**Mechanism design** is a branch of economics, social choice theory, and game theory that deals with designing games (or mechanisms) to implement a given social choice function. Because it starts at the end of the game (the optimal result) and then works backwards to find a game that implements it, it is sometimes called **reverse game theory**.^{[ citation needed ]}

- Intuition
- Foundations
- Mechanism
- Revelation principle
- Implementability
- Highlighted results
- Revenue equivalence theorem
- Vickrey–Clarke–Groves mechanisms
- Gibbard–Satterthwaite theorem
- Myerson–Satterthwaite theorem
- Shapley value
- Examples
- Price discrimination
- Myerson ironing
- See also
- Notes
- References
- Further reading
- External links

Mechanism design has broad applications, including traditional domains of economics such as market design, but also political science (through voting theory) and even networked systems (such as in inter-domain routing).^{ [1] }

Mechanism design studies solution concepts for a class of private-information games. 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."^{ [2] }

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."^{ [3] } The related works of William Vickrey that established the field earned him the 1996 Nobel prize.

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 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.

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 and accordingly the space of types ). 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 that grants an outcome 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 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 would happen under full information. Define a ** 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 and a money transfer )

A proposed mechanism constitutes a Bayesian game (a game of private information), 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^{ [4] } 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 (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 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

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 (e.g. profit)

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** (or just "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** (IC) 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 (individual rationality) constraint is sometimes added if agents have the option of not playing.

Consider a setting in which all agents have a type-contingent utility function . Consider also a goods allocation that is vector-valued and size (which permits number of goods) 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 (MRS) 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 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 (e.g. butter for margarine). Multiple-good mechanisms are an ongoing problem in mechanism design theory.

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 (a exists that can implement it). 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 .

Vickrey ( 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.

The Vickrey (1961) auction model was later expanded by Clarke ( 1971 ) 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 number of agents have quasilinear utility with private valuations where the currency 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.

Gibbard ( 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 most-favored goods allocation,

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

Myerson andSatterthwaite ( 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.

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.^{ [5] } A symmetrical statement is similarly valid for utility-sharing games with convex utility functions.

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 single-good, single-agent setting in which the agent has quasilinear utility with an unknown type parameter

and in which the principal has a prior CDF over the agent's type . 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 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,

Integrating,

where is some index type. Replacing the incentive-compatible in the maximand,

after an integration by parts. This function can be maximized pointwise.

Because is incentive-compatible already the designer can drop the IC constraint. If the utility function satisfies the Spence–Mirrlees condition then a monotonic 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.

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 satisfying the first-order conditions has a single interior peak at and a single interior trough at , illustrated at right.

- Following Myerson (1981) flatten it by choosing satisfying where is the inverse function of x mapping to and is the inverse function of x mapping to . That is, returns a before the interior peak and returns a after the interior trough.
- If the nonmonotonic region of borders the edge of the type space, simply set the appropriate 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.

The proof uses the theory of optimal control. It considers the set of intervals in the nonmonotonic region of over which it might flatten the schedule. It then writes a Hamiltonian to obtain necessary conditions for a 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 satisfying the optimal control problem reconnects to the schedule in the original problem at the interval boundaries (no jumps). Any 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

where 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 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 such that its inverse image maps to a interval satisfying the condition above.

- ↑ Penna, Paolo; Ventre, Carmine (July 2014). "Optimal collusion-resistant mechanisms with verification".
*Games and Economic Behavior*.**86**: 491–509. doi:10.1016/j.geb.2012.09.002. ISSN 0899-8256. - ↑ L. Hurwicz & S. Reiter (2006) Designing Economic Mechanisms, p. 30
- ↑ "The Sveriges Riksbank Prize in Economic Sciences in Memory of Alfred Nobel 2007" (Press release). Nobel Foundation. October 15, 2007. Retrieved 2008-08-15.
- ↑ In unusual circumstances some truth-telling games have more equilibria than the Bayesian game they mapped from. See Fudenburg-Tirole Ch. 7.2 for some references.
- ↑ Phillips, Matthew; Marden, Jason R. (July 2018). "Design Tradeoffs in Concave Cost-Sharing Games".
*IEEE Transactions on Automatic Control*.**63**(7): 2242–2247. doi:10.1109/tac.2017.2765299. ISSN 0018-9286. S2CID 45923961.

In mathematics and physics, **Laplace's equation** is a second-order partial differential equation named after Pierre-Simon Laplace, who first studied its properties. This is often written as

The **likelihood function** is the joint probability mass of observed data viewed as a function of the parameters of a statistical model. Intuitively, the likelihood function is the probability of observing data assuming is the actual parameter.

The **Navier–Stokes equations** are partial differential equations which describe the motion of viscous fluid substances. They were named after French engineer and physicist Claude-Louis Navier and the Irish physicist and mathematician George Gabriel Stokes. They were developed over several decades of progressively building the theories, from 1822 (Navier) to 1842–1850 (Stokes).

In statistics, **maximum likelihood estimation** (**MLE**) is a method of estimating the parameters of an assumed probability distribution, given some observed data. This is achieved by maximizing a likelihood function so that, under the assumed statistical model, the observed data is most probable. The point in the parameter space that maximizes the likelihood function is called the maximum likelihood estimate. The logic of maximum likelihood is both intuitive and flexible, and as such the method has become a dominant means of statistical inference.

In statistics, a statistic is * sufficient* with respect to a statistical model and its associated unknown parameter if "no other statistic that can be calculated from the same sample provides any additional information as to the value of the parameter". In particular, a statistic is sufficient for a family of probability distributions if the sample from which it is calculated gives no additional information than the statistic, as to which of those probability distributions is the sampling distribution.

In mathematics and physical science, **spherical harmonics** are special functions defined on the surface of a sphere. They are often employed in solving partial differential equations in many scientific fields. A list of the spherical harmonics is available in Table of spherical harmonics.

In probability theory and statistics, the **gamma distribution** is a versatile two-parameter family of continuous probability distributions. The exponential distribution, Erlang distribution, and chi-squared distribution are special cases of the gamma distribution. There are two equivalent parameterizations in common use:

- With a shape parameter k and a scale parameter θ
- With a shape parameter and an inverse scale parameter , called a rate parameter.

**Linear elasticity** is a mathematical model of how solid objects deform and become internally stressed due to prescribed loading conditions. It is a simplification of the more general nonlinear theory of elasticity and a branch of continuum mechanics.

In probability and statistics, an **exponential family** is a parametric set of probability distributions of a certain form, specified below. This special form is chosen for mathematical convenience, including the enabling of the user to calculate expectations, covariances using differentiation based on some useful algebraic properties, as well as for generality, as exponential families are in a sense very natural sets of distributions to consider. The term **exponential class** is sometimes used in place of "exponential family", or the older term **Koopman–Darmois family**. Sometimes loosely referred to as "the" exponential family, this class of distributions is distinct because they all possess a variety of desirable properties, most importantly the existence of a sufficient statistic.

In information geometry, the **Fisher information metric** is a particular Riemannian metric which can be defined on a smooth statistical manifold, *i.e.*, a smooth manifold whose points are probability measures defined on a common probability space. It can be used to calculate the informational difference between measurements.

In multivariable calculus, the **implicit function theorem** is a tool that allows relations to be converted to functions of several real variables. It does so by representing the relation as the graph of a function. There may not be a single function whose graph can represent the entire relation, but there may be such a function on a restriction of the domain of the relation. The implicit function theorem gives a sufficient condition to ensure that there is such a function.

In mathematical statistics, the **Fisher information** is a way of measuring the amount of information that an observable random variable *X* carries about an unknown parameter *θ* of a distribution that models *X*. Formally, it is the variance of the score, or the expected value of the observed information.

In mathematical statistics, the **Kullback–Leibler (KL) divergence**, denoted , is a type of statistical distance: a measure of how one probability distribution P is different from a second, reference probability distribution Q. A simple interpretation of the KL divergence of P from Q is the expected excess surprise from using Q as a model when the actual distribution is P. While it is a measure of how different two distributions are, and in some sense is thus a "distance", it is not actually a metric, which is the most familiar and formal type of distance. In particular, it is not symmetric in the two distributions, and does not satisfy the triangle inequality. Instead, in terms of information geometry, it is a type of divergence, a generalization of squared distance, and for certain classes of distributions, it satisfies a generalized Pythagorean theorem.

In mathematics, the **Helmholtz equation** is the eigenvalue problem for the Laplace operator. It corresponds to the linear partial differential equation:

In mathematics, the **Ornstein–Uhlenbeck process** is a stochastic process with applications in financial mathematics and the physical sciences. Its original application in physics was as a model for the velocity of a massive Brownian particle under the influence of friction. It is named after Leonard Ornstein and George Eugene Uhlenbeck.

In statistics, **M-estimators** are a broad class of extremum estimators for which the objective function is a sample average. Both non-linear least squares and maximum likelihood estimation are special cases of M-estimators. The definition of M-estimators was motivated by robust statistics, which contributed new types of M-estimators. However, M-estimators are not inherently robust, as is clear from the fact that they include maximum likelihood estimators, which are in general not robust. The statistical procedure of evaluating an M-estimator on a data set is called **M-estimation**.

In statistics, the **monotone likelihood ratio property** is a property of the ratio of two probability density functions (PDFs). Formally, distributions and bear the property if

**Frontogenesis** is a meteorological process of tightening of horizontal temperature gradients to produce fronts. In the end, two types of fronts form: cold fronts and warm fronts. A cold front is a narrow line where temperature decreases rapidly. A warm front is a narrow line of warmer temperatures and essentially where much of the precipitation occurs. Frontogenesis occurs as a result of a developing baroclinic wave. According to Hoskins & Bretherton, there are eight mechanisms that influence temperature gradients: horizontal deformation, horizontal shearing, vertical deformation, differential vertical motion, latent heat release, surface friction, turbulence and mixing, and radiation. Semigeostrophic frontogenesis theory focuses on the role of horizontal deformation and shear.

In fluid dynamics, the **Oseen equations** describe the flow of a viscous and incompressible fluid at small Reynolds numbers, as formulated by Carl Wilhelm Oseen in 1910. Oseen flow is an improved description of these flows, as compared to Stokes flow, with the (partial) inclusion of convective acceleration.

**Projection filters** are a set of algorithms based on stochastic analysis and information geometry, or the differential geometric approach to statistics, used to find approximate solutions for filtering problems for nonlinear state-space systems. The filtering problem consists of estimating the unobserved signal of a random dynamical system from partial noisy observations of the signal. The objective is computing the probability distribution of the signal conditional on the history of the noise-perturbed observations. This distribution allows for calculations of all statistics of the signal given the history of observations. If this distribution has a density, the density satisfies specific stochastic partial differential equations (SPDEs) called Kushner-Stratonovich equation, or Zakai equation. It is known that the nonlinear filter density evolves in an infinite dimensional function space.

- Clarke, Edward H. (1971). "Multipart Pricing of Public Goods" (PDF).
*Public Choice*.**11**(1): 17–33. doi:10.1007/BF01726210. JSTOR 30022651. S2CID 154860771. - Gibbard, Allan (1973). "Manipulation of voting schemes: A general result" (PDF).
*Econometrica*.**41**(4): 587–601. doi:10.2307/1914083. JSTOR 1914083. - Groves, Theodore (1973). "Incentives in Teams" (PDF).
*Econometrica*.**41**(4): 617–631. doi:10.2307/1914085. JSTOR 1914085. - Harsanyi, John C. (1967). "Games with incomplete information played by "Bayesian" players, I-III. part I. The Basic Model".
*Management Science*.**14**(3): 159–182. doi:10.1287/mnsc.14.3.159. JSTOR 2628393. - Mirrlees, J. A. (1971). "An Exploration in the Theory of Optimum Income Taxation" (PDF).
*Review of Economic Studies*.**38**(2): 175–208. doi:10.2307/2296779. JSTOR 2296779. Archived from the original (PDF) on 2017-05-10. Retrieved 2016-08-12. - Myerson, Roger B.; Satterthwaite, Mark A. (1983). "Efficient Mechanisms for Bilateral Trading" (PDF).
*Journal of Economic Theory*.**29**(2): 265–281. doi:10.1016/0022-0531(83)90048-0. hdl: 10419/220829 . - Satterthwaite, Mark Allen (1975). "Strategy-proofness and Arrow's conditions: Existence and correspondence theorems for voting procedures and social welfare functions".
*Journal of Economic Theory*.**10**(2): 187–217. CiteSeerX 10.1.1.471.9842 . doi:10.1016/0022-0531(75)90050-2. - Vickrey, William (1961). "Counterspeculation, Auctions, and Competitive Sealed Tenders" (PDF).
*The Journal of Finance*.**16**(1): 8–37. doi:10.1111/j.1540-6261.1961.tb02789.x.

- Chapter 7 of Fudenberg, Drew; Tirole, Jean (1991),
*Game Theory*, Boston: MIT Press, ISBN 978-0-262-06141-4 . A standard text for graduate game theory. - Chapter 23 of Mas-Colell; Whinston; Green (1995),
*Microeconomic Theory*, Oxford: Oxford University Press, ISBN 978-0-19-507340-9 . A standard text for graduate microeconomics. - Milgrom, Paul (2004),
*Putting Auction Theory to Work*, New York: Cambridge University Press, ISBN 978-0-521-55184-7 . Applications of mechanism design principles in the context of auctions. - Noam Nisan. A Google tech talk on mechanism design.
- Legros, Patrick; Cantillon, Estelle (2007). "What is mechanism design and why does it matter for policy-making?". Centre for Economic Policy Research.
- Roger B. Myerson (2008). "Mechanism Design,"
*The New Palgrave Dictionary of Economics Online, Abstract.* - Diamantaras, Dimitrios (2009),
*A Toolbox for Economic Design*, New York: Palgrave Macmillan, ISBN 978-0-230-61060-6 . A graduate text specifically focused on mechanism design.

- Eric Maskin "Nobel Prize Lecture" delivered on 8 December 2007 at Aula Magna, Stockholm University.

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