Theoretical Economics Letters, 2012, 2, 385-390
http://dx.doi.org/10.4236/tel.[phone] Published Online October 2012 (http://www.SciRP.org/journal/tel)
Anonymous Giving as a Vice: An Application of
Image Motivation
Masaoki Tamura
Graduate School of Economics, University of Tokyo, Tokyo, Japan
Received June 6, 2012; revised July 2, 2012; accepted August 1, 2012
ABSTRACT
While some donors boast about their giving, others give anonymously. A novel feature of this paper is that anonymity is
endogenously con trolled by dono rs themselves, not exogenously controlled by fundraisers. Is anonymous giving really
a virtue, as is generally recognised? Paradoxically, this p aper proves that anonymity is a vice for fun draisers even if it is
what donors desire. If many altruists (a good type) switch from known to anonymous giving, it relatively lowers the
group reputation of known donors and enhances that of non-donors. These effects dilute the incentive for other indi-
viduals to become known donors, if they have psychological “image motivation”. I suggest a practical method to con-
trol the expected number of anonymous donors: fundraisers remove the “check boxes” from their web sites.
Keywords: Giving; Anonymity; Altruism; Image; Fundraising
1. Introduction
Many studies, including [1-3], show that if experimenters
control the anonymity of examinees, the examinees
change their prosocial behaviour. The more anonymity
experimenters provide, the less altruistic the examinees
become. A novel feature of this paper is that anonymity
is endogenously determined by the donors themselves.
While some donors boast about their giving, others give
anonymously. Is anonymous giving worth the same as
known giving to fundraisers? Is anonymous giving really
a virtue, as is generally recognised? Paradoxically, this
paper proves that anonymous giving is a vice, even if the
donors themselves desire anonymity.
We extend Benabou and Tirole’s image signalling frame-
work to the case in which “anonymity” is redefined and
individuals themselves choose anonymity level. Since
people care about their social reputations, even non-al-
truists donate to conceal their selfishness and enhance
their reputations. The key lies in this “hypocritical” be-
haviour of donors. Such behaviour is possible because
altruism is private information and the donation amount
is what others observe; giving enables non-altruists to
mimic altruists and achieve recogn ition as altru ists. How-
ever, if we allow donors an option to anonymous giving,
it is clear that not all donors choose the same anonymity
level. It is essential to identify wh o donates anonymous ly,
because it determines whether anonymous giving is in-
dependent of or related to the hypocritical behaviour. Our
result is that those who prefer anonymous giving are the
best “target” for the non-altruists to mimic, and the exis-
tence of anonymous donors negatively affects the hypo-
critical behaviour of non-altruists and the total amount of
donation.
Practically, fundraisers should control the expected
number of anonymous donors. However, we can find one
example that exaggerates the existence of anonymous
donors. Currently, many fundraisers solicit contributions
via the Internet in addition to normal fundraising activi-
ties. On their Web sites, some fundraisers place check
boxes for donors to select anonymous or known dona-
tions. Figure 1 shows how a typical such site looks. On
the page, donors fill out not only the information about
the amount of donation, name, address, and credit card
number but also the check box “I prefer to make this do-
nation anonymously”. By providing the check boxes,
these fundraisers intend to make it convenient to donate
anonymously. The problem, however, is that, when fac-
ing the choice, donors expect that fundraisers provided
the check box in response to many requests from a sub-
stantial number of anonymous donors. In other words,
the check box exaggerates the number of anonymous do-
nors. The contribution of this paper is threefold. First and
foremost, it suggests how fundraising campaigns can be
more efficiently designed: fundraisers should all ow anony -
mous donation only as an exception and should not exag-
gerate the number of anonymous donors. Specifically, we
can identify inefficiency in some fundraising Web sites
and suggest an improvement. Such sites can impose a
small “foot cost” on anonymous giving. Second, this is
the first paper that studies a model in which an onymity is
C
opyright © 2012 SciRes. TEL

M. TAMURA 386
Figure 1. Online donation page.
endogenously determined. Here, anonymity is controlled
not by the fundraisers but by the donors themselves. Thus,
we can study the case in which donors choose perfect
anonymity, namely by donating anonymously; almost no
economic studies have addressed this case in detail. Third,
the study gives weight to the heterogeneity of image
motivation among individuals, in contrast to [4]. Thus, we
can identify who donates anonymously
2. Structure of the Model
Let us begin our analysis by specifying the utility function
of the individuals who have both an image motivation
and a warm-glow preference1.



ˆˆ ,,= 1lnlnln
ii iiiiiii
Ucxc x I

   (1)
i
c denotes individual i’s consumption for private goods,
i
x
denotes individual i’s donation, i

denotes the
altruism parameter, and i

denotes the self-conscious-
ness of consumer i. Individuals are heterogeneous in the
parameters i

and i

. Because we assume that
individual i’s parameters i

and i

are private in-
formation and unobservable to all other individuals,
others form beliefs about i’s parameters. ˆi

denotes the
belief of other individuals about i’s altruism, i

.
Individuals derive utility directly from this belief of
others, namely, the extent to which others consider an
individual altruistic. This corresponds to the concept of
self-image or image motivation. The function


I

re-
presents this image motivation of the individuals. We
assume that and that . People who
are considered socially altruistic have a high i

>0
'
I

0=1Iˆ

and
hence high utility. The more altruistic they are con-
sidered the more utility they gain. Here, the hetero-
geneit y lies in altruis m i

and self-consciousness ,
which we specify below. i
beta
The budget constraint for is i
=,
ii i
cx y (2)


=1
iiki ia
.
i
x
DxD x (3)
i is the endowment of individual i. People allocate
their income between consumption and donation.
y
k
x
is
known donation, a
x
is anonymous donation, and i D
is a dummy variable th at takes the value if individual
chooses known over anonymous donation, and takes
the value if individual chooses anonymous over
known donation. In short, in this model, when in-
dividuals want to donate, they must choose between a
known and an anonymous donation. For simplicity, we
assume that individuals cannot make both anonymous
and known donations at the same time. We must draw
attention to the implied assumption that the choice be-
tween anonymous and known donation does not directly
affect the utility: both choices yield the same warm-glow
utility. However, the choice does matter when individuals
form beliefs about others’
1
i0i

values: ˆi

. ˆi

is formed
by
ˆki y=
ii
Ex  ,
i
.



 (4)
The beliefs about others’ altruism are formed based on
income and known donation. In other words, individual’s
decision variables except known donation are unob-
servable to the others. It must be noted that individuals
form beliefs about i

based on ik
x
, not i
x
. This result
is owing to the definitio n of anon ymou s donatio n : anon y-
mous donation is unobservable to others, while known
donation is observable. It is important that anonymous
donors (0
a
x

and k) and non-donors (
and ) are considered the same by others.
=0x=0
a
x
=0
k
Next, we specify the parameter values and the dis-
tribution of the individual types in th is economy. Table 1
summarises the information about the four individual
types of our model.
x
3. The Equilibrium
3.1. Who Gives Anonymously?
The first task is to determine who donates anonymously.
The answer is the “pure altruists”. There are two reasons
for this. First, because anonymous donation, by defini-
tion, cannot be observed by others, anonymous donors
and non-donors are considered the same (i.e. the same
ˆi

). Second, the types who ar e concern ed with their own
reputation (0


) want to be seen as altruistic (i.e., as
donors); consequently, they never prefer anonymous to
known donation. Then, the “pure altruists” (who are
altruistic but not self-conscious) is the only type who
may make anonymous donations. Pure altruists must be
1Theoretical and empirical backgrounds for these two motivations are
given in [4-10].

M. TAMURA 387
Table 1. Individual Types.
Type


Population
Selfish 0 0
s
N
Pure Altruist

0
p
a
N
Hypocrite 0

h
N
Impure Altruist


ia
N
indifferent to whether a donation is known or anony-
mous.
3.2. Equilibrium Actions
From now on, we examine the behaviour of each type to
find the Nash equilibrium of our model. It should be
noted that individuals interact only through the image
(ˆi

) of the types to which they belong. Thus, people
without self-consciousness (i.e., =0
i

) behave without
considering others’ actions. We first limit our attention to
these types: the selfish and the pure altruists.
3.2.1. Selfis h : & =0α0β=
People without self-consciousness, by definition, simply
solve their classical optimisation problems with respect
to donation and private good, but not their images. In
short, it is the simple warm-glow setup we see above. It
is obvious that the selfish spend all of their income on
private goods. For the selfish, i
and . =
i
cy =0
i
x
3.2.2. Pure Altrui st: α=α & 0β=
Pure altruists also face their classical optimisation prob-
lems without image motivation. The solution to this
problem is

=1
ii
cy   and =
ii
x
y

. Because ˆi

does not appear in their optimisation problems, pure
altruists are indifferent to the choice between a and
k; both options yield the same utility. Here, we assume
that a fraction of the pure altruists choose
anonymous over known donation, and thus, 1
x
x

0,1A

A

of
them choose known over anonymous donation. As a
result, the introduction of the option of anonymous
giving makes some pure altruists switch from known to
anonymous giving. We should not overlook that the total
donation amount of pure altruists is independent of
A
.
3.2.3. Hypocrite: & =0αβ=

It is most important to examine the conditions under
which even hypocrites make donations in spite of their
selfish nature. By making donations, th ey mimic the pure
altruists (a good type) to enhance their reputations. Note
that the pure altruists are the best “target” for the hypo-
crites to mimic, becau se the impure altruists are averse to
mimicry as is described later.
It can be checked easily that there are only two options
for hypocrites to choose, or
=0
i
x=
ii
x
y

. If they
choose , then they are in the same group as the
selfish. Conversely, if they choose
=0x=i
x
y

, then they
are in the same group as the pure altruists. Comparing the
two options, the condition for hypocrites to choose
=i
x
y

over is2 =0x



1
lnln1
>lnln.
pa
ii hp
pa
ihpas
AN
yy I
NAN
AN
yI
NAN N
a



 
 








(5)
The left hand side represents the utility when hypo-
crites join the group of known donors (=
ii
x
y

), while
the right hand side represents the utility when they join
the group of non-donors (). In conclusion, the
more anonymous donors there are, the less likely hypo-
crites are to donate. Let us mathematically confirm this
result and explain the intuition behind it. The second
term on the left hand side of (5) is decreasing in
=0
i
x
A
,



1
ln 1<0,
pa
hpa
AN
INAN
A

 






(6)
and the right hand side of (5) is increasing in
A
,
ln
>0.
pa
hpas
AN
INANN
A






 (7)
The interpretation of these two inequalities is the core
of this paper. We refer to (6) as the “decrease effect”, and
(7) as the “blend effect”. We first note that pure altruists
are thought to be a “good” type compared with the
selfish and the hypocrites because pure altruists have
higher altruism. Next, the group reputation is formed
according to the ratio of “good” (altruistic) group mem-
bers. To interpret (6), suppose that some of the pure
altruists (a “good” ty pe) switch from known
(




,=0,
ak i
x
x  y) to anonymous donation
(




,= ,0
ak i
xx y 
). Such a switch implies a decrease of
a “good’ type in the group of known donors (=
ki
x
y

).
Then, to the hypocrites, joining the group of known
donors (=
ki
x
y

) becomes less attractive. This decrease
effect corresponds to inequality (6). The second in-
equality (7) represents the blend effect. Because anony-
mous donation is unobservable, people now think that
some perceived non-donors are actually anonymous do-
nors. Some fraction of non-donors is of the “good” type (
2For simplicity, we consider the case in which all hypo crites behave as
a group. This does not change the result much as long as we focus on
the symmetric equilibrium.

M. TAMURA 388
ˆ=>
pa
hpas
AN
NAN N

 0
do not donate.
3.2.4. Impure Altruist: =

 & β=

on the left hand side of (7)). As a result, the existence of
anonymous donors enhances the reputation of non-do-
nors, and joining the group of non-donors () be-
comes more attractive to the hypocrites. Here, anony-
mous donors blend into non-donors. This blend effect
corresponds to inequality (7).
=0
k
x
If hypocrites mimic pure altruists, the reputations of the
hypocrites improve, as is described above. In contrast,
the reputations of the pure altruists worsen because they
are now in the same group as hypocrites (a “bad type”).
Even then, the equilibrium action of pure altruists is un-
changed because they do not derive utility from the re-
putation (i.e., =0

.)
A numerical simulation is shown in Figure 2, “Utility
of Hypocrites”3. The “Join Donors” curve corresponds to
the left hand side of (5) for each
A
. The “Join Non-Donors”
curve corresponds to the right hand side of (5) for each
A
. We refer to the value of
A
for which the left hand
side equals the right hand side as *
A
. If there are
anonymous donors above , then hypocrites
*=17.7%A
Contrary to pure altruists, impure altruists care about
their own reputation (i.e., =
 0). Impure altruists
are averse to be mimicry by hypocrites. First, when
A
is
below *
A
, hypocrites mimic pure altruists. If
A
is
increasing from 0, then pure altruists become increasingly
less attractive to mimic than impure altruists. In response,
Figure 2. Numerical simulation.
3The values used in o u r numerical simulations are =0.1

, =0.5  , =10
i
yi

, and


ˆˆ
=3 20
ii
I

 .

M. TAMURA 389
impure altruists want to become less attractive because
they are averse to mimicry by hypocrites. Then, impure
altruists gradually increase their donations as
A
be-
comes large because larger-amount donors are less at-
tractive for hypocrites to mimic. Secondly, when
A
surpasses *
A
, hypocrites are non-donors. If
A
is in-
creasing from *
A
, for hypocrites, being non-donors
yields increasing utility compared with mimicking im-
pure altruists. In response, impure altruists can decrease
their donation amounts to their original ideal level,
=
ki
x
y

.
3.3. Total Amount of Donation
While impure altruists are averse to be mimicry by hypo-
crites, pure altruists do not care about it. Thus, the hypo-
crites mimic the pure altruists. The existence of anony-
mous donors means 1) the decrease of the best “target”
for the hypocrites to mimic and 2) the increase of the
“good” type in the non-donor group. The total amount of
donation of these four types is shown in Figure 2. In this
numerical example, . The sharp decrease at
is due to the hypocrites’ behaviour: they no
longer mimic pure altruists. In conclusion, for fundraisers
to raise as much money as possible, it is crucial that
*=0.177A
*=0.177A
A
does not exceed *
A
, though a small fraction of anony-
mous donors is not harmful.
4. Policy Implication and Discussions
In the preceding section, we examine how the total dona-
tion amount varies according to an exogenously deter-
mined
A
. This result raises the question, can fundraisers
control
A
and expected
A
? Inthis section, we discuss
one practical method to control both
A
and expected
A
.
Currently, many fundraisers solicit contributions via
the Internet in addition to normal fundraising activities.
On their Web sites, some fundraisers including Anna
Marie’s Alliance, the Minnesota Aids Project, and Net-
work for Good place check boxes for donors to select
either anonymous or known donations. Figure 1 shows
how a typical such site looks. Donors fill out not only
information about the donation amount, name, address,
and credit card number but also a check box, “I prefer to
make this donation anonymously”. However, some or-
ganisations, such as the American Cancer Society, the
American Red Cross, and Doctors Without Borders do
not provide such a ch eck box on th eir Web sites. Witho ut
a special request, only known donation is available for
donors. We see that not a few fundraisers explicitly offer
opportunities for anonymous giving. By providing the
check boxes, these fundraisers intend to make it conven-
ient to donate anonymously. The problem, however, is
that, when facing the choice, donors expect that fund-
raisers provided the check box in response to many re-
quests from a substantial number of anonymous donors.
In other words, the check box exaggerates the number of
anonymous donors. Here,
A
is expected higher than
that it actually is.
To keep both
A
and expected
A
below *
A
, fund-
raisers can remove this type of check box. Instead, they
can implement some type of small foot cost on anony-
mous giving and accept anonymous donations only as an
exception. For instance, to give anonymously, donors
have to send an e-mail to fund raisers in addition to filling
out the personal information form. Then, those who re-
quire anonymity choose anonymous giving with a small
effort, while those who are indifferent to whether the
donation is anonymous choose known giving.
An important experimental finding is reported in [1].
They show that, if examinees give subjects an option to
donate anonymously, subjects increase their giving. Note
that the subjects mainly increase known giving, not ano-
nymous giving. Their finding is not inconsistent with our
theoretical results, because their finding corresponds to
the case of *
A
A

in our model. Our model predicts
that if
A
is small, impure altruists increase known giv-
ing above their ideal amount. This is what [1] observes.
We also predict that if we would make
A
and expected
A
sufficiently large, then the subjects would decrease
their known giving. Fundraisers should keep the actual
A
and expected
A
at a low level and consequently
maximise the total donation amount.
5. Acknowledgements
I would like to thank Masayuki Otaki for his helpful com-
ments. I am also grateful to Sakuya Tamura for her sup-
port and fruitful discussions.
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