Mechanism design is a field in economics and game theory that takes an objectives-first approach to designing economic mechanisms or incentives, toward desired objectives, in strategic settings, where players act rationally. Because it starts at the end of the game, then goes backwards, it is also called reverse game theory. It has broad applications, from economics and politics in such fields as
market design
Market design is a practical methodology for creation of markets of certain properties, which is partially based on mechanism design. In some markets, prices may be used to induce the desired outcomes — these markets are the study of auction the ...
,
auction theory
Auction theory is an applied branch of economics which deals with how bidders act in auction markets and researches how the features of auction markets Incentivisation, incentivise predictable outcomes. Auction theory is a tool used to inform the ...
and
social choice theory
Social choice theory or social choice is a theoretical framework for analysis of combining individual opinions, preferences, interests, or welfares to reach a ''collective decision'' or ''social welfare'' in some sense.Amartya Sen (2008). "Soci ...
to networked-systems (internet interdomain routing, sponsored search auctions).
Mechanism design studies
solution concept
In game theory, a solution concept is a formal rule for predicting how a game will be played. These predictions are called "solutions", and describe which strategies will be adopted by players and, therefore, the result of the game. The most comm ...
s for a class of private-information games.
Leonid Hurwicz
Leonid Hurwicz (; August 21, 1917 – June 24, 2008) was a Polish-American economist and mathematician, known for his work in game theory and mechanism design. He originated the concept of incentive compatibility, and showed how desired outcomes ...
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.' So, two 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
The 2007
Nobel Memorial Prize in Economic Sciences
The Nobel Memorial Prize in Economic Sciences, officially the Sveriges Riksbank Prize in Economic Sciences in Memory of Alfred Nobel ( sv, Sveriges riksbanks pris i ekonomisk vetenskap till Alfred Nobels minne), is an economics award administered ...
was awarded to
Leonid Hurwicz
Leonid Hurwicz (; August 21, 1917 – June 24, 2008) was a Polish-American economist and mathematician, known for his work in game theory and mechanism design. He originated the concept of incentive compatibility, and showed how desired outcomes ...
,
Eric Maskin
Eric Stark Maskin (born December 12, 1950) is an American economist and mathematician. He was jointly awarded the 2007 Nobel Memorial Prize in Economic Sciences with Leonid Hurwicz and Roger Myerson "for having laid the foundations of mechanism d ...
, and
Roger Myerson
Roger Bruce Myerson (born March 29, 1951) is an American economist and professor at the University of Chicago. He holds the title of the David L. Pearson Distinguished Service Professor of Global Conflict Studies at The Pearson Institute for the ...
"for having laid the foundations of mechanism design theory".
Intuition
In an interesting class of
Bayesian game
In game theory, a Bayesian game is a game that models the outcome of player interactions using aspects of Bayesian probability. Bayesian games are notable because they allowed, for the first time in game theory, for the specification of the soluti ...
s, 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 can 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 mechanism design, and particularly the
revelation principle
The revelation principle is a fundamental principle in mechanism design. It states that if a social choice function can be implemented by an arbitrary mechanism (i.e. if that mechanism has an equilibrium outcome that corresponds to the outcome of ...
, the principal only needs to 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 , 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
)
:
Revelation principle
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
In game theory, a Bayesian game is a game that models the outcome of player interactions using aspects of Bayesian probability. Bayesian games are notable because they allowed, for the first time in game theory, for the specification of the soluti ...
. 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 (subject to an
incentive compatibility
A mechanism is called incentive-compatible (IC) if every participant can achieve the best outcome to themselves just by acting according to their true preferences.
There are several different degrees of incentive-compatibility:
* The stronger d ...
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 ''is 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 (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
Rational choice theory refers to a set of guidelines that help understand economic and social behaviour. The theory originated in the eighteenth century and can be traced back to political economist and philosopher, Adam Smith. The theory postula ...
) 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
(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
In economics, the marginal rate of substitution (MRS) is the rate at which a consumer can give up some amount of one good in exchange for another good while maintaining the same level of utility. At equilibrium consumption levels (assuming no exte ...
(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
Butter is a dairy product made from the fat and protein components of churned cream. It is a semi-solid emulsion at room temperature, consisting of approximately 80% butterfat. It is used at room temperature as a spread, melted as a condiment ...
for
margarine
Margarine (, also , ) is a spread used for flavoring, baking, and cooking. It is most often used as a substitute for butter. Although originally made from animal fats, most margarine consumed today is made from vegetable oil. The spread was orig ...
). Multiple-good 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
single-crossing condition In monotone comparative statics, the single-crossing condition or single-crossing property refers to a condition where the relationship between two or more functionsThe property need not only relate to continuous functions but can also similarly des ...
, 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
.
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
# 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
In probability theory and statistics, a probability distribution is the mathematical Function (mathematics), function that gives the probabilities of occurrence of different possible outcomes for an Experiment (probability theory), experiment. ...
# 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
The Vickrey (1961) auction model was later expanded by 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
Tragedy (from the grc-gre, τραγῳδία, ''tragōidia'', ''tragōidia'') is a genre of drama based on human suffering and, mainly, the terrible or sorrowful events that befall a main character. Traditionally, the intention of tragedy ...
"—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–Satterthwaite theorem
and give an impossibility result similar in spirit to
Arrow's impossibility theorem
Arrow's impossibility theorem, the general possibility theorem or Arrow's paradox is an impossibility theorem in social choice theory that states that when voters have three or more distinct alternatives (options), no ranked voting electoral syste ...
. 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–Satterthwaite theorem
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
There are two fundamental theorems of welfare economics. The first states that in economic equilibrium, a set of complete markets, with complete information, and in perfect competition, will be Pareto optimal (in the sense that no further exchange ...
.
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 The Price of Anarchy (PoA) is a concept in economics and game theory that measures how the efficiency of a system degrades due to selfish behavior of its agents. It is a general notion that can be extended to diverse systems and notions of effici ...
), and then secondly optimizes the best-case outcomes (the
price of stability
In game theory, the price of stability (PoS) of a game is the ratio between the best objective function value of one of its equilibria and that of an optimal outcome. The PoS is relevant for games in which there is some objective authority that ...
), is precisely the Shapley value cost-sharing rule.
A symmetrical statement is similarly valid for utility-sharing games with convex utility functions.
Examples
Price discrimination
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
In economics and consumer theory, quasilinear utility functions are linear in one argument, generally the numeraire. Quasilinear preferences can be represented by the utility function u(x_1, x_2, \ldots, x_n) = x_1 + \theta (x_2, \ldots, x_n) whe ...
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
: