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Published in: Social Choice and Welfare 3/2022

23-09-2021 | Original Paper

The pivotal mechanism versus the voluntary contribution mechanism: an experimental comparison

Authors: Tatsuki Homma, Ryosuke Iba, Junyi Shen, Takuma Wakayama, Hirofumi Yamamura, Takehiko Yamato

Published in: Social Choice and Welfare | Issue 3/2022

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Abstract

We conduct an experimental comparison of two well-known mechanisms for undertaking 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 observe that the voluntary contribution mechanism works better than the pivotal mechanism from the perspectives of Pareto efficiency, individual rationality, and surplus maximization. On the other hand, there is no significant difference between the two mechanisms in terms of decision efficiency. These results suggest that the voluntary contribution mechanism is more favorable than the pivotal mechanism in practical applications.

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Appendix
Available only for authorised users
Footnotes
1
In our context, the VCM is often called the provision point mechanism (Bagnoli and Lipman 1989). This mechanism, for example, is deployed on online crowdfunding platforms such as Kickstarter (Reischmann and Oechssler 2018). The PM is a special case of the so-called Vickrey-Clarke-Groves (VCG) mechanism (Vickrey 1961; Clarke 1971; Groves 1973). Facebook uses the VCG mechanism for allocating ad space (Varian and Harris 2014; Haeringer 2017).
 
2
See Sect. 3.3 for a precise definition of twice iteratively undominated Nash equilibrium and some justifications for this equilibrium concept.
 
3
In a society with fewer participants, each participant often knows the other participants’ private information. Such a situation can be considered as a complete information environment. Even under this environment, the literature on mechanism design usually assumes that the social planner is unaware of the participants’ private information. See Moore (1992) and Maskin and Sjöström (2002) for surveys on the literature of mechanism design in environments with complete information.
 
4
As mentioned above, the rule of the PM is more complicated than that of the VCM. To hold the degree of transparency constant, we also conduct experiments presenting the payoff table to subjects instead of describing the rule of the mechanism to subjects. See Section 6 for discussion about the differences in the results between the treatments with payoff tables and the treatments without payoff tables.
 
5
For the same reason, some experimental studies on the provision of public goods have used only two subjects in each group. See, for example, Cason et al. (2002), Van Dijk et al. (2002), Cason et al. (2004), Cason et al. (2006), and Yamakawa et al. (2016).
 
6
See, for example, Ledyard (1995) for a survey on this topic. Several experimental studies have found devises that mitigate free-riding behaviors in the standard VCM. For example, communication plays a part in lessening free-riding behaviors (Isaac and Walker 1988) and the use of costly punishments is effective for lessening free-riding behaviors (Fehr and Gächter 2000). Further, Bochet et al. (2006) experimentally compared communication with costly punishments and found that some types of communication increased the rate of contribution rather than costly punishments.
 
7
Bagnoli and McKee (1991) also conducted the experiments with a group of five subjects and observed that the ratio at which decision efficiency was satisfied was 86.7% and that the ratio of the Pareto efficient equilibrium strategies that subjects played was 54.1%. It seems that these experimental results support the theoretical prediction. However, Bagnoli and McKee’s experimental method had some problems. For example, several groups of experiments were conducted in the same room, all groups’ payments were announced, and each subject was not given a sheet recording the amount paid and the gain at each session.
 
8
There are other laboratory experimental studies that have examined the effects of changing details of the VCM, such as refund rules (Isaac et al. 1989; Bagnoli and McKee 1991), rebate rules (Marks and Croson 1998), incomplete information on the number of players or the cost of the public project (Rondeau et al. 1999), and simultaneous versus sequential contributions (Coats et al. 2009). See also Chen (2008) and Croson (2008) for surveys on experimental results of the VCM.
 
9
Strictly speaking, before Attiyeh et al. (2000), there were some experimental studies on the PM (e.g., Scherr and Babb (1975) and Tideman (1983)). However, these studies were not well-controlled tests of the PM.
 
10
See Chen (2008) and Chen and Ledyard (2008) for excellent surveys on these experimental results.
 
11
Our definition of individual rationality follows that of Bagnoli and Lipman (1989).
 
12
This is often called “autarkic individual rationality” (Saijo 1991), which is slightly stronger than the notion of individual rationality commonly used in the literature on mechanism design.
 
13
This definition depends on the number of agents. When there are three or more agents, the core is generally smaller than the set of allocations satisfying Pareto efficiency and individual rationality.
 
14
To address this criticism, Sjöström (1994) considered the double implementation in undominated Nash equilibrium and the notion of iterated elimination of weakly dominated strategies.
 
15
Instead of A1, Bagnoli and Lipman (1989) used the following weaker assumption: \(\omega _{1} + \omega _{2} > c\) and for each \(v \in V\) and each \(i \in \{1, 2\}\), \(\omega _i > v_i\).
 
16
This is equivalent to stating that the VCM implements a sub-correspondence of the core in twice iteratively undominated Nash equilibria. Moreover, we could argue that the VCM “almost” fully implements the core in twice iteratively undominated Nash equilibria because the following facts hold: for each \(v \in V\), (i) if \(v_1 + v_2 > c\), then \(C(v) \setminus {\mathcal {A}}(\Gamma ^{\mathrm {VC}}, v) = \{(1, (-v_1, v_1 - c)), (1, (v_2 - c, -v_2))\}\); (ii) if \(v_1 + v_2 = c\), then \(C(v) \setminus {\mathcal {A}}(\Gamma ^{\mathrm {VC}}, v) = \{(1, (-v_1, -v_2))\}\); and (iii) if \(v_1 + v_2 < c\), then \(C(v)={\mathcal {A}}(\Gamma ^{\mathrm {VC}}, v)\).
 
17
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 VCM. However, it is an open question whether a similar implementation result holds for his notion of perfection.
 
18
Specifically, the set of bad Nash equilibria is \(\left\{ (s_{1},s_{2})\in S:s_{1}\in \{0,\dots ,7\}\ \text{ and }\ s_{2}\in \{0,\dots ,11\}\right\} \).
 
19
The set of bad Nash equilibria after eliminating these weakly dominated strategies of two players from the original payoff table coincides with that of the “bad” Nash equilibria of the original payoff table mentioned in Footnote 18.
 
20
The strategy profile \((s_1, s_2) = (8,12)\) is also a unique one surviving after a twice iterated elimination of weakly dominated strategies in Table 2. In other words, the unique undominated Nash equilibrium coincides with the unique strategy profile surviving after twice iterated elimination of weakly dominated strategies under our experimental setting. However, in general, the set of twice iteratively undominated Nash equilibria may differ from the set of strategy profiles surviving after twice iterated elimination of weakly dominated strategies. Thus, we require the concept of twice iteratively undominated Nash equilibria to obtain our implementation result (Proposition 1). See Appendix B for more details.
 
21
This is because the best response function of each type has a “flat” structure. For instance, (i) when Type 2 selects \(s_2 = 7\), the payoffs of Type 1 are “high” (\(u_1 = 20\)) if Type 1 chooses \(s_1 \le 12\), and those are “low” (\(u_1 = 16\)) otherwise; (ii) when Type 2 chooses \(s_2 = 11\), the payoffs of Type 1 are the same (\(u_1 = 19\)) for all Type 1’s strategies; and (iii) when Type 2 selects \(s_2 = 16\), the payoffs of Type 1 are “low” (\(u_1 = 14\)) if Type 1 chooses \(s_1 \le 3\), and “high” (\(u_1 = 19\)) 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 choosing “bad” strategies, or b) the payoffs are the same for all strategies.
 
22
Secure implementability requires strategy-proofness, and further requires that all Nash equilibrium outcomes coincide with the socially desirable outcome. As shown in Table 3, some Nash equilibrium of the PM realizes an outcome that is not the socially desirable one. Hence, the PM violates secure implementability.
 
23
See Appendices G and H for English translations of the instructions for our experiments.
 
24
We used the z-Tree program (Fischbacher 2007).
 
25
In Appendix E, we also provide a table that shows the frequency of distribution of all data in Treatment V and a figure that displays the frequency distribution of the last 10 periods in Treatment V.
 
26
In Appendix E, we also provide a table that shows the frequency of distribution of all data in Treatment P and a figure that displays the frequency distribution of the last 10 periods in Treatment P.
 
27
See Appendix F for the experimental procedure and detailed experimental results of Treatments VT and PT.
 
28
Detailed frequencies of these equilibria are presented in Observations 1 and 2 in Sections 5.1 and 5.2 for Treatment V and Treatment P, and Observations F.1 and F.2 in Sections F.2.1 and F.2.2 for Treatment VT and Treatment PT, respectively.
 
29
See Appendix D for discussing the behavioral differences between the two treatments.
 
30
See, for example, Bagnoli and McKee (1991), Marks and Croson (1999), and Attiyeh et al. (2000).
 
31
For example, Jackson and Moulin (1992) constructed a simple extensive form mechanism that implements a Pareto efficient allocation in undominated Nash equilibria or subgame-perfect equilibria.
 
32
See, for example, Fehr and Schmidt (1999), Bolton and Ockenfels (2000), and Charness and Rabin (2002).
 
33
Let \(a, b \in {\mathbb {R}}\) be such that \(a \le b\). Then, we denote by \([a, b ]\) and \(]a, b [\) the closed interval from a to b and the open interval from a to b, respectively. We also denote by \([a, b[\) and ]ab] the half-open intervals from a to b.
 
34
Note that if Type 2 chooses a dominant strategy \(s_{2}\in \{12,13\}\) and Type 1 switches his or her strategy from a dominant strategy \(s_{1}=8\) (resp. \(s_{1}=9\)) to \(s_{1}\ge 10\), then Type 2’s payoff increases from 21 (resp. 22) to 23, whereas Type 1’s payoff remains the same level at 19 (see Table 3). These “overbidding” behaviors are consistent with best response behavior and with other-regarding behavior to increase social welfare (e.g., see Charness and Rabin (2002)).
 
35
If Type 1 chooses a dominant strategy \(s_{1}\in \{8,9\}\) and Type 2 switches his or her strategy from a dominant strategy \(s_{2}\in \{12,13\}\) to \(s_{2}\le 10\), the project is not undertaken, and no pivotal tax is imposed. As a result, Type 2’s payoff decreases from 21 to 20, whereas Type 1’s payoff increases from 19 to 20 (see Table 3). Such “underbidding” behavior can be explained by the theory of social preferences developed by Fehr and Schmidt (1999). See Sect. 5.3 for details.
 
36
If Type 1 chooses a dominant strategy \(s_{1}\in \{8,9\}\) and Type 2 switches his or her strategy from a dominant strategy \(s_{2}\in \{12,13\}\) to \(s_{2}\ge 14\), neither Type 1’s nor Type 2’s payoff changes (see Table 3 ).
 
37
We thank an anonymous referee for suggesting the discussion of this section.
 
38
Note that this argument holds whenever \(v_1+v_2 \ge c\) and \(\omega _1=\omega _2\). We can easily verify that if \(v_1+v_2 \ge c\) and \(\omega _1= \omega _2\), then \((s_1,s_2)= \left( \frac{v_1-v_2+c}{2}, \frac{v_2-v_1+c}{2} \right) \) is the unique twice iteratively undominated Nash equilibrium in which the sum of two agents’ payoffs is maximized and both agents’ payoffs are equal.
 
39
To see this, we consider any dominant strategy equilibrium in Treatment P. Without loss of generality, we focus on the strategy profile (9, 13), which is a dominant strategy equilibrium in Treatment P. Now consider any “underbidding” \(s_2 \in \{0, \dots , 10\}\). We can easily verify that \(U_{2}(9,13) = 20\) and \(U_{2}(9, s_2) = 22- 3\beta _2\). Therefore, we have \(U_{2}(9, s_2) > U_{2}(9,13)\) whenever \(\beta _{2} > \frac{2}{3}\). That is, under the inequality aversion model, Type 2 has an incentive to “underbid” in Treatment P when he or she has a high degree of guilt.
 
40
See Appendix I for an English translation of the instruction for Treatments VT and PT.
 
41
See Figs. 12, 13, 14, 15 in Appendix I.
 
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Metadata
Title
The pivotal mechanism versus the voluntary contribution mechanism: an experimental comparison
Authors
Tatsuki Homma
Ryosuke Iba
Junyi Shen
Takuma Wakayama
Hirofumi Yamamura
Takehiko Yamato
Publication date
23-09-2021
Publisher
Springer Berlin Heidelberg
Published in
Social Choice and Welfare / Issue 3/2022
Print ISSN: 0176-1714
Electronic ISSN: 1432-217X
DOI
https://doi.org/10.1007/s00355-021-01350-x

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