Mechanism design


Mechanism design is a branch of economics and game theory. It studies how to construct rules—called mechanisms or institutions—that produce good outcomes according to some predefined metric, even when the designer does not know the players' true preferences or what information they have. Mechanism design thus focuses on the study of solution concepts for a class of private-information games.
Mechanism design has broad applications, including traditional domains of economics such as market design, but also political science. It is a foundational component in the operation of the internet, being used in networked systems, e-commerce, and advertisement auctions by Facebook and Google.
Because it starts with the end of the game, then works backwards to find a game that implements it, it is sometimes described as reverse game theory. Leonid Hurwicz explains that "in a design problem, the goal function is the main given, while the mechanism is the unknown. Therefore, the design problem is the inverse of traditional economic theory, which is typically devoted to the analysis of the performance of a given mechanism."
The 2007 Nobel Memorial Prize in Economic Sciences was awarded to Leonid Hurwicz, Eric Maskin, and Roger Myerson "for having laid the foundations of mechanism design theory." The related works of William Vickrey that established the field earned him the 1996 Nobel prize.

Description

One person, called the "principal", would like to condition his behavior on information privately known to the players of a game. For example, the principal would like to know the true quality of a used car a salesman is pitching. He cannot learn anything simply by asking the salesman, because it is in the salesman's interest to distort the truth. However, in mechanism design, the principal does have one advantage: He may design a game whose rules influence others to act the way he would like.
Without mechanism design theory, the principal's problem would be difficult to solve. He would have to consider all the possible games and choose the one that best influences other players' tactics. In addition, the principal would have to draw conclusions from agents who may lie to him. Thanks to the revelation principle, the principal only needs to consider games in which agents truthfully report their private information.
Mechanism design has been described by Noam Nisan as a way to escape the Gibbard–Satterthwaite theorem. While the theorem is traditionally presented as a result about voting systems, it can also be seen as an important result of mechanism design, which deals with a broader class of decision rules. Here is the quote:
The GS theorem seems to quash any hope of designing incentive-compatible social-choice functions. The whole field of Mechanism Design attempts escaping from this impossibility result using various modifications in the model.
The main idea of these "escape routes" is to modify a model to allow for a broader class of mechanisms, similar to the escape routes from Arrow's impossibility theorem in the case of ranked voting.

Foundations

Mechanism

A game of mechanism design is a game of private information in which one of the agents, called the principal, chooses the payoff structure. Following, the agents receive secret "messages" from nature containing information relevant to payoffs. For example, a message may contain information about their preferences or the quality of a good for sale. We call this information the agent's "type". Agents then report a type to the principal that can be a strategic lie. After the report, the principal and the agents are paid according to the payoff structure the principal chose.
The timing of the game is:
  1. The principal commits to a mechanism that grants an outcome as a function of reported type
  2. The agents report, possibly dishonestly, a type profile
  3. The mechanism is executed
In order to understand who gets what, it is common to divide the outcome into a goods allocation and a money transfer, where stands for an allocation of goods rendered or received as a function of type, and stands for a monetary transfer as a function of type.
As a benchmark the designer often defines what should happen under full information. Define a social choice function mapping the type profile directly to the allocation of goods received or rendered,
In contrast a mechanism maps the reported type profile to an ''outcome''

Revelation principle

A proposed mechanism constitutes a Bayesian game, and if it is well-behaved the game has a Bayesian Nash equilibrium. At equilibrium agents choose their reports strategically as a function of type
It is difficult to solve for Bayesian equilibria in such a setting because it involves solving for agents' best-response strategies and for the best inference from a possible strategic lie. Thanks to a sweeping result called the revelation principle, no matter the mechanism a designer can confine attention to equilibria in which agents truthfully report type. The revelation principle states: "To every Bayesian Nash equilibrium there corresponds a Bayesian game with the same equilibrium outcome but in which players truthfully report type."
This is extremely useful. The principle allows one to solve for a Bayesian equilibrium by assuming all players truthfully report type. In one blow it eliminates the need to consider either strategic behavior or lying.
Its proof is quite direct. Assume a Bayesian game in which the agent's strategy and payoff are functions of its type and what others do,. By definition agent i's equilibrium strategy is Nash in expected utility:
Simply define a mechanism that would induce agents to choose the same equilibrium. The easiest one to define is for the mechanism to commit to playing the agents' equilibrium strategies for them.
Under such a mechanism the agents of course find it optimal to reveal type since the mechanism plays the strategies they found optimal anyway. Formally, choose such that

Implementability

The designer of a mechanism generally hopes either
  • to design a mechanism that "implements" a social choice function
  • to find the mechanism that maximizes some value criterion
To implement a social choice function is to find some transfer function that motivates agents to pick. Formally, if the equilibrium strategy profile under the mechanism maps to the same goods allocation as a social choice function,
we say the mechanism implements the social choice function.
Thanks to the revelation principle, the designer can usually find a transfer function to implement a social choice by solving an associated truthtelling game. If agents find it optimal to truthfully report type,
we say such a mechanism is truthfully implementable. The task is then to solve for a truthfully implementable and impute this transfer function to the original game. An allocation is truthfully implementable if there exists a transfer function such that
which is also called the incentive compatibility constraint.
In applications, the IC condition is the key to describing the shape of in any useful way. Under certain conditions it can even isolate the transfer function analytically. Additionally, a participation 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. Consider also a goods allocation that is vector-valued and size and assume it is piecewise continuous with respect to its arguments.
The function is implementable only if
whenever and and x is continuous at. 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 increases as a function of the type,
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,
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, 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 "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. Multiple-good mechanisms are an area of continuing research in mechanism design.

Sufficiency

Mechanism design papers usually make two assumptions to ensure implementability:
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 MRS is increasing in type.
This is a technical condition bounding the rate of growth of the MRS.
These assumptions are sufficient to provide that any monotonic is implementable. In addition, in the single-good setting the single-crossing condition is sufficient to provide that only a monotonic is implementable, so the designer can confine his search to a monotonic.

Highlighted results

Revenue equivalence theorem

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
  1. The buyers have identical valuation functions
  2. The buyers' types are independently distributed
  3. The buyers types are drawn from a continuous distribution
  4. The type distribution bears the monotone hazard rate property
  5. 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.