 Mechanism design

Mechanism design (sometimes called reverse game theory^{[1]}) is a field in game theory studying solution concepts for a class of private information games. The distinguishing features of these games are:
 that a game "designer" chooses the game structure rather than inheriting one
 that the designer is interested in the game's outcome
Such a game is called a "game of mechanism design" and is usually solved by motivating agents to disclose their private information. 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".^{[2]}
Contents
Intuition
In an interesting class of Bayesian games, one player, called the “principal,” would like to condition his behavior on information privately known to other players. 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 his interest to distort the truth. Fortunately, in mechanism design the principal does have one advantage. He may design a game whose rules can influence others to act the way he would like.
Absent 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 mechanism design, and particularly the revelation principle, the principal need only consider games in which agents truthfully report their private information.
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 Harsanyi (1967), 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" (usually noted θ and accordingly the space of types Θ). Agents then report a type to the principal (usually noted with a hat ) 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:
 The principal commits to a mechanism y() that grants an outcome y as a function of reported type
 The agents report, possibly dishonestly, a type profile
 The mechanism is executed (agents receive outcome )
In order to understand who gets what, it is common to divide the outcome y into a goods allocation and a money transfer, where x stands for an allocation of goods rendered or received as a function of type, and t stands for a monetary transfer as a function of type.
As a benchmark the designer often defines what would happen under full information. Define a social choice function f(θ) mapping the (true) type profile directly to the allocation of goods received or rendered,
In contrast a mechanism maps the reported type profile to an outcome (again, both a goods allocation x and a money transfer t)
Revelation principle
Main article: Revelation principleA proposed mechanism constitutes a Bayesian game (a game of private information), and if it is wellbehaved 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' bestresponse 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^{[3]} confine attention to equilibria in which agents truthfully report type. The revelation principle states: "For any 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 (subject to an incentive compatibility constraint). 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 s(θ_{i}) 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 y(θ) such that
Implementability
The designer of a mechanism generally hopes either
 to design a mechanism y() that "implements" a social choice function
 to find the mechanism y() that maximizes some value criterion (e.g. profit)
To implement a social choice function f(θ) is to find some t(θ) transfer function that motivates agents to pick outcome x(θ). 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 t(θ) 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 (or just "implementable"). The task is then to solve for a truthfully implementable t(θ) and impute this transfer function to the original game. An allocation x(θ) is truthfully implementable if there exists a transfer function t(θ) such that
which is also called the incentive compatibility (IC) constraint.
In applications, the IC condition is the key to describing the shape of t(θ) 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 typecontingent utility function u(x,t,θ). Consider also a goods allocation x(θ) that is vectorvalued and size k (which permits k number of goods) and assume it is piecewise continuous with respect to its arguments.
The function x(θ) is implementable only if
whenever x = x(θ) and t = t(θ) and x is continuous at θ. This is a necessary condition and is derived from the first and secondorder conditions of the agent's optimization problem assuming truthtelling.
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 IC 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 lowtype 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 multiplegood 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). Multiplegood mechanisms are an ongoing problem in mechanism design theory.
Sufficiency
Mechanism design papers usually make two assumptions to ensure implementability:
This is known by several names: the singlecrossing condition, the sorting condition and the SpenceMirrlees 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 x(θ) is implementable (a t(θ) exists that can implement it). In addition, in the singlegood setting the singlecrossing condition is sufficient to provide that only a monotonic x(θ) is implementable, so the designer can confine his search to a monotonic x(θ).
Highlighted results
Revenue equivalence theorem
Main article: Revenue equivalence theoremVickrey (1961) 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
 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 article: Vickrey–Clarke–Groves auctionThe Vickrey (1961) auction model was later expanded by Clarke (1971) and Groves (1973) 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 I number of agents have quasilinear utility with private valuations v(x,t,θ) where the currency t 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
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
which sums the distortion in the utilities of the other agents (and not his own) caused by one agent reporting.
GibbardSatterthwaite theorem
Main article: GibbardSatterthwaite theoremGibbard (1973) and Satterthwaite (1975) 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 f() is dictatorial if one agent always receives his mostfavored goods allocation,
The theorem states that under general conditions any truthfully implementable social choice function must be dictatorial,
 X finite and contains at least three elements
 Preferences are rational
 f(Θ) = X
MyersonSatterthwaite theorem
Main article: MyersonSatterthwaite theoremMyerson and Satterthwaite (1983) 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.
Examples
Price discrimination
Mirrlees (1971) 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 singlegood, singleagent setting in which the agent has quasilinear utility with an unknown type parameter θ
 u(x,t,θ) = V(x,θ) − t
and in which the principal has a prior CDF over the agent's type P(θ). The principal can produce goods at a convex marginal cost c(x) and wants to maximize the expected profit from the transaction
subject to IC and IR conditions
The principal here is a monopolist trying to set a profitmaximizing 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 enough 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,
Integrating,
where θ_{0} is some index type. Replacing the incentivecompatible t(θ) = V(x(θ),θ) − U(θ) in the maximand,
after an integration by parts. This function can be maximized pointwise, a fantastic result because it dispenses with the need to use the calculus of variations.
Because U(θ) is incentivecompatible already the designer can drop the IC constraint. If the utility function satisfies the SpenceMirrlees condition then a monotonic x(θ) 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 sufficiency, above) is satisfied everywhere along the allocation and fee schedules. If not, then the designer must use Myerson ironing.
Myerson ironing
In some applications the designer may solve the firstorder 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 SpenceMirrlees 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 typespecific contract.
Consider a monopolist principal selling to agents with quasilinear utility, the example above. Suppose the allocation schedule x(θ) satisfying the firstorder conditions has a single interior peak at θ_{1} and a single interior trough at θ_{2} > θ_{1}, illustrated at right.
 Following Myerson (1981) flatten it by choosing x satisfying
 where ϕ_{1}(x) is the inverse function of x mapping to and ϕ_{2}(x)is the inverse function of x mapping to . That is, ϕ_{1} returns a θ before the interior peak and ϕ_{2} returns a θ after the interior trough.
 If the nonmonotonic region of x(θ) borders the edge of the type space, simply set the appropriate ϕ(x) 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 in the nonmonotonic region of x(θ) over which it might flatten the schedule. It then writes a Hamiltonian to obtain necessary conditions for a x(θ) 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 x(θ) satisfying the optimal control problem reconnects to the schedule in the original problem at the interval boundaries (no jumps). Any x(θ) 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
and use a Hamiltonian to do it, with shadow price ν(θ)
where x is a state variable and the control. As usual in optimal control the costate evolution equation must satisfy
Taking advantage of condition 2, note the monotonicity constraint is not binding at the boundaries of the θ interval,
meaning the costate variable condition can be integrated and also equals 0
The average distortion of the principal's surplus must be 0. To flatten the schedule, find an x such that its inverse image maps to a θ interval satisfying the condition above.
See also
 Algorithmic mechanism design
 Contract theory
 Implementation theory
 Incentive compatibility
 Revelation principle
 Smart market
 Metagame
Notes
 ^ "Multiagent Systems", from the University of Amsterdam.
 ^ "The Sveriges Riksbank Prize in Economic Sciences in Memory of Alfred Nobel 2007" (Press release). Nobel Foundation. October 15, 2007. http://nobelprize.org/nobel_prizes/economics/laureates/2007/press.html. Retrieved 20080815.
 ^ In unusual circumstances some truthtelling games have more equilibria than the Bayesian game they mapped from. See FudenburgTirole Ch. 7.2 for some references.
References
 Chapter 7 of Fudenberg, Drew; Tirole, Jean (1991), Game Theory, Boston: MIT Press, ISBN 9780262061414, http://www.amazon.com/GameTheoryDrewFudenberg/dp/0262061414/ref=sr_1_1?ie=UTF8&s=books&qid=1245092433&sr=81. A standard text for graduate game theory.
 Chapter 23 of MasColell; Whinston; Green (1995), Microeconomic Theory, Oxford: Oxford Press, ISBN 9780195073409, http://www.amazon.com/MicroeconomicTheoryAndreuMasColell/dp/0195073401/ref=sr_1_1?ie=UTF8&s=books&qid=1245092701&sr=11. A standard text for graduate microeconomics.
 Milgrom, Paul (2004), Putting Auction Theory to Work, New York: Cambridge University Press, ISBN 9780521551847, http://www.amazon.com/PuttingAuctionChurchillLecturesEconomics/dp/0521536723/ref=sr_1_1?ie=UTF8&s=books&qid=1245365630&sr=81. Applications of mechanism design principles in the context of auctions.
 Noam Nisan. A Google tech talk on mechanism design.
 Roger B. Myerson (2008). "mechanism design," The New Palgrave Dictionary of Economics Online, Abstract.
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