Foundations
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 1967, the agents get secret "messages" from breed containing information applicable to payoffs. For example, a message may contain information about their preferences or the nature of a value for sale. We known this information the agent's "type" usually listed 
and accordingly the space of types 
. Agents then description a type to the principal usually pointed 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:
In order to understand who gets what, it is for 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 social choice function 
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 offered mechanism constitutes a Bayesian game a game of private information, and whether it is well-behaved the game has a Bayesian Nash equilibrium. At equilibrium agentstheir reports strategically as a function of type
It is difficult to solve for Bayesian equilibria in such(a) a established because it involves solving for agents' best-response strategies and for the best inference from a possible strategic lie. Thanks to a sweeping statement called the revelation principle, no matter the mechanism a designer can confine attention to equilibria in which agents truthfully description 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 any 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) a mechanism the agents of course find it optimal to reveal type since the mechanism plays the strategies they found optimal anyway. Formally, select 
such that
The designer of a mechanism broadly hopes either
To implement a social choice function is to find some 
transfer function that motivates agents to pick . Formally, whether 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 commonly 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 precondition is the key to describing the shape of in any useful way. Underconditions it can even isolate the transfer function analytically. Additionally, a participation individual rationality constraint is sometimes added if agents work the option of not playing.
Consider a defining in which all agents realise a type-contingent improvement function 
. Consider also a goods allocation that is vector-valued and size 
which gives number of goods and assume it is piecewise continual with respect to its arguments.
The function is implementable only if
whenever 
and 
and x is non-stop 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 number one 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 advertising 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 constituent 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 normally make two assumptions to ensure implementability:
This is invited 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 .