The intuition for (ii) is more involved. An increase in the precision hj of the prior
distribution of beliefs about ηj has two effects on the first-order condition that determines
aj (conditions (6) and (7), respectively). First, it lowers the rate h h h at which a marginal
increase in yj∙ — α*, and hence a marginal increase in αj∙ given anticipated effort α*, improves
j’s posterior reputation as given in (2). This is a standard effect in learning models, which
ceteris paribus predicts that higher precision leads to less effort. Second, an increase in
hj makes the distribution ψj (∙) more precise and thereby also affects ψj (0), the (prior)
density of identical posterior reputations; ψj (0) captures the anticipated probability that
the contest will have a close outcome at the end of the nomination period. Since a marginal
change in effort is more likely to alter the contest outcome if posterior reputations are
similar with a higher probability, any increase in ψj (0) raises effort incentives. A priori
the sign of this second effect is ambiguous however. If prior reputations are identical
(∆ = 0), then ψj (0) is the density at the mean zj∙ = 0, which indeed rises as (the normal
distribution) ψj becomes more precise.16 For ∆ > 0, the sign of the second effect depends
on the size of the difference between the prior reputations relative to the variance σ2
of ψj. If mk and mj∙ are sufficiently close given σ2, or σ2 is sufficiently large given ∆,
then the sign of the second effect is positive; otherwise it is negative. Since ∂∂σh < 0 and
limhj →0 σ2 = ∞, the effect is always positive for hj sufficiently close to 0; it also always
dominates the first effect in this case. For large enough hj , however, the first effect always
dominates, as the rate of updating approaches zero.
Hoffler and Sliwka (2003) use a similar model to analyze the incentives of a firm owner
to replace an incumbent manager by someone with less information about the agents’
past performances. The manager’s task is to select the most skillful of two agents for a
promotion at the end of every period. Replacing the manager can increase the agents’
effort incentives by starting a fresh race for the valuable promotion. Unlike the analysis
presented in this section, Hoffler and Sliwka (2003) do not allow for heterogeneity in the
levels of precision of the prior distributions about agents’ skills, nor do they any derive
comparative statics results with respect to these parameters.
So far we treated the nomination prize Wi as an exogenous parameter in order to
16 Rosen (1986) identifies a similar effect in ladder tournaments, where winning probabilities at each
stage are a given function of talents and effort levels, and players form opinions about their own and their
opponents’ talents. In equilibrium the prior reputations of two players meeting for a match are always
identical. When there are only two possible talent levels, the marginal effect of effort on winning is then
increasing in the precision with which players assess their talents.
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