<p>By definition, a direct mechanism is a mechanism that asks all agents for their types and then produces some outcome. Formally, it is a mechanism <span class="math-container">$\langle M,g\rangle$</span> in which <span class="math-container">$M_i$</span> is wlog equal to <span class="math-container">$i$</span>'s type space, usually denoted by <span class="math-container">$\Theta_i$</span>.</p>

<p>In your example, there is no such restriction. However, for any indirect mechanism there exists an equivalent direct mechanism. This result is called the revelation principle.</p>

CC BY-SA 4.0
2019-12-05T19:47:23.010Z
Specifically, the difference is that in a direct mechanism setup the agent is asked to report their preferences to the principal, while in an indirect mechanism setup the agent sends a signal regarding their preferences.
corran_horn
CC BY-SA 4.0
2019-12-05T17:01:19.470Z
2019-12-05T17:01:19.470Z
Bayesian
<p>From L. Hurwicz's work and book "Designing Economic Mechanisms," I cannot figure out whether <em>a resource allocation problem in mechanism design is a direct or indirect mechanism.</em></p>

<p>I think the answer is yes since a resource allocation mechanism can be described as a tuple of two components, i.e., <span class="math-container">$\langle M, g \rangle$</span>, where <span class="math-container">$M = (M_1, \ldots, M_n)$</span> and <span class="math-container">$M_i$</span> defines the set of possible messages of agent <span class="math-container">$i$</span>. By writing the agents' complete message space as</p>

<p><span class="math-container">$$\mathcal{M} = M_1 \times M_2 \times \cdots \times M_n$$</span></p>

<p>we can define the outcome function <span class="math-container">$g$</span> as</p>

<p><span class="math-container">$$g : \mathcal{M} \to \mathcal{O},$$</span></p>

<p>where <span class="math-container">$\mathcal{O}$</span> is the output space defined by</p>

<p><span class="math-container">$$\mathcal{O} = \{(\theta_1, \ldots, \theta_n), (t_1, \ldots, t_n) \; | \; \theta_i \in \mathbb{R}_{ &gt; 0}, \; t_i \in \mathbb{R} \}.$$</span></p>

<p>Then, the outcome function <span class="math-container">$g$</span> determines the outcome, namely <span class="math-container">$g(\mu)$</span> for any given message profile <span class="math-container">$\mu = (m_1, \ldots, m_n) \in \mathcal{M}$</span> and the payment function is defined as</p>

<p><span class="math-container">$$t_i : \mathcal{M} \to \mathbb{R}$$</span></p>

<p>which determines the monetary payment made or received by agent <span class="math-container">$i \in \mathcal{I}$</span>.</p>

CC BY-SA 4.0
2019-11-27T22:34:31.657Z
What exactly do the $\theta$'s represent? Agent types or? Could you provide a concrete reference on where this problem is defined in the book/paper?
Walrasian Auctioneer
CC BY-SA 4.0
2019-11-30T17:34:08.013Z
@WalrasianAuctioneer Yes, the $\theta$'s represent the types of the agents. The formulation I have written in my question comes (not directly) from Hurwicz's book, Chapter 1.
johnny09
<microeconomics><mechanism-design>
Is a resource allocation problem in mechanism design a direct or indirect mechanism?
johnny09
# Is a resource allocation problem in mechanism design a direct or indirect mechanism?

(Asked by: username_0 on 2019-11-27)

From L. Hurwicz's work and book "Designing Economic Mechanisms," I cannot figure out whether *a resource allocation problem in mechanism design is a direct or indirect mechanism.*

I think the answer is yes since a resource allocation mechanism can be described as a tuple of two components, i.e., $\langle M, g \rangle$, where $M = (M\_1, \ldots, M\_n)$ and $M\_i$ defines the set of possible messages of agent $i$. By writing the agents' complete message space as

$$\mathcal{M} = M\_1 \times M\_2 \times \cdots \times M\_n$$

we can define the outcome function $g$ as

$$g : \mathcal{M} \to \mathcal{O},$$

where $\mathcal{O}$ is the output space defined by

$$\mathcal{O} = \{(\theta\_1, \ldots, \theta\_n), (t\_1, \ldots, t\_n) \; | \; \theta\_i \in \mathbb{R}\_{ > 0}, \; t\_i \in \mathbb{R} \}.$$

Then, the outcome function $g$ determines the outcome, namely $g(\mu)$ for any given message profile $\mu = (m\_1, \ldots, m\_n) \in \mathcal{M}$ and the payment function is defined as

$$t\_i : \mathcal{M} \to \mathbb{R}$$

which determines the monetary payment made or received by agent $i \in \mathcal{I}$.

**Tags:** microeconomics, mechanism-design
> **Comments:**
> **username_1**: What exactly do the $\theta$'s represent? Agent types or? Could you provide a concrete reference on where this problem is defined in the book/paper?  
> **username_0**: @WalrasianAuctioneer Yes, the $\theta$'s represent the types of the agents. The formulation I have written in my question comes (not directly) from Hurwicz's book, Chapter 1.  

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## 1 answers

### Answer by username_2 on 2019-12-05. Score: 2 | Accepted answer

By definition, a direct mechanism is a mechanism that asks all agents for their types and then produces some outcome. Formally, it is a mechanism $\langle M,g\rangle$ in which $M\_i$ is wlog equal to $i$'s type space, usually denoted by $\Theta\_i$.

In your example, there is no such restriction. However, for any indirect mechanism there exists an equivalent direct mechanism. This result is called the revelation principle.

> **Comments:**
> **username_3**: Specifically, the difference is that in a direct mechanism setup the agent is asked to report their preferences to the principal, while in an indirect mechanism setup the agent sends a signal regarding their preferences.  

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