# Stochastic Mechanisms

We consider here the framework of section 2.10 with a general cost function C(q, θ).  Let  us  rewrite  the  principal’s  problem  as

When  S(·) is  concave  and  C(·) is  convex,  the  principal’s  objective  function is  concave  in  (q, q¯, U , U¯).  Neglecting  constraints  (2.78)  and  (2.79)  as  usual,  the remaining constraints (2.77) and (2.80) define a convex set in (q, q¯, U , U¯) if is convex in q. Then the optimal mechanism cannot be stochastic. To show this result suppose, on the contrary, that this mechanism is stochastic. A random direct reve- lation mechanism is then a probability measure on the set of possible transfers and outputs, which is conditional on the agent’s report of his type. Let  be such a random stochastic mechanism. We can replace this stochastic mecha- nism by the deterministic mechanism constructed with the expectations of those variables, namely , where E(·) denotes the expectation operator. Since the principal’s objective function is strictly concave in q, this new mechanism gives a higher expected utility to the principal by Jensen’s inequality. Similarly,  when  yj·k is  convex,  Jensen’s  inequality  also  implies  that  ,  so  that  the  new  deterministic  mechanism  expands  the  feasible  set defined by the constraints in (2.77) and (2.80). The principal could thus achieve a higher utility level with the new deterministic mechanism, which is a contra-diction. Therefore, a sufficient condition to ensure the deterministic nature of the optimal  contract  is  convex  or,  equivalently, Cqqθ > 0.

Let  us  explore  briefly  what  happens  if  the  assumption  Cqqθ > 0  is  no  longer satisfied everywhere. Substituting (2.77) and (2.80) into the principal’s objective function,  and  taking  into  account  that  qSB = q (where  again , the principal’s problem amounts to maximizing an objective function

which may no longer be strictly concave in q¯ everywhere.

When this strict concavity is not satisfied, (2.81) may have several maximizers among which the principal can randomize. Note that the randomness of con- tracts only affects outputs. From risk neutrality, with the principal and the agent’s objective functions being linear in transfers, the randomness of transfers is use- less because any lottery of transfers can be replaced by its expected value without changing the principal and the agent’s payoffs in any state of nature.

The lack of concavity of (2.81) in fact captures a deeper property: the possi- ble lack of convexity of the set of incentive feasible allocations. To illustrate this phenomenon, note that, for contracts such that (2.80) is binding and such that q = q, (2.77) can then be written as

Figure 2.8 represents the set of implementable allocations in the (U , q¯) space and shows  that  this  set  may  not  be  convex  when is  nonconvex.  Points  A  and  B are two possible deterministic maximizers of the principal’s (reduced) objective function,

However, the principal can obtain a greater payoff by committing himself to randomize among incentive feasible allocations. Using such random direct reve- lation mechanisms leads indeed to a convexification of the set of incentive feasi- ble allocations as shown in figure 2.9.35 With the objective function (2.83) being strictly  concave  in  (U , q¯),  a  unique  maximizer  to  the  principal’s  problem  exists, and it is now given by point C.

Figure 2.8: Multiple Maximizers

By being able to commit to a randomization through a stochastic mechanism, the principal can achieve a payoff that is strictly greater than what he obtains with deterministic mechanisms. Of course, the difficulty may come from the fact that this randomization has to be verifiable by a court of law before it can be employed in contracting. Ensuring this verifiability is a more difficult problem than ensuring that a deterministic mechanism is enforced, because any deviation away from a given randomization can only be statistically detected once a sufficient number of realizations of the contracts have been observed. This suggests that such a deviation can only be detected in a repeated relationship framework or when the principal is involved in many bilateral one-shot principal-agent relationships and always deviates in the same way. The enforcement of such stochastic mechanisms in a bilateral one-shot relationship is thus particularly problematic. This has led scholars to give up those random mechanisms or, at least, to focus on economic settings where they are not optimal.

Stochastic mechanisms have been sometimes suggested in the insur- ance, nonlinear pricing, and optimal taxation literatures (see Stiglitz 1987, Arnott and Stiglitz 1988, and Maskin and Riley 1984).

Figure 2.9: Unique Maximizer with Randomization

Source: Laffont Jean-Jacques, Martimort David (2002), The Theory of Incentives: The Principal-Agent Model, Princeton University Press.