Voting behavior in the presence of

reputational concerns

Charles Louis-SidoisMay 19, 2014

Mémoire présenté pour le Master Economics and Public PolicyPhd track

Directeur du mémoire : Emeric Henry

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Abstract

This paper studies the voting behavior of agents who take their reputation into account. We build a

model with heterogeneously motivated agents based on a two-stages game. Players contribute to a public

good in the second period and vote for a sanction to punish freeriders in the first period.

We first analyze a setup where only the contribution is observed. In this context, voting is anonymous

and reputation cannot depend on it. We analyze how reputational concerns can provide further incentives

to vote for the sanction or on the contrary undermine the willingness to implement it. We study how the

result of the vote evolves with the level of the proposed sanction and with the motivation of agents. We

show that when the vote is private, agents vote for the alternative that provides them the highest utility

in the contribution game.

When the vote is public, we conclude that agents tend to vote more often for the sanction when the

number of voters is large. We also show all agents pooling on the same voting decision is always an

equilibrium.

Finally, we discuss the social benefits of voting and study how a benevolent social planner should design

the voting process in order to maximize global welfare.

Introduction

Much evidence suggests that people have reputational concerns when they vote. Knowing someone’s votepartly reveals what he thinks and who he is. Image concerns have been studied for long: people generallyadapt their behavior to be perceived as nice. However, to my knowledge, there as been no attempt to intro-duce reputational concerns in voting theory.

Even if not all votes involve reputational concerns, in most elections one can identify a virtuous alterna-tive. For example, when MPs have to vote for or against a law proposal aiming to improve public welfare,public opinion is likely to consider that yes-voters are more pro-social than no-voters. In a meeting of a firm’scommittee, those who vote for a pay rise of low skilled workers will be better considered than those who voteto keep the profit for the members of the committee. We could go on with infinity of examples. One couldeven dare to extend such reasoning to classical political elections opposing right wing to left wing candidates.Because left wing candidates generally have a more pro-social program, those who vote for those candidatesmight be trying to show their generosity.

Voting is however not a cheap talk game: knowing that final decision is conditioned to the result of thevote, individuals’ decisions are payoff relevant when the voter is pivotal. If a voter is decisive in an election,his decision will be implemented and image concerns will not be the only factor determining his vote. Agentswill therefore trade off between their reputational payoff and direct consequences of their vote, meaning thechange in the probability that the law proposal is implemented.

Our study will try to answer two main questions. The first scope of the paper is to determine how peopleadapt their voting strategy when reputation matters. This part can be considered as an extension of strategicvoting with a specific utility function accounting for the seeking of social approval. This point is expectedto shed more light on how people take their voting decisions under different voting designs when they careabout their image.

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The second question our study will focus on is how the voting process should be designed in order tomaximize global welfare. More specifically we will try to determine if it is better for the principal to letagents vote or to impose the sanction by force. We will also study which type of voting procedure shouldthe principal choose: does he have interest to implement anonymous or public voting (i.e. decide if votingdecisions are observed or not)? Which is the optimal level of sanction he should submit to the vote? Shouldhe disclose the detailed results of the vote or should he pretend not to have all information. . . ?

Using a game theoretic framework, we build a model based on a two stages game where agents take theirreputation into account. We use a slightly modified version of the utility function of Bénabou and Tirole(2011). First stage of the game is a voting phase where agents vote for or against a proposed sanction topunish free riders. Second stage is a public good game which rule depends on the result of the vote.

I present the literature review in the next section. In Section 2, I describe the model. Section 3 analysesthe anonymous voting game with a continuum of agents. In Section 4 we introduce uncertainty about thedistribution by considering a finite number of agents. Finally, I study the public voting case in Section 5.

1 Literature review

Our topic is halfway between two major fields in economic theory which are social norm and voting. Iwill briefly present related literature in those two topics.

1.1 Social norm

Many experimental studies have shown that individuals want to act accordingly to what they think thenorm is. In a public good setup where contributions are observed, agents are strongly influenced by whatothers do. Bicchieri and Xiao (2010) led an experiment where they showed that in a standard dictator game,donations can be significantly increased when dictators think that other dictators are generous. In theirstudy, they split dictators in two groups and selectively disclose information. To one group, they pretendthat in previous experiments dictators acted generously and to the other they mention selfish behaviors inprevious experiments. Results show that positive information about others’ generosity significantly increasesdonations. Authors interpret it as a willingness to comply with the norm.

Compliance with the norm has to be linked with the literature on reputation. A way to see reputation ishow “good” someone appears to others. In a public good game, we can consider this idea of “good” as howmuch someone is ready to give. In standard setup, it is optimal for agents not to give but if they decideto do so others will benefit from this sacrifice. But why some people decide to contribute while others donot? Possible answer is to assume that agents have different generosity or taste for the task. Assuming thatbest agents are those who are the most generous, players will try to act as if they were generous. Manyinterpretations have been proposed: in an evolutionary context, it can be seen as a signaling game wherebest individuals mate (Seabright 2004, Gintis 2006). In a career concern setup, agents will try to signal tothe principal that he is motivated for the task (Bénabou and Tirole 2003).

In a more sophisticated model, Bénabou and Tirole (2006) introduce multidimensional heterogeneity.

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Agents differ on their generosity, but also on their greediness (how much they value the incentive) and ontheir taste for glory (how much they value their reputation). In this setup, there is noise in the signal sentby agents when they contribute: did they act because they were generous, because they wanted the moneyof the subvention or because they were trying to look good?

All those models aim to study how people adapt their behavior when they take their reputation intoaccount. Political implication of this literature is to determine how policy makers should design incentivesand sanctions in a world where people care about reputation. Global conclusion is that some agents willcontribute in order to pretend that they are motivated even though they are not.

Less intuitive, most of those papers (especially Bénabou and Tirole 2006) show that a crowding out ef-fect can occur because of the noise induced by the sanction in the signal sent by contributors. Consider anexample where only very motivated people contribute when there is no incentive to do so. Participants willunambiguously show that they have the highest motivation. When principal implements a reward, greedyagents will also contribute. In such case, one cannot tell if agents contribute because they are motivated orbecause they are greedy. If the first participants do not care about money, their motivation is crowded outand they can stop contributing when the incentive is implemented.

This wide theoretical literature is supported by an even wider experimental corpus plus common wisdom.Bowles (2006) and Fehr and Falk (2002) provide good surveys of experiments about the link between rep-utation and incentives. Ariely and al. (2009) present an experiment where students had to exert an effort(click as many times as possible on a button during a given interval of time) in order to earn money forcharity. Students were split in two groups: in the first group, results were anonymous while in the secondperformances were publicly announced. Results showed that when results were public, students clicked onaverage significantly more. Convincing explanation is to say that people do care about what others thinkabout them and that they are ready to pay (here by a physical effort) in order to be identified as good. Itjustifies the need to study modified utility functions designed to take this taste for popularity into account.

Bénabou and Tirole (2011) focus on how a social planner can manipulate the norm by choosing the levelof financial incentives for contributors in order to maximize participation. Agents receive a financial incentivey if they contribute. They have different intrinsic motivations, which can be seen as their taste for the task.They want to signal that they are motivated. In this model, utility of agents is:

U = (V + y � c)a+ eA+ mE[V |a, y]

Agents take into account direct cost of participation and reputation. Direct cost (V +y�c)a is how costly it isfor the agent to contribute. He pays a cost c, but he receives a bonus y plus the satisfaction of having behavedgenerously V . His reputation E[V |a, y] is the expectation of agent’s generosity/motivation V depending on thecontribution chosen and on the level of incentives y. eA is simply the externality from aggregate contribution.

When society is on average motivated for a given cause, agents have more incentives to contribute evenif they are not personally deeply concerned. It is the case because when an individual expects others tocontribute, he knows that freeriding will unambiguously signal him as the black sheep of the group. In themodel, agents are not informed perfectly about the distribution of the types. More specifically, the support

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of the distribution might have been shifted to the right by a given parameter. Agents know the probabilityof the shift and their own motivation and from there they form prior belief about the actual distribution.It implies that more motivated agents think that the distribution is more likely to be shifted, as they willrationally expect their motivation to have been drawn from a higher distribution.

In this set-up, Bénabou and Tirole show that there are two options for a benevolent social planner toincrease participation. The “classic” way is to increase financial incentives, i.e. to increase y. Reputationalconcerns allow for a second option: try to manipulate social norm. If social planner can make people believethat the distribution of types has been shifted, possible free riders might be deterred from this option be-cause of the threat of bearing large stigma. It will increase social pressure on agents and will push them tocontribute. They provide many examples where communication about social norm improved participationsignificantly.

Literature on optimal taxation concluded that optimal level of incentives/sanctions depended on agents’motivations. Generally, the more motivated the agents are, the smaller the need to subsidize. In a set-upwhere agents take their reputation into account and where social planner has interest to manipulate thenorm, Bénabou and Tirole show that planner will be tempted to reduce the incentives in order to let agentsthink that others are motivated (as in such a case large incentive is not needed).

Our model will introduce reputational concerns in the voting process. We will study how social imageimpacts decision making in a voting game. Our topic is therefore also related to the literature on voting.

1.2 Voting

Many voting models have studied how rational agents behave in a two-candidates election. Feddersen andPesendorfer (1996) and Feddersen and Pesendorfer (1997) as well as Austen-Smith and Banks (1996) providebenchmarks of how rational agents with different information and different preferences for the outcome of thevote should vote strategically. In those models, a key point is that agents know that they can only changethe result of the election if they are pivotal.

In Feddersen and Pesendorfer (1996) some agents are perfectly informed and some are not. For unin-formed voters, strategic voting can imply to vote for a given option (or to abstain) regardless of any signalpossibly received in order to let informed voters take the decision.

In these models, voters only consider the result of the vote. They do not take into account potential repu-tational outcome or a possible stage of the game played after the vote. Our model will adapt strategic votingin a context where agents not only care about the results of the election but also consider the consequencesof their voting decision on their reputation.

In a public good setup, free riding is negatively connoted as free riders favor their own utility to thedetrimental of the community. If an election is organized to adopt a sanction for punishing free riders, wecan expect that if we could observe the voting decision of an individual it would give us an information abouthow much this agent is motivated for the task. In such a case, when voting is publicly observed, agents willtake into account the signal sent by their voting decision. This will add a new strategic dimension in the

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voting process, which can change the decision taken by some agents. In the set-up we study, there will bea trade off between the signal an agent wants to send to the community about himself via his vote and thedecision he wants to implement.

Levy (2007) is to my knowledge the most closely related to our problematic. In his model, experts arehired by a firm to choose one of two alternatives. One decision is better for the firm, but experts do notdirectly receive utility from the option chosen. Each expert has private information about which alternativeis the best. The accuracy of information varies across experts. They have career concerns: experts wantto show that they are competent (i.e. their information is accurate). Experts’ utility will then depend onemployer’s beliefs on experts’ quality of information.

In Levy (2007), there is no such thing a behavior after the vote: experts are only concerned by theirreputation after the first phase. Moreover, experts are judged on their ability and not on their generosity.To my knowledge, there is no voting model where the norm is taken into account: there is a right decisionbut no such thing as a “moral” decision from a fairness point of view.

Even though no theoretic work has been realized on the link between vote and norm, an experiment ledby Tyran and Feld (2006) provides us with behavioral evidence on the topic. In this study, players are splitin groups of three. They receive an initial endowment of 20 ECU that they can invest in a public good.Players get what they have kept plus one half of the sum of investments. Noting a

i

2 [0, 20] the investmentof player i, payoffs can be written:

⇡

i

= (20� a

i

) +1

2(a

i

+X

a

j

)

In this setup, dominant strategy is to freeride (ai

= 0 ). It implies a payoff of 20 for all players. However,higher payoff can be achieved with cooperation: if players invest all their endowment in the public good, theyreceive a payoff of 30.

On the top of the game, a sanction can be implemented to punish those who did not contribute enough tothe public good. This sanction can be applied through different channels: it is whether exogenously imposed,whether endogenously chosen by a vote within the group. Results show that when sanction is endogenouslyadopted, it raises participation much more than when it is exogenously imposed. On the contrary, whensanction is rejected contribution drops.

One can interpret these results as a consequence of updated beliefs about social norms. If altruistic peopleare expected to vote yes, players can interpret a positive result of the vote as evidence that agents are onaverage very altruistic and that the norm is to contribute. Combining with willingness to comply with thenorm, this experiment could show that people indeed interpret the results of a vote as an indicator of the norm.

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2 Model

2.1 General Framework

We model the voting process by a two stages game. The second stage of the game is a classic public goodgame where agents can contribute or freeride. A sanction is proposed for punishing freeriders. In the firststage of the game, agents vote for or against the implementation of the sanction. The key feature of ourmodel is that agents take their reputation into account.

There are N players indexed by i = 1, 2, ..., N . We will consider an infinity of players in Section 4 but inSections 5 and 6 N will be an odd number bigger than 3. Each agent has to choose two actions: the votingdecision a1 2 {0, 1} (where a1 = 1 when the agent votes for the sanction) and a2 2 {0, 1} (a2 = 1 when theagent contributes). Each agent is of type v

i

where all the v

i

are i.i.d. from the distribution of types f(.). v

i

is the motivation of agent i. It can represent his taste for the task, his generosity... Players with high v

i

willbe considered as good agents, they will therefore try to show that they have a high motivation. The payofffunction we use is a modified version of Bénabou and Tirole (2011):

U

i

= (vi

� c)a2,i � s(1� a2,i)1[sanction] + e⇥P

j

a2,j

N

+ µ⇥ E

j

[vi

]

There is a cost c for contributing, it can be interpreted as a financial cost (contribution to the publicgood) or as the physical effort of contribution, time it takes to contribute. . . We assume that the cost isthe same for all agents. There is a positive externality e from a high contribution rate to the public good.This externality represents the taste of agents for a nice environment. Last term represents the reputationof agents. µ is how much agents care about reputation. It can be the visibility of the contribution or simplythe taste of agents for glory. For sake of simplicity, we assume that all agents have the same preference forreputation. Reputation of agent i is what others think his motivation is.

This being said, we still have to specify the information available to players. What we allow players toknow depends on the rule of the voting process.

2.2 Voting process

The rules of the voting phase are common knowledge. First of all, we have to specify how the outcomeof the vote is chosen. We assume that there is a fraction q = K

N

of yes-voters needed for the sanction to beaccepted.

What do agents know about the distribution of types? We will always assume that each player observeshis own type. Agents know the distribution f(.) from which types are drawn. In Section 4, we will assumethat N is infinity so the actual distribution of types will be f(.). In Sections 5 and 6 however, there will beuncertainty because of the finite number of agents, each one drawn randomly from the distribution. In sucha case, the actual distribution (i.e. the realization of the draws) is not known.

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The strategy of player i depends on his type and on the result of the vote R:

S

i

:v

i

! a1

v

i

⇥R ! a2

In the game, a full contingent plan implies that before he discovers his type, each player defines a strategyfor each possible type and each possible outcome of the vote. Of course, a1 can only be conditioned on thetype of player i because the result of the vote is revealed after.

What do players know about the result of the vote? If only the decision of the vote is revealed,R 2 {accepted, rejected}. If detailed results are available, players know the share y 2 [0, 1] of yes votersand can specify a strategy for each possible outcome.

Finally we have to tell what is available for the formation of reputation. Each alternative represents adifferent voting process. In Section 4, we will analyze anonymous voting: only the contribution decisionis observed. It is what happens in most political elections: people vote secretly in voting booth but theirbehavior afterwards can be observed. In such a case, reputation can only be conditioned on the secondaction: E[v

i

|a2]. In Section 6, only the vote is observed, which implies that reputation is only based on thefirst action E[v

i

|a1]. It is the case for example in votes by show of hands when the contribution is madeafterwards and privately. Each case will imply different strategic behaviors that we will discuss.

2.3 Social planner

Who decides the rules of the game? In this study we will consider that the setup is imposed. In the endof section 4, we will consider that a social planner has to choose the optimal level of sanction submitted tothe vote in order to maximize global welfare.

It would be interesting to study a canonical game where the planner can also choose the rules of thevoting process.

3 Perfect information, anonymous voting

In this part we will consider a simple setup where voting is anonymous. This setup represents most po-litical elections: people vote secretly in voting booth and voting decision is never observed. One can alwaysclaim that he voted for a given decision but he will not be able to prove it. In this section, we consider thatthere is an infinity of voters. This ensures that actual distribution of types will correspond to the theoreticaldistribution f(.). In such a case, inference about type can only be based on the contribution decision andreputation simplifies to E[v

i

|a2].

3.1 Participation decision

Under these assumptions, we can solve the agents’ decision by backward induction: as the distributionis known, agents will correctly anticipate the average behavior of other agents in the second phase with andwithout the sanction. When sanction is not implemented, agents will participate in the second period if andonly if: E[U

i

|a = 1] � E[Ui

|a = 0]. Like in Bénabou and Tirole (2011), there is a cut-off v

⇤ such that agents

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participate if and only if vi

> v

⇤ 1. For this value, agent is indifferent between participation and abstention:

v

⇤ = c� µ4(v⇤)

Where 4(v⇤) = E[vi

|vi

> v

⇤]� E[vi

|vi

< v

⇤]

Participation is driven by two elements. First, direct cost of participation is c� v. Second, µ4(v⇤) is thesocial pressure. It depends on how much people care about their reputation µ and on the difference of repu-tation between the two possible behaviors. This signaling problem depends on the shape of the distributionof types.

Similarly, there exists another cutoff v

⇤s

when the sanction is implemented:

v

⇤s

= c� µ4(v⇤s

) � s

If the sanction is implemented, participation is increasing in s meaning that a higher sanction will leadto a higher participation2. This implies that sanction always increases participation: v

⇤ � v

⇤s

.

Assume that types are normally distributed v s N(0, 1). Figure 1 shows the social pressure as a functionof the equilibrium cutoff. What we observe is very classical and already discussed in Bénabou and Tirole.They distinguish two types of acts: normal acts and heroic acts.

Figure 1

As we can see in Figure 1, social pressure is strong for heroic acts and for normal acts if we assume thattypes are normally distributed. However, this is not necessarily the case for all density functions. For normalacts, most agents contribute, even without the sanction. This is the case when the cost to contribute is lowor when the distribution of types is high (agents are very motivated). Good agents do not reap much honoras their expected motivation is close to the middle of the distribution. However, bad agents suffer from greatstigma as they are known to be the black sheep.

On the other hand, only a few agents contribute for heroic acts. In such a case, the cost to contributeis very high (think for example to kidney donation). There is not much stigma for not contributors but

1All proofs in the appendix

2See the appendix for further analysis of the link between participation and level of sanction.

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participants are perceived as extremely generous and reap great honor.

Finally, social pressure is the weakest for modal acts because in such cases contribution and freeriding areboth common so there is little information revealed about agents’ types.

We can distinguish three categories of agents for a given level of sanction:v

i

< v

⇤s

are the agents who never participate,v

⇤s

< v

i

< v

⇤ are the agents who condition their participation decision to the result of the vote. Theycontribute if and only if the sanction is implemented,

v

i

> v

⇤ are the agents who always participate.

Composition of the group is expected to evolve with the level of sanction as in Figure 2. The size of thegroup of always participants is fixed and equal to 1� F (v⇤). The number of not participants decreases withs. For large sanctions, almost all agents would prefer to contribute if the sanction is implemented: this iswhy we observe that the number of never participants goes to zero when the sanction becomes large. In thiscase, most agents are intermediate: if sanction is implemented, they contribute because it is too expansiveto pay the fine.

Figure 2

3.2 Voting decision

In the voting part, agents know the equilibrium of the contribution phase with and without the sanction(the two cut-offs are known for a given level of sanction). They also know their own participation decisionin both cases.

We will only consider an equilibrium where agents vote honestly (i.e. for the option that gives them thehighest utility). Honest voting is here only a weakly dominant strategy: atomicity of agents ensures thatthere is no chance to change the decision so in theory agents are indifferent between the two voting options.We could have an infinity of Nash equilibria where agents vote for the option that would hurt them if it

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was implemented because they know that their choice will not change the outcome of the vote. We rule outthose equilibria because although compatible with a pure game theoretic approach they make little sense ina concrete interpretation.

For always participants, utility when the sanction is implemented is

e(1� F (v⇤s

)) + µE[vi

|vi

> v

⇤s

]

ande(1� F (v⇤)) + µE[v

i

|vi

> v

⇤]

when sanction is rejected.

As a result, these agents will vote for the sanction if and only if:

e[F (v⇤)� F (v⇤s

)] � µ(E[vi

|vi

> v

⇤]� E[vi

|vi

> v

⇤s

])

Always participants have to trade between externality gains and reputational concerns.

Similarly, agents who never contribute will vote for the sanction if and only if:

e[F (v⇤)� F (v⇤s

)] � µ(E[vi

|vi

< v

⇤]� E[vi

|vi

< v

⇤s

]) + s

On the top of the two first effects (externality gains and reputational concerns), never participants haveto pay the sanction s, which reduces the incentive to vote for.

For the intermediate group, it becomes:

e[F (v⇤)� F (v⇤s

)] � µ(E[vi

|vi

< v

⇤]� E[vi

|vi

> v

⇤s

]) + c� v

i

For them, voting is an arbitrage between externality and reputational gains on the one hand and the realcost to contribute (c� v

i

) on the other hand.

Note that intrinsic motivation does not appear for always participants nor for never participants. Itimplies our first proposition.

Proposition 1:

All agents in the always participants group take the same voting decisions.

All agents in the never participants group take the same voting decisions.

An analytic comparison of the voting constraints leads to propositions 2.

Proposition 2:

When the vote is anonymous, there exists a voting cutoff V

⇤such that agents vote for the sanction if and

only if their reputation is higher than V

⇤

If this cutoff is in the intermediate group, it can be expressed:

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V

⇤ = c+ µ(E[vi

|v < v

⇤]� E[vi

|v > v

⇤s

])� e[F (v⇤)� F (v⇤s

)]

When V

⇤> v

⇤, V

⇤ = +1 and all agents oppose to the sanction.

When V

⇤< v

⇤s

, V

⇤ = �1 and all agents vote for the sanction.

Proposition 2 states that whenever an agent of type v

i

votes for the sanction, all agents of type v

j

> v

i

also vote for it. Conversely, if v

i

opposes to the sanction, all agents with v

j

< v

i

will also vote against.Combining with the first proposition, it implies that when a never participant votes for the sanction, allagents must vote for it. Furthermore, if an always participant opposes to the sanction, then all agents willvote against it.

For all agents, voting decision depends positively on the externality e[F (v⇤) � F (v⇤s

)] (left hand side ofthe three equations). This term is composed of two elements: the importance of the externality e and thesize of the intermediate group F (v⇤) � F (v⇤

s

). When this group is big, implementing the sanction leads toa huge increase in participation (and then to a nicer environment). As v

⇤s

is decreasing with respect to thesanction, externality depends positively on the level of s. Through this channel, we could imagine that forhigh concerns for externality, a huge sanction might give strong incentives to vote for it, even for poorlymotivated agents.

Recall that we have imposed that sanction increases participation. For sufficiently high concerns for ex-ternality (i.e. large e), agents will always be willing to vote for the sanction, whatever their participationdecision. Let’s take the example of dog pollution. If the sanction is expected to make dog owners much morecareful about their pets, even a very reluctant dog owner who would not change his behavior if the sanctionwere implemented could vote for in order to benefit from a cleaner sidewalk. This remark implies our thirdproposition.

Proposition 3:

For each agent and every possible value of the sanction, there exists a value of concerns for externality e

such that agent votes for implementation if e � e.

Reputational concerns undermine the willingness to adopt the sanction for always participants and neverparticipants. When the sanction is implemented, participation increases. Therefore, good agents cannot bedistinguished anymore from intermediate agents as they both participate (recall that for the time being thevote is not observed). Those who do not participate in the second period signal themselves as the least mo-tivated when the sanction is implemented as they are for sure in the never participants group. Moreover, asthe size of the intermediate group increases with s, those reputational losses increase with the sanction, whichundermines the positive incentive from externality highlighted above. When reputational concerns mattermuch more than externality, we can imagine that even very good agents might vote against the sanction inorder to keep their reputational benefits.

Reputational concerns of the intermediate group are a bit more subtle. For this group, a positive vote isa commitment to participate in the contribution phase. If the sanction is implemented, they would also get apositive reputational benefit as they would pool on the same contribution decision than always participants.

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This reputational incentive is decreasing with the sanction because when sanction is large, intermediate groupis big and expected motivation of participants if the sanction is implemented decreases.

Combining reputational losses of always participants with Proposition 2, we express our fourth result.

Proposition 4:

For each agent and every possible value of the sanction, there exists a taste for reputation µ such that

agent votes against implementation if µ � µ.

Comparing Proposition 3 and Proposition 4 yields the following conclusions. Sanction will be easily im-plemented when externality gains matter a lot while when reputational concerns are important, sanction ismore likely to be rejected. Those two propositions should therefore be interpreted looking at the relative sizeof e and µ.

Proposition 4 has an interesting implication. Consider cases where externality is neglectible with respectto reputation. It can represent charities aiming to help far away countries. For society (meaning for thecountry where money is collected) expected spillovers are close to zero. One can only give in order to boosthis self esteem or to show others that he is generous. Suppose that people have to vote for a law to favordonation (typically a fine for not contributors).

If sanction is implemented, more people will contribute and donation will become a common behavior.Our model predicts that in such a case, always participants will only suffer from the reputational loss. Propo-sition 2 ensures that when always participants vote no, all agents vote no and sanction is rejected. Everyonewould be opposed to a law proposal aiming to encourage people to give money to this kind of charity. Evenpeople who are involved in such associations will vote against the law: they prefer the status quo where theycan signal themselves as very generous and they do not want others to join their charity.

3.3 Impact of the sanction on the vote

To see how the level of the sanction impacts the result of the vote, we have to determine how the thevoting cutoff V

⇤ evolves with respect to s.

Thanks to Proposition 2, we know that we only have to compute the motivation such that an agent isindifferent between the two voting options if he is in the intermediate group. Then we will check if thismotivation is indeed in the intermediate group.

First, let’s define a function f(s) such that:

f(s) = c+ µ(E[vi

|v < v

⇤]� E[vi

|v > v

⇤s

])� e[F (v⇤)� F (v⇤s

)]

f(s) is simply the expression of the voting cutoff provided it is in the intermediate group.

In order to provide a graphical analysis of the result of the vote, we plot f(s), v⇤s

and v

⇤ in function of sin Figure 3.

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This graph is obtained with a normal distribution of types with mean 0 and variance 1. We choosearbitrarily the following parameters to solve the model with Octave:

µ = 1 e = 4

c = 3.3 s 2 [0, 4]

Figure 3

Thanks to Proposition 2, we know that if f(s) > v

⇤, all agents will oppose to the sanction while iff(s) < v

⇤s

all agents will vote for it. From Figure 3, we can therefore easily recover the number of yes-voters,which is simply 1� F (V ⇤).

Figure 4

14

To provide further interpretation of how the level of the sanction affects the voting decision, we can ana-lyze how s impacts the payoffs of the different groups.

First, consider always participants. We can write their gains from implementation as:

e[F (v⇤)� F (v⇤s

)]� µ(E[vi

|vi

> v

⇤]� E[vi

|vi

> v

⇤s

])

To see how it evolves with respect to the level of the sanction, we compute the derivative of the gainswith respect to s:

�@v

⇤s

@s

f(v⇤s

)[e� µ

v

max

� v

⇤s

�´v

max

v

⇤s

F (v)dv

(1� F (v⇤s

))2]

�@v

⇤s

@s

f(v⇤s

) represents participation gains. It is positive because of the restrictions we imposed. Sign ofthe derivative only depends on the second term.

The bigger e, the more likely a marginal increase of sanction will give further incentives to vote for. Thisis not a surprise and confirms Proposition 3.

More interesting is the reputation loss �µ

v

max

�v

⇤s

�´v

max

v

⇤s

F (v)dv

(1�F (v⇤s

))2 . It is always negative.3 We have to knowin which part of the distribution reputation loss is the highest. Provided that µ is sufficiently bigger than e,derivative can be negative and marginal increase in sanction can reduce the incentive of always participantsto vote for.

Similarly, we can rewrite the participation constraints for other groups:For intermediate agents:

e[F (v⇤)� F (v⇤s

)]� µ(E[vi

|vi

> v

⇤s

]� E[vi

|vi

< v

⇤])� c+ v

i

Terms depending on s are the same that for always participants sot the derivative is unchanged. Nowthe constant is more tricky as it includes v

i

. Indeed, we have one constant per individual in the intermediategroup, but those constants are clearly ordered with respect to motivations: whenever left hand side functionis above the constant of a given v

i

, it will also be above constraints for all intermediate agents with v

j

< v

i

,which is consistent with Proposition 2.

For never participants:

e[F (v⇤)� F (v⇤s

)]� µ(E[vi

|vi

< v

⇤s

]� E[vi

|vi

< v

⇤])� s

And derivative of left hand side gives:

�@v

⇤s

@s

f(v⇤s

)[e� µ

´v

⇤s

v

min

F (v)dv

(1� F (v⇤s

))2]� 1

3Denominator is always positive. Numerator is also positive: it graphically represents the area between the cdf and the line

y = 1, which is positive.

15

Here �1 is the marginal cost of sanction: recall that never participants always have to pay it so marginalincrease in sanction is always more likely to hurt them. Here also, agents suffer from reputational loss. Thistime, e has to be big enough to compensate for both reputational losses and marginal cost of sanction.

Depending on the shape of the distribution and relative values of e and µ, we could find examples whereagents are yes-voters for small levels of sanctions and then change their mind while the sanction gets bigger.Typically, think about the case where for a marginal increase in the sanction, even worst agents will decide tocontribute in order to avoid to pay the fine if sanction is implemented. Expected motivation for participantswill decrease a lot so always participants might be tempted to vote against.

Overall, the number of yes voters in function of s is therefore not necessarily monotonous. Knowing thatalways participants and never participants vote in group, it can be discontinuous if those groups happen tochange their voting decisions. To illustrate this prediction, consider the distribution function defined by:

f(x) =

8>>><

>>>:

12 if x 2 [�1.5,�0.5]

12 if x 2 [0.5, 1.5]

0 otherwise

We take the following parameters:

µ = .7 e = 1

c = 2 s 2 [0, 2]

We therefore have a “double-picked” pdf. Agents are split in two groups: one is very motivated to con-tribute and the other very reluctant.

As long as v

⇤s

is above 0.5, marginally convinced agents belong to the motivated group. Reputation de-creases slowly and externality gains dominate. Always participants and some intermediate agents are willingto vote for the sanction. When v

⇤s

is in [�0.5, 0.5], s increases but no agent is marginally convinced. Motiva-tion of the first group does not change: reputation and externality remain constant. When v

⇤s

is smaller than�0.5, marginally convinced agents are in the least motivated group. Because the reputation of those agentsis very low, average motivation of participants will decrease a lot and externality gains will not be sufficientto compensate for this loss. In this case, the number of yes voters will be increasing as long as v

⇤s

is in thefirst group and decreasing when it is in the poorly motivated group.

Last remark, if the derivative of the gains changes sign, it does not necessarily imply that agents immedi-ately switch their voting decision. For example, if sanction is just above the optimal level for a given agent,his gains are likely to be still positive even though they are not at the maximum and he will vote for thesanction. However, if sanction is implemented in two steps, agent can refuse the second increase in s. Socialplanner should take care of this remark as it can reduce his ability to correct the level of sanction if the finehe first proposed happens to be not optimal. It could be an issue in a multi period game where social planneris not perfectly informed of the distribution.

16

3.4 Shifted distribution

We now assume that the distribution has been shifted to the right by a parameter ✓. Other characteris-tics of the distribution remain unchanged. This shift may represent an aggregate shock in the preferences ofcommunity affecting similarly all agents. This can be for example because of a campaign designed to makepeople more sensitive to a specific problem, because of an environmental disaster... The aim of this sectionis to understand how this shift would affect our voting constraints: are more motivated agents necessarilymore likely to vote for the sanction?

First, the shift modifies participation cutoffs. We note v

⇤✓

the cutoff with the shift if the sanction isrejected and v

⇤s,✓

if it is implemented.

v

⇤✓

= c� µ�(v⇤✓

� ✓)

v

⇤s,✓

= c� µ�(v⇤s,✓

� ✓)� s

Under the restrictions that we imposed to have an increasing participation in the level of the sanction,we show in the appendix that participation increases when the distribution is shifted. This seems natural aswe can expect more motivated agents to contribute more.

When the distribution is shifted, we can express the gains from implementation for a given level of sanctionas follows:

v

i

� v

⇤✓

e[F (v⇤✓

� ✓)� F (v⇤s,✓

� ✓)] + µ(E[vi

|vi

> v

⇤s,✓

� ✓]� E[vi

|vi

> v

⇤✓

� ✓])

v

⇤s,✓

v

i

v

⇤✓

e[F (v⇤✓

� ✓)� F (v⇤s,✓

� ✓)] + µ(E[vi

|vi

< v

⇤s,✓

� ✓]� E[vi

|vi

> v

⇤✓

� ✓])� c+ v

i

v

i

v

⇤s,✓

e[F (v⇤✓

� ✓)� F (v⇤s,✓

� ✓)] + µ(E[vi

|vi

< v

⇤s

� ✓]� E[vi

|vi

< v

⇤ � ✓])� s

For a given level of sanction, we want to know if a higher shift implies that agents are more motivated. Wetook the derivative in order to see how it evolved with the shift. Final expressions are given in the appendix.As they look a bit cumbersome, we will focus on the interpretation.

Whatever the shift, voting decision is a tradeoff between externality gains, reputational losses and directparticipation cost. We have to compare the gains for agents if the distribution is shifted and if it is not.This comparison is not clear cut and depends on the distribution and on the parameters. For example, agiven sanction can induce smaller externality gains if the distribution is shifted because most agents alreadycontribute.

Implementation of the sanction therefore does not depend on how motivated agents are but on the utilitygains provided by the sanction in a given context. There is no systematic effect of the shift on the resultof the vote. In some cases, more motivated agents can therefore reject a sanction that would have beenimplemented by the unshifted distribution.

Consider the double-picked density function presented in the previous section with the same parameters.We assume that s = 1 and that the distribution can be shifted by a parameter ✓ = 0.5. Previously, a sanctionof 1 was accepted because it only convinced agents in the motivated group. When the distribution is shifted,meaning that all agents are more motivated, a sanction of the same magnitude would convince agents in thesecond group. In such a case, reputation losses dominate and the sanction is rejected.

17

3.5 Problem of the planner

Now we assume that a benevolent social planner tries to maximize the sum of utilities of all agents. Hechooses the optimal level of sanction in order to maximize:

v

⇤sˆ

�1

��sf(v)dv +

+1ˆv

⇤s

(v + e� c)f(v)dv

Where � is the transaction cost. We assume that the social planner collects the money from the sanctionand can redistribute it. Therefore, it is not a real loss from a global point of view. The transaction costrepresents the cost of implementing and enforcing the sanction.

In our setup, reputation is a zero-sum game. Wherever the cutoff is, the average reputation is the sameand then we can neglect it when we compute the sum of utilities.

If we do not take into account the voting constraints, the optimal level of sanction is given by:

�@v

⇤s

@s

[f(v⇤s

)(e+ �s+ v

⇤s

� c)] = �F (v⇤s

)

The interpretation is very close to Bénabou and Tirole (2011). The global benefit brought by themarginally convinced agents (e+�s+v

⇤s

� c) times the number of marginal agents is equal to the deadweightloss due to an increase in the sanction (right hand side). We can recognize the social multiplier �@v

⇤s

@s

whichhas been interpreted in the appendix.

The interesting case arises when we take the voting constraints into account. Let’s consider that sanctionis implemented if and only if there is at least a fraction q of yes voters. In a simple majority voting process,we would have q = 1

2 . We have two cases.

First recall that v

⇤ does not depend on s. Therefore, it might be the case that agents who alwaysparticipate form a q�majority, and this regardless of the value of s. It is the case when 1 � F (v⇤) � q. Aswe have shown, the constraint for those individuals is easier to satisfy than the constraint for other groups.In this case, it will be sufficient for the planner to ensure that the constraint for high type agents is satisfied.

max

s

´v

⇤s

�1 ��sf(v)dv +´ +1v

⇤s

(v + e� c)f(v)dv

s.t. e[F (v⇤)� F (v⇤s

)] � µ(E[vi

|vi

> v

⇤]� E[vi

|vi

> v

⇤s

])

If 1� F (v⇤) < q, the social planner must also convince agents from other groups. As we have shown, v⇤s

is continuously decreasing in s. It implies that for a given value of s , we have F (v⇤s

) = q. We define thisvalue of s as s

lim

.As long as s < s

lim

, the size of the intermediate group is not large enough so the planner must convinceall agents. For s > s

lim

principal has to convince agents in the intermediate group in order to have aK�majority of agents in favor of the sanction. By proposition 1, principal has to convince agents in theintermediate group up to the agent i such that 1� F (v

i

) = q (as all agents with higher motivations will alsovote for if the median agent votes for). Implicitly, the planner will maximize global utility in each case and

18

implement the sanction that provides the maximum utility:

max

s2[0,slim

]

´v

⇤s

�1 ��sf(v)dv +´ +1v

⇤s

(v + e� c)f(v)dv

s.t. e[F (v⇤)� F (v⇤s

)] � µ(E[vi

|vi

< v

⇤s

]� E[vi

|vi

< v

⇤]) + s

and

max

s2[slim

,+1]

´v

⇤s

�1 ��sf(v)dv +´ +1v

⇤s

(v + e� c)f(v)dv

s.t. e[F (v⇤)� F (v⇤s

)] � µ(E[vi

|vi

< v

⇤]� E[vi

|vi

> v

⇤s

]) + c� v

0

F (v0) = 1� q

merging the two constraints we get:

e[F (v⇤)� F (v⇤s

)] � µ(E[vi

|vi

< v

⇤]� E[vi

|vi

> v

⇤s

]) + c� F

�1(1� q)

The planner solves those two maximizations and implements the sanction that provides the highest globalutility. If the constraint is not binding at the optimum, the planner can implement the first best.

When the constraint is binding, global utility will be strictly lower than if the planner could have im-plemented the sanction without the vote. In the perfect information case, social planner should never bindhis hands proposing agents to vote. When he can impose the sanction, he will not ask agents for their consent.

Social planner is however often bound to let agents vote. For a State, it can be because of internationalagreements that forbid the implementation of a given law without the consent of the population. For a firm,the rules of procedure can require the approval of workers or shareholders for some decisions. In such cases,voting constraints will have to be satisfied.

In perfect information and anonymous voting setup, vote seems to be counterproductive: a benevolentdictatorial social planner would do at least as well as a democratic social planner!

4 Anonymous voting with uncertainty

There are now N agents, N being an odd number bigger than 3. The pdf f(.) and the number of agentsare common knowledge. Main difference with previous setup is that now actual distribution of types is notknown. More specifically, a given agent observes his own type but not the N � 1 other draws.

Such a game can for example represent municipal voting in small towns. People might have access to thetheoretical distribution of types through national surveys but they cannot infer from it the local result. Wewill see how players adapt their strategy in this context.

4.1 Participation decision

Because of anonymous voting, participation game is the same as before. No information is revealed onindividuals type. The cutoffs v

⇤ and v

⇤s

are the same as before.

19

We still consider an equilibrium with a voting cutoff V

⇤. When agents play accordingly to this votingcutoff, 1 � F (V ⇤) is the probability that an agent with a randomly drawn v

i

votes yes. Therefore, when0 < F (V ⇤) < 1 there is always a positive probability for an agent to decide the outcome of the vote.

We want to know the probability that a given voter is decisive in the election. We note K the minimumnumber of yes voters needed to implement the sanction. Under majority rule, we would simply replace K

by N+12 . We express the probability that each agent is pivotal. It arises when there are exactly K � 1 yes

voters and N �K no voters excepting agent i. In such a case, the vote of agent i determines the outcome ofthe vote. We note this probability ⇡:

⇡ = (1� F (V ⇤))K�1 ⇥ (F (V ⇤))N�K ⇥

K � 1

N � 1

!

We can also compute the probabilities that the sanction is accepted whatever the vote of agent i. It isthe case when they are at least K yes voters excluding i:

P (A) =N�K�1X

i=0

(1� F (V ⇤))K+i ⇥ (F (V ⇤))N�K�1�i ⇥

K + i

N � 1

!

Similarly, we express the probability that the sanction is rejected regardless of agent i’s decision:

P (R) =K�2X

i=0

(1� F (V ⇤))K�2�i ⇥ (F (V ⇤))N�K+1+i

K � 2� i

N � 1

!

Knowing those probabilities, we can express the voting problem of agents. As before, we split agents in 3groups.

In this setup, agents can infer additional information on the actual distribution if they happen to bepivotal. Provided that other agents vote accordingly to the voting cutoff V

⇤, an agent knows that if hehappens to be pivotal there are exactly K � 1 yes voters and N �K no voters except from him.

We assume for the time being that the voting cutoff is in the intermediate group. We will check later onif this guess is consistent with the equilibrium.

Under this assumption, yes voters must be always participants or intermediate agents. It follows that ifagent i votes for the sanction and is pivotal, all K � 1 yes-voters will contribute. Moreover, no-voters areintermediate agents with probability F (V ⇤)�F (v⇤

s

)F (V ⇤) and in this case they contribute if sanction is implemented.

Expected participation of others if an agent is pivotal and votes for is:

(K � 1) + (N �K)⇥ F (V ⇤)� F (v⇤s

)

F (V ⇤)

When he votes no, sanction is rejected. All no-voters being never participants or intermediate agents willfreeride. Among yes-voters, there is a fraction 1�F (v⇤)

1�F (V ⇤) of always participants who contribute even if the

20

pivot votes against. Assuming that agent i is pivot and votes against the sanction, expected participation is:

(K � 1)⇥ 1� F (v⇤)

1� F (V ⇤)

Providing that an agent is pivot, the expected participation rate gain from his vote is therefore:

4(P |Pivot) = 1N

{(K � 1)⇥ (1� 1�F (v⇤)1�F (V ⇤) ) + (N �K)⇥ F (V ⇤)�F (v⇤

s

)F (V ⇤) }

= 1N

{(K � 1)⇥ F (v⇤)�F (V ⇤)1�F (V ⇤) + (N �K)⇥ F (V ⇤)�F (v⇤

s

)F (V ⇤) }

4.2 Voting decision

Now we can express the voting problem for each group defined as in previous section by the participationcutoffs. Always participants face the following voting problem:

Y es ! P (A)⇥ (vi

� c

0 + eE[part|accepted, notpivot] + µE[vi

|vi

> v

⇤s

])

+⇡ ⇥ {vi

� c

0 + e[(K � 1) + (N �K)⇥ F (V ⇤)�F (v⇤s

)F (V ⇤) ] + µE[v

i

|vi

> v

⇤s

]}+P (R)⇥ (v

i

� c

0 + eE[part|rejected, notpivot] + µE[vi

|vi

> v

⇤])

No ! P (A)⇥ (vi

� c

0 + eE[part|accepted, notpivot] + µE[vi

|vi

> v

⇤s

])

+⇡ ⇥ {vi

� c

0 + e[(K � 1)⇥ 1�F (v⇤)1�F (V ⇤) ] + µE[v

i

|vi

> v

⇤]}+P (R)⇥ (v

i

� c

0 + eE[part|rejected, notpivot] + µE[vi

|vi

> v

⇤])

Because we have relaxed the assumption of atomicity of agents, aggregate participation is now endoge-nous. We take this into account by defining c

0 ⌘ c� e

N

. This notation implies that E(Participation) is theaggregate participation excluding agent i; it is independent from his own participation decision.

This way of writing the problem takes into account the additional information brought by the hypotheticalpivotal position. When agent is pivotal, his decision impacts both reputational payoff and externality gainsas it determines the rules of the participation game. Because of anonymous voting, agent will be strictlyindifferent between the two options when he is not pivotal. In this case his vote does not change neither thesecond period participation nor the reputation which is only determined by the decision of the second period.

It implies that always participants will vote for the sanction if and only if:

µ(E[vi

|vi

> v

⇤s

]� E[vi

|vi

> v

⇤]) + e4(P |Pivot) � 0

Proceding similarly for intermediate agents, we can show that they will vote for the sanction when:

v

i

� c

0 + µ(E[vi

|vi

> v

⇤s

]� E[vi

|vi

< v

⇤]) + e4(P |Pivot) � 0

And for the never participants:

µ(E[vi

|vi

< v

⇤s

]� E[vi

|vi

< v

⇤]) + e4(P |Pivot)� s � 0

When we take the difference between the utilities of the two outcomes, only the case when agent is pivotremains. Voting decision therefore does not depend on the probability of being pivotal but on the information

21

revealed when it is the case. In previous section, there was no chance for an agent to be pivotal because ofthe continuum. He could not infer information from his decisive position in the vote.

All our propositions of the first part hold in this setup. The expression of the voting cutoff V

⇤ now writes:

V

⇤ = c

0 + µ(E[vi

|vi

< v

⇤]� E[vi

|vi

> v

⇤s

])� e4(P |Pivot)

Equilibrium with voting cutoff will only hold if the cutoff is in the intermediate group. If V ⇤< v

⇤s

, allagents will contribute while if V ⇤

> v

⇤ no agent contributes.4

Keeping the share of yes voters needed for implementation constant, how does this cutoff evolve with thenumber of players? We develop the expressions of c0 and 4(P |Pivot) and we substitute the number of yesvoters needed for implementation K by its expression in N (K = qN):

V

⇤ = c+ µ(E[vi

|vi

< v

⇤]� E[vi

|vi

> v

⇤s

])

� e

N

� e

N

{(qN � 1)⇥ F (v⇤)�F (V ⇤)1�F (V ⇤) +N(1� q)⇥ F (V ⇤)�F (v⇤

s

)F (V ⇤) }

Notice that asymptotically (when N goes to infinity), V ⇤ does not converge to the voting cutoff of the firstpart. Even though the probability of being pivotal goes to zero, agents will still consider the infinitely smallprobability of such situation in order to make their voting choice. Depending on the shape of the theoreticaldistribution, on the number of agents and on the number of yes-voters needed for implementation, we couldcompare the expected participation gains if agent is pivotal to the absolute participation gains we consideredin the previous part.

The effect of N and q on this fixed point equation is not clear cut. To provide a deeper analysis, we haveto assume the shape of the distribution.

4.3 Selection effect

In this setup, suppose that we propose a given sanction to several groups of players independently drawnfrom the same distribution. We assume that the voting cutoff is in the intermediate group so there are yesvoters and no voters. In groups where the sanction is adopted, there are at least K agents with highermotivation than V

⇤. When it is rejected, there are at most K�1 agents more motivated than the voting cut-off. In expectation, the actual distribution when the sanction is implemented is higher than when it is rejected.

Through this channel, there will be a selection effect: only the most motivated groups will implementthe sanction. We will therefore observe a high participation rate when the sanction is implemented partlybecause the groups where the sanction is implemented are the most motivated. Conversely, in groups wheresanction is rejected, we should observe a low participation also because of reverse selection.

Imagine now that the sanction is imposed exogenously to all groups. Average participation with sanctionwill be lower than previously because this process does not allow for the selection effect. Level of participationwill decrease on average because we will also observe groups with low motivation. This selection effect canbe an explanation to the results of Tyran and Feld (2006) presented in the literature review. They found

4Demonstration is similar to Proposition 2

22

that the sanction increased participation significantly more when it was accepted through a vote than whenit was exogenously imposed.

In their article, Tyran and Feld mention a possible selection effect and pretend to rule it out with thefollowing reasoning: if always participants happen to vote for unconditionally on the result, we should observethat yes voters contribute more whatever the result of the vote. They provide evidence showing that it is notthe case. However, this explanation is not fully satisfying. Suppose that the proportion of always participantsis very low and that most agents are in the intermediate group. In such a case, we have a selection effectthrough the number of agents above the voting cutoff but we will observe that yes voters do not contributewhen sanction is rejected simply because they condition their contribution to the result of the vote. Thisselection effect would not be detected by the test proposed by Tyran and Feld.

5 Anonymous contribution

We now turn to cases where the vote is public. In this section, we will assume that the contributiondecision is private. Reputation therefore only depends on the vote a1. We still consider that there are N

players as in the previous section. Actual distribution is also unknown. This setup can represent manysituations, for example votes by show of hands in the general meeting of a firm. In this part, we assume thatcontribution is anonymous. Consider the vote of MPs in a parliament on a law about domestic life, say binrecycling. Their vote will be published in the official journal but it will be very hard to check if they indeedrecycle their wastes. Even if dustmen report a bad behavior, freeriding MPs can pay the sanction secretlyand hope for remaining anonymous.

It is very likely that in such a case voting decision is interpreted as a hint of someone’s motivation, but italso implies that agents will try to manipulate others’ beliefs. When the vote is public, reputation incentivescan lead agents to vote for a decision that hurts them in order to boost their image. For example, if he thinksthat his vote will not change the result, even a MP very reluctant to recycling can vote for and pretend hehas an eco-friendly behavior. We will study possible equilibria in this setup. We will analyze for which valuesof the sanction they can be sustained.

We also solve the model backward. As reputation is not taken into account in the vote, participationcutoff are simpler to express.

If sanction is implemented, agents face the following problem:

a2 = 1 ! v

i

� c

0 + E(A)

a2 = 0 ! �s+ E(A)

The cutoff when sanction is implemented is therefore:

v

⇤s

= c

0 � s

23

If sanction is rejected, the participation cutoff is:

v

⇤ = c

0

Second period strategy is then defined and known by all agents. Note that for given parameters, thosecutoffs are bigger than before because there is no social pressure to contribute. We can now solve the votingphase. We will only consider symmetric equilibria in pure strategy.

5.1 Equilibria with voting cutoff

We still look for an equilibrium where agents vote accordingly to a voting cutoff. In such a case, an agentwith motivation v

i

will vote for if and only if vi

� V

⇤. We are going to characterize this equilibrium and seewhen it exists.

Assuming that all agents vote accordingly to the voting cutoff, we can compute the probability that anagent is pivotal as in the previous section. Probabilities that sanction is accepted or rejected independentlyof player i’s decision are also unchanged given the cutoff. Expected participation gain conditional on thepivotal position will have the same expression in V

⇤.

For always participants, the problem can be written:

Y es ! P (A)⇥ (vi

� c

0 + eE[part|accepted, notpivot])+⇡ ⇥ {v

i

� c

0 + e[(K � 1) + (N �K)⇥ F (V ⇤)�F (v⇤s

)F (V ⇤) ]}

+P (R)⇥ (vi

� c

0 + eE[part|rejected, notpivot])+µE[v

i

|vi

> V

⇤]

No ! P (A)⇥ (vi

� c

0 + eE[part|accepted, notpivot])+⇡ ⇥ {v

i

� c

0 + e[(K � 1)⇥ 1�F (v⇤)1�F (V ⇤) ]}

+P (R)⇥ (vi

� c

0 + eE[part|rejected, notpivot])+µE[v

i

|vi

< V

⇤]

They will vote yes if and only if:

⇡e4(P |Pivot) + µ4(V ⇤) � 0

When only the vote is observed, agents can signal themselves with respect to the voting cutoff V

⇤. Pro-vided that such an equilibrium is reached, agents who vote for the sanction will be identified as being moremotivated than the cutoff while those who oppose have a smaller motivation than the cutoff.

Like in the previous section, knowing that an agent is pivotal will give him additional information aboutthe actual distribution of types.

In this setup, reputational payoff is certain while material considerations are conditioned on the pivotalposition of the agent. Therefore, the lower the probability of being pivotal and the less important materialconcerns will be.

24

Looking at the voting constraint of always participants, we note that all terms in the left hand side arepositive. They will always vote for the sanction in an equilibrium with a voting cutoff. This is the case becausenow, reputation does not depend on the contribution phase. Recall that in the first section, always partic-ipants could vote against the sanction in order to keep their reputational benefits, which is no longer the case.

Condition for intermediate agents to vote for is:

⇡(vi

� c

0 + e4(P |Pivot)) + µ4(V ⇤) � 0

From this expression, we can derive a candidate for the voting cutoff. Providing that agent i is interme-diate, he will be indifferent between voting for and voting against if:

v

i

= c

0 � e4(P |Pivot)� µ

⇡

4(V ⇤) = V

⇤

Which is simply the condition for indifference of agent i between the two voting options.

Finally, we solve the voting problem for never participants and we find that they vote for the sanctionwhen:

e4(P |Pivot) +µ

⇡

4(V ⇤) � s

It implies that we only have an equilibrium with a voting cutoff under some conditions, what we formulatein Proposition 5:

Proposition 5:

When only the vote is observed, there is an equilibrium with a voting cutoff V

⇤:

V

⇤ = c

0 � e4(P |Pivot)� µ

⇡

4(V ⇤)

if and only if it satisfies

e4(P |Pivot) +µ

⇡

4(V ⇤) s

This proposition states that under the condition that never participants oppose to the sanction, we havean equilibrium where agents vote for if and only if v

i

� V

⇤ and contribute according to the voting cutoffs.

When the cost to contribute is high, voting cutoff is also high implying that there are more no voters.On the other hand, gains from externality decrease the voting cutoff: people are more willing to implementthe sanction as it would imply a better environment. When reputational concerns matter (µ4(V ⇤)is high),agents feel more pressure to vote for and the voting cutoff decreases.

Interestingly, as the chance to be pivotal tends to zero when the number of players become large, agentswill consider only their reputation when the number of players becomes high enough. It implies the followingproposition.

25

Proposition 6:

When only the vote is observed and provided that the conditions for the existence of the voting cutoff are

satisfied, agents will tend to vote more often for the the virtuous alternative when the number of voters is large.

The proposition states that provided the equilibrium with voting cutoffs exists, we can decrease the votingcutoff for a sufficiently big number of agents.

When only the vote is observed and when agents vote accordingly to a voting cutoff, a large group willbe more likely adopt the sanction. Knowing that implementation of the sanction raises the participation inthe second stage, we can conclude that a large group will behave more generously than a small one.

Condorcet Jury Theorem states that a large group has more chances to take the right decision. However,this result shows that when reputation is taken into account, a large group has more chances to take the“good” decision from a moral point of view. Sanction can nevertheless be not optimal for social welfare.When for example a too large sanction is adopted, it can push very reluctant agents to contribute. Whenthe net cost of the contribution of these agents (c � v

i

) is larger than the externality of their contributionand when the social planner weights equally the utilities of all agents, it is socially optimal to let them freeride (and possibly make them pay a fine).

Very recent illustration of this example is the result of the 2014 Eurovision song contest. Austrian con-testant was presented as the choice of tolerance and open-mindedness so voting for him could be interpretedas a signal of a high v

i

. In this contest, there are more than a hundred voters, mostly celebrities, and votesare public. As the probability of being pivotal was close to zero, Austrian candidate won easily the contest.

This could have important consequences for policy makers. When voting phase is public and the numberof voters is large and provided that we are in an equilibrium with a voting cutoff, it will be sufficient to presentthe sanction as the choice of fairness in order to make it implemented. However, this type of equilibrium isnot always achievable.

5.2 Equilibria with no pivotal agents

5.2.1 All agents vote for the sanction

If the strategy of all agents is to vote for whatever their type and to contribute according to the cutoffs,sanction is implemented independently of the vote of agent i. In such a case the difference of utility betweenyes and no is the same for all agents and equal to:

E[U |a1 = 1]� E[U |a1 = 0] = E[vi

|a1 = 1]� E[vi

|a1 = 0]

This is the case because the vote of each agent has no chance to change the outcome of the second phase.The only difference is the reputation conditional on the signal sent through the vote. If a1 = 0 is never playedin equilibrium, expectation of type conditional on voting against the sanction relies on out of equilibriumbeliefs. All beliefs for E[v

i

|a1 = 0] are therefore consistent. Knowing that all agents vote for, all agents willhave the same reputation. We must have E[v

i

|a1 = 1] = E[vi

] and the reputation of all agents is simply the

26

mean of the distribution. If we impose:E[v

i

|a1 = 0] E[vi

]

Then no agent will have interest to deviate and sanction will be unanimously adopted. We could showthat it is a sequential equilibrium.

Proposition 7:

When only the vote is observed, there always exists an equilibrium where all agents vote for the sanction.

Proposition holds for all possible parameters of the model. This equilibrium is very intuitive: if othersseem enthusiastic about the law, no agent will dare to show that he is opposed to it, even if he has a very lowmotivation. Consider for example a parliament who votes by show of hands. Even if a large number of MPsare corrupted, they might vote unanimously an anti-corruption law if a deviation is perceived as a proof ofcorruption. When back in their circonscriptions, corrupted MPs can continue their illicit activities and takethe risk of being caught even though they have adopted the law themselves.

5.2.2 All agents vote against the law

Conversely, there is always an equilibrium where all agents vote against the law. In this case also, thereis no chance that an agent changes the outcome of the vote thanks to his decision. As previously, onlyreputational concerns will be taken into account. The condition for it to be an equilibrium will be:

E[vi

|a1 = 1] E[vi

]

Implying that no voters are better perceived than yes voters. As before, this condition depends on outof equilibrium beliefs: any value of E[v

i

|a1 = 0] smaller than the mean of vi

would fit and would be consistent.

Proposition 8:

When only the vote is observed, there always exists an equilibrium where all agents vote against the sanc-

tion.

All no-voter equilibria might represent the fact that people who try to distinguish themselves from themajority are always dodgy. In a general meeting, it might be the reason why it is often difficult to be the firstone to raise the hand. If this equilibrium is reached, note that there will be no hope to implement the sanc-tion through a vote (unless the voting process requires the unanimity against the sanction in order to reject it).

27

Conclusion

This study is a first step in the rational analysis of voting when voters have reputational concerns. Westudied how agents trade off between externality gains, reputational concerns and direct cost of the sanction.

When the vote is anonymous, we found that even very motivated agents could oppose to the sanction inorder to keep a high reputation. With a finite number of agents, we have shown that there is a selectioneffect: only the most motivated groups will implement the sanction. This can partly explain why the law ismore respected when people consent to it. When the vote is public, our main finding is that agents will betempted to vote for the moral alternative regardless of their preferences. This is even more the case whenthe number of voters becomes large.

Our results suggest that if a social planner wants the law to be implemented, he should present it as thevirtuous alternative and make agents vote publicly for it. Our model therefore does not tell why so manyelections are private. Possible answer is to assume that the social planner is not perfectly informed of thedistribution of types. Assume that he proposes a sanction that is optimal only if agents happen to be highlymotivated. In this setup, principal can implement anonymous voting in order to let agents reject the sanctionif they are poorly motivated.

In further research, we could analyze deeper this type of setup where the principal is not perfectly in-formed about the distribution of types. Vote could then be a way to limit the risk of choosing a not optimallevel of sanction. I think that this paper can be extended to provide a benchmark of voting design for policymakers. For this purpose, it would be attractive to generalize the planner’s problem. More specifically, wecould build a canonical game where the planner chooses the voting process: should the vote be public orprivate? What is the optimal sanction he should submit to the vote? What is the number of yes votersneeded for implementation?

Answering these questions requires to compare the voting cutoffs we found and possibly voting cutoffsresulting from other setups. For example, I would like to allow reputation to be based on the two decisions(voting and contribution). This should imply interesting interactions between the two periods. We shouldthen compare the evolution of the voting cutoffs with different distributions and see how they evolve withthe number of agents and the share of yes voters needed for implementation. I regret I did not push thecomparison between the voting cutoffs in the different setups far enough in this paper.

In this study, we have only considered global norms: reputation was built using the theoretical distri-bution of types. I believe that additional effects can arise if we take a local approach of the norm sinceplayers can be willing to be perceived as generous compared to other players in their group. In this case,reputation would be based on the actual distribution of types. Under this assumption, we can expect theresult of the vote to be norm revealing in the sense that players will update their beliefs on the reference scaletheir reputation is built on. Assuming that the social pressure to contribute is higher when others are moti-vated, this effect can be a second explanation to the higher compliance to the law when it is enacted by a vote.

Finally, our theory needs to be empirically confirmed. To my knowledge, only the experiment of Tyranand Feld (2006) presents related results, even though it was not the point of the experiment. However, many

28

natural experiments can already be used to test our predictions. An interesting example is the link betweenthe image of a political party and the bias in the forecast of the results before the election. As surveysinvolve less anonymity than the real vote, we can expect people to pretend to be more moderated when theyanswer to a sounder. The fact that the score of the Front National is often underestimated can be seen asan evidence of this claim.

Our model suggests that people agree on what virtue is. This is likely to be not the case as individualsoften differ on their sense of moral, which can be a serious issue in many possible applications of our model.We can nevertheless easily test some of our predictions in laboratory. We can expect people to agree on thefact that giving in a public good game is a proof a generosity. We could let agents play the two stages gameof the model under different voting rules and see how voting and participation evolve in the different setups.

29

Appendix

3-1 Participation Decision

Participation cutoffs

When sanction is not implemented, agent get:

v

i

� c+ eA+ µE[vi

|a = 1]

if they participate and:

eA+ µE[vi

|a = 0]

if they do not.Where A is the aggregate participation. Atomicity of agents implies that it can be considered as exogenous

for agent i. Agents contribute if and only if vi

> v

⇤ with:

v

⇤ = µ(E[vi

|a = 0]� E[vi

|a = 1]) + c

Similarly, we can derive the equilibrium cutoff sanction is implemented. Agents get vi

�c+eA+µE[vi

|a =

1] when they contribute and eA+ µE[vi

|a = 0]� s when they do not, equilibrium cutoff is:

v

⇤s

= µ(E[vi

|vi

< v

⇤s

]� E[vi

|vi

> v

⇤s

]) + c� s

We get two fixed point equations that we can solve if we specify the distribution of types.

Social multiplier

Demonstration is similar to Bénabou and Tirole. First note that global supply (i.e. participation) is givenby:

P (s) = 1� F (v⇤s

)

Implying that:@P (s)

@s

= �@v

⇤s

@s

f(v⇤s

)

Where �@v

⇤s

@s

is the social multiplier. It can be interpreted as social pressure to contribute.Like in Bénabou and Tirole (2011), we express the social multiplier as:

�@v

⇤s

@s

= � @

@s

[�µ4(v⇤s

) + c� s]

= 1 + µ

@v

⇤s

@s

[40(v⇤s

)]

, �@v

⇤s

@s

= 11+µ40(v⇤

s

)

It gives the increase in participation for a marginal increase of the sanction (given that sanction is imple-mented). As long as this multiplier is positive, participation increases when the sanction increases. This is

30

the case when µ40(v⇤s

) > �1 which always holds for small values of µ. We assume that it is always true.

3-2 Voting Decision

Proof of proposition 2

There exists a voting cutoff V

⇤such that agents vote for the sanction if and only if their reputation is

higher than V

⇤

If this cutoff is in the intermediate group, it can be expressed:

V

⇤ = c+ µ(E[vi

|v < v

⇤]� E[vi

|v > v

⇤s

])� e[F (v⇤)� F (v⇤s

)]

When V

⇤> v

⇤, V

⇤ = +1 and all agents oppose to the sanction.

When V

⇤< v

⇤s

, V

⇤ = �1 and all agents vote for the sanction.

Suppose that an intermediate agent with motivation v

i

is indifferent between voting for and voting against.We have:

v

i

= c+ µ(E[vi

|vi

< v

⇤]� E[vi

|vi

> v

⇤s

])� e[F (v⇤)� F (v⇤s

)]

Note that all intermediate agents with motivation v

j

< v

i

will vote against the sanction while intermediateagents with motivation v

j

> v

i

will vote for it.It is also the case for the limit agent with motivation v

⇤ as v

⇤ � v

i

. Knowing that this agent votes for,we get:

v

⇤ � c+ µ(E[vi

|vi

< v

⇤]� E[vi

|vi

> v

⇤s

])� e[F (v⇤)� F (v⇤s

)]

, e[F (v⇤)� F (v⇤s

)] � c+ µ(E[vi

|vi

< v

⇤]� E[vi

|vi

> v

⇤s

])� v

⇤

, e[F (v⇤)� F (v⇤s

)] � c+ µ(E[vi

|vi

< v

⇤]� E[vi

|vi

> v

⇤])� v

⇤ + µ(E[vi

|vi

> v

⇤]� E[vi

|vi

> v

⇤s

])

, e[F (v⇤)� F (v⇤s

)] � v

⇤ � v

⇤ + µ(E[vi

|vi

> v

⇤]� E[vi

|vi

> v

⇤s

])

, e[F (v⇤)� F (v⇤s

)] � µ(E[vi

|vi

> v

⇤]� E[vi

|vi

> v

⇤s

])

which implies that always participants vote for the sanction.

On the other hand, v⇤s

v

i

so limit agent with motivation v

⇤s

will oppose to the sanction:

v

⇤s

c+ µ(E[vi

|vi

< v

⇤]� E[vi

|vi

> v

⇤s

])� e[F (v⇤)� F (v⇤s

)]

, e[F (v⇤)� F (v⇤s

)] c+ µ(E[vi

|vi

< v

⇤]� E[vi

|vi

> v

⇤s

])� v

⇤s

, e[F (v⇤)� F (v⇤s

)] c+ µ(E[vi

|vi

< v

⇤s

]� E[vi

|vi

> v

⇤s

])� s� v

⇤s

+ µ(E[vi

|vi

< v

⇤]� E[vi

|vi

< v

⇤s

]) + s

, e[F (v⇤)� F (v⇤s

)] v

⇤s

� v

⇤s

+ µ(E[vi

|vi

< v

⇤]� E[vi

|vi

< v

⇤s

])

, e[F (v⇤)� F (v⇤s

)] µ(E[vi

|vi

< v

⇤]� E[vi

|vi

< v

⇤s

])

which implies that never participants vote against the sanction.

It shows that if an intermediate agent v

i

is indifferent between the two voting options, then all agentswith higher motivations will vote for the sanction and all agents with lower motivations will oppose. We then

31

have a voting cutoff V

⇤ = v

i

. We have completed the first part of the proof.

Assume now that V

⇤> v

⇤. All intermediate agents will oppose to the sanction. Substituting V

⇤ and v

⇤

by their expression, we get:

c+ µ(E[vi

|v < v

⇤]� E[vi

|v > v

⇤s

])� e[F (v⇤)� F (v⇤s

)] > c+ µ(E[vi

|v < v

⇤]� E[vi

|v > v

⇤])

, �e[F (v⇤)� F (v⇤s

)] > µ(E[vi

|v > v

⇤s

]� E[vi

|v > v

⇤])

, e[F (v⇤)� F (v⇤s

)] < µ(E[vi

|v > v

⇤]� E[vi

|v > v

⇤s

])

It implies that always participants vote against the sanction. As V

⇤> v

⇤ implies V

⇤> v

⇤s

, we can showsimilarly that never participants vote against the sanction. It completes the proof of the second claim of theproposition.

Finally, we consider the case V

⇤< v

⇤s

. All intermediate agents will vote for the sanction. We rewrite V

⇤

and v

⇤ to get:

c+ µ(E[vi

|v < v

⇤]� E[vi

|v > v

⇤s

])� e[F (v⇤)� F (v⇤s

)] < c+ µ(E[vi

|v < v

⇤s

]� E[vi

|v > v

⇤s

])� s

, �e[F (v⇤)� F (v⇤s

)] < µ(E[vi

|v < v

⇤s

]� E[vi

|v < v

⇤])� s

, e[F (v⇤)� F (v⇤s

)] > µ(E[vi

|v < v

⇤]� E[vi

|v < v

⇤s

]) + s

Which shows that never participants vote for the sanction. As V

⇤< v

⇤s

implies V

⇤< v

⇤ a similar proofshows that always participants also vote for the sanction. It completes the proof of the last claim of theproposition.

Proof of proposition 4

For each agent and every possible value of the sanction, there exists a value µ such that agent votes against

implementation if µ � µ.

From Proposition 2, we know that if always participants oppose to the sanction then all agents voteagainst it.

We also know that always participants endure a reputational loss from implementation.It follows that if µ is high enough, always participants will vote against the sanction in order to save their

reputation. In such a case, all agents vote against the sanction, which implies Proposition 4.

3-4 Shifted distribution

Participation Cutoffs

When the distribution is shifted, the participation cutoff without sanction becomes:

v

⇤✓

= c+ µ(E✓

[vi

|vi

< v

⇤✓

]� E

✓

[vi

|vi

> v

⇤✓

])

Where E

✓

[vi

|vi

< v

⇤✓

] can be seen as E[vi

|vi

< v

⇤✓

� ✓] because an individual with intrinsic motivation v

✓i

in the shifted distribution is recognized as an agent who had motivation v

i

= v

✓i

� ✓ before the shift. We

32

rescale reputation as we consider that it should always sum to the same amount (the idea is that averageagent should not be perceived as better in a shifted distribution).

v

⇤✓

= c+ µ(E[vi

|vi

< v

⇤✓

� ✓]� E

✓

[vi

|vi

> v

⇤✓

� ✓])

= v

⇤✓

= c� µ�(v⇤✓

� ✓)

Impact of the shift on participation

If sanction is rejected, participation is 1�F (v⇤✓

� ✓). We take the derivative with respect to ✓ to see howparticipation is affected by the shift:

@(1�F (v⇤✓

�✓))@✓

= (1� @v

⇤✓

@✓

)f(v⇤✓

� ✓)

With

@v

⇤✓

@✓

= @(c�µ�(v⇤✓

�✓))@✓

= (1� @v

⇤✓

@✓

)µ�0(v⇤✓

� ✓)

, @v

⇤✓

@✓

= µ�0(v⇤✓

�✓)1+µ�0(v⇤

✓

�✓)

Plugging this expression:

@(1� F (v⇤✓

� ✓)) =1

1 + µ�0(v⇤✓

� ✓)f(v⇤

✓

� ✓)

We recognize an expression similar to the social multiplier of part 1. Restrictions on the distributions weimposed (µ40(v) > �1 ) then yields that participation is increasing with the shift.

Participation with a given level of sanction is also bigger when the distribution is shifted.

Similarly, participation with sanction is 1� F (v⇤s,✓

� ✓). Taking the derivative, we get:

@(1� F (v⇤s,✓

� ✓))

@✓

=1

1 + µ�0(v⇤s,✓

� ✓)f(v⇤

s,✓

� ✓)

And participation is also bigger when the distribution is shifted if the sanction is implemented.

Derivative of the constraints with respect to the shift

For always participants, we rewrite their voting constraint as:

e[F (v⇤✓

� ✓)� F (v⇤s,✓

� ✓)] + µ(E[vi

|vi

> v

⇤s,✓

� ✓]� E[vi

|vi

> v

⇤✓

� ✓]) � 0

Only the left hand side is affected by ✓. We take the derivative of this term with respect to ✓. Aftercalculation, we find:

33

e[ 11+µ�0(v⇤

s,✓

�✓)f(v⇤s,✓

� ✓)� 11+µ�0(v⇤

✓

�✓)f(v⇤✓

� ✓)]

+µ{� (✓�v

⇤s,✓

)f(v⇤s,✓

�✓)(1�F (v⇤s,✓

�✓))�[vmax

�(v⇤s,✓

�✓)F (v⇤s,✓

�✓)�z(vmax

)+z(v⇤s,✓

�✓)]f(v⇤s,✓

�✓)

(1+µ�0(v⇤s,✓

�✓))(1�F (v⇤s,✓

�✓))2

+ (✓�v

⇤✓

)f(v⇤✓

�✓)(1�F (v⇤✓

�✓))�[vmax

�(v⇤✓

�✓)F (v⇤✓

�✓)�z(vmax

)+z(v⇤✓

�✓)]f(v⇤✓

�✓)(1+µ�0(v⇤

✓

�✓))(1�F (v⇤s,✓

�✓))2 }

Where z(.) is a primitive of F (.).For intermediate agents:

e[F (v⇤✓

� ✓)� F (v⇤s,✓

� ✓)] + µ(E[vi

|vi

< v

⇤s,✓

� ✓]� E[vi

|vi

> v

⇤✓

� ✓]) � c� v

i

Taking derivative of the left hand side:

e[ 11+µ�0(v⇤

s,✓

�✓)f(v⇤s,✓

� ✓)� 11+µ�0(v⇤

✓

�✓)f(v⇤✓

� ✓)]

+µ{� (✓�v

⇤s,✓

)f(v⇤s,✓

�✓)(1�F (v⇤s,✓

�✓))�[vmax

�(v⇤s,✓

�✓)F (v⇤s,✓

�✓)�z(vmax

)+z(v⇤s,✓

�✓)]f(v⇤s,✓

�✓)

(1+µ�0(v⇤s,✓

�✓))(1�F (v⇤s,✓

�✓))2

+ (z(v⇤✓

�✓)�z(vmin

))f(v⇤✓

�✓)(1+µ�0(v⇤

✓

�✓))(F (v⇤✓

�✓))2 }

For never participants:

e[F (v⇤✓

� ✓)� F (v⇤s,✓

� ✓)] + µ(E[vi

|vi

< v

⇤s

� ✓]� E[vi

|vi

< v

⇤ � ✓]) � s

Derivation gives:

e[ 11+µ�0(v⇤

s,✓

�✓)f(v⇤s,✓

� ✓)� 11+µ�0(v⇤

✓

�✓)f(v⇤✓

� ✓)]

+µ[� (z(v⇤s,✓

�✓)�z(vmin

))f(v⇤s,✓

�✓)

(1+µ�0(v⇤s,✓

�✓))(F (v⇤s,✓

�✓))2 + (z(v⇤✓

�✓)�z(vmin

))f(v⇤✓

�✓)(1+µ�0(v⇤

✓

�✓))(F (v⇤✓

�✓))2 ]

5- Anonymous Contribution

Voting decisions for intermediate agents and never participants when only the

vote is observed

Intermediate agents:

Y es ! P (A)⇥ (�s+ eE[part|accepted, notpivot])+⇡ ⇥ {�s+ e[(K � 1) + (N �K)⇥ F (V ⇤)�F (v⇤

s

)F (V ⇤) ]}

+P (R)⇥ eE[part|rejected, notpivot]+µE[v

i

|vi

> V

⇤]

No ! P (A)⇥ (�s+ eE[part|accepted, notpivot])+⇡ ⇥ {e[(K � 1)⇥ 1�F (v⇤)

1�F (V ⇤) ]}+P (R)⇥ eE[part|rejected, notpivot]

+µE[vi

|vi

< V

⇤]

34

Never participants:

Y es ! P (A)⇥ (vi

� c

0 + eE[part|accepted, notpivot])+⇡ ⇥ {v

i

� c

0 + e[(K � 1) + (N �K)⇥ F (V ⇤)�F (v⇤s

)F (V ⇤) ]}

+P (R)⇥ eE[part|rejected, notpivot]+µE[v

i

|vi

> V

⇤]

No ! P (A)⇥ (vi

� c

0 + eE[part|accepted, notpivot])+⇡ ⇥ {e[(K � 1)⇥ 1�F (v⇤)

1�F (V ⇤) ]}+P (R)⇥ eE[part|rejected, notpivot]

+µE[vi

|vi

< V

⇤]

Proof of proposition 5

When only the vote is observed, there is an equilibrium with a voting cutoff V

⇤:

V

⇤ = c

0 � e4(P |Pivot)� µ

⇡

4(V ⇤)

if and only if it satisfies

e4(P |Pivot) +µ

⇡

4(V ⇤) s

Condition for intermediate agent with motivation v

i

to vote for is:

⇡(vi

� c

0 + e4(P |Pivot)) + µ4(V ⇤) � 0

From this expression, we can derive a candidate for the voting cutoff. Providing that agent i is interme-diate, he will be indifferent between voting for and voting against if:

v

i

= c

0 � e4(P |Pivot)� µ

⇡

4(V ⇤) = V

⇤

Moreover, we have to show that this candidate is in the intermediate group. We have to check thatV

⇤ 2 [v⇤s

, v

⇤]:v

⇤s

V

⇤ v

⇤

, c

0 � s c

0 � e4(P |Pivot)� µ

⇡

4(V ⇤) c

0

, �s �e4(P |Pivot)� µ

⇡

4(V ⇤) 0

Notice that the middle term is necessarily negative. It implies that the voting cutoff cannot be biggerthan v

⇤. We only have to check:e4(P |Pivot) +

µ

⇡

4(V ⇤) s

Which is the condition such that never participants oppose to the sanction. When this condition is sat-isfied, the voting cutoff candidate is in the intermediate group and we have an equilibrium with a voting cutoff.

When the condition such that never participants vote for is satisfied, the candidate for the voting cutoffis smaller than v

⇤s

and is not in the intermediate group. In such a case, there is a degenerated equilibriumwith a voting cutoff in which all agents vote for the sanction. In such a case, ⇡ has to be 0 and we get the

35

all yes voters case.

Sketch of the proof of proposition 6

When only the vote is observed and provided that the conditions for the existence of the voting cutoff are

satisfied, agents will tend to vote more often for the the virtuous alternative when the number of voters is large.

We haveV

⇤ = c

0 � e4(P |Pivot)� µ

⇡

4(V ⇤)

Proposition 6 relies on the fact that lim

N!1⇡ = 0. When the number of players is big, there is veryfew chances of being pivotal. It implies that lim

N!1 � µ

⇡

4(V ⇤) = �1 while other terms are finite. For N

sufficiently large, reputational concerns become infinitely bigger than other terms. Increase the number ofplayers therefore decreases the voting cutoff on the long run.

Note that because of the term e4(P |Pivot) can be decreasing in N (or at least is not necessarilymonotonous), we cannot generalize of Proposition 6 and say that V

⇤ is decreasing in N . This result onlyholds asymptotically.

36

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