Discussion Paper No. 736

A COORDINATION GAME MODEL OFCHARITABLE GIVING AND
SEED MONEY EFFECT

Kentaro Hatsumi

April 2009
Revised September 2009
Secondly revised May 2011

The Institute of Social and Economic Research
Osaka University
6-1 Mihogaoka, Ibaraki, Osaka 567-0047, Japan

A Coordination Game Model of Charitable Giving
and Seed Money Eﬀect∗
Kentaro Hatsumi†First Version: December 24, 2007
This Version: May 26, 2011

Abstract: If potential donors for a charity project possess the warm-glow
properties in their preferences, we can represent their behavior with a coordination game. Accordingly, we construct a simultaneous incomplete information game model of charitable giving based on a simple global coordination
game. We demonstrate that merely by the eﬀect of seed money to shift thethreshold requirement of the donations for project success downwards, theproportion of donors and the total amount of donations strictly and continuously increase with the amount of seed money. This result is partially compatible with the ﬁeld experimental evidence in List and Lucking-Reiley [List,
J. A., Lucking-Reiley, D., 2002. The Eﬀects of Seed Money and Refunds
on Charitable Giving: Experimental Evidence from a University CapitalCampaign. Journal of Political Economy 110 (1), 215-233].
Keywords: charitable fund-raising, global game, leadership giving, philanthropy
JEL Classiﬁcation Numbers: C72, D82, H41, L31∗

I would like to thank an associate editor, two anonymous referees, Masaki Aoyagi,
Atsushi Kajii and Shigehiro Serizawa for their helpful suggestions. I am also gratefulto Kazuhiko Hashimoto, Daisuke Oyama, Shuhei Morimoto, Stephen Morris, Tadashi
Sekiguchi, Hiroshi Uno, as well as other participants at the 2009 Econometric Society Far
East and South Asia Meeting, the 2009 Japanese Economic Association Spring Meeting,and the 2008 Kyoto Game Theory Workshop for their helpful comments and discussions.
The ﬁnancial support from the Japan Society for the Promotion of Science via the Research
Fellowship for Young Scientist is gratefully acknowledged.
†
Corresponding to: Institute of Economic Research, Kyoto University, Yoshida,
Sakyoku, Kyoto, 606-8501, Japan. Email: kentaro.hatsumi at fx5.ecs.kyoto-u.ac.jp1

1

Introduction

1.1

Charitable Giving and Coordination Game

Charitable giving is a topic that has attracted the attention of many public
economists for a long time and has been studied in various aspects.1 This
paper presents a new view to this topic that charitable behavior is well
represented as a coordination game. Further, we claim that if the numberof potential donors is large and they have little information about otherpotential donors’ valuations for the charity project, the global game, which
is a recently developed coordination game with incomplete information, isquite suitable to represent the situation and brings several insights to thestudy of charitable giving. In this ﬁrst subsection of Introduction, we explain
the reason for this.
In recent years, it has been revealed that when considering charitable behavior, especially in large economies, the warm-glow property of preference
is important. A preference with warm-glow property is typically represented
by a utility function that possesses a three-dimensional domain: the private
consumption, the total amount of donations from all contributors to a charity, and her own contribution to a charity. This contrasts with the classicalpreference represented by a utility function with only a two-dimensional domain: the private consumption, and the total amount of donations fromall contributors to a charity. Warm-glow is the property that increases adonor’s utility as a result of her own contribution.2 This property has beendeveloped theoretically by Andreoni (1988, 1989, 1990), and supported empirically by studies such as Palfrey and Prisbrey (1996, 1997) and Ribar andWilhelm (2002).3 In particular, Andreoni (1988) and Ribar and Wilhelm(2002) suggest that in a large economy, the warm-glow property is crucial
to collect adequate amounts of charitable contributions.
A simple simultaneous game of a capital campaign for charity conﬁrms
their ﬁnding as follows. A capital campaign constitutes a funding drive for acharity that incurs a large ﬁxed cost to ensure project success.4 A campaign
to collect funds to build a hospital is a case in point. If the ﬁxed costfor the project success is very large and there are many potential donors,
each contribution from each donor has almost no eﬀect on project success.
Assume a potential donor in a large economy has two choices Donate and
Not Donate, and possesses a simple classical preference that depends on her
own private consumption and whether the project is a success or a failure.
The latter condition is equivalent to whether the total amount of donations
exceeds the ﬁxed cost or not. Assume that even if the project fails, the1

See Andreoni (2006b) for a recent survey on the studies of charitable giving.
Warm-glow is also called impure-altruism or joy-of-giving.
3
Andreoni (2006b) contains a summary and discussions on the warm-glow property.
4
We owe this deﬁnition of a capital campaign to Andreoni (1998).
2

2

donations are nonrefundable. Also assume that if a suﬃciently large numberof potential donors choose Donate, the project succeeds. Then her payoﬀ isrepresented as follows.
DonateNot Donate

Success
Happy
Very Happy

FailureUnhappy
Neutral(1)

Given that a potential donor cares only for her own private consumption
and project success and she has no eﬀect on project success, she is betteroﬀ choosing Not Donate if the project succeeds. Of course, Not Donate also
yields greater happiness if the project fails. From the Payoﬀ Table (1), Not
Donate is the strictly dominant strategy and each potential donor chooses
this action. It is a prisoner’s dilemma game and represents the classicalfree-riding problem.
However, if people possess the warm-glow property in their preferences,what is the payoﬀ table in the same capital campaign game for a large
number of potential donors? It is natural to consider that the payoﬀ table
is given as follows.
DonateNot Donate

Success
Very Happy
Happy

FailureUnhappy
Neutral(2)

A contribution makes a donor with a warm-glow property happier than when
not contributing if the project is a success.5 Then it has two pure strategy
Nash equilibria; that is, either everybody chooses Donate, or everybody
chooses Not Donate. This is a typical coordination game.
The coordination game is one of the most classical problems in gametheory and has signiﬁcantly improved in recent years due to Carlsson andvan Damme (1993). They consider a simultaneous incomplete information
coordination game where each agent has little knowledge about the type
distribution, and they show the existence of a unique equilibrium with few
assumptions concerning the strategies of agents. Even in the Bayesian coordination game where the type distribution is common knowledge, a uniqueequilibrium is diﬃcult to obtain. Consequently, their work has attracted
much attention. Their game model, known as the global game, has become
popular with game theorists and has been extended in a number of diﬀerent directions. Thanks to Morris and Shin (1998, 2003), it has now been
extended to encompass a coordination problem in a large economy.6
5

Ribar and Wilhelm (2002) ﬁnd that in a large economy, the warm-glow property has
a much stronger eﬀect than the altruistic property concerned only with project successor failure in this example. Following this, the payoﬀ to choose Not Donate with projectsuccess is even represented by Neutral.
6
Well-known applications of the global game include debt pricing (Morris and Shin,2004) and bank runs (Goldstein and Pauzner, 2005). Both are typical coordination problems.

3

If there are a suﬃciently large number of potential donors with the warmglow property of preference, donation behavior for a charity project becomesa coordination game. In the situation of simultaneous donations, the incompleteness of information on the valuation of each potential donor for the charity project is also appropriate. These are the reasons why we consider thata global game is suitable for modeling charitable behavior. We construct asimultaneous incomplete information game model of charitable giving basedon a simple global coordination game, and we characterize a unique equilibrium of the game. By means of comparative static analysis, we show thatthe model is compatible with the empirical studies of charitable behavior,especially the ﬁeld experimental evidence in List and Lucking-Reiley (2002).
We discuss this in the following subsection.
Finally, in this subsection, we note that this model has the same structure
as the voluntary provision of an excludable threshold public good. If acontribution suﬃcient to exceed a threshold is necessary to produce a public
good, and if the beneﬁt for each contributor from the public good depends
on how much she pays, then this has the same representation as the capital
campaign to donors with the strong warm-glow property.7 Hence, the resultsin this paper also apply to the problem of producing an excludable thresholdpublic good under incomplete information.

1.2

Seed Money

Seed money is a preliminary fund for charity publicly announced at thetime when the project is announced. Leaders directly solicited to donate
before the public announcement sometimes provide this seed money. Even
before Andreoni (1998) introduced this problem to public economists, it waswell known among charity fund-raisers that if seed money is granted, thedonations from general contributors increase.
The ﬁeld experimental evidence in List and Lucking-Reiley (2002) supports this knowledge among charity fund-raisers.8 List and Lucking-Reiley(2002) divide 3,000 potential donors into six groups. For three groups, eachwith 500 potential donors, they solicit the donation of a laboratory for environmental studies to purchase a computer at a ﬁxed cost of $3,000 with
diﬀerent levels of seed money.9 To the ﬁrst group, they announce that theseed contribution is $300, to the second group, $1000, and to the thirdgroup, $2,000. Table 1 provides the result. As shown, List and LuckingReiley (2002) clearly demonstrate that the total amount of contributions,7

To represent the excludability of the public good in the Payoﬀ Table (2), we need onlychange the payoﬀ of choosing Not Donate with project success to Neutral.
8
Other empirical studies on seed money include List and Rondeau (2003) and Potters
et al. (2005).
9
The other three groups of potential donors are assigned to an experiment concerning
refunds.

4

the proportion of donors who actually contribute, and the per capita amountof contributions strictly and continuously increase according to the amountof seed money.

[Table 1 about here.]

By drawing on a global coordination game, our model demonstrates thatthe proportion of actual donors and the total amount of donations strictly
and continuously increase with the amount of seed money. The result iscompatible with List and Lucking-Reiley’s (2002) ﬁeld experimental work,
and it supports the view that charitable giving actually involves a coordination game. The key to deriving this result is the threshold shift eﬀectof seed money. This is the eﬀect of seed money shifting the minimal requirement of donations for project success from the cost of the project itself
to the amount of the cost minus the seed money. Under the global gamesetting, each potential donor forecasts the distribution of other potential
donors’ valuations for the project based on her own valuation. Once thethreshold for project success is given, she predicts the proportion of actual
contributors and calculates her expected payoﬀs in both cases of donating
and not donating. When seed money shifts the threshold downwards, forsome proportion of potential donors, the expected payoﬀ by donating becomes greater than when not donating. This makes both the proportion of
actual donors and the total amount of donations increase. This behavior
can be observed only under global game setting of incomplete information.
In an alternative Bayesian game where the distribution of potential donors’
valuations is common knowledge, they often together decide to donate or not
to donate in an equilibrium. This is inconsistent with the above empirical
data by List and Lucking-Reiley (2002).
While this paper is the ﬁrst to model charitable giving as a coordination
game and to capture the threshold shift by seed money, there is a considerable theoretical literature on the eﬀect of seed money. Andreoni (1998)
considers the seed money as a contribution from leaders, and he shows thatthe voluntary contribution from leaders cancels the zero-contribution equilibrium and leaves only the equilibrium with a positive amount of contributions. His discussion bases on a complete information game suitable tocharitable giving among a small number of potential donors, while we construct a game with a large number of potential donors under incomplete
information. Vesterlund (2003) and Andreoni (2006a) direct their attention
to the quality signal eﬀect of seed money. They explain that seed money
donated by leaders indicates the high quality of the charity and therefore
increases contributions from remaining donors. If there is a quality signal
eﬀect from seed money, it is easy to derive the strict increase in donations
5

corresponding to the amount of seed money.10 In contrast, our model showsthat even without the quality signal eﬀect, we can derive the positive eﬀectof seed money. Indeed, the threshold shift eﬀect and the quality signal effect are not contradictory but rather are complementary in increasing total
contributions. Our study is meaningful even if we take the quality signal
eﬀect of seed money into account.11

1.3

The Structure of the Paper

The structure of the paper is as follows. Section 2 examines the basic modelof charitable giving without seed money. In Subsection 2.1, we construct amodel based on a simple global coordination game. We derive the equilibrium in Subsection 2.2, and examine the conditions for the eﬃciency andsuccess of a project in Subsection 2.3. In Section 3, we consider the effect of seed money. In Subsection 3.1, we introduce seed money with thethreshold shift eﬀect, and we observe the impact on the behavior of potential
donors. In Subsection 3.2, we calculate the minimal seed money needed toachieve success. Section 4 discusses remaining inconsistencies with List andLucking-Reiley (2002) and their conceivable solutions. Section 5 concludes.
All proofs of propositions and remarks are in the Appendix.

2

Basic Model and Results

In this section, we present a model of charitable giving that provides afoundation for future comparative static analysis. The eﬀect of seed money
is not yet considered. We present the detailed game structure in Subsection
2.1, derive a unique equilibrium in Subsection 2.2, and provide remarks on
the eﬃciency and success of the project in Subsection 2.3.

2.1

Preliminaries

There is a charity project and n ∈ N potential donors (agents). The project
has a threshold level of cost C ∈ R++ , and if the total provision fromagents exceeds the threshold C, then the quality of the project is signiﬁcantly
improved. Imagine, for example, a charitable campaign to promote child10

An interesting aspect of Vesterlund (2003) and Andreoni (2006a) is that the leaders
donating the seed money and the amount of seed money donated are determined endogenously.
11
There are several other studies on the eﬀect of seed money. Romano and Yildirim(2001) show that in a two stage game of public good provision, the announcement of thecontribution in the ﬁrst stage sometimes brings about larger total contributions becauseof the warm-glow property. Bac and Bag (2003) consider a similar question about when
it is beneﬁcial for a charity fund-raiser to reveal the number of potential donors who are
actually willing to donate. Bag and Roy (2008) explain the eﬀect of seed money by theuncertainty regarding other donors’ valuations of the project.

6

health care in a rural area of a developing country. If the charity collects
suﬃcient donations to build a hospital, the quality of the project increases
substantially. Refer to the project as a success (in the ex-post sense) if the
donations collected from agents are suﬃcient to cover C, and a failure (inthe ex-post sense) if the total donations are less than C.
We consider cases where the number of agents are suﬃciently large.12
We adopt the assumption of a continuum of agents. To simplify the analysis,
we normalize the length of the continuum of agents to 1. Let c = Cn . Under
the normalized setting, c plays a role as the threshold cost instead of C.
Thus we also refer to c as the threshold cost.
The charity announces the threshold cost c and solicits agents for donation. Through the announcement, each agent receives a signal of how
worthy the project is in the case of both project success and failure. Basedon this signal, each agent makes a decision about whether to donate. To
simplify the analysis, we assume that her decision is merely a choice fromher action set {Donate, Not Donate} and that the amount of donation on
the basis of her signal is given exogenously.13
Let vi ∈ R be the signal of an agent i’s degree of warm-glow for thisproject. vi > 0 means that that if i decides to donate and the projectsucceeds, she is delighted because of her warm-glow property. vi ≤ 0 meansthat she thinks that the project is not worthwhile and even if it succeeds,
she does not gain anything from her own contribution.
Assume that an agent i with vi donates x(vi ) when she makes a decision
Donate, where x : R → R++ is the same among all agents, weakly increasing,continuous and there exists an upper bound x ∈ R++ such that for all vi ∈ R,
x(vi ) ≤ x. This upper bound x can be interpreted as the constraint of theirincomes. Also assume simply that vi works as the coeﬃcient of x(vi ) in her
payoﬀ when the project succeeds.
Any agent is assumed to take zero payoﬀ if she does not donate, regardless of whether the project is a success or a failure. Because of thewarm-glow property discussed in Introduction, when the project is a success, the payoﬀ not to donate is smaller than that of donating. Thus weadopt the assumption of a zero payoﬀ without donation for simplicity.
The table below summarize the payoﬀs.

12

In each ﬁeld experiment in List and Lucking-Reiley (2002), the number of potential
donors (people who received solicitation letters) was 500.
13
It is widely adopted as a simpliﬁcation technique to limit the elements of each agent’s
action set to binary in the literature of global games. See, for example, Morris and Shin(2003). Moreover, even in classical models of discrete public good provision, such as
Palfrey and Rosenthal (1984) and Gradstein (1994), the elements of the action set arebinary limited to reduce complexity.

7

DonateNot Donate

Successvi · x(vi )0Failure−x(vi )
0(3)

Based on this ex-post payoﬀ table, each agent is assumed to calculate
her own von Neumann-Morgenstern expected interim payoﬀ, and to decide
whether to donate.14
Let us introduce the signal distribution structure and information setting
for agents. Assume that the signal for each agent vi follows a normal distribution with mean θ and standard deviation σ independently and identically.
Given that the density function for a normal distribution approximates awide class of symmetric density functions with a peak, this assumption isadequate.15 Let F (·; θ) denote the normal cumulative distribution function
with mean θ and standard deviation σ, and let f (·; θ) denote its probability
density function.
Following Morris and Shin (2003), we assume that it is common knowledge among agents that each signal vi for an agent i is independently, identically, and normally distributed with standard deviation σ. However, eachagent i has no information about the mean θ. She has to predict the mean
θ based on her realized signal vi . We assume that her prior belief for θ isuniformly distributed over the real line R. This prior belief is improper as
the total probability mass is inﬁnite; however, the posterior belief through
Bayesian estimation is well deﬁned.16 She believes, based on her signal vi ,that θ follows the normal distribution with mean vi and standard deviation
σ. This is to say, from her point of view, F (·; vi ) is the cumulative distribution function of the actual mean θ, and f (·; vi ) is its probability density
function. It is an interesting aspect of this improper uniform prior belief
that the same function is available to represent both the distribution of real
signals and the distribution of the predictable mean based on the belief ofa single agent.
14

We can extend the model to consider the case where an agent who decides to donate
takes a refund if the project is in failure. For example, let r ∈ (0, 1] and assume −r · x(vi )as i’s payoﬀ when she decides to donate and the project is in failure. r represents the rateof her losses including those incurred by waist of time and negative emotional inﬂuence.
Even with this change in payoﬀ, the structure of the following investigation and theconsequences are unchanged. A detailed explanation is given in the Supplementary Noteattached to the discussion paper version of this study (Hatsumi, 2011). However, notethat in the ﬁeld experiments in List and Lucking-Reiley (2002), the refund policy did notaﬀect the proportion of actual contributors.
15
In the ﬁeld experiments in List and Lucking-Reiley (2002), all potential donors met
two criteria: (i) the household’s annual income was above $70,000, and (ii) the household
was known to have previously given to a charity. In this case, the standard deviation σ isrelatively small.
16
See Morris and Shin (2003) for a detailed explanation. They discuss the improper
uniform prior belief with respect to the philosophy of Laplace.

8In comparison with a Bayesian game wherein the signal distribution iscommon knowledge among agents, the assumption that the posterior beliefof signal distribution for each agent is slightly diﬀerent from those of othersis quite natural. This is because of the following reasons. First, in a large
economy where a continuum can approximate the total agents, if the distribution of signals is common knowledge, it almost represents a complete
information game. This appears unrealistic. Second, especially when agentsare relatively homogeneous (i.e., σ is small) as in the ﬁeld experiments in
List and Lucking-Reiley (2002), it is reasonable that each agent predicts thesignal distribution on the basis of her own signal.
A strategy for an agent i is a function mapping her receiving signal
vi to her action set {Donate, Not Donate}. If both actions occasion theequivalent expected payoﬀ for i, we assume that she selects Not Donate. This
assumption is just to simplify the description, and the opposite assumptionis, of course, available.
At this point, we deﬁne a Bayesian type of incomplete information game.
Let G denote this game. In the next subsection, we consider the equilibriumin this game.

2.2

The Equilibrium

We are interested in Bayesian Nash equilibria of the game deﬁned in theprevious subsection. In this subsection, we characterize a unique BayesianNash equilibrium under the assumption that all strategies of agents havesymmetry and switching properties. A strategy is symmetric if all agents
follow the same strategy. A strategy is a switching strategy around some
cutoﬀ point k ∈ R if there exists a unique cutoﬀ point k such that if anagent’s receiving signal vi is above k, she donates; otherwise, she does notdonate. A symmetric switching strategy s : R → {Donate, Not Donate} isformally written as follows.
{
Donateif vi > ks(vi ) =Not Donate if vi ≤ kFirst, let us introduce two functions that play important roles in thefollowing investigation.
∫∞f (v; θ)dv

(4)

x(v)f (v; θ)dv

(5)

k
∫ ∞
k

(4) represents the proportion of agents whose signal is larger than k whenthe mean signal is θ. This function is useful no matter what strategies are
adopted. (5) represents the total donation when the mean of the signals
is θ and any agent receiving a signal larger than the cutoﬀ point k decides9

to donate. (5) is meaningful only when any agent adopts the switchingstrategy.
At the beginning of the investigation, notice the remark below.
Remark 1. For any agent i with vi ≤ 0, Not Donate is the strictly dominantaction.
This remark is straightforward from the payoﬀ tables of the previous
section. Hereafter, we focus on the actions of agents with signals larger
than 0. Note that the interim payoﬀ of an agent i with a positive signal
vi > 0 is represented by
Pr(Success)(vi · x(vi )) + (1 − Pr(Success))(−x(vi )).

(6)

Thus, agent i decides to donate if(6) > 0⇐⇒ Pr(Success) >1.
vi + 1

Let a function p : R++ → (0, 1) be such that p(vi ) = vi1+1 . p(vi ) represents
the threshold probability for i with vi > 0 in the sense that if the probabilityof project success is larger than p(vi ), Donate leads to a higher expected
payoﬀ than Not Donate. On the other hand, if the probability of projectsuccess is less than p(vi ), Not Donate guarantees a higher expected payoﬀ
than Donate. Then p(·) has the following properties obviously.
Remark 2. p(·) is strictly decreasing and continuous in vi > 0.

[Figure 1 about here.]Let θ̂ : R++ → R be a function such that for vi > 0,
∫ ∞f (v; vi )dv = p(vi ).17θ̂(vi )

See Figure 1 for an illustration. Remember that an agent i with a signal
vi believes the mean of normally distributed signals to be vi itself. Thus
the equation represents that i considers that the probability of the mean ofsignal distribution being larger than θ̂(vi ) is p(vi ) (i.e., i considers Pr(θ ≥θ(vi )) = p(vi )). Since p(·) is strictly decreasing in vi > 0 by Remark 2, wehave the remark below.
In the explicit form, θ̂(vi ) = F −1 (1 − p(vi ); vi ), where F −1 (·; vi ) is the inverse functionof F (·; vi ). This expression is rarely used in the following investigation.
17

10Remark 3. (i) θ̂(vi ) is strictly increasing and continuous in vi > 0. (ii)
For vi0 > vi > 0, θ̂(vi0 ) − θ̂(vi ) > vi0 − vi . (iii) limv→+∞ θ̂(v) = +∞. (iv)limv→0 θ̂(v) = −∞.
We introduce two more functions and discuss their properties.
∫ ∞x(v)f (v; θ̂(vi ))dv(7)k(7) is a special case of (5), and the mean of signals is θ̂(vi ). Since θ̂(vi ) isstrictly increasing in vi > 0, the following remark is straightforward.
Remark 4. (7) is strictly increasing and continuous in vi > 0 and strictly
decreasing and continuous in k > 0.
∫ ∞x(v)f (v; θ̂(vi ))dv

(8)

vi

(8) is a special case of (7). The mean of the signals is θ̂(vi ) and the cutoﬀ
point k = vi . Remark 3 induces the next remark.
Remark 5. (8) is strictly increasing and continuous in vi > 0, and as vi → 0,(8) → 0.
Now, we characterize a cutoﬀ point and the equilibrium in a game G.
Given a threshold cost c, let k ∗ ∈ R++ be such that
∫ ∞x(v)f (v; θ̂(k ∗ ))dv = c.
k∗Note that by Remark 5, k ∗ is uniquely determined.
Proposition 1. Let a symmetric switching strategy s∗ (·) be such that{
Donateif vi > k ∗∗s (vi ) =Not Donate if vi ≤ k ∗ .
Then, in a game G, s∗ (·) is a unique symmetric switching strategy constituting a Bayesian Nash equilibrium.
Note that from the deﬁnition of k ∗ , the actual mean of the signal distribution θ has no eﬀect in determining the equilibrium cutoﬀ point k ∗ . In
contrast, θ̂(·) is quite important in determining k ∗ . The example below
illustrates this proposition.
Example 1. Let a threshold level of cost c = 6.18 Let the standard deviationof the signal distribution σ = 3. Let for all vi ∈ R, x(vi ) = 10. Then, we
calculate θ̂(k ∗ ) = 2.26 and k ∗ = 1.50, and have Table 2.
18

In each ﬁeld experiment in List and Lucking-Reiley (2002), the threshold level ofproject cost for success is $3,000 and the number of potential donors is 500 (i.e., C = 3, 000
and n = 500). In this case, c = C/n = 6.

11

[Table 2 about here.]
Note that the proportion of actual contributors is represented by∫ ∞f (v; θ)dvk∗and the total donation is represented by∫ ∞x(v)f (v; θ)dv.
k∗

(9)

(10)

The project is a success if (10) is larger than c.
As in Example 1, the equilibrium cutoﬀ point k ∗ is strictly larger than0. This is because of a particular assumption of the global game that eachagent possesses a diﬀerent belief about the signal distribution. Under aBayesian game model where the signal distribution and the strategies ofother agents constitute common knowledge, all agents with signals larger
than 0 decide together whether to donate or not depending on the actual
mean of the signals θ in an equilibrium. Under the same setting as Example
1, this alternative Bayesian game analysis provides this in an equilibrium;if θ > 2.26, agents with signals larger than 0 together decide to donate, andif θ ≤ 2.26, any agent does not donate and the total amount of donation isequal to zero. This prediction is incompatible with the empirical researchsuch as List and Lucking-Reiley (2002). It suggests the eﬀectiveness of global
game analysis in the study of charitable giving.19

2.3

Conditions for Eﬃciency and SuccessIn this subsection, we ﬁrst consider the condition that achieves the ex-post
Pareto eﬃciency of the donation for agents. Next, we consider the condition
for the success of the project.
We refer to a project as eﬃcient if in the equilibrium, the actions ofagentsderive ex-post Pareto eﬃciency. Formally, this is represented when∫∞x(v)f(v; θ)dv ≥ c, any agent i with signal vi > 0 chooses Donate, and0
any agent j with signal vj ≤ 0 chooses Not Donate, otherwise, all agents
select Not Donate. This is achievable in an equilibrium of the alternative
Bayesian Game discussed in the last paragraph of the previous subsection.
However, it is never obtained under the more realistic global game setting.
19

In the literature of global games, the strategy proﬁle surviving the iterated eliminationof interim strictly dominated strategies is widely used as the equilibrium concept. See,for example, Morris and Shin (2003). However, in the present model, to derive the sameequilibrium to Proposition 1 under this stronger equilibrium concept, we need an additional outlandish assumption that any agent with suﬃciently large signal has the strictly
dominant action to donate. A detailed explanation is given in the Supplementary Noteattached to the discussion paper version of this study (Hatsumi, 2011).

12

Proposition 2. Any project is ineﬃcient.
The intuition for this proposition is quite simple. Even though the
project is a failure, some agents with signals larger than k ∗ choose Donate.
This implies that the project is not eﬃcient. If the project is a success, anagent i with 0 < vi < k ∗ chooses Not Donate, which brings about ineﬃciency. Therefore, there is no condition that occasions eﬃciency.
As already mentioned in the previous subsection, a project is a successif∫ ∞x(v)f (v; θ)dv ≥ c.
(11)k∗This may be the most important matter for a charity fund-raiser. The
proposition below states the condition for project success.
Proposition 3. A project is a success if and only if θ ≥ θ̂(k ∗ ).
∫∞∫∞Since k∗ x(v)f (v; θ̂(k ∗ ))dv = c, θ ≥ θ̂(k ∗ ) induces k∗ x(v)f (v; θ)dv ≥ c
∫
∞
∗
∗
and
∫ ∞ the project succeeds. If θ < θ̂(k ), then k∗ x(v)f (v; θ̂(k ))dv = c >k∗ x(v)f (v; θ)dv. This induces project failure.
In the next section, we introduce seed money. The main purpose for
introducing seed money is to obtain the success of a project. We show how
to calculate the minimal amount of seed money for success.

3

Seed Money

We exogenously introduce seed money. We see the threshold shift eﬀect of
the seed money on the donation behavior of agents in Subsection 3.1. We
also see how to obtain the success of the project by announcing seed money
in Subsection 3.2.

3.1

The Threshold Shift Eﬀect

Let L ∈ [0, C) denote the amount of seed money. L = 0 represents the state
where no seed money is granted. If seed money L is granted, the thresholdlevel for project success shifts from C to C − L. Assume that information
about the project never changes except concerning the existence and amountof seed money. We consider that all other structures of the game except forthe shift of the threshold for project success remain as before.
Let ` denote the normalized version of seed money; i.e., ` = Ln . Consequently, the normalized version of the shifted threshold is c − `.
Let G(`) denote a Bayesian game with the same structure as that in
Subsection 2.1 with the shift of threshold level from c to c − `. Then we
derive a unique equilibrium strategy s∗ (·; `) in a game G(`) as a corollary of
Proposition 1.

13

Corollary 1. In a game G(`), a unique symmetric switching strategy s∗ (·; `)constituting a Bayesian Nash equilibrium is such that{
Donateif vi > k ∗ (`)∗
s (vi ; `) =Not Donate if vi ≤ k ∗ (`),
where k ∗ (`) ∈ R++ is such that
∫ ∞x(v)f (v; θ̂(k ∗ (`)))dv = c − `.
k∗ (`)

According to Corollary 1, we have the remark below.
Remark 6. k ∗ (`) is strictly decreasing and continuous in `, and lim`→c k ∗ (`) =0.
From Remark 6, we derive a proposition below that is quite compatible
with the ﬁeld experiments in List and Lucking-Reiley (2002).
Proposition 4. If a granted amount of seed money ` strictly increases, thenboth the proportion of agents who decide to donate and the total amount of
donations strictly increase.
The example below illustrates this proposition.
Example 2. Similarly to Example 1, let a threshold level of cost c = 6, a
standard deviation of signal distribution σ = 3, and for all vi ∈ R, x(vi ) =10. Let the actual mean of signal distribution θ = 0. Consequently, we havethe results in Table 3.

[Table 3 about here.]

The mechanism behind Proposition 4 is quite simple and only depends
on Remark 6. If k ∗ (`) decreases, it is obvious that the number of agentswith signals larger than the cutoﬀ point k ∗ (`) strictly increases, and thenthe proportion of agents who decide to donate and the total amount of
donations also strictly increase. As discussed in the last part of Subsection
2.2, the emergence of the cutoﬀ point k ∗ (`) strictly larger than 0 is due to theglobal game assumption that the actual signal distribution is not common
knowledge but is noisily observed and guessed by each agent i on the basisof her own signal vi . This realistic setting of the global game is the key tothe derivation of Proposition 4.

14

3.2

Minimal Seed Money for Success

The main purpose for introducing seed money is to obtain the eﬃciency or
success of a project that is not achieved in its absence. Unfortunately, by
Proposition 2, eﬃciency is never obtained by introducing any amount of seedmoney. Hence, in this subsection, we only focus on the minimal amount ofseed money to success.
From the viewpoint of a charity fund-raiser, the attainment of projectsuccess is crucial. Even though we do not explicitly consider the cost of
introducing seed money, charity fund-raisers actually require a considerable
eﬀort to collect seed money and are better oﬀ if the amount of seed money
needed for success is as small as possible.
The project is not a success without seed money if θ < θ̂(k ∗ (0)). In thiscase, in order to achieve success, we have the following proposition.
Proposition 5. When θ < θ̂(k ∗ (0)), the minimal amount of seed money forsuccess `∗ is such that θ̂(k ∗ (`∗ )) = θ.
Note that from Remarks 3 and 6, θ̂(k ∗ (`)) is continuously and strictly
decreasing in ` ∈ [0, `] and θ̂(k ∗ (`)) → −∞ as ` → c. Thus `∗ is uniquelydetermined. In the case of Example 2, `∗ = 1.91.
To calculate the minimal seed money for success, a charity fund-raiser
needs to know the actual mean of the distribution θ. Indeed, the collectionof the actual signal distribution is quite diﬃcult for a charity fund-raiser. It
involves a considerable cost for preliminary surveys on the potential donors
to approach the actual mean θ, and it is very beneﬁcial.

4

Discussion

In the previous section, we establish Proposition 4, which is quite compatible
with the empirical result in List and Lucking-Reiley (2002). In this section,
we discuss the still remaining inconsistency of our model with List andLucking-Reiley (2002). In our model, the per-capita amount of donations
weakly decreases along with the increase of seed money, whereas the percapita amount of donations strictly increases corresponding to the increaseof seed money in List and Lucking-Reiley (2002). We discuss the source of
this inconsistency and propose some ideas for its resolution.
The mechanism behind this inconsistency is quite simple. We assume
that the amount of donation x(vi ) of agent i with signal vi is exogenously
given and never changes to the amount of seed money. From Remark 6, if
the amount of seed money increases, the threshold signal k ∗ (`) decreases.
Therefore, an agent who changes her action from Not Donate to Donate
because of the increase in seed money has a smaller signal than agents who
continue to choose to Donate. Then, since the amount of donation x(vi )15

weakly increases in her signal vi , the per-capita amount of donation should
weakly decrease in the amount of seed money.20
To remove this inconsistency in the extended version of our model, we
need to increase x(vi ) according to the increase of seed money. Two approaches are considerable for this purpose.
The ﬁrst approach is to render x(vi ) endogenously determined as an
optimal value corresponding to the probability of success. This maintains
the assumption that the eﬀect of seed money is only the threshold shift eﬀect.
As the probability of success depends on the amount of donation obtained
from other agents, this optimization problem is highly complicated. At
present, we are unsure about whether this approach makes sense.
The second approach entails the introduction of the quality signal eﬀectof seed money. The increase of x(vi ) corresponding to the increase in seedmoney can be considered as the consequence of the quality signal eﬀect.
For example, consider the simple introduction of the quality signal eﬀect
as follows. Let θ be the mean of each agent’s signal, similarly to Section
2. Let θ : [0, c) → R be a strictly increasing and continuous function suchthat θ(0) = θ. Assume that when seed money ` is granted, the mean ofsignal θ is shifted to θ(`). Moreover, assume that the standard deviation σremains the same. We interpret this shift of θ to θ(`) as indicating that an
agent i receiving a signal vi when no seed money is granted receives a signal
vi (`) = vi + (θ(`) − θ) when seed money ` is granted. It is obvious that inthis model, the inconsistency regarding the per-capita amount of donationis resolved.21
The second explanation also suggests the possibility of constructing amodel in which the threshold shift eﬀect and the quality signal eﬀect arecomplementary. Which eﬀect actually works remains an interesting question
in both the theoretical and empirical work.

5

Concluding Remarks

In this paper, we construct a global coordination game model of charitablegiving. We show that by merely considering the threshold eﬀect of seedmoney, the proportion of agents who decide to donate and the total amountof donations strictly and continuously increase according to the increase in
seed money.
We construct a simultaneous model for comparison with the empirical
work in List and Lucking-Reiley (2002). However, the actual capital campaigns of charities usually allow agents to donate in a certain period, and20
The assumption that the function x(·) is weakly increasing is crucial for the uniquenessof the equilibrium. See the proof of Remark 5.
21
It may be more sophisticated in the global game framework to represent the quality
signal of seed money by the public signal, which is a signal that all agents can observe inthe same way. See Morris and Shin (2003) for the details.

16possibly represented by dynamic games. Even in dynamic models, our viewthat charitable donation is indeed a coordination game plays an importantrole.22 We hope that this paper encourages further study of donation behavior and thereby assists future charity projects.

Appendix
In the Appendix, we provide the proofs of the Propositions and Remarks.
We omit the proofs of Remarks 1, 2 and 4 since they are straightforward.
∫∞
Proof of Remark 3. Remember that θ̂(vi ) f (v; vi )dv = p(vi ) for vi > 0.
Since p(·) is strictly decreasing in vi > 0 by Remark 2, for vi0 > vi > 0,
∫ ∞
∫ ∞
0
f (v; vi )dv = p(vi ) > p(vi ) =f (v; vi0 )dv.
θ̂(vi0 )θ̂(vi )

The continuity of p(·) implies (i) θ̂(vi ) is strictly increasing and continuousin vi > 0.
Note that if p(vi ) > p(vi0 ), θ̂(vi0 ) − θ̂(vi ) > vi0 − vi . (See Figure 2 for an
illustration.) This implies (ii) for vi0 > vi > 0,∫ θ̂(vi0 ) − θ̂(vi ) > vi0 − vi .
∞limv→+∞ p(v) = 0, limv→0 p(v) = 1, and θ̂(vi ) f (v; vi )dv = p(vi ) implies(iii) limv→+∞ θ̂(v) = +∞ and (iv) limv→0 θ̂(v) = −∞.

[Figure 2 about here.]

Proof of Remark 5. Continuity of (8) ∫is obvious. Let v 0 ∫> v > 0. Since∞∞θ̂(vi0 ) − θ̂(vi ) > vi0 − vi by Remark 3, vi f (v; θ̂(vi ))dv < v0 f (v; θ̂(vi0 ))dv.
i
Since x(·) is a weakly increasing function, it implies that∫ ∞∫ ∞x(v)f (v; θ̂(vi ))dv <x(v)f (v; θ̂(vi0 ))dv.
vi0

vi

Thus (8) is strictly increasing in v > 0. When vi → 0, by Remark 3,
θ̂(vi ) → −∞. Since x(v) has an upper bound x, (8) → 0 as vi → 0.

22

Famous dynamic models of investments in projects include Admati and Perry (1991)
and Marx and Matthews (2000). Dynamic models of investment in projects in the global
game framework include, for example, Heidhues and Melissas (2006), Dasgupta (2007),and Barbieri and Mattozzi (2009).

17

Proof of Proposition 1. First, we show that s∗ (·) is a symmetric switchingstrategy constituting a Bayesian Nash equilibrium. Suppose that all agents
except i follow the strategy s∗ (·). By the payoﬀ matrix of this model, if
vi ≤ 0, it is obvious that she has no incentive to donate. We focus on thecase of vi > 0.
Note that for any θ > θ̂(k ∗ ),∫ ∞∫ ∞x(v)f (v; θ)dv >x(v)f (v; θ̂(k ∗ ))dv = c,
(12)
k∗

k∗

and for any θ0 ≤ θ̂(k ∗ ),
∫ ∞
∫ ∞
0
x(v)f (v; θ )dv ≤x(v)f (v; θ̂(k ∗ ))dv = c.
k∗

k∗

(13)

This means that under the assumption of that all agents except i follow
the strategy s∗ (·), if the actual mean of the signals is larger than θ̂(k ∗ ), theproject is a success, and if it is smaller than θ̂(k ∗ ), the project is a failure.
Also note that for any vi > k ∗ ,∫ ∞∫ ∞f (v; vi )dv >f (v; k ∗ )dv = p(k ∗ ),(14)
θ̂(k∗ )

and for any vi0 ≤ k ∗ ,∫ ∞θ̂(k∗ )

θ̂(k∗ )

f (v; vi0 )dx ≤∫ ∞θ̂(k∗ )

f (v; k ∗ )dv = p(k ∗ ).

(15)

This means that i with a signal vi > k ∗ forecasts Pr(θ > θ̂(k ∗ )) > p(k ∗ ),and that with a signal vi0 ≤ k ∗ forecasts Pr(θ ≤ θ̂(k ∗ )) ≤ p(k ∗ ).
Thus, (12) and (14) together imply that an agent i with signal vi > k ∗has an incentive to donate, and (13) and (15) together imply that an agent
i with signal vi0 ≤ k ∗ has no incentive to donate.
Therefore, s∗ (·) surely constitutes an Bayesian Nash equilibrium.
Next, we show the uniqueness of s∗ (·) under the assumption that strategies have symmetric and switching properties.
First, suppose, on the contrary, that a symmetric switching strategys(·) with cutoﬀ point k >∫ k ∗ constitutes a Bayesian Nash equilibrium. In∞
this case, by Remark 5, k x(v)f (v; θ̂(k))dv > c. Thus, by that θ̂(·) is
continuous and Remark 4, for a small ² > 0,
∫ ∞x(v)f (v; θ̂(k − ²))dv > c.
(16)kThis means that if the realized mean of signals θ ≥ θ̂(k − ²) and any agent
with a signal larger than k donates, the project is a success. Note that
∫ ∞
∫ ∞
²
f (v; k − )dv >f (v; k − ²)dv
(17)
2
θ̂(k−²)θ̂(k−²)18

= p(k − ²)²
> p(k − ).
2(by the deﬁnition of θ̂(·))(by Remark 2)

This means that an agent with signal k − 2² predicts Pr(θ ≥ k −²) > p(k − 2² ).
Therefore, (16) and (17) together imply that an agent with signal vi = k − 2²
has an incentive to donate. This is a contradiction.
Second, suppose, on the contrary, that a symmetric switching strategys(·) with a cutoﬀ point k < k ∗ constitutes
a Bayesian Nash equilibrium.
∫∞
In this case, by Remark 5, k x(v)f (v; θ̂(k))dv < c. Thus, by Remark
4, for a small ² > 0,
∫ ∞x(v)f (v; θ̂(k + ²))dv < c.
(18)kThis means that if the realized mean of the signals θ ≤ θ̂(k + ²) and any
agent with a signal smaller than k does not donate, the project is a failure.
Note that∫ ∞∫ ∞²f (v; k + )dv <f (v; k + ²)dv
(19)
2
θ̂(k+²)θ̂(k+²)= p(k + ²)²
< p(k + ).
2

(by the deﬁnition of θ̂(·))(by Remark 2)

This means that an agent with signal k + 2² predicts Pr(θ ≥ k +²) < p(k + 2² ).
Therefore, (18) and (19) together imply that an agent with signal k + 2² has
no incentive to donate. This is a contradiction.
By the above two cases, we have that s∗ (·) with cutoﬀ point k ∗ is aunique symmetric switching strategy for a Bayesian Nash equilibrium.
Proof of Proposition 2. The proof is given in the paragraph after this proposition.
Proof of Proposition 3. The proof is given in the paragraph after this proposition.
Proof of Remark 6. Let a small ² > 0. Then, from Corollary 1, we have
∫ ∞x(v)f (v; θ̂(k ∗ (`)))dv = c − `(20)k∗ (`)and∫ ∞k∗ (`+²)x(v)f (v; θ̂(k ∗ (` + ²)))dv = c − (` + ²).

(21)

By subtracting both sides of (21) from those of (20), we have
∫ ∞
∫ ∞
∗
x(v)f (v; θ̂(k (`)))dv −x(v)f (v; θ̂(k ∗ (` + ²)))dv = ² > 0. (22)k∗ (`)k∗ (`+²)

19Remark 5 and (22) imply k ∗ (`) > k ∗ (` + ²). Hence, k ∗ (`) is strictly
decreasing and continuous in `.
Proof of Proposition 4. Note
∫ ∞ that the proportion of agents who decide to
donate is represented by k∗ (`) f (v; θ)dv and the total amount of donations∫∞
is given by k∗ (`) x(v)f (v; θ)dv. Since both are strictly decreasing in k ∗ (`),
Remark 6 induces this proposition.
Proof of Proposition 5. Let ` ≥ `∗ . Remark 6 implies that k ∗ (`) ≤ k ∗ (`∗ ).
Thus we have θ̂(k ∗ (`)) ≤ θ̂(k ∗ (`∗ )) = θ. Since Corollary 1 states that
∫ ∞x(v)f (v; θ̂(k ∗ (`)))dv = c − `,k∗ (`)

θ̂(k ∗ (`)) ≤ θ implies that∫ ∞k∗ (`)

x(v)f (v; θ)dv ≥ c − `.

This occasions
project success. Let `0 < `∗ . Then the symmetric reasoning∫∞
brings k∗ (`0 ) x(v)f (v; θ)dv < c − `0 . This induces project failure.

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20

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22

Table 1: a Result of the Field Experiment by List and Lucking-Reiley (2002)
A. Experimental Design
Number of solicitations mailed
Seed money (%)
Seed money ($)
B. Result
Number of contributions
Participation rate
Total contributions
Mean amount given
Standard error of mean amount

500
10%
$300

500
33%
$1,000500
67%$2,000

17
3.40%
$202
$11.88
$2.27

33
6.60%
$805
$24.39
$2.50

42
8.40%
$1,485$35.36$2.26

Source: Table 1 of List and Lucking-Reiley (2002)

23

Table 2: the Results of Example 1
θ
5
2.5
0
–2.5

proportion of donors
0.88
0.63
0.31
0.09

total donation
8.78
6.31
3.09
0.91

success/failuressff

Table 3: the Results of Example 2
`
0
0.5
1
1.5
2

k∗
1.50
1.22
1.00
0.82
0.67

θ̂(k ∗ )
2.26
1.60
1.00
0.44
–0.09

proportion of donors
0.31
0.34
0.37
0.39
0.41

24

total donation
3.09
3.42
3.69
3.99
4.12

success/failure
f
f
f
f
s

Figure 1: The relationship between vi , p(vi ), and θ̂(vi )Figure 2: Illustration for the proof of Remark 3

Supplementary Note for “A Coordination Game
Model of Charitable Giving and Seed Money Effect”
Kentaro Hatsumi∗
This Version: May 26, 2011

In this supplementary note, (I) we first show that an alternative payoff
structure that captures a refund policy instead of that in Payoff Table (3)is available with the same consequences. (II) Next, we show that withadditional two assumptions, the equilibrium in Proposition 1 can be derived
as a unique strategy profile surviving the iterated elimination of interim
strictly dominated strategies.
(I) Assume that the payoff table isDonateNot Donate

Successvi · x(vi )0Failure−r · x(vi )0(23)where r ∈ (0, 1]. r represents loss ratio when a refund is allowed. r > 0means that even if 100% refund is possible, disutility occurs bacause of waistof time or negative emotional influence.
To derive the equivalent result to Proposition 1, it is sufficient to show
the equivalent result to Remark 2 because this is the key to the uniquenessof the equilibrium.
Given Payoff Table (23), an agent i with a positive signal vi > 0 chooses
to donate if
Pr(Success)(vi · x(vi )) + (1 − Pr(Success)(−r · x(vi )) > 0r⇐⇒ Pr(Success) >.
vi + rLet a function p : R+ → R++ be such that p(vi ) = vir+r . Then obviously
the following remark holds.
Remark 2’. p(·) is strictly decreasing and continuous in vi ≥ 0.
∗

Corresponding to: Institute of Economic Research, Kyoto University. Yoshida,
Sakyoku, Kyoto 606-8501, Japan. Email: kentaro.hatsumi at fx5.ecs.kyoto-u.ac.jp

1Because of Remark 2’, for this alternative payoff table, we can derive theequivalent results to Remarks 3 to 5 and Proposition 1 by using the same
proofs.
(II) Next, we show that the strategy s∗ (·) defined in Proposition 1 constitutes a unique strategy profile surviving the iterated elimination of interim
strictly dominated strategies. Instead of the assumption that symmetry andswitching properties of agents’ strategies are common knowledge, we need
two additional assumptions.
We require a similar property to Remark 1 that for any agent i with asufficiently large signal vi , Donate is the strictly dominant action.
Remember that we assume the number of agents to be large and approximated by a continuum of agents with length 1. Therefore, an agent
regards her contribution itself as zero even if she donates a positive amountof money. However, an agent with so large a signal that her own contribution
covers the whole project cost C (i.e., an agent i with signal vi > 0 such thatx(vi ) > C) may not consider her contribution to be zero. Note that it is not
the normalized cost c but rather the original C. As her actual contribution
covers the whole cost C, she has no reason to fear project failure.
Assumption 1 below represents the above discussion. Let V ∈ R++ besuch that x(V ) = C.
Assumption 1. For any agent i with vi > V , Donate is the strictly dominant action.23
We add one more assumption related to Assumption 1.
R∞
Assumption 2. V x(v)f (v; θ̂(V ))dv > cAssumption 2 states that if the mean of signals is θ̂(V ) and agents withsignals more than V donate, the total donation from agents with signals
larger than V exceeds c and the project is a success. As already discussed
when we introduced Assumption 1, the signal V is so large that x(V ) =C. Given that θ̂(V ) is also large in accordance with V , Assumption 2 is
supported.
Then we have the following proposition.
Proposition 6. In a game G with Assumptions 1 and 2, s∗ (·) defined in
Proposition 1 is a unique strategy constituting the strategy profile surviving
the iterated elimination of interim strictly dominated strategies.
Proof of Proposition 6. Proposition 2.1 of Morris and Shin (2003) provides
the equilibrium of a general global game where the (ex-post) payoff of anagent depends on her action, the proportion of other agents’ strategies,
23

It is important that an agent with a large enough signal has the strictly dominantaction to donate. The value V itself is not essential.

2and her own signal. Even though in our game G, the payoff of an agent
depends on her action, the total amount of donation, and her own signal,
our Proposition 6 is essentially the same as their Proposition 2.1. We owe
Morris and Shin (2003) and originally Carlsson and van Damme (1993) their
proof techniques.
First, we construct a sequence of recursive values v 0 , v 1 , v 2 , · · · such thatfor agents with signals larger than each value, Not Donate is the interim
strictly dominated action.
By Assumption 1, for any agent i with vi > V , Donate is the strictly
dominant action, and Not Donate is strictly dominated. Let v 0 = V .
Let t ∈ N ∪ {0}. Assume as an induction hypothesis that for any agent iwith vi > v t , Not Donate is the interim strictly dominated action, and theychoose to Donate. Let v t+1 be such thatZ ∞x(v)f (v; θ̂(v t+1 ))dv = c.
(24)vt

Suppose, at first, that all the other agents with signals smaller than v tchoose Not Donate. Then, (24) represents that if the mean signal is θ̂(v t+1 ),the total amount of donation equals c. Thus if the mean signal is larger
than θ̂(v t+1 ), the project is a success.
Note that for an agent i with vi > v t+1 , Pr(θ ≥ θ̂(v t+1 )) > Pr(θ ≥θ̂(vi )) = p(vi ) since θ̂(vi ) > θ̂(v t+1 ). This means that from i’s point of view,
the probability of success is larger than p(vi ), and Donate is more profitable
than Not Donate.
Notice that if we drop the supposition that all other agents with signals
smaller than v t choose Not Donate, the probability of success for i withvi > v t+1 is higher than that with the supposition. Thus, for i with vi > v t+1 ,Not Donate is the interim strictly dominated action.
Next, we show that k ∗ < v t+1 < v t ≤ V .
First, notice k ∗ < V = v 0 by Assumption 2 and Remark
R ∞5.
Assume as an induction hypothesis that k ∗ < v t . Then k∗ x(v)f (v; θ̂(k ∗ ))dv =c by the definition of k ∗ and Remark 5 imply thatZ ∞x(v)f (v; θ̂(v t ))dv > c.
(25)
vt

(24),
4 imply that v t > v t+1 . Similarly, since k ∗ < v t ,R ∞ (25) and Remark∗
t+1 > k ∗ . Hencek∗ x(v)f (v; θ̂(k ))dv = c, (24) and Remark 4 imply v
k ∗ < v t+1 < v t ≤ V .
By recursive means, we have that the sequence v 0 , v 1 , v 2 , · · · is weakly
decreasing and has an lower bound k ∗ . Thus it has the limit value. Let v
be the limit value. (i.e., limt→+∞ v t R= v.) Since (7) is continuous in both∞k and vi by Remark 4, we have that v x(v)f (v; θ̂(v))dv = c. This implies
v = k∗ .
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(B) Similarly, we can construct a sequence of recursive values v 0 , v 1 , v 2 , · · ·such that for agents with signals smaller than each value, Donate is the interim strictly dominated action.
Let v 0 = 0. Given t ∈ N ∪ {0}, let v t+1 be such thatZ ∞x(v)f (v; θ̂(v t+1 ))dv = c.
vt

Then, Remark 1 and the symmetric reasoning of (A) guarantee that foragents with signals smaller than each value, Donate is the interim strictly
dominated action.
Similarly to (A), we have that 0 ≤ v t < v t+1 < k ∗ for any t ∈ N ∪ {0},and limt→+∞ v t = k ∗ .
By discussions (A) and (B), we have the statement of Proposition 6.

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