Effort and Performance in Public-Policy Contests

Proposition 2: In cases (i), (iv) and (v),

∂² Pr                       ∂ x^*i

( ^a ) I^{f      i} = ^0, then   Sign ⅛") = Sign (ηi ) .

∂x_i ∂x                      ∂ I

ij

and

∂² Pr_i       -

. η j n 7

^∂xi         ^<

By Proposition 2 (a), if the contestants are symmetric in equilibrium in terms of their
abilities , then the strategic rival’-stake (“substitution”) effects vanish (efforts are
independent) and the positive strategic own-stake (“income”) effect solely determines
the direct effect of a change in I on a contestant’s effort. In the perfectly symmetric
case where, ∀x and x , Pr (x ,x ) = 1 -Pr (x ,x ) and n = n = n , there
H   L HLH    HHL    H L

∂² Pr_L

exists a symmetric pure strategy Nash equilibrium, x*H = x *L, and -----—

∂x ∂x

LH

∂² Pr

-----— = 0 , see Dixit (1987). Hence, by Proposition 2 (a),
∂x ∂x

HL

Corollary 2.1: If, ∀x_H and x_L, Pr_H (x_L,x_H) = 1 - Pr_H (x_H,x_L) and n_H = n_L = n , then

∂**

xh   ∂xL

-------= ------- 0 .

∂n ∂n

This corollary generalizes a similar result established by Nti (1999), assuming a
particular contest success function of the logit form.

Proposition 2(b) can be used to determine the sensitivity of the contestants’
efforts in all possible situations corresponding to the three types of policy reforms
∂² Pr_i

affecting both stakes and------≠ 0. Consider for example a type (i) policy reform

∂x_i ∂x
ij

∂² Pr_H

and suppose that -----— < 0, that is, the effort of the HB player is a strategic

∂x ∂x

HL

substitute of the effort of the LB player. By Proposition 2 (b), in such a case,

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