3.3 The problem of information aggregation
3.3.3 Information Aggregation under I-IC
The resolution of aggregate uncertainty requires to figure out the true state of the economy, i.e. the fractionpof individuals with a high taste parameter. By definition, the I-IC constraints ensure that no individual has an incentive to hide the own characteristics for a given cross-section distribution of char- acteristics. However, under aggregate uncertainty, an incentive compatible allocation rule is used in addition for the purpose of information aggregation as the mechanism designer has to deduce the actual distribution of charac- teristics from the profile of individual announcements. To make this more explicit, it is instructive to think of the revelation game in the following sequential manner:
Sequence 3.1
Stage 1: The mechanism designer specifies a provision rule Q accompanied by equal cost sharing. I.e. there is a distinct level of public good provi- sion Q(p) and a distinct payment obligation K(Q(p)) for each possible state p of the economy.
Stage 2: The mechanism designer collects all individual data and uses this information to deduce the actual value pˆfor the state of the economy.
Stage 3: According to the specified rule, the amount Q(ˆp) of the public good is provided. Individuals make the corresponding contributionsK(Q(ˆp)).
Multiple equilibria
As outlined in Section 3.2, the problem of aggregate uncertainty in large economies has not been addressed yet by the literature. An exception is Hammond (1979). Based on his work, one could take the view that the problem of information aggregation is resolved trivially as a corollary of I- IC. Suppose, with no loss of generality, that, on stage 2, the planner can communicate with all agents, asks them to report their characteristics13, and is able to measure the fraction
ˆ
p=µ({i∈I |θˆi =θH})
of high valuation reports among the population14. The allocation rule, pro- posed on stage 1, exhibits equal cost sharing and specifies the level of public
13
The revelation principle tells that any achievable allocation can be attained by such a direct mechanism.
14
The problem of measurability is discussed, for example, by Judd (1985) or Al-Najjar (2004), showing that an appropriate measure µ can be constructed. More details are provided in Chapter 4.
good provision as a function of the perceived state of the economy ˆp. Since the economy is large, no single agent iis able to influence the planner’s per- ception ˆp by his report ˆθi. Hence, no individual has a payoff relevant move.
This implies that revealing the own characteristics is a best response. Of course, the same can be said about any reporting behavior.
In other words, there are multiple equilibria and one faces a problem of
equilibrium selection. From the angle just outlined, this problem could be solved trivially by just breaking the agents’ indifference in favor of truth- telling. Hence, the planner would learn the true state of the worldpand the problem of information aggregation would disappear. It would be resolved as a byproduct of I-IC.
However, we doubt this view and aim at offering an alternative one. To illustrate the problem, we first explore the consequences of the perspective that the need of information aggregation does not create additional incentive problems, once I-IC has been taken into account and the costs of public good provision are covered via equal cost sharing.
The optimal utilitarian allocation under I-IC
Given this notion of implementability we can solve for the provision rule which is optimal from a utilitarian perspective. For givenpand an arbitrary provision rule Q, utilitarian welfare under equal cost sharing is given by
(pθH + (1−p)θL)Q(p)−λK(Q(p)) with λ:= ¯ w Z w ¯ f(w) w dw.
The parameterλcan be interpreted asthe shadow cost of public funds.15 We may equivalently assume that the utilitarian objective is to maximize
¯
v(p)Q(p)−K(Q(p)) with v¯(p) := pθH+ (1−p)θL
λ
and refer to the term ¯v(p) as the effective utilitarian valuation or as the
effective aggregate valuation of the public good if the state of the economy is
p.
Now suppose that I-IC is a sufficient condition to guarantee that anyone will truthfully report his taste. Using his prior beliefs on p being uniformly
15
From a utilitarian welfare perspective, equal cost sharing as implied by I-IC, is a distortion. A utilitarian planner would prefer that only individuals with earning ability ¯w
3.3. THE PROBLEM OF INFORMATION AGGREGATION 79 distributed, the planner assesses expected utilitarian welfare to equal
EW :=λ
1
Z
0
{v¯(p)Q(p)−K(Q(p))}dp.
The provision rule which maximizes this expression is denoted Q∗ and will
serve as a benchmark in the following. In combination with equal cost sharing it forms the optimal feasible allocation rule that satisfies the I-IC constraints.
Q∗ is characterized by the following version of the Samuelson rule for public
good provision.
Proposition 3.2 The optimal feasible provision ruleQ∗ :p7→Q∗(p) under
I-IC is characterized by a continuum of first order conditions,
K′(Q∗(p)) = ¯v(p) for p∈[0,1].
Q∗ is strictly increasing and continuously differentiable with derivative Q∗′. Proof. The properties of Q∗ follow immediately from the planners maxi-
mization problem taking into account the assumptions onK.
2
The need of a refinement
We now illustrate that the admissibility of the optimal utilitarian allocation
Q∗ requires that agents use actions which are not robust in the following
sense: Agents are assumed to behave in a way that would be incompatible with implementation in dominant strategies as soon as they were given an arbitrary small influence on the perceived state of the world. Hence, for the problem at hand, the notion of individual incentive compatibility lacks plausibility.
To see this, note first that the utility realized by an individual under provision rule Q∗ is a function of the individual’s characteristics (θ, w) and
the unknown parameterp. This indirect utility function is henceforth written as
U∗(p, θ, w) = θQ∗(p)− K(Q∗(p))
w .
Its partial derivative with respect top equals
U1∗(p, θ, w) =
1
wQ
∗′(p)[θw
−v¯(p)].
Consequently, U∗ is increasing inpas long as θw >v¯(p), i.e. the individual’s
utilitarian planner. Analogously,U∗ is decreasing inpifθw falls short of the
utilitarian valuation.
Now suppose, for the sake of concreteness, that pis such that16
θLw >¯ v¯(p)> θL λ.
This implies that there exists a critical value ˆw∈]w
¯,w¯[ such that all individ- uals with θi = θ
L and wi < wˆ have an effective valuation θLwi which falls
short of the utilitarian planner’s effective valuation ¯v(p) and hence, according to their indirect utility function, would prefer a slightly lower perceived value of p. Analogously, individuals with θi = θ
L and wi > wˆ have an effective
valuationθLwi exceeding the one of the utilitarian planner. Therefore, they
would prefer a slightly larger perceived value of p.
Despite those conflicting interests, under the truth-telling assumption, individuals in both sets are assumed to behave the same way, namely to re- veal their low valuation of the public good. In particular, an individual with
θLw > v¯(p) is assumed not to exaggerate when reporting the own taste for
the public good, even though this individual would be happy if the utilitarian planner could be induced to believe that the effective aggregate valuation of the public good was in fact higher.
In more abstract terms, it has been noted above that, in a revelation game with equal cost sharing, any kind of behavior constitutes a best response as no individual is able to affect with the own announcement the planner’s perception of the aggregate valuation ¯v(p). At the same time individuals are not indifferent regarding this perception. They just have no direct influence on it. Still, they might want to marginally increase their indirect utility by strategic reports in favor of their preferred state perception. Moreover, being involved in the revelation game forces any individual to subscribe either to the group of individuals with a high taste parameter or to the group of individuals with a low taste parameter. Hence, the assumption that under provision rule Q∗ any individual tells the truth, amounts to the postulate
that some individuals do not subscribe to the group whose size they would be willing to support, but instead join the ‘wrong’ group. In that sense, a solution that takes only the I-IC constraints into account is not robust to the (illusion of) indirect individual influence on the perceived aggregate valuation.
Note that this is not only a concern of whether or not it is acceptable to break individual indifference in favor of truth-telling in the presence of
16 As ¯v(p) is a convex combination of θH λ and θL λ, for anyx∈[ θL λ, θH
λ ] there existspsuch
3.4. ROBUSTNESS TO SAMPLING AS A REFINEMENT 81