Coordination Failure in Foreign Aid
Maija HalonenAkatwijuka
University of Bristol
February 9, 2004
Abstract
This paper analyzes the allocation of foreign aid to various sectors
in a recipient developing country. Donors tend to favor social sectors
over other public expenditure programs. Due to incomplete infor
mation, donors may concentrate too much on priority sectors, leaving
lowerpriority yet important sectors lacking funds. Alternatively there
may be gaps in services in priority areas because of the information
problem. The more similar preferences the donors have, the more
scope there is for coordination failure. Therefore improving informa
tion is particularly important when the parties have similar priorities.
A joint database on planned projects and budget allocations in each
recipient country would provide such information. The point of the
paper is that such databases should have both information on current
projects and forwardlooking information on the planned activities
needed to improve aid coordination. It also analyzes the aid fungi
bility problem in an incomplete information setting, and finds that
incomplete information reduces the fungibility problem. On the other
hand, incomplete information introduces coordination failure and the
allocation can be inferior for both the recipient and the donor.
This paper was prepared as a background paper for the World Development Report
2004: Making Services Work for Poor People. I would like to thank Ritva Reinikka for
discussions and Steve Knack, John Mackinnon, Sebastien Mitraille, and InUck Park for
comments. The findings, interpretations, and conclusions expressed in this paper are those
of the author and do not necessarily represent the views of the World Bank, its Executive
Directors, or the governments they represent. Working papers describe research in progress
by the author and are published to elicit comments and to further debate.
Maija.Halonen@bristol.ac.uk, Department of Economics, University of Bristol, 8
Woodland Road, Bristol BS8 1TN, U.K.
1
1 Introduction
Recent evidence demonstrates that aid is effective in raising economic growth
and reducing income poverty in "good" environments but ineffective in "poor"
environments; the result holds both when the environment is defined nar
rowly as fiscal, monetary, and trade policies, and more broadly in terms of
a wide range of policies and public institutions (Burnside and Dollar 2000;
and Collier and Dollar 2002). This evidence also shows that the policy and
institutional environment of the recipient country has only a limited impact
on the present allocation of aid. Because donors have multiple objectives
for their aid, such as strategic and historical considerations,1 the actual al
location of aid differs significantly from, for example, the povertyefficient
allocation that would target poor countries with good policies and institu
tions. According to one calculation, the present allocation of aid lifts 10
million people out of poverty annually, while its povertyefficient allocation
would double the effect of aid on growth and poverty (Collier and Dollar
2002). Hence, in line with this evidence, aid effectiveness could be much en
hanced if donors improved their allocation of aid between recipient countries
toward those that are poor and have good policies and institutions.
This paper takes the aid effectiveness debate beyond country allocations
and contributes to a theoretically underexplored area. It argues that, once
aid has been allocated to a poor recipient country with good policies, over
coming the coordination failure in aid can further enhance its effectiveness.
Donors allocate their funds between regions, sectors and activities in a recip
ient country. When there is incomplete information about the other donors'
aid, coordination failure can occur. The resulting allocation can be such that
all donors regret it ex post. Donors tend to favor social sectors (health and ed
ucation) over other public expenditure programs (e.g., transportation). Due
to incomplete information, they may all concentrate too much on the pri
ority sectors, leaving the lowerpriority yet important sectors lacking funds.
Alternatively there may be gaps in services in the priority areas because of
the information problem.2
The information problem often extends to the recipient countries as well;
many recipients are not aware of donor activities in any detail ex ante (or even
1Alesina and Dollar (2000).
2See Mackinnon (2003) for an argument of misallocation of donations that does not de
pend on information problems. Donors underfund activities they agree about and overfund
each donor's idiosyncratic schemes.
2
ex post). For example, in Vietnam 25 official bilateral donors, 19 official mul
tilateral donors and about 350 international NGOs (nongovernmental orga
nizations) were operating in 2002. They accounted for over 8,000 projects.3
With such a great number of projects, it is difficult to keep track of the
planned (or even the past) activities at the aggregate level. This is true
for the recipient government and for the donors. For the donors the infor
mation problem is exacerbated by the fact that they operate in numerous
developing countries. For example, bilateral donors assist on average 107
recipient countries each.4 Fragmentation imposes high transaction costs on
the recipient countries, not least because the donors have different report
ing requirements. Great emphasis has recently been placed on harmonizing
donor practices to reduce transaction costs. This paper points to a less dis
cussed problem arising from fragmentation. Fragmentation makes acquiring
information about other donors' planned activities and budget allocations
complex. This incomplete information can lead to inefficient allocation de
cisions and coordination failure among donors.
There are often genuine differences between donors and recipients about
how to allocate public funds in the recipient developing country. This leads
to the wellknown aid fungibility problem. The donors aim to increase fund
ing in their priority sectors but the government responds to aid by shifting
funding away from those sectors. Therefore aid in effect finances indirectly
some other, potentially unproductive activities. The paper analyzes the
aid fungibility problem in an incomplete information setting, which to my
knowledge is novel. The analysis finds that incomplete information reduces
the fungibility problem because the recipient government does not have the
information to fully respond to donors' decisions. On the other hand, in
complete information introduces coordination failure and the allocation can
be inferior to both the recipient and the donor.
The analysis of three versions of the allocation game with increasingly dif
ferent preferences finds that the more similar the preferences are, the more
scope there is for coordination failure. When similar agents pursue the com
mon good, misallocation due to incomplete information can be considerable.
When the agents have very different priorities about their activities in the
recipient developing country, information problems are minimal.
Information problems arise because donors run their own projects in the
3Acharya, Fuzzo de Lima, and Moore (2003).
4Acharya, Fuzzo de Lima, and Moore (2003).
3
recipient developing countries. One way of overcoming the information prob
lem (and many other inefficiencies arising from the project approach5) is for
donors to pool their funds within a sector in a recipient developing country.
Alternatively the donors can channel the aid as budget support to the recip
ient country so that the government makes the allocation decisions. There
are, however, strong motives for the donor agencies to maintain the project
approach, e.g., to demonstrate visible results to the taxpayers in the donor
country and to accomplish the type of projects the donor country prefers.
Another way to overcome the coordination failure is to simply improve in
formation about the donors' planned activities and budgets. According to
the paper's results, improving information is particularly important when
the parties have similar priorities. A joint database on planned projects and
budget allocations in each recipient country would provide such information.
Such databases should not have just information on the current projects,
but also forwardlooking information on the planned activities to improve
aid coordination.
The rest of the paper is organized as follows. Section 2 sets up the
model of two donors and a passive recipient. Section 3 analyzes coordination
failure among donors. Section 4 introduces an active recipient with different
preferences from the donor and analyzes how coordination failure can add to
the aid fungibility problem. Section 5 concludes.
The literature on voluntary provision of public goods and the freerider
problem is related. (e.g., Olson 1965; Chamberlin 1974, 1976; McGuire 1974).
See Kemp (1984) on an application to foreign aid. In this literature the level
of contributions is endogenous, while the focus here is on allocating a fixed
budget to a twodimensional public good.
The structure of the model has similarities with networks, in particular
Farrell and Saloner (1985). There are network externalities when the utility a
consumer derives from a product depends on the number of other consumers
with the same product. Consumers have a fixed budget (as in our model) and
choose the product from two possible ones. There is incomplete information
about the other consumers' preferences and coordination failure can occur:
in equilibrium the consumers can choose an inferior good. In this paper's
framework, there are network externalities as the donor's utility depends on
the number of projects funded in each sector by the other donor. However,
the incomplete information is not about the other donor's preferences but
5For other inefficiencies, see Kanbur, Sandler, and Morrison (1999).
4
about his budget size. Therefore the mechanics of the model and the insights
arising from it are different.
Finally, there is an empirical literature on foreign aid fungibility (for a
review, see Devarajan and Swaroop 1998). These studies analyze whether
foreign aid for specific categories of expenditure is shifted by the recipient
government. If the recipient government shifts funding as a response to
foreign aid, aid is in effect financing indirectly some other, potentially un
productive activities. This paper analyzes the aid fungibility problem in an
incomplete information setting, which to my knowledge is novel.
2 The model
Two donors, 1 and 2, operate in a recipient developing country in two sec
tors/activities/regions, 1 and 2. The following refers to sectors. The donors'
objective is to increase welfare in the recipient developing country.
Each donor has a fixed budget. The focus is on the question how the
donors allocate the budget between the sectors. This is the practical problem
the donor agencies face. Discrete projects are analyzed assuming that donor
i can finance ni projects where ni {0, 1,2, 3} . Most of the foreign aid is
channelled as project aid. The size of the budget is private information to
the donor. Donor i attaches prior probability pn for donor j's budget to equal
n. 3 pn = 1.
n=0
The donors' utility depends on the total number of projects financed in
each sector, u (n11 + n12, n21 + n22) , where nji is the number of projects donor i
finances in sector j. In other words, the donors are providing public goods.
The donors have the same preferences. In Section 4 we analyze agents
with different preferences. The utility is strictly increasing in the number of
projects implying that the donors always exhaust their budget.
Assumption 1. (i) u( + 1,) > u(,) and
(ii) u (, + 1) > u (, ) .
The donors view sector 1 as contributing more to welfare than sector 2.
Donors tend to favor social sectors (health or education) over other public
expenditure programs (e.g., transportation). For example five out of the
5
eight Millennium Development Goals  agreed by the international donor
community to reduce poverty  concern health and education.6
We define u (, ) u ( + 1,  1)  u (, ). u (, ) gives the
change in utility when one project is reallocated to the priority sector 1
given the original allocation (, ) . Assumption 2 signs u (,).
Assumption 2. (i) u(,) > 0 if and only if .
(ii) u (, ) < 0 if and only if > .
Property (i) means that the donors prefer to allocate more projects to
ward sector 1 while property (ii) states that the sectors are complementary
and too much inequality in the resources is welfare reducing. These two
properties order all reallocations by one project. The frequently used terms
are given below.
u(0,1) > 0, u(1,1) > 0, u(2,2) > 0, u(3,3) > 0
u(2,1) < 0, u(3,2) < 0, u(4,2) < 0
We assume that the preferences are common knowledge. In reality allo
cation of foreign aid is not a oneshot game (although at the first instance
we model it as such) but the same donors operate in the recipient coun
tries over the years. The type of aid each donor prefers to give is revealed
over time while their budgets vary each year. Therefore we have incomplete
information about the budgets, not about the preferences. 7
The recipient country is a passive player in this version of the model. The
recipient does not have any funds or any bargaining power on how to allocate
the donors' funds. In Section 4 we extend the analysis to an active recipient
who has different preferences from the donor.
6The Millennium Development Goals are: (1) Eradicate extreme poverty and hunger.
(2) Achieve universal primary education. (3) Promote gender equality and empower
women. (4) Reduce child mortality. (5) Improve maternal health. (6) Combat HIV/AIDS,
malaria, and other diseases. (7) Ensure environmental sustainability. (8) Develop a global
partnership for development.
7One interpretation of the model is that there is incomplete information about the
number of other donors operating in the recipient country in a particular year. But
strictly speaking we are analyzing two donors.
6
3 Coordination failure among donors
3.1 Complete information
We first analyze the benchmark case of complete information. Each donor
knows not only his own budget but also that of the other donor, that is
they know what is the aggregate budget. Assumption 2 gives the optimal
allocation of the aggregate budget which is:
(1,0) if and only if n1 + n2 = 1
(2,0) if and only if n1 + n2 = 2
(2,1) if and only if n1 + n2 = 3
(3,1) if and only if n1 + n2 = 4
(3,2) if and only if n1 + n2 = 5
(4,2) if and only if n1 + n2 = 6
Suppose n1 = 1 and n2 = 2. Table 1 reports possible allocation strategies
for each donor and the resulting aggregate allocation. Donor 1's strategy is
given vertically and donor 2's strategy horizontally.
Table 1. Aggregate Allocation
(2, 0) (1, 1) (0, 2)
(1, 0) (3, 0) (2, 1) (1, 2)
(0, 1) (2, 1) (1, 2) (0, 3)
There are two Nash equilibria: (i) donor 1 funds one project in sector 1
and donor 2 funds one project in each sector and (ii) donor 1 invests one
project in sector 2 and donor 2 invests all the funds in sector 1. Both Nash
equilibria lead to the same aggregate allocation (2,1). This allocation is
efficient. This is true for any allocation game. The aggregate allocation is
efficient and for any n1 +n2 > 2 there are multiple Nash equilibria to achieve
that allocation. The donors are contributing toward public goods over which
they have similar preferences. That is why the Nash equilibrium allocation
with complete information is efficient.
Table 2 shows the equilibrium allocation. Donor 1's budget is vertical
and donor 2's budget is horizontal.
Table 2. Complete Information
7
0 1 2 3
0 (0,0) (1,0) (2,0) (2,1)
1 (1,0) (2,0) (2,1) (3,1)
2 (2,0) (2,1) (3,1) (3,2)
3 (2,1) (3,1) (3,2) (4,2)
The outcome is naturally biased toward sector 1 since both donors view
it as more important.
3.2 Incomplete information
Then we turn to incomplete information. The donors know their own budget
but not the size of the other donor's budget. They only know the probability
distribution of the other donor's budget. It is the incomplete information
about the budget size that leads to coordination failure. Coordination failure
means that the equilibrium allocation is Pareto dominated; the resulting
allocation is such that both donors regret it ex post. The coordination failure
can be of two types. First, the less important sector 2 does not get enough
resources as both donors concentrate too much on priority sector 1. Second,
in an attempt at a more balanced allocation, the donors invest too much in
sector 2 and priority sector 1 does not get enough resources.
We describe the allocation decisions as sharing strategies, sharing pointing
to how much resources are allocated to the lower priority sector 2. A sharing
strategy si = (,, ) denotes that given budget size 1 the donor allocates
to sector 2, from budget size 2 and from budget size 3 projects are
funded in sector 2. E.g. si = (0, 0, 0) means that donor i invests all the
funds in sector 1 whatever the budget and si = (1, 2, 3) denotes that the
donor allocates all the funds in sector 2.8
In what follows we find the various types of equilibria that can exist in
this allocation game. After examining each equilibrium in detail we analyze
the general pattern that emerges from them.
We start by examining no sharing strategies.
Proposition 1 It is not a Bayesian equilibrium for both donors to invest all
their budget in sector 1.
Proof. Suppose that donor 2's equilibrium sharing strategy is s2 =
(0, 0, 0) , i.e. donor 2 invests the whole budget in sector 1. Given s2 is donor
8From the complete information case we know that the donors never wish to have more
than 2 projects in sector 2. Therefore si = (1, 2, 3) cannot be an equilibrium strategy.
8
1 better off with strategy s1 = (0,0, 1) or with s1 = (0, 0, 0)? These strategies
differ only for n1 = 3. To find which strategy gives donor 1 a higher utility
we only need to check the outcome for the large budget 3. Donor 1 is better
off with s1 = (0, 0, 1) if and only if:
p0u (2, 1) + p1u (3, 1) + p2u (4, 1) + p3u (5, 1) (1)
> p0u (3, 0) + p1u (4, 0) + p2u (5,0) + p3u (6,0)
Equation(1) simplifies to:
p0u(2,1) + p1u(3, 1) + p2u(4,1) + p3u(5, 1) < 0 (2)
Equation (2) clearly holds since each term is negative by Assumption
2. Therefore it is not a Bayesian equilibrium for both donors to choose
si = (0, 0, 0). Q.E.D.
Proposition 1 shows that if donor i invests all the budget in the priority
sector 1, i.e. chooses si = (0, 0,0) , it is not optimal for donor j to do the
same. If donor j also invests all the budget in sector 1 the resulting allocation
would be too unequal. Therefore it is not a Bayesian equilibrium for both
donors to choose si = (0, 0,0) .
Now that we have established that there is sharing in equilibrium we pro
ceed to ask how much. In the absence of donor j, donor i would focus on the
more important sector 1 and only invest one project in sector 2 if the bud
get is 3. There is an equilibrium where both donors follow this "monopoly"
strategy.
Proposition 2 A symmetric Bayesian equilibrium si = (0,0,1) for i = 1,2
exists if
(i) p0u(0, 1) + p1u (1, 1) + p2u (2, 1) + p3u (2, 2) > 0
(ii) p0u(1, 1) + p1u (2, 1) + p2u(3, 1) + p3u (3, 2) > 0 and
+ + +
(iii) p0u(1, 2) + p1u(2, 2) + p2u(3, 2) + p3u(3, 3) > 0.
+
+ +  +
Proof. See the Appendix.
9
If the conditions in Proposition 2 are satisfied9 both donors choose si =
(0, 0, 1) . We denote the sign of each term under it to ease reading. Condition
(ii) is the most stringent; only the first term has the same sign as the whole
equation. This condition is for the donor with budget size 2 to prefer investing
it all in sector 1. Given sj = (0, 0,1) and ni = 2 donor i wishes to invest all
the funds in sector 1 if and only if there is a high probability that nj = 0.
For nj > 0 this strategy would result in overfunding of sector 1.
Corollary 1 gives an intuitive sufficient condition for all the conditions in
Proposition 2 to be satisfied.
Corollary 1 A symmetric Bayesian equilibrium si = (0,0,1) for i = 1,2
exists if p0 1.
Proof. Straightforward given Proposition 2.
When it is very likely that the other donor has a zero budget each donor
behaves as if he were the only donor active in the recipient country and
follows the "monopoly" strategy. The resulting allocation is given in Table
3.
Table 3. Symmetric minimal sharing equilibrium: s1 = (0, 0,1), s2 =
(0, 0, 1)
0 1 2 3
0 (0,0) (1,0) (2,0) (2,1)
1 (1,0) (2,0) (3,0)* (3,1)
2 (2,0) (3,0)* (4,0)* (4,1)*
3 (2,1) (3,1) (4,1)* (4,2)
In this equilibrium there is too little sharing. There is coordination failure
in five eventualities marked by * . Ex post both donors regret that too much
of the aggregate funds are invested in the priority sector 1. The coordination
failure occurs for budget size 2.
To solve the above coordination failure, the donors could share also from
budget size 2. Proposition 3 establishes when this is a Bayesian equilibrium.
9To derive both necessary and sufficient conditions we would need to consider also
reallocations by two projects and we would need more structure to the utility functions
than we currently have. E.g. we would need to rank u(2, 2) and u (4, 0). We have chosen
not to proceed this way and therefore most of the propositions give sufficient conditions.
10
Proposition 3 A symmetric Bayesian equilibrium si = (0,1,1) for i = 1,2
exists if and only if
p0u(1, 1) + p1u (2, 1) + p2u (2, 2) + p3u (3, 2) < 0
+  + 
Proof. See the Appendix.
The condition in Proposition 3 is for the donor preferring to share a
budget of 2 equally between the two sectors. Given ni = 2 and sj = (0,1, 1)
donor i prefers sharing if (p1 + p3) is large enough. For nj = 1 donor j invests
one project in sector 1. Additional two projects would lead to too unequal
allocation. Thus donor i prefers sharing. With nj = 3 donor j invests two
projects in sector 1 and again additional two projects by donor i would mean
that sector 1 is overfunded.
Corollary 2 gives sufficient conditions for this symmetric equilibrium to
exist.
Corollary 2 A symmetric Bayesian equilibrium si = (0,1,1) for i = 1,2
exists if (i) p1 1 or (ii) p3 1.
Proof. Straightforward given Proposition 3.
The other donor is likely to have a larger budget than in the previous
corollary. Therefore the expected aggregate budget is higher and more of it
should go to sector 2. This can be achieved by sharing strategy si = (0, 1,1) .
The allocation of funds is given in Table 4.
Table 4. Symmetric sharing equilibrium: s1 = (0, 1, 1), s2 = (0, 1,1)
0 1 2 3
0 (0,0) (1,0) (1,1)** (2,1)
1 (1,0) (2,0) (2,1) (3,1)
2 (1,1)** (2,1) (2,2)** (3,2)
3 (2,1) (3,1) (3,2) (4,2)
This time we have oversharing in equilibrium in three eventualities marked
by **. Both sectors are funded equally in the three eventualities while the
donors would get higher utility from reallocating one project to the priority
sector. The coordination failure occurs again for budget size 2.
To balance the two types of coordination failure (undersharing and over
sharing) donor i could follow si = (0,0, 1) and donor j could choose sj =
11
(0, 1, 1) . The following proposition establishes the sufficient conditions for
this to be a Bayesian equilibrium.
Proposition 4 An asymmetric Bayesian equilibrium si = (0,0,1) and sj =
(0, 1, 1) exists if
(i)  p2u+(2, 2) < p0u+(1, 1) + p1u(2, 1) + p3u(3, 2) < p2u(3, 1)
(ii) p0u(0, 1) + p1u (1, 1) + p2u (2, 1) + p3u (2, 2) > 0 and
(iii) p0u(1, 2) + p1u (2, 2) + p2 (3, 2) + p3u (3, 3) > 0.
+ + +
u
+ +  +
Proof. See the Appendix.
None of the conditions in Proposition 4 is particularly stringent. But
for this equilibrium we cannot find a simple probability distribution that
would satisfy all the conditions. Uniform distribution with some additional
restrictions fulfills the conditions as the following corollary shows.
Corollary 3 An asymmetric Bayesian equilibrium si = (0,0,1) and sj =
(0, 1, 1) exists if
(i) p0 = p1 = p2 = p3,
(ii) u (2, 2) < u(1, 1) + u (2, 1) + u (3, 2) < u (3, 1) and
(iii) u(1, 2) + u (2, 2) + u (3, 3) + u (3, 2) > 0.
+ + 
+ + + 
Proof. See the Appendix.
When all the budget sizes are equally likely it is optimal to balance over
sharing and undersharing. The resulting equilibrium is in Table 5.
Table 5. Asymmetric sharing equilibrium: s1 = (0, 0, 1), s2 = (0,1, 1)
0 1 2 3
0 (0,0) (1,0) (1,1)** (2,1)
1 (1,0) (2,0) (2,1) (3,1)
2 (2,0) (3,0)* (3,1) (4,1)*
3 (2,1) (3,1) (3,2) (4,2)
Asymmetric sharing strategies eliminate two out of the three oversharing
eventualities of Table 4 and three out of the five undersharing eventualities
of Table 3.
12
Alternatively the donors can specialize in different sectors. With special
ization donor i invests all his budget in sector 1 and donor j specializes in
sector 2. Donor j does not fund more than two projects in sector 2 as we
know from the complete information case that the donors never wish to have
more than two projects in sector 2. Proposition 5 shows when specialization
is a Bayesian equilibrium.
Proposition 5 Specialization si = (0,0,0) and sj = (1,2,2) is a Bayesian
equilibrium if
(i) p0u(2, 1) + p1u (2, 2) + p2u (2, 3) + p3u (3, 3) > 0,
(ii) p0u(0, 1) + p1u (1, 1) + p2u (2, 1) + p3u (3, 1) < 0,
+ + +
(iii) p0u(1, 1) + p1u (2, 1) + p2 (3, 1) + p3 (4, 1) < 0,
+ +
u u
(iv) p0u(0, 2) + p1u(1, 2) + p2u(2, 2) + p3u(3, 2) < 0 and
+
(v) p0u(1, 2) + p1u (2, 2) + p2u (3, 2) + p3u(4, 2) < 0.
+ + +
+ +  
Proof. See the Appendix.
Out of the five conditions in Proposition 5 only (i) is for donor i. Given
donor j is specializing in the low priority sector it is obvious that donor i
has very strong incentives to specialize in sector 1 whatever the probability
distribution of donor j's budget. The only case when donor i would not
allocate all the funds in sector 1 is when ni = 3 and donor j is very likely to
have a zero budget. Then donor i would fund one project in sector 2. But if
p0 is low enough si = (0, 0,0) is best response to sj = (1, 2, 2) as condition
(i) requires.
It is clearly more difficult for a donor to specialize in the lower priority
sector. Conditions (ii)  (v) in Proposition 5 have to be fulfilled for special
ization in sector 2 to be optimal. The most stringent of them is condition
(iv). This condition is for donor j to prefer to invest all his budget nj = 2
in sector 2 given si = (0,0, 0). Donor j gives higher priority to sector 1 and
only wishes to invest two projects in sector 2 if sector 1 receives more than
that from donor i. This is the case when donor i is very likely to have a
budget of 3.
Corollary 4 shows that not only condition (iv) but all the conditions in
Proposition 5 are satisfied for p3 large.
13
Corollary 4 Specialization equilibrium si = (0,0,0) and sj = (1,2,2) exists
if p3 1.
Proof. Straightforward given Proposition 5.
Why would a donor ever specialize in the low priority sector? When
he knows that the other donor is funding the priority sector and has large
enough budget to be able to deal with that on his own, then he is happy to
concentrate on the less important sector. This is the case when p3 is large.
Furthermore, when the expected aggregate budget is large (when p3 is large)
the lower priority sector 2 should be relatively well funded and accordingly
one donor specializing in sector 2 is optimal. Allocation with specialization
is demonstrated in Table 6.
Table 6. Specialization equilibrium: s1 = (0,0, 0), s2 = (1, 2, 2)
0 1 2 3
0 (0,0) (0,1)** (0,2)** (1,2)**
1 (1,0) (1,1)** (1,2)** (2,2)**
2 (2,0) (2,1) (2,2)** (3,2)
3 (3,0)* (3,1) (3,2) (4,2)
There is oversharing in seven eventualities and in four of them the lower
priority sector 2 is actually better funded than sector 1. Additionally there
is one eventuality of undersharing. If the donor specializing in the priority
sector has a low budget, specialization leads to oversharing.
Much of the coordination failure in Table 5 occurs when the donor spe
cializing in sector 2 has a budget of 2. If that donor specializes less in sector
2 and chooses sj = (1, 1, 2) some of this coordination failure is eliminated.
In equilibrium this strategy is chosen in combination with si = (0, 0,1): now
also donor i specializes only partially in sector 1.
14
Proposition 6 Partial specialization si = (0,0,1) and sj = (1,1,2) is a
Bayesian equilibrium if
(i) p0u(0, 1) + p1u (1, 1) + p2u (2, 1) + p3u (2, 2) < 0,
(ii) p0u(1, 1) + p1u (2, 1) + p2 (3, 1) + p3u (3, 2) < 0 and
+ + +
u
(iii) p0u(1, 2) + p1 (2, 2) + p2 (3, 2) + p3 (3, 3) < 0.
+
u u u
+ +  +
Proof. See the Appendix.
Conditions (i) and (iii) in Proposition 6 are stringent. Condition (i) is
for donor j with nj = 1 to prefer investing in sector 2 given si = (0, 0,1) . If
p2 is large, donor i is likely to invest two projects in sector 1. Then donor j
is better off funding one project in the lower priority sector 2 or else sector
1 is overfunded. Condition (iii) is for donor j with nj = 3 to prefer funding
two projects in sector 2. If p2 is large, donor i is likely to invest two projects
in the priority sector 1. When donor j funds one project in sector 1 and two
projects in sector 2, the aggregate allocation is in favor of the priority sector
1 but not too unbalanced.
It turns out that also condition (ii) is satisfied for p2 large as Corollary 5
proves.
Corollary 5 Partial specialization with si = (0,0,1) and sj = (1,1,2) is a
Bayesian equilibrium if p2 1.
Proof. Straightforward given Proposition 6.
The other donor is likely to have a fairly large budget and therefore
nearly be able to deal with one sector. Specialization is not complete: for
large budget the donors allocate some funds to the other sector. Table 7
shows the resulting allocation.
Table 7. Partial specialization equilibrium: s1 = (0,0, 1), s2 = (1, 1, 2)
0 1 2 3
0 (0,0) (0,1)** (1,1)** (1,2)**
1 (1,0) (1,1)** (2,1) (2,2)**
2 (2,0) (2,1) (3,1) (3,2)
3 (2,1) (2,2)** (3,2) (3,3)**
15
Compared to specialization equilibrium partial specialization eliminates
two out of the seven oversharing eventualities and reduces coordination fail
ure in one eventuality. The one case of undersharing is eliminated as well.
On the other hand, two new cases of oversharing occur.
The five equilibria analyzed above are the only pure strategy Bayesian
equilibria that can exist in this allocation game. We have restricted analysis
to weakly increasing sharing strategies, i.e. si = (, , ) such that
.
The pattern that emerges from these results is that:
· When the other donor is likely to have a small budget (0 or 1) or the
distribution is uniform both donors concentrate on the priority sector
and share marginally.
· When the other donor is likely to have a large budget (2 or 3) the
donors specialize.10
If it is very likely that the other donor has a large budget, the donors
specialize. From large aggregate budget quite a lot should go to the lower
priority sector 2 making it feasible for one donor to specialize in the lower
priority sector. With a large budget the donor is able to deal with one sector
on his own.
If it is very likely that the other donor has a small budget, each donor
concentrates on the priority sector and invests a little in sector 2. From
small aggregate budget most funds should go to the priority sector 1. This
is achieved by both donors focusing on the priority sector.
10For p3 1 two equilibria exist: specialization and symmetric sharing equilibrium.
16
3.3 Discussion of the results
The following factors cause the coordination failure:
· complementary projects
· incomplete information about the other donor's budget
· simultaneous decisions
· discrete investments
The first three factors are needed for the coordination failure to occur
and the fourth factor exacerbates it.
The donors' projects are complementary. The donor's utility depends
also on the number of projects financed by the other donor. If each donor
cared only about their own projects and not about the general state of the
sector, there would be no need for coordination and no coordination failure.
If the donors have complete information about the other donor's budget,
the Nash equilibrium allocation (Table 1) is efficient (from the donors' point
of view). In reality the donors are not fully aware of the other donors'
activities and budgets and therefore complete information is not the relevant
case to be analyzed.
If the donors move sequentially rather than simultaneously, there is no
coordination failure. If donor 1 takes leadership and goes first and donor
2 invests only after observing donor 1's decisions, the allocation would be
efficient. However, if the donors have different preferences sequential deci
sions are not a complete answer. Also in practise there is often incomplete
information about the other donors' activities even ex post and therefore
sequentially would not solve the information problem.
If the funds were fully divisible and the donor had CobbDouglas util
ity functions with no sufficiency level for either sector, allocation would be
efficient even with incomplete information. With CobbDouglas utility the
optimal allocation is to spend a certain proportion of the budget in each
sector. Then if each donor allocates funds exactly in proportion to their
priorities, the aggregate allocation would be efficient. If there is a minimum
amount of funding needed in either sector (which is quite realistic in a de
veloping country) or the donors utility functions have some other functional
form for which the optimal allocation is not proportional to the budget, co
ordination failure occurs even if funds are fully divisible. In any case the
17
lumpiness of funds exacerbates the coordination failure. Margins are large.
In our model coordination failure is not just marginal but allocations like
(4,0) and (0,2) occur in equilibrium and with a larger number of donors the
misallocation can be much worse. In reality most of foreign aid is channeled
as project aid which is discrete.
How would learning in a dynamic game help the coordination failure? The
donors know each other's preferences and can calculate equilibrium sharing
strategies and there is no need for learning the other donor's strategy. Budget
size varies in every round of the game and therefore there is limited scope
for learning (although the donors will learn who are the big players and who
are the small players). What they can learn is the extent of misallocation
after each round. Say the equilibrium of the first round is such that sector
2 does not get any funds although the aggregate budget is large. Ex post
the donors learn the aggregate allocation and agree that there is a gap in
services: sector 2 should get more resources. In the next round the donors
have revised priorities and wish to allocate more funds towards sector 2.
The only difference to the first round of the game is that the priorities are
different. Coordination failure (of a different type) can occur again. Dynamic
game does not solve the coordination failure.
4 Active recipient
In this section we analyze allocation game between the recipient country, R,
and one donor, D. We modify the model to allow for heterogeneous agents.
R's utility function is denoted by uR (,) and D's by uD (,). Similarly
the probability distribution for R's budget is denoted by pR and D's by pD
n n
3
where pin = 1 for i = R, D.
n=0
D has the same preferences as in Section 3, that is D prefers sector 1. R's
preferences are described by . = 0 denotes that R has similar preferences
to D. = 1 means that R values the sectors equally. = 1 denotes that
2
R prefers sector 2 and has the opposite preferences to D, i.e. R prefers
sector 2 to sector 1 as strongly as D prefers sector 1 to sector 2. As Section
3 analyzed allocation games between parties that have similar preferences
we now concentrate on = 1 (different preferences) and = 1 (opposite
2
preferences). Recipient government may not agree with donors about the
actions that will promote welfare among its population or they may not
prioritize 'propoor' spending because the poor have weak political voice.
18
The properties of R's utility functions are formalized in Assumptions 3
and 4.
Assumption 3. For = 1
2 (i) uR (, ) < 0 if and only if .
(ii) uR (, ) = 0 if and only if =  1.
(iii) uR (, ) > 0 if and only if <  1.
Assumption 4. For = 1 (i) uR (,) < 0 if and only if  < 3.
(ii) uR (, ) > 0 if and only if  3.
According to Assumption 3 when R values the sectors equally reallocating
a project to sector 1 increases R's utility if and only if it makes the allocation
more equal. Assumption 4 says that when the agents have the opposite
preferences reallocating a project to R's lower priority sector 1 reduces R's
utility (property (i)) unless the original allocation was too much in favour of
sector 2 (property (ii)).11
Allocation decisions are sequential. D allocates funds to the sectors first
and R moves second. We analyze firstly complete information where R ob
serves D's choice before his decision. Secondly, we analyze incomplete infor
mation where R moves without knowing D's action or budget. This can be
modelled as a simultaneous move game with incomplete information.
4.1 Different preferences
4.1.1 Complete information
We start the analysis from = 1 and complete information. D prefers sector
2
1 while R values the sectors equally.
Suppose D invests in sector 1 and the remaining (nD  ) in sector
2. R observes D's choice and then aims to balance the resources in the
sectors. R does not have the funds to equalize the resources if and only if
(nD  )   > nR. Then R invests all of nR in the sector with less funds.
D would naturally choose to fund sector 1 more generously and then R invests
all his funds in sector 2. The resulting allocation is (,nR + nD  ) . Given
11Notice that although R prefers sector 2 to sector 1 as strongly as D prefers sector 1 to
sector 2 Assumption 2 and 4 look somewhat different. The reason is that we have defined
ui (,) as reallocation of one project towards D's priority but R's lower priority sector
1.
19
this D chooses to maximize his utility and thus chooses = nD (unless
the budgets are (3,0)). The subgame perfect Nash equilibrium allocation is
(nD, nR) if and only if nD > nR and (nD  nR) < 3. For nD = 3 and nR = 0
the Nash equilibrium allocation is (2,1).
R has the funds to balance the resources if and only if nD nR. If D has
funded projects in sector 1, R will fund projects in sector 1 so that
+ = nD + nR   = nD + nR  2
2
for even (nD + nR) . The resulting allocation is nD+nR nD+nR
, . For odd
2 2
(nD + nR) R is indifferent between choosing = nD+nR21 and =
2
nD+nR2+1 . The resulting allocation is either , or , nR .
2 nD+nR+1 nD+nR1
2 2 nD+nR1 nD+2 +1
2
When nD nR the subgame perfect Nash equilibrium allocation is the same
whatever D chooses. Therefore there are multiple equilibria though the
equilibrium aggregate allocation is unique (or in the case of odd aggregate
budget there are two equilibrium allocations).
The subgame perfect Nash equilibrium allocations are given in Table 8.
D's budget is given vertically and R's budget horizontally.
Table 8. Complete Information
0 1 2 3
0 (0,0) (1,0),(0,1) D (1,1) D (2,1),(1,2) D
1 (1,0) (1,1) D (2,1), (1,2) D (2,2) D
2 (2,0) R (2,1) (2,2) D (3,2), (2,3) D
3 (2,1) (3,1) R (3,2) (3,3) D
When R has a larger budget than D, D cannot affect the final allocation.
Whatever D does, R will balance the resources between the sectors. Without
aid R would share his budget equally between the two sectors. D wishes to
raise funding in his priority sector 1 but cannot do that because R responds by
shifting funds away from sector 1. This is called the aid fungibility problem.
When a donor builds e.g. a hospital, the recipient  who would have built
that hospital anyhow  shifts his funding away from the health sector and
builds a road instead or at worst increases military expenses. Therefore, the
donor is not financing a hospital at the margin. In this case D could as well
give the aid in the form of budget support to R and allow R to make all the
allocation decisions.
When D has a larger budget than R, the agents specialize. D invests all
his funds in his priority sector 1. Although R values the sectors equally, he
20
only invests in sector 2. This is because D has such a strong preference for
sector 1. Without aid R would have funded the sectors equally but given the
aid R only invests in sector 2. Again aid is fungible. In reality often when
the donors start funding the health and the education sectors, the recipient
government moves away from these sectors and shifts funds to other sectors.
D in Table 8 denotes that there is misallocation from D's point of view
(respectively for R). The resulting allocation is unfavorable to D when R has a
relatively large budget and can match D's investment, in total 9 eventualities.
Only in two eventualities is there Rmisallocation when D has a large budget
relative to R.
4.1.2 Incomplete information
With incomplete information R does not observe D's allocation or budget
when making his choice. Therefore we have a simultaneous move game with
incomplete information. Incomplete information introduces coordination fail
ure between R and D. Not only is aid fungible but equilibrium allocation can
be such that both D and R regret it ex post and would wish to change the
allocation in the same way.
Complete specialization is the first type of Bayesian equilibrium we have.
D allocates all the funds to his priority sector 1 and R invests only in sector
2. Sharing strategy si = (,, ) is defined as in the previous section and
denotes how many projects i allocates in sector 2.
Proposition 7 Complete specialization sD = (0,0,0) and sR = (1,2,3) is a
Bayesian equilibrium if
(i) pRuD(2, 1) + pRuD (2, 2) + pRuD (2, 3) + pRuD (2, 4) > 0,
0 1 2 3
 + + +
(ii) pDuR(0, 2) + pDuR (1, 2) + pDuR (2, 2) + pDuR (3, 2) < 0 and
0 1 2 3
+ 0
(iii) pDuR(0, 3) + pDuR (1, 3) + pDuR (2, 3) + pDuR (3, 3) < 0.
 
0 1 2 3
+ + 0 
Proof. See the Appendix.
It is obvious that D has very strong incentives to specialize in his priority
sector 1 given R only funds sector 2. The only eventuality when D would
invest in sector 2 is when he has a budget of 3 and R has a zero budget. If the
probability of this eventuality, pR, is small enough, D specializes completely
0
in sector 1 as condition (i) in Proposition 7 states.
21
R aims for equal resources in both sectors. D investing all the funds in
sector 1 gives good incentives for R to specialize in sector 2. Only if R has a
large budget relative to D, would R invest in sector 1. If D is likely to have a
large budget (2 or 3), he is able to deal with sector 1 on his own and R can
concentrate on sector 2 as conditions (ii) and (iii) in Proposition 7 require.
Corollary 6 gives sufficient conditions for the conditions in Proposition 7
to be satisfied.
Corollary 6 Complete specialization sD = (0,0,0) and sR = (1,2,3) is a
Bayesian equilibrium if pR 0 and pD = pD 0.
0 0 1
Complete specialization occurs when R is unlikely to have a zero budget
and D is likely to have a large budget (2 or 3). We have explained above the
intuition for this result.
The resulting allocation is given in Table 9. D's budget is given vertically
and R's budget horizontally.
Table 9. Complete Specialization
0 1 2 3
0 (0,0) (0,1) D (0,2)** (0,3)**
1 (1,0) (1,1) D (1,2) D (1,3) **
2 (2,0) R (2,1) (2,2) D (2,3) D
3 (3,0)* (3,1) R (3,2) (3,3) D
Compared to complete information we have coordination failure in four
eventualities. Coordination failure means that there is misallocation from
both D's and R's point of view and they would like to change the allocation
to the same direction. Sector 1 is overfunded in one eventuality marked by *
and underfunded in three eventualities marked by **. Coordination failure
occurs when one party has a large budget compared to the other party, in
particular when R's budget is large compared to D's budget.
With complete specialization most of the coordination failure occurs be
cause R invests too much in sector 2. If R specializes less in sector 2 this
coordination failure could be eliminated. The next proposition shows when
this is an equilibrium.
22
Proposition 8 sD = (0,0,0) and sR = (1,2,2) is a Bayesian equilibrium if
(i) pRuD(2, 1) + pRuD (2, 2) + pRuD (2, 3) + pRuD (3, 3) > 0,
0 1 2 3
 + + +
(ii) pDuR(0, 2) + pDuR (1, 2) + pDuR (2, 2) + pDuR (3, 2) < 0 and
0 1 2 3
+ 0
(iii) pDuR(0, 3) + pDuR (1, 3) + pDuR (2, 3) + pDuR (3, 3) > 0.
 
0 1 2 3
+ + 0 
Proof. See the Appendix.
Now D specializes completely in sector 1 as in Proposition 7 while R shifts
one project to sector 1 when he has a budget of 3. None of the conditions in
Proposition 8 is particularly stringent. The main change from Proposition 7
is that condition (iii) changes the sign.12 R prefers to invest one project in
sector 1 when his budget is 3 if D is unlikely to have a budget of 3. If D had
a budget of 3, he would invest it all in sector 1 and an additional project by
R would make sector 1 overfunded from R's point of view.
Corollary 7 gives sufficient conditions for all the conditions in Proposition
8 to be satisfied.
Corollary 7 sD = (0,0,0) and sR = (1,2,2) is a Bayesian equilibrium if
pR 0 and pD = pD 0.
0 0 3
D has now an intermediary budget (1 or 2) and smaller than in Corollary
6. Therefore R, aiming at balance, shifts some funds to sector 1 as D is less
able to deal with it on his own. R's changed strategy changes the outcome
in the following way:
Table 10
0 1 2 3
0 (0,0) (0,1) D (0,2)** (1,2)
1 (1,0) (1,1) D (1,2) D (2,2) D
2 (2,0) R (2,1) (2,2) D (3,2)
3 (3,0)* (3,1) R (3,2) (4,2) R
Compared to complete specialization two eventualities of coordination
failure are eliminated.
One more eventuality of coordination failure could be eliminated if R
invested in sector 1 also from nR = 2. Proposition 9 finds the sufficient
conditions for this equilibrium.
12Condition (ii) is the same as in Proposition 7 and (i) differs only by the last term.
23
Proposition 9 sD = (0,0,0) and sR = (1,1,2) is a Bayesian equilibrium if
(i) pRuD(2, 1) + pRuD (2, 2) + pRuD (3, 2) + pRuD (3, 3) > 0,
0 1 2 3
 +  +
(ii) pDuR(0, 2) + pDuR (1, 2) + pDuR (2, 2) + pDuR (3, 2) > 0 and
0 1 2 3
+ 0
(iii) pDuR(0, 3) + pDuR (1, 3) + pDuR (2, 3) + pDuR (3, 3) > 0.
 
0 1 2 3
+ + 0 
Proof. See the Appendix.
The main change compared to Proposition 8 is that condition (ii) changes
the sign.13 Condition (ii) is for R to prefer sharing budget 2 between the
sectors. R prefers sharing unless D has a large budget (2 or 3). D invests all
the funds in sector 1 and if this investment is large R prefers not to share the
resources between the sectors as it would leave sector 2 underfunded from
R's point of view.
Corollary 8 gives sufficient conditions for (i)  (iii) in Proposition 9 to
hold.
Corollary 8 sD = (0,0,0) and sR = (1,1,2) is a Bayesian equilibrium if
pR = pR 0 and pD = pD 0.
0 2 2 3
Compared to Corollary 7 D has an even smaller budget (0 or 1) and
therefore R shifts even more funds towards sector 1 to equalize resources.
Corollaries 6  8 show that when D specializes in his priority sector 1, R
concentrates on funding sector 2 but gives the more resources to sector 1 the
smaller is D's expected budget.14
What is different to Corollaries 6 and 7 is that in addition to pR also pR
0 2
is required to be negligible. To understand this we examine condition (i) in
Proposition 9 which is for D to prefer investing 3 projects in sector 1. We
already know why it does not hold for pR large. Why it does not hold for pR
0 2
large is the following. If it is very likely that R has a budget of 2 which he
13Condition (iii) is the same as in Proposition 8 and (i) differs by the third term.
14In fact, we could continue this line of analysis for the smallest possible budget for
D, i.e. pD 1. Then R would follow his "monopoly" strategy. R shares the resources
0
evenly between the sectors if his budget is even. If R's budget is odd, he is indifferent
between which sector gets the additional project. R is then indifferent between strategies
sR = (0, 1, 1) , sR = (1, 1, 1) , sR = (0, 1, 2) and sR = (1, 1, 2) . We could then check when
these strategies form a Bayesian equilbrium with sD = (0, 0, 0). (Proposition 9 already
deals with sR = (1, 1, 2).) This is unnecessarily complicated and does not add much to
our results. We therefore do not proceed with this.
24
would share between the sectors and D funds 3 projects in sector 1, sector 1
would be overfunded even from D's point of view. Therefore sD = (0,0, 0) is
not a best reply to sR = (1, 1, 2) if pR is large.
2
The allocation resulting from the equilibrium of Proposition 9 is given
in Table 11. Compared to Table 10 one event of coordination failure is
eliminated but one event of overfunding of sector 1 is introduced (the event
explained in the previous paragraph with budgets (3,2)).
Table 11
0 1 2 3
0 (0,0) (0,1) D (1,1) D (1,2) D
1 (1,0) (1,1) D (2,1) (2,2) D
2 (2,0) R (2,1) (3,1) R (3,2)
3 (3,0)* (3,1) R (4,1) * (4,2) R
Now we move on to examining equilibria where D does not fully specialize
in his priority sector 1 but invests one project in sector 2 from a large budget,
i.e. D follows a sharing strategy sD = (0,0, 1). We can prove that we can
have only two types of Bayesian equilibria with sD = (0, 0, 1) : one where
sR = (1, 2, 2) and another where sR = (1, 1, 2) .
Proposition 10 sD = (0,0,1) and sR = (1,2,2) is a Bayesian equilibrium
if
(i) pRuD(2, 1) + pRuD (2, 2) + pRuD (2, 3) + pRuD (3, 3) < 0 and
0 1 2 3
 + + +
(ii) pDuR(0, 2) + pDuR (1, 2) + pDuR (2, 2) + pDuR (2, 3) < 0.
0 1 2 3
+ 0  0
Proof. See the Appendix.
In this equilibrium the agents specialize in different sectors and invest one
project in the other sector from a large budget of 3. Condition (i) is stringent;
only one of the four terms has the same sign as the whole expression. It is
for D to prefer investing one project in sector 2 when his budget is 3. D
will do that only if it is very likely that otherwise sector 2 does not get any
resources, which is the case when R has a zero budget. Corollary 9 gives
the sufficient conditions for both of the conditions in Proposition 10 to be
fulfilled.
Corollary 9 sD = (0,0,1) and sR = (1,2,2) is a Bayesian equilibrium if
pR 1 and pD 0.
0 0
25
When it is very likely that R has a zero budget, D is willing to share
when his budget is 3. Condition (ii) in Proposition 10 is for R to invest all
of budget 2 in sector 2. As D is concentrating on sector 1, R aiming for
balanced resources can invest all of budget 2 in sector 2 unless D has no
resources, i.e. if pD is negligible. Table 12 reports the resulting allocations.
0
Table 12
0 1 2 3
0 (0,0) (0,1) D (0,2)** (1,2) D
1 (1,0) (1,1) D (1,2) D (2,2) D
2 (2,0) R (2,1) (2,2) D (3,2)
3 (2,1) (2,2) D (2,3) D (3,3) D
There is only one eventuality of coordination failure where sector 2 is
overfunded.
The second possible equilibrium where D shares marginally is proved in
Proposition 11.
Proposition 11 sD = (0,0,1) and sR = (1,1,2) is a Bayesian equilibrium
if
(i) pRuD(2, 1) + pRuD (2, 2) + pRuD (3, 2) + pRuD (3, 3) < 0 and
0 1 2 3
 +  +
(ii) pDuR(0, 2) + pDuR (1, 2) + pDuR (2, 2) + pDuR (2, 3) > 0.
0 1 2 3
+ 0  0
Proof. See the Appendix.
Compared to Proposition 10 R is investing less in sector 2. Condition
(i) is nearly the same as in Proposition 10: only the third term is different
and as it is now negative the condition is easier to satisfy. D is more willing
to invest one project in sector 2 (when his budget is 3) when R invests less
in sector 2. The terms in condition (ii) are the same as in Proposition 10,
only now the expression is required to be positive so that R rather shares a
budget of 2 between the sectors than invests it all in sector 2. The condition
is satisfied unless pD is very large. Then D would invest 2 projects in sector
2
1 and R would prefer to match the investment in sector 2.
Corollary 10 again gives the sufficient conditions for (i) and (ii) in Propo
sition 11 to be fulfilled.
26
Corollary 10 sD = (0,0,1) and sR = (1,1,2) is a Bayesian equilibrium if
pR = pR 0 and pD 0.
1 3 2
Corollary 10 shows that D is willing to share when R's budget is either
0 or 2. It is clear why R's zero budget gives D incentives to share. Why
nR = 2 induces D to share is the following. R would share his budget of 2
between the sectors and an additional three projects in sector 1 would lead
to too unbalanced allocation even from D's point of view. Table 13 gives the
resulting allocation.
Table 13
0 1 2 3
0 (0,0) (0,1) D (1,1) D (1,2) D
1 (1,0) (1,1) D (2,1) (2,2) D
2 (2,0) R (2,1) (3,1) R (3,2) D
3 (2,1) (2,2) D (3,2) (3,3) D
In this equilibrium coordination failure is eliminated completely. It does
not, however, implement the complete information allocation as one out
come is turned unfavorable to D (budgets (3,1)) and one outcome is turned
unfavorable to R (budgets (2,2)).
We have checked that the equilibria examined above in Propositions 7 
11 (see also footnote 11) are the only pure strategy Bayesian equilibria that
can exist in this allocation game. We have restricted analysis to weakly
increasing sharing strategies. We observe from these results that only sD =
(0, 0, 0) or sD = (0, 0, 1) can arise in a pure strategy Bayesian equilibrium.
In other words, D maximally invests one project in his lower priority sector.
4.1.3 Summary
When D prefers sector 1 and R values the sectors equally D concentrates on
his priority sector. D specializes fully in sector 1, i.e. follows sD = (0, 0,0) ,
when R is able to deal with sector 2 on his own (pR is negligible). While
0
R shifts the more funds to sector 1 the lower are D's expected resources. D
shares marginally, i.e. follows sD = (0, 0, 1) , when R may not be able to deal
with sector 2 on his own (pR > 0). Then only two equilibria are possible and
0
there is not as clear pattern but R again shifts funds between the sectors so
that expected allocation would be equalized.
27
Aid fungibility problem exists also with incomplete information: R re
sponds to D's strategy. However, incomplete information gives some power
to D. With complete information and nD nR D was indifferent between
any sharing strategy. His choice had no effect on the final allocation be
cause R could match his investments since he has a larger budget. Now
D's choice matters because of incomplete information R cannot fully match
D's investment. D can therefore influence the final allocation. On the other
hand, incomplete information introduces coordination failure and the final
allocation can be unfavorable to both D and R.
4.2 Opposite preferences
4.2.1 Complete information
Finally we analyze = 1. D prefers sector 1 and R prefers sector 2. With
complete information D invests all the funds in sector 1 unless budgets are
(3,0) in which case D invests one project in sector 2. Similarly R invests all
the funds in sector 2 unless the budgets are (0,3). The resulting allocations
are:
Table 14. Complete Information, Opposite Preferences
0 1 2 3
0 (0,0) (0,1) D (0,2) D (1,2) D
1 (1,0) R (1,1) D,R (1,2) D (1,3) D
2 (2,0) R (2,1) R (2,2) D,R (2,3) D
3 (2,1) R (3,1) R (3,2) R (3,3) D,R
When the parties have completely opposite preferences, there is always
misallocation from someone's point of view. The party with a relatively large
budget is able to enforce an allocation that is favorable to him. Timing of
the moves makes no difference. Simultaneous and sequential decisions lead
to the same allocation with complete information. When the resources are
equally divided between the sectors both D and R view it as a misallocation.
However, it is not coordination failure as D prefers to reallocate towards
sector 1 and R prefers to reallocate towards sector 2.
4.2.2 Incomplete information
From the complete information case we know that D and R will not share
from budget sizes 1 and 2 and wish to invest at most one project in the lower
28
priority sector when their budget size is 3. Therefore the only equilibria that
can exist are:
(i) Symmetric sharing equilibrium: both D and R invest one project in
the lower priority sector from budget size 3.
(ii) Symmetric no sharing equilibrium (complete specialization): both D
and R invest all the funds in their most preferred sector.
(iii) Asymmetric equilibria: D shares but R does not, R shares but D
does not.
With similar and different preferences a variety of equilibria can exist
but with the opposite preferences only these three equilibria are possible.
Therefore we can derive both the necessary and the sufficient conditions,
which we do in Propositions 13  15.
Proposition 12 sD = (0,0,0) and sR = (1,2,3) is a Bayesian equilibrium if
and only if pR < pR and pD < pD where pR = pR1 uD(2,2)+pRuD(2,3)+pRuD(2,4)
2 3
0 0 0 0 0 uD(2,1)
and pD = pD1 uR(1,3)+pDuR(2,3)+pDuR(3,3)
2 3 .
0 uR(0,3)
Proposition 13 (i) sD = (0,0,1) and sR = (1,2,3) is a Bayesian equilib
rium if and only if pR > pR and pD < pD where pD = pD1 uR(1,3)+pDuR(2,3)+pDuR(2,4)
2 3 .
0 0 0 0 0 uR(0,3)
(ii) sD = (0, 0, 0) and sR = (1, 2, 2) is a Bayesian equilibrium if and only
if pR < pR and pD > pD where pR = pR1 uD(2,2)+pRuD(2,3)+pRuD(3,3)
2 3 .
0 0 0 0 0 uD(2,1)
Proposition 14 sD = (0,0,1) and sR = (1,2,2) is a Bayesian equilibrium
if and only if pR > pR and pD > pD.
0 0 0 0
Proofs. See the Appendix.
Propositions 12  14 help us to construct Figure 1. It is intuitive to
assume that pi0 > pi0. For D this is implied by uR (3, 3) < uR (2,4). In
other words, shifting resources away from R's priority sector 2 isreducing

R's welfare less when sector 2 is originally better funded than sector 1 than
when the sectors are equally funded. pR > pR requires uD (2,4) > uD (3, 3).
0 0
+ +
When we shift resources towards D's priority sector 1 D's utility increases
more when sector 1 is originally funded less than sector 2 than when we start
from equal funding.
29
In Figure 1 we observe that for very high pj0 > pj0 agent i shares whatever
is j's strategy. If it is very likely that agent j is not able to fund his priority
sector, then agent i will share to avoid too unbalanced allocation.
For somewhat lower pj0, pj0 < pj0 < pj0, agent i shares only if agent j
shares too. When agent j shares, the expected funds in i's priority sector
are increased. To achieve a more balanced allocation agent i in return shifts
funds towards j's priority sector since it is fairly likely that j has a zero
budget. This explains the multiplicity of equilibria: either both agents share
or neither does in three regions in Figure 1.
Finally, if both pD and pR are very low neither agent shares as both agents
0 0
can fund their priority sectors.
We have therefore identified two forces that give incentives for sharing
when the agents have the opposite preferences:
(i) If it is very likely that agent j is unable to fund his priority sector, i.e.
pj0 is high, then agent i will share.
(ii) If agent j shares, agent i's incentives to share are increased as expected
funds are shifted from j's priority sector towards i's priority sector.
Tables 1518 report the equilibrium allocations with opposite preferences.
Some of the allocations are the same as in the last subsection. However, com
plete information case differs and it is the comparison to that that determines
what is coordination failure. Note that the definition of coordination failure
is that there is an allocation that Pareto dominates the equilibrium alloca
tion. There are allocations that both D and R view as misallocations but it
is not coordination failure as each would like to change it in favor of their
preferred sector.
Table 15. Complete Specialization
0 1 2 3
0 (0,0) (0,1) D (0,2) D (0,3)*
1 (1,0) R (1,1) D,R (1,2) D (1,3) D
2 (2,0) R (2,1) R (2,2) D,R (2,3) D
3 (3,0)* (3,1) R (3,2) R (3,3) D,R
Table 16. D Sharing Equilibrium
0 1 2 3
0 (0,0) (0,1) D (0,2) D (0,3)*
1 (1,0) R (1,1) D,R (1,2) D (1,3) D
2 (2,0) R (2,1) R (2,2) D,R (2,3) D
3 (2,1) R (2,2) D,R (2,3) D (2,4) D
30
Table 17. R Sharing Equilibrium
0 1 2 3
0 (0,0) (0,1) D (0,2) D (1,2) D
1 (1,0) R (1,1) D,R (1,2) D (2,2) D,R
2 (2,0) R (2,1) R (2,2) D,R (3,2) R
3 (3,0)* (3,1) R (3,2) R (4,2) R
Table 18. Symmetric Sharing Equilibrium
0 1 2 3
0 (0,0) (0,1) D (0,2) D (1,2) D
1 (1,0) R (1,1) D,R (1,2) D (2,2) D,R
2 (2,0) R (2,1) R (2,2) D,R (3,2) R
3 (2,1) R (2,2) D,R (2,3) D (3,3) D,R
With the opposite preferences there are at most two cases of coordination
failure. In an equilibrium where both agents share coordination failure is
eliminated completely.
Although coordination failure occurs in very few cases allocations do
change from the complete information case. The agent who shares does that
not only when the other agent's budget is 0 but for the other budget sizes
too. Therefore he shares too much (from complete information perspective)
and turns some allocations unfavorable to himself.
4.2.3 Summary
When the agents have the opposite preferences, they each specialize in their
priority sector and invest maximally one project in the lower priority sector.
Some coordination failure can occur in equilibrium but its extent is minimal.
5 Conclusions
This paper has analyzed three versions of a foreign aid allocation game with
increasingly different preferences between the agents. It first analyzed an
allocation game with two identical donors who prioritize sector 1, and found
that incomplete information can lead to coordination failure. There are two
types of coordination failure. First, the lower priority sector does not get
enough funds as both donors concentrate too much on the priority sector
31
(undersharing). Second, there may be gaps in services in the priority sector
due to overfunding of the lowerpriority sector (oversharing). When the other
donor is expected to have a small budget, both donors concentrate on the
priority sector and share marginally. This can lead to either undersharing
or oversharing. With high expected budgets, the donors specialize. One
donor fully or partially specializes in the priority sector while the other donor
mainly invests in the lowerpriority sector, resulting in oversharing in some
eventualities.
The analysis then introduced an active recipient country with equal pref
erences for both sectors. In the allocation game, the donor concentrates on
funding his priority sector. The recipient mainly funds the donor's lower
priority sector and shifts the funds between the sectors according to the
donor's ability to finance his priority sector. In other words, aid is fungible.
However, with incomplete information, the recipient cannot fully match the
donor's investment and the donor can affect the final allocation. Therefore in
some sense the aid fungibility problem is reduced by incomplete information.
On the other hand, incomplete information introduces coordination failure so
that the final allocation can be inferior to both the recipient and the donor.
The paper finally analyzed a case where the recipient prefers the donor's
lowerpriority sector, that is, the agents have opposite preferences. With
these preferences each agent focuses on his priority sector and shares mar
ginally, and there was minimal coordination failure.
Comparing the extent of coordination failure with three types of pref
erences (see Tables 218) shows that the more similar the preferences are,
the more scope there is for coordination failure. With similar preferences,
coordination failure occurs in 3 to 8 eventualities, with different preferences
in 0 to 4 eventualities and with opposite preferences in only 0 to 2 even
tualities. When similar agents pursue the common good, misallocation due
to incomplete information can be considerable. When the agents have very
different priorities about their activities in the recipient developing country,
informational problems are minimal.
The focus of this paper has been on coordination failure among agents
who run projects in a recipient developing country. The paper has not
evaluated the arising equilibrium against a social welfare function but sim
ply examined the extent of Pareto inefficient allocations among providers.
The government of the recipient country may for example represent partic
ular constituencies where the poor may not be well represented. Therefore
the recipient government's utility function is not the social welfare function.
32
Likewise although the donors are interested in the welfare of the recipient
country, they also have interests of their own (e.g., their own country or bu
reaucracy). Therefore we are not saying that social welfare is higher when
agents have more preferences. We are simply saying that the inefficiency aris
ing from incomplete information is worse when preferences are more similar.
An interesting direction to extend the analysis is to take into account that
the donor's utility depends not only on the aggregate amount of funding in
each sector, but also on his own contribution. Aid agencies have a need for
visibility to justify their activities and therefore get more utility from their
own donations. This paper's polar case is one step toward understanding the
allocation of foreign aid. Furthermore, in this model even an active recipient
has no power over how the donor's projects are allocated but the donor can
unilaterally make decisions about his funds. A recipient with more bargaining
power would be another interesting extension.
33
6 Appendix
6.1 Proofs for Section 3
We first lay out the basic structure of the proofs which will then be applied
to prove each proposition.
Suppose donor j's equilibrium sharing strategy is sj = (, , ) where
{0,1}, {0,1,2}, {0,1,2}. From the complete information case
we know that the donors never wish to have more than two projects in sector
2. sj implies donor j allocates (1  ,2  , 3  ) projects in sector 1.
What is donor i's best response? We analyze each budget size for donor i
separately.
If ni = 1 donor i is better off investing it in sector 1 if and only if:
p0u (1,0) + p1u (2  , ) + p2u (3  , ) + p3u (4  , ) (3)
> p0u (0, 1) + p1u (1  , 1 + ) + p2u (2  , 1 + ) + p3u (3  ,1 + )
Equation (3) simplifies to:
p0u(0,1)+p1u (1  , 1 + )+p2u (2  , 1 + )+p3u (3  , 1 + )(4) > 0
The first entry in donor i's best response is zero, i.e. si = (0, , ) , if and
only if equation (4) holds. The first entry is one, si = (1, , ), if and only
if equation (4) does not hold.
Given ni = 2 donor i is better off investing all the funds in sector 1 rather
than sharing the funds between the sectors if and only if:
p0u(1,1)+p1u (2  , 1 + )+p2u (3  , 1 + )+p3u (4  , 1 + )(5) > 0
Given ni = 2 donor i prefers sharing the funds between the sectors to invest
ing all the funds in sector 2 if and only if:
p0u(0,2)+p1u (1  , 2 + )+p2u (2  , 2 + )+p3u (3  , 2 + )(6) 0 >
34
The second entry in donor i's best response is zero, i.e. si = ( ,0, ) , if both
(5) and (6) hold. The best response is si = ( , 1, ) if and only if (5) does
not hold and (6) holds. Finally, the second entry is 2, i.e. si = ( ,2, ) , if
neither (5) nor (6) hold.
Finally, we analyze ni = 3. Donor i prefers allocating 3 rather than 2
projects in sector 1 if and only if:
p0u(2,1)+p1u (3  , 1 + )+p2u (4  , 1 + )+p3u (5  , 1 + )(7) 0 >
Donor i prefers funding 1 rather than 2 projects in sector 2 if and only if:
p0u(1,2)+p1u (2  , 2 + )+p2u (3  , 2 + )+p3u (4  , 2 + )(8) 0 >
Accordingly, the third entry is zero if both (7) and (8) hold. The best re
sponse is si = ( , , 1) if and only if (7) does not hold and (8) holds. Finally,
the third entry is 2, i.e. si = ( , , 2), if neither (7) nor (8) hold.
We use equations (4)  (8) in all the proofs by inserting the appropriate
values for , and .
Proof of Proposition 2.
si = (0,0, 1) for i = 1,2 is Bayesian equilibrium if equations (4), (5), (6)
and (8) hold and (7) does not hold for = 0, = 0 and = 1.
p0u(0, 1) + p1u (1, 1) + p2u (2, 1) + p3u (2, 2) > 0 (9)
+ +  +
p0u(1, 1) + p1u (2, 1) + p2u (3, 1) + p3u (3, 2) > 0 (10)
+   
p0u(0, 2) + p1u (1, 2) + p2u (2, 2) + p3u (2, 3) > 0 (11)
+ + + +
p0u(2, 1) + p1u (3, 1) + p2u (4, 1) + p3u (4, 2) < 0 (12)
   
p0u(1, 2) + p1u (2, 2) + p2u (3, 2) + p3u (3, 3) > 0 (13)
+ +  +
35
(11) and (12) hold unambiguously. Therefore si = (0, 0, 1) for i = 1, 2 is
Bayesian equilibrium if equations (9) , (10) and (13) hold.
Q.E.D.
Proof of Proposition 3.
si = (0,1, 1) for i = 1,2 is Bayesian equilibrium if and only if equations
(4) , (6) and (8) hold and (5) and (7) do not hold for = 0, = 1 and = 1.
p0u(0, 1) + p1u (1, 1) + p2u (1, 2) + p3u (2, 2) > 0 (14)
+ + + +
p0u(1, 1) + p1u (2, 1) + p2u (2, 2) + p3u (3, 2) < 0 (15)
+  + 
p0u(0, 2) + p1u (1, 2) + p2u (1, 3) + p3u (2, 3) > 0 (16)
+ + + +
p0u(2, 1) + p1u (3, 1) + p2u (3, 2) + p3u (4, 2) < 0 (17)
   
p0u(1, 2) + p1u (2, 2) + p2u (2, 3) + p3u (3, 3) > 0 (18)
+ + + +
(14) , (16) , (17) and (18) hold unambiguously. Therefore si = (0, 1, 1) for
i = 1, 2 is Bayesian equilibrium if and only if equation (15) holds. Q.E.D.
Proof of Proposition 4.
Donor i's best response to sj = (0, 1, 1) is si = (0,0, 1) if (14), (16)(18)
hold (and they hold unambiguously) and
p0u(1, 1) + p1u (2, 1) + p2u (2, 2) + p3u (3, 2) > 0 (19)
+  + 
Donor j s best response to si = (0,0, 1) is sj = (0,1, 1) if and only if
(9) , (11)  (13) hold and
p0u(1, 1) + p1u (2, 1) + p2u (3, 1) + p3u (3, 2) < 0 (20)
+   
Equations (11) and (12) hold unambiguously.
si = (0, 0,1) and sj = (0, 1, 1) is a Bayesian equilibrium if equations
(9) , (13) , (19) and (20) and hold. Equations (19) and (20) can be combined
as:
36
p2u+(2,2) < p0u+(1,1) + p1u(2,1) + p3u(3,2) < p2u(3,1) (21)
Notice that p2u+(2, 2) < p2u(3, 1) and therefore (21) is feasible. The
sufficient conditions are therefore (9) , (13) and (21) .
Q.E.D.
Proof of Corollary 3.
Equations (9) ,(13), (19) and (20) and are satisfied if the probability
distribution is uniform (p0 = p1 = p2 = p3) and the following conditions hold:
u(1,1) + u(2,2) > u(2,1)  u(3,2) (22)
+ +  
u(0,1) + u(1,1) + u(2,2) > u(2,1) (23)
+ + + 
u(1,1) < u(2,1)  u(3,1)  u(3,2) (24)
+   
u(1,2) + u(2,2) + u(3,3) > u(3,2) (25)
+ + + 
We have moved all the negative terms to the righthandside of the equations.
(22) implies (23) since we can rewrite (23) as:
u(1,1) + u(2,2) > u(2,1)  u(0,1) (26)
+ +  +
and the righthandside of (22) is larger than the righthandside of (26) .
Equations (22) and (24) are satisfied if
u(2,2) < u(1,1) + u(2,1) + u(3,2) < u(3,1) (27)
Accordingly, if equations (27) and (25) hold si = (0, 0, 1) and sj = (0,1, 1)
is a Bayesian equilibrium. Q.E.D.
Proof of Proposition 5.
Donor i's best response to sj = (1,2, 2) is si = (0, 0,0) if
p0u(0, 1) + p1u (0, 2) + p2u (0, 3) + p3u (1, 3) > 0 (28)
+ + + +
37
p0u(1, 1) + p1u (1, 2) + p2u (1, 3) + p3u (2, 3) > 0 (29)
+ + + +
p0u(0, 2) + p1u (0, 3) + p2u (0, 4) + p3u (1, 4) > 0 (30)
+ + + +
p0u(2, 1) + p1u (2, 2) + p2u (2, 3) + p3u (3, 3) > 0 (31)
 + + +
p0u(1, 2) + p1u (1, 3) + p2u (1, 4) + p3u (2, 4) > 0 (32)
+ + + +
Equations (28)  (30) and (32) hold unambiguously.
Donor j s best response to si = (0, 0,0) is sj = (1, 2,2) if
p0u(0, 1) + p1u (1, 1) + p2u (2, 1) + p3u (3, 1) < 0 (33)
+ +  
p0u(1, 1) + p1u (2, 1) + p2u (3, 1) + p3u (4, 1) < 0 (34)
+   
p0u(0, 2) + p1u (1, 2) + p2u (2, 2) + p3u (3, 2) < 0 (35)
+ + + 
p0u(2, 1) + p1u (3, 1) + p2u (4, 1) + p3u (5, 1) < 0 (36)
   
p0u(1, 2) + p1u (2, 2) + p2u (3, 2) + p3u (4, 2) < 0 (37)
+ +  
Equation (36) holds unambiguously.
si = (0, 0, 0) and sj = (1,2, 2) is a Bayesian equilibrium if equations (31) ,
(33)  (35) and (37) hold. Q.E.D.
Proof of Proposition 6.
Donor i's best response to sj = (1,1, 2) is si = (0, 0,1) if
p0u(0, 1) + p1u (0, 2) + p2u (1, 2) + p3u (1, 3) > 0 (38)
+ + + +
p0u(1, 1) + p1u (1, 2) + p2u (2, 2) + p3u (2, 3) > 0 (39)
+ + + +
p0u(0, 2) + p1u (0, 3) + p2u (1, 3) + p3u (1, 4) > 0 (40)
+ + + +
38
p0u(2, 1) + p1u (2, 2) + p2u (3, 2) + p3u (3, 3) < 0 (41)
 +  +
p0u(1, 2) + p1u (1, 3) + p2u (2, 3) + p3u (2, 4) > 0 (42)
+ + + +
Equations (38)  (40) and (42) hold unambiguously.
Donor j s best response to si = (0, 0,1) is sj = (1, 1,2) if
p0u(0, 1) + p1u (1, 1) + p2u (2, 1) + p3u (2, 2) < 0 (43)
+ +  +
p0u(1, 1) + p1u (2, 1) + p2u (3, 1) + p3u (3, 2) < 0 (44)
+   
p0u(0, 2) + p1u (1, 2) + p2u (2, 2) + p3u (2, 3) > 0 (45)
+ + + +
p0u(2, 1) + p1u (3, 1) + p2u (4, 1) + p3u (4, 2) < 0 (46)
   
p0u(1, 2) + p1u (2, 2) + p2u (3, 2) + p3u (3, 3) < 0 (47)
+ +  +
Equations (45)and (46) hold unambiguously. Equation (47) implies (41) as
they differ only in the first term and the first term is negative in (41) and
positive in (47).
si = (0, 0, 1) and sj = (1,1, 2) is a Bayesian equilibrium if equations (43) ,
(44) and (47) are satisfied. Q.E.D.
6.2 Proofs for Section 4
6.2.1 Proofs for different preferences
The structure of the proofs is very similar to Section 3. We only need to
modify to have different utility functions and probability distributions for D
and R. The equivalent of equations (4)  (8) for D are:
pRuD(0, 1)+pRuD (1  ,1 + )+pRuD (2  , 1 + )+pRuD (3 (48) ,1 + ) > 0
0 1 2 3
39
pRuD(1, 1)+pRuD (2  ,1 + )+pRuD (3  , 1 + )+pRuD (4 (49)1 + ) > 0 ,
0 1 2 3
pRuD(0, 2)+pRuD (1  ,2 + )+pRuD (2  , 2 + )+pRuD (3 (50) ,2 + ) > 0
0 1 2 3
pRuD(2, 1)+pRuD (3  ,1 + )+pRuD (4  , 1 + )+pRuD (5 (51) ,1 + ) > 0
0 1 2 3
pRuD(1, 2)+pRuD (2  ,2 + )+pRuD (3  , 2 + )+pRuD (4 (52) ,2 + ) > 0
0 1 2 3
For R we also have to take into account that he may invest 3 projects in
sector 2 (equation (58)) and we have:
pDuR(0, 1)+pDuR (1  ,1 + )+pDuR (2  , 1 + )+pDuR (3 (53)1 + ) > 0 ,
0 1 2 3
pDuR(1, 1)+pDuR (2  ,1 + )+pDuR (3  , 1 + )+pDuR (4 (54) ,1 + ) > 0
0 1 2 3
pDuR(0, 2)+pDuR (1  ,2 + )+pDuR (2  , 2 + )+pDuR (3 (55)2 + ) > 0 ,
0 1 2 3
pDuR(2, 1)+pDuR (3  ,1 + )+pDuR (4  , 1 + )+pDuR (5 (56) ,1 + ) > 0
0 1 2 3
pDuR(1, 2)+pDuR (2  ,2 + )+pDuR (3  , 2 + )+pDuR (4 (57) ,2 + ) > 0
0 1 2 3
pDuR(0, 3)+pDuR (1  ,3 + )+pDuR (2  , 3 + )+pDuR (3 (58) ,3 + ) > 0
0 1 2 3
40
We use these equations in all the proofs by substituting in the other
agent's sharing strategy and changing the sign of the equations appropriately.
Proof of Proposition 7.
D's best response to sR = (1, 2, 3) is sD = (0, 0, 0) if
pRuD(0, 1) + pRuD (0, 2) + pRuD (0,3) + pRuD (0, 4) > 0 (59)
0 1 2 3
+ + + +
pRuD(1, 1) + pRuD (1, 2) + pRuD (1,3) + pRuD (1, 4) > 0 (60)
0 1 2 3
+ + + +
pRuD(0, 2) + pRuD (0, 3) + pRuD (0,4) + pRuD (0, 5) > 0 (61)
0 1 2 3
+ + + +
pRuD(2, 1) + pRuD (2, 2) + pRuD (2,3) + pRuD (2, 4) > 0 (62)
0 1 2 3
 + + +
pRuD(1, 2) + pRuD (1, 3) + pRuD (1,4) + pRuD (1, 5) > 0 (63)
0 1 2 3
+ + + +
Equations (59)  (61) and (63) hold unambiguously.
R invests all the funds in sector 2 given sD = (0, 0,0) if
pDuR(0, 1) + pDuR (1, 1) + pDuR (2,1) + pDuR (3, 1) < 0 (64)
0 1 2 3
0   
pDuR(1, 1) + pDuR (2, 1) + pDuR (3,1) + pDuR (4, 1) < 0 (65)
0 1 2 3
   
pDuR(0, 2) + pDuR (1, 2) + pDuR (2,2) + pDuR (3, 2) < 0 (66)
0 1 2 3
+ 0  
pDuR(2, 1) + pDuR (3, 1) + pDuR (4,1) + pDuR (5, 1) < 0 (67)
0 1 2 3
   
41
pDuR(1, 2) + pDuR (2, 2) + pDuR (3,2) + pDuR (4, 2) < 0 (68)
0 1 2 3
0   
pDuR(0, 3) + pDuR (1, 3) + pDuR (2,3) + pDuR (3, 3) < 0 (69)
0 1 2 3
+ + 0 
Equations (64) , (65), (67)and (68) hold unambiguously.
Therefore complete specialization is an equilibrium if (62), (66) and (69)
are satisfied. Q.E.D.
Proof of Proposition 8.
sD = (0, 0, 0) is a best reply to sR = (1, 2,2) if
pRuD(0, 1) + pRuD (0, 2) + pRuD (0,3) + pRuD (1, 3) > 0 (70)
0 1 2 3
+ + + +
pRuD(1, 1) + pRuD (1, 2) + pRuD (1,3) + pRuD (2, 3) > 0 (71)
0 1 2 3
+ + + +
pRuD(0, 2) + pRuD (0, 3) + pRuD (0,4) + pRuD (1, 4) > 0 (72)
0 1 2 3
+ + + +
pRuD(2, 1) + pRuD (2, 2) + pRuD (2,3) + pRuD (3, 3) > 0 (73)
0 1 2 3
 + + +
pRuD(1, 2) + pRuD (1, 3) + pRuD (1,4) + pRuD (2, 4) > 0 (74)
0 1 2 3
+ + + +
Equations (70)  (72) and (74) hold unambiguously.
sR = (1, 2, 2) is a best reply to sD = (0, 0, 0) if equations (64) (68)
hold and
pDuR(0, 3) + pDuR (1, 3) + pDuR (2,3) + pDuR (3, 3) > 0 (75)
0 1 2 3
+ + 0 
42
Equations (64) , (65) , (67)and (68) hold unambiguously.
sD = (0,0, 0) and sR = (1, 2, 2) is a Bayesian equilibrium if (66) , (73)
and (75) are satisfied. Q.E.D.
Proof of Proposition 9.
sD = (0, 0, 0) is best response to sR = (1, 1,2) if
pRuD(0, 1) + pRuD (0, 2) + pRuD (1,2) + pRuD (1, 3) > 0 (76)
0 1 2 3
+ + + +
pRuD(1, 1) + pRuD (1, 2) + pRuD (2,2) + pRuD (2, 3) > 0 (77)
0 1 2 3
+ + + +
pRuD(0, 2) + pRuD (0, 3) + pRuD (1,3) + pRuD (1, 4) > 0 (78)
0 1 2 3
+ + + +
pRuD(2, 1) + pRuD (2, 2) + pRuD (3,2) + pRuD (3, 3) > 0 (79)
0 1 2 3
 +  +
pRuD(1, 2) + pRuD (1, 3) + pRuD (2,3) + pRuD (2, 4) > 0 (80)
0 1 2 3
+ + + +
Equations (76)  (78)and (80) hold unambiguously.
sR = (1,1, 2) is best response to sD = (0, 0,0) if equations (64) , (65) ,
(67) , (68) and (75) hold and
pDuR(0, 2) + pDuR (1, 2) + pDuR (2,2) + pDuR (3, 2) > 0 (81)
0 1 2 3
+ 0  
Equations (64) , (65) ,(67)and (68) hold unambiguously.
Therefore sD = (0,0, 0) and sR = (1,1, 2) is a Bayesian equilibrium if
(75) , (79) and (81) hold. Q.E.D.
43
Proof of Proposition 10.
sD = (0, 0,1) is a best response to sR = (1, 2,2) if (70)  (72) and (74)
hold (and they hold unambiguously) and
pRuD(2, 1) + pRuD (2, 2) + pRuD (2,3) + pRuD (3, 3) < 0 (82)
0 1 2 3
 + + +
sR = (1,2, 2) is a best response to sD = (0,0, 1) if
pDuR(0, 1) + pDuR (1, 1) + pDuR (2,1) + pDuR (2, 2) < 0 (83)
0 1 2 3
0   
pDuR(1, 1) + pDuR (2, 1) + pDuR (3,1) + pDuR (3, 2) < 0 (84)
0 1 2 3
   
pDuR(0, 2) + pDuR (1, 2) + pDuR (2,2) + pDuR (2, 3) < 0 (85)
0 1 2 3
+ 0  0
pDuR(2, 1) + pDuR (3, 1) + pDuR (4,1) + pDuR (4, 2) < 0 (86)
0 1 2 3
   
pDuR(1, 2) + pDuR (2, 2) + pDuR (3,2) + pDuR (3, 3) < 0 (87)
0 1 2 3
0   
pDuR(0, 3) + pDuR (1, 3) + pDuR (2,3) + pDuR (2, 4) > 0 (88)
0 1 2 3
+ + 0 +
Equations (83) , (84) and (86) (88) hold unambiguously.
Therefore sD = (0,0, 1) and sR = (1,2, 2) is a Bayesian equilibrium if
(82) and (85) are satisfied. Q.E.D.
Proof of Proposition 11.
sD = (0, 0, 1) is best response to sR = (1,1, 2) if equations (76) (78)
and (80) hold (and they hold unambiguously) and
44
pRuD(2, 1) + pRuD (2, 2) + pRuD (3,2) + pRuD (3, 3) < 0 (89)
0 1 2 3
 +  +
sR = (1,1, 2) is best response to sD = (0, 0, 1) if equations (83) , (84) and
(86)  (88) hold (and they hold unambiguously) and
pDuR(0, 2) + pDuR (1, 2) + pDuR (2,2) + pDuR (2, 3) > 0 (90)
0 1 2 3
+ 0  0
sD = (0, 0, 1) and sR = (1, 1,2) is a Bayesian equilibrium if (89) and (90)
are satisfied. Q.E.D.
6.2.2 Proofs for the opposite preferences
D and R maximally invest one project in the less preferred sector when
ni = 3. Therefore we need to check only one condition for each agent.
D invests all the funds in sector 1 if and only if
pRuD(3,0) + pRuD (4  , ) + pRuD (5  , ) + pRuD (6  , )
0 1 2 3
> pRuD(2,1) + pRuD (3  , 1 + ) + pRuD (4  , 1 + ) + pRuD (5  , 1 + )
0 1 2 3
pRuD(2, 1)+pRuD (3  ,1 + )+pRuD (4  , 1 + )+pRuD (5  , 1 + ) > 0
0 1 2 3
(91)
R invests all the funds in sector 2 if and only if
pDuR(0,3) + pDuR (1  , 3 + ) + pDuR (2  , 3 + ) + pDuR (3  , 3 + )
0 1 2 3
> pDuR(1,2) + pDuR (2  , 2 + ) + pDuR (3  , 2 + ) + pDuR (4  , 2 + )
0 1 2 3
45
pDuR(0, 3)+pDuR (1  ,3 + )+pDuR (2  , 3 + )+pDuR (3  , 3 + ) < 0
0 1 2 3
(92)
Proof of Proposition 12.
Using equations (91) and (92) we can prove that sD = (0, 0, 0) and sR =
(1, 2, 3) is a Bayesian equilibrium if and only if:
pRuD(2,1) + pRuD (2, 2) + pRuD (2, 3) + pRuD (2,4) > 0
0 1 2 3
 + + +
pR <  pRuD (2, 2) + pRuD (2, 3) + pRuD (2, 4)
1 2 3
0 uD(2,1) pR0
and
pDuR(0,3) + pDuR (1, 3) + pDuR (2, 3) + pDuR (3,3) < 0
0 1 2 3
+   
pD <  pDuR (1, 3) + pDuR (2, 3) + pDuR (3, 3)
1 2 3
0 uR(0,3) pD0
Q.E.D.
Proof of Proposition 13.
sD = (0, 0, 1) and sR = (1,2, 3) is a Bayesian equilibrium if and only if:
pRuD(2,1) + pRuD (2, 2) + pRuD (2, 3) + pRuD (2,4) < 0
0 1 2 3
 + + +
pR > pR
0 0
and
pDuR(0,3) + pDuR (1, 3) + pDuR (2, 3) + pDuR (2,4) < 0
0 1 2 3
+   
46
pD <  pDuR (1, 3) + pDuR (2, 3) + pDuR (2, 4)
1 2 3
0 uR(0,3) pD0
sD = (0, 0, 0) and sR = (1,2, 2) is a Bayesian equilibrium if and only if:
pRuD(2,1) + pRuD (2, 2) + pRuD (2, 3) + pRuD (3,3) > 0
0 1 2 3
 + + +
pR <  pRuD (2, 2) + pRuD (2, 3) + pRuD (3, 3)
1 2 3
0 uD(2,1) pR0
and
pDuR(0,3) + pDuR (1, 3) + pDuR (2, 3) + pDuR (3,3) > 0
0 1 2 3
+   
pD > pD
0 0
Q.E.D.
Proof of Proposition 14.
sD = (0, 0, 1) and sR = (1,2, 2) is a Bayesian equilibrium if and only if:
pRuD(2,1) + pRuD (2, 2) + pRuD (2, 3) + pRuD (3,3) < 0
0 1 2 3
 + + +
pR > pR
0 0
and
pDuR(0,3) + pDuR (1, 3) + pDuR (2, 3) + pDuR (2,4) > 0
0 1 2 3
+   
pD > pD
0 0
Q.E.D.
47
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49
Figure 1.
S,SR S S
~D
p0
N,S N,S S
p^0
D
N N,S S,SD
0
p^0
R ~R
p0 1
N : sD = (0,0,0),sR = (1,2,3)
S : sD = (0,0,1),sR = (1,2,2)
SD : sD = (0,0,1),sR = (1,2,3)
SR : sD = (0,0,0),sR = (1,2,2)