The name is absent

(which yields nearly the same level of dispersion CV ≈ 0.2828) is added the difference
between AA and ET becomes even more profound: EAA ≈ 0.4522 versus E*_et ≈ 0.4038
and EAA — EE_t ≈ 0.0483. Note also that both policies induce higher total effort in
comparison with the three player example. However, if the fourth contestant is highly
discriminated (β4 = 10) this would imply a decline of total effort in the case of AA:
EAA ≈ 0.3938. This decline is less intense in case of ET because here the fourth player
will not participate. As only the first three contestants will be active under ET, the
result is identical to the three player contest game considered before, i.e. E*_et = 0.4.
Comparing both values shows that for this four player constellation the result of the
policy analysis has been reversed because now EAA < E*_et.

This example demonstrates that the key factor for the outcome of the policy compari-
son is the distribution of the discrimination parameter in combination with the number
of contestants. In general it can be stated that either a low number of contestants or a
sufficient low dispersion makes it more probable that AA will induce more total effort
than ET because then the set of active contestants tends to be similar for both poli-
cies.²¹ The exact relation between the distribution of discrimination parameters and
the number of players is described by the inequality²² in Proposition 3 in combination
with the characterization of the active set in Eq. (11).

An additional remark with respect to the relation between Proposition 1 and 3 should
be in order. Applying Proposition 3 to a two-player contest game would yield the same
result as Proposition 1 because condition (13) holds irrespective of the distribution of
cost parameters in the two-player case: For the optimally designed vector of policy
parameters both contestants will exert positive equilibrium effort, i.e. set M and N
coincide. Therefore, Proposition (3) is satisfied without further restriction because
condition (13) can be reduced to β > βH which is always true (comp. the proof for
Proposition 4).

For the two-player contest game Proposition 1 also contained a statement that com-

²¹ The observation that affirmative action might imply a distortion of the participation decision of
individuals (which could finally dominate the effect of increased competitive pressure) has also
empirical relevance: In an econometric analysis of bid preferences in highway procurement auctions
(Marion 2007), it is shown that preferential treatment implies a decline in competitive pressure
because non-preferred bidders switched to procurement auctions without bid preference program.

²²Note, that the left hand side of condition (13) is lower than one for m small and larger than one
for m large (where m is determined according to condition (11)). Inspection of the right hand
side reveals that it is always lower or equal to one. This confirms the qualitative statement that
condition (13) is likely to hold if the number of contestants is relatively small or the distribution
is not too dispersed.

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