Solve the below:
Q: Consider the exchange problem in an organizational system including a single agent and a single principal.
Construct an incentive-compatible mechanism
π(s) = (x1(s), x2(s)), assuming that the principal has incomplete awareness of the agent's type
r ∈ [rmin, rmax], rmin > 0. Take f0(x1, x2) = x2 - x1 and
f1 ( x1, x2, r ) = x1 - ( x22 /2r ) as the goal functions of the principal and agent, respectively.
The principal's problem is to maximize the expected utility of the exchange: Ef0(π(s)) → max.p( s )
The principal knows that the agent's types possess a uniform probability distribution on the segment
[r , r ] : F (r ) = r - rmin .
min max
rmax - rmin
The whole resource of the first (second) type is kept by the principal (by the agent, respectively); both resources are unlimited [83].