Editing Mechanism design (section)

==Highlighted results==

===Revenue equivalence theorem===
{{main|Revenue equivalence}}

{{harvs|txt|last=Vickrey|year=1961|author-link=William Vickrey}} gives a celebrated result that any member of a large class of auctions assures the seller of the same expected revenue and that the expected revenue is the best the seller can do. This is the case if
# The buyers have identical valuation functions (which may be a function of type)
# The buyers' types are independently distributed
# The buyers types are drawn from a [[Continuous distribution#Continuous probability distribution|continuous distribution]]
# The type distribution bears the monotone hazard rate property
# The mechanism sells the good to the buyer with the highest valuation

The last condition is crucial to the theorem. An implication is that for the seller to achieve higher revenue he must take a chance on giving the item to an agent with a lower valuation. Usually this means he must risk not selling the item at all.

===Vickrey–Clarke–Groves mechanisms===
{{main|Vickrey–Clarke–Groves mechanism}}

The Vickrey (1961) auction model was later expanded by {{harvs|txt|last=Clarke|year=1971|author-link=Edward H. Clarke}} and Groves to treat a public choice problem in which a public project's cost is borne by all agents, e.g. whether to build a municipal bridge. The resulting "Vickrey–Clarke–Groves" mechanism can motivate agents to choose the socially efficient allocation of the public good even if agents have privately known valuations. In other words, it can solve the "[[tragedy of the commons]]"—under certain conditions, in particular quasilinear utility or if budget balance is not required.

Consider a setting in which <math>I</math> number of agents have quasilinear utility with private valuations <math>v(x,t,\theta)</math> where the currency <math>t</math> is valued linearly. The VCG designer designs an incentive compatible (hence truthfully implementable) mechanism to obtain the true type profile, from which the designer implements the socially optimal allocation
:<math> x^*_I(\theta) \in \underset{x\in X}{\operatorname{argmax}} \sum_{i \in I} v(x,\theta_i) </math>

The cleverness of the VCG mechanism is the way it motivates truthful revelation. It eliminates incentives to misreport by penalizing any agent by the cost of the distortion he causes. Among the reports the agent may make, the VCG mechanism permits a "null" report saying he is indifferent to the public good and cares only about the money transfer. This effectively removes the agent from the game. If an agent does choose to report a type, the VCG mechanism charges the agent a fee if his report is '''pivotal''', that is if his report changes the optimal allocation ''x'' so as to harm other agents. The payment is calculated
:<math> t_i(\hat\theta) = \sum_{j \in I-i} v_j(x^*_{I-i}(\theta_{I-i}),\theta_j) - \sum_{j \in I-i} v_j(x^*_I (\hat\theta_i,\theta_I),\theta_j) </math>
which sums the distortion in the utilities of the other agents (and not his own) caused by one agent reporting.

===Gibbard–Satterthwaite theorem===

{{main|Gibbard–Satterthwaite theorem}}

{{harvs|txt|last=Gibbard|year=1973|author-link=Allan Gibbard}} and {{harvs|txt|last=Satterthwaite|year=1975|author-link=Mark Satterthwaite}} give an impossibility result similar in spirit to [[Arrow's impossibility theorem]]. For a very general class of games, only "dictatorial" social choice functions can be implemented.

A social choice function <math>f(\cdot)</math> is '''dictatorial''' if one agent always receives his most-favored goods allocation,
:<math>\text{for } f(\Theta)\text{, } \exists i \in I \text{ such that } u_i(x,\theta_i) \geq u_i(x',\theta_i) \ \forall x' \in X</math>

The theorem states that under general conditions any truthfully implementable social choice function must be dictatorial if,
# ''X'' is finite and contains at least three elements
# Preferences are rational
# <math>f(\Theta) = X</math>

===Myerson–Satterthwaite theorem===
{{main|Myerson–Satterthwaite theorem}}

{{harvs|txt|last=Myerson|last2=Satterthwaite|year=1983|author-link=Roger Myerson}} show there is no efficient way for two parties to trade a good when they each have secret and probabilistically varying valuations for it, without the risk of forcing one party to trade at a loss. It is among the most remarkable negative results in economics—a kind of negative mirror to the [[fundamental theorems of welfare economics]].

=== Shapley value ===
{{main|Shapley value}}

Phillips and Marden (2018) proved that for cost-sharing games with concave cost functions, the optimal cost-sharing rule that firstly optimizes the worst-case inefficiencies in a game (the [[price of anarchy]]), and then secondly optimizes the best-case outcomes (the [[price of stability]]), is precisely the Shapley value cost-sharing rule.<ref>{{Cite journal|last1=Phillips|first1=Matthew|last2=Marden|first2=Jason R.|date=July 2018|title=Design Tradeoffs in Concave Cost-Sharing Games|journal=IEEE Transactions on Automatic Control|language=en-US|volume=63|issue=7|pages=2242–2247|doi=10.1109/tac.2017.2765299|issn=0018-9286|s2cid=45923961}}</ref> A symmetrical statement is similarly valid for utility-sharing games with convex utility functions.

===Price discrimination===

{{harvs|txt|last=Mirrlees|year=1971|author-link=James Mirrlees}} introduces a setting in which the transfer function ''t''() is easy to solve for. Due to its relevance and tractability it is a common setting in the literature. Consider a single-good, single-agent setting in which the agent has [[quasilinear utility]] with an unknown type parameter <math>\theta</math>
:<math>u(x,t,\theta) = V(x,\theta) - t</math>
and in which the principal has a prior [[Cumulative distribution function|CDF]] over the agent's type <math>P(\theta)</math>. The principal can produce goods at a convex marginal cost ''c''(''x'') and wants to maximize the expected profit from the transaction
:<math>\max_{x(\theta),t(\theta)} \mathbb{E}_\theta \left[ t(\theta) - c\left(x(\theta)\right) \right]</math>
subject to IC and IR conditions
:<math> u(x(\theta),t(\theta),\theta) \geq u(x(\theta'),t(\theta'),\theta) \ \forall \theta,\theta' </math>
:<math> u(x(\theta),t(\theta),\theta) \geq \underline{u}(\theta) \ \forall \theta </math>
The principal here is a monopolist trying to set a profit-maximizing price scheme in which it cannot identify the type of the customer. A common example is an airline setting fares for business, leisure and student travelers. Due to the IR condition it has to give every type a good enough deal to induce participation. Due to the IC condition it has to give every type a good enough deal that the type prefers its deal to that of any other.

A trick given by Mirrlees (1971) is to use the [[envelope theorem]] to eliminate the transfer function from the expectation to be maximized,
:<math>\text{let } U(\theta) = \max_{\theta'} u\left(x(\theta'),t(\theta'),\theta \right)</math>
:<math>\frac{dU}{d\theta} = \frac{\partial u}{\partial \theta} = \frac{\partial V}{\partial \theta}</math>
Integrating,
:<math>U(\theta) = \underline{u}(\theta_0) + \int^\theta_{\theta_0} \frac{\partial V}{\partial \tilde\theta} d\tilde\theta</math>
where <math>\theta_0</math> is some index type. Replacing the incentive-compatible <math>t(\theta) = V(x(\theta),\theta) - U(\theta)</math> in the maximand,
:<math>\begin{align}& \mathbb{E}_\theta \left[ V(x(\theta),\theta) - \underline{u}(\theta_0) - \int^\theta_{\theta_0} \frac{\partial V}{\partial \tilde\theta} d\tilde\theta - c\left(x(\theta)\right) \right] \\
&{} =\mathbb{E}_\theta \left[ V(x(\theta),\theta) - \underline{u}(\theta_0) - \frac{1-P(\theta)}{p(\theta)} \frac{\partial V}{\partial \theta} - c\left(x(\theta)\right) \right]\end{align}</math>
after an integration by parts. This function can be maximized pointwise.

Because <math>U(\theta)</math> is incentive-compatible already the designer can drop the IC constraint. If the utility function satisfies the Spence–Mirrlees condition then a monotonic <math>x(\theta)</math> function exists. The IR constraint can be checked at equilibrium and the fee schedule raised or lowered accordingly. Additionally, note the presence of a [[hazard rate]] in the expression. If the type distribution bears the monotone hazard ratio property, the FOC is sufficient to solve for ''t''(). If not, then it is necessary to check whether the monotonicity constraint (see [[Mechanism design#Sufficiency|sufficiency]], above) is satisfied everywhere along the allocation and fee schedules. If not, then the designer must use Myerson ironing.

===Myerson ironing===

[[Image:Myerson ironing.png|thumb|325px| It is possible to solve for a goods or price schedule that satisfies the first-order conditions yet is not monotonic. If so it is necessary to "iron" the schedule by choosing some value at which to flatten the function.]]

In some applications the designer may solve the first-order conditions for the price and allocation schedules yet find they are not monotonic. For example, in the quasilinear setting this often happens when the hazard ratio is itself not monotone. By the Spence–Mirrlees condition the optimal price and allocation schedules must be monotonic, so the designer must eliminate any interval over which the schedule changes direction by flattening it.

Intuitively, what is going on is the designer finds it optimal to '''bunch''' certain types together and give them the same contract. Normally the designer motivates higher types to distinguish themselves by giving them a better deal. If there are insufficiently few higher types on the margin the designer does not find it worthwhile to grant lower types a concession (called their [[information rent]]) in order to charge higher types a type-specific contract.

Consider a monopolist principal selling to agents with quasilinear utility, the example above. Suppose the allocation schedule <math>x(\theta)</math> satisfying the first-order conditions has a single interior peak at <math>\theta_1</math> and a single interior trough at <math>\theta_2>\theta_1</math>, illustrated at right.

* Following Myerson (1981) flatten it by choosing <math>x</math> satisfying <math display="block"> \int^{\phi_1(x)}_{\phi_2(x)} \left( \frac{\partial V}{\partial x}(x,\theta) - \frac{1-P(\theta)}{p(\theta)} \frac{\partial^2 V}{\partial \theta \, \partial x}(x,\theta) - \frac{\partial c}{\partial x}(x) \right) d\theta = 0</math> where <math>\phi_1(x)</math> is the inverse function of x mapping to <math>\theta \leq \theta_1</math> and <math>\phi_2(x)</math> is the inverse function of x mapping to <math>\theta \geq \theta_2</math>. That is, <math>\phi_1</math> returns a <math>\theta</math> before the interior peak and <math>\phi_2</math> returns a <math>\theta</math> after the interior trough.
* If the nonmonotonic region of <math>x(\theta)</math> borders the edge of the type space, simply set the appropriate <math>\phi(x)</math> function (or both) to the boundary type. If there are multiple regions, see a textbook for an iterative procedure; it may be that more than one troughs should be ironed together.

====Proof====

The proof uses the theory of optimal control. It considers the set of intervals <math>\left[\underline\theta, \overline\theta \right] </math> in the nonmonotonic region of <math>x(\theta)</math> over which it might flatten the schedule. It then writes a Hamiltonian to obtain necessary conditions for a <math>x(\theta)</math> within the intervals
# that does satisfy monotonicity
# for which the monotonicity constraint is not binding on the boundaries of the interval
Condition two ensures that the <math>x(\theta)</math> satisfying the optimal control problem reconnects to the schedule in the original problem at the interval boundaries (no jumps). Any <math>x(\theta)</math> satisfying the necessary conditions must be flat because it must be monotonic and yet reconnect at the boundaries.

As before maximize the principal's expected payoff, but this time subject to the monotonicity constraint
:<math>\frac{\partial x}{\partial \theta} \geq 0</math>
and use a Hamiltonian to do it, with shadow price <math>\nu(\theta)</math>
:<math>H = \left( V(x,\theta) - \underline{u}(\theta_0) - \frac{1-P(\theta)}{p(\theta)} \frac{\partial V}{\partial \theta}(x,\theta) - c(x) \right)p(\theta) + \nu(\theta) \frac{\partial x}{\partial \theta} </math>
where <math>x</math> is a state variable and <math>\partial x/\partial \theta</math> the control. As usual in optimal control the costate evolution equation must satisfy
:<math> \frac{\partial \nu}{\partial \theta} = -\frac{\partial H}{\partial x} = -\left( \frac{\partial V}{\partial x}(x,\theta) - \frac{1-P(\theta)}{p(\theta)} \frac{\partial^2 V}{\partial \theta \, \partial x}(x,\theta) - \frac{\partial c}{\partial x}(x) \right) p(\theta) </math>
Taking advantage of condition 2, note the monotonicity constraint is not binding at the boundaries of the <math>\theta</math> interval,
:<math>\nu(\underline\theta) = \nu(\overline\theta) = 0</math>
meaning the costate variable condition can be integrated and also equals 0
:<math>\int^{\overline\theta}_{\underline\theta} \left( \frac{\partial V}{\partial x}(x,\theta) - \frac{1-P(\theta)}{p(\theta)} \frac{\partial^2 V}{\partial \theta \, \partial x}(x,\theta) - \frac{\partial c}{\partial x}(x) \right) p(\theta) \, d\theta = 0 </math>
The average distortion of the principal's surplus must be 0. To flatten the schedule, find an <math>x</math> such that its inverse image maps to a <math>\theta</math> interval satisfying the condition above.