DP2018-23

The Pivotal Mechanism Versus the
Voluntary Contribution
M e c h a n i s m : An E x p e r i m e n t a lComparison
Tatsuki HOMMA
Junyi SHEN
Takuma WAKAYAMA
Hirofumi YAMAMURA
Takehiko YAMATO

November 5, 2018

The pivotal mechanism versus
the voluntary contribution mechanism:
An experimental comparison∗
Tatsuki Homma†

Junyi Shen‡

Takuma Wakayama§

Hirofumi Yamamura¶ Takehiko Yamato∥
November 5, 2018

Abstract
We conduct an experimental comparison of two well-known mechanisms forundertaking a binary public project: the pivotal mechanism and the voluntary
contribution mechanism. We compare the two mechanisms under complete information in which each subject knows the other subjects’ payoﬀs. We then observethat the voluntary contribution mechanism works better than the pivotal mechanism from the perspectives of Pareto eﬃciency and individual rationality. On
the other hand, we observe that the pivotal mechanism works better than thevoluntary contribution mechanism in light of decision eﬃciency. These results
suggest that whenever we insist on either Pareto eﬃciency or individual rationality, the voluntary contribution mechanism is more favorable than the pivotal
mechanism in practical applications.
Keywords: laboratory experiment; pivotal mechanism; voluntary contribution
mechanism
JEL codes: C92; D71; D78; H41∗

This work was supported by JSPS KAKENHI Grant Numbers JP22730165, JP22330061,
JP25380244, JP26285045, JP15H03328, and JP16K03567.
†
Department of Social Engineering, Graduate School of Decision Science and Technology, Tokyo
Institute of Technology. E-mail: homma.t.aa@gmail.com‡
Research Institute for Economics and Business Administration, Kobe University. E-mail:shen@rieb.kobe-u.ac.jp§
Faculty of Economics, Ryukoku University. E-mail: wakayama@econ.ryukoku.ac.jp¶
Department of Law and Economics, School of Economics, Hokusei Gakuen University. E-mail:h-yamamura@hokusei.ac.jp∥
Department of Industrial Engineering and Economics, School of Engineering, Tokyo Institute ofTechnology. E-mail: yamato.t.aa@m.titech.ac.jp1

1

Introduction

We consider an economy with a public good in which agents are facing a decision ofwhether to undertake a project that produces a ﬁxed level of the public good. The
public project should be undertaken if the sum of agents’ true values of the project islarger than or equal to the cost of undertaking the project, but not otherwise. However,this condition of decision eﬃciency is diﬃcult to satisfy since the mechanism designer
does not know each agent’s true value of the public project, and the agent may have an
incentive to “free-ride” payments from other agents and undertake the project whilelying about their own values.
There are two well-known mechanisms to solve this free-rider problem theoretically:
the pivotal mechanism and the voluntary contribution (or provision point) mechanism.
In the pivotal mechanism due to Clark (1971), reporting their true value of the projectis a dominant strategy for everyone. Therefore, this mechanism satisﬁes the decisioneﬃciency condition. However, the rule of the mechanism is complicated and dominant
strategy equilibrium allocations of the mechanism may be neither Pareto eﬃcient nor
individually rational.
On the other hand, the rule of the voluntary contribution mechanism is simple,
and the mechanism implements the core when the equilibrium concept is undominatedperfect equilibrium, which is a reﬁnement of Nash equilibrium (Bagnoli and Lipman,1989). Then, all equilibrium allocations of the mechanism are Pareto eﬃcient andindividually rational. The mechanism also satisﬁes decision eﬃciency. We should however note that this equilibrium concept requires complete information among agents:
everyone knows the payoﬀs of all other agents.
The above theoretical predictions suggest that the voluntary contribution mechanism would perform better than the pivotal mechanism under complete information.
This motivates us to investigate whether these predictions are correct in a laboratory
experiment. Indeed, several papers have examined how one of the two mechanisms
works in laboratory experiments.1 However, these experimental studies cannot specify
which of these two mechanisms is better because they focused either on the pivotal
mechanism or on the voluntary contribution mechanism. To the best of our knowledge, no paper experimentally compares the two mechanisms in the same economic
environment with a public project.
The purpose of this paper is to investigate which mechanism, the pivotal mechanism or the voluntary contribution mechanism, should be used in the provision of
1

See Chen (2008) and Chen and Ledyard (2010) for excellent surveys on these experimental results.

2

a binary public project by conducting an experimental comparison. In designing anexperiment to compare the two mechanisms, we pay close attention to the following
two points. First, the comparison should be made under the same economic environmental conditions on the number of subjects, the true values that subjects receive fromthe project, the cost of undertaking the project, and the informational condition. We
compare the two mechanisms under complete information in which each subject knows
the true values of all subjects.
The second point we are concerned with in comparing the two mechanisms is to holdthe complexity constant. Clearly, the rule of the pivotal mechanism is more complicatedthan that of the voluntary contribution mechanism. Attiyeh et al. (2000) and Kawagoe
and Mori (2001) argued that subjects suﬀered from confusion due to the complexityof the pivotal mechanism and the non-transparency of the dominant strategy. To
eliminate such confusion, following Cason et al. (2006), we focused on the two-agent
case and simply presented to each subject a payoﬀ table describing the relation between
outcomes and choices by the subject and the others without explaining the rule of thepivotal mechanism. Similarly, for the voluntary contribution mechanism experiment,
we used payoﬀ tables only and gave no explanation on the rule of the mechanism.
The use of payoﬀ tables allows for a comparison of the pivotal mechanism with thevoluntary contribution mechanism holding the degree of transparency constant.
We compare the two mechanisms from four diﬀerent normative criteria: the Paretoeﬃciency index, the individual rationality index, the decision eﬃciency index, and thesurplus index. Here, the Pareto eﬃciency index is the ratio of Pareto eﬃcient outcomes.
The individual rationality index is the ratio of individually rational outcomes in which
each subject receives a payoﬀ at least as large as their own before participating in themechanism, that is, the payoﬀ the subject receives from the initial endowment of a
private good without any public project. The decision eﬃciency index is the ratio of
decision-eﬃcient outcomes in which the public project is undertaken because the sumof the subjects’ true values of the project exceeds the cost of undertaking the project inour experimental setting. The surplus index is a measure of the gains subjects obtainby participating in a mechanism, which is a percentage of the highest possible payoﬀ
that could be achieved in the mechanism.
We observed that the Pareto eﬃciency index, the individual rationality index, andthe surplus index for the voluntary contribution mechanism were signiﬁcantly higher
than those for the pivotal mechanism. On the other hand, the decision eﬃciency index for the voluntary contribution mechanism was signiﬁcantly lower than that for thepivotal mechanism. These results suggest that whenever we insist on either Pareto3

eﬃciency or individual rationality, the voluntary contribution mechanism is more favorable than the pivotal mechanism in practical applications.
The remainder of the paper is organized as follows. Section 2 presents a brief reviewof the laboratory evidence on the pivotal and voluntary contribution mechanisms. In
Section 3, we describe the model of a public project and the deﬁnitions of mechanisms.
We also propose a new equilibrium concept and provide theoretical prediction resultsthat are useful in examining our experimental observations. Section 4 describes our
experimental design. We explain the experimental results in Section 5. Section 6
provides concluding remarks. The appendix contains the proofs of our theoreticalresults and our experimental instructions.

2

Related literature

Attiyeh et al. (2000) conducted laboratory experiments on the pivotal mechanism andfound that few subjects reported their true values for the public project when only therule of the pivotal mechanism was explained to groups of ﬁve or ten subjects. Less
than 10% of the bids were truth-telling in incomplete information settings in which
each subject knew their own true value but not the others’ values in the same group.
Kawagoe and Mori (2001) also found that the ratio of the truth-telling bids signiﬁcantly
increased from 16% to 47% when payoﬀ tables were given to subjects in addition toan explanation of the rule of the pivotal mechanism. Moreover, Cason et al. (2006)
experimentally compared the Groves mechanism, that is “secure” in the sense thatthe set of dominant strategy equilibrium outcomes coincides with the set of Nashequilibrium outcomes, with the pivotal mechanism that is not secure. They observedthat the ratio of dominant strategies that subjects chose was signiﬁcantly higher in the
secure Groves mechanism than in the non-secure pivotal mechanism.
For the voluntary contribution mechanism with a single unit of public good, laboratory experimental studies found limited support for the theoretical prediction thateﬃcient outcomes are achieved. For example, Marks and Croson (1999) observed thatthe ratio that decision eﬃciency was satisﬁed was 48% under complete information and
54% under incomplete information, and that the ratio of Pareto eﬃcient equilibrium
strategies subjects played was 4% under complete information and 6% under incomplete
information, even though group contributions converged toward the equilibrium level
over time both under complete information and under incomplete information. There
are other laboratory experimental studies examining the eﬀects of changing details of
the voluntary contribution mechanism such as refund rules, rebate rules, incomplete in-4formation on the number of players or the cost of the public project, and simultaneous
versus sequential contributions (see Chen (2008) for a survey).
Healy (2006) conducted an experimental comparison of ﬁve public good mechanisms, the voluntary contribution, proportional taxation, Groves-Ledyard, Walker, andVickrey-Clarke-Groves (VCG) mechanisms, in the same economic environment with
ﬁve-person groups under incomplete information. He found that the VCG mechanism
was the most eﬃcient among the ﬁve mechanisms. In his experimental setting, thelevel of the public good is chosen from a continuum, and each subject knows theirown preference parameters but not the preference parameters of others. On the other
hand, the public project choice is binary in our experiment, and we investigate thecomplete information case. Moreover, we compare the voluntary contribution mechanism with the pivotal mechanism which is the VCG mechanism with a binary publicproject from the viewpoints of decision eﬃciency and individual rationality as well as
Pareto eﬃciency.

3

Preliminaries

3.1

Model

We consider a two-agent economy with a binary public project. Two agents face adecision whether to undertake a public project. Let y ∈ {0, 1} be a public project. In
our model, y = 1 if the public project is undertaken, and y = 0 otherwise. The cost ofundertaking a public project is c > 0. Let t = (t1 , t2 ) ∈ R2 be a transfer vector, whereti denotes the payment for agent i. We denote the set of feasible allocations by{}
A = (y, t) ∈ {0, 1} × R2 : − (t1 + t2 ) ≥ c · y .
For each i ∈ {1, 2}, let
Ai = {(y, ti ) ∈ {0, 1} × R : − ti ≥ c · y}be the set of i-feasible allocations. Let ωi > 0 be agent i’s initial endowment.
Each agent i ∈ {1, 2} has a quasi-linear preference relation over {0, 1} × R, which
is described by the value of the public project vi ≥ 0. Denote the set of such valuesof agent i by Vi . That is, given any i ∈ {1, 2} and any vi ∈ Vi , i’s preference relationis represented by the function ui ( · , · ; vi ) : {0, 1} × R → R deﬁned for each (y, ti ) ∈5

{0, 1} × R, byui (y, ti ; vi ) = vi · y + ωi + ti .
Let v = (v1 , v2 ) ∈ V ≡ V1 × V2 be a value proﬁle.

3.2

Properties of allocations

We introduce some standard properties of allocations. An allocation (y, t) ∈ A is Paretoeﬃcient for v ∈ V if there is no allocation (y ′ , t′ ) ∈ A such that ui (y ′ , t′i ; vi ) ≥ ui (y, ti ; vi )for each i ∈ {1, 2} with a strict inequality for some j ∈ {1, 2}. We denote the set of
Pareto eﬃcient allocations for v by P (v). The following fact characterizes the set of
Pareto eﬃcient allocations.
Fact 1. Let v ∈ V . An allocation (y, t) ∈ A is Pareto eﬃcient for v if and only if

u1 (y, t1 ; v1 ) + u2 (y, t2 ; v2 ) =v + v − c + ω + ω

if v1 + v2 ≥ cω + ω

otherwise.

1

2

1

1

2

2

An allocation (y, t) ∈ A is decision eﬃcient for v ∈ V if y ∈ arg maxy′ ∈{0,1} (v1 +v2 ) · y ′ − c · y ′ . The following fact states that decision eﬃciency is weaker than Paretoeﬃciency.
Fact 2. Let v ∈ V . If an allocation (y, t) ∈ A is Pareto eﬃcient for v, then it isdecision eﬃcient for v.
An allocation (y, t) ∈ A is individually rational for v ∈ V if ui (y, ti ; vi ) ≥ ui (y ′ , t′i ; vi )for each i ∈ {1, 2} and each (y ′ , t′i ) ∈ Ai . We denote the set of individually rational
allocations for v by I(v). The following fact characterizes the set of individually rational
allocations.
Fact 3. Let v ∈ V . An allocation (y, t) ∈ A is individually rational for v if and only
if for each i ∈ {1, 2},ωiui (y, ti ; vi ) ≥v − c + ωiiif c > vi(1)otherwise.

The core for v ∈ V is the set of allocations that are Pareto eﬃcient and individually6rational for v.2 We denote the core for v by C(v). That is, C(v) = P (v) ∩ I(v) for each
v ∈V.

3.3

Equilibrium concepts

A mechanism is a pair Γ = (S, g), where S ≡ S1 × S2 , Si is strategy space of agent i,and g : S → A is an outcome function. Given v ∈ V , a pair (Γ, v) constitutes a (normalform) game.
We now introduce several equilibrium concepts.
• Dominant strategy equilibrium. A strategy proﬁle (s1 , s2 ) ∈ S is a dominant
strategy equilibrium of the game (Γ, v) if(i) for each s′1 ∈ S1 and each s′2 ∈ S2 , u1 (g1 (s1 , s′2 ); v1 ) ≥ u1 (g1 (s′1 , s′2 ); v1 ); and(ii) for each s′1 ∈ S1 and each s′2 ∈ S2 , u2 (g2 (s′1 , s2 ); v2 ) ≥ u2 (g2 (s′1 , s′2 ); v2 ).
• Nash equilibrium. A strategy proﬁle (s1 , s2 ) ∈ S is a Nash equilibrium of thegame (Γ, v) if(i) for each s′1 ∈ S1 , u1 (g1 (s1 , s2 ); v1 ) ≥ u1 (g1 (s′1 , s2 ); v1 ); and(ii) for each s′2 ∈ S2 , u2 (g2 (s1 , s2 ); v2 ) ≥ u2 (g2 (s1 , s′2 ); v2 ).
• Undominated Nash equilibrium. Given any i ∈ {1, 2}, we say that an agent
i’s strategy si ∈ Si is weakly dominated in the game (Γ, v) if there is s′i ∈ Si suchthat for each sj ∈ Sj , ui (gi (s′i , sj ); vi ) ≥ ui (gi (s); vi ) with a strict inequality forsome s′j ∈ Sj . Denote the set of i’s strategies that are not weakly dominated in
the game (Γ, v) by U (Si ; (Γ, v)). A strategy proﬁle (s1 , s2 ) ∈ S is an undominatedNash equilibrium of the game (Γ, v) if (i) it is a Nash equilibrium of (Γ, v) and(ii) si ∈ U (Si ; (Γ, v)) for each i ∈ {1, 2}.
• Twice iteratively undominated Nash equilibrium. Let U (S; (Γ, v)) ≡U (S1 ; (Γ, v))×U (S2 ; (Γ, v)) and Γ(U ) ≡ (U (S; (Γ, v)), g). A strategy proﬁle (s1 , s2 )∈ S is a twice iteratively undominated Nash equilibrium of the game (Γ, v) if it
is an undominated Nash equilibrium of the game (Γ(U ), v). Let E(Γ, v) andA(Γ, v) be the set of twice iteratively undominated Nash equilibria for (Γ, v) andthe set of twice iteratively undominated Nash equilibrium allocations for (Γ, v),
respectively.
2

This deﬁnition depends on the number of agents. When there are three or more agents, the coreis generally smaller than the set of allocations satisfying Pareto eﬃciency and individual rationality.

7

3.4

Voluntary contribution mechanism

We consider two mechanisms that play central roles in the literature. Our ﬁrst mechanism is the following simple mechanism that is widely used in practice: each agent
oﬀers a voluntary contribution. If the sum of the contributions covers the cost of the
public project, it is undertaken; otherwise, the contributions are returned. Formally:
The voluntary contribution mechanism, ΓVC = (S VC , g VC ). For each i ∈ {1, 2},
SiVC = [0, ωi ]; and for each s ∈ S VC ,(0, (0, 0))
if s1 + s2 < c
VC
g VC (s) = (y VC (s), (tVC
(s),
t
(s)))
=
1
2
(1, (−s , −s )) otherwise.
1
2This subsection now makes the following assumption:
A1. For each v ∈ V and each i ∈ {1, 2}, ωi ≥ c > vi .
A1 says that each agent’s initial endowment exceeds the cost of undertaking a publicproject, and each agent’s value is less than the cost of undertaking a public project. A1
also says that no agent would like to contribute more than their initial endowment.3
We conduct laboratory experiments in the environments that satisfy A1.
Under A1, Facts 1 and 3 together imply that for each v ∈ V ,






0
≤
−t
≤
v
,
0
≤
−t
≤
v
,
and
1
1
2
2


(1, t) ∈ A :if v1 + v2 > c−(t1 + t2 ) = c
C(v) =

{(0, (0, 0)), (1, (−v1 , −v2 ))}if v1 + v2 = c{(0, (0, 0))}if v1 + v2 < c.
The next proposition states that the twice iteratively undominated Nash equilibrium allocations of the voluntary contribution mechanism are in the core for true
preferences.4
3

Instead of A1, Bagnoli and Lipman (1989) used the following weaker assumption: ω1 + ω2 > c
and for each v ∈ V and each i ∈ {1, 2}, ωi > vi .
4This is equivalent to stating that the voluntary contribution mechanism implements a subcorrespondence of the core in twice iteratively undominated Nash equilibria. Moreover, we could
argue that the voluntary contribution mechanism “almost” fully implements the core in twice iteratively undominated Nash equilibria because the following facts hold: for each v ∈ V , (i) if
v1 + v2 > c, then C(v) \ A(ΓVC , v) = {(1, (−v1 , v1 − c)), (1, (v2 − c, −v2 ))}; (ii) if v1 + v2 = c,then C(v) \ A(ΓVC , v) = {(1, (−v1 , −v2 ))}; and (iii) if v1 + v2 < c, then C(v) = A(ΓVC , v).

8

Proposition 1. Assume A1. Let v ∈ V . Then,
A(ΓVC , v) ⊆ C(v).
The voluntary contribution mechanism is simple and, thus, it can be easily implemented in practice. However, to obtain allocations that are in the core via thevoluntary contribution mechanism, agents are required to have a higher degree of rationality, and the mechanism designer must understand that a value proﬁle is common
knowledge between agents.
Remark 1. Our notion of twice iteratively undominated Nash equilibrium is diﬀerent
from the notion of iterated elimination of weakly dominated strategies. The reason
we do not employ the notion of iterated elimination of weakly dominated strategies
is that the voluntary contribution mechanism cannot implement the core (and any
sub-correspondence of the core) in iterated elimination of weakly dominated strategies;that is, in the voluntary contribution mechanism, some allocations obtained by iterated
elimination of weakly dominated strategies do not belong to the core. See Appendix B
for a more detailed discussion.
♦
Remark 2. To show that the voluntary contribution mechanism implements the core,Bagnoli and Lipman (1989) used another reﬁnement of Nash equilibrium called “undominated perfect equilibrium” that applies the notion of (trembling-hand) perfection
to the game after eliminating dominated strategies of the original game. However, they
did not directly apply the notion of perfection to the inﬁnite game associated with thevoluntary contribution mechanism.5 Instead, they deﬁned an undominated perfectequilibrium in the inﬁnite game associated with the voluntary contribution mechanism
as the limit of some sequence of undominated perfect equilibria of approximating ﬁnite versions of the voluntary contribution mechanism. Thus, it is an interesting open
question whether a similar implementation result can be obtained when perfection is
directly deﬁned on the inﬁnite game associated with the voluntary contribution mechanism.
♦

5

Recently, Carbonell-Nicolau (2011) provided a notion of perfection in inﬁnite games. By invoking
his notion of perfection, he examined the existence of undominated perfect equilibrium in the inﬁnite
game associated with the voluntary contribution mechanism. However, it is an open question whether
a similar implementation result holds for his notion of perfection.

9

3.5

Pivotal mechanism

Our second mechanism determines the allocation as follows: each agent i ∈ {1, 2}
reports their value vi of the public project simultaneously. If v1 + v2 ≥ c, then the
public project is undertaken, and it is not undertaken otherwise. If the public projectis undertaken, then both agents pay for an equal share of the project’s cost. Moreover,whether the public project is undertaken or not, each agent i ∈ {1, 2} must pay thepivotal tax pi deﬁned as follows:
 (
c
c)


if v1 + v2 ≥ c and vj − < 0
−
v
−
j


2
2

c
c
pi = v j −if v1 + v2 < c and vj − > 0


2
2


0
otherwise.
The pivotal tax is the actual net beneﬁt the other agent enjoys minus the maximal net
beneﬁt the agent can enjoy under the equal share cost. Thus, the pivotal tax is thesocial cost of considering agent i’s preference. Formally:
The pivotal mechanism, ΓP = (S P , g P ). For each i ∈ {1, 2}, SiP = Vi ; and foreach v ∈ V = S P , we denote g P (v) = (y P (v), (tP1 (v), tP2 (v))), wherey P (v) ∈ arg max (v1 + v2 ) · y + c · y,y∈{0,1}()(c · y)c · y P (v)c · y P (v)PP+ v2 · y (v) −− max+ v2 · y ,t1 (v) = −y∈{0,1}2
22()(c · y)c · y P (v)c · y P (v)PP
t2 (v) = −+ v1 · y (v) −− max+ v1 · y .
y∈{0,1}2
2
2
The pivotal mechanism might appear less simple than the voluntary contribution
mechanism. However, it is well known that under the pivotal mechanism, truth-tellingis a dominant strategy for everyone. Therefore, in contrast to the voluntary contribution mechanism, in the pivotal mechanism, agents need not care about the exact valueof the other agent when considering their own action. Moreover, although the dominant strategy equilibrium allocations of this mechanism may not be Pareto eﬃcient or
individually rational, they are always decision eﬃcient.

10

4

Experimental design

Our experiment studies the voluntary contribution mechanism and the pivotal mechanism under complete information. Treatment V implements the voluntary contribution
mechanism for a two-agent group. Treatment P implements the pivotal mechanism fora two-agent group. The public project problem to be solved is the same for both
treatments. There are two types of agents, 1 and 2. The initial endowment vectorof the private good is given by (ω1 , ω2 ) = (20, 20). The true value proﬁle (v1 , v2 ) isequal to (9, 13). The cost of undertaking the public project is c = 20. By decisioneﬃciency introduced in Section 3.2, the public project should be undertaken sincev1 + v2 − c = 2 > 0. Note that this environment satisﬁes A1 mentioned in Section 3.4.

4.1

Treatment V

Let the strategy space of each type be the set of integers from 0 to 20. According to
the rules of the voluntary contribution mechanism explained in Section 3.4, we canconstruct the payoﬀ matrices of Types 1 and 2. The payoﬀ tables that we employ inTreatment V are Tables 1 and 2 whose structures are the same as the original payoﬀ
tables with the exception that a linear transformation of the valuation functions isemployed: 10ui − 40 for each type i ∈ {1, 2}.
Table 1 (resp. Table 2) is a payoﬀ table of Type 1 (resp. Type 2) with both types’
payoﬀs displayed: the lower left-hand number is Type 1’s (resp. Type 2’s) payoﬀ, andthe upper right-hand number is Type 2’s (resp. Type 1’s) payoﬀ in each cell. Table 1
also speciﬁes the Nash equilibria and the unique twice iteratively undominated Nashequilibrium.6 In Table 1, there are three “good” Nash equilibria, (7, 13), (8, 12), and(9, 11), in the sense that the public project is undertaken and, hence, decision eﬃciency
is met, whereas there are ninety-six “bad” Nash equilibria in the sense that the publicproject is not undertaken and, thus, decision eﬃciency is not satisﬁed.7 Among these
Nash equilibria, only the pair (s1 , s2 ) = (8, 12) is a twice iteratively undominated Nashequilibrium. Note that each Type 1’s strategy s1 ∈ {9, . . . , 20} is weakly dominated by
s1 = 8, whereas each strategy s1 ∈ {0, . . . , 7} is not weakly dominated by any strategy.
Similarly, each Type 2’s strategy s2 ∈ {13, . . . , 20} is weakly dominated by s2 = 12,
whereas each strategy s2 ∈ {0, . . . , 11} is an undominated strategy.
Eliminating these weakly dominated strategies of two players from Table 1 of the
6

In the payoﬀ table that we distributed to subjects, there was no tag and highlighting indicating
the equilibria as in Table 1.
7
The set of “bad” Nash equilibria is {(s1 , s2 ) ∈ S : s1 ∈ {0, . . . , 7} and s2 ∈ {0, . . . , 11}}.

11

Table 1: Payoﬀ table of Type 1 in Treatment V.

12

Table 2: Payoﬀ table of Type 2 in Treatment V.

13

Table 3: Payoﬀ table obtained by eliminating the weakly dominated strategies of two players
from Table 1.

original 21 × 21 payoﬀ table, we have Table 3 that is the 8 × 12 payoﬀ table consistingof eight strategies of Type 1, s1 ∈ {0, . . . , 7} and twelve strategies of Type 2, s2 ∈{0, . . . , 11}. In Table 3, there is one “good” Nash equilibria, (s1 , s2 ) = (8, 12), whereas
there are ninety-six “bad” Nash equilibria.8 Notice that each Type 1’s strategy s1 ∈{0, . . . , 7} is weakly dominated by s1 = 8, and each Type 2’s strategy s2 ∈ {0, . . . , 11}is weakly dominated by s2 = 12 in Table 3. Thus, (8, 12) is a unique undominated Nashequilibrium in Table 3 and a unique twice iteratively undominated Nash equilibriumin Table 1.9
Moreover, this equilibrium payoﬀ vector (170, 170) at (8, 12) is Pareto eﬃcient andindividually rational and, hence, it is a core payoﬀ vector (Proposition 1). The othergood Nash equilibrium payoﬀ vectors in Table 1, (180, 160) and (160, 180), are also
8

The set of “bad” Nash equilibria after eliminating these weakly dominated strategies of two players
from the original payoﬀ table coincides with that of “bad” Nash equilibria of the original payoﬀ table
mentioned in Footnote 8.
9Additionally, (8, 12) is a unique pair of strategies surviving after a twice iterated eliminationof weakly dominated strategies in Table 1. In other words, the unique undominated Nash equilibrium coincides with the unique pair of strategies surviving after twice iterated elimination of weakly
dominated strategies under our experimental setting. In general, however, the set of twice iteratively
undominated Nash equilibria may diﬀer from the set of pairs of strategies surviving after twice iterated
elimination of weakly dominated strategies. We need the concept of twice iteratively undominatedNash equilibria to obtain the core implementation result in Proposition 1. See Appendix B for moredetails.

14

Pareto eﬃcient and individually rational, and they are core payoﬀ vectors. All bad
Nash equilibrium payoﬀ vectors in Table 1 are the same, (160, 160), and they are
individually rational but not Pareto eﬃcient.

4.2

Treatment P

Let the strategy space of each type be the set of integers from 0 to 20. According to therules of the pivotal mechanism described in Section 3.5, we can construct the payoﬀ
tables of Types 1 and 2. The payoﬀ tables that we use in Treatment P are Tables 4
and 5 whose structures are the same as the original payoﬀ tables except that a linear
transformation of the utility functions is employed: 10ui − 40 for each type i ∈ {1, 2}.
Table 4 (resp. Table 5) is a payoﬀ table of Type 1 (resp. Type 2) with both types’
payoﬀs displayed. Table 4 also speciﬁes the dominant strategy equilibria and the otherNash equilibria.10 Type 1’s dominant strategies are 8 and 9, and Type 2’s dominant
strategies are 12 and 13. The two dominant strategies are equivalent for each type inthat their own payoﬀs are identical for every possible strategy played by the other type.
In this sense, there is an essentially unique dominant strategy for each type. However,the payoﬀs of Type 2 are diﬀerent depending on Type 1’s choices: Type 2’s payoﬀ is
170 when Type 1 selects 8 and Type 2 chooses 12 or 13, whereas Type 2’s payoﬀ is 180
when Type 1 selects 9 and Type 2 chooses 12 or 13.
There is a huge set of Nash equilibria in Table 4.11 The lower-right region of Nashequilibria is “good” in the sense that the public project is undertaken and, hence,
decision eﬃciency is satisﬁed. The upper-left region of Nash equilibria is “bad” in thesense that the public project is not undertaken and, thus, decision eﬃciency is notsatisﬁed. The number of good Nash equilibria is 137, whereas the number of bad Nashequilibria is 96. Implementation is clearly not secure in the sense of Cason et al. (2006).
Notice that the payoﬀ vector (u1 , u2 ) is Pareto eﬃcient if u1 + u2 = 340, and it
is individually rational if u1 ≥ 160 and u2 ≥ 160 in the current environment. The
dominant strategy equilibrium payoﬀs, (150, 170) and (150, 180), are neither Pareto
10

Again, in the payoﬀ table that we actually distributed to subjects, there was no tag and highlighting indicating the equilibria, as in Table 4 .
11This is because the best response function of each type has a “ﬂat” structure. For instance, (i)when Type 2 selects 7, the payoﬀs of Type 1 are “high” (160) if Type 1 chooses less than or equal
to 12, and those are “low” (120) otherwise; (ii) when Type 2 chooses 11, the payoﬀs of Type 1 are
the same (150) for all Type 1’s strategies; and (iii) when Type 2 selects 16, the payoﬀs of Type 1 are“low” (100) if Type 1 chooses less than or equal to 3, and Type 1’s payoﬀs are “high” (150) otherwise.
In other words, given each strategy of the other type, either a) the payoﬀs of each type are divided
into just two “tiers”: a “high” payoﬀ obtained by choosing “good” strategies and a “low” payoﬀ by“bad” strategies, or b) the payoﬀs are the same for all strategies.

15

Table 4: Payoﬀ table of Type 1 in Treatment P.

16

Table 5: Payoﬀ table of Type 2 in Treatment P.

17

eﬃcient nor individually rational. This is because the pivotal mechanism generates
a wasteful budget surplus, that is, the tax revenue exceeds the cost of undertaking
the public project. It is easy to check that the ratio of Pareto eﬃcient payoﬀs among
the good Nash equilibrium payoﬀs is 80.3%(= 110), but no good Nash equilibrium
137
payoﬀ is individually rational. The ratio of individually rational payoﬀs among the88
bad Nash equilibrium payoﬀs is 91.7%(= 96), but no bad Nash equilibrium payoﬀ isPareto eﬃcient.
Although no Nash equilibrium payoﬀ vector belongs to the core, the pivotal mechanism is designed speciﬁcally to achieve decision eﬃciency rather than Pareto eﬃciency
or individual rationality. Since decision eﬃciency has played a central role in the literature, we check whether the outcome is consistent with decision eﬃciency in our
experiment.

4.3

Procedures

We conducted two sessions in each of the two treatments at Tokyo Institute of Technology during January and June of 2013 and July of 2014. Twenty subjects participatedin each session (eighty separate subjects in total). We recruited the student subjects bycampus-wide advertisement. These students were told that there would be an opportunity to earn money in a research experiment. None of them had prior experience ina public project experiment. No subject attended more than one session. Each session
took approximately two hours to complete. The mean payoﬀ per subject was $32.57($1 = 100 yen) in Treatment P and $33.56 in Treatment V.
In each session, twenty subjects were seated at computer stations that were separated with visual partitions in the Experimental Economics Laboratory at Tokyo Institute of Technology. We made ten pairs out of twenty subjects and conducted twenty
periods. In every period, each of Type 1 subjects was paired with one of Type 2 subjects. The pairings were anonymous and were determined in advance by experimenters
to pair the same two subjects just two times. Each subject was informed that the person the subject was paired with was randomly chosen by the experimenters and thatthe person the subject was paired with changed over twenty periods. Each subject
could not know which person the subject was paired with at each period.
Each subject received written instructions, a record sheet, and a payoﬀ table. Eachsubject chose an integer number between 0 and 20 by looking at their own payoﬀ table
only. In each subject’s payoﬀ table, both their own payoﬀs and the payoﬀs of the othertype were shown in Treatments P and V.

18

After deciding which number to choose, each subject typed in that number into thecomputer. We used the z-Tree program (Urs Fischbacher, 2007). After the calculationof payoﬀs, the following information was displayed on each subject’s computer screen:• the subject’s chosen number;• the other’s chosen number;• the subject’s own payoﬀ; and• the other’s payoﬀ.
Each subject was asked to ﬁll out these values as well as the reasons why they had
chosen a particular number. These steps were repeated for twenty periods. No information or decisions regarding the other pairs were shown on the computer screen. No
communication among the subjects was allowed, and we declared that the experiment
would be stopped if we observed any communication among the subjects. This did not
happen.
Before the real twenty periods, the subjects had an opportunity to practice in
two periods using a payoﬀ table that diﬀered from the table employed in the actual
experiment. In these non-monetary periods, the numbers to be chosen were decided
in advance by the experimenters. We allowed the subjects ten minutes to examine thepayoﬀ table before the real periods started.

5

Results

Since each period had 20 pairs of subjects and each session had 20 periods, there were
400 pairs of data. We denote each pair by (s1 , s2 ), where si is a strategy chosen by asubject of Type i ∈ {1, 2}.

5.1

Treatment V

Figure 1 shows the frequency of distribution of all data in Treatment V. The maximum
frequency pair was the unique twice iteratively undominated Nash equilibrium (8, 12)with 313 pairs out of 400, the second frequency pair was (8, 10) with 31 pairs, the third
frequency pair was (8, 11) with 11 pairs, and the fourth frequency pair was (7, 12)with 6 pairs. The total frequency of Nash equilibria other than (8, 12) was 8. Allof them were bad Nash equilibria that failed to achieve decision eﬃciency. The total

19

Figure 1: All pairs choices in Treatment V.

frequency of strategy proﬁles that achieved decision eﬃciency was 316. The maximum
frequency strategy chosen by Type 1 subjects was s1 = 8 with 377 choices out of 400,
the second maximum frequency strategy was s1 = 7 with 8 choices, and the third
maximum frequency strategy was s1 = 4 with 6 choices. The maximum frequency
strategy chosen by Type 2 subjects was s2 = 12 with 328 choices out of 400, the second
was s2 = 10 with 32 choices, and the third was s1 = 11 with 14 choices.
Figure 2 displays the rates that Type 1 and Type 2 subjects chose s1 = 8 ands2 = 12, respectively. After period 3, the ratios of Type 1 subjects who chose s1 = 8
were at least 90%. The ratios of Type 2 subjects who chose s2 = 12 were at least 80%
after period 4. Figure 2 also demonstrates the rates that pairs of subjects played theunique twice iteratively undominated Nash equilibrium (8, 12) for all periods. These
equilibrium rates were at least 70% after period 4. By conducting the Wilcoxon signed
rank tests period by period, we found that Type 1 subjects’ median choice was not
signiﬁcantly diﬀerent from 8 in 18 out of 20 periods (periods 3–20 at the 5% signiﬁcancelevel) and that Type 2 subjects’ median choice was not signiﬁcantly diﬀerent from 12
in 18 periods (periods 2 and 4–20 at the 5% signiﬁcance level).
These results are summarized by the following observations.
Observation 1.
(i) The frequency of the unique twice iteratively undominated Nash equilibrium was20

Figure 2: Rates that Types 1 and 2 subjects, respectively, chose s1 = 8 and s2 = 12 in
Treatment V.

313
78% (= 400) across all periods in Treatment V.

) across all(ii) The frequency that Type 1 subjects chose s1 = 8 was 94% (= 377
400
periods in Treatment V.
(iii) The frequency that Type 2 subjects chose s2 = 12 was 82% (= 328) across all
400
periods in Treatment V.

5.2

Treatment P

Figure 3 shows the frequency of distribution of all data in Treatment P. The maximum
frequency pair was (8, 13) with 95 pairs out of 400, the second was (8, 12) with 91 pairs,
the third was (9, 12) with 30 pairs, the fourth was (10, 12) with 29 pairs, and the ﬁfth
was (9, 13) with 26 pairs. The total frequency of dominant strategy equilibria was 242.
The total frequency of Nash equilibria was 370. All Nash equilibria were good. The
maximum frequency strategy chosen by Type 1 subjects was s1 = 8 with 224 choices
out of 400, the second was s1 = 9 with 72 choices, and the third was s1 = 10 with 70
choices. The total frequency of dominant strategies selected by Type 1 subjects was
296. The maximum frequency strategy chosen by Type 2 was s2 = 12 with 167 choices
out of 400, the second was s2 = 13 with 160 choices, and the third was s1 = 20 with 15

21

Figure 3: All pairs choices in Treatment P.

choices. Hence, the total frequency of dominant strategies selected by Type 2 subjects
was 327.
Figure 4 displays the rates that each type of subjects chose dominant strategies
separately as well as the rates that pairs of subjects played dominant strategy equilibria
for all periods.
These results are summarized by the following observations.
Observation 2.
(i) The frequency of dominant strategy equilibria was 61% (= 242) across all periods
400
in Treatment P.
(ii) The frequency of good Nash equilibria was 93% (= 370) across all periods in
400
Treatment P.
(iii) The frequency that Type 1 subjects chose dominant strategies was 74% (= 296)
400
across all periods in Treatment P.
(iv) The frequency that Type 2 subjects chose dominant strategies was 82% (= 327)
400
across all periods in Treatment P.

22

Figure 4: Rates that subjects chose dominant strategies in Treatment P.

5.3

Comparing the two mechanisms

We compare the performances of the voluntary contribution mechanism and those of
the pivotal mechanism based on the following four indices:• Pareto eﬃciency index: We deﬁne the Pareto eﬃciency index at each period
as the ratio of Pareto eﬃcient outcomes at that period. The sum of two subjects’
payoﬀs in one pair is equal to 340 at any Pareto eﬃcient outcome.
• Individual rationality index: We deﬁne the individual rationality index ateach period as the ratio of individually rational outcomes at that period suchthat each subject receives a payoﬀ at least as large as their own payoﬀ before
participating in the mechanism (160), that is, the payoﬀ the subject receives fromtheir initial endowment of the private good with no public project.
• Surplus index: We deﬁne the surplus index by
Surplus Index =average of sums of two subjects’ payoﬀs − reference payoﬀ,
Pareto eﬃcient payoﬀ − reference payoﬀ

where the Pareto eﬃcient payoﬀ is the sum of two subjects’ payoﬀs at any Paretoeﬃcient outcome, which equals 340, and the reference payoﬀ is the sum of two
subjects’ payoﬀs that they receive before participating in the mechanism, which23equals 320. This index measures how much gain subjects obtain by participating
in a mechanism.
• Decision eﬃciency index: We deﬁne the decision eﬃciency index at eachperiod as the ratio of outcomes at which the public project is undertaken at that
period.
Figure 5 displays the mean Pareto eﬃciency index, the mean individual rationality
index, the mean surplus index, and the mean decision eﬃciency index at each periodin Treatments V and P. As shown in this ﬁgure, we observed that the mean Paretoeﬃciency index, the mean individual rationality index, and the surplus index for Treatment V were higher than those for Treatment P in each period (except for the ﬁrst
period for the surplus index). We also observed that the mean decision eﬃciency index
for Treatment P was higher than that for Treatment V in 18 out of 20 periods (except
for periods 8 and 11), although the mean decision eﬃciency index for Treatment V
attained at least 80% in 16 out of 20 periods (except for periods 1–3 and 20).
To formally test these graphical observations, we ran the panel regressions. Table 6
reports the panel regression results of the pooled data. Among the four regressions, therandom eﬀects panel probit model was applied in the regressions of the Pareto eﬃciency
index, the individual rationality index, and the decision eﬃciency index, while the random eﬀects panel model was conducted for the regression of the surplus index. Besides
the continuous variable of Period controlling for round eﬀects, the dummy variable ofTreatment P was included in the four regressions as the independent variable, leaving
Treatment V as the base of comparison. As presented in the table, the dummy variable of Treatment P was estimated with signiﬁcantly negative sign in the regressionsof the Pareto eﬃciency index, the individual rationality index, and the surplus index,
implying that the voluntary contribution mechanism is superior to the pivotal mechanism based on these criteria. However, this variable was signiﬁcantly and positively
estimated in the regression of the decision eﬃciency index, which exhibits a better
performance of the pivotal mechanism than the voluntary contribution mechanism inthe sense of decision eﬃciency.
These results are summarized by the following observations.
Observation 3.
(i) The Pareto eﬃciency index for Treatment V was signiﬁcantly higher than the
Pareto eﬃciency index for Treatment P.

24

(a) Pareto eﬃciency index(b) Individual rationality index(c) Surplus index(d) Decision eﬃciency index

Figure 5: Comparing the voluntary contribution mechanism and the pivotal mechanism
based on four indices.

(ii) The individual rationality index for Treatment V was signiﬁcantly higher than
the individual rationality index for Treatment P.
(iii) The surplus index for Treatment V was signiﬁcantly higher than the surplus index
for Treatment P.
(iv) The decision eﬃciency index for Treatment V was signiﬁcantly lower than thedecision eﬃciency index for Treatment P.

25

IndexParetoIndividual
Surpluseﬃciency rationality−2.012∗∗−4.662∗∗ −0.494∗∗
Treatment P(0.448)(0.300)(0.069)∗0.046−0.0120.013∗
Period(0.020)(0.022)(0.006)0.411
2.969∗∗
0.640∗∗
Constant(0.461)(0.408)(0.112)Log-likelihood −365.671 −70.720
R2 (overall)
0.248
Observations
800
800
800

Decisioneﬃciency
0.747∗∗(0.262)0.049∗(0.023)0.333(0.461)−289.310
800

Notes: ∗∗ p < 0.001, ∗ p < 0.01. Standard errors in the parentheses are clustered in session.

Table 6: Panel regressions on the four indices6

Concluding remarks

In our experiment, we used the payoﬀ tables to simplify the presentation of the two
mechanisms for subjects. We provided no explanation regarding the rules of the mechanisms or how the payoﬀ tables were constructed. The rule of the pivotal mechanism
is more complicated than that of the voluntary contribution mechanism. Using payoﬀ
tables only allows for a comparison of the two mechanisms holding their degree of transparency constant. We selected maximally-transparent conditions as a ﬁrst step in thisinitial experiment, but payoﬀ tables are somewhat unrealistic for potential applicationsof these mechanisms in the ﬁeld. Hence, we need to conduct another experiment tocheck whether the voluntary contribution mechanism performs better than the pivotal
mechanism when the rules of the two mechanisms are explained. Since the rule ofthe pivotal mechanism is more diﬃcult to understand, it is natural to expect that thevoluntary contribution mechanism would perform better than the pivotal mechanism
when the rules of the mechanisms are explained. This is left for future experiments.
Several other questions remain to be answered. We do not consider incomplete information in which all agents know their own true values, but not the values to others.
In contrast to the complete information setting, the theoretical predictions suggest thatthe pivotal mechanism might perform better than the voluntary contribution mechanism under incomplete information. Thus, it would be worth investigating the performances of these mechanisms under incomplete information by economic experiments.
Moreover, there were three or more agents in each group in some existing experiments26

(e.g., Bagnoli and McKee (1991), Marks and Croson (1999), and Attiyeh et al. (2000)),
whereas there were two subjects in each group in our experiment. Therefore, it isworthwhile to compare experimentally the performance of the voluntary contribution
mechanism and that of the pivotal mechanism when there are more than two subjects
in each group. We hope to address these issues in future work.

27

A

Appendix: Proofs

A.1

Proof of Fact 1

The “if” part. Suppose, by contradiction, that (y, t) ∈/ P (v). Then, there is (y ′ , t′ ) ∈A such that ui (y ′ , t′i ; vi ) ≥ ui (y, ti ; vi ) for each i ∈ {1, 2} with a strict inequality forsome i ∈ {1, 2}. There are two cases.
• Case 1: v1 + v2 ≥ c. Then,u1 (y ′ , t′1 ; v1 ) + u2 (y ′ , t′2 ; v2 ) > u1 (y, t1 ; v1 ) + u2 (y, t2 ; v2 ) = v1 + v2 − c + ω1 + ω2 .
If y ′ = 1, thenv1 + v2 + ω1 + ω2 + t′1 + t′2 = u1 (y ′ , t′1 ; v1 ) + u2 (y ′ , t′2 ; v2 )> u1 (y, t1 ; v1 ) + u2 (y, t2 ; v2 )= v1 + v2 − c + ω1 + ω2 ,which implies −(t′1 + t′2 ) < c, a contradiction. If y ′ = 0, then
ω1 + ω2 + t′1 + t′2 = u1 (y ′ , t′1 ; v1 ) + u2 (y ′ , t′2 ; v2 )> u1 (y, t1 ; v1 ) + u2 (y, t2 ; v2 )= v1 + v2 − c + ω1 + ω2 ,which implies −(t′1 + t′2 ) < c − (v1 + v2 ) ≤ 0, a contradiction.
• Case 2: v1 + v2 < c. Then,u1 (y ′ , t′1 ; v1 ) + u2 (y ′ , t′2 ; v2 ) > u1 (y, t1 ; v1 ) + u2 (y, t2 ; v2 ) = ω1 + ω2 .
If y ′ = 1, thenv1 + v2 + ω1 + ω2 + t′1 + t′2 = u1 (y ′ , t′1 ; v1 ) + u2 (y ′ , t′2 ; v2 )> u1 (y, t1 ; v1 ) + u2 (y, t2 ; v2 )= ω1 + ω2 ,which implies −(t′1 + t′2 ) < v1 + v2 < c, a contradiction. If y ′ = 0, then
ω1 + ω2 + t′1 + t′2 = u1 (y ′ , t′1 ; v1 ) + u2 (y ′ , t′2 ; v2 )28

> u1 (y, t1 ; v1 ) + u2 (y, t2 ; v2 )= ω1 + ω2 ,which implies −(t′1 + t′2 ) < 0, a contradiction.
The “only if” part. Let (y, t) ∈ P (v). There are three cases.
• Case 1: v1 + v2 > c. In this case, it suﬃces to show that y = 1, because this
together with (y, t) ∈ P (v) implies that u1 (y, t1 ; v1 )+u2 (y, t2 ; v2 ) = v1 +v2 −c+ω1 +ω2 .
To show this, suppose by contradiction that y = 0. Then, u1 (y, t1 ; v1 ) = ω1 + t1 ,u2 (y, t2 ; v2 ) = ω2 + t2 , and −(t1 + t2 ) ≥ 0. Without loss of generality, assume t1 ≥ t2 .
Let (y ′ , t′ ) ∈ A be such that y ′ = 1, −(t′1 + t′2 ) = c, and t′1 = t1 − v1 . Since v1 + v2 > c,
v1 + v2 − c = v1 + v2 + (t′1 + t′2 ) > 0 ≥ t1 + t2 , which implies that v2 + t′2 > t2 . It thenfollows thatu1 (y ′ , t′1 ; v1 ) = v1 + t′1 + ω1 = t1 + ω1 = u1 (y, t1 ; v1 ),u2 (y ′ , t′2 ; v2 ) = v2 + t′2 + ω1 > t2 + ω2 = u2 (y, t2 ; v2 ),which contradict (y, t) ∈ P (v).
• Case 2: v1 + v2 < c. In this case, it suﬃces to show that y = 0, because this
together with (y, t) ∈ P (v) implies that u1 (y, t1 ; v1 ) + u2 (y, t2 ; v2 ) = ω1 + ω2 . To
show this, suppose by contradiction that y = 1. Then, u1 (y, t1 ; v1 ) = v1 + ω1 + t1 ,u2 (y, t2 ; v2 ) = v2 + ω2 + t2 , and −(t1 + t2 ) ≥ c > 0. Without loss of generality, assume
v1 + t1 ≥ v2 + t2 . Let (y ′ , t′ ) ∈ A be such that y ′ = 0, t′1 + t′2 = 0, and t′1 = v1 + t1 .
Since v1 + v2 < c, t′1 + t′2 = 0 > v1 + v2 − c ≥ v1 + v2 + (t1 + t2 ), which implies thatt′2 > v2 + t2 . It then follows thatu1 (y ′ , t′1 ; v1 ) = ω1 + t′1 = v1 + t1 + ω1 = u1 (y, t1 ; v1 ),u2 (y ′ , t′2 ; v2 ) = ω1 + t′2 > v2 + t2 + ω2 = u2 (y, t2 ; v2 ),which contradict (y, t) ∈ P (v).
• Case 3: v1 + v2 = c. Then, either y = 1 or y = 0. If y = 1, then it follows from(y, t) ∈ P (v) that u1 (y, t1 ; v1 ) + u2 (y, t2 ; v2 ) = v1 + v2 − c + ω1 + ω2 . If y = 0, then it
follows from (y, t) ∈ P (v) that v1 + v2 = c, that u1 (y, t1 ; v1 ) + u2 (y, t2 ; v2 ) = ω1 + ω2 =v1 + v2 − c + ω1 + ω2 .
□

29

A.2

Proof of Fact 2

We prove this fact by contraposition. Suppose that an allocation (y, t) ∈ A is notdecision eﬃcient for v. Then, y ∈/ arg maxy′ ∈{0,1} (v1 + v2 ) · y ′ − c · y ′ . There are twocases.
• Case 1: y = 1. Then,
0 > v1 + v2 − c.

(2)

Let (y ′ , t′ ) ∈ A be such that y ′ = 0 and for each i ∈ {1, 2},t′i ≡ ti + vi +c − (v1 + v2 ).
2

Note that by −(t1 + t2 ) ≥ c,t′1 + t′2 = t1 + t2 + v1 + v2 + c − (v1 + v2 )= t1 + t2 + c≤ 0.
By (2), we havec − (v1 + v2 )> ω1 + t1 + v1 = u1 (y, t1 ; v1 ),2c − (v1 + v2 )u2 (y ′ , t′2 ; v2 ) = ω2 + t′2 = ω2 + t2 + v2 +> ω2 + t2 + v2 = u2 (y, t2 ; v2 ).
2

u1 (y ′ , t′1 ; v1 ) = ω1 + t′1 = ω1 + t1 + v1 +These imply that (y, t) ∈/ P (v).
• Case 2: y = 0. Then,
v1 + v2 − c > 0.
Let (y ′ , t′ ) ∈ A be such that y ′ = 1 and for each i ∈ {1, 2},t′i ≡ ti − vi +v1 + v2 − c.
2Note that by −(t1 + t2 ) ≥ 0,
t′1 + t′2 = t1 + t2 − v1 − v2 + v1 + v2 − c= t1 + t2 − c

30

(3)

≤ −c.
By (3), we havev1 + v2 − c> ω1 + t1 = u1 (y, t1 ; v1 ),2
v1 + v2 − cu2 (y ′ , t′2 ; v2 ) = v2 + ω2 + t′2 = ω2 + t2 +> ω2 + t2 = u2 (y, t2 ; v2 ).
2

u1 (y ′ , t′1 ; v1 ) = v1 + ω1 + t′1 = ω1 + t1 +These imply that (y, t) ∈/ P (v).

A.3□

Proof of Fact 3

The “if” part. Suppose, by contradiction, that (y, t) ∈ A is not individually rational
for v. Then, there are i ∈ {1, 2} and (y ′ , t′i ) ∈ Ai such that ui (y ′ , t′i ; vi ) > ui (y, ti ; vi ).
There are two cases.
• Case 1: vi ≥ c. Then, ui (y ′ , t′i ; vi ) > ui (y, ti ; vi ) ≥ vi − c + ωi . If y ′ = 1, thenvi + ωi + t′i > vi − c + ωi , which implies −t′i < c, a contradiction. If y ′ = 0, thenωi + t′i > vi − c + ωi , which implies −t′i < c − vi < 0, a contradiction.
• Case 2: vi < c. Then, ui (y ′ , t′i ; vi ) > ui (y, ti ; vi ) ≥ ωi . If y ′ = 1, then vi +ωi +t′i > ωi ,which implies −t′i < vi < c, a contradiction. If y ′ = 0, then ωi + t′i > ωi , which implies−t′i < 0, a contradiction.
The “only if” part. We prove this by contraposition. Suppose that there are (y, t) ∈A and i ∈ {1, 2} such that (1) does not hold. If vi ≥ c, then ui (y, ti ; vi ) < vi − c + ωi .
Let (y ′ , t′i ) ∈ Ai be such that y ′ = 1 and t′i = −c. Then, ui (y, ti ; vi ) < ui (y ′ , t′i ; vi ),which implies (y, t) ∈/ I(v). If vi < c, then ui (y, ti ; vi ) < ωi . Let (y ′ , t′i ) ∈ Ai be suchthat y ′ = 0 and t′i = 0. Then, ui (y, ti ; vi ) < ui (y ′ , t′i ; vi ), which implies (y, t) ∈/ I(v). □A.4

Proof of Proposition 1

Let v ∈ V . We proceed in two steps.12
Step 1: For each i ∈ {1, 2}, U (SiVC ; (ΓVC , v)) = [0, vi[. Let i ∈ {1, 2}. Without
loss of generality, assume i = 1.
Let a, b ∈ R be such that a ≤ b. Then, we denote by [a, b] and ]a, b[ the closed interval from ato b and the open interval from a to b, respectively. We also denote by [a, b[ and ]a, b] the half-openintervals from a to b.
1231◦ Substep 1-1: Any s1 ∈ [v1 , ω1 ] is weakly dominated. Let s1 ∈ [0, v1 [. Since ω2 ≥ c by
VC
A1, there is s∗2 ∈ S2VC such that s1 + s∗2 = c. Let s∗∗
be such that s′1 + s∗∗
2 ∈ S22 = c.
∗∗∗′
VC
Note that s2 < s2 . Let s2 ∈ S2 . There are three cases.
VC• Case 1: s′2 ∈ [0, s∗∗(s1 , s′2 ) = g VC (s′1 , s′2 ) = (0, (0, 0)), which implies
2 [. Then, g

u1 (g1VC (s2 , s′2 ); v1 ) = ω1 = u1 (g1VC (s′2 , s′2 ); v1 ).
∗VC• Case 2: s′2 ∈ [s∗∗(s1 , s′2 ) = (0, (0, 0)) and g VC (s′1 , s′2 ) = (1, (−s′1 , −s′2 )).
2 , s2 [. Then, g
Since s′1 ≥ v1 ,u1 (g1VC (s1 , s′2 ); v1 ) = ω1 ≥ v1 + ω1 − s′1 = u1 (g1VC (s′1 , s′2 ); v1 ).
• Case 3: s′2 ∈ [s∗2 , ω2 ]. Then, g VC (s1 , s′2 ) = (1, (−s1 , −s′2 )) and g VC (s′1 , s′2 ) =(1, (−s′1 , −s′2 )). Since s′1 > s1 ,u1 (g1VC (s1 , s′2 ); v1 ) = v1 + ω1 − s1 > v1 + ω1 − s′1 = u1 (g1VC (s′1 , s′2 ); v1 ).
By Cases 1–3, we can conclude that s′1 is weakly dominated.
◦ Substep 1-2: Any s1 ∈ [0, v1 [ is not weakly dominated. Note that there is s∗2 ∈ S2VC
be such that s1 + s∗2 = c. Let s′1 ∈ S1VC \ {s1 }. There are two cases.
• Case 1: s1 > s′1 . Then g VC (s1 , s∗2 ) = (1, (−s1 , −s∗2 )) and g VC (s′1 , s∗2 ) = (0, (0, 0)).
Since v1 > s1 ,u1 (g1VC (s1 , s∗2 ); v1 ) = v1 + ω1 − s1 > ω1 = u1 (g1VC (s′1 , s∗2 ); v1 ).
• Case 2: s1 < s′1 . Then, g VC (s1 , s∗2 ) = (1, (−s1 , −s∗2 )) and g VC (s′1 , s∗2 ) = (1, (−s′1 , −s∗2 )).
Since s′1 > s1 ,u1 (g1VC (s1 , s∗2 ); v1 ) = v1 + ω1 − s1 > v1 + ω1 − s′1 = u1 (g1VC (s′1 , s∗2 ); v1 ).
By Cases 1–2, s1 is not weakly dominated.
Step 2: A(ΓVC , v) ⊆ C(v). There are two cases.
• Case 1: v1 + v2 > c. For each i ∈ {1, 2}, letsi ≡ inf {si ∈ [0, vi [ : si + sj = c for some sj ∈ [0, vj [} .

32

Note that under A1, for each si ∈ [0, si ] and each sj ∈ [0, vj [, g VC (si , sj ) = (0, (0, 0)).
◦ Substep 2-1: For each i ∈ {1, 2} and each s′i ∈ [0, si ], s′i is weakly dominated in
the game (ΓVC (U ), v). Let i ∈ {1, 2}. Without loss of generality, assume i = 1. Let
s1 , s′1 ∈ [0, v1 [ be such that s′1 ≤ s1 < s1 . Then, there is s∗2 ∈ [0, v2 [ such that s1 +s∗2 = c.
It follows that g VC (s1 , s2 ) = (0, (0, 0)) if s2 < s∗2 and g VC (s1 , s2 ) = (1, (−s1 , −s2 ))otherwise. Let s2 ∈ [0, v2 [. We also distinguish two subcases.
1. If s2 < s∗2 , then g1VC (s1 , s2 ) = g1VC (s′1 , s2 ) = (0, (0, 0)), which implies
u1 (g1VC (s1 , s2 ); v1 ) = ω1 = u1 (g1VC (s′1 , s2 ); v1 ).
2. If s2 ≥ s∗2 , then g1VC (s1 , s2 ) = (1, (−s1 , −s2 )) and g1VC (s′1 , s2 ) = (0, (0, 0)). Since
v1 > s 1 ,
u1 (g1VC (s1 , s2 ); v1 ) = v1 + ω1 − s1 > ω1 = u1 (g1VC (s′1 , s2 ); v1 ).
Hence s′1 is weakly dominated.
◦ Substep 2-2: For each i ∈ {1, 2} and each si ∈ ]si , vi [, si is not weakly dominated in
the game (ΓVC (U ), v). Let i ∈ {1, 2}. Without loss of generality, assume i = 1. Let
s1 ∈ ]s1 , v1 [. Note that there is s∗2 ∈ [0, v2 [ such that s1 + s∗2 = c. Let s′1 ∈ [0, v1 [ \{s1 }.
We also distinguish two subcases.
1. If s1 > s′1 , then g1VC (s1 , s∗2 ) = (1, (−s1 , −s∗2 )) and g1VC (s′1 , s∗2 ) = (0, (0, 0)). Since
v1 > s 1 ,
u1 (g1VC (s1 , s∗2 ); v1 ) = v1 + ω1 − s1 > ω1 = u1 (g1VC (s′1 , s∗2 ); v1 ).
2. If s1 < s′1 , then g1VC (s1 , s∗2 ) = (1, (−s1 , −s∗2 )) and g1VC (s′1 , s∗2 ) = (1, (−s′1 , −s∗2 )).
Since s1 < s′1 ,
u1 (g1VC (s1 , s∗2 ); v1 ) = v1 + ω1 − s1 > v1 + ω1 − s′1 = u1 (g1VC (s′1 , s∗2 ); v1 ).
Hence s1 is not weakly dominated.
◦ Substep 2-3: E(ΓVC , v) = {(s1 , s2 ) ∈ [0, v1 [ × [0, v2 [ : s1 + s2 = c}. We ﬁrst show that
E(ΓVC , v) ⊆ {(s1 , s2 ) ∈ [0, v1 [ × [0, v2 [ : s1 + s2 = c}. Let s ∈ E(ΓVC , v). Suppose, bycontradiction, that s1 + s2 ̸= c. If s1 + s2 > c, there is i ∈ {1, 2} with si > 0. Let ε > 033

be such that si + sj > c + ε. Let s′i = si − ε. Then,ui (giVC (s′i , sj ); vi ) = vi + ωi − si + ε > vi + ωi − si = ui (giVC (si , sj ); vi ),a contradiction. If s1 + s2 < c, by Substeps 2.1 and 2.2, then s1 > s1 and s2 > s2 . Leti ∈ {1, 2}. Then, there is s′i such that s′i + sj = c. Since vi > si ,ui (giVC (s′i , sj ); vi ) = vi + ωi − si > ωi = ui (giVC (si , sj ); vi ),a contradiction.
We next show that {(s1 , s2 ) ∈ [0, v1 [ × [0, v2 [ : s1 + s2 = c} ⊆ E(ΓVC , v). Let
s′ = (s′1 , s′2 ) ∈ {(s1 , s2 ) ∈ [0, v1 [ × [0, v2 [ : s1 + s2 = c}. Then, for each i, j ∈ {1, 2} andeach s′′i (> si ),ui (giVC (s′i , s′j ); vi ) = vi + ωi − si ≥ ui (giVC (s′′i , s′j ); vi ).
Moreover, since s′1 + s′2 = c, both s′1 and s′2 are not weakly dominated. Thus, s′ ∈E(ΓVC , v).
◦ Substep 2-4: Concluding. It follows from Substep 2-3 that A(ΓVC , v) ⊆ C(v).
• Case 2: v1 + v2 ≤ c. To show A(ΓVC , v) ⊆ C(v), we ﬁrst show that E(ΓVC , v) =[0, v1 [ × [0, v2 [. It is obvious that E(ΓVC , v) ⊆ [0, v1 [ × [0, v2 [. Therefore, it suﬃces toshow that [0, v1 [ × [0, v2 [ ⊆ E(ΓVC , v). Let (s1 , s2 ) ∈ [0, v1 [ × [0, v2 [. Since s1 + s2 < c,g VC (s) = (0, (0, 0)). It follows that for each i ∈ {1, 2}, si is not weakly dominated in thegame (ΓVC (U ), v) and, moreover, (s1 , s2 ) is a Nash equilibrium of the game (ΓVC (U ), v).
Hence (s1 , s2 ) ∈ E(ΓVC , v). It then follows that A(ΓVC , v) = {(0, (0, 0))} ⊆ C(v). □B

Appendix: Implementation in iterated elimination of weakly dominated strategies

This section shows that the voluntary contribution mechanism cannot implement any
sub-correspondence of the core in iterated elimination of weakly dominated strategies.
To see this, suppose that ω1 = ω2 = c = 8.
We now consider the case where v = (7, 7). Let Ui1 ≡ U (SiVC ; (ΓVC , v)) and U 1 ≡U11 × U21 . From the proof of Proposition 1 (Step 2), we obtain the following facts:• For each i ∈ {1, 2}, Ui1 = [0, 7[;34• For each i ∈ {1, 2} and each si ∈ [0, 1[, si is weakly dominated in the game(ΓVC (U 1 ), v); and• For each i ∈ {1, 2} and each si ∈ Ui1 \ [0, 1[ = ]1, 7[, si is not weakly dominatedin the game (ΓVC (U 1 ), v).
Therefore, for each i ∈ {1, 2}, U12 ≡ U (Ui1 ; (ΓVC (U 1 ), v)) = ]1, 7[. We now show that foreach i ∈ {1, 2} and each si ∈ Ui2 , si is not weakly dominated in the game (ΓVC (U 2 ), v).
Let i ∈ {1, 2}. Without loss of generality, assume i = 1. Let s1 ∈ U12 . Note that thereis s∗2 ∈ U22 such that s1 + s∗2 = c = 8. Let s′1 ∈ U12 \ {s1 }. There are two cases.
• Case 1: s1 > s′1 . Then, g1VC (s1 , s∗2 ) = (1, (−s1 , −s∗2 )) and g1VC (s′1 , s∗2 ) = (0, (0, 0)).
Since v1 = 7 > s1 ,u1 (g1VC (s1 , s∗2 ); v1 ) = v1 + ω1 − s1 = 8 + (7 − s1 ) > 8 = ω1 = u1 (g1VC (s′1 , s∗2 ); v1 ).
• Case 2: s1 < s′1 . Then, g1VC (s1 , s∗2 ) = (1, (−s1 , −s∗2 )) and g1VC (s′1 , s∗2 ) = (1, (−s′1 , −s∗2 )).
Since s1 < s′1 ,
u1 (g1VC (s1 , s∗2 ); v1 ) = v1 + ω1 − s1 > v1 + ω1 − s′1 = u1 (g1VC (s′1 , s∗2 ); v1 ).
From Cases 1–2, we can conclude that s1 is not weakly dominated. This fact
implies that the set of strategies that survive iterated elimination of weakly dominated strategies is ]1, 7[ × ]1, 7[. We now consider the strategy proﬁle (2, 2). Then,(2, 2) ∈ ]1, 7[ × ]1, 7[ but g VC (2, 2) = (0, (0, 0)) ∈/ C(v). This implies that the voluntary contribution mechanism fails to implement any sub-correspondence of the core in
iterated elimination of weakly dominated strategies.

C

Appendix: Experimental instructions

In this experiment, please remember that you cannot talk or communicate with othersubjects. If there is communication, the experiment will be stopped at that point.
First, please conﬁrm that the following items are on your desk. If any of the items
are missing, please contact an experimenter.
• Instructions (this set of papers)• Payoﬀ table for practice35• Record sheet for practice• Pencil and eraser

C.1

Overview

In this experiment, at the beginning of each period, an experimenter will choose theperson you are paired with from the other subjects at random. The person you are pairedwith will change each period. This experiment consists of 20 periods. As explainedin more detail below, at the beginning of each period, you will make decisions using
a computer assigned to you based on the “Payoﬀ table”. You will choose one integer
number from 1 to 20 in each period. No subject knows who they have been pairedwith either during or after the experiment.
The rewards you receive after the experiment is complete are proportional to thetotal payoﬀs you earn throughout the 20 periods of the experiment. A detailed explanation of the rewards you receive will be provided later in Section C.4 (Rewards).

C.2

Details

You will choose one integer number from 1 to 20 in each period. Your payoﬀ in eachperiod is determined by the “Outcome” of the period. The outcome is determined by
the number you choose as well as the number the other person chooses. Your “Payoﬀ
table” describes the relation between outcomes and the payoﬀs you earn.
Your “Payoﬀ table” displays both your payoﬀ and the other person’s payoﬀ when
the number you choose and the number the other person chooses are determined. The
following table is a part of the “Payoﬀ table for practice”.

The vertical line of the above table shows the number you choose, and the horizontal
line is the number the other person chooses. The lower left-hand red number is your36

payoﬀ, and the upper right-hand blue number is the other person’s payoﬀ in each cell.
The numbers in each cell are displayed in yen.
For example, suppose that the number you choose is “3”, and the number the otherperson chooses is “1”. In this case, your payoﬀ is “140” when the vertical line is 3 andthe horizontal line is 1. Then, the other person’s payoﬀ is “100”.
The person you are paired with is chosen from the other subjects at random by anexperimenter. The person you are paired with has a payoﬀ table in which the verticalline and the horizontal line of your payoﬀ table are reversed.

C.3

Operation

This experiment consists of 20 periods. Here, we will explain how to operate thecomputer you will use in period 1 of the experiment. The operation after period 1 isthe same as the operation in period 1.
1. At the beginning of period 1, you will be paired with the person an experimenter
will choose from the other subjects at random. Then, “Input one integer number
from 1 to 20” is displayed on the screen of your computer. The following ﬁgure
is an example of the screen.

37

2. Please look at the “Payoﬀ table” and conﬁrm the relation between the numbers
you and the other person choose and the payoﬀs.
3. After deciding which number to choose, you input that number in the cell of “The
number you chose”. Then, click the “OK” button at the bottom of the screen.
4. After inputting the number you chose on your computer, ﬁll out the number you
chose in the column “the number you chose” in the record sheet. Moreover, pleaseﬁll out why you chose that number in the column “Reason for your decision” inthe record sheet.
5. After all subjects click the “OK” button, period 1 is complete. Then, “The
number you chose”, “The number the other person chose”, and “Your payoﬀ”
are displayed on the screen. We ask that you transcribe this information in therecord sheet.
6. After the transcription, please the “NEXT” button at the bottom of the screen.
Once all subjects click the “NEXT” button, period 2 will start. At the beginningof period 2, you will be paired with the person an experimenter will choose from theother subjects at random. The operation after period 1 is the same as the operationin period 1. The experiment is complete when period 20 is complete.

C.4

Rewards

Your rewards are the sum of your payoﬀs over all 20 periods. For example, if the sumof your payoﬀs are 2,580, then your rewards are 2,580 yen. If you earn more payoﬀs ineach period, then you receive more rewards.
This concludes our explanation. Next, you can practice making decisions using the“Payoﬀ table for practice”. After practicing, you will have some time to look at the
payoﬀ table for the actual experiment before we begin the experiment. If you have any
questions, please raise your hand.

38

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