GCOE Discussion Paper Series
Global COE Program

Human Behavior and Socioeconomic Dynamics
Discussion Paper No.50

A Coordination Game Model of Charitable Giving
and Seed Money EffectKentaro Hatsumi

May 2009; revised September 2009GCOE Secretariat
Graduate School of Economics

OSAKA UNIVERSITY

1-7 Machikaneyama, Toyonaka, Osaka, 560-0043, Japan

A Coordination Game Model of Charitable Giving
and Seed Money Eﬀect∗
Kentaro Hatsumi†First Version: December 24, 2007
This Version: September 15, 2009

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 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 Capital Campaign.
Journal of Political Economy 110 (1), 215-233].
Keywords: charity, seed money, global game, coordination game, excludable threshold public goodJEL Classiﬁcation Numbers: C72, D82, H41, L31∗
I would like to thank Atsushi Kajii and Shigehiro Serizawa for their helpful suggestions. I am also grateful to Tadashi Sekiguchi, Daisuke Oyama, Kazuhiko Hashimoto,
Shuhei Morimoto, as well as other participants at the 2009 Econometric Society Far Eastand South Asia Meeting, the 2009 Japanese Economic Association Spring Meeting, the2008 Kyoto Game Theory Workshop and seminars at Osaka University for their helpfulcomments 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 Social and Economic Research, Osaka University, 6-1,
Mihogaoka, Ibaraki, 567-0047, Japan. Email: hatsumi at iser.osaka-u.ac.jp1

1

Introduction

1.1

Charitable Giving and Coordination GameWhy should economists study charity? We present two responses to thisquestion. The ﬁrst is the considerable impact of charitable behavior on
the economy. For example, the Giving USA Foundation (2006) estimatesthat in the United States, total charitable donations in 2005 were $260.28
billion, of which $199.07 billion (76.5% of the total) comprised individual
contributions. The size of these donations constitutes an important topic ofresearch. The second is that charity is one of areas for which economists still
do not possess a suitable theory. Andreoni (2006b) argues “philanthropy
is one of the greatest puzzles for economics.” Mainstream economics has
assumed that people are extremely selﬁsh and only interested in their selfbeneﬁcial consumptions. Economists have constructed models based on thisassumption. However, the assumption of selﬁshness is incompatible withcharitable behavior. Even today, the ﬁeld of philanthropy remains a puzzle
and requires focused scholarly attention.
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.1 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).2 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.3 A campaign
to collect funds to build a hospital is a case in point. If the ﬁxed cost forthe project success is very large and there are many potential contributors,
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
1

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

2

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, thedonations 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.4 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 thetype distribution, and they show the existence of a unique equilibrium withfew assumptions concerning the strategies of agents. Even in the standard
Bayesian coordination game where the type distribution is common knowledge, a unique equilibrium 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
4

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.

3

of diﬀerent directions. Thanks to Morris and Shin (1998, 2003), it has now
been extended to encompass a coordination problem in a large economy.5
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.6 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.7 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.8 To the ﬁrst group, they announce that the5Well-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.
6
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.
7
Other empirical studies on seed money include List and Rondeau (2003) and Potters
et al. (2005).
8
The other three groups of potential donors are assigned to an experiment concerning4

seed 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,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
that the proportion of 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. Merely because of thisshift, for at least some proportion of potential donors, the expected payoﬀ
by donating becomes greater than when not donating. This behavior can beobserved only under global game setting of incomplete information. In thestandard Bayesian game where the type distribution is common knowledge,
all potential donors often together decide to donate or not to donate in an
equilibrium. This is inconsistent with the above empirical data by List andLucking-Reiley (2002).
While our model is the ﬁrst to capture the feature that the threshold shiftby seed money causes a continuous increase in the proportion of donors andthe total amount of donations, there is a considerable theoretical literatureon the eﬀect of seed money. Andreoni (1998) considers the seed money asa contribution from leaders, and He shows that the voluntary contribution
from leaders cancels the zero-contribution equilibrium and leaves only theequilibrium with a positive amount of contributions. His discussion bases
on a complete information game suitable to charitable giving among a small
number of potential donors, while we construct a game with a large numberof potential donors under incomplete information. Vesterlund (2003) andAndreoni (2006a) direct their attention to the quality signal eﬀect of seedmoney. They explain that seed money donated by leaders indicates the highquality of the charity and therefore increases contributions from remaining
donors. If there is a quality signal eﬀect from seed money, it is easy toderive the strict increase in donations corresponding to the amount of seedrefunds.

5money.9 In contrast, our model shows that even without the quality signal
eﬀect, we can derive the positive eﬀect of seed money. Indeed, the threshold
shift eﬀect and the quality signal eﬀect are not contradictory but rather arecomplementary in increasing total contributions. Our study is meaningfuleven if we take the quality signal eﬀect of seed money into account.10

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 the conditions for the eﬃciency and success 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 the threshold shift eﬀect,
and we observe the impact on the behavior of potential donors. In Subsection 3.2, we calculate the minimal seed money needed to achieve eﬃciency
and 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 nonconvex technology. There exists a threshold level of its cost C ∈R++ , and if the total provision from agents exceeds the threshold C, thenthe quality of the project is signiﬁcantly improved. Imagine, for example,
a charitable campaign to promote child health care in a rural area of a9

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.
10
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

developing country. If the charity collects suﬃcient donations to build ahospital, the quality of the project increases substantially. Refer to theproject as a success (in the ex-post sense) if the donations collected fromagents are suﬃcient to cover C, and a failure (in the ex-post sense) if thetotal donations are less than C.
We consider cases where the number of agents are suﬃciently large.11
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.12
Let vi ∈ R be the signal of the project value for an agent i in the caseof project success (ex-post sense). vi > 0 means that i believes that the
project value is positive and that it is worth donating a positive amount ofmoney to if it succeeds. vi ≤ 0 means that she thinks that the project is not
worthwhile and that donating money to it would be wasteful. In the case ofvi ≤ 0, consider that the absolute value |vi | represents the degree of dislike
for the project.
Assume that if vi > 0, she donates x(vi ) when she makes a decision
Donate, where x : R+ → R+ is the same function among all agents, andpossesses the properties of twice diﬀerentiability, and for all v > 0, 0 <x(v) < v, x0 (v) > 0 and x00 (v) ≤ 0 (i.e., x(·) is a weakly concave function).
Furthermore, assume that limv→0 x(v) = 0, limv→+∞ x(v) = +∞ and there
exists α ∈ (0, 1) such that limv→0 x0 (v) = α. Note that v(·) represents awide class of functions including a linear function x : R+ → R+ such thatfor all v > 0, x(v) = αv.
Moreover, assume that if vi ≤ 0, she donates ² > 0 if she decides to
Donate. In the case of vi ≤ 0, it is natural to consider that in equilibria,
she never decides to donate. The amount ² > 0 is a technical assumption.
We assume throughout the paper that if the project is a failure, any agent
i believes that in the ex-post, the project is not attractive and the amount11

In each ﬁeld experiment in List and Lucking-Reiley (2002), the number of potential
donors (people who received solicitation letters) was 500.
12
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

of donation with a minus sign itself is her payoﬀ.
Also assume that any agent has zero payoﬀ if she does not donate, regardless of whether the project is a success or a failure. Because of thewarm-glow property discussed earlier, when the project is a success, thepayoﬀ not to donate is smaller than that of donating. Thus we adopt theassumption of a zero payoﬀ without donation for simplicity. Note that evenif we allow some positive payoﬀ for an agent who does not donate when
the project is a success, the structure of the following investigation and theconsequences are unchanged.13
The tables below summarize the payoﬀs.
If vi > 0,

DonateNot Donate

Successvi − x(vi )0Failure−x(vi )
0

(3)

DonateNot Donate

Successvi − ²0

Failure−²
0(4)

If vi ≤ 0,

Based on these ex-post payoﬀ tables, each agent is assumed to calculate her
own von Neumann-Morgenstern expected interim payoﬀ, and the to decide
whether to donate.
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.14 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, each
13
The following setting is also available. If the project is a success, an agent i with vi > 0
who does not donate receives a payoﬀ b(vi ), where b : R+ → R+ is a diﬀerentiable function
such that for all v > 0, 0 ≤ b(v) ≤ v − x(v), b0 (v) ≥ 0, {b(v)/x(v)}0 ≤ 0, and there existsβ ∈ (0, 1) with limv→0 b0 (v) = β. A detailed explanation is given in the SupplementaryNote.
14
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.

8

agent 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.15 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.
In comparison with a standard Bayesian game wherein the signal distribution is common knowledge among agents, the assumption that the posterior belief of signal distribution for each agent is slightly diﬀerent from thoseof others is quite natural. This is because of the following reasons. First, ina 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.
In the next subsection, we consider the equilibrium in this game.

2.2

The Equilibrium

We are interested in Bayesian Nash equilibria of the game deﬁned in the previous subsection. If possible, a unique equilibrium is preferable especiallyfor the purpose of comparative statics. The global game analysis makes it
possible to provide a unique equilibrium with a few natural additional assumptions to the strategies of agents. Furthermore, it is the strategy proﬁlesurviving the iterated elimination of interim strictly dominated strategies.
Since this concept does not rely on the assumption that the strategies ofother agents are common knowledge, it is a weaker condition than BayesianNash equilibrium.
15

See Morris and Shin (2003) for a detailed explanation. They discuss the improper
uniform prior belief with respect to the philosophy of Laplace.

9

In this subsection, we characterize a unique strategy proﬁle by the iterated elimination of interim strictly dominated strategies as the equilibrium.
The equilibrium strategy possesses two interesting properties: symmetry
and switching properties. A strategy is symmetric if all agents follow thesame strategy. A strategy is a switching strategy around some cutoﬀ pointk ∈ R if there exists a unique cutoﬀ point k such that if an agent’s receiving
signal vi is above k, she donates; otherwise, she does not donate. A symmetric switching strategy s : R → {Donate, Not Donate} is formally written as
follows.
{
Donateif vi > ks(vi ) =Not Donate if vi ≤ kNote that the symmetry and switching properties of the equilibrium are notassumptions but are obtained endogenously.
We introduce several functions that play important roles in the following
investigation.
∫∞f (v; θ)dv

(5)

x(v)f (v; θ)dv

(6)

k
∫ ∞
k

(5) 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. (6) represents the total donation when the mean of the signals
is θ and any agent receiving a signal larger than the cutoﬀ point k decides
to donate. (6) 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, and it has an important role in the global game analysis. We requirea similar property that for any agent i with a suﬃciently 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, it is not natural that an agent with so large a signalthat her own contribution covers the whole project cost C (i.e., an agent iwith signal vi > 0 such that x(vi ) > C) would 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 tofear project failure.
10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 dominantaction.16
We add one more assumption related to Assumption 1.
∫∞
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 , it is also a natural
assumption.
Assumptions 1 and 2 may appear somewhat outlandish. The existenceof an agent i with a high signal vi > C may be especially diﬃcult to accept.
However, we note that the model relies very little on these assumptions in
two ways.
First, under the normal distribution of signals, the contribution of agentswith signals larger than C is actually quite small. In fact, it is almost zeroand relatively unimportant. What is important is not the existence of anagent i with such a high signal vi > C but that of an agent j with vj < vi
who believes that an i with such a high signal vi > C exists. Alternatively,
instead of j, we only need an agent h with vh < vj who thinks such j exists.
This type of expectation of a certain proportion of existing agents is the key
to deriving the equilibrium in the global game analysis.
Second, if we assume that any agent takes a symmetric threshold strategy
and this is common knowledge, then we can derive the same equilibrium as
an interim Bayesian Nash equilibrium without Assumptions 1 and 2. See
the Supplementary Note for more details.
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 )).

(7)

Thus, agent i decides to donate if(7) > 0⇐⇒ Pr(Success) >x(vi ).
vi

Since limvi →0 x(vi ) = 0 and limvi →0 x0 (vi ) = α, we have
lim

vi →0

x(vi )=αvi(by l’Hospital’s rule.)

16

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

11Let a function p : R+ → R++ be such that{x(vi )
if vi > 0vip(vi ) =α
if vi = 0.
p(vi ) represents the threshold probability such that i with vi > 0 considers
that if the probability of project success is larger than p(vi ), Donate leads
to a higher expected payoﬀ than Not Donate. On the other hand, if theprobability of project success is less than p(vi ), Not Donate guarantees ahigher expected payoﬀ than Donate. The reason that p(0) is deﬁned as thelimit value at vi → 0 is merely to simplify the following descriptions. Sincex(·) is weakly concave and diﬀerentiable in vi > 0, we have the remark
below.
Remark 2. p(·) is weakly 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 )). Note that since p(0) = α ∈ (0, 1), θ̂(0) ∈ R. Since p(·) is
weakly decreasing in vi ≥ 0, we have the remark below.
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) = +∞.
We introduce two more functions and discuss their properties.
∫ ∞x(v)f (v; θ̂(vi ))dv(8)k(8) is a special case of (6), and the mean of signals is θ̂(vi ). Since θ̂(vi ) isstrictly increasing in vi ≥ 0, the following remark is straightforward.
Remark 4. (8) is strictly increasing and continuous in vi ≥ 0 and strictly
decreasing and continuous in k ≥ 0.
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

12

∫ ∞
x(v)f (v; θ̂(vi ))dv

(9)

vi

(9) is a special case of (8). The mean of the signals is θ̂(vi ) and the cutoﬀ
point k = vi . Remark 3 induces the next remark.
Remark 5. (9) is strictly increasing and continuous in vi ≥ 0.
Let G denote the Bayesian type of incomplete information game deﬁned
in Subsection 2.1 with Assumption 1 and 2. Now, we characterize a cutoﬀ
point and the equilibrium. Given a threshold cost c, let{∫∞
0 if 0 x(v)f (v; θ̂(0))dv ≥ c∗k =κ otherwise,
where κ ∈ R++ satisﬁes∫ ∞x(v)f (v; θ̂(κ))dv = c.
κNote that since (9) diverges to inﬁnity when v∫i goes to inﬁnity (i.e., (9) →∞+∞ (vi → +∞)), κ is uniquely determined if 0 x(v)f (v; θ̂(0))dv < c.
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∗ (·) constitutes a unique strategy proﬁle surviving theiterated elimination of interim strictly dominated strategies.
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 the proposition.
Example 1. Let a threshold level of cost c = 6.18 Let the standard deviationof the signal distribution σ = 10. Let for all vi > 0, x(vi ) = 21 vi . Then, forall vi > 0, p(vi ) = 21 and θ̂(vi ) = vi . We calculate k ∗ = 4.02115 and have
Table 2.

[Table 2 about here.]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.

13

Note that the proportion of actual contributors is represented by∫ ∞f (v; θ)dvk∗and the total amount of donation is represented by∫ ∞x(v)f (v; θ)dv.
k∗

(10)

(11)

The mean amount of donation from actual contributors is (11)/(10), andthe project is a success if (11) is larger than c.
As in Example 1, the equilibrium cutoﬀ point k ∗ is often larger than 0.
This is because of a particular assumption of the global game that each agent
possesses a diﬀerent belief about the signal distribution. Under a standard
Bayesian 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, the standard Bayesian game analysis provides this in an equilibrium; if
θ > 4.02115, agents with signals larger than 0 together decide to donate, andif θ ≤ 4.02115, any agent does not donate and the total amount of donation
is equal 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.

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 action of
agentsderives 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 easily achievable in a equilibrium of the standard
Bayesian Game discussed in the last paragraph of the previous subsection.
However, it is more diﬃcult to obtain it under the more realistic global gamesetting. In a game G, the following proposition holds.
∫∞
Proposition 2. A project is eﬃcient if and only if k ∗ = 0 and 0 x(v)f (v; θ)dv ≥c.
Actually, this is equivalent to the condition for success when k ∗ = 0.
The intuition for this proposition is quite simple. Even though the projectis a failure, some agents with signals larger than k ∗ choose Donate. This14

implies that the project is not eﬃcient. If the project is a success andk ∗ > 0, an agent i with 0 < vi < k ∗ chooses Not Donate, which brings
about ineﬃciency. Therefore, the condition that occasions eﬃciency is only
described in Proposition 2.
As already mentioned in the previous subsection, a project is a successif∫ ∞x(v)f (v; θ)dv ≥ c.
(12)k∗This may be the most important matter for a charity fund-raiser. Note that
if a project is eﬃcient, then the project is a success. Besides, even in thecase of k ∗ > 0, the project is a success as the proposition below states.
Proposition 3. A project is a success if and only if it is eﬃcient or θ ≥θ̂(k ∗ ).
∫∞∫∞If k ∗ > 0, k∗ x(v)f (v; θ̂(k ∗ ))dv = c. Thus θ ≥ θ̂(k ∗ ) induces k∗ x(v)f (v; θ)dv ≥c and the project succeeds.
In the next section, we introduce seed money. The main purpose for
introducing seed money is to obtain the success or eﬃciency of a project.
We show how to calculate the minimal amount of seed money for both
eﬃciency and success.

3

Seed Money

We exogenously introduce seed money. We see the threshold shift eﬀectof the seed money on the donation behavior of agents in Subsection 3.1.
We also see how to obtain the success and the eﬃciency of the project byannouncing 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 − `. We
also rewrite the additional assumptions in Subsection 2.2 as follows. Let
V (`) ∈ R++ be such that x(V (`)) = C − L.
Assumption 1’. For any agent i with vi > V (`), Donate is the strictly
dominant action.
∫∞
Assumption 2’. V (`) x(v)f (v; θ̂(V (`)))dv ≥ c − `.
15

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 − ` and Assumptions 1’ and 2’. Then we derive a unique equilibrium strategy s∗ (·; `) in agame G(`) as a corollary of Proposition 1.
Corollary 1. In a game G(`), a symmetric switching strategy s∗ (·; `) suchthat{
Donateif vi > k ∗ (`)
s∗ (vi ; `) =Not Donate if vi ≤ k ∗ (`),
where

{
0
k (`) =
κ(`)∗∫∞
if 0 x(v)f (v; θ̂(0))dv ≥ c − `otherwise,and κ(`) ∈ R++ is such that
∫ ∞x(v)f (v; θ̂(κ(`)))dv = c − `κ(`)

constitutes a unique strategy proﬁle surviving the iterated elimination of interim strictly dominated strategies.
∫∞
Hereafter, we consider the case of 0 x(v)f (v; θ̂(0))dv < c. Otherwise,
not only success but also eﬃciency is obtained without seed
∫ ∞ money, and theintroduction of seed money is then redundant. Let ` = c− 0 x(v)f (v; θ̂(0))dv.
According to Corollary 1, we have the remark below.
Remark 6. k ∗ (`) is strictly decreasing and continuous in ` < `, and k ∗ (`0 ) = 0
for `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 as long as ` < `.
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 = 10, and for all vi > 0,x(vi ) = 21 vi . Consequently, we have the results in Table 3.
[Table 3 about here.]

16

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.

3.2

Minimal Seed Money for Eﬃciency and 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. Especially fromthe viewpoint of a charity fund-raiser, the attainment of project success is
crucial. Even though we do not explicitly consider the cost of introducing
seed money, charity fund-raisers actually require a considerable eﬀort tocollect seed money and are better oﬀ if the amount of seed money neededfor eﬃciency or success is as small as possible. In this subsection, we derive
the minimal amount of seed money to attain each of eﬃciency and success.
First, consider the minimal seed money for eﬃciency.
∫ ∞ Note that without
seed money, the project is not eﬃcient if k ∗ (0) > 0 or 0 x(v)f (v; θ)dv < c.
In this case, we have the following proposition.
Proposition 5. The minimal amount of seed money for eﬃciency ` is such
that
{
∫∞
c − 0 x(v)f (v; θ̂(0))dv if θ ≥ θ̂(0)∫∞`=c − 0 x(v)f (v; θ)dv
otherwise.
This proposition is straightforward from Proposition 2. If θ ≥ θ̂(0), theamount of seed money to guarantee that agents with signals larger than 0choose Donate is suﬃcient. If θ < θ̂(0), we need enough seed money ` sothat the shifted threshold∫ ∞ c − ` is weakly smaller than the possible total
amount (i.e., c − ` ≤ 0 x(v)f (v; θ)dv).
Next, we consider the minimal seed money for success. The project is
not a success without seed money if it is not eﬃcient and θ < θ̂(k ∗ (0)). In
this case, in order to achieve success, we have the following proposition.
Proposition 6. The minimal amount of seed money for success `∗ is such
∗
∗ ∗
∗
that
∫ ∞ if θ̂(0) ≤ θ < θ̂(k (0)), θ̂(k (` )) = θ, and if θ < θ̂(0), ` = c −0 x(v)f (v; θ)dv.
Note that from Remarks 3 and 6, in the case of θ̂(0) ≤ θ < θ̂(k ∗ (0)),θ̂(k ∗ (`)) is continuously and strictly decreasing in ` ∈ [0, `]. Thus `∗ is17

uniquely determined. In this case, the minimal seed money for success isstrictly smaller than that for eﬃciency, and this suits our intuition. In
Example 2, in the case of θ = 0, both ` = `∗ = 2.01058. However, in thecase of θ = 2, ` = 2.01058 and `∗ = 1.01058.
To calculate the minimal seed money for eﬃciency and success, a charity
fund-raiser needs to know the actual mean of the distribution θ. Indeed, thecollection of the actual signal distribution is quite diﬃcult for a charity
fund-raiser. It involves a considerable cost for preliminary surveys on thepotential 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
strictly 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.
In the previous section, Example 2 demonstrates that the per-capita
amount of donation is strictly decreasing with the amount of seed money.
The mechanism is quite simple. We assume that the amount of donation
x(vi ) of agent i with signal vi is exogenously given and never changes tothe amount of seed money. From Remark 6, if the amount of seed money
increases, the threshold signal k ∗ (`) decreases. Note that an agent whochanges her action from Not Donate to Donate because of the increase in
seed money has a smaller signal and a smaller donation than agents who continue to choose to Donate. This induces the per-capita amount of donation
to decrease.
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 eﬀect18

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.
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, andpossibly represented by dynamic games. Even in dynamic models, our viewthat charitable donation is indeed a coordination game plays an importantrole.19 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 and 4 since they are straightforward.
Proof of Remark 2. The continuity and diﬀerentiability of p(·) are obvious.
Note that for vi > 0,p0 (vi ) =−x(vi ) + vi x0 (vi )
.
vi2

19

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 (2008).

19

Since x(·) is weakly concave in vi > 0, −x(vi ) + vi x0 (vi ) ≤ 0. It implies thatfor vi > 0, p0 (vi ) ≤ 0, and p(vi ) is weakly decreasing in vi ≥ 0.
∫∞
Proof of Remark 3. Remember that θ̂(vi ) f (v; vi )dv = p(vi ) for vi > 0.
Since p(·) is weakly 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 )

Thus the continuity of p(·) implies (i) θ̂(vi ) is strictly increasing and continuous in vi ≥ 0.
Note that if p(vi ) = p(vi0 ), θ̂(vi0 ) − θ̂(vi ) = vi0 − vi and if p(vi ) > p(vi0 ),θ̂(vi0 ) − θ̂(vi ) ≥ vi0 − vi . (See Figure 2 for an illustration.) They imply (ii) for
vi0 > vi ≥ 0, θ̂(vi0 ) − θ̂(vi ) ≥ vi0 − vi .
∫∞Since θ̂(0) ∈ R, the construction of θ̂(vi ) as θ̂(vi ) f (v; vi )dv = p(vi )implies (iii) limv→+∞ θ̂(v) = +∞.

[Figure 2 about here.]

0
0
0
Proof
∫ ∞of Remark 5. Let∫ v∞ > v > 0.0 Since θ̂(vi ) − θ̂(vi ) ≥ vi − vi by Remark
3, vi f (v; θ̂(vi ))dv ≤ v0 f (v; θ̂(vi ))dv. Since x(·) is a strictly increasingifunction, it implies that∫ ∞∫ ∞x(v)f (v; θ̂(vi0 ))dv.
x(v)f (v; θ̂(vi ))dv <vi0

vi

We have the statement of the remark.
Proof of Proposition 1. 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,
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 1 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.
∫∞
Case 1: 0 x(v)f (v; θ̂(0))dv < c.
(A) First, we construct a sequence of recursive values v 0 , v 1 , v 2 , · · · suchthat for 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 .
20

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 that
∫ ∞x(v)f (v; θ̂(v t+1 ))dv = c.
(13)vt

Suppose, at ﬁrst, that all the other agents with signals smaller than v tchoose Not Donate. Then, (13) 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 κ ≤ v t+1 ≤ v t ≤ V .
First, notice κ ≤ V = v 0 by Assumption 2 and Remark∫ ∞5.
tAssume as an induction hypothesis that κ ≤ v . Then κ x(v)f (v; θ̂(κ))dv =c by the deﬁnition of κ and Remark 5 imply that
∫ ∞x(v)f (v; θ̂(v t ))dv ≥ c.
(14)
vt(14) and Remark 4 imply that v t ≥ v t+1 . Similarly, since κ ≤ v t ,∫(13),∞
t+1 ≥ κ. Hence κ ≤κ x(v)f (v; θ̂(κ))dv = c, (13) and Remark 4 imply vtt+1v≤v ≤V.
By recursive means, we have that the sequence v 0 , v 1 , v 2 , · · · is weakly
decreasing and has an lower bound κ. Thus it has the limit value. Let v
be the limit value. (i.e., limt→+∞ v t ∫= v.) Since (8) is continuous in both∞k and vi by Remark 4, we have that v x(v)f (v; θ̂(v))dv = c. This impliesv = κ.
(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 that
∫ ∞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.
21

Similarly to (A), we have that 0 ≤ v t < v t+1 < κ for any t ∈ N ∪ {0},and limt→+∞ v t = κ.
By discussions (A) and (B), we have the statement of Case 1.
∫∞
Case 2: 0 x(v)f (v; θ̂(0))dv ≥ c.
(A) We can use the same reasoning as (A) in Case 1 with only one
modiﬁcation. Instead of (13), let{∫∞
0
if vt x(v)f (v; θ̂(0))dv ≥ ct+1
v=
wt+1 otherwise,∫∞
where wt+1 ∈ R+ is such that vt x(v)f (v; θ̂(wt+1 ))dv = c.
Then we have that 0 ≤ v t+1 ≤ v t ≤ V for any t ∈ N ∪ {0}, andlimt→+∞ v t = 0.
(B) Since Remark 1 says that for any agent i with vi < 0, Not Donate
is a strictly dominant action.
By discussions (A) and (B), we have the statement for Case 2.
Proof of Proposition 2. The if part is obvious. The only if part is explainedin the paragraph after this proposition.
Proof of Proposition 3. if part : It is obvious that eﬃciency implies projectsuccess. The explanation for the case of θ ≥ θ̂(k ∗ ) is given in the paragraphafter this proposition. only∫ if part : We show the contraposition.
Suppose∫∞∞θ < θ̂(k ∗ ) and (k ∗ > 0 or 0 x(v)f (v; θ)dv < c). If 0 x(v)f (v; θ)dv < c,∫∞the projectis never a success. If θ < θ̂(k ∗ ) and k ∗ > 0, then k∗ x(v)f (v; θ̂(k ∗ ))dv =∫∞c > k∗ x(v)f (v; θ)dv. This induces project failure.
Proof of Remark 6. In the case of ` ≥ `, k ∗ (`) = 0 is directly induced from
Corollary 1. We consider the case of ` < `. Let a small ² > 0. Then, from
Corollary 1, we have
∫ ∞x(v)f (v; θ̂(k ∗ (`)))dv = c − `(15)k∗ (`)and∫ ∞k∗ (`+²)x(v)f (v; θ̂(k ∗ (` + ²)))dv = c − (` + ²).

(16)

By subtracting both sides of (16) from those of (15), we have
∫ ∞
∫ ∞
∗
x(v)f (v; θ̂(k (`)))dv −x(v)f (v; θ̂(k ∗ (` + ²)))dv = ² > 0. (17)k∗ (`)k∗ (`+²)

Remark 5 and (17) imply k ∗ (`) > k ∗ (` + ²). Hence, k ∗ (`) is strictly
decreasing and continuous in ` < `.
22

Proof of Proposition 4. Let ` < `. Note that, under this ∫condition, 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. Sinceboth are strictly decreasing in k ∗ (`), Remark 6 induces the statement.
of Proposition ∫5. First consider the case of θ ≥ θ̂(0). Let ` ≥ `. Then
∫Proof∞∞x(v)f(v; θ)dv ≥ 0 x(v)f (v; θ̂(0))dv ≥ c − `. This occasions project0
success∫ and k ∗ (`) = 0. Thus it is eﬃcient. On the other hand, let `0 < `.
∞Then 0 x(v)f (v; θ̂(0))dv = c − ` < c − `0 . Then k ∗ (`0 ) > 0 and this brings
about ineﬃciency.
Next,∫ ∞ we consider the other∫ ∞ case, in which θ < θ̂(0). Let ` ≥ `.
Then 0 x(v)f (v; θ̂(0))dv ≥ 0 x(v)f (v; θ)dv ≥ c − `. This occasions
success and k ∗ (`) = 0. Thus it is eﬃcient. Let `0 < `. Then
∫project∞00 x(v)f (v; θ)dv = c − ` < c − ` . This occasions project failure and ineﬃciency.
Proof of Proposition 6. First consider the case of θ̂(0) ≤ θ < θ̂(k ∗ (`∗ )). 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.
For the case of θ < θ̂(0), the proof is exactly the same as for the case ofθ < θ̂(0) in Proposition 5.

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Linear Public Goods Experiments. Journal of Public Economics 61 (3),409-427.
Palfrey, T. R., Prisbrey, J. E., 1997. Anomalous Behavior in Public Goods
Experiments: How Much and Why? American Economic Review 87(5), 829-846.
Palfrey, T. R., Rosenthal, H., 1984. Participation and the Provision of Discrete Public Goods: a Strategic Analysis. Journal of Public Economics
24 (2), 171-193.
Potters, J., Sefton, M., Vesterlund, L., 2005. After You - Endogenous Sequencing in Voluntary Contribution Games. Journal of Public Economics 89 (8), 1399-1419.
Ribar, D. C., Wilhelm, M. O., 2002. Altruistic and Joy-of-giving Motivationsin Charitable Behavior. Journal of Political Economy 110 (2), 425-457.
Romano, R., Yildirim, H., 2001. Why Charities Announce Donations: a
Positive Perspective. Journal of Public Economics 81 (3), 423-447.
Vesterlund, L., 2003. The Informational Value of Sequential Fundraising.
Journal of Public Economics 87 (3-4), 627-657.

25

Table1: Results of the Field Experiment by List and Lucking-Reiley (2002)
A. Experimental Design
Number of solicitations mailed

500

500

500

Seed money (%)

10%

33%

67%

Seed money ($)$300$1,000$2,000173342Participation rate

3.40%

6.60%

8.40%

Total contributions

$202

$805

$1,485

Mean amount given$11.88$24.39$35.36

Standard error of mean amount

$2.27

$2.50

$2.26

B. Result
Number of contributions

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

Table 2: the Results of Example 1
θproportion of donors

total donation

mean amount of
donation

success/failure

50.538988

6.66530

12.3663s

00.343800

3.67958

10.7027f-50.183498

1.73830

9.47316f-100.080441

0.688448.55843f

Table 3: the Results of Example 2
θ

l

k

*

proportion ofdonorstotal donation

2

0
1

4.02115
2.021150.419913
0.499156

4.74859
4.98773

11.30851
9.99233ff

2
30.0211500.578432
0.57926

5.06894
5.06895

8.763248.75073ss

04.02115

0.343800

3.67958

10.7027f

1
2
3

2.02115
0.0211500.419913
0.4991560.500000

3.90876
3.989413.989429.30850
7.992317.97884ffs

0mean amountsuccess /failureof donation

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: September 15, 2009

In this supplementary note, (i) we ﬁrst show that some alternative payoﬀ
structures other than that in Payoﬀ Tables (3) and (4) are available withthe same consequences. (ii) Next, we show that without Assumptions 1 and
2, the equilibrium in Proposition 1 can be derived as the unique BayesianNash equilibrium under the symmetric switching strategy assumption.
(i) In the case of vi > 0, instead of Payoﬀ Table (3), we assume that the
payoﬀ table isDonateNot Donate

Successvi − x(vi )b(vi )

Failure−x(vi )
0(18)

where b : R+ → R+ is a diﬀerentiable function such that for all v > 0,0 ≤ b(v) ≤ v − x(v), b0 (v) ≥ 0, {b(v)/x(v)}0 ≤ 0, and there exists β ∈ (0, 1)
with limv→0 b0 (v) = β.
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 (18), an agent i with a positive signal vi > 0 chooses
to donate if
Pr(Success)(vi − x(vi )) + (1 − Pr(Success)(−x(vi )) > Pr(Success)b(vi )
x(vi ).
⇐⇒ Pr(Success) >vi − b(vi )Since limvi →0 x(vi ) = 0, limvi →0 vi − b(vi ) = 0, limvi →0 x0 (vi ) = α andlimv→0 b0 (v) = β, we havelimx(vi )

vi →0 vi − b(vi )

=

α
1−β

(by l’Hospital’s rule.)∗

Corresponding to: Institute of Social and Economic Research, Osaka University. 6-1,
Mihogaoka, Ibaraki, 567-0047, Japan. Email: hatsumi at iser.osaka-u.ac.jp

1

Let a function p : R+ → R++ be such that
{ x(v )ip(vi ) =vi −b(vi )α1−βif vi > 0
if vi = 0.

Remark 2’. p(·) is weakly decreasing and continuous in vi ≥ 0.
Proof of Remark 2’. The continuity and diﬀerentiability of p(·) are obvious.
Note that for vi > 0,p0 (vi ) =−x(vi ) + vi x0 (vi ) + x(vi )b0 (v) − x0 (vi )b(v).
(vi − b(vi ))2Since x(·) is weakly concave in vi > 0, −x(vi ) + vi x0 (vi ) ≤ 0. Furthermore,by {b(v)/x(v)}0 ≤ 0, x(vi )b0 (v)−x0 (vi )b(v) ≤ 0. These imply that for vi > 0,p0 (vi ) ≤ 0, and p(vi ) is weakly decreasing in vi ≥ 0.
Because of Remark 2’, for this alternative payoﬀ setting, we can derive
the equivalent results to Remarks 3 to 5 and Proposition 1 by using thesame proof.
(ii) Next, we show that even without Assumptions 1 and 2, the strategy s∗ (·)
deﬁned in Proposition 1 constitutes a Bayesian Nash equilibrium. To derive
equilibrium, instead of Assumptions 1 and 2, we assume that agents takesymmetric threshold strategies and these constitute common knowledge. We
also show that s∗ (·) is a unique Bayesian Nash equilibrium under thesealternative assumptions.
Proposition 7. s∗ (·) is a unique symmetric switching strategy constituting
a Bayesian Nash equilibrium.
Proof of Proposition 7. 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 v∫i > 0.
∞
Case 1: 0 x(v)f (v; θ̂(0))dv ≥ c.
Note that for any θ ≥ θ̂(0),∫ ∞∫ ∞x(v)f (v; θ)dv >x(v)f (v; θ̂(0))dv ≥ c.
(19)
0

0

This means that if the actual mean θ is larger than θ̂(0) and all agents except
i follow the strategy s∗ (·), the project is always a success. Also note thatfor any vi > 0,∫ ∞∫ ∞f (v; vi )dv >f (v; 0)dv = p(0)
(20)
θ̂(0)

θ̂(0)

2

This means that i with a signal vi > 0 forecasts Pr(θ > θ̂(0)) > p(0).
Thus, (19) and (20) together imply that i with a signal vi > 0 considers
that the probability of project success is larger than p(0), and she has anincentive∫ to donate.
∞
Case 2: 0 x(v)f (v; θ̂(0))dv < c.
Note that for any θ > θ̂(k ∗ ),∫ ∞∫ ∞x(v)f (v; θ)dv >x(v)f (v; θ̂(k ∗ ))dv = c,
(21)
k∗

k∗

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

k∗

(22)

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 ∗ ),(23)
θ̂(k∗ )

and for any vi0 ≤ k ∗ ,∫ ∞θ̂(k∗ )

θ̂(k∗ )

f (v; vi0 )dx ≤∫ ∞θ̂(k∗ )

f (v; k ∗ )dv = p(k ∗ ).

(24)

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, (21) and (23) together imply that an agent i with signal vi > k ∗has an incentive to donate, and (22) and (24) together imply that an agent
i with signal vi0 ≤ k ∗ has no incentive to donate.
In both Cases 1 and 2, s∗ (·) surely constitutes an Bayesian Nash equilibrium.
Next,∫ we show the uniqueness of s∗ (·).
∞
Case 1: 0 x(v)f (v; θ̂(0))dv ≥ c.
First, suppose, on the contrary, that a symmetric switching strategys(·) with cutoﬀ point k >∫ ∞0 constitutes a Bayesian Nash equilibrium. In
this case, by Remark 5, k x(v)f (v; θ̂(k))dv > c.19 Thus, by that θ̂(·) is
continuous and Remark 4, for a small ² > 0,
∫ ∞x(v)f (v; θ̂(k − ²))dv > c.
(25)
k
19

Note that to derive Remarks 1 to 5, we do not need Assumptions 1 and 2.

3

This 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
(26)
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, (25) and (26) 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 cutoﬀ point k < 0 constitutes a Bayesian Nash equilibrium. Immediately, an agent i with a signal k + ² < 0 has an incentive not to donate.
This is a∫ contradiction.
∞
Case 2: 0 x(v)f (v; θ̂(0))dv < c.
First, suppose, on the contrary, that a symmetric switching strategy s(·)∗
with a cutoﬀ∫ ∞ point k > k∗ constitutes a Bayesian Nash
∫ ∞ equilibrium. In thiscase, by k∗ x(v)f (v; θ̂(k ))dv = c and Remark 5, k x(v)f (v; θ̂(k))dv > c.
We derive a contradiction similarly to Case 1.
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.
(27)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
(28)
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, (27) and (28) 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 switching strategy for a Bayesian Nash equilibrium.

4