additional assumption that the n contestants belong either to group A or B that each
consists of nA ≥ 2 and nB ≥ 2 members. The members of each group are assumed
to be identical, i.e. face the same marginal cost parameter which is normalized for
the non-discriminated group A such that βi = βA = 1 for all i ∈ A and βi = βB for
all i ∈ B where βB > 1. It should be emphasized that this specification is already
covered by the model of section 4 which implies that αiET = αET and αiAA = βi
for all i ∈ N . The main objective is therefore another clarification of Proposition 3
and the interplay between total effort and the active set of contestants. Additionally,
the simplified model presented here can be considered as the starting point of the
generalized model in the next section.
At first, the active set under the optimally designed vector parameters for the ET
policy has to be determined (for AA all contestants will always be active). Denote
the number of active contestants of A by mA , and mB for group B . Starting with
the less discriminated group A, it is obvious that all members of A are active because
condition (11) reduces to 1 < mm-1 which is trivially satisfied for all mA ≤ nA. Hence,
all members of group A will be active under ET.
Considering the members of group B, condition 11 becomes в B < n A + m B-B1 which can
be simplified to:
βB < -aa-1 . (15)
nA - 1
Notice that the last condition does not depend on mB anymore which implies that this
condition either holds for all or for none of the members of B . Based on the number
of group A-members and the marginal cost parameter of group B the following two
cases are possible:
1. If condition (15) is satisfied both groups are active under ET.
2. Otherwise, only members of group A are active under ET.
Based on these two cases the aggregated equilibrium effort level under the optimal
designed vector of policy parameters under policy AA and ET can now be compared.
In case 1 all contestants are active such that Proposition 4 can be used directly to
conclude that AA induces higher aggregated effort than ET. The same proposition
gives conditions for each discrimination level under which AA induces more individual
equilibrium effort than ET. As в A = 1 and в B > β this implies that в A ∈
βB ∈ ^βn n-ι/ŋ, and that there exists no contestant i such that βi ∈
1, -lγ β ,
n-1
[ П-i β,β].
However, it remains to be checked whether the first interval is non-empty, i.e. if
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