ECONOMY

494 voting and efficient public good mechanisms
The third alternative definition asserts that a mechanism is interim incentive efficient
if and only if it is ex ante incentive efficient for all affine transformation of the
utility functions. 32 It is this fact that convinces me, at least for a Bayesian analysis,
that interim incentive efficiency is the right concept to use when trying to identify
mechanisms that will survive and be used.
3.2 A Bayesian Characterization of Interim Incentive
Efficient Mechanisms
In Ledyard and Palfrey (1999), we were able to characterize the class of interim
incentive efficient mechanisms for the very simple Bayesian environment. In these,
consumers announce their values, y is chosen according to a virtual cost–benefit rule,
and taxes are computed using a rule discovered by d’Aspremont and Gerard-Varet
(1979). We called these mechanisms virtual cost–benefit mechanisms.
Definition 3: The VCB (virtual cost–benefit) mechanism, for given functions Î i (v i ),
is given by m i ∈ M i =V i , an output rule
y ∗ (v) =1if ∑ i
w i (v i ) ≥ K
y ∗ (v) =0otherwise,
where
w i (v i )=v i +{[F i (v i ) − i (v i )]/ f i (v i )},
and a tax rule
i (v i )=
t ∗i (v) =ky ∗ (v) − [T i (v i ) − Q i (v i )] +
where
Q i (v i )=
∫ v
i
∫ v
i
Î i (s )ds
( ) 1 { ∑
}
[T j (v j ) − Q j (v j )] .
N − 1
j ≠i
y ∗ (x)d F (x|v i )
and
T i (v i )=
∫ v
i
sdQ i (s).
³² “Ex ante” refers to the analysis that is done prior to anyone knowing anything other than the
common knowledge. A mechanism is ex ante efficient iff there are Î i > 0suchthat[y ∗ (·), t ∗ (·)] solves
max ∫ ∑ Î i [v i y(v) − t i (v)]dF(v) subject to feasibility and incentive compatibility. Compare this with
the definition of interim efficiency to see that the difference is that the Î i donotdependonthev i . An
affine transformation of utilities for each v i yields u ′i = a i (v i )[v i y(v) − t i (v)] + b i (v i ). This does not
change either the feasibility or incentive constraints. But it does change the objective function to
∫ ∑ Î i a i (v i )[v i y(v) − t i (v)]dF(v). So, letting Î i (v i )=Î i a i (v i ), it is easy to see that a mechanism is ex
ante efficient for all affine transformation of utilities iff it is interim efficient.