# Are direct mechanisms always truthful?

I am very new to mechanism design so I am a bit lost among the concepts, even the most basic ones.

One thing that is not clear to me is the concept of direct mechanism and its relationship with the revelation principle.

From what I got a direct mechanism is a mechanisms, in which, instead of messages, the agents report their preference/type. Are we assuming this is a truthful report?

I am asking that because the revelation principle is stated as "anything that can be accomplished by any mechanism can actually be accomplished by a direct revelation mechanism that is individual rational and incentive compatible" in this notes but as "a social choice function can be implemented by an arbitrary mechanism (i.e. if that mechanism has an equilibrium outcome that corresponds to the outcome of the social choice function), then the same function can be implemented by an incentive-compatible-direct-mechanism (i.e. in which players truthfully report type) with the same equilibrium outcome (payoffs)." on Wikipedia.

So I don't get if truthfulness is implied by the direct mechanism or if it is an additional requirement.

Take an arbitrary (direct or indirect) mechanism in which all participants $$i \in I$$ can take one or more actions from some action space and together all players' actions determine some outcome. A strategy in this game maps a private type $$t_i$$ into actions $$\sigma_i (t_i)$$. Hence, a type profile $$t = (t_i)_{i \in I}$$ leads to an equilibrium action profile $$\sigma(t)=(\sigma_i(t_i))_{i \in I}$$ which then leads to an outcome determined by the mechanisms outcome function $$o_{indirect} (\sigma(t))$$.
Instead of letting the participants play some complicated game, we can ask all participants for their type and then "let the mechanism play the game for them". That is, each player reports her type $$t_i$$ and the mechanism takes the type profile $$t$$ and maps it into an outcome $$o_{direct}(t) = o_{indirect} (\sigma(t))$$ for all $$t$$. We have replicated the indirect mechanism. Is it incentive compatible to report the TRUE type? Yes, it is, because $$\sigma$$ is an equilibrium in the original mechanism. If a participant had an incentive to report some type $$t'_i \neq t_i$$, there would also be a profitable deviation $$\sigma_i (t'_i) \neq \sigma_i (t_i)$$ in the indirect mechanism. A contradiction to $$\sigma$$ being an equilibrium. Note that this equilibrium does not have to be in dominant strategies.