Research InstituteDiscussion Paper Series№04
Title：

Securely Implementable Social Choice Functions in
Divisible and Non-Excludable Public Good Economies
with Quasi-Linear Utility Functions

NISHIZAKI Katsuhiko ／ 西﨑 勝彦
ka-nishi@andrew.ac.jp

2017 年 9 月

http://www.andrew.ac.jp/soken/

Securely Implementable Social Choice Functions
in Divisible and Non-Excludable Public Good Economies
with Quasi-Linear Utility Functions ∗Katsuhiko Nishizaki †‡May 19, 2017

Abstract
This paper studies the possibility of secure implementation (Saijo, T., T. Sjöström, and T. Yamato (2007) “Secure Implementation,” Theoretical Economics 2, pp.203-229) in divisible and nonexcludable public good economies with quasi-linear utility functions. In this economies, although
Saijo, Sjöström, and Yamato (2007) showed that the Groves mechanisms (Groves, T. (1973) “Incentives in Teams,” Econometrica 41, pp.617-631) are securely implementable when the valuation
functions of the public good are (a) differentiable and concave and (b) fixed and the agents are identified with their parameters respectively, this paper presents the following negative result: securely
implementable social choice functions are dictatorial or constant when the valuation functions of the
public good are strictly increasing and strictly concave.
Keywords: Secure Implementation, Dominant Strategy Implementation, Nash Implementation,
Strategy-Proofness, Non-Excludable Public Good.
JEL Classification: C72, D61, D63, D71, H41.

∗ The author wishes to thank the editor, Rabah Amir, and an anonymous referee for their valuable comments and efforts. This

paper is based on Chapter 2 of the author’s Ph.D. thesis submitted to the Graduate School of Economics at Osaka University.
The author is grateful to Tatsuyoshi Saijo, Shigehiro Serizawa, and Yukihiro Nishimura for their helpful advices on the thesis.
This paper is also a product of research which was financially supported (in part) by the Kansai University Subsidy for Supporting Young Scholars (2014) “Robustness of Secure Implementation” and the research grant from the Research Institute forSocionetwork Strategies at Kansai University (2015-2017) “Implementation Theory and Rationality: Secure Implementation
Reconsidered”. Any errors in this paper are entirely the responsibility of the author.
† Graduate School of Economics, Momoyama Gakuin University, 1-1 Manabino, Izumi, Osaka 594-1198, Japan. TEL: +81
725 54 3131 (main phone number). FAX: +81 725 54 3202. E-mail: ka-nishi@andrew.ac.jp‡ Research Institute for Socionetwork Strategies, Kansai University, 3-3-35, Yamate, Suita, Osaka , 564-8680, Japan. TEL:
+81 6 6368 1228. FAX: +81 6 6330 3304.

1

Introduction

This paper considers divisible and non-excludable public good economies in which n ≥ 2 agents collectively decide (a) how much of the public good (e.g., seawalls, protection forests, and storm sewers)
should be provided and (b) how the cost should be shared among the agents. 1 These decisions are
made to achieve a goal characterized by a social choice function which associates an outcome withthe agents’ preferences. This paper studies strategy-proof social choice functions in such public good
economies with quasi-linear utility functions. Strategy-proofness requires that truthful revelation is aweakly dominant strategy for the agent.
Although strategy-proofness is a desirable property, some experimental studies have questioned theperformance of strategy-proof mechanisms. 2 On the basis of these observations, Saijo, Sjöström, andYamato (2007) introduced secure implementation which is defined as double implementation in dominant strategy equilibria and Nash equilibria. 3 They showed that the social choice function is securely
implementable if and only if it satisfies strategy-proofness and the rectangular property (Saijo, Sjöström,
and Yamato, 2007). In addition, they showed that the rectangular property is in general equivalent tothe combination of strong non-bossiness (Ritz, 1983) and the outcome rectangular property (Saijo,
Sjöström, and Yamato, 2007). Strong non-bossiness requires that the agent cannot change the allocationby changing the agent’s revelation while maintaining the agent’s utility. The outcome rectangular property requires that the allocation does not change by changing all the agents’ revelations, each of whom
does not change the allocation. On the basis of these characterizations, some researchers have studied
the possibility of secure implementation in several environments and illustrated the difficulty of finding
securely implementable social choice functions with desirable properties. 4
In divisible and non-excludable public good economies with quasi-linear utility functions, which arealso considered in this paper, Saijo, Sjöström, and Yamato (2007) showed that the Groves mechanisms(Groves, 1973) are securely implementable when the valuation functions of the public good are (a) differentiable and concave and (b) fixed and the agents are identified with their parameters respectively.
Because this assumption of the valuation functions is so restrictive, this paper considers the following
more reasonable assumption: the valuation functions of the public good are strictly increasing and strictly
concave. Under this assumption of the valuation functions, this paper demonstrates that securely implementable social choice functions are dictatorial or constant in divisible and non-excludable public good
economies with quasi-linear utility functions.
The main result presented here depends on the results of Barberà and Peleg (1990). In a voting
environment in which the set of alternatives is a metric space and each agent’s preference is represented
by a continuous utility function, they showed that if the social choice function satisfies strategy-proofness
and it’s range contains at least three alternatives, then it is dictatorial. Because secure implementability
1 See Clarke (1971), Moulin (1994), and Serizawa (1996, 1999) for non-excludable public good economies.
2 See Chen (2008) for a survey of experimental studies on strategy-proof mechanisms in non-excludable public good

economies.
3 Cason, Saijo, Sjöström, and Yamato (2006) conducted experiments on secure implementation and suggested that it might
be a benchmark for constructing a mechanism which works well in practice.
4 See Mizukami and Wakayama (2005), Saijo, Sjöström, and Yamato (2003, 2007), Fujinaka and Wakayama (2008, 2011),
Berga and Moreno (2009), Bochet and Sakai (2010), Kumar (2013), and Nishizaki (2011, 2012, 2013, 2014) for theoreticalresults on secure implementation.

1

reduces the problem of providing a divisible and non-excludable public good with the cost shares to sucha voting environment in the model presented here, this paper presents the main result on the basis of theresults of Barberà and Peleg (1990).
The remainder of this paper is organized as four sections. Section 2 introduces the model presented
here and Section 3 the main properties of social choice functions. Section 4 presents the main result andthe relationship between it and some previous studies. Section 5 concludes this paper. Some preliminary
results and the proof of the main result of this paper are presented in Appendix.

2

Model

Let I ≡ {1, . . . , n} be the set of agents, where n ≥ 2. Let Y ⊆ R+ ≡ {r ∈ R|r ≥ 0} be a convex set of
production levels of the public good and c : Y → R+ be the cost function. In the model presented here,a production level of the public good is equal to consumption of the public good for each agent. For eachi ∈ I, let (y, xi ) ∈ Y × R+ be a consumption bundle for agent i, where xi ∈ R+ is a cost share of the
public good for agent i. Let (y, x) be an allocation, where x ≡ (xi )i∈I is a profile of cost shares of the
public good, and Z ≡ {(y, x) ∈ Y × Rn+ |c(y) ≤ ∑i∈I xi } be the set of feasible allocations.
This paper assumes that each agent’s preference is represented by a quasi-linear utility function. For
each i ∈ I, let ui : Y × R+ → R be an utility function for agent i such that there is vi : Y → R, called avaluation function of the public good for agent i, and for each (y, xi ) ∈ Y × R+ , ui (y, xi ) = vi (y) − xi .
This paper also assumes that each valuation function of the public good is strictly increasing and strictly
concave. For each i ∈ I, let Vi be the set of all valuation functions of the public good for agent i with suchcharacteristics. Let v ≡ (vk )k∈I be a profile of valuation functions of the public good and V ≡ ∏k∈I Vk
be the set of the profiles. For each i ∈ I, let v−i ≡ (vk )k∈I\{i} be a profile of valuation functions of the
public good other than agent i and V−i ≡ ∏k∈I\{i} Vk be the set of the profiles. For each i, j ∈ I, letv−i, j ≡ (vk )k∈I\{i, j} be a profile of valuation functions of the public good other than agents i and j. For
each S, S′ , S′′ ⊆ I, where these sets are mutually disjoint and S ∪ S′ ∪ S′′ = I, and each v, v′ , v′′ ∈ V , let(vS , v′S′ , v′′S′′ ) be the profile of valuation functions of the public good, where agent i ∈ S has vi , agent i ∈ S′
has v′i , and agent i ∈ S′′ has v′′i .
Let f : V → Z be a social choice function. For each v ∈ V , let (y(v), x(v)) ∈ Z be the allocation under
the social choice function f at the profile of valuation functions of the public good v and (y(v), xi (v)) be
the consumption bundle for agent i ∈ I at the allocation (y(v), x(v)).

3

Properties of Social Choice Functions

Saijo, Sjöström, and Yamato (2007, Theorem 1) characterized securely implementable social choicefunctions by strategy-proofness and the rectangular property (Saijo, Sjöström, and Yamato, 2007).
Strategy-proofness requires that truthful revelation is a weakly dominant strategy for the agent.
Definition 1. The social choice function f satisfies strategy-proofness if and only if for each v, v′ ∈ Vand each i ∈ I, vi (y(vi , v′−i )) − xi (vi , v′−i ) ≥ vi (y(v′i , v′−i )) − xi (v′i , v′−i ).
In addition, Saijo, Sjöström, and Yamato (2007, Proposition 3) showed that the rectangular property
is in general equivalent to the combination of strong non-bossiness (Ritz, 1983) and the outcome rect2

angular property (Saijo, Sjöström, and Yamato, 2007). Strong non-bossiness requires that if the agent
does not change the agent’s “utility” by changing the agent’s revelation, then the allocation also does notchange by the change of the revelation. The outcome rectangular property requires that if each agent
cannot change the “allocation” by changing the agent’s revelation, then the allocation does not change
by changing all the agents’ revelations.
Definition 2. The social choice function f satisfies strong non-bossiness if and only if for each v, v′ ∈ Vand each i ∈ I, if vi (y(vi , v′−i )) − xi (vi , v′−i ) = vi (y(v′i , v′−i )) − xi (v′i , v′−i ), then (y(vi , v′−i ), x(vi , v′−i )) =(y(v′i , v′−i ), x(v′i , v′−i )).
Definition 3. The social choice function f satisfies the outcome rectangular property if and only if
for each v, v′ ∈ V , if (y(vi , v′−i ), x(vi , v′−i )) = (y(v′i , v′−i ), x(v′i , v′−i )) for each i ∈ I, then (y(v), x(v)) =(y(v′ ), x(v′ )).
Theorem 1 (Saijo, Sjöström, and Yamato, 2007). The social choice function is securely implementable if
and only if it satisfies strategy-proofness, strong non-bossiness, and the outcome rectangular property.
This paper shows that securely implementable social choice functions are dictatorial or constant indivisible and non-excludable public good economies with quasi-linear utility functions whose valuation
functions of the public good are strictly increasing and strictly concave.
Definition 4. The social choice function f is dictatorial if and only if there is i ∈ I such that for eachv, v′ ∈ V , vi (y(v)) − xi (v) ≥ vi (y(v′ )) − xi (v′ ). 5
Definition 5. The social choice function f is constant if and only if for each v, v′ ∈ V , (y(v), x(v)) =(y(v′ ), x(v′ )).

4 Main Result
Saijo, Sjöström, and Yamato (2007, Lemma 3) showed that the Groves mechanisms are securely implementable in some of divisible and non-excludable public good economies with quasi-linear utility
functions. A major difference between the model presented here and those of Saijo, Sjöström, and Yamato (2007) is the assumption of valuation functions of the public good. Although this paper assumes thatthe valuation functions of the public good are strictly increasing and strictly concave, Saijo, Sjöström,
and Yamato (2007) assumed that they are (a) differentiable and concave and (b) fixed and the agents are
identified with their parameters respectively (e.g., for each i ∈ I and each y ∈ Y , vi (y) = θi b(y), whereθi ∈ R and b : Y → R is differentiable and concave). 6 Under this assumption of Saijo, Sjöström, andYamato (2007), the Groves mechanisms satisfy strong non-bossiness and the outcome rectangular property in addition to strategy-proofness. In the model presented here, the Groves mechanisms do not satisfy
both properties. The following example shows a social choice function satisfying strategy-proofness andthe outcome rectangular property, but not strong non-bossiness.
5 Note that this dictatorship is required on the range of the social choice function, but not on the set of all feasible allocations.
6 Because this assumption does not require the strict increasingness, single-peaked valuation functions of the public goodare also considered by Saijo, Sjöström, and Yamato (2007). In addition, they assumed that the cost function is differentiable
and convex, the set of maximizers of the sum of all the agents’ benefits from the consumption is singleton, and the element isin the interior of the set of production levels of the public good.

3

y

v1c

v2c v1

v2
1 1
c
2

y (vi , vc i )y (vi , vi )y (vic, vc i )xiFigure 1: A violation of the outcome rectangular property under the conservative equal cost sharing
mechanism, where n = 2 and c is a liner cost function
Example 1. Let f be the following social choice function: there is y ∈ Y such that for each v ∈ V ,y(v) = y and xi (v) = −{∑k∈I\{i} vk (y(v)) − c(y(v))} for each i ∈ I. We find that f satisfies strategyproofness and the outcome rectangular property. In addition, we find that f does not satisfy strong
non-bossiness because the agent can change other agents’ cost shares of the public good by changing theagent’s revelation while maintaining the agent’s utility.
Although the cost shares of the public good under the social choice function in Example 1 are contained in those of the Groves mechanisms, the consumption of the public good does not maximize thesum of all the agents’ benefits from the consumption. Even if the consumption maximizes it, then thesocial choice function does not satisfy strong non-bossiness. Together with Theorem 1, this implies that
the Groves mechanisms are not securely implementable in the model presented here.
The conservative equal cost sharing mechanism (Moulin, 1994) is a well-known cost sharing scheme
satisfying strategy-proofness and non-bossiness (Satterthwaite and Sonnenschein, 1981) in non-excludable
public good economies. 7 For each i ∈ I, let ti : Y → R+ be a cost sharing function for agent i. The
social choice function f is a cost sharing scheme if and only if there are cost sharing functions t1 , · · · ,tnsuch that for each v ∈ V and each i ∈ I, xi (v) = ti (y(v)). Given a cost sharing scheme, for each v ∈ Vand each i ∈ I, let Bi (vi ,ti , y(V )) ≡ {y ∈ y(V )|vi (y) −ti (y) ≥ vi (y′ ) −ti (y′ ) for each y′ ∈ y(V )} be the set
of utility maximizers for agent i in the range of consumption of the public good y(V ) at the profile ofvaluation functions of the public good v and bi (vi ,ti , y(V )) ≡ max Bi (vi ,ti , y(V )). The social choice function f is the conservative equal cost sharing mechanism if and only if f is a cost cost sharing schemesuch that for each v ∈ V , ti (y(v)) = c(y(v))/n for each i ∈ I, where y(v) = min{bi (vi ,ti , y(V ))}i∈I .
In the model presented here, if the cost function c is convex, then the conservative equal cost sharing
mechanism satisfies strong non-bossiness, but not the outcome rectangular property (see Figure 1). 8 Together with Theorem 1, this implies that the conservative equal cost sharing mechanism is not securely
implementable.
It is well-known that a social choice function satisfying strategy-proofness and non-bossiness is a
7 The

social choice function f satisfies non-bossiness if and only if for each v, v′ ∈ V and each i ∈ I, if(y(vi , v′−i ), xi (vi , v′−i )) = (y(v′i , v′−i ), xi (v′i , v′−i )), then (y(vi , v′−i ), x(vi , v′−i )) = (y(v′i , v′−i ), x(v′i , v′−i )).
8 In this figure, ((y(v), x(v))) is the allocation induced by the unique weakly dominant strategy((y(v′ ), x(v′ ))) is an allocation induced by a “bad” Nash equilibrium.

4

equilibrium andcost sharing scheme (Lemma 5 in Appendix). Because strong non-bossiness is in general stronger thannon-bossiness, a social choice function satisfying strategy-proofness and strong non-bossiness is also a
cost sharing scheme. In addition, secure implementability implies that the range of consumption of the
public good is closed (Proposition 3 in Appendix) and convex (Proposition 4 in Appendix). These imply
that the problem of providing a divisible and non-excludable public good with the cost shares is reduced
to a voting environment in which the set of alternatives is equivalent to the range of consumption of
the public good, which is a closed interval by secure implementability. In such a voting environment,
Barberà and Peleg (1990, Theorem 3.1) showed the following negative result.
Theorem 2 (Barberà and Peleg, 1990). In a voting environment in which the set of alternatives is ametric space and each agent’s preference is represented by a continuous utility function defined overthe set, if the social choice function satisfies strategy-proofness and it’s range contains at least threealternatives, then it is dictatorial.
Note that Theorem 2 implies that any strategy-proof social choice function is constant if it is notdictatorial and it’s range is a closed interval in a voting environment considered by Barberà and Peleg(1990).
In the model presented here, each utility function induces a continuous preference defined over therange of consumption of the public good associated with secure implementability (Remark 2 in Appendix). 9 On the basis of Theorem 2, the following main result is presented.
Theorem 3. If the social choice function f satisfies strategy-proofness, strong non-bossiness, and theoutcome rectangular property, then f is dictatorial or constant. 10
Theorem 3 is tight. The argument about the conservative equal cost sharing mechanism showed thenecessity of the outcome rectangular property and Example 1 the necessity of strong non-bossiness. In
addtion, the following example shows the necessity of strategy-proofness.
Example 2. Let f be the following social choice function: there is y ∈ Y such that for each v ∈ V , y(v) = y
and xi (v) = −{∑k∈I vk (y(v)) − c(y(v))} for each i ∈ I. We find that f satisfies strong non-bossiness andthe outcome rectangular property. In addition, we find that f does not satisfy strategy-proofness because
the agent benefits from untruthful revelation that changes the agent’s cost share of the public good in theagent’s favor.
Together with Theorem 1, Theorem 3 implies the following negative result on secure implementation in divisible and non-excludable public good economies with quasi-linear utility functions whose
valuation functions of the public good are strictly increasing and strictly concave.
Corollary. If the social choice function is securely implementable, then it is dictatorial or constant.

5

Conclusion

This paper studies the possibility of secure implementation in divisible and non-excludable public good
economies with quasi-linear utility functions. In this economies, although Saijo, Sjöström, and Yamato
9 It is unclear whether the preferences are also single-peaked. Together with the result of Saijo, Sjöström, and Yamato (2007,

Theorem 8), we have the same result as Theorem 3 even if the preferences are single-peaked.
10 See Appendix for the formal proof of this theorem.

5(2007) showed that the Groves mechanisms are securely implementable when the valuation functions of
the public good are (a) differentiable and concave and (b) fixed and the agents are identified with theirparameters respectively, the results presented here implied that securely implementable social choicefunctions are dictatorial or constant when the valuation functions of the public good are strictly increasing
and strictly concave. This result depends on the results of Barberà and Peleg (1990) because secure
implementability reduces the problem of providing a divisible and non-excludable public good with thecost shares to a voting environment considered by Barberà and Peleg (1990).
Investigating securely implementable social choice functions in divisible and “excludable” public
good economies is an interesting research topic because there are non-trivial ones (e.g. a convex cost
sharing mechanism under which the convexity of the cost sharing functions is established on the range ofconsumption of the public good and each agent is assigned the consumption bundle which maximizes theagent’s utility according to the agent’s cost sharing function) although the serial cost sharing mechanism(Molin, 1994) is not securely implementable. 11 In addition, Saijo, Sjöström, and Yamato (2007) andKumar (2013) showed a positive result on secure implementation in the problems of providing a divisible
private good with monetary transfers and Nishizaki (2014) in pure exchange economies with Leontief
utility functions. These environments suggest further research on secure implementation.

Appendix: Some Preliminary Results and Proof of Theorem 3This paper demonstrates some preliminary results on strategy-proofness, strong non-bossiness, and theoutcome rectangular property. These results specify the characteristics of the option sets, the cost sharesof the public good, and the range of consumption of the public good under a securely implementablesocial choice function.
For each i ∈ I and each v′−i ∈ V−i , let Oi (v′−i ) ≡ {y ∈ Y |there is vi ∈ Vi such that y(vi , v′−i ) = y} bethe option set for agent i at v−i under the social choice function f , that is, the set of consumption of
the public good, which the agent can induce given f and v−i and Oi (V−i ) ≡ ∪v′−i ∈V−i Oi (v′−i ). In addition,
let y(V ) ≡ {y ∈ Y |there is v ∈ V such that y(v) = y} be the range of consumption of the public goodunder the social choice function f , that is, the set of consumption of the public good, which all theagents can induce given f . By definition, y(V ) ⊇ Oi (V−i ) for each i ∈ I. Lemma 1 shows that both setsare equivalent. 12
Lemma 1. For each i ∈ I, y(V ) = Oi (V−i ).
The cost sharing scheme f is (a) strictly increasing if and only if for each i ∈ I, each v′−i ∈ V−i ,and each y, y′ ∈ Oi (v′−i ), where y < y′ , ti (y) < ti (y′ ), (b) lower semi-continuous if and only if for eachi ∈ I, each v′−i ∈ V−i , each y ∈ Oi (v′−i ), and each ε > 0, there is a neighborhood U ⊆ Oi (v′−i ) of ysuch that ti (y′ ) ≥ ti (y) − ε for each y′ ∈ U, (c) upper semi-continuous if and only if for each i ∈ I andeach v′−i ∈ V−i , each y ∈ Oi (v′−i ), and each ε > 0, there is a neighborhood U ⊆ Oi (v′−i ) of y such that
′ ∈V , tti (y′ ) ≤ ti (y) + ε for each y′ ∈ U, and (d) continuous if and only if for each i ∈ I and each v−i
−i i
′
13
is upper semi-continuous and lower semi-continuous on Oi (v−i ).
11 Secure implementability of the serial cost sharing mechanism was noted by Yuji Fujinaka.
12 See Nishizaki (2016) for a proof of this lemma.
13 Note that these properties are required on the option sets, but not on the set of consumption of the public good.

6

A1. Results on Strategy-Proofness
Lemma 2 shows the strict increasingness of cost sharing schemes satisfying strategy-proofness. 14 On
the other hand, Lemma 3 shows the lower semi-continuity of cost sharing schemes satisfying strategyproofness on the basis of the continuity of valuation functions of the public good.
Lemma 2. If the cost sharing scheme satisfies strategy-proofness, then it is strictly increasing.
Lemma 3. If the cost sharing scheme f satisfies strategy-proofness, then it is lower semi-continuous.
Proof. To the contrary, we suppose that f is not lower semi-continuous. This implies that there are i ∈ I
and v′−i ∈ V−i such that ti is not lower semi-continuous on Oi (v′−i ). In addition, there is vi ∈ Vi suchthat ti is not lower semi-continuous at y(vi , v′−i ). This implies that there is ε ∈ R+ such that for eachneighborhood U ⊆ Oi (v′−i ) of y(vi , v′−i ),ti (y′ ) < ti (y(vi , v′−i )) − ε(1)

for some y′ ∈ U. By the continuity of valuation functions of the public good, we can take the neighborhood to satisfy the following condition:vi (y(vi , v′−i )) − vi (y′ ) < ε .

(2)Because U ⊆ Oi (v′−i ), there is v′i ∈ Vi such that y(v′i , v′−i ) = y′ and we find that vi (y(vi , v′−i )) −vi (y(v′i , v′−i )) <ε < ti (y(vi , v′−i )) −ti (y(v′i , v′−i )) by (1) and (2). This implies that vi (y(vi , v′−i )) −xi (vi , v′−i ) < vi (y(v′i , v′−i )) −′ ) and contradicts strategy-proofness.
xi (v′i , v−i
Proposition 1 shows the closedness of the option sets under a cost sharing scheme satisfying strategyproofness on the basis of Lemma 2 and the continuity and strict increasingness of valuation functions of
the public good.
Proposition 1. Suppose that the cost sharing scheme f satisfies strategy-proofness. For each i ∈ I andeach v′−i ∈ V−i , Oi (v′−i ) is closed.
Proof. To the contrary, we suppose that there are i ∈ I and v′−i ∈ V−i such that Oi (v′−i ) is not closed.
This implies that we can take y ∈ Ō(v′−i ) \ Oi (v′−i ), where Ō(v′−i ) is the closure of Oi (v′−i ). We have thefollowing three situations according to the relationship between y and Oi (v′−i ).
′ )Situation 1. y = inf Oi (v−i

Let xiH ≡ inf{xi ∈ R+ |there is vi ∈ Vi such that ti (y(vi , v′−i )) = xi }. By Lemma 2, the definition of xiH ,and the continuity and strict increasingness of valuation functions of the public good, we can take vi ∈ Visuch that vi (y) − vi (y(v′′i , v′−i )) > xiH −ti (y(v′′i , v′−i )) for each v′′i ∈ Vi . 15 This implies that vi (y(vi , v′−i )) −ti (y(vi , v′−i )) < vi (y) − xiH . Together with the supposition of y and the definition of xiH , this implies
that we can take v′i ∈ Vi such that vi (y(vi , v′−i )) − ti (y(vi , v′−i )) < vi (y(v′i , v′−i )) − ti (y(v′i , v′−i )), that is,vi (y(vi , v′−i )) − xi (vi , v′−i ) < vi (y(v′i , v′−i )) − xi (v′i , v′−i ). This contradicts strategy-proofness.
14 See Nishizaki (2016) for a proof of this lemma.
15 Note that we cannot take such a valuation function of the public good by the supposition of y and the strict increasingness ofvaluation functions of the public good if xiH −ti (y(v′′i , v′−i )) = 0 for each v′′i ∈ Vi . By Lemma 2, we find that xiH −ti (y(v′′i , v′−i )) <0 for each v′′i ∈ Vi because y(v′′i , v′−i ) = y and we have a contradiction to the definition of y if xiH = ti (y(v′′i , v′−i )) for some v′′i ∈ Vi .

7

′ )Situation 2. y = sup Oi (v−i

Let xiL ≡ sup{xi ∈ R+ |there is vi ∈ Vi such that ti (y(vi , v′−i )) = xi }. By Lemma 2, the definitionof xiL , and the continuity and strict increasingness of valuation functions of the public good, we cantake vi ∈ Vi such that vi (y) − vi (y(v′′i , v′−i )) > xiL − ti (y(v′′i , v′−i )) for each v′′i ∈ Vi . 16 This implies thatvi (y(vi , v′−i )) −ti (y(vi , v′−i )) < vi (y) − xiL . Together with the supposition of y and the definition of xiL , this
implies that we can take v′i ∈ Vi such that vi (y(vi , v′−i )) −ti (y(vi , v′−i )) < vi (y(v′i , v′−i )) −ti (y(v′i , v′−i )), thatis, vi (y(vi , v′−i )) − xi (vi , v′−i ) < vi (y(v′i , v′−i )) − xi (v′i , v′−i ). This contradicts strategy-proofness.
Situation 3. Otherwise
Let xiH ≡ inf{xi ∈ R+ |there is vi ∈ Vi such that ti (y(vi , v′−i )) = xi and y(vi , v′−i ) > y} and xiL ≡ sup{xi ∈R+ |there is vi ∈ Vi such that ti (y(vi , v′−i )) = xi and y(vi , v′−i ) < y}. By the supposition of y, we have thefollowing three cases according to whether xiH and xiL are induced by some valuation function of the pubH ′lic good or not: (i) there is vLi ∈ Vi such that ti (y(vLi , v′−i )) = xiL , but not vHi ∈ Vi such that ti (y(vi , v−i )) =
L
L ′
H
L
H ′
xiH , (ii) there is vH
i ∈ Vi such that ti (y(vi , v−i )) = xi , but not vi ∈ Vi such that ti (y(vi , v−i )) = xi , and
H
H ′
L ′
L
(iii) there are no vLi , vH
i ∈ Vi such that ti (y(vi , v−i )) = xi and ti (y(vi , v−i )) = xi . In the case (i), we
know that y ̸= y(vLi , v′−i ). Together with Lemma 2, the definition of xiH , and the continuity and strict
increasingness of valuation functions of the public good, this implies that we can take vi ∈ Vi such thatvi (y(vi , v′−i )) − ti (y(vi , v′−i )) < vi (y) − xiH and have a contradiction by arguments similar to Situations 1
and 2. Similarly, we have a contradiction in the cases (ii) and (iii).
Lemma 4 shows that the two valuation functions of the public good, whose “peaks” on the option set
are equal, induce the same consumption of the public good if the cost sharing scheme satisfies strategyproofness and Lemma 5 is a well-known result on strategy-proofness and non-bossiness. 17
Lemma 4. Suppose that the cost sharing scheme f satisfies strategy-proofness. For each v, v′ ∈ V andeach i ∈ I, if v′i (y(v′′i , v′−i )) − ti (y(v′′i , v′−i )) < v′i (y(vi , v′−i )) − ti (y(vi , v′−i )) for each y(v′′i , v′−i ) ∈ Oi (v′−i ) \{y(vi , v′−i )}, then y(vi , v′−i ) = y(v′i , v′−i ).
Lemma 5. If the social choice function satisfies strategy-proofness and non-bossiness, then it is a cost
sharing scheme.

A2. Results on Strong Non-Bossiness
Lemma 6 shows the uniqueness of the agent’s utility maximizer in the agent’s option set under a social
choice function satisfying strategy-proofness and strong non-bossiness. 18
Lemma 6. Suppose that the social choice function f satisfies strategy-proofness and strong nonbossiness. For each v, v′ ∈ V and each i ∈ I, if y(vi , v′−i ) ̸= y(v′i , v′−i ), then vi (y(vi , v′−i )) − xi (vi , v′−i ) >vi (y(v′i , v′−i )) − xi (v′i , v′−i ).
16 Note that we can take such a valuation function of the public good even if xL − t (y(v′′ , v′ )) = 0 for each v′′ ∈ V because
i
i
i
i −i
i
0 < vi (y) − vi (y(v′′i , v′−i )) for each v′′i ∈ Vi by the supposition of y and the strict increasingness of valuation functions of the

public good.
17 See Nishizaki (2016) for a proof of these lemmas.
18 See Nishizaki (2016) for a proof of this lemma.

8

By Proposition 1, we know that strategy-proofness implies the closedness of the option sets under acost sharing scheme. By imposing strong non-bossiness in addition to strategy-proofness, we also have
Proposition 2 showing the convexity.
Proposition 2. Suppose that the cost sharing scheme f satisfies strategy-proofness and strong nonbossiness. For each i ∈ I and each v′−i ∈ V−i , Oi (v′−i ) is convex.
Proof. Let i ∈ I and v′−i ∈ V−i . It is sufficient to demonstrate that λ y + (1 − λ )y′ ∈ Oi (v′−i ) for eachy, y′ ∈ Oi (v′−i ), where y ̸= y′ , and each λ ∈ (0, 1). To the contrary, we suppose that there are y, y′ ∈′ ), where y ̸= y′ , and λ ∈ (0, 1) such that λ y + (1 − λ )y′ ̸∈ O (v′ ). Together with Proposition 1,
Oi (v−ii −iLHthis implies that we can take vi , vi ∈ Vi such that
′
′
L ′
H ′
[y(vLi , v′−i ),y(vH
i , v−i )] ∩ Oi (v−i ) = {y(vi , v−i ), y(vi , v−i )},′y(vLi , v′−i ) < λ y + (1 − λ )y′ < y(vH
i , v−i ).

(3)
(4)

By (3) and Lemma 2, we can take v∗i ∈ Vi such that (iii-2a) v∗i (y(vLi , v′−i )) −v∗i (y(v′′i , v′−i )) ≥ ti (y(vLi , v′−i )) −′′ ′∗′ti (y(v′′i , v′−i )) for each v′′i ∈ Vi , where y(v′′i , v′−i ) ≤ y(vLi , v′−i ), and (iii-2b) v∗i (y(vHi , v−i )) −vi (y(vi , v−i )) ≥
′
′′ ′
′′
′′ ′
H ′
ti (y(vHi , v−i )) − ti (y(vi , v−i )) for each vi ∈ Vi , where y(vi , v−i ) ≥ y(vi , v−i ). Together with strategy′
proofness, these imply that v∗i (y(vLi , v′−i )) −ti (y(vLi , v′−i )) = v∗i (y(v∗i , v′−i )) −ti (y(v∗i , v′−i )) = v∗i (y(vHi , v−i )) −
′
∗
L ′
L ′
∗
∗ ′
∗ ′
∗
H ′
H ′
ti (y(vHi , v−i )), that is, vi (y(vi , v−i )) −xi (vi , v−i ) = vi (y(vi , v−i )) −xi (vi , v−i ) = vi (y(vi , v−i )) −xi (vi , v−i ).
′Together with Lemma 6, this implies that y(viL , v′−i ) = y(v∗i , v′−i ) = y(vH
i , v−i ) and contradicts (4).
By Lemma 3, we know that strategy-proofness implies the lower semi-continuity of the cost sharing
scheme. By imposing strong non-bossiness in addition to strategy-proofness, we have Lemma 7 showing
the continuity.
Lemma 7. If the cost sharing scheme f satisfies strategy-proofness and strong non-bossiness, then it iscontinuous.
Proof. To the contrary, we suppose that f is not continuous. This implies that there are i ∈ I and v′−i ∈ V−isuch that ti is not continuous on Oi (v′−i ). In addition, there is vLi ∈ Vi such that ti is not continuous aty(vLi , v′−i ). Together with Lemma 3, this implies that ti is not upper semi-continuous at y(vLi , v′−i ) andthere is ε ∈ R+ such that for each neighborhood U ⊆ Oi (v′−i ) of y(vLi , v′−i ), ti (y′ ) > ti (y(vLi , v′−i )) + ε forsome y′ ∈ U. Because U ⊆ Oi (v′−i ), this implies that y(vLi , v′−i ) < y′ by Lemma 2. On the basis of theabove argument, let yH ∈ (y(vLi , v′−i ), y′ ) be such that we can take vi ∈ Vi which satisfies the following
condition: vi (y(vLi , v′−i )) − vi (λ y(vLi , v′−i ) + (1 − λ )yH ) > ti (y(vLi , v′−i )) −ti (λ y(vLi , v′−i ) + (1 − λ )yH ) forH ′Heach λ ∈ (0, 1). 19 Because (y(vLi , v′−i ), y′ ) ⊆ Oi (v′−i ), there is vH
i ∈ Vi such that y(vi , v−i ) = y and wefind that′y(vLi , v′−i ) < y(vH
(5)
i , v−i )by the definition of yH . On the basis of the definition of yH and the continuity and strict increasingness ofvaluation functions of the public good, we can take v∗i ∈ Vi such that (a) v∗i (y(vLi , v′−i )) − v∗i (y(v′′i , v′−i )) ≥′′ ′∗′ti (y(vLi , v′−i )) −ti (y(v′′i , v′−i )) for each v′′i ∈ Vi , where y(v′′i , v′−i ) ≤ y(vLi , v′−i ), (b) v∗i (y(vHi , v−i )) −vi (y(vi , v−i )) ≥19 Note that we can take such a valuation function of the public good by letting yH be sufficiently close to y(vL , v′ ). Thisi −irequirement is introduced to respect the strict concavity of valuation functions of the public good.

9

′
′′ ′
′′
′′ ′
H ′
∗
L ′
ti (y(vHi , v−i )) − ti (y(vi , v−i )) for each vi ∈ Vi , where y(vi , v−i ) ≥ y(vi , v−i ), and (c) vi (y(vi , v−i )) −
′
L ′
L ′
H ′
v∗i (λ y(vLi , v′−i ) + (1 − λ )y(vHi , v−i )) > ti (y(vi , v−i )) − ti (λ y(vi , v−i ) + (1 − λ )y(vi , v−i )) for each λ ∈(0, 1). Together with strategy-proofness, these imply that v∗i (y(vLi , v′−i )) −ti (y(vLi , v′−i )) = v∗i (y(v∗i , v′−i )) −
′
H ′
∗
L ′
L ′
∗
∗ ′
ti (y(v∗i , v′−i )) = v∗i (y(vHi , v−i )) − ti (y(vi , v−i )), that is, vi (y(vi , v−i )) − xi (vi , v−i ) = vi (y(vi , v−i )) −
′
H ′
L ′
∗ ′
xi (v∗i , v′−i ) = v∗i (y(vHi , v−i )) −xi (vi , v−i ). Together with Lemma 6, this implies that y(vi , v−i ) = y(vi , v−i ) =′y ( vHi , v−i ) and contradicts (5).

A3. Results on Outcome Rectangular Property
On the basis of Lemma 8 showing that the outcome rectangular property can be considered within therange of consumption of the public good under a social choice function satisfying strategy-proofness andnon-bossiness, we have Proposition 3 showing the closedness of the range of consumption of the public
good under the cost sharing scheme by imposing non-bossiness and the outcome rectangular property inaddition to strategy-proofness. 20
Lemma 8. Suppose that the social choice function f satisfies strategy-proofness, non-bossiness, andthe outcome rectangular property. For each v, v′ ∈ V , if y(vi , v′−i ) = y(v′i , v′−i ) for each i ∈ I, theny(v) = y(v′ ).
Proposition 3. If the cost sharing scheme f satisfies strategy-proofness, non-bossiness, and the outcome
rectangular property, then y(V ) is closed.
Proof. To the contrary, we suppose that y(V ) is not closed. This implies that we can take y ∈ ȳ(V ) \y(V ),where ȳ(V ) is the closure of y(V ). We have the following three situations according to the relationshipbetween y and y(V ).
Situation 1. y = inf y(V )By Proposition 1, we can take v ∈ V such that y(vi , v−i ) = min Oi (v−i ) for each i ∈ I. 21 In addition,
we can take v′ ∈ V such thaty < y ( v′ ) < y ( v )(6)by the supposition of y. For each i ∈ I, we have the following two cases according to the position of y(v′ )in Oi (v′−i ) by Proposition 1: (i) y(v′i , v′−i ) = max Oi (v′−i ) and (ii) y(v′i , v′−i ) < max Oi (v′−i ). In addition,
we consider the following two subcases of the case (ii) according to the relationship between y(vi , v−i )and y(v′i , v−i ) on the basis of Lemma 4 and the definition of y(vi , v−i ) : (ii-1) y(vi , v−i ) = y(v′i , v−i ) and(ii-2) y(vi , v−i ) < y(v′i , v−i ). Let I(i) ⊆ I be the set of agents belonging to the case (i), I(ii-1) ⊆ I be theset of agents belonging to the subcase (ii-1), and I(ii-2) ⊆ I be the set of agents belonging to the subcase(ii-2).
For each i ∈ I(i) , we can take v∗i ∈ Vi such that y(vi , v−i ) = y(v∗i , v−i ) and y(v′i , v′−i ) = y(v∗i , v′−i )by Lemma 4 and an argument similar to Proposition 2 because y(vi , v−i ) = min Oi (v−i ) (see the left
hand side of Figure 2). For each i ∈ I(ii-1) , we know that y(vi , v−i ) = y(v′i , v−i ) by definition. For each
∗∗ ′
′ ′
∗∗
i ∈ I(ii-2) , we can take v∗∗i ∈ Vi such that y(vi , v−i ) = y(vi , v−i ) and y(vi , v−i ) = y(vi , v−i ) by Lemma
20 See Nishizaki (2016) for a proof of this lemma.
21 Note that we can take such a profile of valuation functions of the public good by letting the slope of v on y(V ) bei

sufficiently low for each i ∈ I.

10

y

y

vi* ( y, x )viiy (vi , vi )y (vic, vi )y (vi , vi )y (vic, vc i )y (vi , vc i ) O (vc )i
i

Oi (vi )y (vi , vi )y (vi , vi )y (vic, vc i )y (vi , vc i )Oi (vc i )y (V )( y, xi )Oi (vi )y (V )yyCase (i)xiSubcase (ii-2)

xi

Figure 2: Proof of Situation 1 in Proposition 3

4 and an argument similar to Proposition 2 because y(vi , v−i ) = min Oi (v−i ) (see the right hand side of
Figure 2). Let v′′ ≡ (v′′I(i) , v′′I(ii-1) , v′′I(ii-2) ) be such that (v′′I(i) , v′′I(ii-1) , v′′I(ii-2) ) = (v∗I(i) , v′I(ii-1) , v∗∗I(ii-2) ). These imply that
′
′′
′′
′
′
y(vi , v−i ) = y(vi , v−i ) and y(vi , v−i ) = y(vi , v−i ) for each i ∈ I. Together with Lemma 8, this implies thaty(v) = y(v′′ ) = y(v′ ) and contradicts (6).
Situation 2. y = sup y(V )By Proposition 1, we can take v ∈ V such that y(vi , v−i ) = max Oi (v−i ) for each i ∈ I. In addition, we
can take v′ ∈ V such that y(v) < y(v′ ) < y by the supposition of y. For each i ∈ I, we have the following
two cases according to the position of y(v′ ) in Oi (v′−i ) by Proposition 1: (i) y(v′i , v′−i ) = min Oi (v′−i )and (ii) y(v′i , v′−i ) > min Oi (v′−i ). In addition, we consider the following two subcases of the case (ii)according to the relationship between y(vi , v−i ) and y(v′i , v−i ) on the basis of Lemma 4 and the definitionof y(vi , v−i ) : (ii-1) y(vi , v−i ) = y(v′i , v−i ) and (ii-2) y(vi , v−i ) > y(v′i , v−i ). By an argument similar to
Situation 1, we have a contradiction.
Situation 3. Otherwise
Let U ⊆ Y be a neighborhood of y such that U ∩ y(V ) is convex. This implies that there are thefollowing two cases according the relationship between y and consumption of the public good in U ∩y(V ): (i) y < y′′ for each y′′ ∈ U ∩ y(V ) and (ii) y > y′′ for each y′′ ∈ U ∩ y(V ).
In the case (i), we can take v ∈ V such that y(v) ∈ U and y(vi , v−i ) = min Oi (v−i ) for each i ∈ I by
Proposition 1. In addition, we can take v′ ∈ V such that y(v′ ) ∈ U and y < y(v′ ) < y(v) by the suppositionof y. By an argument similar to Situation 1, we have a contradiction.
In the case (ii), we can take v ∈ V such that y(v) ∈ U and y(vi , v−i ) = max Oi (v−i ) for each i ∈ I by
Proposition 1. In addition, we can take v′ ∈ V such that y(v′ ) ∈ U and y(v) < y(v′ ) < y by the suppositionof y. By an argument similar to Situation 2, we have a contradiction.
Remark 1. In non-excludable public good economies with classical preferences, Serizawa (1996, Lemma)
showed the closedness of the range of consumption of the public good under a social choice function satisfying strategy-proofness, non-bossiness, individually rationality, budget-balancedness, and nonexploitation. In other directions, Serizawa (1999, Fact 1) showed it by strategy-proofness, symmetry, andbudget-balancedness.

11







y

viH

L
( y, xi ) vi vi

Oi (vHi )y (vi , vHi )y (viH , vHi )Oy  (1  O ) ycy (vi , vLi ) y (viL , vLi )Oi (vLi )y (V )

xi

Figure 3: Proof of Proposition 4

By imposing the outcome rectangular property in addition to strategy-proofness and strong nonbossiness, we have Proposition 4 showing the convexity of the range of consumption of the public goodunder the cost sharing scheme.
Proposition 4. If the cost sharing scheme f satisfies strategy-proofness, strong non-bossiness, and theoutcome rectangular property, then y(V ) is convex.
Proof. Let y, y′ ∈ y(V ) and λ ∈ [0, 1]. We have the following three cases according to the value of λ : (i)λ = 0; (ii) λ = 1; and (iii) λ ∈ (0, 1). In the case (i), we know that λ y + (1 − λ )y′ = y′ ∈ y(V ). In thecase (ii), we know that λ y + (1 − λ )y′ = y ∈ y(V ). In the case (iii), we have the following two subcases
according to the relationship between y and y′ : (iii-1) y = y′ and (iii-2) y ̸= y′ . In the subcase (iii-1), we
know that λ y + (1 − λ )y′ ∈ y(V ).
The remainder of this proof demonstrates that λ y + (1 − λ )y′ ∈ y(V ) in the subcase (iii-2). To thecontrary, we suppose that λ y + (1 − λ )y′ ̸∈ y(V ). On the basis of Proposition 3, we can take vL , vH ∈ Vsuch that for each i ∈ I,[y(vL ), y(vH )] ∩ Oi (vL−i ) = {y(vL )},(7)

H[y(vL ), y(vH )] ∩ Oi (vH−i ) = {y(v )},(8)y(vL ) < λ y + (1 − λ )y′ < y(vH ).

(9)

By (7), (8), and Lemma 2, we can take v∗ ∈ V such that for each i ∈ I, (iii-2a) v∗i (y(vLi , vL−i )) −v∗i (y(v′′i , vL−i )) >Hti (y(vLi , vL−i )) − ti (y(v′′i , vL−i )) for each y(v′′i , vL−i ) ∈ Oi (vL−i ) \ {y(vLi , vL−i )} and (iii-2b) v∗i (y(vHi , v−i )) −
H H
′′ H
′′ H
H
H H
v∗i (y(v′′i , vH−i )) > ti (y(vi , v−i )) −ti (y(vi , v−i )) for each y(vi , v−i ) ∈ Oi (v−i ) \ {y(vi , v−i )} (see Figure 3).
H H
Together with Lemma 4, this implies that y(v∗i , vL−i ) = y(vLi , vL−i ) and y(v∗i , vH−i ) = y(vi , v−i ) for eachi ∈ I. Together with Lemma 8, this implies that y(vL ) = y(v∗ ) = y(vH ) and contradicts (9).
Remark 2. The combination of Lemmas 1 and 7 and Proposition 4 implies the continuity of a cost sharing scheme satisfying strategy-proofness, strong non-bossiness, and the outcome rectangular property on
the range of consumption of the public good.

A4. Proof of Theorem 3
Let f be a social choice function satisfying strategy-proofness, strong non-bossiness, and the outcome
rectangular property. By Lemma 5, we know that f is a cost sharing scheme. By Propositions 3 and12

4, we know that the range of consumption of the public good is closed and convex. These imply thatthe problem of providing a divisible and non-excludable public good with the cost shares is reduced to avoting environment in which the set of alternatives is equivalent to the range of consumption of the public
good, which is a closed interval. In addition, we know the continuity of f on the range of consumptionof the public good, as stated in Remark 2. This implies that each utility function induces a continuous
preference defined over the range of consumption of the public good. Together with the result of Barberà
and Peleg (1990, Theorem 3.1), these imply that f is dictatorial if the range of consumption of the public
good contains at least three alternatives. If not, then f is constant because the range of consumption of
the public good is closed and convex.

References
Barberà, S. and B. Peleg (1990) “Strategy-Proof Voting Schemes with Continuous Preferences,” Social
Choice and Welfare 7, pp.31-38.
Berga, D. and B. Moreno (2009) “Strategic Requirements with Indifference: Single-Peaked versus
Single-Plateaued Preferences,” Social Choice and Welfare 32, pp.275-298.
Bochet, O. and T. Sakai (2010) “Secure Implementation in Allotment Economies,” Games and Economic
Behavior 68, pp.35-49.
Cason, T., T. Saijo, T. Sjöström, and T. Yamato (2006) “Secure Implementation Experiments: Do
Strategy-Proof Mechanisms Really Work?” Games and Economic Behavior 57, pp.206-235.
Chen, Y. (2008) “Incentive-Compatible Mechanisms for Pure Public Goods: A Survey of Experimental
Literature,” in Handbook of Experimental Economics Results, edited by C. R. Plott and V. L. Smith,
North-Holland.
Clarke, E. H. (1971) “Multipart Pricing of Public Goods,” Public Choice 11, pp.17-33.
Fujinaka, Y. and T. Wakayama (2008) “Secure Implementation in Economies with Indivisible Objects
and Money,” Economics Letters 100, pp.91-95.
Fujinaka, Y. and T. Wakayama (2011) “Secure Implementation in Shapley-Scarf Housing Markets,” Economic Theory 48, pp.147-169.
Groves, T. (1973) “Incentives in Teams,” Econometrica 41, pp.617-631.
Kumar, R. (2013) “Secure Implementation in Production Economies,” Mathematical Social Sciences 66,
pp.372-378.
Mizukami, H. and T. Wakayama (2005) “Bossiness and Implementability in Pure Exchange Economies,”
RIMS Kokyuroku 1461, pp.126-140.
Moulin, H. (1994) “Serial Cost-Sharing of Excludable Public Goods,” Review of Economic Studies 61,
pp.305-325.

13

Nishizaki, K. (2011) “Secure Implementation in Discrete and Excludable Public Good Economies,” Osaka Economic Papers 61, pp.48-56.
Nishizaki, K. (2012) “Secure Implementation in Queueing Problems,” Theoretical Economics Letters 2,
pp.561-565.
Nishizaki, K. (2013) “An Impossibility Theorem for Secure Implementation in Discrete Public Good
Economies,” Economics Bulletin 33, pp.300-308.
Nishizaki, K. (2014) “An Equivalence of Secure Implementability and Full Implementability in Truthful
Strategies in Pure Exchange Economies with Leontief Utility Functions,” Review of Economic Design
18, pp.73-82.
Nishizaki, K. (2016) “Securely Implementable Social Choice Functions in Divisible and Non-Excludable
Public Good Economies with Quasi-Linear Utility Functions,” RISS Discussion Paper Series, No. 38,
Research Institute for Socionetwork Strategies, Kansai University.
Ritz, Z. (1983) “Restricted Domains, Arrow-Social Welfare Functions and Noncorruptible and NonManipulable Social Choice Correspondences: The Case of Private Alternatives,” Mathematical Social
Sciences 4, pp.155-179.
Saijo, T., T. Sjöström, and T. Yamato (2003) “Secure Implementation: Strategy-Proof Mechanisms Reconsidered,” RIETI Discussion Paper 03-E-019.
Saijo, T., T. Sjöström, and T. Yamato (2007) “Secure Implementation,” Theoretical Economics 2, pp.203229.
Satterthwaite, M. A. and H. Sonnenschein (1981) “Strategy-Proof Allocation Mechanisms at Differentiable Points,” Review of Economic Studies 48, pp.587-597.
Serizawa, S. (1996) “Strategy-Proof and Individually Rational Social Choice Functions for Public Good
Economies,” Economic Theory 7, pp.501-512.
Serizawa, S. (1999) “Strategy-Proof and Symmetric Social Choice Functions for Public Good
Economies,” Econometrica 67, pp.121-145.

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