# Is a resource allocation problem in mechanism design a direct or indirect mechanism?

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 $$i \in \mathcal{I}$$.

• 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 Nov 27 '19 at 22:34
• @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 Nov 30 '19 at 17:34

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 $$\Theta_i$$.