Editing Mechanism design (section)

===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 {{harvs|txt|last=Harsanyi|year=1967|author-link=John Harsanyi}}, 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 <math>\theta</math> and accordingly the space of types <math>\Theta</math>). Agents then report a type to the principal (usually noted with a hat <math>\hat\theta</math>) 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 <math>y()</math> that grants an outcome <math>y</math> as a function of reported type
# The agents report, possibly dishonestly, a type profile <math>\hat\theta</math>
# The mechanism is executed (agents receive outcome <math>y(\hat\theta)</math>)

In order to understand who gets what, it is common to divide the outcome <math>y</math> into a goods allocation and a money transfer, <math>y(\theta) = \{ x(\theta), t(\theta) \}, \ x \in X, t \in T </math> where <math>x</math> stands for an allocation of goods rendered or received as a function of type, and <math>t</math> stands for a monetary transfer as a function of type.

As a benchmark the designer often defines what should happen under full information. Define a [[social choice function]] <math>f(\theta)</math> mapping the (true) type profile directly to the allocation of goods received or rendered,

:<math>f(\theta): \Theta \rightarrow Y</math>

In contrast a '''mechanism''' maps the ''reported'' type profile to an ''outcome'' (again, both a goods allocation <math>x</math> and a money transfer <math>t</math>)
:<math>y(\hat\theta): \Theta \rightarrow Y</math>