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HCEO **WORKING** PAPER SERIES

Working Paper

The University of Chicago

1126 E. 59th Street Box 107

Chicago IL 60637

www.hceconomics.org

Guilt in Voting and Public Good Games ∗

Dominik Rothenhäusler, Nikolaus Schweizer and Nora Szech

November 2016

Abstract

This paper analyzes how moral costs affect individual support of

morally difficult group decisions. We study a threshold public good

game with moral costs. Motivated by recent empirical findings, we

assume that these costs are heterogeneous and consist of three parts.

The first one is a standard cost term. The second, shared guilt, decreases

in the number of supporters. The third hinges on the notion of being

pivotal. We analyze equilibrium predictions, isolate the causal effects

of guilt sharing, and compare results to standard utilitarian and nonconsequentialist

approaches. As interventions, we study information

release, feedback, and fostering individual moral standards.

JEL Classification: D02, D03, D23, D63, D82.

Keywords: Moral Decision Making, Committee Decisions, Diffusion

of Responsibility, Shared Guilt, Being Pivotal, Division of Labor, Institutions

and Morals.

∗ This manuscript previously circulated under the title “Institutions, Shared Guilt, and

Moral Transgression.” We would like to thank Peter Dürsch, Armin Falk, Werner Hildenbrand,

Steffen Huck, Philipp Lergetporer, Clemens Puppe, Frank Rosar, Patrick Schmitz,

Avner Shaked, Joel Sobel, Justin Valasek, and Christoph Vanberg. Dominik Rothenhäusler,

Seminar for Statistics, ETH Zürich, Rämistr. 101, CH-8092 Zürich, email: rothenhaeusler@stat.math.ethz.ch,

phone: +41 44 6325319. Nikolaus Schweizer, Department of

Econometrics and OR, Tilburg University, P.O.box 90153, NL-5000 LE Tilburg, email:

n.f.f.schweizer@uvt.nl. Nora Szech, Chair of Political Economy, ECON Institute, Karlsruhe

Institute of Technology, Fritz-Erler-Str. 1-3, D-76128 Karlsruhe, email: nora.szech@kit.edu,

phone: +49 721 60843809.

1 Introduction

Many morally problematic acts and decisions require the support of several

people to become implementable, and often, morally difficult tasks are delegated

to groups instead of one individual alone. Extreme examples are so-called

“execution teams,” juries deciding about death penalties, and group military

activity such as shooting squads.

There may be different reasons for these

arrangements. Certainly, they causally affect moral responsibility in people

involved. An expert in military psychology, Dave Grossman, stresses that individual

barriers towards morally problematic activity often break when people

become part of teams or groups. 1

In a related vein, studies from social psychology

(compare the overviews in Bandura 1999 and 2016) document that

diffusion of responsibility and shared guilt reduce moral conscience in people.

In addition, a diffused notion of being pivotal can decrease moral feelings

(compare Falk and Szech 2014). Various studies from economics and related

fields further document that people are heterogeneous in their moral behavior.

Some personalities adhere to higher moral standards than others, even across

different institutional contexts. 2

Motivated by these findings, we analyze how moral costs affect individual support

and outcomes in morally difficult group decisions. For this purpose, we

study a threshold public good game with moral costs. We assume that agents

are heterogeneous, and that moral costs consist of three terms. The first one

is a standard cost term. The second, shared guilt, decreases in the number

of supporters. The third hinges on the notion of being pivotal. We analyze

equilibrium predictions, isolate the causal effects of guilt sharing, and compare

results to standard utilitarian and non-consequentialist approaches. As

interventions, we study information release, feedback, and fostering individual

moral standards.

We first prove existence and uniqueness of a symmetric Bayesian Nash equilibrium

in which transgression can happen with positive probability. We focus

1 Compare Grossman (1996): “The individual is not a killer, but the group is” (p. 149).

2 See O’Fallon and Butterfield (2005), Loe, Ferrell and Mansfield (2000), Deckers, Falk,

Kosse and Szech (2016) for reviews of questionnaire and hypothetical scenario studies, and

e.g. Albrecht, Krämer and Szech (2016) and Deckers et al. (2016) for evidence from economic

settings. A related economic study focusing on social behavior is e.g. Bruhin, Fehr and

Schunk (2016).

2

on this equilibrium throughout. Its outcome Pareto dominates the outcome of

the trivial equilibrium in which all agents remain passive. We find that this

Bayesian Nash equilibrium takes the form of a threshold equilibrium. Up to a

certain moral type, agents stay out, while agents above the threshold support

moral transgression.

An increase in individual moral standards, e.g. through training moral virtues,

unambiguously prevents moral transgression. Yet releasing additional information

to the agents, such that moral views become more spread out, can have

different effects. If selfish benefits from transgression are high, information

release can fight transgression. The opposite is true if selfish benefits are low.

The reason behind is that information release shifts probability mass both

into the upper and into the lower tail of the distribution of the agents’ moral

types. Morally “good” agents become better, and “bad” agents become worse.

A transgression that is only appealing to the morally worst agents can thus

materialize more likely if information is released, while a transgression that

requires the support of high moral types can become prevented more easily.

Information on the exact number of fellow supporters affects notions of being

pivotal, and thereby feelings of moral responsibility. In order to isolate the

role of feeling pivotal, we also analyze the variation of the model in which such

precise feedback is not provided. Again, there exist non-trivial, symmetric

Bayesian equilibria. These are also in threshold strategies. We find that the

thresholds are smaller than those in the basic model with feedback. Transgression

becomes less likely as agents cannot clear their conscience from learning

that the transgression ‘would have happened anyway.’

We further contrast our findings with the case in which guilt does not diffuse in

the number of agents. The quantitative predictions differ drastically. The impact

of guilt diffusion is roughly comparable to multiplying the selfish benefit

by the number of required supporters: We observe a kind of ‘strategic equivalence’

between the game in which guilt diffuses in the number of supporters,

and the comparison game with a much smaller selfish benefit. Precisely, the

selfish benefit is divided by the number of supporters needed, one agent less is

required, and the total group size is one agent less.

Finally, we compare our predictions to those of the companion model in which

3

agents’ moral reasoning follows a non-consequentialist instead of a consequentialist

approach. What matters for feelings of guilt is now the intention, not

the outcome. If agents act as supporters, they face moral costs – regardless of

whether transgression materializes or not. In this model, symmetric equilibria

in which transgression occurs with positive probability only exist if the selfish

benefit from transgression is sufficiently high. If the benefit is large, there

exist multiple equilibria, all in threshold strategies. The equilibrium with the

highest transgression probability Pareto dominates.

1.1 Related Literature

Many previous applications of discrete public good games to morally relevant

action have studied the behavior of bystanders. Bystanders are people who

observe a crime or accident. Often they do not help the person in distress.

A rich empirical literature has documented that helping becomes less likely

if others could help as well. Helping drastically decreases in the number of

bystanders, see e.g. Darley and Latané (1968), Latané and Nida (1981), and

Fischer et al. (2011) for an overview of the literature on bystading. For a

victim, in order to receive help, it is often much better to have just one or

few bystanders than many. This finding has been coined as the “bystander

effect.” In social psychology and sociology, effects of shared and diffused guilt

and reduced feeling of breaking a norm have been put forward as explanations

for the bystander effect (see also Zimbardo 2007). Diffusion of guilt has also

been found to operate in committee situations (compare Bandura, Underwood

and Fromson 1975, Bandura 1999, 2016). In a related vein, empirical studies

have shown that a reduced notion of being pivotal decreases feelings of moral

responsibility, in individual decision contexts and in committees (Falk and

Szech 2014). 3

In the game-theoretic literature on bystanding, such as Harrington (2001), Osborne

(2004), and Crettez and Deloche (2011), mixed equilibrium effects have

been put forward as alternative explanations. If coordination is difficult, a high

individual likelihood of helping could result in very many bystanders helping.

This would be an unneeded overprovision of costly help. Therefore, in mixed

equilibrium, individual willingness to help is small, and drastically decreases

3 Sobel (2010) analyzes perceptions of being pivotal in market trading.

4

in the number of bystanders. These approaches assume that individual costs

of helping and not helping are independent of the number of other bystanders.

The standard game-theoretic approaches work well in a situation in which bystanders

cannot coordinate. They cannot explain the many empirical findings

as also overviewed in Fischer et al. (2011) in which bystanders see the behavior

of others such that an overprovision of help becomes unlikely. Yet also in these

situations, the bystanding effect emerges.

This motivates us to enrich the standard game-theoretic approach. Coordination

remains an important factor. Agents hope that others provide support

instead of themselves. Yet moreover, costs of behaving in morally problematic

ways can be shared and thereby diffuse, and notions of being pivotal affect feelings

of guilt. Our model builds on the benchmark model due to Palfrey and

Rosenthal (1984). We extend this model by including incomplete information 4

about other agents’ types and a richer model of how both, own behavior and

the behavior of others affect moral costs. This enables us to study the effects

of information provision and feedback. Moreover, we can analyze the impact

and interplay of different types of moral costs. The complete information game

possesses pure equilibria with perfect coordination as well as mixed equilibria.

In the incomplete information case, the pure equilibria break away while

the mixed equilibria survive.

relevant.

This renders the mixed equilibria specifically

We first analyze the consequentialist moral perspective.

Costs for immoral

activity only arise if transgression materializes. In Section 5.2, we contrast this

with a principle-based (Kantian) moral perspective under which moral costs do

not hinge on the actual outcome. This part of the analysis is related to Huck

and Konrad (2005) who consider mixed equilibria of a similar voting game with

complete information. They study a committee situation in which agents vote

about confiscating foreign investment. Confiscation leads to a selfish benefit

to the group yet voting for confiscation comes along with Kantian costs of

guilt. Guilt is thus independent of the voting outcome, and they assume that

it cannot be shared with others. They find that transgression, i.e. confiscation,

becomes less likely if the group size becomes larger.

4 There exists a literature on public good games with incomplete information, e.g. Nitzan

and Romano (1990). Yet we are not aware of any contribution which addresses moral costs

or cost sharing in these settings.

5

Dufwenberg and Patel (2015) also assume a complete information environment

and focus on pure equilibria. They add reciprocity preferences and cost-sharing

(similar to our guilt diffusion) to Palfrey and Rosenthal’s model. With this

combination (and in the absence of private information), they show that there

can be efficient equilibria which do not require coordination. An earlier application

of “diffusive” costs in complete information public good games is found

in Weesie and Franzen (1998) who study the “volunteer’s dilemma,” i.e. the

special case in which only one single agent needs to contribute.

Diffusion of moral costs further plays a crucial role in the analysis of Lindbeck,

Nyberg and Weibull (1999). They consider a continuum situation such that

coordination is always feasible. Agents need to decide if they want to work and

thereby support others, or if they want to free-ride on others’ contributions.

Opting for the latter comes with social stigma and thus with a morally relevant

cost, yet this cost decreases (and thus diffuses) the larger the fraction of freeriders

is. Agents are heterogeneous in their types such that the appeal of

working varies across them. Lindbeck et al. analyze the equilibria of this

model and demonstrate that sorting is feasible such that agents for whom

working is less costly work, while others free-ride. In a sense, these equilibria

have parallels to our threshold equilibrium in which agents with low moral cost

types support transgression, while others stay out.

2 Model and Basic Properties

This section introduces the morally relevant threshold public good game we are

going to study. Motivated by recent empirical findings, we assume that moral

costs are heterogeneous and consist of three parts. The first one is a standard

cost term. The second one is diffusive in the number of people sharing in. The

third one hinges on the notion of being pivotal.

We prove existence and uniqueness of a non-trivial, symmetric equilibrium.

We will rely on this equilibrium for all predictions.

6

2.1 The Model

We consider n ≥ 1 agents who each face the decision to support a morally

difficult action. One may want to think about voting Yes for going to war

instead of No, or for taking some morally shaky business endeavor instead of

playing fair. The action (which we will also call transgression) requires the

support of at least k of these agents in order to materialize. We are thus in a

threshold public good game with moral costs.

We assume that if transgression happens, all agents receive a benefit V > 0

each (e.g. the country is defended, the group becomes richer etc.). Otherwise

each agent receives 0. Transgression is associated with moral costs. These

only accrue in the active supporters, i.e. in those who voted Yes instead of

No. Motivated by recent empirical research from economics, business ethics,

and social and military psychology, such as Bandura (1999, 2016), Deckers et

al. (2016), Falk and Szech (2013, 2014), Grossman (1996), Loe et al. (2000),

and Zimbardo (2007), these moral costs are heterogeneous and consist of three

terms. The first term is a standard cost from the transgression happening. The

second term captures guilt sharing and decreases in the number of supporters.

The third term hinges on the notion of being pivotal.

Each agent has a private moral type x i . Types are drawn independently from

a commonly-known, continuous distribution function F with F (0) = 0 whose

density function f is strictly positive over the support (0, a), a ∈ (0, ∞] of

F . The three components of moral costs are parametrized by the nonnegative

constants α, β and γ. The first one, α · x i , accrues if the agents

successfully supports the transgression. We will sometimes call it the nondiffusive

part in order to differentiate it from the second cost term. The second

moral cost term captures the concept of guilt sharing, β · x i /Y . Y denotes the

total number of agents who support the transgression. Moral costs become

smaller the more supporters guilt can be shared with. Third, the agent faces a

cost of γ·x i which materializes if he is pivotal, i.e., if the transgression would not

have occurred without his support. If the transgression does not materialize,

we assume that agents do not feel guilty. We thus follow a consequentialist

approach. 5 Agents who do not support the transgression do not face any moral

costs either.

5 See Section 5.2 for a discussion of the corresponding non-consequentialist approach.

7

The collective decision process is modeled as a threshold public good game.

Agents simultaneously opt either for Yes or No. If at least k agents choose Yes,

transgression happens. k ∈ {1, . . . , n} is commonly known. This game can be

interpreted as the result of a decision rule that was prescribed or agreed upon

beforehand. Alternatively, it can be thought of as a game of volunteering to

participate in an immoral action, and k as the minimum number of volunteers

needed to carry it out successfully.

To sum up, the realized utility of agent i from supporting immoral activity,

i.e. opting for Yes, is

(

))

β

(V − x i α +

1 {Y−i ≥k−1} − γx i 1 {Y−i =k−1} (1)

1 + Y −i

where Y −i denotes the number of agents other than i who act as supporters.

Realized utility from choosing No is

V 1 {Y−i ≥k}. (2)

We assume throughout that at least one of the guilt parameters α, β and γ is

positive.

2.2 Equilibrium Analysis

The solution concept we employ is Bayesian Nash equilibrium. We focus on

equilibria which are symmetric in the sense that agents with the same type

take the same moral decision.

For k > 1 there exists a pooling equilibrium in which all agents choose No

independent of their type. All agents receive a utility of 0 in this equilibrium.

As is common, e.g., in the voting-games and matching literatures, we ignore

this equilibrium in the following and focus on the Pareto-superior equilibria

in which the morally difficult action is taken with positive probability. 6

As a

first main result, we find that there always exists exactly one equilibrium of

6 As agents can guarantee themselves a non-negative payoff by choosing No, all agents

must earn a non-negative payoff in any equilibrium. Thus any agent strictly preferring Yes

in an equilibrium in which the morally difficult action is taken with positive probability,

must earn a strictly positive expected payoff.

8

this type.

Theorem 1. There exists a unique symmetric Bayesian equilibrium in which

transgression happens with positive probability. In this equilibrium, agent i opts

for Yes if x i ≤ θ k,n , and for No if x i > θ k,n . θ k,n is the unique solution of

V b(n − 1, k − 1, F (θ k,n ))

( ∑n−1

(

=θ k,n α +

j=k−1

β

1 + j

)

)

b(n − 1, j, F (θ k,n )) + γb(n − 1, k − 1, F (θ k,n ))

(3)

with

b(n, j, p) =

( n

j)

p j (1 − p) n−j .

If k < n, θ k,n lies in the interior of the support of F , F (θ k,n ) ∈ (0, 1) such that

the equilibrium is a separating equilibrium. 7

Equation (3) has a straightforward interpretation in terms of the costs and

benefits of a pivotal agent. The left hand side consists of the gains in utility

if the agent is pivotal, i.e. if exactly k − 1 other agents are willing to take

action. On the right hand side we find the expected costs of an agent with

threshold-type θ k,n .

Uniqueness holds up to the decision of agents with type θ k,n . These are indifferent

between Yes and No. We ignore this technicality in the following as it

only becomes relevant with a probability of zero.

Throughout the paper, we focus on three measures of moral transgression.

θ k,n captures individual willingness to support morally difficult activity.

Morally difficult behavior can increase on an individual level if institutional

design lowers moral costs, e.g.

via introducing better possibilities for guilt

sharing. We measure such institutional effects on the individual by the quantity

θ k,n , i.e. by the threshold-type who is indifferent between Yes and No.

P k,n refers to the likelihood of transgression materializing. N k,n is the num-

7 We refer to equilibria in which agents’ actions depend on their types as “separating

equilibria.” Strictly speaking, they are only “semi-separating” as we cannot infer an agent’s

type from his action because there are infinitely many types and only two actions.

9

er of agents supporting transgression. P k,n and N k,n are thus outcome-based

measures, derived from the individual thresholds. The equilibrium probability

P k,n with which transgression occurs is given by

P k,n =

n∑

b(n, j, F (θ k,n )),

j=k

and the according equilibrium expected number of supporters is N k,n = n F (θ k,n ).

3 Comparative Statics and Limit Behavior

In this section, we establish the comparative statics for the game, as well as

the convergence behavior as the number of agents grows large.

We analyze the comparative statics in the benefit V , the moral cost parameters

α, β, and γ, the number of supporters necessary k, and the overall group size

n. To this end, it proves useful to rewrite the equilibrium condition (3) into

the form

with

H(F, α, β, γ, k, n) = γ +

V

θ k,n

= H(F (θ k,n ), α, β, γ, k, n) (4)

∑n−1

j=k−1

(

b(n − 1, j, F )

α +

β )

. (5)

b(n − 1, k − 1, F ) 1 + j

The left hand side of (4) is strictly decreasing from +∞ to 0 in θ while

H(F (θ), α, β, γ, k, n) is increasing in θ with

H(0, α, β, γ, k, n) = α + β k + γ.

The two functions thus intersect once, at the equilibrium threshold θ k,n . Therefore,

if we wish to determine the comparative statics, e.g.

the effect of an

increase in V on equilibrium thresholds, we can argue as follows. Increasing

V shifts the function V/θ upwards. The point of intersection and thus θ k,n

accordingly move to the right. Increasing the benefit V therefore increases the

equilibrium threshold and thus induces more agents to support the transgression.

The next proposition collects further basic properties of the equilibrium

threshold θ k,n and the associated outcome-based measures P k,n and N k,n .

10

Proposition 1.

(i) θ k,n , P k,n and N k,n are increasing in the benefit V and decreasing in the cost

parameters α, β and γ.

(ii) If α + β > 0, θ k,n and N k,n are increasing in k.

The monotonicity behavior of θ k,n and N k,n in k can be interpreted as follows.

If k increases, i.e. if the number of supporters necessary increases, more agents

become willing to support the transgression both individually and overall, in

expectation. There are thus fewer agents who hope to free-ride on others with

lower moral standards who could support the transgression instead.

In the case α + β = 0, in which guilt purely depends on the notion of being

pivotal, it holds that

θ k,n = V γ . (6)

Thus, thresholds are independent of the voting rule and of the size of the group.

The reason behind is that agents compare their selfish benefit from transgression

only to the cost that emerges if they turn out to be pivotal. Therefore,

there are no incentives to free-ride on others in this case. Accordingly, the

number of supporters N k,n = nF (V/γ) is also independent of k, while P k,n ,

the probability of finding at least k supporters decreases with k.

The dependence of the transgression probability P k,n on k is in general more

complex as demonstrated in Figure 1. The Figure shows that in a setting with

purely diffusive guilt (α = γ = 0), the probabilities form a U-shape. Transgression

probability is smallest for an intermediate level of k. The intuition

is as follows. If k is small, it is likely that there exist some agents with very

low moral types. These will implement the transgression. For larger k, higher

moral types would have to give their support. This only becomes an option

if guilt diffuses well, i.e. if very many agents are required. For intermediate

values of k neither of the two mechanisms helps agent much to overcome

their moral concerns and, accordingly, immoral action materializes only with

comparatively small probability. 8

8 See Rothenhäusler, Schweizer and Szech (2015) for further discussion and some analytical

results on this U-shape.

11

1

0.98

0.96

P k,25

0.94

0.92

0.9

5 10 15 20 25

k

Figure 1: P k,25 as function of k for n = 25, V = 1, β = 1, α = γ = 0 and the

exponential distribution F (x) = 1 − exp(−x).

These effects still play a role in the general model in which guilt can only partly

be shared, thereby precluding an easy comparative statics analysis of P k,n . Yet

there is one crucial difference. If guilt has either a non-diffusive component

(α > 0) or there are costs from being pivotal (γ > 0) (and when the support of

F is unbounded), there will be some types who do not support transgression

regardless of k and n. These very moral agents render transgression for k close

to n less likely. The following lemma summarizes the results.

Lemma 1.

(i) If α + γ > 0, individual thresholds are bounded,

θ k,n ≤

V

α + γ .

In particular, if the support of F is unbounded, there exists a positive mass of

agent types with high moral values who never support transgression, independently

of k and n.

(ii) If α = γ = 0, i.e. if guilt is fully diffusive as n grows large, for any type x

in the support of F there exist k and n such that x < θ k,n . Thus any type x

12

can become a supporter of transgression.

Finally, let us consider the limit results if the population size grows large.

Proposition 2. Suppose that α + β > 0. Then the following claims hold.

(i) lim n→∞ θ k,n = 0.

(ii) lim n→∞ N k,n = ∞.

(iii) lim n→∞ P k,n = 1.

While individual thresholds generally converge to 0, the expected number of

supporters N k,n becomes arbitrarily large. We thus find an overshooting effect.

In the limit, instead of the required k, infinitely many agents become

supporters. This holds even if only one supporter is necessary, and also if only

α is strictly positive and β is 0, i.e. if guilt does not diffuse. Instead of one,

infinitely many agents join in. Accordingly, the transgression probability P k,n

converges to 1. The reason behind is that the number of agents with very

low moral standards grows that fast in n that it overrules the counteractive

effect from the threshold which decreases in n. There is uncertainty about

whether there are indeed many enough low types to free-ride on. This makes

many more agents think about joining in than necessary. Coordination is impossible,

and the gains from a transgression that is supported by too many

are higher than the losses from the transgression not happening. In the case

α + β = 0, guilt only depends on notions of being pivotal. As shown in (6),

the thresholds θ k,n are positive and independent of k and n. Also in that case,

(ii) and (iii) are satisfied. 9

4 Interventions

In this section, we study the impact of three different interventions: training

agents’ individual moral virtues, increasing the precision of their information,

and varying whether they receive feedback about the number of supporters (as

in the basic model) or not.

9 Each agent participates with a probability p = F (V/θ) that does not depend on n. This

implies N k,n = np → ∞ and leads to P k,n → 1 far more directly than in the general case.

13

4.1 Trained Virtues

Possibilities for training moral virtues have already been put forward by Aristoteles

(see Barnes 1984). The idea of nudging agents towards more moral behavior

has been discussed extensively in recent public debate, see e.g. Thaler

and Sunstein (2008).

The next proposition demonstrates that an intervention

which induces an upwards shift in the distribution of moral types indeed

reduces all three measures of transgression. To this end, we compare two versions

of our model with respective distributions F and G while we keep all

other parameters constant. We indicate the distribution dependence through

a superscript on the thresholds, θk,n F and θG k,n , and analogously for all other

quantities.

Proposition 3. Suppose that G dominates F in terms of first order stochastic

dominance, G(x) ≤ F (x) for all x ≥ 0, so that G can be interpreted as a

(non-linear) upwards shift of F . Then it holds that θ G k,n ≥ θF k,n , N G k,n ≤ N F k,n ,

and P G k,n ≤ P F k,n .

An upwards shift in the distribution of moral types thus establishes higher

moral standards in equilibrium. According to all three morally relevant measures,

transgression decreases.

4.2 Information Release

We next investigate the effect of information release on moral outcomes. We

now interpret F as the distribution of (conditionally expected) moral costs,

based on coarse information. Through releasing information, a designer can

induce a more spread-out distribution G in which each agent has learned his

moral costs more precisely. 10 To formalize the notion of G being more spread

out than F we rely on the following single-crossing criterion. G is more spread

out in the sense of the single-crossing criterion, G ≽ SC F , if there exists x 0

such that F (x) ≤ G(x) for x ≤ x 0 and G(x) ≤ F (x) for x ≥ x 0 . If F and

G share the same mean – as is the case if agents employ Bayesian updating –

single-crossing is a sufficient condition for G being a mean-preserving spread

of F . 11

10 This is in line with Ganuza and Penalva (2010)’s model of information release.

11 See Diamond and Stiglitz (1974) for further discussion of this relation. We do not need

the assumption of F and G sharing the same mean and thus do not impose it.

14

The next proposition demonstrates that the effect of information release crucially

depends on the potential benefits of the moral transgression, i.e. on the

level of V .

Proposition 4. Fix all parameters except for V and suppose that G is more

spread out than F in single-crossing precision, G ≽ SCF . Then there exists

v 0 ≥ 0 with the following properties.

For V ≥ v 0 , we have θk,n G ≥ θF k,n , N k,n G ≤ N k,n F , and P k,n G ≤ P k,n F .

For V ≤ v 0 , we have θk,n G ≤ θF k,n , N k,n G ≥ N k,n F , and P k,n G ≥ P k,n F .

Thus, additional information decreases transgression if stakes are high, but

increases transgression if stakes are low. This happens because information

release shifts probability mass both into the upper and into the lower tail.

Morally “good” agents become better, and “bad” agents become worse. If

stakes are low, only the worst agents in the population act as supporters. The

moral costs of these agents tend to decrease as the distribution becomes more

spread out. Therefore, transgression becomes more likely.

Conversely, if stakes are high, even good agents with high costs are near the

threshold of becoming supporters. As the upper tail of the distribution becomes

more spread out, these agents tend to end up above the threshold and

thus do not become supporters. Hence transgression becomes less likely.

4.3 Feedback

In the baseline model, we assume that supporters of the transgression observe

the precise number of other supporters and share their guilt with these. In the

following, we contrast this baseline setting with the situation in which agents

only observe whether the transgression has occurred. Thus, they do not learn

the precise number of supporters. We call this counterpart model the game

without feedback.

In the game without feedback, utility from voting Yes is given by

V 1 {Y−i ≥k−1} − x i

((α +

)

)

β

1 {Y−i ≥k−1} + γ 1 {Y−i =k−1} ,

1 + E[Y −i |Y −i ≥ k − 1]

where the conditional expectation is taken under the distribution induced by

equilibrium behavior. For instance, in an equilibrium in which agents vote Yes

15

whenever their type is below the threshold ˜θ k,n , this conditional expectation is

given by

E[Y −i |Y −i ≥ k − 1] =

∑ n−1

j=k−1 j b(n − 1, j, F (˜θ k,n ))

∑ n−1

j=k−1 b(n − 1, j, F (˜θ k,n ) .

The next proposition shows that symmetric equilibria of the game without

feedback are equilibria in threshold strategies, just like in the baseline model.

Such an equilibrium always exists, and the moral thresholds ˜θ k,n in any such

equilibrium are higher than the thresholds θ k,n in the corresponding baseline

model. Withholding feedback thus makes diffusion of guilt less effective, rendering

moral transgression less likely. 12

Proposition 5. The game without feedback possesses a symmetric equilibrium

in which transgression happens with positive probability. All such equilibria are

equilibria in threshold strategies where the thresholds ˜θ k,n solve

with

V

˜θ k,n

= ˜H(F (˜θ k,n ), α, β, γ, k, n) (7)

B(n − 1, k − 1, F )

˜H(F, α, β, γ, k, n) = α

b(n − 1, k − 1, F ) + γ

+ β

B(n − 1, k − 1, F )

b(n − 1, k − 1, F )

∑ n−1

j=k−1

1

b(n−1,j,F )

(1 + j)

B(n−1,k−1,F )

and

B(n − 1, k − 1, F ) =

∑n−1

j=k−1

b(n − 1, j, F ).

Moreover, any solution ˜θ k,n to (7) satisfies ˜θ k,n ≥ θ k,n where θ k,n is the unique

equilibrium threshold of the baseline model with feedback.

Accordingly, the transgression probability ˜P k,n and the expected number of

supports Ñk,n are smaller in this model. The lack of precise feedback makes it

harder for agents to excuse their behavior.

12 We do not show uniqueness of equilibrium yet numerical results suggest that it holds.

We prove existence and a comparison between any equilibrium and the unique equilibrium

of the baseline model.

16

5 Diffusion of Guilt

In this section, we further explore the effect of diffusion of guilt. Section 5.1

discusses the role of diffusion of guilt compared to the other components of

moral costs. Section 5.2 describes how predictions change if agents adopt a

non-consequentialist perspective, thus experiencing guilt whenever they act as

supporters – regardless of whether transgression actually happens.

5.1 Isolating the Effect of Diffusion

We will see that there is an almost substitutive relation between the diffusive

and the non-diffusive components of moral costs. By and large, if guilt is

completely diffusive (α = 0) predictions are quite comparable to the case in

which guilt is completely non-diffusive (β = 0) but where the benefit V from

transgression is almost k times larger. Theorem 2 specifies this relationship in

all precision.

Theorem 2. Denote by θ k,n (α, β, γ, V ) the equilibrium threshold of the game

with parameters α,β,γ,V , k and n as defined in (4).

(i) Fix w ∈ [0, 1] and α 0 > 0 and define θ 1 and θ 2 as the equilibrium thresholds

of the game without diffusion (β = 0) for (k, n) and for (k + 1, n + 1),

θ 1 = θ k,n (α 0 , 0, γ, V ) and θ 2 = θ k+1,n+1 (α 0 , 0, γ, V ).

Then it holds that

min(θ 1 , θ 2 ) ≤ θ k,n (wα 0 , k(1 − w)α 0 , γ, V ) ≤ max(θ 1 , θ 2 ).

(ii) For any δ > 0, we have θ k,n (0, δ, 0, V ) = θ k+1,n+1 (δ, 0, 0, k · V ).

(iii) We have the bound

θ k,n (α, β, γ, V ) ≤ θ k,n (α, 0, γ, V ) ≤

(

1 + β )

θ k,n (α, β, γ, V ).

kα

The theorem demonstrates a kind of strategic equivalence between the game

with n agents out of whom k are required for transgression to happen and guilt

diffuses in the number of supporters, and the comparison game in which guilt

does not diffuse. The number of agents needs to be mildly adjusted by one,

17

oth for the group size and for the number of agents required. The benefit

from transgression needs to be inflated by k.

Part (i) illustrates that up to adjusting between (k, n) and (k+1, n+1), decreasing

the non-diffusive guilt term still leads to the same equilibrium thresholds

if the diffusive term is increased by k-times that amount. Part (ii) focuses

on the case in which notions of being pivotal do not affect guilt, γ = 0. Up

to adjusting k and n by one, if guilt diffuses, (α, β) = (0, δ), equilibrium

thresholds are identical to those of the game with perfectly non-diffusive guilt,

(α, β) = (δ, 0) if the benefit V is inflated by k. 13

Part (iii) assesses quantitatively

how the diffusive component of guilt affects equilibrium thresholds.

Equilibrium thresholds of the two games with β > 0 and β = 0, lie within a

factor 1 + β of each other. This holds independently of the distribution of

kα

moral types, F . For fixed α and β, the impact of the diffusive guilt component

on equilibrium behavior vanishes quickly as k increases. Guilt sharing

between agents is then so strong that this costs component has little impact

on the agents’ behavior.

5.2 A Non-Consequentialist View

So far, guilt hinged on consequences of actions instead of intent. In this section,

we focus on the latter. Under this non-consequentialist (or Kantian) moral

perspective, agents face moral costs from choosing Yes, independent of the

outcome. 14 As before, we assume that agents share guilt with other supporters,

while notions of being pivotal play not role anymore as consequences do not

affect guilt.

The following dichotomy emerges. For small values of the selfish benefit V ,

the equilibrium in which all agents opt against transgression is the unique

symmetric equilibrium. For larger V , there are several symmetric equilibria in

which the transgression occurs with strictly positive probability. The equilibrium

with the highest transgression probability Pareto-dominates.

13 This is, of course, not a strategic equivalence in the strictest sense as the number of

agents mildly varies across the two games.

14 In the terminology of Admati and Perry (1991), we compare our basic model, a subscription

game, to the associated contribution game.

18

Moral costs for agent i from opting for Yes are now given by

(

))

β

(V − x i α +

. (8)

1 + Y −i

We thus dropped the term that captured notions of being pivotal which was

inherently consequentialist. Realized utility from voting No is given by

V 1 {Y−i ≥k}.

The following proposition shows that – under mild conditions – transgression

occurs in equilibrium only if V is sufficiently large.

Proposition 6.

(i) For sufficiently large V , there exist symmetric equilibria in which transgression

occurs with positive probability.

Equilibrium thresholds solve

These are in threshold strategies.

V

¯θ k,n

= ¯H(F (¯θ k,n ), α, β, k, n) (9)

with

¯H(F, α, β, k, n) =

∑n−1

j=0

(

b(n − 1, j, F )

α +

b(n − 1, k − 1, F )

β )

.

1 + j

¯θ

(1)

k,n

(2) (1)

If two thresholds and ¯θ

k,n

> ¯θ

k,n

equilibria. The equilibrium associated with the larger threshold

dominates the other.

both solve (9), they both correspond to

¯θ

(2)

k,n

Pareto

(ii) Assume f(0) < ∞ and k > 2. Then, for sufficiently large V , there exist

at least two symmetric equilibria in which transgression occurs with positive

probability.

(iii) Assume f(0) < ∞ and k > 1. Then, for small V , the unique symmetric

equilibrium is the one in which all agents choose No.

Thus, in equilibrium, small temptations V do not cause moral transgression

– but sufficiently large ones do. This is due to a “two-sided” coordination

problem. For an agent, in addition to the possibility of free-riding on other

agents’ Yes-votes, opting for No has another appeal. It avoids the risk of facing

19

moral costs without reaping any benefits. This two-sided coordination problem

also causes the multiplicity of equilibria in case (ii).

6 Conclusion

We analyzed how moral costs affect individual support of morally difficult

group decisions, studying a threshold public good game. Empirical studies

from economics, psychology, sociology and business ethics motivated us to

assume that agents are heterogeneous in their moral costs, and that these

costs consist of three parts: a standard cost term, a shared guilt term that

decreases in the number of supporters, and a cost term that hinges on the

notion of being pivotal.

We proved existence and uniqueness of a symmetric non-trivial equilibrium,

and analyzed the comparative statics of the model. We found that increasing

the number of supporters necessary for transgression to materialize does not

necessarily reduce transgression. As supporters can share their guilt, transgression

can instead become more likely if many supporters become active.

Furthermore, while an increase in individual moral types prevents transgression,

information release can have counteractive effects if selfish benefits from

transgression are small. Moreover, the provision of feedback can make it easier

for agents to support transgression. The chance to learn that they were not

pivotal reduces their feelings of moral responsibility. This effect can be strong

enough to increase the likelihood of moral transgression.

Economic models of group activity in morally relevant contexts are rare. Notable

exceptions are Huck and Konrad (2005) looking into committee decisions

on confiscating foreign investments, Sobel (2010) analyzing moral responsibility

in market activity, and Huesmann and Wambach (2015) studying mechanisms

for rewarding dangerous services of moral relevance. If people engage in

“psychosocial manoeuvres – often aided by the institutions [...], which absolve

them from moral responsibility for harmful acts” (Haidt and Kesebir, 2010, p.

812), further research is needed to understand and disentangle the mechanisms

behind institutionally driven moral transgression. This paper contributes to

this endeavor. Speaking with the social psychologist Albert Bandura (1999,

p. 297), “...societies cannot rely entirely on individuals to deter human cru-

20

elty. Civilized life requires, in addition to human personal codes, effective

social safeguards against the misuse of power for exploitative and destructive

purposes.” We believe that economic research can and should contribute to

shaping such social safeguards.

A

Proofs

Proof of Theorem 1. We first show that all equilibria, symmetric or not, in

which transgression happens with positive probability must be equilibria in

threshold strategies. For each agent i, there is a threshold θ k,n,i ∈ [0, ∞] such

that i votes Yes if x i < θ k,n,i and No if x i > θ k,n,i . To see this, fix the strategies

of the other agents. In an equilibrium in which transgression happens with

positive probability, at least k − 1 of the other agents must vote Yes with positive

probability. Assume agent i weakly prefers voting Yes over No for some

type x i . Comparing the expectations of (1) and (2) implies that this preference

must be strict for all lower types. Likewise, if agent i weakly prefers voting No

over Yes for some type x i , this preference must be strict for all higher types.

Thus, all best responses to the other agents’ strategies are threshold strategies.

Therefore, all equilibria in which transgression happens with positive probability

are equilibria in threshold strategies. 15

For k < n, we now show that there exists a unique threshold θ k,n > 0 with the

property that if all agents play a threshold strategy with θ k,n , an agent with

type θ k,n is indifferent between voting Yes and No. We treat the case k = n

separately at the end.

Consider an agent with type x and assume that all

other agents follow a threshold strategy with θ k,n > 0. Expected payoff from

voting Yes is then given by

∑n−1

j=k−1

( (

b(n − 1, j, F (θ k,n )) V − α +

β ) )

x − b(n − 1, j, F (θ k,n )) γ x (10)

1 + j

where the Binomial distribution arises as the other agents independently vote

15 As the equilibrium in which all agents always vote No is also in threshold strategies, it

follows that in fact all symmetric equilibria are in threshold strategies.

21

Yes with probability F (θ k,n ). Expected payoff from voting No is given by

∑n−1

b(n − 1, j, F (θ k,n )) V. (11)

j=k

Equilibria are characterized by thresholds θ k,n for which the two expressions

coincide for x = θ k,n . Equating (10) and (11) for x = θ k,n yields condition (3)

as stated in the proposition. It remains to show existence of a unique (positive)

solution. To this end, we write (3) as

V

θ k,n

= H(F (θ k,n ), α, β, γ, k, n) (12)

with

H(F, α, β, γ, k, n) = γ +

∑n−1

j=k−1

(

b(n − 1, j, F )

α +

β )

.

b(n − 1, k − 1, F ) 1 + j

The left hand side of (12) is strictly decreasing in θ k,n , diverging to ∞ at 0

and becoming arbitrarily small for large θ k,n . Writing the right hand side as

γ +

∑n−1

j=k−1

(

α +

β

1 + j

) ( )

n−1

j

)

( n−1

k−1

( ) j−k+1 F (θk,n )

,

1 − F (θ k,n )

we see that it strictly increases from α + β k + γ to ∞ as θ k,n moves through

the support of F . This proves the existence of a unique solution θ k,n in the

interior of the support.

It remains to study the case k = n. Like in the case of k < n, an interior

separating equilibrium must be characterized by (12) which becomes

and thus

V

θ n,n

= α + β n + γ (13)

θ n,n =

V

α + β n + γ . (14)

In the case where this choice of θ n,n lies above the support of F (and only

then), all agents voting Yes regardless of their type is an equilibrium. 16

16 The separating equilibrium thus degenerates to a pooling equilibrium.

22

Proof of Proposition 1. By (12), θ k,n is characterized as the intersection between

the decreasing function V/θ k,n and the function H which is increasing

in θ k,n via F . Increasing the decreasing function V/θ k,n by increasing V necessarily

shifts the point of intersection to the right, thus increasing θ k,n . Similarly,

θ k,n decreases in α, β and γ: Increasing any of these parameters increases the

function H, thus shifting the intersection to the left. The claims about N k,n

and P k,n in (i) follow from the behavior of θ k,n : N k,n and P k,n can be seen

as the mean and the probability of observing a realization above k − 1 for a

binomial distribution with “success probability” F (θ k,n ). Both quantities increase

in this probability. To prove (ii), we first show the following alternative

representation of the function H:

H(F, α, β, γ, k, n) = α

∑n−1

j=k−1

b(n − 1, j, F )

b(n − 1, k − 1, F ) + β k

n∑

j=k

b(n, j, F )

+ γ. (15)

b(n, k, F )

To see this, we rewrite the part of H which depends on β. Observe first the

following connection between binomial distributions for (k−1, n−1) and (k, n):

=

= 1 n

= 1 n

∑n−1

j=k−1

∑n−1

j=k−1

= 1

nF

∑n−1

j=k−1

n∑

l=k

1

b(n − 1, j, F )

1 + j

( ) n − 1

F j (1 − F ) n−j−1 1

j

1 + j

( ) n

F j (1 − F ) n−j−1

j + 1

( ) n

F l−1 (1 − F ) n−l

l

n∑

b(n, l, F ).

l=k

Combining this identity with b(n − 1, k − 1, F ) = k b(n, k, F ) yields

nF

β

∑ n−1

1

j=k−1

b(n − 1, j, F )

1+j

b(n − 1, k − 1, F )

= β k

∑ n

j=k

b(n, j, F )

b(n, k, F )

and thus (15). From (15), we can conclude that H is decreasing in k. This

23

follows immediately from the fact that both

∑ n−1

j=k−1

b(n − 1, j, F )

b(n − 1, k − 1, F )

and

∑ n

j=k

b(n, j, F )

b(n, k, F )

are reciprocals of the failure rate of the Binomial distribution which is increasing

in k, see Johnson, Kotz and Kemp (1992), Chapter 3. Consequently, θ k,n is

increasing in k as an increase in k decreases H and thus shifts the intersection

of V/θ and H to the right. As N k,n = nF (θ k,n ) depends on k only through

θ k,n , it increases in k as well.

Proof of Lemma 1. To see (i), note that H(F, α, β, γ, k, n) ≥ α + γ.

Thus,

(12) implies V/θ k,n ≥ α + γ. To see (ii), note that by (14) the threshold θ n,n

becomes θ n,n = nV/β for α = γ = 0. θ n,n thus becomes arbitrarily large as n

increases.

Proof of Proposition 2. The proof of (i) proceeds in two steps. We first show

that it suffices to prove the result for the case α = γ = 0. Then we prove (i)

for this case. Let θ ′ k,n be the equilibrium threshold with parameters α′ = 0,

γ ′ = 0 and β ′ = α + β. By the definition of H, we have

H(F, α, β, γ, k, n) ≥ H(F, 0, α + β, 0, k, n).

If θ ′ k,n < θ k,n, then by this equation we would have

V

θ ′ k,n

= H(F (θ ′ k,n), 0, α + β, 0, k, n) ≤ H(F (θ k,n ), α, β, γ, k, n) = V

θ k,n

,

which is a contradiction. Hence, we have the bound θ ′ k,n ≥ θ k,n and θ ′ k,n → 0

implies θ k,n → 0.

We prove θ ′ k,n → 0 by contradiction. Suppose (i) is violated for θ′ k,n , i.e.

suppose that there exists a subsequence n j of N and ε > 0 such that θ k,n ′ j

≥

ε > 0 for all j ∈ N. Note that the sum

n∑

i=k

( n

i

)

p i (1 − p) n−i

24

is monotonically increasing in p for p ∈ [0, 1]. Thus,

n j

∑

i=k

(

nj

i

)

F (θ ′ k,n j

) i (1 − F (θ ′ k,n j

)) n j−i

≥

n j

∑

i=k

(

nj

i

∑k−1

= 1 −

i=0

)

F (ε) i (1 − F (ε)) n j−i

(

nj

i

(16)

)

F (ε) i (1 − F (ε)) n j−i .

Since ε > 0 implies F (ε) > 0, the right hand side converges to 1: ( n j

)

i grows

polynomially in n j and (1 − F (ε)) nj−i decays exponentially in n j . Now we

rewrite the equilibrium condition (15) into

β ′ ∑ nj

( nj

i=k i

( nj

k

)

F (θ

′

k,nj ) i (1 − F (θ ′ k,n j

)) n j−i

)

F (θ

′

k,nj

) k (1 − F (θ ′ k,n j

)) n j−k

= kV

θ ′ k,n j

and then into

β ′

n j

∑

i=k

(

nj

= kV (

nj

θ

k,n ′ j

k

≤

kV ε

(

nj

k

)

F (θ ′ k,n j

) i (1 − F (θ ′ k,n j

)) n j−i

i

)

F (θ k,n ′ j

) k (1 − F (θ k,n ′ j

)) n j−k

)

(1 − F (ε))) n j−k .

(17)

As F (ε) > 0, and as ( n j

)

i grows polynomially in nj and (1 − F (ε)) nj−i decays

exponentially in n j , the final term converges to 0. This is a contradiction as we

proved in (16) that the left hand side of (17) converges to 1. This completes

the proof of (i).

The proof of (ii) proceeds in two similar steps. We first argue that we can

focus on the case in which the diffusive guilt term is absent, and then prove

the result for this case. Let θ k,n ′ now be the equilibrium threshold for the case

α ′ = α + β, γ ′ = γ and β ′ = 0. By definition,

H(F, α, β, γ, k, n) ≤ H(F, α + β, 0, γ, k, n).

25

If θ ′ k,n > θ k,n, then by this inequality

V

θ k,n

= H(F (θ k,n ), α, β, γ, k, n) ≤ H(F (θ ′ k,n), α + β, 0, γ, k, n) = V

θ ′ k,n

which is a contradiction. Hence, we have θ ′ k,n ≤ θ k,n, and it suffices to show that

nF (θ ′ k,n ) → ∞. To this end, consider a subsequence n l such that n l F (θ k,nl ) ′

converges to a number in c ∈ [0, ∞). Now we use that

V

θ ′ k,n

= γ +

∑n−1

j=k−1

b(n − 1, j, F (θ k,n ′ )) (α + β) . (18)

b(n − 1, k − 1, F (θ

k,n ′ ))

Calculating the limits for l → ∞ results in

lim b(n l − 1, k − 1, F (θ k,n ′ l l

))

( )

nl − 1

= lim F (θ k,n ′ l k − 1

l

) k−1 (1 − F (θ k,n ′ l

)) n l−k

( nl

)

−1

k−1

= lim lim

l l

=

and, analogously,

n k−1

l

n k−1

l

F (θ ′ k,n l

) k−1 lim

l

(1 − F (θ ′ k,n l

)) n l−k

1

(k − 1)! ck−1 lim

l

(1 − F (θ ′ k,n l

)) n l

= ck−1

(k − 1)! exp(−c)

∑n−1

j=k−1

∑k−2

b(n − 1, j, F (θ k,n)) ′ = 1 − b(n − 1, j, F (θ k,n))

′

→ 1 −

j=0

∑k−2

j=0

c j

j! exp(−c).

Hence, along the subsequence, the right hand side of (18) converges to

γ + (α + β)

(

exp(c) −

∑k−2

j=0

)

c j (k − 1)!

.

j! c k−1

But, as shown in part (i) of the proposition, the left hand side of (18) diverges

to ∞. This is a contradiction. Hence there exists no such subsequence n l .

26

Thus nF (θ ′ k,n ) → ∞.

It remains to prove (iii). For this purpose, fix k and V and let κ > 0

be arbitrary. By part (ii) of the proposition, nF (θ k,n ) → ∞. Thus, for n

sufficiently large, nF (θ k,n ) ≥ κ. As

n∑

i=k

( n

i

)

p i (1 − p) n−i

is increasing in p for p ∈ [0, 1], we have for sufficiently large n,

P k,n =

n∑

i=k

We also know that for fixed i,

( n

i

)

F (θ k,n ) i (1 − F (θ k,n )) n−i ≥

i=0

Thus, for n → ∞, (19) converges to

which converges to 1 as κ → ∞.

n∑

( ) n (κ i (

1 −

i n) κ ) n−i

n

i=k

∑k−1

( ) n (κ i (

= 1 −

1 −

i n)

n) κ n−i

. (19)

( ) n 1

lim

n→∞ i n = lim 1 ∏i−1

n − j

i n→∞ i! n

j=0

1 −

∑k−1

i=0

κ i

i! e−κ

= 1 i! .

Proof of Proposition 3. Throughout the proof, we drop the dependence of H

on (α, β, γ, k, n), writing just H(F ). In the proof of Theorem 1, we saw that

H(F ) is increasing. By assumption, we have H(F (θ)) ≥ H(G(θ)) and thus

V

θ F k,n

= H(F (θ F k,n)) ≥ H(G(θ F k,n)),

implying that θk,n F lies to the left of the unique intersection of V/θ and H(G(θ)).

27

Accordingly, θ F k,n ≤ θG k,n . Moreover,

H(F (θ F k,n)) = V

θ F k,n

≥ V

θ G k,n

= H(G(θ G k,n))

implies F (θk,n F ) ≥ G(θG k,n ) by the monotonicity of H.

implies the results for N k,n and P k,n .

The latter inequality

Proof of Proposition 4. The proof extends the one of Proposition 3. Note that

the functions H(F (θ)) and H(G(θ)) inherit the single-crossing property of F

and G. Denote by x 0 > 0 a point of intersection of F and G and define

v 0

= x 0 H(F (x 0 )) such that v 0 /θ intersects both H(F (θ)) and H(G(θ)) for

θ = x 0 . Consequently, for V = v 0 we have θ F k,n = θG k,n = x 0. For V > v 0 , it

follows that both, θ F k,n and θG k,n lie in [x 0, ∞]. For θ ∈ [x 0 , ∞], we know by the

single-crossing condition that H(F (θ)) ≥ H(G(θ)). For V > v 0 , it thus follows

as in the proof of Proposition 3 that θk,n F ≤ θG k,n and F (θF k,n ) ≥ G(θG k,n ), implying

the results about N k,n and P k,n . The argument for V < v 0 is analogous, using

that the ranking of F and G is reversed on [0, x 0 ].

Proof of Proposition 5. Arguing as in the proof of Theorem 1, best responses

to any non-trivial strategies of the other agents must be threshold strategies.

This implies that equilibria must be in threshold strategies. Consider agent i

of type x i who faces opponents who play a threshold strategy with threshold

θ k,n . In determining a symmetric equilibrium in threshold strategies, the sole

difference to the proof of Theorem 1 lies in the diffusive part of the expected

cost from voting Yes. We now obtain

P (Y −i ≥ k − 1)

β x i

E[1 + Y −i |Y −i ≥ k − 1]

= B(n − 1, k − 1, F (θ k,n ))

∑ n−1

j=k−1

β x i

b(n−1,j,F (θ k,n ))

B(n−1,k−1,F (θ k,n

(1 + j).

))

Modifying the equilibrium condition (12) accordingly shows that equilibria are

now characterized by

V

˜θ k,n

= ˜H(F (˜θ k,n ), α, β, γ, k, n) (20)

28

with

B(n − 1, k − 1, F )

˜H(F, α, β, γ, k, n) = α

b(n − 1, k − 1, F ) + γ (21)

+ β

B(n − 1, k − 1, F )

b(n − 1, k − 1, F )

∑ n−1

j=k−1

The function ˜H(F, α, β, γ, k, n) is continuous in F and satisfies

1

(22)

b(n−1,j,F )

(1 + j).

B(n−1,k−1,F )

˜H(0, α, β, γ, k, n) = α + β k + γ.

Moreover, it is bounded from below by α + β + γ > 0. The last claim follows

n

from (21) because of the estimates b(n − 1, k − 1, F ) ≤ B(n − 1, k − 1, F ) and

j + 1 ≤ n. As V/θ is arbitrarily large for small θ and arbitrarily small for large

θ, it follows that (20) possesses at least one solution ˜θ k,n > 0. To complete

the proof of the proposition, we need to show that any solution ˜θ k,n satisfies

˜θ k,n ≥ θ k,n . To prove this, we show that ˜H(F, α, β, γ, k, n) ≤ H(F, α, β, γ, k, n).

The result then follows from the monotonicity of V/θ in θ. To compare ˜H and

H, it suffices to compare the β-dependent diffusive terms in (5) and (21).

Applying Jensen’s inequality with the convex function h(x) = 1/x and the

probability distribution with probabilities b(n − 1, j, F )/B(n − 1, k − 1, F ) on

k, . . . , n, we find that

B(n − 1, k − 1, F )

b(n − 1, k − 1, F )

B(n − 1, k − 1, F )

≤

b(n − 1, k − 1, F )

=

∑n−1

j=k−1

∑ n−1

j=k−1

∑n−1

j=k−1

b(n − 1, j, F ) 1

b(n − 1, k − 1, F ) 1 + j

and thus ˜H(F, α, β, γ, k, n) ≤ H(F, α, β, γ, k, n).

1

b(n−1,j,F )

(1 + j)

B(n−1,k−1,F )

b(n − 1, j, F ) 1

B(n − 1, k − 1, F ) 1 + j

Proof of Theorem 2. We begin with claim (i). To simplify the notation, we

define functions h 1 (θ) and h 2 (θ) by

h 1 (θ) =

B(n − 1, k − 1, F (θ))

b(n − 1, k − 1, F (θ))

and h 2 (θ) =

B(n, k, F (θ))

b(n, k, F (θ)) ,

29

and notice that due to (15), the following identity holds:

H(F (θ), α, β, γ, k, n) = αh 1 (θ) + β k h 2(θ) + γ. (23)

The thresholds θ i , i = 1, 2 defined in the proposition solve L i (θ i ) = 0 where

the strictly increasing functions L i are given by

L i (θ) = α 0 h i (θ) + γ − V θ .

It follows that for the strictly increasing function

L w (θ) = wL 1 (θ) + (1 − w)L 2 (θ), w ∈ (0, 1),

there exists a unique θ w with the properties that L w (θ w ) = 0 and

min(θ 1 , θ 2 ) < θ w < max(θ 1 , θ 2 ),

i.e. the unique zero of a convex combination of the strictly increasing functions

h 1 and h 2 lies between the unique zeros of these functions. It thus remains to

show that θ w = θ k,n (wα 0 , k(1 − w)α 0 , γ, V ). This follows from the observation

that by (23),

L w (θ) = wL 1 (θ) + (1 − w)L 2 (θ)

= wα 0 h 1 (θ) + k(1 − w)α 0

h 2 (θ) + γ − V k

θ

= H(wα 0 , k(1 − w)α 0 , γ, V, k, n) − V θ .

To see (ii), observe that it follows from (23) that

H(F (θ), 0, δ, 0, k, n) = k · H(F (θ), δ, 0, 0, k + 1, n + 1)

which immediately leads to the result. We finally prove (iii). Define the short

notation θ = θ k,n (α, β, γ, V ) and θ ′ = θ k,n (α, 0, γ, V ). By Proposition 1, we

have θ ≤ θ ′ . Note further that

H(F, α, β, γ, k, n) ≤ H(F, α + β/k, 0, γ, k, n). (24)

30

Hence,

V

θ

= H(F (θ), α, β, γ, k, n)

≤ H(F (θ), α + β/k, 0, γ, k, n)

(

≤ 1 + β )

H(F (θ), α, 0, γ, k, n)

kα

(

≤ 1 + β )

H(F (θ ′ ), α, 0, γ, k, n)

kα

(

≤ 1 +

kα) β V

θ ′

which yields θ ≤ θ ′ ≤ ( 1 + β

kα)

θ as required.

Proof of Proposition 6. Throughout the proof, we do not use the exact functional

form of how guilt depends on the total number y of Yes votes. Instead we

assume that shared guilt takes the form of x/s(y) where x denotes an agent’s

type and s is an increasing, positive function in y. This shortens the notation

and illustrates that the result holds also in this more general setting. In our

case, we have

1

s(y) = α + β y

⇔ s(y) = 1 .

α + β y

Let us prove the first assertion. We look for θ > 0 where the expected payoff

of voting No is equal to the expected payoff of voting Yes, i.e. we look for

solutions θ > 0 of the equation

V

i.e.

∑n−1

b(n−1, j, F (θ)) = V

j=k

∑n−1

j=k−1

V b(n − 1, k − 1, F (θ)) =

∑n−1

θ

b(n−1, j, F (θ))−

b(n−1, j, F (θ)),

s(1 + j)

∑n−1

j=0

j=0

(25)

θ

b(n − 1, j, F (θ)). (26)

s(1 + j)

The right hand side is nonegative, 0 for θ = 0, and diverges to ∞ for θ → ∞.

The left hand side is bounded, continuous in θ and positive for θ > 0. This

implies that for large V there exists at least one nontrivial solution θ > 0 to

this equation. A solution θ of equation (25) is the threshold of a symmetric

31

Bayesian equilibrium in which each agent i votes Yes if x i ≤ θ and No if x i > θ

as the expected utility of voting Yes decreases in x i .

Now let us turn to the second assertion. Analogously as in Proposition 1, all

symmetric Bayesian equilibria must be in threshold strategies. The case in

which the threshold is ∞ is trivial. Let us look again at (26). We want to

show that this equation has no solution θ > 0 for small V . For large θ,

V b(n − 1, k − 1, F (θ)) <

∑n−1

j=0

θ

b(n − 1, j, F (θ)).

s(1 + j)

Thus, by continuity in θ, it suffices to show that for small θ > 0 and small V ,

First note that

V b(n − 1, k − 1, F (θ)) <

∑n−1

j=0

θ

b(n − 1, j, F (θ)).

s(1 + j)

with

θ

s(1) (1 − F ∑n−1

θ

(θ))n−1 <

b(n − 1, j, F (θ)) (27)

s(1 + j)

j=0

V b(n − 1, k − 1, F (θ)) = V

( ) n − 1

F (θ) k−1 (1 − F (θ) n−k ≤ V

k − 1

( ) n − 1

F (θ)

k − 1

(28)

as k > 1. Now choose V sufficiently small such that V ( n−1

k−1)

F ′ (0) < 1/s(1)

which implies that

V

( n − 1

)F (θ) <

k − 1

θ(1 − F (θ))n−1

s(1)

for small θ as F (θ)/θ → F ′ (0) and F (θ) → 0 for θ → 0. This proves the claim.

Now let us turn to the third assertion. Similarly as in (27) and (28), one can

show that

V b(n − 1, k − 1, F (θ)) <

∑n−1

j=0

θ

b(n − 1, j, F (θ))

s(1 + j)

32

for a fixed (arbitrary large) V if θ > 0 is small enough. Hence, by continuity

and positivity of both terms for θ > 0, for large V there are at least two

positive solutions to (26) with at least one solution satisfying F (θ) < 1. Thus,

the two solutions correspond to two distinct threshold strategies.

Note that

∑n−1

j=0

1

b(n − 1, j, F (θ)) (29)

s(1 + j)

is weakly decreasing in θ. By linearity, and as the function s(1 + j) is weakly

increasing, this claim can be reduced to the well-known fact that

m∑

b(n − 1, j, F (θ))

j=0

is weakly decreasing in θ for m ≤ n − 1.

Now let us denote by

¯θ

(1)

k,n

< ¯θ

(2)

k,n

two solutions of (25) which correspond to

(2)

) < F (¯θ ). We show that the

two different threshold strategies, i.e. F (¯θ (1)

k,n k,n

equilibrium associated with the second threshold strategy strictly Pareto dominates

the first. This argument also covers the trivial equilibrium

¯θ

(1)

k,n = 0.

Consider an agent of type x ≤

the expected payoffs

(1) ¯θ

k,n

. In both equilibria, voting Yes leads to the

V

∑n−1

j=k−1

n−1

b(n − 1, j, F (¯θ (i)

k,n )) − ∑ x

(i)

b(n − 1, j, F (¯θ

k,n

s(1 + j) ))

j=0

for i ∈ {1, 2}.

The expected payoff is larger in the second equilibrium as the first term is

increasing whereas the second term is weakly decreasing, see (29). The case

¯θ (2)

(2)

(2)

k,n

< x can be treated similarly. In the case ¯θ

k,n

< x ≤ ¯θ

k,n

, we have to

compare the expected payoffs

V

∑n−1

b(n − 1, j, F (¯θ (1)

k,n ))

j=k

33

and

V

∑n−1

j=k−1

n−1

b(n − 1, j, F (¯θ (2)

k,n )) − ∑ x

(2)

b(n − 1, j, F (¯θ

k,n

s(1 + j) )).

j=0

We use F (¯θ (1)

(2)

k,n

) < F (¯θ

k,n

) to show that

V

≥V

=V

>V

∑n−1

j=k−1

∑n−1

j=k−1

∑n−1

j=k

∑n−1

j=k

n−1

b(n − 1, j, F (¯θ (2)

k,n )) − ∑ x

(2)

b(n − 1, j, F (¯θ

k,n

s(1 + j) ))

j=0

n−1

b(n − 1, j, F (¯θ (2)

k,n )) − ∑ ¯θ (2)

k,n

(2)

b(n − 1, j, F (¯θ

k,n

s(1 + j) ))

b(n − 1, j, F (¯θ (2)

k,n ))

b(n − 1, j, F (¯θ (1)

k,n ))

j=0

which proves the claim.

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