Construct by defining the functions q and ti as required by item (ii) in Proposition 3.1. We
Question:
Construct by defining the functions q and ti as required by item (ii) in Proposition
3.1. We can prove the result by showing that truth telling is a Bayesian equilibrium of
the game. Suppose it were not. If type ?i prefers to report that her type is ?
i , then the
same type ?i prefers to deviate from ?, and to play the strategy that ? prescribes for
?
i in . Hence ? is not a Bayesian equilibrium of . -
Proposition 3.1 shows that in the setup that we have described we can, without
loss of generality, restrict our attention to the case in which the seller chooses a di-
rect mechanism and proposes to agents that they report their types truthfully. Note,
however, that it is crucial to this construction that we have neglected problems of mul-
tiple equilibria by assuming that agents follow the seller's proposal provided that it is
an equilibrium and provided that it gives them expected utility of at least zero. This
is crucial because the equivalent direct mechanism that is constructed in the proof of
Proposition 3.1 might have Bayesian Nash equilibria other than truthtelling, and there
is no reason why these equilibria should be equivalent to any Bayesian Nash equilib-
rium of the indirect mechanism . Depending on how equilibria are selected, one or
the other mechanism might be strictly preferred by the seller in that case.
The revelation principle greatly simplifies our search for optimal mechanisms. We
can restrict our attention to direct mechanisms in which it is a Bayesian equilibrium
that everyone always reports their type truthfully and in which every type's expected
utility is at least zero. We want to define these properties of a direct mechanism
formally. For this we introduce additional notation.
We denote by ?-i the vector of all types except player i's type. We define
-i ? [ ?, ?]N-1. We denote by F-i the cumulative distribution of ?-i, and we denote
by f-i the density of F-i. Given a direct mechanism, we define for each agent i ? I a
function Q i : [ ?, ?] ? [0, 1] by setting
Q i(?i) = -
-i
qi(?i, ?-i)f-i(?-i) d?-i. (3.1)