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

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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 $S_i = Θ_i$ ?

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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.

Together with the revelation principle, direct mechanisms are very powerfull, see my post here.

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