<p>A direct revelation mechanism is one in which a player's type space is also their action space (<span class="math-container">$X_i=T_i$</span> for all <span class="math-container">$i$</span>) and the outcome function is the same as the social choice function (<span class="math-container">$a(t)=f(t)$</span> for all <span class="math-container">$t\in T_1\times\cdots\times T_n$</span>).</p>

CC BY-SA 4.0
2021-11-08T04:10:40.380Z
I should have been more clear - in the definition of a direct revelation mechanism, $f$ maps from $V_1\times\dots\times V_n$ to $A$, and each $v_i\in V_i$ is a function $A\to\mathbf R$. Whereas in the definition of a general mechanism, one of the parameters is a set of $n$ (predefined) preferences $v_i:T_i\times A\to\mathbf R$. How should we reconcile these two definitions? I've fixed my question to account for this.
andrew
CC BY-SA 4.0
2021-11-08T06:36:21.987Z
Each $t_i$ induces a preference $\succeq$ over $A$, so $T_i$ and $V_i$ can be identified by one another.
Walrasian Auctioneer
CC BY-SA 4.0
2021-11-08T15:58:47.133Z
@andrew: I think the difference is probably due to sloppy notations. Another possible reconciliation is that $v_i:A\to \mathbf R$ is defined with the assumption of "truth telling" (as if types can be observed), whereas $v_i:T_i\times A\to \mathbf R$ explicitly models the incompleteness of information. This interpretation is supported by the open paragraph of Sec 9.4, which says: "*The mechanisms considered so far extract information from the different players by motivating them to "tell the truth."*"
Herr K.
CC BY-SA 4.0
2021-11-08T02:25:56.527Z
2021-11-08T02:25:56.527Z
Herr K.
<p>In <em>Algorithmic Game Theory</em> by Noam Nisan, a (general) mechanism for <span class="math-container">$n$</span> player is defined as</p>
<p><span class="math-container">$$
M=(\{T_i\},\{X_i\},A,\{v_i\},a,\{p_i\}),\tag1
$$</span></p>
<p>where <span class="math-container">$T_1,\dots,T_n$</span> are the players' type spaces (private information), <span class="math-container">$X_1,\dots,X_n$</span> are the players' action spaces, <span class="math-container">$A$</span> is the set of alternatives, each <span class="math-container">$v_i:T_i\times A\to\mathbf R$</span> is a valuation for player <span class="math-container">$i$</span>, <span class="math-container">$a:\times X_i\to A$</span> is the outcome function, and each <span class="math-container">$p_i:\times X_i\to\mathbf R$</span> is the amount that player <span class="math-container">$i$</span> pays. (pg. 224 <a href="https://www.cs.cmu.edu/%7Esandholm/cs15-892F13/algorithmic-game-theory.pdf" rel="nofollow noreferrer">here</a>).</p>
<p>On the other hand, a <em>direct revelation mechanism</em> (pg. 218) is defined as</p>
<p><span class="math-container">$$
(f,p_1,\dots,p_n),
$$</span></p>
<p>where <span class="math-container">$f:V_1\times\dots\times V_n\to A$</span> is a social choice function from a set <span class="math-container">$V_i$</span> of individual preferences <span class="math-container">$v_i:A\to\mathbf R$</span> to an alternative, and each <span class="math-container">$p_i:V_1\times\dots\times V_n\to\mathbf R$</span> is the amount that player <span class="math-container">$i$</span> pays.</p>
<p><strong>Question:</strong> How can I recover a direct revelation mechanism from the definition of a general mechanism? More specifically, what should the parameters in <span class="math-container">$(1)$</span> be set to, in order to obtain a direct revelation mechanism?</p>
<p>Thank you.</p>

<microeconomics><game-theory><mechanism-design><social-choice>
Definition of direct revelation mechanism
andrew
# Definition of direct revelation mechanism

(Asked by: username_0 on 2021-11-08)

In *Algorithmic Game Theory* by Noam Nisan, a (general) mechanism for $n$ player is defined as

$$
M=(\{T\_i\},\{X\_i\},A,\{v\_i\},a,\{p\_i\}),\tag1
$$

where $T\_1,\dots,T\_n$ are the players' type spaces (private information), $X\_1,\dots,X\_n$ are the players' action spaces, $A$ is the set of alternatives, each $v\_i:T\_i\times A\to\mathbf R$ is a valuation for player $i$, $a:\times X\_i\to A$ is the outcome function, and each $p\_i:\times X\_i\to\mathbf R$ is the amount that player $i$ pays. (pg. 224 [here](https://www.cs.cmu.edu/%7Esandholm/cs15-892F13/algorithmic-game-theory.pdf)).

On the other hand, a *direct revelation mechanism* (pg. 218) is defined as

$$
(f,p\_1,\dots,p\_n),
$$

where $f:V\_1\times\dots\times V\_n\to A$ is a social choice function from a set $V\_i$ of individual preferences $v\_i:A\to\mathbf R$ to an alternative, and each $p\_i:V\_1\times\dots\times V\_n\to\mathbf R$ is the amount that player $i$ pays.

**Question:** How can I recover a direct revelation mechanism from the definition of a general mechanism? More specifically, what should the parameters in $(1)$ be set to, in order to obtain a direct revelation mechanism?

Thank you.

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

---

## 1 answers

### Answer by username_1 on 2021-11-08. Score: 2

A direct revelation mechanism is one in which a player's type space is also their action space ($X\_i=T\_i$ for all $i$) and the outcome function is the same as the social choice function ($a(t)=f(t)$ for all $t\in T\_1\times\cdots\times T\_n$).

> **Comments:**
> **username_0**: I should have been more clear - in the definition of a direct revelation mechanism, $f$ maps from $V_1\times\dots\times V_n$ to $A$, and each $v_i\in V_i$ is a function $A\to\mathbf R$. Whereas in the definition of a general mechanism, one of the parameters is a set of $n$ (predefined) preferences $v_i:T_i\times A\to\mathbf R$. How should we reconcile these two definitions? I've fixed my question to account for this.  
> **username_2**: Each $t_i$ induces a preference $\succeq$ over $A$, so $T_i$ and $V_i$ can be identified by one another.  
> **username_1**: @andrew: I think the difference is probably due to sloppy notations. Another possible reconciliation is that $v_i:A\to \mathbf R$ is defined with the assumption of "truth telling" (as if types can be observed), whereas $v_i:T_i\times A\to \mathbf R$ explicitly models the incompleteness of information. This interpretation is supported by the open paragraph of Sec 9.4, which says: "*The mechanisms considered so far extract information from the different players by motivating them to "tell the truth."*"  

---

