Editing Mechanism design (section)

===Implementability===
{{Main|Implementability (mechanism design)}}
The designer of a mechanism generally hopes either
* to design a mechanism <math>y()</math> that "implements" a social choice function
* to find the mechanism <math>y()</math> that maximizes some value criterion (e.g. profit)

To '''implement''' a social choice function <math>f(\theta)</math> is to find some transfer function <math>t(\theta)</math> that motivates agents to pick <math>f(\theta)</math>. Formally, if the equilibrium strategy profile under the mechanism maps to the same goods allocation as a social choice function,
:<math>f(\theta) = x \left(\hat\theta(\theta) \right)</math>
we say the mechanism implements the social choice function.

Thanks to the revelation principle, the designer can usually find a transfer function <math>t(\theta)</math> to implement a social choice by solving an associated truthtelling game. If agents find it optimal to truthfully report type,
:<math>\hat\theta(\theta) = \theta</math>
we say such a mechanism is '''truthfully implementable'''. The task is then to solve for a truthfully implementable <math>t(\theta)</math> and impute this transfer function to the original game. An allocation <math>x(\theta)</math> is truthfully implementable if there exists a transfer function <math>t(\theta)</math> such that
: <math>u(x(\theta),t(\theta),\theta) \geq u(x(\hat\theta),t(\hat\theta),\theta) \ \forall \theta,\hat\theta \in \Theta</math>
which is also called the '''incentive compatibility''' (IC) constraint.

In applications, the IC condition is the key to describing the shape of <math>t(\theta)</math> in any useful way. Under certain conditions it can even isolate the transfer function analytically.  Additionally, a participation ([[individual rationality]]) constraint is sometimes added if agents have the option of not playing.

====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.

====Sufficiency====

Mechanism design papers usually make two assumptions to ensure implementability:

<math>\frac{\partial}{\partial \theta} \frac{\partial u / \partial x_k}{\left|\partial u / \partial t\right|} > 0 \ \forall k</math>

This is known by several names: the [[single-crossing condition]], the sorting condition and the Spence–Mirrlees condition. It means the utility function is of such a shape that the agent's [[Marginal rate of substitution|MRS]] is increasing in type.{{Clarification|date=July 2024}}

<math>\exists K_0, K_1 \text{ such that } \left| \frac{\partial u / \partial x_k}{\partial u / \partial t} \right| \leq K_0 + K_1 |t|</math>

This is a technical condition bounding the rate of growth of the MRS.

These assumptions are sufficient to provide that any monotonic <math>x(\theta)</math> is implementable (a <math>t(\theta)</math> exists that can implement it). In addition, in the single-good setting the single-crossing condition is sufficient to provide that only a monotonic <math>x(\theta)</math> is implementable, so the designer can confine his search to a monotonic <math>x(\theta)</math>.