Editing Mechanism design (section)

====Necessity====

Consider a setting in which all agents have a type-contingent utility function <math>u(x,t,\theta)</math>. Consider also a goods allocation <math>x(\theta)</math> that is vector-valued and size <math>k</math> (which permits <math>k</math> number of goods) and assume it is piecewise continuous with respect to its arguments.

The function <math>x(\theta)</math> is implementable only if
:<math> \sum^n_{k=1} \frac{\partial}{\partial \theta} \left( \frac{\partial u / \partial x_k}{\left|\partial u / \partial t\right|} \right) \frac{\partial x}{\partial \theta} \geq 0 </math>
whenever <math>x=x(\theta)</math> and <math>t=t(\theta)</math> and ''x'' is continuous at <math>\theta</math>. This is a necessary condition and is derived from the first- and second-order conditions of the agent's optimization problem assuming truth-telling.

Its meaning can be understood in two pieces. The first piece says the agent's [[marginal rate of substitution]] (MRS) increases as a function of the type,
:<math>\frac \partial {\partial \theta} \left( \frac{\partial u / \partial x_k}{\left|\partial u / \partial t\right|} \right) = \frac{\partial}{\partial \theta} \mathrm{MRS}_{x,t}</math>
In short, agents will not tell the truth if the mechanism does not offer higher agent types a better deal. Otherwise, higher types facing any mechanism that punishes high types for reporting will lie and declare they are lower types, violating the truthtelling incentive-compatibility constraint. The second piece is a monotonicity condition waiting to happen,{{Clarification|date=July 2024}}
:<math>\frac{\partial x}{\partial \theta} </math>
which, to be positive, means higher types must be given more of the good.

There is potential for the two pieces to interact. If for some type range the contract offered less quantity to higher types <math>\partial x / \partial \theta < 0</math>, it is possible the mechanism could compensate by giving higher types a discount. But such a contract already exists for low-type agents, so this solution is pathological. Such a solution sometimes occurs in the process of solving for a mechanism. In these cases it must be "[[Mechanism design#Myerson ironing|ironed]]". In a multiple-good environment it is also possible for the designer to reward the agent with more of one good to substitute for less of another (e.g. [[butter]] for [[margarine]]). Multiple-good mechanisms are an area of continuing research in mechanism design.