly, but with a greater
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 designed to tackle different aspects of thecooperative 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
8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 amore 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 propose
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 design 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 allocations,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, orin 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
9

problem. Several experiments have been conducted, with the final goal of testing Nashaxioms, or of comparing Nash and Raiffa solutions (see, e.g., Nydegger and Owen,
1975; Rapoport et al., 1977). In addition, there is a large 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 and Wettstein (2001) and Ju andWettstein (2009) to implement solution concepts in the framework 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).4
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 a4

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).

10

probability 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, 2005c), and Fréchette et al.
(2005b) provide an experimental study of demand bargaining.5 However, Fréchette
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 ,5

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• 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”
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 a12single 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 appliedin 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, but it is not necessarily stable. However, if the game is superadditive, the Shapley value is an imputation, and if the game is convex, it belongs to thecore (in particular, it is its barycenter).
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 (Shapley, 1953; Young, 1985; van den Brink,2002). Given a solution ψ : G N → RN , we list here the four axioms that are used in thecharacterization by Shapley (1953), and later be used in analyzing our data.6
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).
6

See Online Appendix IV for other axioms that have been proposed to characterize the Shapley value.

13

Axiom 4 (Null player property): if i is a null player in game v ∈ G N , thenψi (v) = 0.

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
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
14with the rest of the players using the same rule on v restricted on N \ S.
Here, we present the one-period Winter mechanism we consider in our experiment
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 one-period 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:
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;
15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
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 agree 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,16

the game moves to the next stage. The more general mechanism by Hart and Mas-Colell(1996) allows for a proposer, even after a rejection, to remain in the game and join theother to next stage with a given probability. In our experiment, we consider the special
case in which such a probability is zero, and then, if the proposal is rejected, the proposer
leaves the game with his or her individual value and the bargaining continues among therest of the players, with a new player randomly chosen as a new proposer7 .
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: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
7

A first simplified version of the mechanism by Hart and Mas-Colell had already appeared in MasColell (1988).

17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.
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

18

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
185 245=
nisms. The Shapley value of this game is given by the vector φ(v) = 170
,
,
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 theWinter 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)
19

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)55
x3

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.
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.

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

5

The experimental setting

5.1

The games

2,4

We consider the four four-player games shown in Table 2 in our experiment. These
games are chosen to test the properties of the Shapley value that are discussed in Section
3.1. 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 of the H–MC mechanism, whereas all except game 2 respect theassumptions of the Winter mechanism. However, with game 2 being only convex,
we consider that “strict convexity” could be relaxed and the game could still beplayed in such a case;• in games 1 and 4, players 2 and 3 are symmetric;• 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;

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

• 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 and
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 andFebruary 2022 (H–MC mechanism).8 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.9 The experiment
8

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 Online Appendix VI)
did not reflect the H–MC model precisely (we thank an anonymous reviewer for pointing this out), andwe have redone the H–MC experiment to correct this. Online Appendix VI compares the outcomes of thepseudo-H–MC conducted in December 2019 and the (corrected) H–MC conducted in January–February2022.
9
The difference in the number of participants between the two mechanisms is a result of variations inthe show-up rate among experimental sessions.

22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 4321.10 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.11 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.12 The average earnings
10

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.
11
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.
12
Participants received a copy of instruction slides, and a pre-recorded instruction video was played.
Quiz was given on the screen after the explanation of the game. The user interface was explained during
the practice rounds referring to the handout about the computer screen. The quiz was given on thescreen after the explanation of the game. The user interface was explained during the practice rounds,
referring to the handout about the computer screen. See Online Appendix VII for English translations of
the instruction materials and the comprehension quiz. At each decision screen, there was a non-bindingtime limit. The time limit was set to 60 seconds to make a demand (Winter) or a proposal (H-MC) and
30 seconds to choose a coalition (Winter) or decide to approve or reject the proposal (H-MC). Whenthe time limit was reached, the message "please make a decision" appears on the top of the screen toencourage participants to make their decisions. On average (the standard deviation, the max, and themin), participants took 48.62 (25.28, 304, and 2) second for making a demand (n=768) and 20.36 (13.25,
164, and less than 1) seconds to choose a coalition (n=550) in the Winter. For H-MC, they took, on
average (the standard deviation, the max, and the min), 40.83 (23.23, 160, and 7) seconds for making a
proposal (n=257) and 16.18 (11.54, 87, and 2) seconds to approve or reject a proposal (n=531).

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, and also try understanding
the reasons for the discrepancies between the realized allocations and the Shapley values from the failure of four axioms that characterize the Shapley values presented in
Section 3.1. 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 Online Appendix I to VI.

6.1

Grand coalition formation and efficiency

Panel (a) of Figure 2 presents the results concerning the grand coalition formation underthe H–MC and Winter mechanisms pooling the data of all the four games.13 For game 2,
we consider the partition {{1}, {2, 3, 4}} as a realization of the grand coalition for boththe H–MC and Winter mechanisms because player 1 is a null player and, consequently,
the game is only convex and not strictly convex.14
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.
13

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 valueof 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 ofthe estimated coefficients of two treatment dummies. See Online Appendix I for results for four games
separately.
14
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 and efficiency(a) Frequency of Grand Coalition Formation(b) Efficiency

***

**

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 outcomes of H–MC mechanism
and the Winter mechanism are significantly different at the 0.1%, 1%, and 5% significance levels (Wald test), 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 Panel (b) of Figure 2 shows, considering all the four games together, efficiency issignificantly higher under the H–MC mechanism than under the Winter mechanism.15
Therefore, we conclude as follows.
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 proposes an allocation to all
15

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. If weconsider four games separately, however, the efficiency is not statistically significantly different at 5%
significance level in any of the game. See Online Appendix I.

25

the remaining members given the feasibility condition. Conversely, under the Wintermechanism, the players, speaking one after the other, may make unfeasible demands
or form a small coalition early without waiting for others making their demands. As
a result, the formation of the grand coalition under the H–MC mechanism is simply
determined by whether the remaining players choose to accept the proposal or reject it,
whereas under the Winter mechanism, it can be blocked by either by players forming asmaller coalition prematurely or by unfeasibility 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. Indeed,in Online Appendix III.1, we report that the main reason for the failure of grand coalition
formation in Winter in our experiment is a coalition being formed before reaching thefourth player.16

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.
Figure 3 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. The mean
16

In Online Appendix II, we report the frequency of the grand coalition formation and efficiency byseparating 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 someof the games, for both mechanisms. A significantly higher frequency of grand coalition formation andefficiency is observed under the H–MC mechanism than under the Winter mechanism even in the second
half of the experiment.

26

and the standard errors are obtained by running a set of ordinary least squares (OLS)
regressions for the following system of equations:
π1 = a1 g1 + a2 g2 + a3 g3 + a4 g4 + u1
π2 = b1 g1 + b2 g2 + b3 g3 + b4 g4 + u2(1)

π3 = c1 g1 + c2 g2 + c3 g3 + c4 g4 + u3
π4 = d1 g1 + d2 g2 + d3 g3 + d4 g4 + u4
where πi is the payoff of player i, gj is a dummy variable that takes 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.
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 3.
We do observe, however, that in game 2, the average payoff of player 1 under Winteris zero (just as Shapley value) while it is positive under H–MC. Below, we first follow
the approach proposed by Aguiar et al. (2018) and compute the Shapley distance todecompose the reasons behind the deviations of the realized payoff vectors from theShapley values into the failure of its four main properties. We then further investigate
the realized payoff vectors by focusing on those groups that formed the grand coalition.

27

Figure 3: Mean payoffs, all groups

Player 4

Player 1

Player 1

Player 2

Player 3

Player 4

Player 1

Player 2***

WinterPlayer 4

Player 3

***

***

H–MC

Winter

Winter

*

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***H–MC

WinterPlayer 3***

Winter

***

H–MC

***

60
50
40
30
20
10
0

WinterPlayer 2

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

6.2.1

Shapley distance

The approach of Aguiar et al. (2018) we apply is based on the decomposition of the
distance of the payoff vectors from the Shapley value into the failure of efficiency, symmetry, additivity and null player property. The same decomposition has been used in
Chessa et al. (2022). We present the procedure below.
Let π = (π1 , π2 , π3 , π4 ) be the realized vector of payoffs in a game. We firstfind a vector of payoffs closest to π that satisfies the symmetry. Call such a vector
π sym = (π1sym , π2sym , π3sym , π4sym ). Namely, we take the sum of payoffs obtained bysymmetric players s (players 2 and 3 in games 1 and 4) and divide the sum equally28

among them. Thus, in games 1 and 4, πssym =P

s∈{2,3} πs /2, and for non-symmetricplayers k (including all the players in games 2 and 3), πksym = πk .
Next, we find a vector of payoffs satisfying efficiency that is closest to π sym . Callthe new payoff vector π sym,ef f = (π1sym,ef f , π2sym,ef f , π3sym,ef f , π4sym,ef f ). Namely, forP
each player i = 1, 2, 3, 4, πisym,ef f = πisym + [v(N ) − j∈N πj ]/4.
We then find a vector of payoffs satisfying null player property that is closest toπ sym,ef f . Let π sym,ef f,null = (π1sym,ef f,null , π2sym,ef f,null , π3sym,ef f,null , π4sym,ef f,null ) be
the resulting vector of payoffs. Specifically, if player n is a null player (player 1 ingame 2), then her new payoff must be equal to zero, that is, πnsym,ef f,null = 0. Threeother players j in the game (players 2, 3, 4 in game 2) equally share πnsym,ef f of the nullplayer. That is, πjsym,ef f,null = πjsym,ef f + πnsym,ef f /3. When there is no null player inthe game, πisym,ef f,null = πisym,ef f for all i.
Theorem 3 in Aguiar et al. (2018) shows that a vector of payoffs π = (π1 , π2 , π3 , π4 )
obtained when playing game v can be decomposed as follows: π = φ(v)+esym +eef f +enull + eadd . Thus, the Shapley error, eφ = π − φ(v), is eφ = esym + eef f + enull + eadd
where
esym
= πi − πisym for all i,ifeef= πisym − πisym,ef f for all i,ienull= πisym,ef f − πisym,ef f,null = for all i,ieadd= πisym,ef f,null − φi (v) for all i.
iGiven this decomposition, the Shapley distance is given by:||eφ ||2 = ||esym ||2 + ||eef f ||2 + ||enull ||2 + ||eadd ||2 + 2 < eadd , enull >29

Table 4: Result of Shapley distance decomposition. Based on pooling the data of allgroups and all games||esym ||2
H-MC
38.19
(12.73)Winter
85.18(18.97)
No. Obs
352
2
R
0.132
p-value∗0.079||eef f ||2
429.96(53.41)606.81(101.35)
352
0.220
0.167

||enull ||2
63.97(8.27)
7.28(1.87)
352
0.111
0.0003||eadd ||2
270.84(20.70)321.49(17.33)
352
0.369
0.103

||eφ ||2
802.88(62.70)1020.72(72.22)
352
0.421
0.057

Note: Standard errors are corrected for session-level clustering effects
and shown in parentheses. < eadd , enull > are not reported in the table
as they are negligible (the mean values are 0.0093 for H–MC and 0.0026
for Winter).
* p-values for testing H0 : H–MC = Winter (based on the Wald test)

where < ·, · > is the scalar product and for any vector y ∈ Rn , ||y||2 =< y, y >=P
2 17i∈N yi .
We perform the Shapley distance decomposition of each payoff vector and the corresponding Shapley value, and compute the average distance, pooling data of all groups
and all games, to compare between H–MC and Winter. Results are presented in Table 4.
One can observe from the last column of Table 4, indeed, the Shapley distance is(marginally significantly) larger under Winter than under H–MC. And this is because
the distance due to the violations of symmetry, efficiency, and additivity axioms tend
to be lager under Winter than under H–MC (and marginally significantly so for thesymmetry). However, as we have observed in Figure 3, the distance due to the violationof null player property is significantly smaller under Winter than under H–MC. Notethat when decomposed in this way, the distance due to the violation of efficiency axiom
17Differently from the original decomposition by Aguiar et al. (2018), that ensures orthogonalcomponents, with our decomposition, in general, vectors enull and eadd are not orthogonal so that
< eadd , enull > is not equal to zero.

30

is not significantly different between the two mechanisms unlike what we have seen
above. However, this is due to the difference in the definition. The efficiency measureP
used above ( i πi /v(N )) and ||eef f ||2 are negatively correlated (the Pearson correlation
coefficient is -0.78) but not perfectly so.
To further compare the realized allocations between the two mechanisms in light oftheoretical predictions, let us now focus on those groups that formed the grand coalition. The reason for focusing on groups that formed the grand coalition is that different
coalition structures provide a different value to be shared, and therefore a comparison
between the payoff vectors in relation to some theoretical benchmark would not be clear
cut. We complement our analyses in Online Appendix V and report the corresponding
results based on all groups using payoff shares, instead of restricting our attention togroups that formed the grand coalition. The idea behind these additional analyses is tocheck whether payoff shares respect the hierarchy among the players as predicted by
the theory.

6.2.2

Allocations when the grand coalition is formed

Figure 4 shows the average payoffs obtained by each player in the four games conditional on the grand coalition being formed.18 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 at least three out of four players in all four games. This
indicates that provided that the grand coalition is formed, the average payoffs under the18

The mean and the standard errors are obtained by running a system of linear regressions as we havedone to generate Figure 3 but restricting to those cases where the grand coalition is formed.

31

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

Player 3

Player 4

Player 1

Player 2

Player 1

Player 3

Player 4

Player 1

H–MC

Winter

Winter**Player 4

Player 2***

Player 3

Winter

*

H–MC

H–MC

Winter***H–MC

WinterH–MC

WinterH–MC

Winter

H–MC

120
100
80
60
40
20
0

***

Player 2***

Game 4***Winter

H–MC***

Player 3

WinterGame 3
120
100
80
60
40
20
0

**

WinterH–MC***

Winter***H–MC

WinterH–MC

Winter***

H–MC

Player 2

H–MC

Winter*

WinterPlayer 1

H–MC

Winter

H–MC***

Game 2
60
50
40
30
20
10
0

H–MC

Game 1
60
50
40
30
20
10
0

Player 4

Note: The horizontal lines indicate the Shapley values. The error bars show the one standard error range. The symbols ***, **,and * indicate that the average payoff is significantly different from the Shapley values at the 0.1%, 1%, and 5% significance levels(Wald test), respectively.

Winter mechanism are closer to the Shapley values than those under the H–MC mechanism. Furthermore, we report in Online Appendix IV that, provided that the grandcoalition if formed, the realized allocations under the Winter mechanism better satisfy
the axioms characterizing the Shapley values than ones realized under the H–MC mechanism.
Result 2. Provided the grand coalition is formed, the average payoffs follow the Shapley
values more closely under the Winter mechanism than under the H–MC mechanism.

32

6.2.3

Realized allocations in the grand coalition 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 19 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 forthe given game.
Figure 5 shows the mean Dis2N E and the mean Dis2ED for the two mechanisms
in the four games.20 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 inall 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
than under 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 Wintermechanism in games 1 and 4 (in which the equilibrium payoffs are less unequal than19

For the sake of simplicity, we omit the specifications about the considered mechanism and the game.
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 theWald test for the equality of the estimated coefficients of the two treatment dummies. For the difference
between Dis2N E and Dis2ED we compute ∆ = Dis2N E − Dis2ED and run the same regression as
above and test whether the estimated coefficients of treatment dummies are significantly different from
zero.
20

33

Figure 5: Mean of the distances of the realized payoff vectors from the SPNE and theequal division among those groups that formed the grand coalition.
(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 differences are statisticallysignificant at the 0.1%, 1%, and 5% significance levels (Wald test for across treatment differences and t-test for within treatment
difference between Dis2N E and Dis2ED ), respectively.

in games 2 and 3) at the 0.1% level. For games 2 and 3, however, the distance to theequilibrium allocations is not significantly different between the two mechanisms.
Figure 5 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. When the grand coalition is formed, the H–MC mechanism more often resultsin payoffs that are closer to the equal division than to the equilibrium payoffs compared34

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. In Online
Appendix III.2, we show that in H-MC, the proposals become more equal after players
observing rejection of earlier, less equal, proposals. Such dynamics lead to this result.

6.3

Exploitation of the first mover advantage

We investigate now for the strategic behavior of the players in our games. At first,
we observe that a direct comparison between H-MC and Winter mechanisms on thispoint is rather challenging because the strategic behavior to optimally participate in anoffered based vs. a demand based mechanism is much different off the equilibrium path.
After a first proposal has been made, on the one side, and as already noticed, an offer
based mechanism resemble a voting situation, in which each player simply accepts or
reject the proposal. We may observe that, in this case, the best response simply depends
on the set of players who are left in the game and on the offer he or she received,
independently on what has been offered to the other players. Moreover, at a given
period, the only possibility for a player is to accept forming the coalition containing
all the remaining players, or refusing to do that. On the other side, in a demand based
mechanism, the announcement of a player who is not playing first in the period, i.e.,
after the first demand of the remaining players has been declared, not only depends
on the set of players who are left in the game, but also on all precedent demands in the
period which are not yet satisfied. In this case, the choice is not only on whether forminga coalition containing all the remaining players, but also on whether eventually forming
other feasible sub-coalitions. Thus, a direct comparison of the strategic behavior along35

the entire strategic interaction under the two different mechanisms is not possible.
Instead, the two mechanisms are fully comparable when investigating the strategic
behavior of the first mover. In fact, once the first player has been randomly chosen, heor she is theoretically expected to offer him- or herself (under the H-MC mechanism),or to demand (under the Winter mechanism) exactly the same payoff, i.e., his or her
marginal contribution to the coalition formed by the remaining players (which is equal
to the his or her a posteriori equilibrium payoff). For this reason, we focus our analysis
on the behavior of the first mover to analyze the degree to which participants play thegame as predicted by the theory.
When asking his-or her marginal contribution to the coalition formed by the remaining players, the first mover experiences what we call the first mover advantage, i.e.,he or she can strategically ask more than what given by the a priori expected equilibrium payoff, i.e., the Shapley value. We therefore investigate whether the first moversuccessfully exploit this advantage.
In the following, let F Ai be the degree to which the first mover i exploits his or her
first mover advantage. Namely, F Ai = (ai − φi ) / (eqi − φi ), where i is the first moverand ai is either the proposed allocation to i in H-MC or the demand by i in Winter, φi
is i’s Shapley value, and eqi is i’s a posteriori equilibrium payoff. Note that since F Ai
is not defined for the null player (player 1 in Game 2), we exclude the null player fromthe following analysis.21 But we consider all the groups, otherwise.
Table 5 show the average F Ai in two mechanisms for four games. The standard
errors are corrected for session clustering effect. The table is created based on the
21

When the null player was the first mover, the average ai was 35 in H–MC and 16.5 in Winter. The
standard errors adjusted for the session clustering effect were 6.830 and 2.284 in H–MC and in Winter,
respectively. The difference is significant at 5% level (p = 0.037) according to the Wald test for theequality of coefficients of the two treatment dummies in a linear regression where ai is the dependent
variable and two treatment dummies, without the constant, are the only independent variables.

36

Table 5: Exploitation of the first move advantage

H-MC
WinterNo. ObsR2
p-value†Game 10.354∗∗∗(0.022)-0.096(0.152)
88
0.245
0.022

Game 2-0.055(0.156)-0.024(0.166)
62
0.002
0.897

Game 30.674∗(0.264)-0.123(0.104)
88
0.232
0.026

Game 40.406∗∗(0.089)-0.218(0.095)
88
0.238
0.002

Note: Standard errors are corrected for session-level clustering effects and shown in parentheses.
***, **, *: estimated coefficient is significantly different
from zero at 0.1, 1, and 5% significance level.
† p-values for testing H0 : H–MC = Winter (based on theWald test)

estimated coefficients of the linear regression that takes F Ai as the independent variable
and two dummy variables representing the two mechanisms as only dependent variables
without the constant. We observe that, on average, while the first mover advantage is
exploited, although not fully, in H–MC except for Game 2, it is not in Winter. Thus, thefirst movers are acting significantly closer to the a posteriori equilibrium prediction in
H–MC than in Winter. This finding is in line with the result we have seen in Figure 6.2.3,
where we have shown that the H-MC mechanism provides result that are closer to the
SPNE when the grand coalition is formed.

7

Conclusion

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 that the two mechanisms stand out in the literature for their distinctive fea-37

tures, and they are a fundamental contribution on the Nash program. Moreover, they arerecognized for their simplicity, which is a key desideratum when considering possible
applicability of a theoretical mechanism to the real world. These two mechanisms havethe same ex ante equilibrium prediction, but differ mainly in their processes: H–MC
mechanism is based on offers, and the Winter on demands.
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. Instead, we show that our two mechanisms behave very differently, despite the close theoretical predictions. In particular, the H–MC mechanism
results in higher frequencies of the grand coalition 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., fullcooperation in the airport problem (Littlechild and Owen, 1973) results in one single
airport being built instead of many, and this is certainly desirable for environmental reasons). Conversely, provided that the grand coalition is formed, the Winter mechanism
results in average payoffs that are closer to the Shapley values and better satisfy various axioms. We suggest that the Winter mechanism, when it leads to collaboration, isbest 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 as social welfare and symmetry are inescapable (e.g., allocation of resources in health or social care
(Kluge, 2007)). Of course, the major drawback of the Winter mechanism is its failureof reaching the full collaboration.
Our findings suggest that when facing a cost or payoff allocation problem, the choiceof which bargaining procedure to use, one based on offers or on demands, may have38

some unexpected effects, regardless of the theoretical prediction. This should be taken
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.
Theoretically, a player who always has a zero marginal contribution should receive azero payoff, according to Shapley. In accordance with the theoretical prediction, in ademand-based mechanism, non-null players have refused a strictly positive demand by anull player in our experiment. However, we find that non-null players in our experiment
seem to be 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 (see,Chessa et al., 2022, for two periods version), or the H–MC mechanism where the proposer whose offer is refused then leaves the game with a probability strictly smaller
than one) can be compared with our actual results. Comparing the outcomes of the experiments based on noncooperative mechanisms with those of unstructured bargaining
experiments would be an interesting topic for future research.

39

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)(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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Online Appendix
I

Grand coalition formation and efficiency for each game

Figure I.1 shows the frequency of grand coalition formation under H-MC and Winter for
four games separately. We observe that for game 2 and the Winter mechanism, the grandcoalition never forms (because player 1 is a null player and, consequently, the game is
only convex and not strictly convex. 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 Wintermechanisms. Grand coalition is significantly more frequently formed under H-MC thanunder Winter in games 3 and 4.
Figure I.1: 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 different between the two mechanisms at the 0.1%, 1%, and 5% significance levels(Wald test), respectively.

1

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

n.s.

n.s.

1
0.8
0.6
0.4
0.2
0

1
0.8
0.6
0.4
0.2
0
H–MC

H–MC

Winter(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

H–MC

Winter

WinterNote: The error bars show the one standard error range. The symbols ***, **, and * indicate that the efficiency is significantly
different between the two mechanisms, at the 0.1%, 1%, and 5% significance levels (Wald test), respectively.

Figure I.2 shows the average efficiency under H-MC and Winter for four games
separately. Unlike the case where we pool all the data, the efficieny is not significantly
different under two mechanisms in any of the games at 5% significance level.

II

Effect of learning

II.1

Grand coalition formation and efficiency

We have already shown in Section 6.1 that both mechanisms fail to achieve 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 6.1 and with
Result 1, H–MC mechanism forces feasible offers, while Winter mechanism allows
for unfeasible demands or players forming smaller coalitions prematually which, as a2

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 mechanisms 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
small if we separate the data into the first half and second half (see Table II.1), we
investigate only the frequency of grand coalition formation and efficiency.
Figure II.1 shows the frequency of the grand coalition formation (left) and the average efficiency (i.e., the average total payoff / value of the grand coalition, right) for the
first half and the second half (i.e., the first four rounds vs. the second four rounds). We
pool four games. For H–MC, while the frequency of the grand coalition formation is not
significantly different in the early and later rounds, the efficiency is significantly higher
in the later rounds than in the early rounds. For the Winter mechanism, both the fre-

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.

3

game 4912
17
13

Figure II.1: Grand coalition formation and efficiency in early and late rounds (all games
pooled)Grand Coalition Formation
1
0.8
0.6
0.4
0.2
0

**

n.s.

Early

Late

H–MC

Efficiency

n.s.

1
0.8
0.6
0.4
0.2
0

n.s.

Early

Late

Early

Winter

*

***

Late

H–MC

*
n.s.

Early

Late

WinterNote: The error bars show the one standard error range. The symbols ***, **, and * indicate the outcomes shown in two bars are
significantly different at 0.1%, 1%, and 5% significance levels, respectively (Wald test).

quency of the grand coalition formation and the average efficiency are not significantly
different between early and late rounds.
Figures II.2 and II.3 show the frequency of the grand coalition formation and theaverage efficiency for the first half and the second half of each game. For H–MC, thefrequency of the grand coalition formation and the average efficiency 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 the average efficiency are significantly higher in the later rounds only in game 3, with no significant differences inother games.
As conclusion, we report no statistically significant learning effect, when testing
either the H–MC or the the Winter mechanism. This does not rule out the possibility
that by implementing a higher number of repetitions, a significant learning effect could
be observed.

4

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

Game 1
1
0.8
0.6
0.4
0.2
0

n.s.

*

Early

n.s.

1
0.8
0.6
0.4
0.2
0

n.s.

Late

Late

Early

H–MC

Early

Winter

n.s.

n.s.

*

n.s.

Early

Late

H–MC

n.s.

Late

Early

H–MC

Game 3
1
0.8
0.6
0.4
0.2
0

n.s.

Late

WinterGame 4
**

1
0.8
0.6
0.4
0.2
0

**

Early

Late

Early

Winter

n.s.

n.s.

Late

n.s.
n.s.

Early

H–MC

Late

WinterNote: The error bars show the one standard error range. The symbols ***, **, and * indicate the outcomes shown in two bars are
significantly different at 0.1%, 1%, and 5% significance levels, respectively (Wald test).

III

Observed dynamics in Winter and H-MC

In this section, we analyze the coalition formation dynamics of Winter as well as theway proposals evolved in H–MC to better understand the results presented in the maintext. Namely, the reason for failure of the grand coalition formation in Winter and theallocation becoming closer to the equal division solution in H–MC.

III.1

Dynamics of coalition formation in WinterWe first show the low frequency of the grand coalition formation under Winter is due
to participants forming smaller coalitions before reaching the 4th player making thedemand.

5

Figure II.3: Efficiency in early and late rounds
Game 1
1
0.8
0.6
0.4
0.2
0

n.s.

***

Early

Game 2
*
n.s.

Late

Late

Early

H–MC

1
0.8
0.6
0.4
0.2
0

Early

Winter

n.s.

n.s.

1
0.8
0.6
0.4
0.2
0

n.s.

Early

Late

H–MC

n.s.

Late

Early

H–MC

Game 3
n.s.

n.s.

Late

WinterGame 4

***
**

Early

1
0.8
0.6
0.4
0.2
0
Late

Early

Winter

*

n.s.

Late

H–MC

n.s.
n.s.

Early

Late

WinterAs one can observe from Table III.1, out of 176 play of the games where the firstmover was not the null player, the first mover exited the game by belonging to a coalition
without waiting for the fourth player making the demand 77 times. Thus, there is a clear
tendency for participants forming a coalition and exiting the game prematurely.
Among 70 cases in games other than game 2 where the first move belonged to thecoalition formed by the fourth mover, only 50 formed the grand coalition. For game 2,
in 2 out of 2 cases in which the first move belonged to the coalition formed by the fourthmover and 17 out of 18 in which the first move belonged to the coalition formed by thethird mover, the resulting coalition were {2, 3, 4} which, in our analyses, considered as
the grand coalition.
Furthermore, among these 21 cases where the grand coalition (including {2, 3, 4} ingame 2) was not formed even when the coalition to which the first player belonged to6Table III.1: Frequency of the timing of the formation of the coalition to which thefirst mover belongs to.
Formed by
Self
Second mover
Third mover
Fourth mover
None
Total

All All*
4
4
8
8
65
65
72
72
43
27
192 176

Game 1
0
0
15
27
6
48

Game 2
3
2
18
2
23
48

Game 2*
3
2
18
2
7
32

Game 3
0
3
19
17
9
48

Game 4
1
3
13
26
5
48

* Excluding the cases where the null player was the first mover.

was formed by the 4th (or the 3rd in case of game 2) player, the total amount demanded
by four players exceeded v(N ), and thus it was not possible to form the grand coalition,in 14 cases. In remaining 7 cases, the grand coalition was not formed although doing
so was possible. An interesting observation is that in 2 cases in game 2 where the firstmover belonged to the coalition formed by the fourth mover and the coalition {2, 3, 4}
instead of {1, 2, 3, 4} was formed, the sum of the demand including the one by the nullplayer did not exceed v(N ).

III.2

Dynamics of proposals in H–MC

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 resultsof offer-based mechanisms (such as the well-known two-player bargaining over a pie
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.
Let the distance between the proposal and equal devision in Lth proposal (for a7

group), devEQL , be devEQL =P

i |aL,i − ED| where aL,i is the proposed allocation

for player i in Lth proposal (for the group) and ED is the equal division payoff for thegame.
First, we observe the first proposal is less likely to be accepted if its distance fromthe equal division is larger. The estimated coefficient for the devEQ1 is negative andsignificant in a linear regression in which the dependent variable is the dummy variable
that takes value of 1 if the proposal is accepted and 0 otherwise, and the independent
variables are the constant and devEQ1 (-0.005 with the standard error (corrected for thesession clustering effect) being 0.0009 and p-value = 0.008. N = 160. R2 = 0.1352)).
Figure III.1: H–MC mechanism: Distance from equal division for the first and thesecond proposals (top) and the second and the third proposals (bottom)
Game 1

Game 2

Game 3

Game 4������

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n=15
p=0.003������

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

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

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�� �� �� �� ��� ��� ���

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 .

8

In Figure III.2, devEQL =P

i |aL,i − ED| for the first (L = 1, horizontal axis) andthe second (vertical axis) proposals (top) and the second (L = 2, horizontal axis) andthe third (vertical axis) proposals (bottom) for each game. Each dot corresponds to apair of 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.

IV

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.
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 have alreadylisted in Section 3 four axioms that are used in the characterization by Shapley (1953):
efficiency, symmetry, additivity and null player property. Here, we list three additionalcommonly used axioms to provide a characterization.
Axiom 5 (Homogeneity): for all v ∈ G N and a ∈ R, ψ(av) = aψ(v).
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).
Axiom 7 (Fairness): if i, j are symmetric in w ∈ G N , then ψi (v + w) − ψi (v) =9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.
The axiomatization of Young (1985) involves axioms 1, 2, and 6, whereas that ofvan den Brink (2002) involves axioms 1, 4, and 7. Note that axiom 5, even if not directly
involved in any of these axiomatizations, is crucial because, together with axiom 3, it
guarantees the linearity of the solution.1
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 focusing on thegroups that formed grand coalition. These axioms are tested based on the estimatedcoefficients 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 5) require that x3 = x1 + x2 andx4 = 2x1 for x ∈ {a, b, c, d}, respectively.
• Null player property (axiom 4) requires that a2 = 0.
• Strong monotonicity (axiom 6) requires that a1 > a2 and a4 > a3 .
• Fairness (axiom 7) requires that b3 − b2 = c3 − c2 .
1
The equal division solution satisfies 1, 2, and 3, but does not satisfy the null player property in 4.
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).

10

Table IV.1: 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

In Table IV.1, 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 not confirmed in one 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 berejected) 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
11Table IV.2: 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.

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 IV.2 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.

V

Additional results based on payoff shares

In this section, we report the results based on all the groups and we use 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

12

shown by Eq. 1 but using payoff shares as dependent variables.
Figure V.1 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 respect the Shapley
values only in game 3 at the 5% significance level.
Figure V.1: Mean of the normalized payoffs
Game 2

Player 1

Player 2

Player 3

Player 4

Player 1

Player 2

Game 3Player 3

WinterH–MC***

Winter**

Winter

***

H–MC

***

H–MC

WinterH–MC

WinterH–MC

WinterH–MC

WinterH–MC***

Winter

60
50
40
30
20
10
0

H–MC

Game 1
60
50
40
30
20
10
0

Player 4

Game 4
120
100
*

**

80

120
100
80
60
40
20
0

***

*

60
40
20

Player 1

Player 2

Player 3

Player 4

Player 1

Player 2

Player 3

WinterH–MC

WinterH–MC

WinterH–MC

WinterH–MC

WinterH–MC

WinterH–MC

WinterH–MC

WinterH–MC

0Player 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 0.1%,
1%, and 5% significance levels, respectively (Wald test).

13

Tables V.1 and V.2 summarize the results of testing the six axioms. Based on thenormalized payoff, on average, the symmetry, strong monotonicity, and fairness axioms
are now satisfied under the H–MC mechanism. For the Winter mechanism, with normalized 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.
Table V.1: Wald tests for the verification of the symmetry, additivity, homogeneity,strong monotonicity, and fairness axioms for normalized payoffs

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

0.53
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

Figure V.2 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
normalized payoffs under the H–MC mechanism are significantly closer to the equal
14

Table V.2: 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.

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

Figure V.2: Mean of the distances of the normalized payoff vectors from the SPNE andthe equal division(a) Game 1
***

n.s.

40

(b) Game 2

***

n.s.
***

20

***

n.s.

40

***

20

0

0
H–MC

WinterH–MC

Dis2N E

Winter

H–MC

Dis2ED

Winter

Dis2N E(c) Game 3
100
80
60
40
20
0

n.s.

n.s.

H–MC

Winter

Winter

Dis2ED(d) Game 4

***

100
80
60
40
20
0

**

Dis2N E

H–MC

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 statistically significant differences
between the two bars at the 0.1%, 0.5%, and 5% significance levels (Wald test for across treatment differences and t-test for withintreatment difference between Dis2N E and Dis2ED ), respectively.

16

VI

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

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.1 We showthat sometimes, as observed by Fréchette et al. (2005a), 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.2 The average earnings were 2,780 JPY.
1

We thank an anonymous reviewer for pointing this out.
Just as in the H-MC and the Winter, there was a non-binding time limit of 60-seconds to make aproposal and of 30-seconds to accept or reject the proposal. The average (the standard deviation, themaximum, and the minimum) time participants spent to make a proposal is 43.10 (22.31, 111, and 5)
seconds (n=313), while those to accept or reject a proposal is 16.72 (10.83, 61, and 1) seconds (n-782).
2

17

VI.1

Grand coalition formation and efficiency

Figure VI.1 report the results concerning the grand coalition formation and efficiency bypooling four games. There are no significant difference between H–MC and H–MCsim .
Figure VI.1: Proportion of times the grand coalition is formed and efficiency(a) Frequency of Grand Coalition Formation(b) Efficiency

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

WinterNote: The error bars show the one standard error range. The symbols ***, **, and * indicate the outcomes of H–MC mechanism
and the H–MCsim mechanism are significantly different at the 0.1%, 1%, and 5% significance levels (Wald test), respectively.

Figures VI.2 reports the results concerning the grand coalition formation and efficiency for each game separately. The only significant differences reported are forgame 2.

18

Figure VI.2: H–MC and H–MCsim mechanisms, proportion of times the grand coalition
is formed and efficiency
Grand coalition formation(b) Game 2(a) Game 1
n.s.
1
0.8
0.6
0.4
0.2
0

*

1
0.8
0.6
0.4
0.2
0
H–MC

(b’) Game 2, allow (2,3,4)n.s.

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

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

H–MCsim(c) Game 3

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 a statistically significant difference atthe 0.1%, 1%, and 5% significance levels (Wald test), respectively.

19

Figure VI.3: H–MC and H–MCsim mechanisms, mean payoffs all the group

Player 3

Player 1

Player 2

Player 1

Player 2

Player 3

Player 3

Player 4

Player 1

Player 2

***

***

H–MC

H–MCsim

Player 4

Player 3

***

***

H–MC

H–MCsim

H–MCsim

**

H–MC

***

H–MC

120
100
80
60
40
20
0

H–MCsim

***

H–MC

***

H–MCsim

***

H–MC

***

H–MCsim******Game 4

H–MC***

H–MCsim

H–MCsim

***

H–MC

120
100
80
60
40
20
0

H–MC

Game 3***

H–MCsim

***

H–MC

***

H–MCsim**

H–MC

H–MCsim

Player 4***

H–MCsim

***

H–MC

***

H–MC

**

H–MCsim

H–MCsim

Player 2

***

60
50
40
30
20
10
0

H–MCsim

Player 1

H–MC

***

H–MCsim

H–MC**

Game 2

H–MC

Game 1
60
50
40
30
20
10
0

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

VI.2

Allocations

We follow the analyses in the main text by first looking at the average payoffs of eachplayer in each game based on the all the groups.

VI.2.1

Average payoffs

Figure VI.3 shows the average payoffs of each player in each game. The horizontal lines
indicate the Shapley values for each game. As we have seen in the main text comparing
20

Table VI.1: Result of Shapley distance decomposition. Based on pooling the data of allgroups and all games

H-MC
H-MCsimNo. ObsR2
p-value∗||esym ||2
38.19(12.73)25.90(2.28)
328
0.085
0.374

||eef f ||2
429.96(53.41)386.64(40.36)
328
0.170
0.538

||enull ||2
63.97(8.27)
39.55(10.47)
328
0.107
0.110

||eadd ||2
270.84(20.70)317.10(21.81)
328
0.261
0.168

||eφ ||2
802.88(62.70)769.12(55.17)
352
0.378
0.698

Note: Standard errors are corrected for session-level clustering effects
and shown in parentheses. < eadd , enull > are not reported in the table
as they are negligible (the mean values are 0.0093 for H–MC and 0.0074
for H–MCsim ).
* p-values for testing H0 : H–MC = H–MCsim (based on the Wald test)

the Winter and H–MC, because of the failure of forming the grand coalition, the average
payoffs are significantly different from the Shapley values in both H-MC and H-MCsim .
VI.2.2

Shapley distance decomposition

Are there significant differences in terms of the results of Shapley distance decomposition between H-MC and H-MCsim ? The results reported in Table VI.1 suggest that is not
the case. The average Shapley distances and their four components are not significantly
different between H-MC and H-MCsim .

VI.3

Analyses based only on groups that formed the grand coalition

We now focus on those groups that formed grand coalition. Figure VI.4 shows theaverage payoffs obtained by each player in the four games, conditional on the grandcoalition being formed. The horizontal lines indicate the Shapley values for each game.

21

Figure VI.4: H–MC and H–MCsim mechanisms, mean payoffs based only on the groups
that formed the grand coalition

Player 2

Player 2

Player 3

Player 4

Player 1

Player 2**

H–MC

H–MCsim

H–MCsim

Player 4

Player 3******

H–MCsim

***

H–MC

*

H–MCsim***

H–MC

H–MCsim**

H–MCsim

***

120
100
80
60
40
20
0
H–MC

***

H–MC

H–MCsim***

H–MC

H–MCsim

Player 1

Player 3***

Game 4***

H–MCsim

***

H–MC

***

H–MC

Game 3
120
100
80
60
40
20
0

**

H–MC

H–MCsim

Player 1

Player 4***

H–MCsim

*

H–MC

***

H–MC

***

H–MCsim

H–MCsim

Player 3

***

H–MC

***

H–MC

Player 2

*

H–MC

***

Game 2
60
50
40
30
20
10
0

H–MCsim

Player 1***

H–MCsim

H–MCsim

H–MC

**

H–MC

Game 1
60
50
40
30
20
10
0

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

We observe that the two mechanisms perform similarly in that there are players whose
average payoff is significantly different from the Shapley value in all four games under both mechanisms even when we focus on only those groups that formed the grandcoalition.

22

VI.3.1

Realized allocations and a posteriori equilibria

In terms of distance from SPNE or equal division, we observe from Figure VI.5 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 VI.5: 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.
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 a statistically significant differenceat the 0.1%, 0.5%, and 5% significance levels (Wald test for across treatment differences and t-test for within treatment difference
between Dis2N E and Dis2ED ), respectively.

23

VI.3.2

AxiomsFinally, verification of the axioms (comparing Table VI.2 and the left column of Tables IV.1 and IV.2) 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 VI.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
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 axiom thandoes H–MCsim if we focus on the groups that formed grand coalitions.

24

Figure VI.6: Mean of the normalized payoffs

Player 3

Player 4

Player 1

Player 2

Player 2

Player 3

Player 4

Player 1

Player 2

H–MC

H–MCsim

H–MCsim

Player 4

Player 3*****

H–MCsim

H–MCsim

H–MC

H–MC

H–MCsim

H–MC*

H–MCsim

120
100
80
60
40
20
0

***

H–MC

H–MC

H–MCsim***

H–MC

H–MCsim

H–MC***

Player 1

Player 3***

Game 4

H–MCsim

120
100
80
60
40
20
0

***

H–MC

Game 3

**

H–MC

***

H–MCsim**

H–MC

H–MCsim

H–MC

H–MCsim

H–MC

H–MCsim

Player 2***

H–MCsim

Player 1

H–MC

H–MCsim

H–MC***

H–MCsimGame 2
60
50
40
30
20
10
0

H–MC

Game 1
60
50
40
30
20
10
0

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 0.1%, 1%, and 5% significancelevels (Wald test), respectively.

VI.4

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.
Figure VI.6 shows the average normalized payoffs obtained by each player in thefour games under H–MC and H–MCsim based on results of running a set of OLS regressions as in Eq. 1. The horizontal lines indicate the Shapley values for each game.
25

Figure VI.7: Mean of the distances of the normalized payoff vectors from the SPNE andthe equal division(a) Game 1
40

n.s.

***

(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.

n.s.

n.s.

(d) Game 4

***

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 a statistically significant differenceat the 0.1%, 1%, and 5% significance levels (Wald test for across treatment differences and t-test for within treatment difference
between Dis2N E and Dis2ED ), respectively.

We observe that for games 1 and 4 under H–MCsim , the average normalized payoffs ofeach of the four players are not significantly different from the Shapley values at 5%
significance level, while those for H-MC is significantly different at 5% level for at least
for one player in all the four games.

VI.4.1

Normalized payoffs and a posteriori equilibria

In terms of distance from SPNE or equal division, we observe from Figure VI.7 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 difference26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
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.

VI.4.2

AxiomsFinally, verification of the axioms (comparing Table VI.3 and the left column of Tables V.1 and V.2) 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 ).
Table VI.3: 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.

To summarize, even comparing the payoff shares using all the groups, there is no27

systematic difference between the H–MC and the H–MCsim mechanisms, except that
H–MC better satisfies the additivity axiom than H–MCsim .

VIITranslated 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=
028