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I have a few mathematical problems with the paper Moral hazard in teams, by Bengt Hölmstrom 1982. Theorem 4

Denote the conditional distribution of $x$, given the action vector $a$, by $F(x, a)$ and the conditional density function by $f(x, a)$. Assume that the partial derivatives for $a_i$ exist and are given by $F_i(x, a)$ and $f_i(x,a)$

$\bar{x}$ is a critical output level

There are two Assumption:

(1)$F(x,a)$ is convex in $a$

(2)$\frac{ F_i(x,a)}{1-F(x,a)} \rightarrow -\infty$ as $x \rightarrow +\infty$ (or its upper bond)

The scheme that the principal will use pays a bonus $b_i$ , if $x > \bar{x}$ and pays $s_ix(a*)$, (sum $s_i=1$), if $x < \bar{x}$.

Can someone help me to find the definition of $b_i$ that garantees that $a^*$ is a Nash equilibrium?

Here is a photo of the theorem 4, if my question is not clear. Thank you :) Here is a scrennshoot of the paper

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