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]}

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.

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 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.

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 ${\displaystyle \theta }$ and accordingly the space of types ${\displaystyle \Theta }$). Agents then report a type to the principal (usually noted with a hat ${\displaystyle {\hat {\theta }}}$) 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 ${\displaystyle y()}$ that grants an outcome ${\displaystyle y}$ as a function of reported type
2. The agents report, possibly dishonestly, a type profile ${\displaystyle {\hat {\theta }}}$
3. The mechanism is executed (agents receive outcome ${\displaystyle y({\hat {\theta }})}$)

In order to understand who gets what, it is common to divide the outcome ${\displaystyle y}$ into a goods allocation and a money transfer, ${\displaystyle y(\theta )=\{x(\theta ),t(\theta )\},\ x\in X,t\in T}$ where ${\displaystyle x}$ stands for an allocation of goods rendered or received as a function of type, and ${\displaystyle t}$ 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 ${\displaystyle f(\theta )}$ mapping the (true) type profile directly to the allocation of goods received or rendered,

${\displaystyle f(\theta ):\Theta \rightarrow X}$

In contrast a mechanism maps the reported type profile to an outcome (again, both a goods allocation ${\displaystyle x}$ and a money transfer ${\displaystyle t}$)

${\displaystyle y({\hat {\theta }}):\Theta \rightarrow Y}$

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. At equilibrium agents choose their reports strategically as a function of type

${\displaystyle {\hat {\theta }}(\theta )}$

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, ${\displaystyle u_{i}\left(s_{i}(\theta _{i}),s_{-i}(\theta _{-i}),\theta _{i}\right)}$. By definition agent i's equilibrium strategy ${\displaystyle s(\theta _{i})}$ is Nash in expected utility:

${\displaystyle s_{i}(\theta _{i})\in \arg \max _{s'_{i}\in S_{i}}\sum _{\theta _{-i}}\ p(\theta _{-i}\mid \theta _{i})\ u_{i}\left(s'_{i},s_{-i}(\theta _{-i}),\theta _{i}\right)}$

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.

${\displaystyle y({\hat {\theta }}):\Theta \rightarrow S(\Theta )\rightarrow Y}$

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 ${\displaystyle y(\theta )}$ such that

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