Compared to the case with only two agents (ICCBP S), the Incentive Compatibility
Constraint under the Multilateral Punishment StrategyICCM P S changes with respect to
the expected future loss of the benefit from potential corrupt transactions in case of defec- tion of the official. Incentive compatibility has to include all O’s potential transactions with any of B’s fellow sub-group members.25 Since we assume clients to be homoge-
neous, we consider a representative agent (B) in a sub-group of size n in order to derive the ICCM P S. ICCM P S : b−c+ T X i=1 δin(bi−c)≥b+ 0; T → ∞: b−c+ δ 1−δn(b−c)≥b (1.3)
Solving for the minimal incentive compatible bribe yields b∗(δ, c, n)M P S = (1nδ−δ + 1) c.
The larger the sub-group, the lower the equilibrium bribe has to be in order to hinder
O from defection: ∂b∂n∗ = −1−δ
n2 c < 0. For the full effect of n on the level of corruption,
consider B’s new Participation Constraint P CM P S. B will only participate if her profits
are still positive after paying b∗M P S and compensating all her sub-group members for the damage realized by the corrupt transaction.
P CM P S : E−n
iE N −b
∗
M P S ≥0 (1.4)
This yields the minimum return: EM P S∗ (δ, N, c, i, n) = cNnδ1+(δN(n−−in1)). For a given size of the sub-group n and under the technical assumption of N > in, the partial effect of the size of society N is negative: ∂E∂N∗ = −(1+δ(Nδ(−nin−1)))2 i c < 0 if N > i n. The larger
the society, the wider the total damage is spread and the less of it has to be internalized through the compensation of all sub-group members. There are two countervailing effects of sub-group size n onE∗: En∗(n) ≡ ∂E ∗(n) ∂n = cN(i(2 +δ(n−2))n−N(1−δ)) δn2(N −in)2 S0. (1.5)
On the one hand we can identify a positive effect, the Coalition Effect (CE). The larger the sub-group, the more future potential earnings are at stake for O when deciding
25The assumption that all n transactions potentially take place in any of the periods implies sufficient
between cooperation (t) and defection (nt). This decreases the equilibrium bribe b∗M P S
needed for incentive compatibility.
On the other hand, there is a negative effect, the Internalization Effect (IE), which captures the direct internalization of the sum of negative external effects relevant forB’s fellow sub-group members. The number of sub-group members (group size) increases the sum of compensation payments needed for the successful engagement in corruption.
Corruption maximizing group-size
We can show that the relationship betweenE∗andnis U-shaped for all relevant coefficient values. This means that an intermediate group size balances the trade-off between the two countervailing effects.
First, we show that an increase in the size of the sub-group (whose members useMPS) increases the level of corruption for low values of n. Consider the marginal effect of n on
E∗(n) atn= 1: En∗(n= 1) =c δN ((2((1−δ−)iδ−)i(1−−δtδ))2N).
If N > i 2−δ
(1−δ): E
∗
n(n= 1) <0. (1.6)
For group enforcement through MPS to be effective (E∗ is smaller in a group when using MPS than in the case of BPS), the size of society N needs to be large enough compared to the factor of inefficiency of corruption (i) and the discount factor δ.26
Second, we show that, due to the inefficiency of corruption, group enforcement (through MPS) does not provide more stability than BPS when sub-group size n is large relative to the size of society N. Note that in our set-up of perfect information transmission inside a sub-group, sub-group members cannot choose to use a bilateral punishment strategy since information cannot be withheld. This may be rationalized by the inability of group members to hide criminal activities from their peers.27 If sub-group size n reaches values smaller but close to N (N = Ni ) which is, by definition (i > 1), strictly smaller than N, we can show that E∗ approaches infinity under MPS and hence
26Withδ= 0.9 (a depreciation rate of 10%) society must be only ten times larger than the factor of
inefficiency.
27The U-shaped curve of E∗(n) indicates that MPS is individually optimal only up to a certain
no corruption occurs.28 lim n→N i E∗ = lim n→N i cN 1 +δ(iN −1) nδ(N −in) = limz→0 c(1 +δ(Ni −1)) z → ∞. (1.7)
Third, we show that there is a unique group size n∗ that minimizes E∗ and hence maxi- mizes the level of corruption. En∗(n) ≡ ∂E∂n∗(n) = cN(i(2+δnδ(2n(−N2))−inn−)2N(1−δ)) = 0. Solving
for n∗ yields:
n∗(i, δ, N) = i(1−δ) +
p
i2+δ(1−δ)(N −i)i
δi . (1.8)
Since the second derivative is positive (∂2(∂nE∗)(2n) > 0) for the relevant range of n29, sub-
group size n∗(i, δ, N) characterizes the global (and unique) minimum inE∗.
Characteristics of corruption maximizing group size
The size of society N affects the corruption maximizing sub-group size in the following way. The larger N, the broader the spreading of the inefficiency causing a lower per capita damage d. While having no effect on the CE, a bigger society decreases the IE, as less of the total negative effect needs to be internalized under a given sub-group size. Hence the balancing sub-group size n∗ increases with N.
n∗N ≡ ∂n
∗
∂N =
1−δ
2pi2+δ(1−δ)(N −i)i >0 (1.9)
The opposite is true for the factor of the inefficiency of corruption. While the degree of inefficiency is not accounted for in the CE it increases the (total) amount of the internalized damage (IE) and thus shifts n∗ in the opposite direction. The larger the inefficiency relative to the return E, the smaller the size of the sub-group that maximizes corrupt activity. n∗i ≡ ∂n ∗ ∂i =− (1−δ)N 2ipi2+δ(1−δ)(N −i)i <0 (1.10)
Since the function E∗(δ, N, c, i, n) is convex and continuous in n (for at least
28If BPS is feasible, the maximum amount of corruption is determined byE∗
BP S for all groups of size
n > no 29E∗ nn(n)≡ ∂2E∗(n) (∂n)2 = cN((1−δ)(2(N−in)2−in(n−2in))+i2δn3) δn3(N−in)3 >0forn≤ 2N 3i. SpecificallyE ∗ nn(n∗)>0
1 < n < 23Ni)30, decreases in n for small sub-group sizes (n < n∗), has a unique minimum of corrupt activity atn∗ and increases in n forn > n∗, we conclude thatE∗(n)
must be a U-shaped function in n.
It is trivial to check numerically that, for any reasonable set of parameters (e.g. δ≥0.5, discounting is not excessive and i ≥ 1.2, the inefficiency caused by corruption is sig- nificant), the corruption maximizing group size n is small relative to N. In this set of parameters, the corruption maximizing sub-group size is at least ten times smaller than the size of the society (n∗ ≤ N
10). As an example, Figure 1.3 depicts the function E ∗
Figure 1.3: Exemplary form of E∗
over n where E∗ is displayed in multiples of the direct costs of corruption c with the parameter values delta = 0.9, N = 100 and i = 1.5. In this example the minimum return E∗(n), characterizing a maximum number of corrupt transactions, is produced by sub-groups of size n∗ = 4. Using MPS, groups of size n between 2 and 12 produce corruption levels that are higher than those attainable under BPS. If group size is larger than13, we find the same per capita corruption levels as in a totally fractionalized society (n = 1), if BPS is feasible and lower levels if not. The decline ofE∗(n) in n for n < n∗
can be interpreted as the result of the stabilizing effect of increasing the radii of trust, linking more individuals to the sub-group. The section to the right of the minimum (n∗) shows that a further increase in the radii of trust deters individuals from engaging in
30Recall that ∂2E∗(n)
corrupt transactions because of increasing costs of internalization. The U-shaped curve of E∗(n) translates immediately into a hump-shaped curve in the relation between per capita frequency of corruption and sub-group size n.