# Definition of direct revelation mechanism

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).

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.

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 $$t\in T_1\times\cdots\times T_n$$).
• 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. Nov 8, 2021 at 4:10
• Each $t_i$ induces a preference $\succeq$ over $A$, so $T_i$ and $V_i$ can be identified by one another. Nov 8, 2021 at 6:36
• @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."" Nov 8, 2021 at 15:58