<p>A social choice function presumes the individuals' preference parameters <span class="math-container">$\theta_i$</span>'s are observable, whereas in a mechanism, such knowledge is not presupposed. Therefore, in a mechanism, the arguments of the outcome function are strategies of the players, which are observable, not their preference parameters, which, although indirectly determine the players' strategies, are not observable. </p>

<p>As an example, suppose <span class="math-container">$3$</span> individuals are either rich or pool (<span class="math-container">$\Theta_i=\{R,P\}$</span> for <span class="math-container">$i=1,2,3$</span>) and there are two policies, <span class="math-container">$X=\{A,B\}$</span>. Assume rich people prefer <span class="math-container">$A$</span> and the poor prefer <span class="math-container">$B$</span>. A social choice function would take a profile of the three people's wealth status and produce a policy outcome, for instance, 
<span class="math-container">\begin{equation}
f(R,R,P)=A,\quad\text{or}\quad f(P,P,R)=B.
\end{equation}</span>
However, if individual wealth status is private information which cannot be directly observed, and the policy must be chosen by a simple majority voting mechanism, then each individual's strategy space would be <span class="math-container">$S_i=\{A,B\}$</span> and their strategy <span class="math-container">$s_i:\Theta_i\to X$</span> would map from their own type to a policy choice. The mechanism outcome function would turn a profile of the strategies into a policy outcome, e.g. 
<span class="math-container">\begin{equation}
g(A,B,A)=A,\quad\text{or}\quad g(A,B,B)=B.
\end{equation}</span></p>

<p>There is a special mechanism: when each player's strategy space is also their type space, i.e. <span class="math-container">$S_i=\Theta_i$</span>, and the mechanism's outcome function is the same as the social choice function <span class="math-container">$g(\theta)=f(\theta)$</span>, we have what's known as a <em>direct mechanism</em>.</p>

CC BY-SA 4.0
2019-07-22T22:31:53.907Z
2019-07-22T22:58:51.957Z
2019-07-22T22:58:51.957Z
Herr K.
<p>Mas-Colell, Whinston and Green's Microeconomic Theory (3rd edition) defines the social choice function as the following:</p>

<p><a href="https://i.sstatic.net/yQs2J.jpg" rel="nofollow noreferrer"><img src="https://i.sstatic.net/yQs2J.jpg" alt="enter image description here"></a></p>

<p>Later, the mechanism outcome function is also defined:</p>

<p><a href="https://i.sstatic.net/VgqD5.jpg" rel="nofollow noreferrer"><img src="https://i.sstatic.net/VgqD5.jpg" alt="enter image description here"></a></p>

<p>The relationship between a social choice function and a mechanism is then stated as:</p>

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

<p>What is the difference between a social choice function and a mechanism outcome function?</p>

<microeconomics><mathematical-economics><mechanism-design><social-choice>
Difference between social choice function and mechanism outcome function
Incognito
# Difference between social choice function and mechanism outcome function

(Asked by: username_0 on 2019-07-22)

Mas-Colell, Whinston and Green's Microeconomic Theory (3rd edition) defines the social choice function as the following:

[![enter image description here](https://i.sstatic.net/yQs2J.jpg)](https://i.sstatic.net/yQs2J.jpg)

Later, the mechanism outcome function is also defined:

[![enter image description here](https://i.sstatic.net/VgqD5.jpg)](https://i.sstatic.net/VgqD5.jpg)

The relationship between a social choice function and a mechanism is then stated as:

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

What is the difference between a social choice function and a mechanism outcome function?

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


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## 1 answers

### Answer by username_1 on 2019-07-22. Score: 5 | Accepted answer

A social choice function presumes the individuals' preference parameters $\theta\_i$'s are observable, whereas in a mechanism, such knowledge is not presupposed. Therefore, in a mechanism, the arguments of the outcome function are strategies of the players, which are observable, not their preference parameters, which, although indirectly determine the players' strategies, are not observable.

As an example, suppose $3$ individuals are either rich or pool ($\Theta\_i=\{R,P\}$ for $i=1,2,3$) and there are two policies, $X=\{A,B\}$. Assume rich people prefer $A$ and the poor prefer $B$. A social choice function would take a profile of the three people's wealth status and produce a policy outcome, for instance,
\begin{equation}
f(R,R,P)=A,\quad\text{or}\quad f(P,P,R)=B.
\end{equation}
However, if individual wealth status is private information which cannot be directly observed, and the policy must be chosen by a simple majority voting mechanism, then each individual's strategy space would be $S\_i=\{A,B\}$ and their strategy $s\_i:\Theta\_i\to X$ would map from their own type to a policy choice. The mechanism outcome function would turn a profile of the strategies into a policy outcome, e.g.
\begin{equation}
g(A,B,A)=A,\quad\text{or}\quad g(A,B,B)=B.
\end{equation}

There is a special mechanism: when each player's strategy space is also their type space, i.e. $S\_i=\Theta\_i$, and the mechanism's outcome function is the same as the social choice function $g(\theta)=f(\theta)$, we have what's known as a *direct mechanism*.


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