Public Choice 108: 331–354, 2001.© 2001 Kluwer Academic Publishers. Printed in the Netherlands.

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Can the Pivotal mechanism induce truth-telling? Anexperimental study ∗

TOSHIJI KAWAGOE1 & TORU MORI2

1Future University Hakodate, 116-2 Kameda, Hakodate, Nakano-cho / 041-8655, Japan;2Faculty of Economics, Nagoya City University, 1 Yamanohata, Mizuho-cho, Mizuho-ku,Nagoya, 467-8501 Japan

Accepted 17 March 2000

Abstract. In this paper we use a laboratory experiment to examine the Pivotal mechanismwhen applied to a binary decision on a public project of a fixed size. We first point out thatthe well-known incentive compatibility of the Pivotal mechanism is true only in a weak sense;There are always strategies other than truth-telling that do no worse for a subject than truth-telling, in any particular set of circumstances. This weakness of the incentive compatibilityof the Pivotal mechanism makes it difficult for subjects to understand that truth-telling is theunique dominant strategy for the mechanism, unless they have comprehensive understandingof the payoff structure, with the result that subjects often do not play the dominant strategy. Wesuggest that this tendency to depart from the dominant strategy can be overcome by providingsubjects with more information about the payoff structure. We controlled the level of informa-tion provision in the laboratory and verified that the strategies used are closer to the dominantstrategy when more information is provided. Under no information provision conditions wereoutcomes poor, but under “wide” provision conditions, in which each subject experienced avariety of true personal valuations of the project, departure from the dominant strategy wassmaller in magnitude, and under “deep” provision conditions, in which detailed payoff tableswere available to each subject, the rate of use of the dominant strategy increased significantly.

1. Introduction

In 1970s Clarke (1971), Groves (1973), Groves and Loeb (1975) and otherschallenged the task of finding the incentive compatible mechanisms for publicgoods provision in which revelation of true preference is a dominant strategy

∗ We would like to thank Prof. Noriko Soyama of Tenri University for developing thecomputer program used in our experiments. We also owe a great debt to Prof. Shinichi Hirotaof Waseda University and Prof. Shigeru Watari of Ritsumeikan University who helped us torecruite subjects and to arrange for the laboratory. We are grateful to Prof. Tatsuyoshi Saijoof Osaka University and others for their helpful comments on earlier versions of this paperat the 1997 Meeting of the Japanese Economic Association. We are also grateful to Prof. R.Mark Isaac and his co-researchers at University of Arizona for their kindness in sending usthe draft of their own experimental study on the Pivotal mechanism. Finally, we are grateful toan anonymous referee for helpful comments, which greatly helped us in improving this paper.

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for each subject. They succeeded in the independent discovery of such mech-anisms and all the mechanisms they discovered were integrated into the classcalled Groves mechanisms by Green and Laffont (1977).

Although some problems, such as the lack of individual rationality andthe failure to satisfy budget balance conditions, are pointed out as seriousdrawbacks of Groves mechanisms, it is rarely questioned whether the mech-anisms can induce truth-telling. The property of Groves mechanisms thattruth-telling is a dominant strategy has been considered to be enough to givestrong incentives to subjects to reveal their true preferences.

However, in the context of large economies, there has been skepticismabout the assertion that the incentive compatibility is enough to induce sub-jects to tell the truth. Hammond (1979) proved that, in continuum economywith public goods, there is an incentive compatible mechanism which canalso achieve a Pareto efficient allocation. But, as pointed out by Groves andLedyard (1987), in this mechanism truth-telling is not a unique dominantstrategy. Groves and Ledyard (1987) state that, in this situation, “there is noincentive either to lie or tell the truth.” (p. 97)

Even before Hammond (1979) published his paper, Tideman and Tullock(1976) conjectured that, in large economies, the incentive compatibility ofGroves mechanisms is not enough to provide the incentive to tell the truth.Considering the Pivotal mechanism, a version of Groves mechanisms de-signed by Clarke (1971), they conjectured that the magnitude of Clarke taxbecomes smaller as the number of subjects increases.1 Since Clarke tax worksas an engine to make the mechanism incentive compatible, their conjectureimplies that, in large economies, subjects have almost the same payoff eitherby telling the truth or by telling a lie. They state that in such a situation“voters (subjects) have almost no incentive to vote (reveal their preference).”(p. 1156)

Based on Tideman and Tullock’s hypothesis, Brubaker (1983) insistedthat, if the claim to have solved the free-rider problem is to be convincing,a mechanism should satisfy strict incentive compatibility which means that,for any strategies chosen by other subjects, truth-telling leaves the subjectconcerned strictly better off than any other strategies permissible to him.Brubaker (1983) concluded that the lack of strict incentive compatibility isthe reason why incentive compatible mechanisms fail to induce subjects totell the truth in large economies. He wrote, “if a strategy (truth-telling) leavesthe person materially as well off as some another strategy (telling a lie), anyother appreciable reason for the latter will tip the choice to it.”(p. 318)

In this paper, we will point out that the lack of strict incentive compat-ibility makes it difficult for subjects to reveal their true preferences under

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incentive compatible mechanisms, even in small-group decisions. We willalso report the results of our experiments to ascertain this claim.

The mechanism we used in our experiments is the Pivotal mechanism ap-plied to the decision-making problem of determining whether or not a publicproject of fixed size should be realized. The reason we adopted the Pivotalmechanism in such a binary public decision problem is that we can expect themechanism to have a great practical virtue in this environment. Subjects in thereal world may be ready to reveal their valuation of a single public project.But they may be puzzled if they are required to reveal their valuations ofseveral kinds of projects simultaneously, even more so if they are required toreveal their continuous valuation functions over a possible range of divisibleprojects. It is well known that, in decision making by groups with a finitenumber of subjects, the Pivotal mechanism guarantees that truth-telling is aunique dominant strategy for each subject. However, even in a group with afinite number of subjects, the incentive compatibility of the mechanism in thesituation of binary public decision is weak in the sense that a large numberof other strategies leave a subject as well off as by truth-telling for a widerange of strategies chosen by other subjects. This weak incentive compatibil-ity of the Pivotal mechanism makes it difficult for a subject to distinguish thedominant strategy from other best responses, unless he has comprehensiveunderstanding of the payoff structure. This point will be explained in detailin Section 2 after the formal presentation of the Pivotal mechanism in theframework of binary public decision problems.

The design and results of our experiment will be reported in Sections 3 and4, respectively. Recognizing the problem to be due to the lack of strict incent-ive compatibility of the Pivotal mechanism, we conducted our experimentsunder three different conditions regarding information on the payoff structure.The first condition is such that subjects are assigned a fixed value as theirtrue valuation of the public project throughout the experiment, and are onlygiven an explanation about the rules of the Pivotal mechanism. The secondcondition is such that subjects are only given an explanation about the rulesof the Pivotal mechanism, as in the first condition, but are randomly assignedtheir true valuation of the public project in each round of the experiment. Thisdesign allows subjects to look at the payoff structure from a wider viewpointrelative to the fixed assignment of true valuation in the first design. Thus, wepredict that the possibility that subjects reveal their true valuation is higherin the second condition than in the first. In the third condition, subjects aregiven a detailed payoff table in addition to the explanation of the rule. Sincethe detailed payoff table represents the whole structure of payoff, it is mucheasier for subjects to obtain a comprehensive understanding of the payoff

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structure. Thus, we predict that, in the third design, we will have much lessdiscrepancy between true valuation and the revealed value of messages.

Our prediction in each of the three informational conditions stated abovewas verified by the results of our experiments. The rate of attainment of thedominant strategy, i.e., truth-telling, is highest in the third condition, nexthighest in the second condition and lowest in the first. And the magnitudeof deviation from the dominant strategy is lowest in the third condition, nextlowest in the second condition and highest in the first.

In Section 5, we will discuss the future direction of experimental stud-ies of the Pivotal mechanism and the possibility of the practical use of themechanism.

2. Model

2.1. The Pivotal mechanism

The configuration we adopted in our experiment is a class of the Pivotalmechanism applied to the problem of determining whether or not a publicproject of fixed size should be realized. In this mechanism, each subject isasked to reveal his valuation to the project, si, which may be different fromhis true valuation, θi. If the aggregate value of the valuation revealed by eachsubject,

∑ni=1 si (where n is the number of subjects), exceeds the cost of the

project, c, then the project is realized. Otherwise the project is not realized. Inour experiments, without any loss of theoretical property of the mechanism,the cost of the project is always set zero to simplify the subjects’ decision-making. So, the project is realized if si + s−i > 0 and it is not realized ifsi + s−i ≤ 0, where s−i stands for the aggregate valuation of the projectrevealed by all the subjects other than i.

A subject is called as a pivotal agent if his participation in the decision-making reverses the decision based on other members’ valuations. That is,subject i is a pivotal agent if either (1) s−i > 0 and si + s−i ≤ 0, or (2)s−i ≤ 0 and si + s−i > 0. A pivotal agent must pay Clarke tax equal to theabsolute value of the aggregate valuation to the project revealed by the othermembers. For a subject who is not a pivotal agent, the Clarke tax is zero.Thus, by denoting Clarke tax by ti, we get

ti =

|s−i| if (si + s−i) · s−i < 0|s−i| if si + s−i = 0 & s−i > 00 otherwise

In our experimental mechanism described above, the payoff to subject idenoted by ui is equal to θi − ti when the project is realized, and otherwise is

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just equal to −ti. For any level of aggregate valuation revealed by the othermembers, revealing true valuation maximizes the payoff determined in thisway. That is, truth-telling is a dominant strategy. This incentive compatibilityof the mechanism is well known and is clearly explained, for example, byTideman and Tullock (1976).

2.2. Weak incentive compatibility

Since truth-telling is a dominant strategy in the Pivotal mechanism, we alwayshave the inequality ui(θi, s−i) ≥ ui(si, s−i) for any si and s−i, where ui(si, si)

denotes i’s payoff in the case that he reveals si and the aggregate valuationrevealed by the others is s−i. However, in the case of binary public decisionproblems adopted in our experiments, we cannot drop the strict equality fromthe expression above, except in the case that the aggregate message by othershappens to take the value in a specified range. For example, when θi > 0,we have ui(θi, s−i) = 0 > θi − |s−i| = ui(si, s−i) for any si > θi in thecase that −si < s−i ≤ θi, but ui(θi, s−i) = ui(si, s−i) for any si > θi inany other range of s−i. In other words, there is an infinite number of bestresponses other than truth-telling for the wide range of aggregate messagesby other members. We call this incentive property of the Pivotal mechanismweak incentive compatibility.

The weak incentive compatibility makes it difficult for a subject to dis-tinguish the dominant strategy from other best responses. There are twoexplanations to justify this hypothesis. First, if a subject believes that someof the strategies of other subjects are never played, he could gain the sameexpected payoff from best responses as one from the dominant strategy. Tak-ing the same example stated above, if a subject with θi > 0 believes thatothers never play strategies such that −si < s−i ≤ θi, then he has exactly thesame expected payoff from any strategy si larger than θi as from the dominantstrategy θi. Second, even if a subject believes that others may choose anyof their strategies, he is never convinced that truth-telling is the dominantstrategy unless he compares the expected payoff from truth-telling with thosefrom a wide range of strategies permissible to him. Therefore, comprehens-ive understanding of the payoff structure is necessary for each subject torecognize that truth-telling is the dominant strategy.

If a mechanism has strict incentive compatibility in the sense thatui(θi, s−i) > ui(si, s−i) for any si and for any s−i, even though subjects par-tially understand the payoff structure in the mechanism, their best strategyin such a case would coincide with the best strategy under comprehensiveunderstanding of the payoff structure. So, in that mechanism, we can expectthat subjects will recognize truth-telling as the unique dominant strategy moreeasily and they will reveal their true valuation without additional information

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about the payoff structure. However, in the case of the Pivotal mechanism ina discrete public goods environment, due to its weak incentive compatibility,subjects may need some additional information about the payoff structureto play the dominant strategy. This prediction motivates us to conduct ourexperiments in the design detailed in the next section.

3. Design

3.1. Treatments and hypotheses

We designed our experiment to examine whether performance of the Pivotalmechanism depends on the level of subjects’ understanding of the payoffstructure. For this purpose, we introduced some informational conditions, but,without changing the payoff structure of the mechanism. Our informationalcondition has three levels.

Providing no special information (INFO1) is the case of the ordinal in-formational structure taken from the literature. That is, subjects are assigneda fixed value as their true valuation of the public project throughout the ex-periment, and are only told the rules of the Pivotal mechanism. They are toldunder what conditions the public project is realized, in which case Clarke taxis imposed, and how their payoffs are calculated, but they are not given eitherany other additional information about their payoff structure or the chanceto gain experience under different valuations. So subjects need to derive thepayoff structure from the rule. This is our base-line experiment.

Under the second condition (INFO2), each subject is told the rule as inINFO1, but his valuation of the project is randomly changed over the rounds.So they are able to gain experience under different valuations, they can lookat the payoff structure of the Pivotal mechanism from a wider viewpoint.

Under the third condition (INFO3), each subject is told the rule and as-signed a fixed valuation for the project, as in INFO1, but is given a detailedpayoff table. In the detailed payoff table, the entire payoff to each subject isprovided explicitly for any possible combination of his value and the aggreg-ate value of other subjects’ values. So subjects can look at the comprehensivepayoff structure in the mechanism at a glance.

Condition INFO1 imposes the highest burden of the comprehensive taskupon subjects. Under condition INFO2, each subject is assigned differentvaluations for each round, so this provides the opportunity for subjects toobtain a wider viewpoint for the payoff structure of the mechanism. ConditionINFO3 provides the payoff structure of the mechanism for subjects in detail,so it forces subjects to have a deep understanding of the payoff structure of the

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mechanism. Therefore, information condition INFO3 is the most informative,INFO2 is the next, and INFO1 the least.

We controlled these information conditions as a treatment variable in ourexperiment. We expect that information condition is the more informative,the more subjects have comprehensive understanding of the payoff structureand reveal their true valuation as a dominant strategy. Therefore, we set upthe following null hypotheses;

Hypothesis 1: The ratio of the number of subjects attaining the dominantstrategy (truth-telling) to the total number of subjects is same between thethree information conditions.

Hypothesis 2: The ratio of the number of dominant strategy (truth-telling)plays to the total number of plays is same between the three informationconditions.

Hypothesis 3: The distribution of the absolute difference between subjects’revealed value and the true value, a measure of deviation from the dominantstrategy, is same among the three information conditions.

As we can see, Hypothesis 1 is the strongest, Hypothesis 2 the nextstrongest, and Hypothesis 3 the weakest. If these null hypotheses are rejec-ted significantly, we can conclude that (1) there is a significant differencebetween information conditions for subjects to attain a dominant strategy, (2)having experience with different viewpoints for the payoff structure of themechanism has a significant effect on subjects to attain a dominant strategy,(3) giving the detailed payoff table to subjects has a significant effect onsubjects to attain a dominant strategy, and (4) there is a significant differencefor subjects to attain a dominant strategy between having experience with adifferent viewpoint for the payoff structure and looking at the detailed payofftable.

When these hypotheses are all rejected, we can say that subjects cannothave comprehensive understanding of the payoff structure of the mechanismwithout providing some additional information. So our hypothesis on theproblem caused by the lack of strict incentive compatibility of the Pivotalmechanism is verified. That is, it is difficult for subjects to understand thattruth-telling is a dominant strategy under INFO1 conditions in which theyprobably have partial understanding of the payoff structure, and thereforeinformation condition is the more informative, the higher the number ofdominant strategy equilibriums attained.

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3.2. Experiments

3.2.1. ParametersIn our experiments, we adopted the following common experimental settings;(1) There are five subjects in each group for all the experiments. Each subjectwas assigned to each group at random before the experiment, the members ofeach group were fixed throughout the session, and all the subjects were awareof these conditions.(2) Each subject in a group was assigned one of the true valuations for thepublic project from the set of values {−10,−5,−2, 5, 15} mutually exclus-ively. This set was same for all the experiments. Each subject only knew hisown valuation. They did not know the whole set of values.(3) Each subject’s revealing value was restricted in integers ranging from –25to 25.

3.2.2. Instructions and informationWe provided the instructions for the subjects by playing a pre-recorded tape inwhich young women, not including the researchers and experimenters in thisexperiment, read the instructions.2 Written instructions were also distributed.3

The instructions told the subjects that (1) this experiment was about groupdecision-making on the realization of a public project, (2) decision-makingwas done by specific rules which determined the realization of the projectand tax payment to subjects, (3) each group contained five members, groupmembership was fixed throughout the session, and members did not knowwho was in the same group, (4) initial funding was given, (5) monetary rewardfor the total payoff earned by the subjects was paid in cash according to aconversion rate, (6) the experimental session consisted of two practice roundsand actual ten experimental rounds, (7) the environment was always the sameexcept for the initial fund, (8) all communication with the experimenter wasconducted through a computer network and communication between subjectswas forbidden, (9) the true valuation of the project was assigned to eachsubject and their payoff was determined according to their true valuation,(10) the revealing value was restricted in integers ranging from –25 to 25.In addition to the above information, subjects in the INFO3 experiment wereshown how to read the detailed payoff table.4

A computer network was used for providing information relevant to groupdecision-making. The information provided to each subject contained (1) histrue valuation, (2) his revealed value, (3) the sum of the revealed values inhis group, (4) whether or not the project was realized, (5) the sum of therevealed values in his group minus his value, (6) whether or not the projectwas realized when the subject would not participate in the decision-making,(7) his tax, (8) his payoff, (9) the remaining fund.

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3.2.3. SubjectsThe subjects were recruited from undergraduates at various universities inJapan. Twenty undergraduates were recruited from Tezukayama Universityfor the INFO1 and INFO3 experiments in February 1997, ten undergradu-ates from Ritsumeikan University and Setsunan University for the INFO1experiment in July 1997, and ten undergraduates from Saitama Universityfor the INFO2 experiment in July 1997. They all applied to take part in theexperiment, and were randomly assigned to each treatment and each group.5

3.2.4. Rewards and session timeEach subject was paid a monetary reward in cash equal to ten times theamount of their total payoff added to the initial fund. At the start of the exper-iment, each subject was given an initial fund to avoid the risk of bankruptcy,which was different between subjects in the INFO1 and INFO3 experiments,and which was the same between subjects in INFO2 experiment. Subject’sexpected earnings were set at equal to two thousands yen for all subjects inall the experiments. Session time was about two hours for each experimentand half this time was spent in instruction and practice.

4. Results

In this section, we examine the hypotheses described in Section 3.1 us-ing the experimental data obtained. We have taken the INFO1 experimentat Ritsumeikan University (including students from Setsunan University),the INFO2 experiment at Saitama University and the INFO3 experiment atTezukayama University as our data sources.6 All the data obtained in theexperiment are included in Appendix 3.

First, we compared the ratio of the number of subjects attaining a dom-inant strategy to the total number of subjects between the informationconditions. Data are summarized in Table 1. The subject who attained a dom-inant strategy indicates those people who revealed the assigned value (truevalue) for all the rounds or on and after the 5th round, or who revealed theassigned value plus one for all the rounds or on and after the 5th round. Thereason why we included subjects who revealed the assigned value plus one isbecause this strategy gives exactly the same payoff to these subjects as truth-telling for any combinations of the others’ strategies, which is the result ofour conversion of continuous message space of the mechanism into discretemessage space in our experiment.

As shown in Table 1, only a few subjects attained the dominant strategyunder the INFO1 and INFO2 experiments. More subjects played the domin-ant strategy under INFO3 experiment, but these were less than half the total

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Table 1. Comparison of the ratio of the number of subjects attaining the dominant strategy tothe total number of subjects between information conditions

Treatment Number of subjects Number of subjects not Total

attaining dominant strategy attaining dominant strategy

INFO1 1 9 10

INFO2 0 10 10

INFO3 4 6 10

Total 5 25 30

Table 2. Comparison of the ratio of the number of dominant strategy plays to the total numberof plays between information conditions

Treatment Number of dominant Number of non- Total

strategy plays dominant strategy plays

INFO1 17 83 100

INFO2 14 86 100

INFO3 47 53 100

Total 78 222 300

number. This result may, we think, be inevitable because of the weak incent-ive compatibility of the mechanism. As we have already explained, subjectsneed to have comprehensive understanding of the payoff structure in order tofind the dominant strategy under a weak incentive compatible mechanism.But compared with INFO1 and INFO2, subjects could find the dominantstrategy more easily under INFO3, because they could use the detailed payofftable.

We used a Chi-square test to examine Hypothesis 1 in Section 3.1. Sincethe hypothesis was rejected at the 5% significance level, we concluded thatthe number of subjects attaining the dominant strategy was significantly dif-ferent between information conditions7 (χ2 = 6.240, df = 2, P = 0.044∗).In particular, the ratio in INFO3 was significantly higher than in the others.Therefore, Hypothesis 1 had to be rejected.

Further, we compared the ratio of the number of dominant strategy playsto the total number of plays among information conditions. The data aresummarized in Table 2. In this case, we also add plays of revealing true valuesplus one to dominant strategy plays.

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Table 3. Estimates of the coefficients of the logistic regression model.

Variable Coefficient S.E. t Wald df P-value

Constant –1.508 0.375 –4.019 16.153 1 0.000∗∗INFO2 –0.229 0.392 –0.585 0.341 1 0.559

INFO3 1.466 0.333 4.399 19.348 1 0.000∗∗Round –0.014 0.049 –0.292 0.085 1 0.770

In comparing the frequency of dominant strategy plays, we can suspectthat there is some learning effect or correlation during the rounds. So, weshould account for the learning effect when comparing the frequency betweeninformation conditions8. In order to estimate the effects of information andlearning simultaneously, we used a logistic regression model. Our regressionmodel is as follows:

p = F(b1 + b2 INFO2 + b3 INFO3 + b4 Round)

where p is the response equal to 1 if the subject revealed his true valuationor the true valuation plus one and otherwise 0, INFO2 is the dummy variableequal to 1 if the INFO2 condition is presented and otherwise 0, INFO3 is thedummy variable which is equal to 1 if the INFO3 condition is presented andotherwise 0, and F is the logistic function. INFO1 is represented by the statewhen both INFO2 and INFO3 are equal to 0. The estimates of the coefficientsof the model are shown in Table 3.

The coefficients for constant and INFO3 were significant at the 1% level,but others were not significant. That is, the frequency of dominant strategyplays was different between INFO1 and INFO3, but was not between INFO1and INFO2. So, we can say that there is a significant difference between in-formation conditions, and, in particular, INFO3 is very effective in importingthe frequency of dominant strategy plays. Therefore, Hypothesis 2 had to berejected. On the other hand, the coefficient for Round was not significant.This means that there is no learning effect in this experiment.

Next, we compared the magnitude of divergences from the dominantstrategy between information conditions. We took the absolute value of dif-ference between revealed values and true valuation as the magnitude ofdivergence. Summary statistics of divergences are shown in Table 4.

Apparently information condition was the more informative, mean diver-gence was less. This implies that the lack of comprehensive understandingof the payoff structure leads subjects to deviate from the dominant strategy.Figure 1 shows that mean divergence from the dominant strategy was

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Table 4. Descriptive statistics for the absolute difference between true valuation and revealedvalues

Treatment Mean St. dev.

INFO1 10.840 9.318

INFO2 7.680 6.764

INFO3 5.680 7.226

Figure 1. Mean divergences from true value.

significantly different between information conditions throughout all therounds.

In fact, mean divergence was high in the INFO1 experiment, but diver-gence in both the INFO2 and INFO3 experiments was lower than in theINFO1 and was closely distributed each other on and after the 5th round.Therefore, we can predict that the magnitude of divergences from the domin-ant strategy was significantly different between information conditions. Byusing each subject’s divergence in each round as a data point, we testedwhether or not divergence was significantly different between information

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Table 5. Repeated measures ANOVA table comparing divergences between informationconditions.

Source of variation SS df MS F P

Between subjects 9095.097 29

Information (I) 1354.487 2 677.243 2.362 0.113

Subjects within groups (S) 7740.610 27 286.689

Within subjects 10538.700 270

Round (R) 532.430 9 59.159 1.509 0.145

I×R 481.380 18 26.743 0.682 0.828

S×R 9524.890 243 39.197

conditions. But, in this case, it can also be suspected that there is somelearning effect during the rounds, in particular, under INFO2 conditions. Sowe should account for the learning effect when comparing divergence amonginformation conditions. Thus, we used repeated measures ANOVA to identifytreatment and learning effects simultaneously9. The ANOVA table is shownin Table 5.

Neither the between subjects factor (Information) or the within subjectfactor (Round) were significant (P = 0.113 and P = 0.145, respectively).That is, this test shows that the mean divergence was not significantly dif-ferent between information conditions and that there is no learning effect.Therefore, Hypothesis 3 stated in Section 3.1 was not rejected.

Thus, two of the three hypotheses mentioned in Section 3.1 were signific-antly rejected. The hypothesis we cannot reject is the weakest one mentionedin Section 3.1. Although we cannot say that there is a learning effect, as veri-fied by repeated measures ANOVA, we suspect that the performance underINFO2 conditions changed during the rounds. In fact, the mean divergenceunder INFO2 is close to that under INFO1 in the early rounds, but, in turn, itconverges that mean divergence under INFO3. This fact is why we did not re-ject Hypothesis 3. So, both information conditions, INFO2 and INFO3, havethe effect of improving the dominant strategy play in the Pivotal mechanism.That is, the more informative the information condition, the less deviationfrom the dominant strategy.

From the statistical analysis of our experimental data stated above, we canconclude that weak incentive compatibility of the mechanism could matter inthe subject’s understanding of the payoff structure, as we predicted earlier.This implies that we need to provide some information to induce subjects totell the truth in the Pivotal mechanism.

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5. Discussions and conclusion

Through our experiments, we have clarified that, even in small group decisionmaking, the Pivotal mechanism cannot induce subjects to reveal their truepreferences when only an explanation of the rules of the mechanism is given.And we have provided some empirical evidence which implies that perform-ance of the Pivotal mechanism in inducing truth-telling can be improved bysupplying some additional information on payoff structure.

Earlier experimental studies anticipated the first point of our results.Scherr and Babb (1975) conducted laboratory experiments to compare theperformance of the Pivotal mechanism with that of Lindahl mechanism. Sincethe results of their experiments showed that the amount of public goodsprovided under the Pivotal mechanism was less than that under Lindahlmechanism, they questioned the asserted power of the Pivotal mechanismto induce the revelation of the true demand for public goods. Tideman (1983)applied the Pivotal mechanism to the voting in decisions at university fratern-ity meetings. Some of the results of his experiments suggest the possibilitythat subjects might deviate from revealing their true valuation of each choice.

However, as pointed by Clarke (1975) in his comments on Scherr andBabb (1975), their experiments did not conform to the induced value theoryadvocated by Smith (1976), which prescribes that meaningful economic ex-periments should satisfy the conditions, such that subjects should be assignedsome preference function specified by the experimenter and be paid accordingto the payoff they receive in the experiments. Thus, we cannot evaluate theperformance of the Pivotal mechanism directly from the results of Scherrand Babb’s (1975) experiments. Neither did Tideman’s (1983) experimentswas conform to the induced value theory, since his study was a kind of fieldexperiment.

In our experiments, the subjects were assigned specified value as theirtrue valuation of the public project, and received enough monetary rewardswhich were dependent on the payoff they got in the experiments. Thus, ourexperiments conform to the induced value theory.

Recently, Attiyeh et al. (2000) conducted laboratory experiments of thePivotal mechanism which also agree with the induced value theory. The res-ults of their experiments were similar to those of our INFO1 experiment.Although they seems to be puzzled by the results of their experiments, theirnegative result regarding the performance of the Pivotal mechanism in indu-cing truth-telling can be well anticipated by our consideration of the incentiveproperty of the mechanism. That is, we suppose the lack of strict incentivecompatibility of the Pivotal mechanism could also cause subjects to fail toplay the dominant strategy in Attiyeh et al.’s (1997) experiments.

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Although the results of our experiments suggest that the lack of strictincentive compatibility can be a cause of failure of the Pivotal mechanism ininducing truth-telling, such a property of the mechanism may not be the onlyreason why the mechanism fails to induce truth-telling. Complicated pro-cesses to calculate a subject’s payoff from a set of all the subject’s messages,especially the way of determining his Clarke tax, may also deter subjectsfrom understanding that truth-telling is a dominant strategy. In order to seehow seriously a factor other than the lack of strict incentive compatibilitymatters, we need to conduct further experiments of the Pivotal mechan-ism in environments in which the mechanism does satisfy strict incentivecompatibility.

With strict convex preferences, a version of the Pivotal mechanism withdivisible public goods and revelation of continuous marginal demand sched-ules for public goods can provide such an environment, because the lackof strict incentive compatibility is due to indivisibility of public goods ornonconvexity of preferences in the environment we chose.10

Finally, as noted in the first section, application of the Pivotal mechanismto the problem of determining whether or not a public project with fixedsize should be realized, has great practical virtue. This is because, in realworld decision-making, the general public are probably ready to reveal theirvaluation to a public project of fixed size, such as the construction of a com-munity park of a fixed area with given facilities. On the other hand, it cannotbe imagined that people will reveal their demand curve over a whole rangeof possible sizes of public goods. Even revealing their valuations to severalproposed sizes of a project seems to impose a formidable task on them andto puzzle them. If just the environment we adopted in this investigation isrealistic, our results suggesting the failure of the Pivotal mechanism to inducetruth-telling with no information about the structure of the payoff, have veryimportant implications in practice. Should we give up the practical use of thePivotal mechanism in real public decision-making? We think it too precip-itate to discard the Pivotal mechanism as an impractical scheme. Provisionof some additional information, such as the presentation of a moderatelydetailed payoff table, may facilitate the mechanism to induce truth-telling. Ofcourse, we cannot present a true payoff table to each subject in real worlddecision-making, simply because we cannot know his true valuation of apublic project. However, it may be possible to construct some hypotheticalpayoff tables, which would show his payoffs under variable combinations ofrevealed valuations of his own and other community members’ contingent onthe assumption that his true valuation is this or that. Presentation of such hy-pothetical payoff tables may facilitate subjects in the real world to understandthat truth-telling is a dominant strategy in the Pivotal mechanism. Just ex-

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plaining the incentive compatibility of the mechanism may be not enough forreal world subjects to be convinced that truth-telling is a dominant strategy. Toconvince them, we need some means of providing information about payoffstructure, by use of which each subject can understand that truth-telling is adominant strategy as their own judgment. The experiments reported in thispaper suggest that devising such a means for provision of information is ofcritical importance in utilizing the Pivotal mechanism as a practical devicefor achieving efficient public decision-making.

Notes

1. Their conjecture was rigorously proved by Rob (1982) and Mitsui (1983).2. In INFO1 and INFO3 experiments at Tezukayama University, the experimenter read the

instructions aloud in front of the subjects.3. The instructions we used in the experiment are included in Appendix 1.4. The detailed payoff table we used in the experiment is included in Appendix 2.5. Using a questionnaire, we tested subjects, how they understood the explanation of Pivotal

mechanism. Then we compared their scores between the different subject pools using theChi-square test, and verified that there was no significant difference between them.

6. We omitted the INFO1 experiment at Tezukayama University because divergence from thedominant strategy was significantly higher than others. Our conclusion was not changedby omitting these data.

7. We also conducted Fisher’s exact test to examine the same null hypothesis, and, weobtained a rather weak conclusion (P = 0.094+).

8. We are grateful that the referee pointed this out to us.9. Repeated measures ANOVA is described in Winer et al. (1991).

10. It can be proved that the Pivotal mechanism with indivisible public goods, in general,cannot satisfy strict incentive compatibility. Some authors, including Groves and Loeb(1975), Mark Walker (1978, 1980) and Laffont and Maskin (1980, 1982), introduced strictconvex preferences and divisible public goods into their demonstration on Grove mech-anism as an essential feature. We think that these authors were aware of the importanceof strict incentive compatibility.

References

Attiyeh, G., Franciosi, R. and Isaac, R.M. (2000). Experiments with the pivot process forproviding public goods. Public Choice 102: 95–114.

Brubaker, E.R. (1983). On the Margolis thought experiment, and the applicability of demand-revealing mechanisms to large-group decisions. Public Choice 41: 315–319.

Clarke, E.H. (1971). Multipart pricing of public goods. Public Choice 11: 17–33.Clarke, E.H. (1975). Experimenting with public goods pricing: A comment. Public Choice 23:

49–53.Green, J. and Laffont, J.J. (1977). Characterization of satisfactory mechanisms for the

revelation of preferences for public goods. Econometrica 45: 427–438.

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Groves, T. (1973). Incentives in teams. Econometrica 41: 617–631.Groves, T. and M. Loeb (1975). Incentives and public inputs. Journal of Public Economics 4:

211–226.Groves, T. and Ledyard, J. (1987). Incentive compatibility since 1972. In Groves, T., Rad-

ner, R. and Reiter, S. (Eds.), Information, incentives and economic mechanisms, 48–111.University of Minnesota Press.

Hammond, P.J. (1979). Straightforward individual incentive compatibility in large economies.Review of Economic Studies 46: 263–282.

Laffont, J.J. and Maskin, E. (1980). A differential approach to dominant strategy mechanisms.Econometrica 48: 1507–1520.

Laffont, J.J. and Maskin, E. (1982). Nash and dominant strategy implementation in economicenvironments. Journal of Mathematical Economics 10: 17–47.

Margolis, H. (1982). A thought experiment on demand-revealing mechanisms. Public Choice38: 87–91.

Mitsui, T. (1983). Asymptotic efficiency of the pivotal mechanism with general project space.Journal of Economic Theory 31: 318–331.

Rob, R. (1982). Asymptotic efficiency of the demand revealing mechanism. Journal ofEconomic Theory 28: 207–220.

Scherr, B.A. and Babb, E.M. (1975). Pricing public goods: An experiment with two proposedpricing systems. Public Choice 23: 35–48.

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Tideman, T.N. and Tullock, G. (1976). A new and superior process for making social choices.Journal of Political Economy 84: 1145–1159.

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Winer, B.J., Brown, D.R. and Michels, K.M. (1991). Statistical principles in experimentaldesign. Third ed., McGraw-Hill.

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Appendix 1

Instructions

Before the experiment1. Take these “Instructions” and the envelope with your number written at the top.

2. Take a seat in front of the computer on which your number is written.

3. Do not talk to anyone.

4. Do not touch any key on the keyboard until the experimenter tells you to.

5. Take the “Recording sheet” out of your envelope and fill in your name andstudent number.

Read these “Instructions” until the experimenter gives you further instructions.

Group number, your number, and initial fundWrite “Group No.”, “Your No.”, and “Initial Fund” on your “Recording Sheet” whenthese numbers are displayed on your computer screen.

Overview of the experimentIn this experiment, you are a member of a group which is indicated by your “GroupNo.” together with four others. Who else is in your group is not known. What youhave to do in this experiment is to reveal “Your Value” for a project which givesbenefit or loss for all the members of your group. The project is realized (“GO”)when the “Total Value”, which is the sum of the revealed value by all the membersof your group, is positive and is not realized (“STOP”) when the “Total Value” isnegative or zero.

You can earn the amount equal to “Your Profit” each “Round” when the projectis realized. But if “Your Profit” is negative, you will lose the amount equal to “YourProfit” when the project is realized. You get a zero profit when the project is notworked out.

For each “Round”, “Your Net Benefit” is determined by subtracting “Tax” from“Your Profit” when the project is realized, and minus “Tax” when the project is notrealized. “Tax” is the absolute value of the “Total Value” minus “Your Value” whenthe results of “Group Decision” and “Decision by other members” are different, andzero when the results of “Group Decision” and “Decision by other members” is thesame. “Your Net Profit” could be negative, so we will give you “Initial Fund” fromthe start. “Your Fund” is determined by adding “Your Net Profit” to “Initial Fund”when “Your Net Profit” is positive, and by subtracting “Your Net Profit” from “Ini-tial Fund” when “Your Net Profit” becomes negative. All the information describedabove is calculated by computer, and the results will be displayed on the screen.

Your monetary reward will be paid in Japanese Yen, equal to ten times “YourFund” which still remains after the final “Round”.

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Notes1. This experiment will continue until “Round” ten. (Two “Practice Rounds” will

be conducted before the experiment.)2. “Your Profit” is specified and displayed on your computer screen. “Your Profit”

may be same for all “Rounds” or be changed each “Round”. So watch yourscreen carefully. Of course, “Your Profit” is different between the members ofyour group (“Your Profit” is different between the “Practice Round” and the“Round” in the experiment.)

3. The range of “Your Value” is restricted in integers from –25 to 25 (includingzero).

4. You should retain all the information, including the numbers displayed on yourcomputer screen and the numbers you enter via the keyboard. Record the detailson your “Recording Sheet” each time. Do not give this information to anyone.

Practice rounda. Press key “g” (there is no need to press the “Return” key).b. “Practice Round 1” and “(A) Your Profit” is shown on your screen together

with “Group No.”, “Your No.”, etc. “(A) Your Profit” is the same number asthat written in your number column on the first line, called “(A) Your Profit” inTable A1 of these instructions for “Practice Round 1” (other member’s “YourProfit” is, of course, unknown to you).

c. Fill in the space in the first column of “(A) Your Profit” in Table A1: “PracticeRound 1” on your “Recording Sheet”.

d. When “(B) Input Your Value?” is displayed, this prompts you to enter “YourValue”. You should enter the same number as that written in “(B) Your Value”in Table A1 for “Practice Round 1”. Of course, you should enter any integersranging from –25 to 25 to your benefit in the experiment.

e. “Your Input Value =∗∗∗∗ OK? <Y/N>” is then shown. Press “y” if it is OK orotherwise press “n” and re-enter. Record the number you enter in the columnlabeled “(B) Your Value” in “Practice Round 1” on your “Recording Sheet”.

f. When all the members have entered their “Your Value”, “Group No.”, “YourNo.”, and “Practice Round 1” are shown again. Next the following information(A) through (H) is shown written in each column of Table A1, separately foreach subject.(A) Your Profit: Pre-specified value for you.(B) Your Value: The number you enter.(C) Total Value: The sum of item (B) which you and the four other members

of your group enter.(D) Group Decision: “GO” if (C) is positive or “STOP” if (C) is negative or

zero.(E) Sum of Other Members’ Value: The sum of item (B) which the four other

members of your group enter, or this is equal to (C)–(B).(F) Decision by Other Members: “GO” if (E) is positive or “STOP” if (E) is

negative or zero.(G) Tax: Zero if (F) and (D) are same, or the absolute value of (E).

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Table A1. Practice Round 1

Your Number 1 2 3 4 5 Total

(A) Your Profit –10 –15 5 15 10 5

(B) Your Value 5 –25 –5 10 25 10

(C) Total Value 10 10 10 10 10 –

(D) Group Decision GO GO GO GO GO –

(E) Sum of Other Members’ Value 5 35 150 –15 –

(F) Decision by Other Members GO GO GO STOP STOP –

(G) Tax 0 0 0 0 15 –

(H) Your Net Profit –10 –15 5 15 –5 –

Table A2. Practice Round 2

Your Number 1 2 3 4 5 Total

(A) Your Profit –10 –15 5 15 10 5

(B) Your Value –25 5 25 –3 –10 10

(C) Total Value –8 –8 –8 –8 –8 –

(D) Group Decision STOP STOP STOP STOP STOP –

(E) Sum of Other Members’ Value 17 –13 –33 –52 –

(F) Decision by Other Members GO STOP STOP STOP GO –

(G) Tax 17 0 0 0 2 –

(H) Your Net Profit –17 0 0 0 –2 –

(H) Your Net Profit: (A) minus (G) if (D) is “GO” or minus (G) if (D) is“STOP”.

In “Practice Round 1”, these values are shown in (A) through (H) in Table A1.

g. Record (C) through (H) in the appropriate column in “Practice Round 1” onyour “Recording Sheet”. Your monetary reward is the amount equal to ten times(H). In each “Round” “(I)Your Fund” is shown next to (H). This value indicatesthe reward you have earned so far. Record “(I)Your Fund” on your “RecordingSheet”.

h. When you understand how a “Round” works, press key “g” and go to “PracticeRound 2”.

In “Practice Round 2”, proceed in the same order as in “Practice Round 1”.In “Practice Round 2”, “Your Profit” is the same as in “Practice Round 1”, but

you should enter “(B)Your Value “ in Table A2 as “Your Value”. In “Practice Round2”, the following (C) through (H) values are shown after you enter “Your Value”.

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Notes1. Do not talk to anyone and do not show your “Recording Sheet” to anyone during

this experiment.2. The value you enter is restricted to integers ranging from –25 to 25 (including

zero).3. You should enter “Your Value” within thirty seconds.4. To go to the next “Round”, press key “g”.5. You should choose “Your Value” to make “Your Net Profit” as high as possible

(or to reduce it as little as possible if “Your Profit” is negative).

After the experiment1. Take “Debriefing Sheet” from your envelope, fill in “Group No.” and “Your No.”,

and answer the two questions listed.2. Take your “Recording Sheet” and “Debriefing Sheet” to the experimenter, and

receive your monetary reward equal to ten times “Your Fund”. This is the end ofthe experiment.

All participants should take their instructions home.

Press key “g” and go to “Round 1”.

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