112 D.J. Clark, C. Riis J. of Economic Behavior Org. 42 2000 109–124
Assume that the equilibrium effective bribe functions, G
i
v
i
=x
i
, are continuous and strictly increasing except possibly at zero, with upper and lower supports ¯
x
i
and x
i
, and with inverse g
i
x
i
=v
i
.
7
Using the fact that the v
i
are distributed uniformly, we can write: π
1
x
1
= g
2
x
1
− v
2
D
2
v
1
− φ x
1
α
γ
, π
2
x
2
= g
1
x
2
− v
1
D
1
v
2
− φx
2 γ
. 2
The first-order conditions for an interior maximum of Eq. 2 are: g
′ 2
x
1
v
1
D
2
− φγ
α
γ
x
γ − 1
1
= 0,
g
′ 1
x
2
v
2
D
1
− φγ x
2 γ −
1
= 0.
3 Inserting v
i
=g
i
x
i
into Eq. 3 yields a system of two differential equations: g
′ 2
xg
1
x = φD
2
γ α
γ
x
γ − 1
, g
′ 1
xg
2
x = φD
1
γ x
γ − 1
. 4
In Appendix A, it is shown that the unique solution to the system in Eq. 4 is: g
1
x = v
l 1
Z
11+λ
, g
2
x = ¯ v
2
v
l 1
¯ v
1 λ
Z
λ 1+λ
, where v
l 1
= v
1
iff ¯
v
2
v
1
¯ v
1 λ
≥ v
2
, otherwise
v
l 1
= ¯ v
1
v
2
¯ v
2 1λ
, Z ≡
1 + φD
1
1 + λx
γ
¯ v
1
¯ v
2
v
1
¯ v
1 1+λ
, λ ≡
D
2
D
1
α
γ
. 5
3. Allocation efficiency
We are now in a position to look at allocation efficiency in the competitive bribery procedure. Recalling the definition of g
i
x
i
=v
i
, it can easily be calculated from Eq. 5 that:
x
1
, =, x
2
iff v
1
¯ v
1 λ
, =, v
2
¯ v
2
. 6
Eq. 6 depicts the role of the different parameters in determining the underlying selection properties of this asymmetric competitive bribery model. We can isolate several factors or
combinations of these which influence selection: differences in the means and variances of the valuation distributions, the fairness of the competition, as well as the cost of bribery
as depicted by γ . Notice that the shift parameter in the cost function, φ, does not affect the selection properties of the bribery procedure.
Consider first the completely symmetric case in which ¯v
1
= ¯ v
2
, v
1
= v
2
, α = 1; Eq. 6
indicates that the bribery game allocates efficiently since Firm i wins if and only if it has a higher valuation than Firm j. In other cases, allocation will not always be efficient. In the
remainder of the paper we assume, with no loss of generality, that ¯v
1
≥ ¯ v
2
.
7
This assumption is fulfilled in equilibrium.
D.J. Clark, C. Riis J. of Economic Behavior Org. 42 2000 109–124 113
Fig. 1. Erroneous selection; λ1.
The locus of valuation combinations v
1
, v
2
which gives x
1
=x
2
and is defined by v
1
= ¯ v
1
v
2
¯ v
2 1λ
7 is graphed as 0e in Fig. 1, illustrating the case where λ1; above and to the left of this line
8
, we have x
2
x
1
2 wins below and to the right x
1
x
2
1 wins, and exactly on 0e, x
1
=x
2
by assumption 2 wins. Only when the function in Eq. 7 coincides exactly with a 45
◦
line from the origin in the relevant range from v
i
to ¯v
i
, do we have efficiency in allocation. When this is not the case, the firm with the lower valuation may, on occasion, be awarded
the contract. Notice that whenever ¯v
1
6= ¯ v
2
, there are no settings of the other parameters which ensure complete efficiency in allocation in this model.
9
Area ‘abced’ in Fig. 1 represents the combinations of v
1
and v
2
which yield erroneous selection i.e. that 2 is chosen when v
1
v
2
. When α is small, implying that the contest is heavily biased in favor of Firm 2, this area is large. Firm 1, being at a disadvantage, gives
small bribes or, if v
1
is sufficiently small, does not bribe at all. Hence he often loses to less efficient opponents. However, Firm 2, realizing that the opponent behaves in this way, acts
less aggressively since it is likely that it wins the contract with a lower bribe than if Firm 1 were at less of a disadvantage. Consequently, only if v
1
turns out to be very large does Firm 1 actually win the game. As α is increased and Firm 1 is at less of a disadvantage, the
area ‘abced’ shrinks while 0e maintains the same start and end points and the probability of wrong selection falls.
Raising α sufficiently will eventually lead to a case in which 1λ; 0h in Fig. 2 represents the graph of Eq. 7 for this situation. As before, above below 0h, we have x
2
x
1
. Two types of erroneous selection are now possible: i Firm 1 is chosen when v
2
v
1
, represented in Fig. 2 by area ‘jkn’; ii Firm 2 is chosen when v
1
v
2
which is the area ‘nrh’ in Fig. 2. Define v
∗
= [ ¯v
2
¯ v
λ 1
]
11−λ
as the point at which 0h crosses the 45
◦
line. If v
1
and v
2
are
8
0e is a straight line when λ=1.
9
In a different setting, Clark and Riis 1996 investigate a prize structure which would lead to allocation efficiency in this case.
114 D.J. Clark, C. Riis J. of Economic Behavior Org. 42 2000 109–124
Fig. 2. Erroneous selection; λ2.
below v
∗
, then both types of error are made.
10
Otherwise, only a type ii error occurs. For high values of v
2
, a type ii error may occur; Firm 1 tends to underestimate the opponent, expects it to give a small bribe, and consequently loses if v
2
is large. For low values of v
2
, a type i error can occur, so that Firm 1 wins more often than allocation efficiency would
dictate; Firm 2 is at a disadvantage relative to Firm 1, since α is large, and in addition has a low valuation, hence it bribes very little or not at all. Raising α increases the probability of a
type i but simultaneously reduces that of a type ii error. When α is sufficiently large, then further increases in this parameter will raise the probability of choosing the least efficient
firm, as the increase in the chance of a type i error outweighs the fall in the probability that a type ii error will be made.
Let us now consider the result of Lien 1990 that making an unfair contest α6=1 more unfair will increase the probability of choosing a less efficient contractor. We have two cases
to consider: λ1 and 1λ. 3.1. λ1
Take first the case represented by Fig. 1 λ1, where we have already noted that increasing α
reduces the probability of wrong selection. The point is, however, that whether this increase in α represents a move to a more or a less fair contest is parameter specific. If D
1
≥D
2
, then λ
1 implies 1α and hence an increase in α makes the contest more fair and simultaneously reduces the probability of wrong selection. This is consistent with Lien 1990 and extends
his result to a wider range of possibilities.
11
However, if D
2
D
1
, it is permissible within Fig. 1 where λ1 to have α1. Here an increase in α makes the contest less fair i.e.
more biased towards Firm 1 and reduces the probability of erroneous selection. This is the opposite of Lien’s result.
3.2. 1λ Similarly, Lien’s result and its exact opposite can be generated when Fig. 2 1λ is
relevant. Assuming for illustrational simplicity that v
1
= v
2
= 0, and writing ¯v ≡ ¯v
2
¯ v
1
,
10
Since the derivative ∂v
2
∂v
1
from Eq. 7 tends to infinity as v
1
tends to zero in this case, 0h will cross the 45
◦
line.
11
Lien’s assumption of ex ante symmetry obtains if D
1
=D
2
.
D.J. Clark, C. Riis J. of Economic Behavior Org. 42 2000 109–124 115
the probability that the higher cost firm actually wins the contract in this case is θ = η + τ
where, η =
1 − λ 21 + λ
¯ v
1+λ1−λ
τ = λ
1 + λ −
¯ v
2 + η.
Here, the total probability that a mistake is made which is expressed by θ , is the sum of the probabilities of making a type i error, denoted by η, and a type ii mistake, τ . The
comparative static effect of a change in α is given by
∂θ ∂α
= −λγ
α 1 + λ
2
1 + 2v
1+λ1−λ
− 1 +
1 + λ 1 − λ
ln ¯v .
8 Since 1 ≥ ¯v, the term in square brackets in Eq. 8 is negative, making the sign of the whole
expression ambiguous. It is straightforward to determine that ∂θ ∂α0 as ¯v → 1, whilst ∂θ
∂α0 as ¯v → 0. As we approach the situation in which the firms are approximately identical ex ante, increasing α makes mistaken selection more likely. Again this is Lien’s
result. However, when the firms are expected to be very dissimilar ¯v low, the opposite result obtains in which the probability of erroneous selection is decreasing in α. Intuitively,
Firm 1 is expected to be much superior to Firm 2 in this case, so reducing the probability of mistakenly choosing the higher cost firm implies making the contest more biased toward
Firm 1. When ¯v takes intermediate values, the effect of changing α on θ is not monotonic.
3.3. Efficiency loss due to erroneous selection Naturally, one is concerned with the efficiency loss which making a selection error en-
tails, and not the probability of such an occurrence.
12
Normalizing ¯v
1
= 1, retaining the
assumption that ¯v
i
= 0 i=1, 2, and writing v
∗
= ¯ v
11−λ
if 1λ, and v
∗
= 0, otherwise
where v
∗
is the intersection point between the 45
◦
line and the curve at which x
1
=x
2
given in Eq. 7, the expected efficiency loss in the competitive bribery procedure can be calculated as
= Z
v
∗
Z
v
2
v
2
¯ v
1λ
v
2
− v
1
¯ v
dv
1
dv
2
+ Z
¯ v
v
∗
Z
v
2
¯ v
1λ
v
2
v
1
− v
2
¯ v
dv
1
dv
2
= Z
¯ v
Z
v
2
v
2
¯ v
1λ
v
2
− v
1
¯ v
dv
1
dv
2
= λ
22 + λ −
¯ vλ
2λ + 1 +
¯ v
2
6 .
12
As in Lien 1990, we focus in this section solely on losses due to erroneous selection. In our model, however, losses can occur in the transfer of bribes since their value may be different to giver and receiver. This deadweight
loss complicates the calculations, whilst adding little to the analysis; it is, therefore, omitted in what follows. An expression for the expected deadweight loss of the bribery procedure is straightforward to calculate, and is
available from the authors on request.
116 D.J. Clark, C. Riis J. of Economic Behavior Org. 42 2000 109–124
Fig. 3. Efficiency loss due to erroneous selection.
We can illustrate the efficiency effects within a ¯v, λ diagram recall that ¯v ≡ ¯v
2
¯ v
1
; consider Fig. 3 which depicts iso-social-cost curves which are concentric to the global
minimum at 1, 1. The further away from this point the iso-social-cost curve lies, the larger the efficiency loss associated with the competitive bribery procedure. Since we have defined
λ = ¯ vα
γ
, points to the right left of the 45
◦
line reflect α1, whilst along the 45
◦
line, we have α=1.
Given ¯v, the λ which minimizes the social cost of erroneous selection is unique and determined by
∂ ∂λ
= 1
2 + λ
2
− ¯
v 2λ + 1
2
= 0.
9 The locus of points satisfying Eq. 9 is denoted λ
m
¯v in Fig. 3 and always lies above the 45
◦
line. To the right of λ
m
¯v, is monotonically increasing in λ, and thus, the efficiency loss is monotonically decreasing in α. To the left of λ
m
¯v, the efficiency loss is monotonically decreasing in λ increasing in α.
Three areas are of interest in this figure, represented by Q the area bounded by λ
m
¯v and ¯v = 1, R bounded by λ
m
¯v and the 45
◦
line and S below and to the right of the 45
◦
line. The comparative static effect on the efficiency loss in selection of changes in α and γ
are given in Table 1. Notice from Eq. 1 that the firms’ cost functions are identical up to a multiplicative term
1α
γ
which indicates the cost of bribery to Firm 1 relative to Firm 2; when 1α
γ
1, Firm 1 has a larger marginal and average cost relative to Firm 2 for an identical bribe level. The
intuition behind the results in Table 1 can be gleaned by looking at this factor, which we
Table 1 The comparative static effect on the efficiency loss in selection of changes in α and γ
∂ ∂α
∂ ∂γ
Q +
+ R
− −
S −
+
D.J. Clark, C. Riis J. of Economic Behavior Org. 42 2000 109–124 117
shall denote by f. The assumptions which underlie in Fig. 3 imply that λ = ¯vf . Notice that in areas Q and R, we have that α1, whilst 1α in S.
Suppose we fix ¯v = V in Fig. 3, and start from a point where λ is low in Q, and hence f
is small, implying that Firm 1 has a very low marginal cost of bribery relative to Firm 2. This leads to Firm 2 bidding little if at all and losing often to inferior Firm 1 types a type
i error occurs with a large probability. Knowing this, Firm 1 also bribes a small amount, but enough to beat almost all Firm 2 types.
13
Suppose now that we move horizontally from this point from left to right in Fig. 3 for fixed V. This movement can be accomplished by reducing α andor reducing γ . Firm 2
is now at less of a cost disadvantage, and consequently, bids more and loses less often to inferior Firm 1 types. At the same time, it is also possible for a very efficient Firm 2 to
beat inferior Firm 1 types, due to the fact that Firm 1 may tend to ‘slack off’ because it has an initial advantage in its lower relative cost. These two effects pull in opposite directions,
but the net effect in area Q is that the efficiency loss decreases as α and γ fall. Reducing α
andor γ sufficiently to reach the point w in Fig. 3 gives the smallest welfare loss for ¯
v = V due to mistaken selection in the competitive bribery procedure. By definition, any
horizontal move away from w leads to an iso-social cost curve at a lower level of welfare. Thus, a further reduction in α andor γ and a move to the right from w into R increases
the efficiency loss; in Fig. 2, area ‘jkn’ shrinks by less than the enlargement in area ‘nrh’. As the relative marginal cost to Firm 1 becomes larger and larger, it offers successively
smaller bribes and loses more often to inferior Firm 2 types. Simultaneously, Firm 1 beats more efficient opponents less and less often until f becomes so large that this sort of error
does not occur.
Moving further to the right along ¯v = V into area S increases the efficiency loss even more. Note that in area S, since 1α, a horizontal move from left to right is achieved by
reducing α andor increasing γ . One interpretation of increasing γ is as a toughening of the stance against large bribes. As indicated in Table 1, this reduces the expected efficiency loss
in area R. However, in areas Q and S, the efficiency loss increases following this tougher stance; any gains due to a reduced level of bribery activity must thus be weighed up against
the losses due to erroneous selection.
The cases examined by Lien 1990 all lie along the horizontal line ¯v = 1 in Fig. 3, from which it is clear that any move away from the point of a fair contest α=1 implying λ=1
must reduce social welfare; in other words, making an unfair contest more fair will improve welfare.
14
The delineations in Fig. 3 can be used to show the conditions under which this result holds and when it does not in our more general framework. In area Q, it is the case
that α1 and that the efficiency loss is increasing in this parameter; this is consistent with Lien’s result — making an unfair contest even more unfair reduces welfare. In area S, we
have that increasing α reduces the efficiency loss and improves welfare; 1α in this region so increasing this parameter implies making the contest more fair and Lien’s result again
obtains. In area R, however, Lien’s result does not hold. In this region, an increase in α — implying a less fair contest — increases welfare. Notice that when the firms are not
constrained to have the same expected efficiency as depicted by v, it is socially optimal to
13
Recall that, in Fig. 2, the chance of a type ii error is low when λ takes low values.
14
Recall again our and Lien’s narrow definition of social welfare as pertaining to efficiency in selection only.
118 D.J. Clark, C. Riis J. of Economic Behavior Org. 42 2000 109–124
run an unfair contest i.e. set α such that λ=λ
m
¯v in Fig. 3, not α=1, which would imply λ = ¯
v . Lien’s symmetric model is incapable of capturing this fact.
4. Endogenous discrimination
