The Economic Journal, 118 (January), 60–91. Ó The Author(s). Journal compilation Ó Royal Economic Society 2008. Published by
Blackwell Publishing, 9600 Garsington Road, Oxford OX4 2DQ, UK and 350 Main Street, Malden, MA 02148, USA.
REPEATED CHARITABLE CONTRIBUTIONS UNDER
INCOMPLETE INFORMATION*
Parimal Kanti Bag and Santanu Roy
Incomplete information about (independent) private valuations of charities by potential donors
provides an important strategic rationale for announcement of donations during fundraising drives
and explains why donors may add to their initial contributions after learning about contributions
made by others. In a two-stage fundraising drive where potential donors may contribute at either or
both stages, it is shown that under certain conditions, announcement of contributions generates
higher expected total contribution. Contribution announcement plays a similar positive role even
when the charity acquires information about donor valuations prior to actual fundraising and can
take actions to mitigate incomplete information among donors.
Following the devastation caused by tsunami over large parts of South East Asia on
December 26, 2004, the world’s leading countries got together to help the millions
affected by the event. The pledges made by governments of various countries rose to
over $8 billion quickly, by January 12, 2005.1 The US government had first pledged $15
million, then increased it to $35 million and, later on, increased the amount to $350
million.2 A report in the International Herald Tribune soon announced, Japan’s pledge
made the US package the second-largest promised so far.3 The same report also noted
an increase of pledge by Taiwan from an initial figure of $5 million to a subsequent $50
million. Some of the other countries similarly adjusted their pledges upwards.4,5
The above episode is one illustration of the fact that in a large number of fundraising
drives by charities and other non-profit organisations, individual donors may contribute multiple times (often, by adding to their pledges) as they learn about the
contributions made by other donors. Announcement of contributions appears to
increase the incentive for donors to add to their contributions and contribute higher
total amounts to the charity than they would had they not learnt about the contributions made by others. This is somewhat paradoxical as conventional analysis of strategic voluntary contribution to public goods suggests that announcement of
contributions ought to increase the incentive of early donors to free-ride on later
donors and, to that extent, have a negative impact on the funds raised; further, donors
ought to have no incentive to contribute early and repeatedly.
In this article, we argue that incomplete information about the independent private
valuation of charities by individual donors explains why announcement of donations can
* We are grateful to two anonymous referees and the editor, Leonardo Felli, for raising important questions and providing constructive suggestions that made the current version of this article possible. We remain
solely responsible for any shortcomings.
1
http://www.news24.com/News24/World/Tsunami_Disaster/0,,2-10-1777_1646450,00.html.
2
http://www.washingtonpost.com/wp-dyn/articles/A33290-2004Dec29.html; http://www.cnn.com/2004/
US/12/31/us.aid/.
3
http://www.iht.com/articles/2005/01/02/news/aid.php.
4
See a United Nations list of pledges at http://www.msnbc.msn.com/id/6767190/.
5
A similar process of countries adding to their earlier pledges is now (at the time of the article’s writing)
being observed in response to the call for funds for reconstruction of Lebanon.
[ 60 ]
[ J A N U A R Y 2008 ]
REPEATED CHARITABLE . . .
61
generate multiple contributions from individual donors in course of the fund-drive and
increase the expected total funds contributed, thus providing a strategic rationale for
charities, political parties or international bodies involved in fundraising (with a
common objective) to announce donations during the campaign. Besides explaining a
well-observed phenomenon, our argument also implies a normative prescription that
fundraisers for charities and other public goods ought to announce donations – particularly when potential donors are sufficiently dispersed and ignorant of each others
valuations or preferences. It also tangentially sheds some light on the effect of legislation requiring political parties to publicly reveal large donations in fundraising drives;
while such legislation is usually motivated by the desire for political transparency and
for minimising the influence of organised special interests, our analysis shows that it
may actually enable political parties to raise more funds.
Our argument is developed in a Bayesian model of two-stage fundraising where
potential donors may contribute to a charity at either or both stages (endogenous
move). The total contribution at the end of the fund-drive determines the provision of
the charity which is a public good. Potential donors care only about provision of the
charity and their marginal valuation of the charity is private information. From a gametheoretic point of view, announcement of contributions during the fund-drive makes
the contribution game a two-stage simultaneous move game (sometimes, called a
repeated contribution game) while non-announcement effectively reduces it to a oneshot simultaneous move game.
We show that announcement of contributions at the end of the first stage can
generate (perfect Bayesian) equilibria where donors contribute repeatedly and, under
certain conditions, generate higher expected total contribution relative to the fundraising scheme with no announcement. Moreover, under these conditions, no equilibrium of the contribution game with announcement generates lower expected total
contributions compared to the game without announcement.
The main argument is as follows. Announcement of contributions leads to the
possibility that first-stage contributions of high-valuation donors may reveal their true
types and this increases the incentive of contributors to free-ride on them in later
stages of the multistage fund-drive. This, in turn, creates a strategic incentive for highvaluation donors to hide their types in the first stage and indeed, in every equilibrium, no information about types of donors is revealed at the end of the first stage.
The second stage of the fund-drive is therefore almost the same as a one-shot
simultaneous move game except for a certain level of contribution from the first stage
that is now exogenously given. If the pooling contribution level that allows players to
hide their types in the first stage is small, then the outcome of the second-stage game
is simply a modification of the outcome of a one-shot simultaneous move game where
contributions are reduced by exactly the amount of the first-stage total contribution,
i.e., perfect crowding out occurs and the game generates the same outcome as a funddrive with no announcement of contributions. However, if the pooling contribution
level in the first round is high, then the crowding out of donations that it causes in
the second round is imperfect and this can lead to higher expected total contribution
relative to that in fundraising with no announcement. The strategic incentives created
by donors having private information about their valuations play a critical role in this
argument.
Ó The Author(s). Journal compilation Ó Royal Economic Society 2008
62
THE ECONOMIC JOURNAL
[JANUARY
Next, we analyse an extended model where the fundraiser acquires information
about private valuations (or, socio-economic attributes correlated with valuations) of
potential donors through pre-campaign research or previous fundraising experience
and can choose to reveal this information partially or fully to potential donors prior to
actual fundraising. Examples of such information transmission include pre-fundraising
events that bring together potential donors and create publicity and other information
links between donors that eventually allow donors to find out more about each other.
As our main argument rests on incomplete information among donors, it is important
to check whether it can hold even when the charity may, in principle, mitigate the
extent of incomplete information. We show that even when the charity has full information about private valuations of donors, announcement of contributions continues
to be a better strategy for fundraising. In particular, we show that when the fund-drive is
designed as one where contributions are not announced, the optimal information
transmission strategy of the fundraiser will generate full information for all potential
donors prior to the contribution game. In contrast, if the fund-drive is designed as one
where contributions are announced, optimal information revelation strategy of the
fundraiser may not lead to full information among potential donors. Further, the
expected total contribution in the latter case may exceed the former. In other words,
the strategic advantage of contribution announcement continues to hold even when
the fundraiser has the option of mitigating the extent of incomplete information.
There is a sizeable literature on dynamic voluntary contribution to a public good
where players are free to contribute any amount at any point of time and may contribute repeatedly. Fershtman and Nitzan (1991) analyse an infinite duration game
with continuous contributions and flow benefits where the players contributions
accumulate over time and show that the free-riding problem is aggravated when players contributions are conditional on the observable collective contributions. This, in
effect, suggests that charities ought not to announce contributions so that current
contributors cannot free-ride on future contributors.6 In a game of repeated voluntary
contributions to a public good where the total contribution generated in each period is
announced, Marx and Matthews (2000) show that there exist equilibria where intertemporal free-riding tendencies are largely overcome and nearly efficient outcomes
achieved through the use of appropriate punishment strategies for not adhering to
more cooperative behaviour. While their argument does provide a rationale for
announcement of contributions, it is not particularly relevant to fundraising drives by
charities and other organisations where the time span is limited; the usual logic of long
horizon multi-period games for inducing cooperation through punishment strategies
does not readily apply.
Romano and Yildirim (2001) analyse a two-stage game of voluntary contribution to a
public good with endogenous moves7 and show that the announcement of contributions can lead to a higher total contribution if donors care about individual levels of
contributions by players in addition to the total contribution, thus allowing for snob
6
A similar qualitative result is indicated in variations of dynamic or sequential voluntary contribution
games of complete information with pre-determined (exogenous) order of moves; see, among others, Bruce
(1990), Admati and Perry (1991), Varian (1994). Admati and Perry allow for repeat contributions.
7
See Section 4 of their paper. Some of these ideas are contained in Saloner (1987).
Ó The Author(s). Journal compilation Ó Royal Economic Society 2008
2008 ]
REPEATED CHARITABLE CONTRIBUTIONS
63
8
appeal, warm glow etc. A discussion of this kind of effect for a more general class of
accumulation games is contained in Romano and Yildrim (2005). Our argument in this
article complements their explanation and it applies to more conventional contexts
where such psychological effects may not be strong. In our model, donors simply care
about the public good provided by the charity (which depends only on total contributions).
Our article specifically forms a part of the literature on incomplete information
models of dynamic voluntary contribution to public goods. Much of this literature
(Bliss and Nalebuff, 1984; Gradstein, 1992; Vega-Redondo, 1995) focuses on the specific question of inefficiency due to costly delay in contributions resulting from
incomplete information. This is not directly relevant to the question we address in this
article. Andreoni (2006a) considers a contribution game with endogenous donor
moves where the quality of the charity is unknown, may be learnt by incurring a cost
and signalled by an initial donor; this leads to a war-of-attrition type problem and it is
shown that the wealthiest choose to be leaders in equilibrium and that leadership
grants, if announced, can act as signals of quality. The signalling role of announced
leadership grants may be seen as an indirect justification for announcement of contributions. However, it does not explain why many charities tend to announce all
contributions – not just the leadership grants. Also, the problem of the quality of a
charity being unknown does not apply well to programmes of established charities that
have been ongoing for a while. Finally, even when the quality or productivity of the
charity is unknown, it may not be possible for donors to acquire additional information
about the charity at the fundraising stage – for example, when pledges are made by
donors after natural disasters, it is almost impossible to acquire information in advance
about how well the funds are going to be utilised (beyond what is publicly known).
While the argument developed in our article is also based on incomplete information,
we focus on uncertainty about independent private valuations of the charity (rather
than its common value). Further, the process by which announcement generates
higher contribution is not through acquiring or signalling of information.
The existing literature on charitable contributions also contains several other
explanations of announcement of contributions that require potential donors to
contribute in pre-determined order of moves. Announcement of contributions
then generates a sequential game of perfect information among donors while
non-announcement generates a simultaneous move game. In a fundraising situation
with natural leaders and followers among contributors, Andreoni (1998) showed that if
the production technology of the public good exhibits non-convexity such as a minimal
threshold requirement for the public good to be of any value, announcement by the
charity of an initial leadership gift generates higher contributions by follower donors.9
8
Most of their analysis deals with showing that announcement raises the total contribution in a sequential
game where contributors move in a pre-determined sequence. Harbaugh (1998) also modelled warm-glow
and prestige motives, while Glazer and Konrad (1996) suggest a wealth-signalling motive, but neither paper
explains why donation announcements are often made while the fundraising is still in progress and not
necessarily after its completion. See also Andreoni (1990).
9
In a somewhat different approach, Cornelli (1996) notes the importance of announcement of contributions (or sales) in the monopoly provisions of services with fixed costs such as theatres and performing arts
where patrons with relatively high valuations need to subscribe voluntarily to a larger share of the cost burden
in order for the service to be provided.
Ó The Author(s). Journal compilation Ó Royal Economic Society 2008
64
THE ECONOMIC JOURNAL
[JANUARY
This explanation is a powerful one – particularly for capital campaigns and for ventures
with large fixed costs. However, it is not as relevant for campaigns to expand the scope
and outreach of pre-existing programmes and for projects with no significant
non-convexity or threshold requirements. Further, it does not explain why charities
continue to announce donations beyond the initial leadership grants or once the
minimum threshold has been met. Vesterlund (2003) shows that announcement may
signal the unknown common value of a charity if donors can acquire information at a
cost and later donors can learn this information from announced contributions of
earlier donors; this explanation requires not only that there be natural leading donors,
but that they are also able to acquire prior information about the charity at reasonable
cost which, as mentioned earlier, may not be possible in large number of situations.10
More generally, this class of models assumes that the fundraiser can pre-commit not to
accept donations from donors who want to contribute earlier than their assigned move
or from donors who want to add to their contributions later; further, they do not
explain why donors contribute multiple times.11 In contrast, our article shows that even
when donors may freely choose when, how much and how many times to contribute
during the fundraising drive, there is an important role for announcement of contribution in raising the size of the funds generated.
The article is organised as follows. The next Section presents the basic model and the
contribution game. Section 2 reviews the outcomes of the contribution game with no
announcement of donations. Section 3 analyses the equilibria of the contribution game
with announcement of donations. The comparison of the outcomes of these two games
in terms of expected total contribution is assessed in Section 4. Section 5 studies the
issue of information transmission by charities. Section 6 concludes. The proofs are
relegated to an Appendix.
1. Basic Framework
There are two potential contributors, called players and indexed by i ¼ 1, 2. Each
player i divides his wealth Wi > 0 between private consumption vi 0 and a contribution gi 0 to the public good:
vi þ g i ¼ W i :
Each player’s utility depends only on the total amount of the public good created,
G ¼ g1 þ g2 ;
and the amount of private consumption; we do not consider the prestige value or
warm-glow motives of donations. The utility functions are quasi-linear:
Ui ¼ ts V ðGÞ þ vi ¼ ts V ðgi þ gj Þ þ Wi gi ; i 6¼ j;
where V : Rþ ! Rþ is strictly concave and twice continuously differentiable with
V 0 (Æ) > 0 and V 00 (Æ) < 0. Each player’s utility from the public good depends also on his
type s – high-value (H) or low-value (L) – and is parameterised by ts, with tH > tL.
10
Bag and Roy (2004) show that sequential fundraising by announcing donations can be optimal even
when the donors valuations are independent and private.
11
For details, see the survey by Andreoni (2006b).
Ó The Author(s). Journal compilation Ó Royal Economic Society 2008
2008 ]
REPEATED CHARITABLE CONTRIBUTIONS
65
The distribution of player is type is common knowledge: type H occurs with probability pi and type L with probability 1 pi. The realised type of each player is only
known to the player himself. Players maximise expected utility. We assume two-sided
incomplete information:12
0 < pi < 1;
i ¼ 1; 2:
Define the standalone contribution by a player of type s, denoted by xs, to be his
optimal public good contribution if he knows that he is the only contributor:
ts V 0 ðxs Þ ¼ 1;
s ¼ H ; L:
We assume that the players are not wealth-constrained and tLV 0 (0) > 1, so that xss are
well-defined and interior. It is easy to see that xH > xL and no player of type s will make
a positive contribution if he believes that the other player is going to contribute at least
xs with certainty.
We consider a fundraising drive with two rounds of contribution – called Round 1 and
Round 2. Players contribute simultaneously in both rounds. As players may contribute
strictly positive amounts in either or both rounds, this is a multi-stage game of
endogenous moves.
We consider and compare two variants of this fundraising scheme: with and without
public announcement of individual contributions at the end of the first round. It is
assumed that no player directly observes the contribution made by the other player so
that announcement by the fundraiser is the only way a player gets to know about the
action chosen by the other player in the first round. The solution concept used is perfect
Bayesian equilibrium.
As in much of the literature on charitable fundraising, the fundraiser is only interested in the size of total contribution. We assume that the fundraiser is risk-neutral and
therefore ranks alternative campaign designs according to the (ex ante) expected total
contribution of the two players.
2. Contribution Game with No Announcement
In this Section, we consider the game among potential donors generated by a fundraising drive where no contribution is announced at the end of the first round. It is easy
to check that this is equivalent to a one-shot simultaneous move contribution game. In
what follows, we analyse this one-shot contribution game and its Bayes-Nash equilibrium outcomes will be interchangeably referred to as outcomes of fundraising with no
announcement.
Let yis denote the equilibrium contribution of player i, i ¼ 1, 2, of type s, s ¼ H, L,
solving the following expected utility maximisation problem:
max
y0
pj ts V ðy þ yjH Þ þ ð1 pj Þts V ðy þ yjL Þ y;
j 6¼ i;
j ¼ 1; 2:
The net marginal expected utility for player i of type s which determines his incentive
to contribute is given by:
12
The qualitative nature of our analysis largely remains unchanged for one-sided incomplete information
where only one player’s type is known with certainty.
Ó The Author(s). Journal compilation Ó Royal Economic Society 2008
66
THE ECONOMIC JOURNAL
[JANUARY
pj ts V 0 ðy þ yjH Þ þ ð1 pj Þts V 0 ðy þ yjL Þ 1:
It follows that for any player the low type never contributes more, and generally less,
than the high type’s contribution. Also, for any given pair of contributions by (the two
types of) player j, player i s incentive to contribute is decreasing in pj.
Given the nature of the first-order conditions to be written below, in general it is not
possible to guarantee uniqueness of equilibrium contributions or even expected total
contribution in the one-shot game. However, for later analysis it is relevant to note here
that H-type players will always make strictly positive contributions:
Lemma 1. y1H > 0; y2H > 0.
Given Lemma 1, the first-order conditions for an equilibrium can thus be written as:
y1L 0 : p2 tL V 0 ðy1L þ y2H Þ þ ð1 p2 ÞtL V 0 ðy1L þ y2L Þ 1
y1H > 0 : p2 tH V 0 ðy1H þ y2H Þ þ ð1 p2 ÞtH V 0 ðy1H þ y2L Þ ¼ 1
y2L 0 : p1 tL V 0 ðy1H þ y2L Þ þ ð1 p1 ÞtL V 0 ðy1L þ y2L Þ 1
y2H > 0 : p1 tH V 0 ðy1H þ y2H Þ þ ð1 p1 ÞtH V 0 ðy1L þ y2H Þ ¼ 1;
with equalities holding for strictly positive-valued y1L and y2L .
3. Contribution Game with Announcement
We now consider the contribution game where, at the end of the first round, the
fundraiser publicly announces the individual contributions by both players, allowing
each player to further update his belief about the other player’s type before making the
second-round contribution.
3.1. Non-revealing Equilibrium
In this game, one might expect that in equilibrium, the announced first-round
contributions of players might reveal some information about their types so that, after
updating, players are more informed about each others types in the second round
of fundraising. Surprisingly, this does not happen in equilibrium. In every perfect
Bayesian equilibrium, each player’s first-round contribution is independent of his
type. The high-valuation donor has a strong incentive to pool with the low-type
donor in the first round in order to hide his type for otherwise he may have to bear
a very high burden of contribution in the second round. This holds as long as the
contribution of the low-valuation donor is not too large. If, however, the latter is at
a level large enough to dissuade the high-valuation donor from pooling, then it is
the low-valuation donor who wants to deviate and pool with the high-valuation
donor.
Denote the contributions in the first round by wis ; i ¼ 1; 2; s ¼ H ; L.
Proposition 1. In every Perfect Bayesian Equilibrium (PBE) of the two-rounds contribution
L
game with announcement, the first-round contributions satisfy: w H
i ¼ w i ; i ¼ 1; 2.
Ó The Author(s). Journal compilation Ó Royal Economic Society 2008
2008 ]
REPEATED CHARITABLE CONTRIBUTIONS
67
3.2. Construction of Equilibrium
Proposition 1 implies that every equilibrium of the contribution game with
announcement is characterised by a pooling outcome in which both types of each player
contribute an identical amount in Round 1 (the amounts may well differ between the
players) so that at the beginning of Round 2 the beliefs about player types are same as
the original beliefs at the start of Round 1. We now construct a class of such equilibria.
Consider the one-shot contribution game (i.e., that generated by no announcement)
discussed in the previous Section. Let us define
y~ ¼ supfy1L þ y2L : ðy1H ; y2H ; y1L ; y2L Þ is a Bayes-Nash equilibrium of the one-shot gameg: ð1Þ
In other words, y~ is the highest possible total contribution in any equilibrium of
the contribution game with no announcement when both players are of type L.
Observe that13
y~ < xL :
ð2Þ
As before, we denote the contributions in the first round of the contribution game
with announcement by wis ; i ¼ 1; 2; s ¼ H ; L. The contributions in the second round
of this game are denoted by zis ; i ¼ 1; 2; s ¼ H ; L.
A class of equilibria: E
Round 1: 0 wL1 ¼ wH1 ¼ w~1 and 0 wL2 ¼ wH2 ¼ w~2
such that y~ w~1 þ w~2 ¼ w~ xL ;
Round 2: z1L ¼ z2L ¼ 0; z1H > 0; z2H > 0 satisfying for i 6¼ j
pj tH V ðziH þ zjH þ w
ei þ w~j Þ þ ð1 pj ÞtH V ðziH þ w~i þ w~j Þ ziH
¼ max ½pj tH V ðx þ zjH þ w~i þ w~j Þ þ ð1 pj ÞtH V ðx þ w
ei þ w~j Þ x:
x0
Players beliefs at the beginning of Round 2 are same as at the beginning of Round 1.
Off-equilibrium beliefs (and play) are specified as follows. If player i deviates and
contributes an amount w 6¼ w~i in Round 1, then beliefs are that player i is H-type with
probability one and in the continuation play in the second round, equilibrium prescribes zjL ¼ zjH ¼ 0 and player i is expected to contribute maxfxH ðw þ w~j Þ; 0g.14
13
Inequality (2) is obvious if y1L ¼ 0, since y2L < xL . On the other hand, if y1L > 0, then
p2 tL V 0 ðy1L þ y2H Þ þ ð1 p2 ÞtL V 0 ðy1L þ y2L Þ ¼ 1
and since y2L < y2H and V(Æ) is strictly concave, it follows that tL V 0 ðy1L þ y2L Þ > 1 i.e., y1L þ y2L < xL so that (2)
holds. (Why y2L < y2H ? From Section 2 we know that y2L y2H . If y2L ¼ y2H , then the first-order conditions
corresponding to y2H > 0 and y2L > 0 would hold with equality; see Section 2. But since tL < tH, the first-order
equality conditions are not compatible.)
14
A donor can easily compensate for a low level of contribution in the first round while contributing in
the second round. Therefore, a first-round deviation by player i to a contribution level below w~i does not
really say anything about his type; it cannot be argued that a high-valuation player i has less incentive to
undertake this deviation than a low-valuation player i, using arguments like the Intuitive and D1 refinements
in pure signalling games with one sender, as in Cho and Kreps (1987). However, for a first-round deviation by
donor i to a level higher than w~i , there are reasonable grounds for some restrictions on out-of-equilibrium
beliefs – precisely because the donor cannot undo the effect of such contribution in the second stage. Thus, if
a donor contributes an amount exceeding xL, the standalone contribution level for an L-type, it can be argued
that such a contribution could only come from an H-type agent. The out-of-equilibrium beliefs specified in
the paper assign all probability mass to the high-valuation type and therefore meets any such restriction.
Ó The Author(s). Journal compilation Ó Royal Economic Society 2008
68
THE ECONOMIC JOURNAL
[JANUARY
Observe that almost all (of the continuum of) equilibria in the class E are characterised by repeated contributions by both donors; the realised second-round contributions may be less or more than the first-round contributions. It can be shown that
(see Appendix):
Lemma 2. In every (second-round) continuation game following announcement of first-round
contributions, the prescribed strategies in E constitute a Bayes-Nash equilibrium.
It now remains to verify that no one would deviate in Round 1.
Suppose only player i ¼ 1, 2 of type L deviates in Round 1 and contributes w 6¼ w~i .
Then, no contribution comes from player j 6¼ i in Round 2. Thus, player i s deviation
payoff is
tL V ðmaxfxL ; w~j þ wgÞ ½w þ maxf0; xL ðw~j þ wÞg:
ð3Þ
Observe that if w xL w~j , then the deviation payoff in (3) is
¼ tL V ðw~j þ wÞ w
tL V ðxL Þ ðxL w~j Þ
because the function f ðwÞ ¼ tL V ðw~j þ wÞ w is maximised at ðxL w~j Þ. On the
other hand, if w < xL w~j , then the deviation payoff in (3) is
¼ tL V ðxL Þ ðxL w~j Þ:
Next, observe that if player i sticks to equilibrium play, his payoff
¼ pj tL V ðzjH þ w~i þ w~j Þ þ ð1 pj ÞtL V ðw~i þ w~j Þ w~i
pj tL V ðzjH þ w~i þ w~j þ ½xL ðw~i þ w~j ÞÞ þ ð1 pj ÞtL V ðw~i þ w~j þ ½xL ðw~i þ w~j ÞÞ
w~i ðxL w~i w~j Þ
¼ pj tL V ðzjH þ xL Þ þ ð1 pj ÞtL V ðxL Þ ðxL w~j Þ
> tL V ðxL Þ ðxL w~j Þ:
deviation payoff.
Thus, deviation cannot be gainful. (The second inequality above follows from the fact
that, on the equilibrium path, player i of L-type maximises his expected utility by
contributing zero in the second round.)
Suppose only player i ¼ 1, 2 of type H deviates in Round 1 and contributes w 6¼ w~i .
Then, again, no contribution comes from player j 6¼ i in Round 2. Thus, player i s
deviation payoff is
tH V ðmaxfxH ; w~j þ wgÞ ½w þ maxf0; xH ðw~j þ wÞg:
ð4Þ
Observe that if w xH w~j , then the deviation payoff in (4) is
¼ tH V ðw~j þ wÞ w
tH V ðxH Þ ðxH w~j Þ
because the function g ðwÞ ¼ tH V ðw~j þ wÞ w is maximised at xH w~j . On the other
hand, if w < xH w~j , then the deviation payoff in (4) is
Ó The Author(s). Journal compilation Ó Royal Economic Society 2008
2008 ]
REPEATED CHARITABLE CONTRIBUTIONS
69
¼ tH V ðxH Þ ðxH w~j Þ:
Next observe that if player i sticks to equilibrium play, his payoff
¼ pj tH V ðziH þ zjH þ w~i þ w~j Þ þ ð1 pj ÞtH V ðziH þ w~i þ w~j Þ ðziH þ w~i Þ
¼ ½pj tH V ðziH þ zjH þ w~i þ w~j Þ þ ð1 pj ÞtH V ðziH þ w~i þ w~j Þ ziH w~i
½pj tH V ððxH fw~i þ w~j gÞ þ zjH þ w~i þ w~j Þ
þ ð1 pj ÞtH V ððxH fw~i þ w~j gÞ þ w~i þ w~j Þ ðxH fw~i þ w~j gÞ w~i
¼ pj tH V ðxH þ zjH Þ þ ð1 pj ÞtH V ðxH Þ xH þ w~j
> tH V ðxH Þ ðxH w~j Þ
deviation payoff:
Thus, deviation cannot be gainful. (The third inequality above follows from the fact
that, on the equilibrium path, player i of H-type maximises his expected utility by
contributing ziH in the second round.)
Note that in an equilibrium of the class E, the first-round pooling contribution levels
can differ between the two players so that in the first round, it is possible that a lowvaluation donor actually contributes more than the other donor even though the latter
is a high-valuation donor.
4. Comparison of Announcement
In Section 3, we have shown that in every perfect Bayesian equilibrium of the game with
announcement of contributions, the first-round contributions satisfy wiH ¼ wiL ¼
xi ; i ¼ 1; 2. We will refer to x ¼ x1 þ x2 as the total contribution in the first round.
Lemma 3. Assume p1 p2, p1 > 0 and p2 < 1. Consider any perfect Bayesian equilibrium
of the two-rounds game with announcement where the total contribution in the first round, denoted
by x, satisfies
x > y~
and y~ is defined by (1). Then, on the equilibrium path, each low-type player contributes zero in the
second round: ziL ¼ 0; i ¼ 1; 2. Further, the high type of each player i contributes an amount
ziH > 0. Finally, the following holds:
z1H þ x > y1H þ y2L
ð5Þ
z1H þ z2H þ x < y1H þ y2H
ð6Þ
z2H þ x > y2H þ y1L
ð7Þ
where ðy1H ; y2H ; y1L ; y2L Þ is an equilibrium of the one-shot contribution game (no announcement).
Note that Lemma 3 is a characterisation of a subset of equilibria of the game with
announcement of contributions. In Section 3.2, we constructed a class of equilibria E
Ó The Author(s). Journal compilation Ó Royal Economic Society 2008
70
THE ECONOMIC JOURNAL
[JANUARY
(containing a continuum of equilibria) that satisfy the conditions in the hypothesis of
Lemma 3. Our main result, stated next, shows that such equilibria may strictly dominate (in terms of expected total contribution) any equilibrium of the contribution
game with no announcement.
Proposition 2. Suppose that at least one of the following holds: (i) V 0 (Æ) is concave; (ii)
p1 þ p2 1. Consider any perfect Bayesian equilibrium of the game with announcement of
contributions, where the combined total contribution by the players in the first round exceeds ~y
(defined by (1)). Such an equilibrium generates strictly higher expected total contribution than
that generated in any equilibrium of the game with no announcement of contributions.
We like to make two remarks. First, for the dominance of the announcement strategy, neither of the two conditions stated in the hypothesis of Proposition 2 are
necessary. While the conditions help us to compare expected total contributions using
the first-order marginal conditions defining Bayes-Nash equilibrium, it is not difficult
to see from our proof in the Appendix that there is a large class of situations where the
domination holds even though neither of the two conditions hold; they should be seen
as strong sufficient conditions that are easily verifiable and transparent. Second, we can
now specify the precise sense in which announcement of donations is better for the
charity in our framework. Define S1, S2 by
S1 ¼ set of Perfect Bayesian Equilibria of the two-rounds game where
the total contribution in the first round > y~:
S2 ¼ set of Perfect Bayesian Equilibria of the two-rounds game where
the total contribution in the first round y~:
Then, S1 6¼ ; by our equilibrium construction, E, in Section 3.2. Under the conditions of Proposition 2, every equilibrium in S1 strictly dominates all Bayes-Nash equilibria of the game with no announcement. Further, for each equilibrium in S2, it is easy
to verify that there is some equilibrium of the game with no announcement that
generates exactly the same distribution of contributions.15
In particular, if the game with no announcement has a unique equilibrium, then
under the conditions of Proposition 2, the game with announcement weakly dominates the game with no announcement in terms of expected total contribution and
there exist a continuum of equilibria of the game with announcement that generate
strictly higher expected total contributions than the game with no announcement.
Intuitively, the strategy of announcing donations induces a kind of pre-emptive race
that may compel either player to make a positive and non-trivial donation in the first
round so as not to be perceived as a high-value type by the other player. The resulting
total contribution in Round 1 could be so high that in Round 2 complete crowding out
is not possible, and on average (over various possible realisations of types) total contributions in the two-rounds game with announcement turn out to be strictly larger
than in the case of no announcement.
15
Simply set, yis ¼ xi þ zis and using the conditions for an equilibrium in the second round of the
contribution game with announcement, verify that this is a Bayes-Nash equilibrium of the one-shot simultaneous move contribution game.
Ó The Author(s). Journal compilation Ó Royal Economic Society 2008
2008 ]
REPEATED CHARITABLE CONTRIBUTIONS
71
Proposition 2 gives sufficient conditions for the domination of announcement
strategy over non-announcement. We can also state a necessary condition for the
domination to hold.
Proposition 3. The expected total contribution in any PBE of the contribution game with
announcement is strictly higher than that in all equilibria of the game with no announcement only
if the second-round contributions after announcement satisfy zL2 ¼ zL1 ¼ 0.
Proposition 3 implies that if announcement of contributions is followed by a strictly
positive contribution by some low-type player then there is an equilibrium of the game
with no announcement that yields the same expected total contribution. The
announcement strategy yields strictly higher expected total contribution for the charity
only when the first-round contributions are large enough to cause incomplete crowding out in which case the low-type donors must hit the non-negativity constraint on
their contributions.
4.1. Donation Announcement: Non-commitment Case
The above analysis of fundraising with announcement assumes that the charity precommits to announcing all donations independent of the contribution levels. However,
it is quite possible that the charity cannot pre-commit to an announcement strategy. In
that case, would the charity find it optimal to announce contributions?
We argue briefly that charities may announce contributions even when they cannot
pre-commit to doing so. The domination of announcement over non-announcement
may continue to hold even when the charity cannot commit to announcing donations
prior to the contribution game.
Consider the two-rounds contribution game of incomplete information described in
Section 1 and now assume that the charity decides whether or not to announce contributions after observing the actions chosen by donors in the first round. However, for
simplicity, assume that if the charity announces contributions, then it must announce
contributions of both players. We argue that there is an equilibrium where first-round
contributions are always announced and the generated total contributions are identical
to that in equilibrium E (see Section 3.2) of the game where the charity commits ex ante
to an announcement strategy.
To see this, consider the following out-of-equilibrium beliefs:
(*) If the charity does not announce first-round contributions then each player believes
that the total contribution (in the first round) must be at least xH.
This out-of-equilibrium belief is quite intuitive – only when very large amounts have
been contributed in the first stage would the charity have an incentive to hide it so as
not to reduce the incentive to donate in the second round. Given this belief, if the
charity does not announce first-round contributions, each player contributes zero in
the second round which is optimal for the players.
Choose any equilibrium in class E as described in Section 3.2. For the equilibrium we
construct, the details are as follows. Independent of amount contributed, the charity
announces donations. If it deviates and does not announce donations, second-round
Ó The Author(s). Journal compilation Ó Royal Economic Society 2008
72
THE ECONOMIC JOURNAL
[JANUARY
contributions are zero. In Round 1, each player makes the same pooling contributions
w~1 and w~2 and if they are announced, they are followed by the same Round 2 contributions as in the chosen equilibrium from E. When contributions are announced, the
out-of-equilibrium beliefs following a player’s (i.e., a donor’s) unilateral deviation from
the equilibrium contribution in Round 1 and Round 2 play following such deviation
are also the same as in the chosen equilibrium from E. If contributions are not
announced, beliefs are as in (*) and both players contribute zero. It is easy to see that
the expected total contribution is identical to that when the charity commits to
announcing all donations and the chosen equilibrium in class E is played.
5. Information Transmission by Charity
Our analysis of the contribution game has so far been carried out under the assumption of an exogenously given (incomplete) information structure among potential
donors and we have argued that private information about preferences of donors
creates an important incentive for charities to announce contributions during the
fund-drive. We have not allowed for the possibility of deliberate strategic manipulation
of the information structure by the fundraiser. This is quite realistic in situations where
there is no scope for the fundraising charity to be significantly more informed about
the preferences of any potential donor (in addition to what is publicly known to all
other donors).
However, in certain situations, the fundraising charity may acquire superior
information about the private preferences of potential donors which it may, if it wishes,
reveal to other donors. The fundraiser may acquire this information through careful
research prior to the actual campaign, establishing contacts with targeted potential
major donors before soliciting contributions and on the basis of historical interaction
with same donors in the past.16 For example, universities carrying out capital and other
fundraising campaigns are often observed to contact high profile alumni and
important corporate sponsors in their carefully organised pre-fundraising events.17 In
the presence of such superior information, it becomes important to analyse the
incentives of the charity to communicate this information to donors thereby mitigating
the extent of incomplete information. Indeed, if the charity is fully informed and
conveys all its information to all donors, then the contribution game is one of complete
information in which case there is no further role for announcement of contributions
(in our framework).
In this Section, we briefly explore the incentives for strategic transmission of superior
information by charities and its impact on the role of announcement of contributions
during fundraising. To do this, we make the extreme assumption that the fundraising
charity receives full information about the type of each donor (though it is not known
16
We confine our attention to what happens after such information has been gathered. Analysing the
charity’s incentives for costly information acquisition requires a separate analysis.
17
Some instances can be found on the internet. We provide two such links: http://www.instadv.ucsb.edu/
93106/2005/January10/national.html (National fund-raising campaign goes public; Events planned for
potential donors on east, west coasts – the 2005 campaign by the University of California Santa Barbara);
http://giving.mit.edu/who/partnerships.html (the launch by the Massachusetts Institute of Technology in
2000 of a Donor Partnerships initiative to bring together prospective donors who might be interested in
furthering specific goals complementary with the institute’s research initiatives).
Ó The Author(s). Journal compilation Ó Royal Economic Society 2008
2008 ]
REPEATED CHARITABLE CONTRIBUTIONS
73
to other donors) and decides whether or not to reveal this publicly to both donors. In
particular, the charity cannot reveal its information to one donor and not to the other.
We also assume that the fundraiser cannot misrepresent its information; if it chooses to
reveal the types of the donors, it must do so truthfully (to all potential donors). Finally,
we assume that the fundraising charity cannot pre-commit (before observing the realisation of
types) to not reveal any type-related information.18
Our information transmission mechanism can be seen as a reduced form
representation of a process whereby the charity has the option of introducing potential
donors to other prospective donors through a public fund-drive launching event and
other events that facilitate information communication links between potential donors.
During such events the charity may highlight who are the interested sponsors for the
particular cause and may even ask them to make brief statements explaining what it
means to them to be part of the charitable cause.19 In introductions of this nature, it
might be argued, sponsors cannot hide their types by projecting something other than
their true self. Alternatively the charity can disclose, in sophisticated ways, information
relating to the sponsors social identity (for example, economic status or social attributes) that can provide indirect information about the sponsors true valuations. Note
that a prospective donor’s social identity, which may be correctly identified by the
fundraising team, is different from the donor’s common-knowledge identity in the
fundraising game. The charity’s role is to provide the means to expose (or hide) such
type-related information at the beginning of the fund-drive.20
Based on the initial prior, the equilibrium belief about the strategy of the fundraising
charity and the charity’s information disclosure/non-disclosure, each donor updates
his subjective belief about the type of the other donor in a Bayesian fashion. After this,
the donors play the contribution game. As before, we look at two variants of the
contribution game – a two-rounds simultaneous move game with announcement of
contributions at the end of the first round and one with no announcement of contributions which is equivalent to a one-shot simultaneous move game. The charity’s
payoff is the expected total contribution generated while the donors payoffs are as
specified in the previous Sections.
Note that if both donors acquire full information about each other’s types before the
contribution game begins (either because the charity reveals this information or
because the donors can infer it from the charity’s action), then the contribution game
18
Most fundraising charities acquire information about donors through pre-campaign research and
investigation of potential donors and by drawing on historical experience of fundraising in previous campaigns. While some fundraising charities may continue to acquire information during the campaign, it is most
likely that much of the relevant information about donors would have been acquired before the actual launch
of the campaign. In these cases, the question of pre-commitment to revelation or non-revelation of information about donors types before knowing the realised types, is not very relevant. Also note that unless
potential donors insist on complete anonymity with regard to any mention of their involvement with the
particular charitable cause, assurances by the charity not to release donor related information may not be
credible. In fact, preliminary fundraising events that highlight and publicise potential major donors are now
integral parts of most large campaigns.
19
Such events are sometimes called road shows (with celebratory lunches and dinners) but may also be
part of media campaign where charities also outline their objectives/missions. Bringing together potential
key donors and facilitating communication links are parts of a multitude of objectives here.
20
Our model rules out the possibility of the charity finding out information about donor types during the
first round of contributions. See footnote 18.
Ó The Author(s). Journal compilation Ó Royal Economic Society 2008
74
THE ECONOMIC JOURNAL
[JANUARY
is a complete information game in which case the outcome of the contribution games
with and without announcement are identical.
Formally, the game begins with nature choosing a pair of realised types for the two
potential donors that we refer to as states of nature from the set T of all possible states
T ¼ fðL; H Þ; ðH ; LÞ; ðH ; H Þ; ðL; LÞg;
where the first (second) entry in a pair refers to donor 1s (donor 2s) type. The
charity’s information transmission strategy can then be summarised by an announcement
set A T consisting of states that, if realised, will be revealed to donors. If the charity
chooses not to reveal any state, then A ¼ ;, the empty set. Given any announcement set
A, we can talk about a concealment set C T where Ac ¼ C.21
As in the previous Sections, our main purpose of analysing this game is to compare
the generated contributions between two alternative designs of the fund-drive (following information transmission) – one with no announcement of contribution and
the other with announcement of contributions.
Our first result characterises the outcome for the case of no announcement of
contributions.
Proposition 4. Consider the game where the charity does not announce contributions during
the fund-drive. Then, in any Perfect Bayesian equilibrium, the total contribution generated (for
each realisation of types) is exactly identical to that generated under complete information. The (ex
ante) expected total contribution generated is equal to
ð1 p1 Þð1 p2 ÞxL þ ½1 ð1 p1 Þð1 p2 ÞxH :
ð8Þ
The proof is contained in the Appendix. In particular, the proof establishes that
either the equilibrium is one where the donors are able to infer the realised state fully
from the equilibrium strategy of the charity, or it is one where the equilibrium contributions (for each pair of realised types) are exactly what they would be if the donors
knew each other’s types with certainty. At the heart of this result is a process of
informational unravelling whose basic intuition is as follows. For concealment to be
strictly beneficial to the charity ex post (i.e., after knowing the true state), it must be that
the charity wants the donor to believe that the state could be something other than the
true state with positive probability, as such a belief could generate higher contributions.
But when the more beneficial state(s) (beneficial from charity’s point of view) are
realised, the charity is obviously better off announcing the true state(s). Thus the
strategy of ex post concealment tends to unravel. Note that the lack of commitment
power by the charity to withhold favourable information is a crucial factor behind the
informational unravelling.22
21
Independent private communication by the charity to individual players about the other player’s type is
not considered. This rules out finer messages like (L, ;), (H, ;) etc. The use of the coarser message space is
motivated by the need to model mutual introduction of donors in a preliminary fundraising event.
22
A similar type of unravelling argument appears in the literature in various contexts; see Bac and Bag
(2003) for an application in the context of fundraising, and some of the references cited there for other
applications. Okuno-Fujiwara et al. (1990) offer an analysis of when unravelling may or may not occur in a
broadly defined strategic environment. However, their sufficient conditions for full revelation are not met in
our context; in particular, the equilibrium of the contribution game (following information transmission) is
not necessarily unique, which was a basic assumption in Okuno-Fujiwara et al.
Ó The Author(s). Journal compilation Ó Royal Economic Society 2008
2008 ]
REPEATED CHARITABLE CONTRIBUTIONS
75
Proposition 4 assumes common knowledge about the charity’s full knowledge of
player types. If there is some uncertainty in the charity’s acquisition of information
about the correct state, then the unravelling argument breaks down. For instance,
announcing types in order to benefit from a possible gainful deviation from a concealing equilibrium is simply not feasible. Also, the due-scepticism inference that is
often invoked to induce full revelation cannot be applied.23
We now turn to a different extensive form where the contribution game following the
charity’s transmission of information is one where contributions are announced at the end
of the first round. In this game, each donor updates his belief about the type of the
other donor twice: first, after the charity’s information transmission (and prior to
choosing the first-round contribution) and second, after the first-round contributions
have been announced (and prior to choosing the second-round contribution). Interestingly, and in contrast to the game analysed above where donors play a one-shot
simultaneous move contribution game after information transmission by the charity
(no announcement of contributions), the informational unravelling result does not
hold for all equilibria. There are situations where the charity does better by selective
transmission of information and by suppressing some type profiles. We demonstrate
this by constructing an equilibrium with this property.
A partially revealing equilibrium, E1:
The details of the construction are contained in the Appendix. Suppose that xH xL
is large enough (in a sense to be specified in the Appendix). Consider the following
strategies and beliefs.
Charity’s information transmission strategy:
C ¼ fðL; LÞ; ðL; H Þ; ðH ; H Þg; A ¼ fðH ; LÞg:
If information is transmitted by the charity, then the two rounds contribution game
with announcement is effectively reduced to a one-shot simultaneous move game of
complete information. In such a subgame, the equilibrium strategies for donors 1 and
2 are specified as in a Nash equilibrium of the one-shot complete information game
(there could many such Nash equilibria where the types of the two donors are identical) and yield a total contribution of xL if the transmitted state is (L, L) and xH,
otherwise.
If no information is transmitted by the charity, then each donor still believes that the
other donor could be of either H or L type, with strictly positive probabilities, with one
exception. An H-type donor 1 would know for sure that donor 2 must also be an H type;
however, it is not common knowledge that donor 2 is of type H.
The equilibrium path following no information transmission by the charity is as
follows:
First-round contributions:
w1L ¼ w1H ð¼ wÞ ¼ xL þ ;
w2L
¼
w2H
> 0 small;
¼ 0:
23
The principle of due-scepticism simply says that whenever an agent does not report his type, or reports a
non-singleton subset of types containing the true type, the other agents believe that the actual type must be
the worst element (from the reporting agent’s point of view) in the reported type set.
Ó The Author(s). Journal compilation Ó Royal Economic Society 2008
76
THE ECONOMIC JOURNAL
[JANUARY
Beliefs following announcement of first-round contributions:
If w1 ¼ w, posterior p1 > 0;
if w1 6¼ w, posterior p1 ¼ 1;
if w2 ¼ 0, posterior p2 > 0;
if w2 6¼ 0, posterior p2 ¼ 1.
Second-round contributions:
z1L ¼ z1H ¼ 0;
z2L ¼ 0
z2H ¼ xH w
z2H
¼0
if w1 ¼ w
if w1 6¼ w:
In the Appendix we check that E1 is indeed a PBE.
Observe that in the proposed equilibrium, the total contribution is xL þ when type
profile is (L, L), and xH otherwise. Thus, the outcome of the equilibrium constructed
weakly dominates (from charity’s point of view) the outcome when the charity always
reveals player types (which would be the case if, for instance, the charity pre-commits to
not announce the first-round contributions).
The implication of an equilibrium like E1 is that even when the charity has the option
of removing incompleteness of information among donors prior to the actual contribution process, it may choose not to do so if it understands how lack of information
about other donors preferences can create incentives for donors to contribute higher
amounts in multistage fund-drives. Further, fundraising schemes with announcement
of contributions may involve lower incentives for charities to reveal their private
information about preferences of donors and actually generate higher expected total
contribution than fundraising schemes with no announcement of contributions.
6. Conclusion
Announcements of individual donor contributions during the fundraising drive creates
strategic incentives for potential donors with independent private valuation of the
charity to contribute repeatedly and for high valuation contributors to contribute in
early rounds at a level so as to conceal their types. By designing a fund-drive where
contributions are announced after each round, charities may gain in terms of expected
total contribution (relative to non-announcement). The strategic role of contribution
announcement in raising expected contribution continues to hold even when the
charity (or fundraiser) may ensure that the private information of every potential
donors is revealed (to other donors) prior to the contribution game; the extent of
incomplete information under which donors contribute to the charity may be higher in
fundraising with announcement and this can help generate higher contributions by
donors (relative to non-announcement during fund-drives).
Appendix
Proof of Lemma 1. To show that the high-types of both players will make strictly positive
contributions, suppose not. Then, only one of the H-type players contributes. In that case, both
Ó The Author(s). Journal compilation Ó Royal Economic Society 2008
2008 ]
REPEATED CHARITABLE CONTRIBUTIONS
77
Low- and High-types of the other player must be contributing zero. Therefore, the first player’s
High-type must be contributing exactly xH. The maximum contribution of the Low-type of that
player is xL. But then the expected marginal utility at zero contribution by the second player,
when he is of H-type, is strictly greater than 1, implying that zero contribution could not be
(expected) utility maximising.
h
Proof of Proposition 1. We first show that there is no Perfect Bayesian Equilibrium (PBE) of the
two-rounds contribution game with announcement where the first-round contributions satisfy:
L
wH
i 6¼ w i ; i ¼ 1; 2.
Suppose to the contrary that there exists a PBE where the first-round contributions satisfy:
wiH 6¼ wiL ; i ¼ 1; 2. For s ¼ H, L, let csi denote the equilibrium contribution of player i of type s
in the second round if the first-round contributions are ðw1s ; w2s Þ (and therefore both players are
revealed to be of identical type s). It follows that
cs1 þ cs2 ¼ maxfxs ðw1s þ w2s Þ; 0g; s ¼ H ; L:
ð9Þ
Consider the incentive compatibility of an H-type player i. The equilibrium payoff of the player
is ½pj Ei þ ð1 pj ÞE^i where
Ei ¼ tH V ðmaxfxH ; wiH þ wjH gÞ ðwiH þ cH
i Þ
ð10Þ
is the payoff to player i when the other player j is of type H, and
E^i ¼ tH V ðmaxfxH ; wiH þ wjL gÞ ðmaxfxH ; wiH þ wjL g wjL Þ
ð11Þ
is the payoff to player i when the other player j is of type L. Now, consider a deviation by player i
of type H where he contributes wiL in the first round (thus pretending to be an L-type player). If
the state of nature is such that the other player j is revealed to be of type H in the first round, then
in the second round player j (as part of his PBE strategy, given the first-round contributions and
believing player i to be of type L) makes the contribution needed (if at all) to ensure that the
total contribution is at least as large as xH ; knowing this, player i rationally contributes 0 in the
second round, so that player is (deviation) payoff in this state is
Di ¼ tH V ðmaxfxH ; wiL þ wjH gÞ wiL :
ð12Þ
Further, as contributing zero is a best response for player i in the second round, his payoff is at
least as large as what he would obtain if his second-round contribution is set at
maxf0; maxðxH ; wiH þ wjH Þ maxðxH ; wiL þ wjH Þg i.e.,
Di tH V ðmaxfxH ; wiH þ wjH ; wiL þ wjH gÞ wiL
maxf0; maxðxH ; wiH þ wjH Þ maxðxH ; wiL þ wjH Þg:
ð13Þ
From (10) and (12) :
Di Ei ¼ ½tH V ðmaxfxH ; wiL þ wjH gÞ tH V ðmaxfxH ; wiH þ wjH gÞ
L
þ ðwiH þ cH
i wi Þ:
ð14Þ
If the state of nature is such that the other player j is revealed to be of L-type in the first round,
then in the continuation play in the second round, player j contributes cLj (as part of his PBE
strategy, believing player i to be of type L and given the first-round contributions); knowing this,
player i contributes his best response to this; let D^i denote the (deviation) payoff obtained by
player i when he plays this best response. Then D^i must be at least as large as what player i obtains
if he contributes maxf0; ½maxðxH ; wiH þ wjL Þ ðwiL þ wjL þ cLj Þg in the second round. Thus,
Ó The Author(s). Journal compilation Ó Royal Economic Society 2008
78
THE ECONOMIC JOURNAL
[JANUARY
D^i tH V ðmaxfxH ; wiH þ wjL ; wiL þ wjL þ cLj gÞ
½maxfxH ; wiH þ wjL ; wiL þ wjL þ cLj g wjL cLj :
ð15Þ
The expected payoff to player i of type H from the entire deviation strategy is
½pj Di þ ð1 pj ÞD^i .
Let w be the function defined by:
wðxÞ ¼ tH V ðxÞ x:
Note that w(Æ) is strictly concave and attains its maximum at xH.
First, suppose that
wiH > wiL for some i:
ð16Þ
Choose i such that (16) holds. We will show that Di Ei > 0. To see this note that using (16)
and (13), Di ½tH V ðmaxfxH ; wiH þ wjH gÞ wiL maxf0; maxðxH ; wiH þ wjH Þ maxðxH ; wiL þ wjH Þg
so that using (10):
L
H
H
L
H
Di Ei ðwiH þ cH
i Þ ½wi þ maxf0; maxðxH ; wi þ wj Þ maxðxH ; wi þ wj Þg
( H
H
L
H
H
L
H
wi þ ci wi > 0;
if wi þ wj xH ð as wi þ wj < wiH þ wjH Þ;
¼
L
H
if wiL þ wjH < xH < wiH þ wjH .
cH
i wi wj þ xH > 0;
If xH wiL þ wjH < wiH þ wjH , then from (14):
L
Di Ei ¼ tH V ðwiL þ wjH Þ tH V ðwiH þ wjH Þ þ ðwiH þ cH
i wi Þ
¼ wðwiL þ wjH Þ wðwiH þ wjH Þ þ cH
i > 0:
Next, we consider a situation where
w1L þ w2L < xH ;
ð17Þ
w1H < w1L ; w2H < w2L
ð18Þ
but (16) does not hold i.e.,
w1H
w1L ,
w2H
w2L ).
6¼
6¼
We will show that under (18) and (17), Di > Ei
(note that by hypothesis:
for some i. To establish this, we first claim that there exists some i such that:
L
wiH þ cH
i wi > 0:
To see this, suppose not. Then,
L
H
H
L
w1H þ cH
1 w1 þ w2 þ c2 w2 0;
and using (9) and (17) we have
ðw1H þ w2H Þ þ maxfxH ðw1H þ w2H Þ; 0g w1L þ w2L < xH ;
a contradiction. Choose player i such that (19) holds. Then, from (14) we have
L
Di Ei ¼ ½tH V ðmaxfxH ; wiL þ wjH gÞ tH V ðmaxfxH ; wiH þ wjH gÞ þ ðwiH þ cH
i wi Þ
L
L
H
ðwiH þ cH
i wi Þ; as V ðÞ is increasing and wi > wi
> 0; using ð19Þ:
Hence, Di > Ei. Thus, we have established that if either (16) or (17) holds, then
Ó The Author(s). Journal compilation Ó Royal Economic Society 2008
ð19Þ
2008 ]
REPEATED CHARITABLE CONTRIBUTIONS
79
ð20Þ
Di > Ei for some i:
We will now show that if either (16) or (17) holds, then
wiL
wjL
wiH
D^i E^i ;
i ¼ 1; 2:
cLj
maxfxL ; wiL
ð21Þ
wjL g
To see this, first note that
þ
þ
þ
so that if (16) or (17) holds,
then wiL þ wjL þ cLj maxfxH ;
þ wjL g so that from (15) we have: D^i ½tH V ðmaxfxH ; wiH þ wjL gÞ ðmaxfxH ; wiH þ wjL g wjL cLj Þ and combining this with (11), it
follows that D^i E^i cLj 0. This establishes (21). Combining (20) and (21) (and using
0 < pj < 1), we have that if either (16) or (17) holds, there exists some i such that
pj Di þ ð1 pj ÞD^i > pj Ei þ ð1 pj ÞE^i
i.e., (some) agent i of type H has a strict incentive to deviate.
In what follows, we consider the situation where neither (16) nor (17) holds. This implies that
(18) holds and, in addition,
w1L þ w2L xH :
ð22Þ
We will show that some L-type player has a strict incentive to deviate. Let this hypothetical
L-type player be player i. On the equilibrium path, if player j 6¼ i is of type H then in the second
round only player j makes the necessary contribution to raise total contribution (in the two
rounds together) to xH ; player i makes no contribution in the second round, and the payoff to
player i is given by:
ei ¼ tL V ðmaxfxH ; wiL þ wjH gÞ wiL :
ð23Þ
If player j is of type L, then no contribution is made in the second round (use (22) and the fact
that xH > xL) and the payoff to player i is given by
e^i ¼ tL V ðwiL þ wjL Þ wiL :
ð24Þ
Now, suppose player i of type L deviates and pretends to be of H-type by contributing wiH in the
first round. In the state of nature where player j is of type H, in the continuation game (second
round), player j will contribute cH
j (as part of his PBE strategy, given the first-round contributions
and believing player i to be of type H) and, knowing this, player i contributes
maxfxL ðwiH þ wjH þ cH
j Þ; 0g in best response so that his payoff is
H
H
H
H
di ¼ tL V ðmaxfxL ; wiH þ wjH þ cH
j gÞ ðwi þ maxfxL ðwi þ wj þ cj Þ; 0gÞ:
ð25Þ
In the state of nature where player j is of type L, he will make no contribution in the second
round (as part of his PBE strategy, given his belief that player i is of type H), and so in the second
round player i will contribute maxfxL ðwiH þ wjL Þ; 0g and the deviation payoff to player i in
this state is given by:
d^i ¼ tL V ðmaxfxL ; wiH þ wjL gÞ ðwiH þ maxf0; xL ðwiH þ wjL ÞgÞ:
ð26Þ
Let u be the function defined by:
uðxÞ ¼ tL V ðxÞ x:
Note that u is strictly concave and attains its (unique) maximum at xL.
First, we claim that for some i 6¼ j,
H
wiL xH þ cH
j þ wj 0:
ð27Þ
H
H
H
To see (27), suppose not. Then, ðw1L þ w2L Þ 2xH þ cH
1 þ c2 þ w1 þ w2 < 0, and using
L
L
(9) we have: ðw1 þ w2 Þ xH < 0, which contradicts (22).
Ó The Author(s). Journal compilation Ó Royal Economic Society 2008
80
THE ECONOMIC JOURNAL
[JANUARY
Choose i such that (27) holds. Consider player i when he is of type L. Our first step is to show
that
di ei :
ð28Þ
From (23) and (25), we have:
L
H
di ei ¼ tL V ðmaxfxL ; wiH þ wjH þ cH
j gÞ tL V ðmaxfxH ; wi þ wj gÞ
þ ½wiL wiH maxfxL ðwiH þ wjH þ cH
j Þ; 0g:
ð29Þ
We will now establish that
L
H
di ei wiL þ wjH þ cH
j maxfxH ; wi þ wj g:
ð30Þ
wiH þ wjH þ cH
j xL :
ð31Þ
To see this, suppose
Then, from (29) we have:
di ei ¼ ½uðxL Þ uðmaxfxH ; wiL þ wjH gÞ
maxfxH ; wiL þ wjH g þ wiL þ wjH þ cH
j
L
H
wiL þ wjH þ cH
j maxfxH ; wi þ wj g; as xL < xH :
ð32Þ
On the other hand, if (31) does not hold i.e.,
wiH þ wjH þ cH
j > xL ;
ð33Þ
L
H
L
H
di ei ¼ tL V ðwiH þ wjH þ cH
j Þ tL V ðmaxfxH ; wi þ wj gÞ þ wi wi :
ð34Þ
then from (29) we have:
We claim that (33) implies that
L
H
xL < wiH þ wjH þ cH
j maxfxH ; wi þ wj g:
ð35Þ
To see the second inequality in (36) suppose to the contrary that
L
H
wiH þ wjH þ cH
j > maxfxH ; wi þ wj g:
ð36Þ
H
H
H
L
H
so that cH
Then, using (18), wiL þ wjH þ cH
j > wi þ wj þ cj > wi þ wj
j > 0 which
implies wiH þ wjH þ cH
x
that,
in
turn,
contradicts
(36).
Using
(35)
in (34) we have:
H
j
L
H
di ei ¼ uðwiH þ wjH þ cH
j Þ uðmaxfxH ; wi þ wj gÞ
L
H
þ ½wiL þ wjH þ cH
j maxfxH ; wi þ wj g
L
H
0 þ ½wiL þ wjH þ cH
j maxfxH ; wi þ wj g; using ð35Þ:
From (32) and (37), we have that in general, (30) holds. Observe that
L
H
wiL þ wjH þ cH
j maxfxH ; wi þ wj g
( H
cj 0;
if xH wiL þ wjH ;
¼
L
H
H
wi þ wj þ cj xH 0; if xH wiL þ wjH ðuse ð27ÞÞ,
so that (30) implies that (28) holds.
Next, we show that
Ó The Author(s). Journal compilation Ó Royal Economic Society 2008
ð37Þ
2008 ]
81
REPEATED CHARITABLE CONTRIBUTIONS
d^i > e^i :
ð38Þ
From (24) and (26),
d^i e^i ¼ ½tL V ðmaxfxL ; wiH þ wjL gÞ tL V ðwiL þ wjL Þ
þ wiL wiH maxf0; xL ðwiH þ wjL Þg:
ð39Þ
First, suppose that wiH þ wjL xL . Then, from (39) we have
d^i e^i ¼ ½tL V ðxL Þ tL V ðwiL þ wjL Þ xL þ wiL þ wjL
¼ ½uðxL Þ uðwiL þ wjL Þ > 0;
using ð22Þ and xH > xL :
Next, suppose that wiH þ wjL > xL . Then,
d^i e^i ¼ ½tL V ðwiH þ wjL Þ tL V ðwiL þ wjL Þ þ wiL wiH
¼ uðwiH þ wjL Þ uðwiL þ wjL Þ
> 0;
as(using (18)) wiL þ wjL > wiH þ wjL > xL :
Thus, (38) holds. Combining (28) and (38) we have that under (18) and (22),
pj di þ ð1 pj Þd^i > pj ei þ ð1 pj Þ^
ei
i.e., some player i of type L has a strict incentive to deviate. This completes the proof of the fact
that there is no PBE where the first-round contributions satisfy: wiH 6¼ wiL ; i ¼ 1; 2.
Next, we show that there does not exist a PBE where the first-round contributions satisfy:
wiH 6¼ wiL and wjH ¼ wjL , i 6¼ j.
Suppose, to the contrary, that there is a PBE in which, in Round 1, player 1 contributes
w1H 6¼ w1L , and player 2 makes a pooling contribution w2H ¼ w2L ¼ x2 0. It is easy to check
(which we omit) that x2 < xH. We claim that
w1L < xH x2 :
ð40Þ
To see this note that an L-type player 1 can always get a payoff at least
[tLV (maxfxL, x2g) maxfxL x2, 0g] by contributing maxfxL x2, 0g in Round 1 (with
player 2 contributing x2). If an L-type player 1 contributes w1L xH x2 in Round 1, he reveals
his type successfully but does not elicit any contribution from the other player in Round 2 (in
Round 2, both players contribute 0 as existing public good weakly exceeds the standalone levels
for both types) so that his payoff is ½tL V ðw1L þ x2 Þ w1L which is less than
[tLV(maxfxL, x2g) maxfxL x2, 0g], because w1L xH x2 > maxfxL x2 ; 0g and z ¼
maxfxL x2, 0g solves maxz 0 tLV(z þ x2) z.
Next, we claim that if (40) holds then an H-type player 1 has an incentive to deviate and
contribute w1L instead of w1H in the first round. To see this, observe that if an H-type player 1
contributes w1H then his type is revealed. In the second round, player 2 will have no incentive to
contribute whatever be his type as he knows that he can rely on player 1 to contribute
½xH ðw1H þ x2 Þ and the latter is indeed the best response of player 1 of type H in Round 2 if
he expects player 2 to contribute 0 with probability 1.24 Therefore, the payoff to player 1 when he
24
To see that player 2 will free-ride completely in Round 2, suppose not. Suppose z < xH w1H x2 so
that z2H ¼ xH w1H x2 z > 0; also, z2L ¼ maxðxL w1H x2 z; 0Þ. This means total contribution
in the two rounds together is xH if player 2 is of type H, and maxðxL ; w1H þ x2 þ zÞ < xH (the inequality
follows by our contraposition hypothesis) if player 2 is of type L . Now write player 2s first-order condition in
the second round with z 0, as follows:
tH ½p2 V 0 ðz þ w1H þ x2 þ z2H Þ þ ð1 p2 ÞV 0 ðmaxfxL ; w1H þ x2 þ zgÞ 1;
i.e., tH ½p2 V 0 ðxH Þ þ ð1 p2 ÞV 0 ðmaxfxL ; w1H þ x2 þ zgÞ 1:
ðÞ
But, tHV 0 (xH) ¼ 1 and tH V 0 ðmaxfxL ; w1H þ x2 þ zgÞ > 1, so (*) cannot hold for any p2 < 1, a contradiction.
Ó The Author(s). Journal compilation Ó Royal Economic Society 2008
82
[JANUARY
THE ECONOMIC JOURNAL
w1H
w1H
w1H
w1H
contributes
in Round 1 is ½tH V ðmaxfxH ;
þ x2 gÞ maxf0; xH x2 g. On
the other hand, if he mimics his L-type action and chooses w1L in Round 1, then we have a
simultaneous move game in Round 2 where player 2 believes that player 1 is of type L with
probability 1. The continuation payoff for player 1 (in any Bayes-Nash equilibrium of the
simultaneous move game where w1L þ x2 amount of public good is already provided and with the
stated beliefs) must be equal to:
max ftH ½p2 V ðz2H þ w1L þ x2 þ zÞ þ ð1 p2 ÞV ðz2L þ w1L þ x2 þ zÞ ðw1L þ zÞg
z0
tH fp2 V ðz2H þ w1L þ x2 þ maxfxH w1L x2 ; 0gÞ
þ ð1 p2 ÞV ðz2L þ w1L þ x2 þ maxfxH w1L x2 ; 0gÞg
½w1L þ maxfxH w1L x2 ; 0g
¼ tH ½p2 V ðz2H þ xH x2 Þ þ ð1 p2 ÞV ðz2L þ xH x2 Þ ðxH x2 Þ ðusing ð40ÞÞ
> tH V ðxH x2 Þ ðxH x2 Þ:
The last inequality follows from the fact that z2H > 0, i.e., player 2 will make a positive contribution in Round 2 if he is of H-type.25 Thus, player 1 of type H is strictly better off by deviating.
This concludes the proof that player 1s type is never revealed in equilibrium.
h
Proof of Lemma 2. Consider the second-round game for any x 2 ½~y ; xH Þ. We want to show that
H
L L
L
L
H
H
there exists a Bayes-Nash equilibrium ðzH
1 ; z2 ; z1 ; z2 Þ where z1 ¼ z2 ¼ 0 and z1 > 0; z2 > 0. To
i
see this, consider the reaction function R (z) of player i of type H when player j 6¼ i of type H
contributes z and assuming both low types contribute 0. Consider the maximisation problem of
agent i of type H
max
0zi xH x
pj tH V ðzi þ x þ zÞ þ ð1 pj ÞtH V ðzi þ xÞ zi :
ð41Þ
This problem has a unique solution zi ¼ R i ðzÞ for each z and the maximand is continuous in z . From the Maximum Theorem it follows that Ri(z) is continuous in z . It is obvious
that
R i ð0Þ ¼ xH x;
i ¼ 1; 2:
ð42Þ
i
Note that R (0) is identical for i ¼ 1, 2. Let zi be uniquely defined by
pj tH V 0 ð
zi þ xÞ þ ð1 pj ÞtH V 0 ðxÞ ¼ 1:
Then,
R i ðzÞ ¼ 0; z zi :
ð43Þ
Note that zi may differ between i ¼ 1 and i ¼ 2. It is easy to check that
zi > xH x ¼ R j ð0Þ:
ð44Þ
Finally, the strict concavity of V(Æ) implies that the maximisation problem in (41) is strictly
submodular in (zi, z) so that for z 2 ½0; zi Þ; R i ðzÞ is a strictly decreasing function (you can also see
it by differentiating the relevant first-order condition implicitly). This also implies Ri(z) is
invertible on ½0; zi Þ. Choose > 0 small enough. Consider the function f(z) ¼ R1(z) (R2)1(z)
on [, xH x]. Then, f(Æ) is continuous on its domain and one can verify that (if is close
enough to zero)
25
Suppose, to the contrary, that z2H ¼ 0. Then, it must be true that z2L ¼ 0. As player 2 believes that player
1 is an L-type player with probability one, he must rationally expect z1L maxfxL w1L x2 ; 0g. However,
as xL < xH and w1L < xH x2 (as in (40)), it must be that tH V 0 ðz1L þ w1L þ x2 Þ > tH V 0 ðxH Þ ¼ 1 which
contradicts z2H ¼ 0.
Ó The Author(s). Journal compilation Ó Royal Economic Society 2008
2008 ]
83
REPEATED CHARITABLE CONTRIBUTIONS
f ðÞ ¼ R 1 ðÞ ðR 2 Þ1 ðÞ ’ ðxH xÞ z2 < 0; using ð42Þ; ð43Þ and ð44Þ:
On the other hand,
f ðxH xÞ ¼ R 1 ðxH xÞ ðR 2 Þ1 ðxH xÞ
¼ R 1 ðxH xÞ 0; using ð42Þ
> 0; using ð43Þ and the fact that R 1 ðzÞ is strictly decreasing:
Using the intermediate value theorem, there exists z 2 (, xH x) such that f(z ) ¼ 0 i.e.,
R 2 ½R 1 ðz Þ ¼ z
and thus, setting z2H ¼ z ; z1H ¼ R 1 ðz Þ we have the Bayes-Nash equilibrium provided we can also
show that z1L ¼ z2L ¼ 0 is the best response of the L-types given contributions ðz1H ; z2H Þ. Clearly
z2H > 0. Given that xH x < zð¼ ðR 1 Þ1 ð0ÞÞ and R1(z) is strictly decreasing, it follows that
z1H ¼ R 1 ðz Þ > 0.
Now we will argue that z1L ¼ z2L ¼ 0. The first-order conditions corresponding to the BayesNash equilibrium of the one-shot simultaneous contribution game are as follows:
y1L 0 : p2 tL V 0 ðy1L þ y2H Þ þ ð1 p2 ÞtL V 0 ðy1L þ y2L Þ 1
ð45Þ
y1H > 0 : p2 tH V 0 ðy1H þ y2H Þ þ ð1 p2 ÞtH V 0 ðy1H þ y2L Þ ¼ 1
ð46Þ
y2L 0 : p1 tL V 0 ðy1H þ y2L Þ þ ð1 p1 ÞtL V 0 ðy1L þ y2L Þ 1
ð47Þ
y2H > 0 : p1 tH V 0 ðy1H þ y2H Þ þ ð1 p1 ÞtH V 0 ðy1L þ y2H Þ ¼ 1;
ð48Þ
y1L
y2L .
and
with equalities holding (instead of weak inequalities) for strictly positive-valued
For any x 2 ½~
y; xH Þ, the configurations z1L ¼ z2L ¼ 0; z1H > 0; z2H > 0 will be Bayes-Nash
equilibrium in the continuation game in Round 2 if these satisfy:
z1L ¼ 0 : p2 tL V 0 ðx þ z2H Þ þ ð1 p2 ÞtL V 0 ðxÞ 1
ð49Þ
z1H > 0 : p2 tH V 0 ðx þ z1H þ z2H Þ þ ð1 p2 ÞtH V 0 ðx þ z1H Þ ¼ 1
ð50Þ
z2L ¼ 0 : p1 tL V 0 ðx þ z1H Þ þ ð1 p1 ÞtL V 0 ðxÞ 1
ð51Þ
z2H > 0 : p1 tH V 0 ðx þ z1H þ z2H Þ þ ð1 p1 ÞtH V 0 ðx þ z2H Þ ¼ 1:
ð52Þ
Note particularly that of the last four conditions, (50) and (52) hold by our arguments in the
initial part of this proof that z1H > 0; z2H > 0. So our remaining task is to show that the other two
conditions, (49) and (51), also hold.
We claim that x þ z1H y1H þ y2L . Suppose not so that
x þ z1H < y1H þ y2L :
ð53Þ
Then using the two sets of first-order conditions above (assume for the time being that the
second set of first-order conditions also hold) and the fact that V 00 (Æ) < 0, it follows that
x þ z1H þ z2H > y1H þ y2H
ð54Þ
x þ z2H < y2H þ y1L :
ð55Þ
By adding the respective sides of (53) and (55) we obtain
Ó The Author(s). Journal compilation Ó Royal Economic Society 2008
84
THE ECONOMIC JOURNAL
[JANUARY
x þ x þ z1H þ z2H < y1H þ y2H þ y1L þ y2L :
ð56Þ
Given (54), from (56) it must follow that
x < y1L þ y2L ;
contradicting x y~.
Similarly, x þ z2H y2H þ y1L . Now finally use the results that x y1L þ y2L ,
x þ z1H y1H þ y2L along with conditions (45), (46) and (50) and the fact that V(Æ) is strictly
concave to conclude that the condition (49) must be satisfied. Similarly, using the results that
x y1L þ y2L , x þ z2H y2H þ y1L along with conditions (47), (48) and (52) and V(Æ) strictly
concave to conclude that the condition (51) must be satisfied. This completes the argument that
given any x 2 ½~
y; xH Þ, the configurations z1L ¼ z2L ¼ 0; z1H > 0; z2H > 0 constitute a Bayes-Nash
equilibrium in the continuation game in Round 2.
h
Proof of Lemma 3. Let zsi denote the equilibrium contribution by player i of type s in the
second round (on the equilibrium path). If zLi > 0 for some i, then it can be shown that
there exists a Bayes-Nash equilibrium of the one-shot game where the contributions are
^yis ¼ zsi þ x; ^yjs ¼ zsj ; j 6¼ i, and in that equilibrium the total contribution in the one-shot
game when both players are of L-type is strictly greater than x and this violates x > ~y and the
definition of ~y .
Next we show more generally that, in equilibrium, following any total contribution x 0 in
Round 1, in the second round z1H > 0; z2H > 0. First, we argue that x < xH. It is obvious that
x > xH is not possible. If x ¼ xH, suppose without loss of generality that x1 > 0. Then, x2 < xH.
In such perfect Bayesian equilibrium (where x ¼ xH), neither player of any type contributes in
the second round. The equilibrium payoff of player 1 when he is of L-type is:
tL V ðxH Þ x1 :
We show that player 1 of type L benefits by reducing his Round 1 contribution to
maxfxL x2, 0g. To see this, consider the worst scenario in Round 2 that could follow from such
deviation viz., that player 2 of either type contributes zero in Round 2. Then, the deviation payoff
of player 1 when he is of L-type is:
tL V ðmaxfxL ; x2 gÞ maxfxL x2 ; 0g
¼ ½tL V ðmaxfxL ; x2 gÞ maxfxL ; x2 g þ ½maxfxL ; x2 g maxfxL x2 ; 0g
¼ ½tL V ðmaxfxL ; x2 gÞ maxfxL ; x2 g þ x2
> ½tL V ðxH Þ xH þ x2
¼ ½tL V ðxH Þ x1 ;
ðsince x2 < xH Þ
which is his equilibrium payoff. Thus, 0 x < xH. Suppose that contrary to the Proposition,
ziH ¼ 0 for some i. Then clearly ziL ¼ 0 (apply the first-order condition). In response player j
would choose zjL ¼ maxfxL x; 0g; zjH ¼ xH x. Then H-type player 1s expected marginal
utility from choosing ziH ¼ 0 is
pj tH V 0 ðxH Þ þð1 pj Þ tH V 0 ðmaxfxL ; xgÞ > 1;
|fflfflfflfflfflffl{zfflfflfflfflfflffl}
|fflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}
¼1
ðsince 0 < pj < 1Þ
>1
which leads to a contradiction.
Now write the first-order conditions for the H-types in the second round of the two-rounds
contribution game:
z2H > 0 : p1 tH V 0 ðz2H þ z1H þ xÞ þ ð1 p1 ÞtH V 0 ðz2H þ xÞ ¼ 1
ð57Þ
z1H > 0 : p2 tH V 0 ðz1H þ z2H þ xÞ þ ð1 p2 ÞtH V 0 ðz1H þ xÞ ¼ 1:
ð58Þ
Ó The Author(s). Journal compilation Ó Royal Economic Society 2008
2008 ]
REPEATED CHARITABLE CONTRIBUTIONS
85
The first-order conditions for the H-types in the one-shot contribution game are:
y2H > 0 : p1 tH V 0 ðy2H þ y1H Þ þ ð1 p1 ÞtH V 0 ðy2H þ y1L Þ ¼ 1
ð59Þ
y1H > 0 : p2 tH V 0 ðy1H þ y2H Þ þ ð1 p2 ÞtH V 0 ðy1H þ y2L Þ ¼ 1:
ð60Þ
Suppose that (5) does not hold, i.e.,
z1H þ x y1H þ y2L :
ð61Þ
Comparing (58) and (60) and using (61), it follows that
z1H þ z2H þ x y1H þ y2H :
ð62Þ
Then, comparing (57) and (59), it follows that
z2H þ x y2H þ y1L :
ð63Þ
Combining (61) and (63) yields z1H þ z2H þ 2x y1H þ y2H þ y1L þ y2L so that
z1H þ z2H þ x y1H þ y2H þ ðy1L þ y2L xÞ
< y1H þ y2H ; as y~ < x
which contradicts (62). Thus, (5) holds. By comparing (58) and (60), we see that (6) holds.
Further, comparing (57) and (59), it follows that z2H þ xL > y2H þ y1L i.e., (7) holds.
h
Lemma 4. Let S be a convex subset of R and f : S ! R is concave and differentiable with f 0 < 0.
Suppose that for some p 2 [0, 1] and a, b, c, d 2 S, a < b < c < d
pf ðaÞ þ ð1 pÞf ðdÞ ¼ pf ðbÞ þ ð1 pÞf ðcÞ:
ð64Þ
Then,
pa þ ð1 pÞd pb þ ð1 pÞc:
Proof. From (64) we have,
p½f ðaÞ f ðbÞ ¼ ð1 pÞ½f ðcÞ f ðdÞ:
ð65Þ
Since f is concave and differentiable,
f ðaÞ f ðbÞ f 0 ðbÞða bÞ
f ðcÞ f ðdÞ f 0 ðcÞðc dÞ
so that (65) yields,
pf 0 ðbÞða bÞ ð1 pÞf 0 ðcÞðc dÞ
so that
pða bÞ
f 0 ðcÞ
0
1 ðnote c d < 0; f 0 < 0Þ
ð1 pÞðc dÞ f ðbÞ
and this implies (again using c d < 0)
pða bÞ ð1 pÞðc dÞ
i.e.,
pa þ ð1 pÞd pb þ ð1 pÞc:
Ó The Author(s). Journal compilation Ó Royal Economic Society 2008
(
86
THE ECONOMIC JOURNAL
[JANUARY
Proof of Proposition 2. The expected total contribution in the constructed equilibrium of the
two-rounds game equals
x þ p1 z1H þ p2 z2H :
ð66Þ
The expected total contribution in any equilibrium (y1H ; y2H ; y1L ; y2L Þ of the one-shot game
equals
p1 y1H þ p2 y2H þ ð1 p2 Þy2L þ ð1 p1 Þy1L :
ð67Þ
We will first consider the case where (i) holds. Comparing (57) and (59) and using Lemma 4,
the expected total contribution corresponding to (57), p1 ðz2H þ z1H þ xÞ þ ð1 p1 Þðz2H þ xÞ,
must (weakly) exceed the expected total contribution corresponding to (59), p1 ðy2H þ y1H Þ þ
ð1 p1 Þðy2H þ y1L Þ, because V 0 (Æ) is concave and decreasing. A similar relationship between the
expected totals will hold when comparing (58) and (60). In summary,
p1 ðz2H þ z1H þ xÞ þ ð1 p1 Þðz2H þ xÞ p1 ðy2H þ y1H Þ þ ð1 p1 Þðy2H þ y1L Þ
ð68Þ
p2 ðz1H þ z2H þ xÞ þ ð1 p2 Þðz1H þ xÞ p2 ðy1H þ y2H Þ þ ð1 p2 Þðy1H þ y2L Þ:
ð69Þ
Adding the respective sides of (68) and (69) and comparing, yields
ðp1 z1H þ p2 z2H þ xÞ þ ðz1H þ z2H þ xÞ ½p1 y1H þ p2 y2H þ ð1 p2 Þy2L þ ð1 p1 Þy1L þ ðy1H þ y2H Þ:
Using (6), it follows that
p1 z1H þ p2 z2H þ x > p1 y1H þ p2 y2H þ ð1 p2 Þy2L þ ð1 p1 Þy1L ;
which establishes the Proposition.
Next, consider the case where (ii) holds. Observe that
x > y~ y2L þ y1L :
ð70Þ
Multiplying both sides of (5) by p1 and both sides of (7) by p2 and adding them we obtain:
p1 z1H þ p2 z2H þ ðp1 þ p2 Þx > p1 y1H þ p2 y2H þ p1 y2L þ p2 y1L
so that
p1 z1H þ p2 z2H þ x > p1 y1H þ p2 y2H þ p1 y2L þ p2 y1L þ ½1 ðp1 þ p2 Þx
¼ p1 y1H þ p2 y2H þ ð1 p2 Þy2L þ ð1 p1 Þy1L
þ ½1 ðp1 þ p2 Þ½x ðy2L þ y1L Þ
p1 y1H þ p2 y2H þ ð1 p2 Þy2L þ ð1 p1 Þy1L
since p1 þ p2 1 and (70) holds. The proof is complete.
h
Proof of Proposition 3. In the proof of Lemma 3 we already established that ziH > 0; i ¼ 1; 2.
So write the first-order conditions for any equilibrium of the two-rounds contribution game with
x > 0 as the total contribution in Round 1, as follows:
z1L 0 : p2 tL V 0 ðx þ z1L þ z2H Þ þ ð1 p2 ÞtL V 0 ðx þ z1L þ z2L Þ 1
ð71Þ
z1H > 0 : p2 tH V 0 ðx þ z1H þ z2H Þ þ ð1 p2 ÞtH V 0 ðx þ z1H þ z2L Þ ¼ 1
ð72Þ
z2L 0 : p1 tL V 0 ðx þ z1H þ z2L Þ þ ð1 p1 ÞtL V 0 ðx þ z1L þ z2L Þ 1
ð73Þ
Ó The Author(s). Journal compilation Ó Royal Economic Society 2008
2008 ]
REPEATED CHARITABLE CONTRIBUTIONS
z2H
0
> 0 : p1 tH V ðx þ
z1H
þ
z2H Þ
0
þ ð1 p1 ÞtH V ðx þ
z1L
þ
z2H Þ
¼ 1;
87
ð74Þ
where (71) and (73) hold with equality if z1L > 0; z2L > 0 respectively.
The first-order conditions for an equilibrium of the one-shot contribution game are as follows:
y1L 0 : p2 tL V 0 ðy1L þ y2H Þ þ ð1 p2 ÞtL V 0 ðy1L þ y2L Þ 1
ð75Þ
y1H > 0 : p2 tH V 0 ðy1H þ y2H Þ þ ð1 p2 ÞtH V 0 ðy1H þ y2L Þ ¼ 1
ð76Þ
y2L 0 : p1 tL V 0 ðy1H þ y2L Þ þ ð1 p1 ÞtL V 0 ðy1L þ y2L Þ 1
ð77Þ
y2H > 0 : p1 tH V 0 ðy1H þ y2H Þ þ ð1 p1 ÞtH V 0 ðy1L þ y2H Þ ¼ 1;
ð78Þ
with (75) and (77) holding with equality if y1L > 0; y2L > 0 respectively.
Suppose z2L > 0 so that (73) holds with equality. We will argue that for any equilibrium profile
ðx; z1L ; z1H ; z2L ; z2H Þ in the two-rounds game, there is a corresponding equilibrium in the one-shot
game with the same expected total contribution. To see this, set y1L ¼ z1L ; y1H ¼ z1H ;
y2L ¼ z2L þ x; y2H ¼ z2H þ x so that the conditions (71)–(74) translate into exactly the conditions
(75)–(78). Note that y2L > 0 for x > 0 and condition (77) must hold with equality. But
since we assumed z2L > 0, satisfying (77) with equality is guaranteed by the condition (73). By
construction, the equilibrium of the one-shot game yields the same total contribution as the
two-rounds game for each combination of player types. Thus, the two game forms yield identical
expected total contributions. The above argument shows that for the two-rounds contribution
game to yield a strictly higher expected total contribution (compared to the one-shot game),
it must be that z2L ¼ z1L ¼ 0.
h
Proof of Proposition 4. Given the charity’s information transmission strategy, the players updated beliefs (updated from the priors p1 and p2 according to Bayes rule, wherever applicable)
are denoted by p^1 and p^2 . Denote the players contributions following non-announcement of types by
y~is ; i ¼ 1; 2; s ¼ H ; L, and the contributions following announcement of types by yis ; i ¼ 1; 2;
s ¼ H ; L. The introduction strategies are summarised in Table 1. In order for an equilibrium
of the game C to be less than fully revealing, the concealment set must contain at least two
elements. The strategies (C6, A6) and (C11, A11) are clearly revealing. Below we verify that either
the concealing strategies in Table 1 can be eliminated in equilibrium, or when there is a concealing equilibrium it yields the same total contribution compared to the situation where the
charity always reveals information about donor types.
Ruling out (C1, A1). We claim that at least one of the following must hold: y~1L þ y~2L < xL ,
y~1L þ y~2H < xH , y~1H þ y~2L < xH . Suppose not so that y~1L þ y~2L xL and y~1L þ y~2H xH and
y~1H þ y~2L xH . Since y~1L þ y~2L xL > 0, at least one of y~1L ; y~2L must be positive. Then either
L-type player 1s or L-type player 2s first-order condition (or possibly both) must be violated:
p^2 tL V 0 ð~
y1L þ y~2H Þ þ ð1 p^2 ÞtL V 0 ð~
y1L þ y~2L Þ < 1;
y2L þ y~1H Þ þ ð1 p^1 ÞtL V 0 ð~
y2L þ y~1L Þ < 1;
p^1 tL V 0 ð~
because xH > xL and p^1 ¼ p1 > 0; p^2 ¼ p2 > 0; a contradiction. Thus, our claim is established
and the charity benefits by deviating and announcing at least one of the following type
realisations, (L, L), (L, H), (H, L), that results in a higher total contribution. This shows that the
strategy (C1, A1) cannot be part of an equilibrium.
Ruling out (C2, A2). We claim that at least one of the following must hold: y~1L þ y~2H < xH ,
y~1H þ y~2L < xH . Suppose not so that both y~1L þ y~2H xH and y~1H þ y~2L xH . Since in the proposed equilibrium an L-type player 1 knows for sure that he is facing an H-type player 2,
Ó The Author(s). Journal compilation Ó Royal Economic Society 2008
88
THE ECONOMIC JOURNAL
[JANUARY
Table 1
Type Announcement
Introduction strategy
Remark
C1 ¼ f(L, H), (H, L), (H, H), (L, L)g
A1 ¼ ;
C2 ¼ f(L, H), (H, L), (H, H)g
A2 ¼ f(L, L)g
C3 ¼ f(L, H), (H, L), (L, L)g
A3 ¼ f(H, H)g
C4 ¼ f(L, L), (H, L), (H, H)g
A4 ¼ f(L, H)g
C5 ¼ f(L, L), (L, H), (H, H)g
A5 ¼ f(H, L)g
C6 ¼ f(L, H), (H, L)g
A6 ¼ f(H, H), (L, L)g
C7 ¼ f(L, H), (H, H)g
A7 ¼ f(H, L), (L, L)g
C8 ¼ f(H, L), (H, H)g
A8 ¼ f(L, H), (L, L)g
C9 ¼ f(L, H), (L, L)g
A9 ¼ f(H, L), (H, H)g
C10 ¼ f(H, L), (L, L)g
A10 ¼ f(L, H), (H, H)g
C11 ¼ f(H, H), (L, L)g
A11 ¼ f(L, H), (H, L)g
Non-revealing
Fully
revealing
Conceals pl 1s
type
Conceals pl 2s
type
Conceals pl 2s
type
Conceals pl 1s
type
Fully
revealing
y~1L þ y~2H xH must imply that y~1L ¼ 0 (if y~1L > 0, then L-type player 1 is better off lowering
contribution below the proposed equilibrium level of y~1L ), which in turn implies that y~2H xH . By
similar reasoning, y~1H þ y~2L xH must imply y~2L ¼ 0, which in turn implies that y~1H xH > 0.
But then H-type player 1s first-order condition cannot be satisfied:
p^2 tH V 0 ð~
y1H þ y~2H Þ þ ð1 p^2 ÞtH V 0 ð~
y1H Þ < 1;
because y~1H þ y~2H 2xH and y~1H xH and p^2 > 0; a contradiction. Hence our claim is
established and the charity would deviate to announce (L, H) or (H, L) as that would increase
total contribution to xH. Hence (C2, A2) cannot be part of an equilibrium.
Ruling out (C3, A3). We claim that y~1L þ y~2L < xL . Given that in the proposed equilibrium an
H-type player knows for sure that he is facing an L-type of the other player, and an L-type player
never contributes more than xL, it must be that both y~1L þ y~2H ¼ xH and y~1H þ y~2L ¼ xH . Now if
contrary to our claim y~1L þ y~2L xL , then at least one of y~1L ; y~2L must be positive. Then either
L-type player 1 or L-type player 2s first-order condition must be violated:
p^2 tL V 0 ð~
y1L þ y~2H Þ þ ð1 p^2 ÞtL V 0 ð~
y1L þ y~2L Þ <1;
y2L þ y~1H Þ þ ð1 p^1 ÞtL V 0 ð~
y2L þ y~1L Þ <1;
p^1 tL V 0 ð~
because xH > xL and p^1 > 0; p^2 > 0; a contradiction. Hence our claim is established and the
charity would deviate to announce (L, L) as that would increase total contribution to xL. Thus,
(C3, A3) cannot be part of an equilibrium.
(C4, A4). We are going to argue that either (C4, A4) (in equilibrium) yields the same total
contribution for each type-pair realisation as in the case where the charity always reveals the types,
or the strategy (C4, A4) can be ruled out as an equilibrium.
Ó The Author(s). Journal compilation Ó Royal Economic Society 2008
2008 ]
REPEATED CHARITABLE CONTRIBUTIONS
89
y~2L
Suppose
> 0. Then as in the earlier cases analysed above, it must be that one of the
following must hold:
y~1L þ y~2L < xL ;
y~1H þ y~2L < xH ;
y~1H þ y~2H < xH ;
in which case L-type player 2s first-order condition would be violated, implying that (C4, A4)
cannot be part of an equilibrium.
On the other hand, if y~2L ¼ 0 and yet y~1L þ y~2L xL , y~1H þ y~2L xH , y~1H þ y~2L xH (so that a
direct contradiction cannot be achieved), then it must be that y~1H ¼ xH (because y~1H þ y~2L xH
and y~1H xH ), which implies y~2H ¼ 0 (because y~1H þ y~2H xH and y~2H xH ). Finally, y~2L ¼ 0
implies y~1L ¼ xL , because L-type player 1 knows for sure in equilibrium (following nonannouncement) that he is facing an L-type player 2. Thus, if (C4, A4) were to be part of an
equilibrium of C, then it must be that
y~1L þ y~2L ¼ xL ;
y~1H þ y~2L ¼ xH ;
y~1H þ y~2H ¼ xH ;
y~1L þ y~2H ¼ xH ;
yielding the same total contributions as in the case where the charity always reveals the types, and
is thus payoff equivalent.
The case of (C5, A5) is similar to (C4, A4).
(C7, A7). It is easy to see that the contributions ð~
y1L ¼ 0; y~1H ¼ 0; y~2L ¼ 0; y~2H ¼ xH Þ (following
H
non-announcement), and the contributions ðy1 ¼ xH ; y1L ¼ xL ; y2L ¼ 0; y2H ¼ 0Þ (following
announcement) will constitute an equilibrium (with appropriate beliefs), where the charity plays
(C7, A7).26 This equilibrium is payoff equivalent for each type-pair realisation to the equilibrium
in the case where the charity always reveals the types.
On the other hand, suppose y~2H < xH (contrary to the equilibrium just constructed). Then
H
y~1 ¼ xH y~2H and y~1L ¼ maxfxL y~2H ; 0g. But then the charity would deviate and announce
(L, H) (because the total contribution under non-announcement is less than xH), upsetting the
proposed equilibrium (C7, A7).
Thus, either (C7, A7) is part of an equilibrium of C that is payoff equivalent to the case where
the charity always reveals the types, or (C7, A7) is not part of an equilibrium.
The case of (C8, A8) is similar to (C7, A7).
(C9, A9). It is easy to see that the contributions ð~
y1L ¼ 0; y~1H ¼ 0; y~2L ¼ xL ; y~2H ¼ xH Þ and
ðy1H ¼ xH ; y1L ¼ xL ; y2L ¼ 0; y2H ¼ 0Þ will constitute an equilibrium (with appropriate beliefs),
where the charity plays (C9, A9). This equilibrium is payoff equivalent for each type-pair realisation to the equilibrium in the case where the charity always reveals the types.
On the other hand, suppose y~2H < xH (contrary to the equilibrium just constructed). Then
y~1L ¼ maxfxL y~2H ; 0g. But then the charity would deviate and announce (L, H) (because the
total contribution under non-announcement is less than xH), upsetting the proposed equilibrium
(C9, A9).
Suppose now y~2L < xL (another possibility, contrary to the equilibrium constructed above).
Then one cannot have y~1L < xL y~2L because then the charity would deviate and announce type
realisation (L, L), upsetting the equilibrium. So it must be that y~1L ¼ xL y~2L > 0. But then
H-type player 2s best response (who knows that he is facing an L-type player 1, in equilibrium)
would have y~2H < xH , which contradicts that in equilibrium y~2H < xH must hold. Hence, y~2L < xL
is not possible in equilibrium.
Thus, either (C9, A9) is part of an equilibrium of C that is payoff equivalent to the case where
the charity always reveals the types, or (C9, A9) is not part of an equilibrium.
26
For any equilibrium involving (C7, A7) and y~2H ¼ xH , the proposed contributions following
non-announcement are unique.
Ó The Author(s). Journal compilation Ó Royal Economic Society 2008
90
THE ECONOMIC JOURNAL
The case of (C10, A10) is similar to (C9, A9). This proof is now complete.
[JANUARY
h
Verification of the equilibrium E1
Following the discussion in the text, we need to verify that the strategies and beliefs specified in
E1 in Section 5 will form a PBE, so long as xH xL is not too small. It is obvious that there is no
incentive to deviate for the either donor in the contribution game following transmission of the
information. Consider the continuation contribution game when no information is transmitted
by the charity.
If donor 1 of L-type deviates in Round 1, the overall contribution is xL, with donor 1 contributing the entire amount and donor 2 contributing zero; following the deviation, contributing
xL (in the two rounds together) is donor 1s best strategy;
If L-type donor 1 does not deviate in Round 1, the overall contribution is:
(1) w with probability ð1 p2 Þ (where p2 > 0 is the conditional probability that donor 2 is of
type H given donor 1s type);
(2) xH with probability p2 , where donor 2s contribution is xH w.
The fact that donor 2 contributes xH w (which is not too small) with a positive probability
when donor 1 does not deviate, and donor 2 contributes zero otherwise with donor 1 saving only
a small amount (w xL ¼ ), implies that donor 1 is better off not deviating.
If H-type donor 1 deviates in Round 1, the overall contribution is xH (all of which is contributed
by donor 1); if he does not deviate then the overall contribution is xH, but donor 2 (who is known
by donor 1 to be of type H for sure) provides xH w > 0. Clearly, H-type donor 1 will not deviate.
It is easy to check that player 2 will not deviate either in the first round or in the second round.
Finally, consider the incentive of the charity to deviate at its information transmission stage.
Such deviation can take the form of concealing realised state (H, L) or transmitting information
about one of the following states: (L, L), (L, H), (H, H). The charity will not conceal (H, L)
because concealment would yield total contribution equal to xL þ that is less than xH. If the
charity announces any of the states (L, L), (L, H) or (H, H), the generated total contribution is
at most xH and so there is no incentive to deviate. This completes verification of the PBE.
National University of Singapore
Southern Mehodist University
Submitted: 16 July 2004
Accepted: 20 February 2007
References
Admati, A. and Perry, M. (1991). Joint projects without commitment, Review of Economic Studies, vol. 58, pp.
259–76.
Andreoni, J. (1990). Impure altruism and donations to public goods: a theory of warm-glow giving, Economic Journal, vol. 100, pp. 464–77.
Andreoni, J. (1998). Toward a theory of charitable fund-raising, Journal of Political Economy, vol. 106, pp.
1186–213.
Andreoni, J. (2006a). Leadership giving in charitable fund-raising, Journal of Public Economic Theory, vol. 8, pp.
1–22.
Andreoni, J. (2006b). Philanthropy, in (S.-C. Kolm and J. Mercier Ythier, eds.), Handbook of the Economics of
Giving, Altruism and Reciprocity, pp. 1201–69, Amsterdam: Elsevier/North-Holland.
Bac, M. and Bag, P.K. (2003). Strategic information revelation in fundraising, Journal of Public Economics, vol.
87, pp. 659–79.
Bag, P.K. and Roy, S. (2004). Sequential fundraising under incomplete information, mimeo. Department of
Economics, Southern Methodist University, Dallas, Texas.
Bliss, C. and Nalebuff, B. (1984). Dragon-slaying and ballroom dancing, Journal of Public Economics, vol. 25,
pp. 1–12.
Ó The Author(s). Journal compilation Ó Royal Economic Society 2008
2008 ]
REPEATED CHARITABLE CONTRIBUTIONS
91
Bruce, N. (1990). Defence expenditures by countries in allied and adversarial relationships, Defence Economics, vol. 1, pp. 179–95.
Cho, I.-K. and Kreps, D. (1987). Signaling games and stable equilibria, Quarterly Journal of Economics, vol. 102,
pp. 179–222.
Cornelli, F. (1996). Optimal selling procedures with fixed costs, Journal of Economic Theory, vol. 71, pp. 1–30.
Fershtman, C. and Nitzan, S. (1991). Dynamic voluntary provision of public goods, European Economic Review,
vol. 35, pp. 1057–67.
Glazer, A. and Konrad, K.C. (1996). A signaling explanation for charity, American Economic Review, vol. 86, pp.
1019–28.
Gradstein, M. (1992). Time dynamics and incomplete information in the private provision of public goods,
Journal of Political Economy, vol. 100, pp. 581–97.
Harbaugh, W. (1998). What do donations buy? A model of philanthropy based on prestige and warm glow,
Journal of Public Economics, vol. 67, pp. 269–84.
Marx, L. and Matthews, S. (2000). Dynamic voluntary contribution to a public project, Review of Economic
Studies, vol. 67, pp. 327–58.
Okuno-Fujiwara, M., Postlewaite, A. and Suzumura, K. (1990). Strategic information revelation, Review of
Economic Studies, vol. 57, pp. 25–47.
Romano, R. and Yildirim, H. (2001). Why charities announce donations: a positive perspective, Journal of
Public Economics, vol. 81, pp. 423–47.
Romano, R. and Yildirim, H. (2005). On the endogeneity of Cournot-Nash and Stackelberg equilibria: games
of accumulation, Journal of Economic Theory, vol. 120, pp. 73–107.
Saloner, G. (1987). Cournot duopoly with two production periods, Journal of Economic Theory, vol. 42, pp.
183–87.
Varian, H.R. (1994). Sequential contributions to public goods, Journal of Public Economics, vol. 53, pp. 165–86.
Vega-Redondo, F. (1995). Public subscriptions and private contributions, Social Choice and Welfare, vol. 12, pp.
193–212.
Vesterlund, L. (2003). The informational value of sequential fundraising, Journal of Public Economics, vol. 87,
pp. 627–57.
Ó The Author(s). Journal compilation Ó Royal Economic Society 2008