<p>In mechanism design you are free to choose the rules of the game. The designer can determine <span class="math-container">$(S, g)$</span>, i.e., what players can do and what happens when players played some strategy profile <span class="math-container">$s \in S := \times S_i$</span>.</p>

<p>In a direct mechanism, players are simply asked to report their type. Hence, every player <span class="math-container">$i$</span> must have a strategy that corresponds to "I am type <span class="math-container">$\theta_i$</span>" for all <span class="math-container">$\theta_i \in \Theta_i$</span> such that we can, without loss of generality, set <span class="math-container">$S_i = \Theta_i$</span> and each strategy corresponds to a type report. </p>

<p>Together with the revelation principle, direct mechanisms are very powerfull, see my post <a href="https://economics.stackexchange.com/a/27320/11590">here</a>.</p>

CC BY-SA 4.0
2019-07-25T15:41:12.267Z
2019-07-25T15:41:12.267Z
Bayesian
<p>Mas-Colell, Whinston and Green in Microeconomic Theory describe the direct revelation mechanism as it follows:</p>

<p><a href="https://i.sstatic.net/17Uj0.jpg" rel="nofollow noreferrer"><img src="https://i.sstatic.net/17Uj0.jpg" alt="enter image description here"></a></p>

<p>The employed notation is the following:</p>

<p><span class="math-container">$θ_i$</span>: Player i's type</p>

<p><span class="math-container">$Θ_i$</span>: Set of types for player i</p>

<p><span class="math-container">$S_i$</span> Set of strategies for player i</p>

<p>f: Social choice function</p>

<p>g: Mechanism outcome function</p>

<p>X: Set of collective choices</p>

<p>If <span class="math-container">$S_i$</span> is a set of strategies and <span class="math-container">$Θ_i$</span> is a set of types, how can <span class="math-container">$S_i = Θ_i$</span> ?</p>

<microeconomics><mathematical-economics><game-theory><mechanism-design><social-choice>
Direct revelation mechanism's sets of strategies and types
Incognito
# Direct revelation mechanism's sets of strategies and types

(Asked by: username_0 on 2019-07-25)

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

[![enter image description here](https://i.sstatic.net/17Uj0.jpg)](https://i.sstatic.net/17Uj0.jpg)

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 $S\_i = Θ\_i$ ?

**Tags:** microeconomics, mathematical-economics, game-theory, mechanism-design, social-choice

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## 1 answers

### Answer by username_1 on 2019-07-25. Score: 4 | Accepted answer

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](https://economics.stackexchange.com/a/27320/11590).


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