This note constructs an efficient mechanism for finding the best candidate for a committee from a sequence of potential candidates. Committee members have independent private values information about the quality of the candidate. The mechanism selects the best candidate according to the standard utilitarian welfare criterion. Furthermore, the mechanism can be modified to have a balanced budget.Search, Committtees, Voting, Mechanism design, Dynamic pivot mechanism

Dirk Bergemann

Juuso Valimaki

English

Research Papers in Economics

 
 
EFFICIENT SEARCH BY COMMITTEE 
 
 
By 
 
Dirk Bergemann and Juuso Välimäki 
 
 
 
January 2011 
 
 
 
 
 
 
 
COWLES FOUNDATION DISCUSSION PAPER NO. 1782 
 
 
 
 
 
 
 
 
 
 
 
 
 
COWLES FOUNDATION FOR RESEARCH IN ECONOMICS 
YALE UNIVERSITY 
Box 208281 
New Haven, Connecticut 06520-8281 
 
 http://cowles.econ.yale.edu/  Ecient Search by Committee
Dirk Bergemann∗ Juuso V alim aki†
January 2011
Abstract
This note constructs an ecient mechanism for nding the best
candidate for a committee from a sequence of potential candidates.
Committee members have independent private values information about
the quality of the candidate. The mechanism selects the best candi-
date according to the standard utilitarian welfare criterion. Further-
more, the mechanism can be modied to have a balanced budget.
Jel Classification: C72, D43, D82, D83, H41
Keywords: Search, Committtees, Voting, Mechanism Design, Dy-
namic Pivot Mechanism.
Department of Economics, Yale University, New Haven, U.S.A.,
dirk.bergemann@yale.edu.
yDepartment of Economics, Aalto University School of Economics, and HECER, ju-
uso.valimaki@aalto..
11 Introduction
A recent paper by Albrecht, Anderson & Vroman (2010) investigated equi-
libria in models of search by committees. As is natural for many committees,
that paper does not allow side payments to be made during the search pro-
cess. Nevertheless the committee members have often other trade-os even
if outright monetary transfers are not allowed. Consider the example of an
academic job applicant. Dierent research groups may derive dierent pay-
os from hiring the candidate. Often there are other decisions within the
department that can be adjusted to compensate a given group. At the very
least, the department is also likely to hire in coming years. It is possible
to provide some incentives for the committee members by linking current
decisions to future ones.
Motivated by considerations of this type, we make the opposite assump-
tion in this note. We assume that utility is perfectly transferable between the
committee members. In other words, we assume that the preferences of the
committee members are quasilinear in transfers, and we adopt a mechanism
design approach to the committee decisions. It seems natural to require that
the budget be balanced in the committee. In other words, the members can-
not get outside nancial help to resolve their conict and they do not burn
money.
The dynamic pivot mechanism dened in Bergemann & V alim aki (2010)
can be used in the current setting to support the ecient choice of a candi-
date. The transfers required to implement this mechanism are easily com-
puted. In fact, these computations are much easier than the equilibrium com-
putations in the model without transfers. The committee decision problem
has similarities to a public goods provision problem. The chosen candidate
will have an eect on the welfare of all committee members. The opportunity
cost of not continuing the search can be thought of as the cost of the chosen
candidate. Nevertheless, this cost is just an opportunity cost and therefore
not reected when balancing the budget. Hence in contrast to the problem
of providing a public good with a real production cost, we show that the
2dynamic pivot mechanism generates a surplus. By usual arguments, this
surplus can be rebated to the committee members in a manner that balances
the budget.
The insights in this note suggest that even if the committees cannot use
transfers, search can be conducted more eectively than by using simple vot-
ing rules. In more nely tuned deliberations, trade-os between committee
members' preferences are possible. This can be done e.g. by putting less
weight in future deliberations on the opinions of the committee members
that are blocking the selection of the current candidate.
In Section 2, we describe the model. In Section 3, we dene the dynamic
pivot mechanism for this setting and show that an ecient mechanism with
a balanced budget exists. In Section 4, we discuss some extensions of the
result.
2 Model
2.1 Uncertainty and Payos
A committee consisting of N symmetric members i ∈ {1;:::N} must choose
one alternative from a countable sequence of candidates indexed by their time
of appearance t = 0;1;::: All committee members discount future with the
same discount factor . A candidate t yields a payo xi
t ∈ R to committee
member i if t is selected. Following Albrecht, Anderson & Vroman (2010) , we
assume that the xi
t are i.i.d. across the i and t and that they are distributed
according to a distribution function F: Denote the sum of payos amongst
the committee members by
z =
N ∑
i=1
x
i
t:
By independence, z has distribution GN (·) that can be readily computed by
the convolution formula. The sum of the payos of all committee members
except i is denoted by z−i and again by independence, this random variable
3has distribution GN−1 (·) for all values of xi
t: We denote the realization of of
the vector of payos in period t by xt and the sequence of such vectors by
x = (x0;x1;:::).
The candidate selection is denoted by d = (d0;d1;:::) where dt = 1 indi-
cates that candidate t was selected. Since only one candidate can be chosen,
we have
dt = 1 ⇒ dt′ = 0 for all t ̸= t
′:
Given the linear structure of payos, it is without loss of generality to consider
only deterministic allocations.
The committee members can also be asked to make transfers in each
period t. Let p = (p1
t;:::;pN
t ) denote the vector of transfers in period t and
let p = (p0;p1;:::) be the sequence of transfer vectors
The expected payo for player i from an allocation (d;p) is given by
v
i (x;d) − p
i
t := E
∞ ∑
t=0

t[x
i
tdt − p
i
t];
where the expectation is taken over the realizations xi
t.
Social welfare in the model is dened by the utilitarian welfare criterion
on the committee members ignoring the transfers for the moment.
W (x;d) =
N ∑
i=1
v
i (x;d):
2.2 Direct Mechanism
Each xi
t is observed only by committee member i: We construct a direct
revelation mechanism where each committee member reports in each period
her payo xi
t from this candidate. We denote these reports by ri
t: The hiring
decisions dt ∈ {0;1} must be decided based on the vector of reports rt rather
than on the true payos xt:
In order to give the committee members incentives to report truthfully,
i.e. to choose ri
t = xi
t, we assume that they can be asked to make or receive
4transfers pi
t based on the vector of realized reports. We denote the prole of
reports in period t by rt and we let r = (r0;r1;:::):
A public history in period t is a sequence of decisions and reports in
the past periods, ht := (r0;d0;r1;d1;:::;rt−1;dt−1): Each player i also knows
her own past realized types hi
t := (xi
0;r0;d0;xi
1;r1;d1;:::;xi
t−1;rt−1;dt−1): A
reporting strategy of player i is a sequence of functions

i
t : H
i
t × R → R:
Following (Bergemann & V alim aki 2010), a direct dynamic mechanism is
a pair of functions (dt (rt;tt);pt (rt;ht)): The mechanism is ex-post incentive
compatible if for all i, all t; and all xi
t
x
i
t ∈ argmax
ri
t
[
x
i
tdt
(
r
i
t;x
−i
t ;ht
)
− p
i
t
(
r
i
t;x
−i
t ;ht
)]
+ Exs>t
[
∞ ∑
s=t+1

s−tx
i
sds
(
x
i
s;x
−i
s ;hs
)
− p
i
s
(
x
i
s;x
−i
s ;hs
)
| ht;xt
]
:
Notice that here we are assuming that committee members other than
i report truthfully. By the one-shot deviation principle, it is sucient to
check the incentives in period t only for player i: Notice also that in this ex
post criterion, all the values of the xi
t are assumed to be known up to t, but
expectations are taken over these variables for s > t.
3 The Dynamic Pivot Mechanism
In order to compute the dynamic pivot mechanism we need to solve two
social welfare maximization problems for the case of complete information.
Let d∗ solve
max
d
E
[
N ∑
i=1
∞ ∑
s=t

s−t[x
i
sds
(
x
i
s;x
−i
s
)
|ht;xt
]
;
5and let d∗−i solve
max
d
E
[
∑
j̸=i
∞ ∑
s=t

s−t[x
i
sds
(
x
i
s;x
−i
s
)
|ht;xt
]
:
It is a standard exercise in search theory to show that the optimal decision
rule for each of these cases is described by a cuto policy that requires hiring
the rst candidate for whom zt ≥ z∗k; where k is the number of players, and
z∗k solves
(1 − )z
∗k
=
∫ ∞
z−z∗k
(
z − z
∗k)
dG
k (z):
Denote the optimal values by obtained using the optimal cuto strategies
after history ht and at current type prole xt by W (xt;ht) and W −i (
x
−i
t ;ht
)
:1
The marginal contribution of player i is
M
i (xt;ht) := W (xt;ht) − W
−i (
x
−i
t ;ht
)
:
The transfers are ow variables in the sense that they are paid each period
and the quantities on the right hand side are stock variables reecting the
players' payos in the dynamic game. Hence it is clear that the dynamic pivot
mechanism must account for this. For this reason, we dene the dynamic
marginal contributions
mi (xt;ht) := M
i (xt) − EM
i (xt+1;ht+1):
Suppose that the transfers in the model are dened as follows:
v
i (
x
i
t;d
∗
t
)
− p
∗i
t
(
r
i
t;r
−i
t ;ht
)
= mi (rt;ht) (1)
Bergemann & V alim aki (2010) show that the mechanism dened by
(d∗
t (rt;ht);p∗
t (rt;ht)) is ex post incentive compatible and satises ex post
1Even though the optimal decision rule does not depend on histories, we are keeping
it in the notation to cover also the case discussed in the extensions where the committee
members learn about their own distribution of candidate types.
6individual rationality. The payo from not participating in the mechanism is
understood to be the payo to i resulting from the choice rule d∗−i: Therefore
we may use the realized values xi
t for the reported values in the formulas that
follow.
The next step is to calculate the payments in the dynamic pivot mecha-
nism. As in any pivot mechanism, pi
t = 0 if d∗
t = d
∗−i
t : Hence we have to cover
only cases where d∗
t < d
∗−i
t or d∗
t > d
∗−i
t : In the rst, we have by denition
zt < z
∗N and z
−i
t > z
∗N−1:
Therefore
W (xt;ht) = z
∗N and W
−i (xt;ht) = z
−i
t :
Since d∗
t = 0; we have
v
i (
x
i
t;d∗t
)
= 0; EW (xt+1;ht+1) = z
∗N;
and
EW
−i (xt+1;ht+1) = z
∗N−1:
Substituting into equation (1), one can solve
p
∗i
t
(
x
i
t;z
−i
t ;ht
)
= z
−i
t − z
∗N + z
∗N − z
∗N > 0:
The last inequality follows from the assumption that
z
−i
t > z
∗N−1:
Consider next the case where
zt > z
∗N and z
−i
t < z
∗N−1:
Since d∗
t = 1; it is easy to see that
W (xt;ht) = zt and W
−i (xt;ht) = z
∗N−1:
For the next period,
EW (xt+1;ht+1) = EW
−i (xt+1;ht+1) = 0:
7Therefore
p
∗i
t
(
x
i
t;z
−i
t ;ht
)
= zt − z
∗N−1 > 0:
Since the payments are non-negative for all players at all realization of
the xi
t; this calculation shows that the dynamic pivot mechanism generates
a surplus.
4 Discussion
Budget Balance The dynamic pivot mechanism constructed is ex post
incentive compatible, individually rational and results in an ex post bud-
get surplus. In order to balance the budget, the transfers could be modi-
ed to bring the transfers to the level of the associated expected externality
mechanism. This would nevertheless come at a cost. In the dynamic pivot
mechanism, the committee members get as their payo their marginal con-
tribution. In the modied balanced budget mechanism, this would no longer
be true. Hence there could be a potential trade-o between budget balance
and correct incentives to pay private participation costs in the committee.
Correlated Committee Members Since the dynamic pivot mechanism
satises ex post incentive constraints, the same construction remains valid
if the xi
t are correlated across i. The only adjustment needed to cover this
case is to note that the distribution GN(zn) is no longer a convolution of the
marginal distributions.
Learning the Payo Distributions The dynamic pivot mechanism can
be modied to cover situations where the xi
t are correlated across t.. An
example of this would be models where each committee member draws payos
from a conditionally i.i.d. draws from independent distributions F
(
· : 
i)
and
where the they learn in a Bayesian fashion about the 
i: The main dierence
in this setting would be that the optimal decisions now depend in a non-
trivial manner on the histories.
8Interdependent Values Sometimes it is natural to think that the commit-
tee members have independent payo relevant signals yi
t on the candidates.
To satisfy the appropriate monotonicity requirements, it must be assumed
that the signal of member i has a larger impact on i′s payo than on the
payos of the other committee members. For example, we could have:
x
i
t = y
i
t + 
∑
j̸=i
y
j
t for some  ≤ 1:
In this case, we could construct a dynamic version of the generalized VCG
mechanism to implement the ecient decision rule based on the reported
signals yi
t. The main dierence compared to the dynamic pivot mechanism
would be in the calculation of the transfers. In the generalized VCG mech-
anism, the transfer of a pivotal player i that blocks an otherwise acceptable
candidate is calculated based on the societal payos from the highest yi
t that
overturns the decision of the other members. Similarly the transfer of the
pivotal player that forces the acceptance of an otherwise rejected candidate
would be computed based on the lowest signal overturning the decision.
It should perhaps be noted that even in the pure common values setting
where  = 1, voting games perform quite badly whereas the generalized VCG
mechanism implements the ecient allocation without requiring transfers.
More Complicated Committee Decisions The dynamic pivot mech-
anism can be computed for committee decisions when multiple candidates
must be selected. The insights gained from the simplest model give some sug-
gestions for further work along this dimension even in cases where transfers
are ruled out. If the committee must select two candidates from a sequence
of potential candidates, an optimal decision rule can discipline those commit-
tee members that block the appointment of the current candidate by putting
less weight on their reports when deciding on future candidates. With two
candidates, the member that insists on getting her favorite candidate chosen
for the rst slot can be made to pay an implicit price by discounting her
opinions when lling the second slot.
9References
Albrecht, J., A. Anderson & S. Vroman. 2010. \Search by committee." Jour-
nal of Economic Theory 145(4):1386 { 1407.
Bergemann, D. & J. V alim aki. 2010. \The Dynamic Pivot Mechanism."
Econometrica 78(2):771{789.
10

Efficient Search by Committee

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