# Direct revelation mechanism's sets of strategies and types

Mas-Colell, Whinston and Green in Microeconomic Theory describe the direct revelation mechanism as it follows:

The employed notation is the following:

$$θ_i$$: Player i's type

$$Θ_i$$: Set of types for player i

$$S_i$$ Set of strategies for player i

f: Social choice function

g: Mechanism outcome function

X: Set of collective choices

If $$S_i$$ is a set of strategies and $$Θ_i$$ is a set of types, how can $$S_i = Θ_i$$ ?

In mechanism design you are free to choose the rules of the game. The designer can determine $$(S, g)$$, i.e., what players can do and what happens when players played some strategy profile $$s \in S := \times S_i$$.
In a direct mechanism, players are simply asked to report their type. Hence, every player $$i$$ must have a strategy that corresponds to "I am type $$\theta_i$$" for all $$\theta_i \in \Theta_i$$ such that we can, without loss of generality, set $$S_i = \Theta_i$$ and each strategy corresponds to a type report.