Dominant Strategy Implementation with
a Convex Product Space of Valuations
Katherine Cuff1 Sunghoon Hong2 Jesse A. Schwartz3
Quan Wen 2 John A. Weymark2
1 McMaster
University, Canada
2 Vanderbilt
3 Kennesaw
University, USA
State University, USA
November 2009
Introduction
Consider a direct mechanism that assigns an outcome and a
payment vector to each reported characteristic type vector, where
the set of possible types is the Cartesian product of the possible
types for each individual. Suppose that for a given characteristic
type vector, utilities are quasilinear (linear in the payment). For a
given individual i and given types of the other individuals, we can
define a complete directed graph whose nodes are the types of
person i and the length of the directed arc joining type s i to t i is
the gain in the value of the outcome obtained by person i when he
is of type t i if he truthfully reports t i instead of s i . Note that the
payments are being ignored in this construction.
Rockafellar (1970) and Rochet (1987) have shown that the
allocation function assigning outcomes to types is dominant
strategy implementable (DISC) if and only if for each individual i
and characteristic type vector of the other individuals, the
corresponding graph does not contain a cycle with k directed arcs
whose length is negative for any finite positive integer k.
When there are a finite number m outcomes, for a given individual
i and characteristic type vector of the other individuals, we can
i
equivalently describe i’s type t i by the vector v t in Rm whose jth
component is the value of the jth outcome when he is of type t i .
This set of valuation types can be used to define a new graph
whose nodes are the set of alternatives and whose directed arcs
have lengths determined by the lengths in the original graph. The
Rockafellar–Rochet Theorem can be restated in terms of k-cycles
in this graph.
Bikhchandani, Chatterji, Lavi, Mu’alem, Nisan, and Sen (2006)
have identified a fairly abstract domain richness condition on the
sets of these valuation vectors (one set for each individual i and
given characteristic types of the other individuals) for which it is
sufficient for DISC that all 2-cycles in each of the corresponding
valuation graphs have nonnegative length.
The BCLMNS article has lead to a rapidly growing literature that
applies graph-theoretic and linear programming techniques to
analyze dominant strategy implementation. This literature is
surveyed and extended in the lecture notes of Mishra (2009) and
Vohra (2009).
Saks and Yu (2005) have shown that when there is a finite number
of outcomes, if for each individual i and each type vector for the
other individuals, the set of possible valuation vectors for i is
convex, then it is sufficient for DISC that all 2-cycles in each of the
corresponding valuation graphs have nonnegative length.
Extensions and variants of the Saks–Yu Theorem have been
established by Archer and Kleinberg (2008) and Monderer (2008).
We strengthen the assumptions of Saks and Yu by assuming that
the set of possible valuation vectors for given i and given types of
the other individuals is the product of convex intervals of R and
that this set satisfies a regularity condition that ensures that for
each outcome a, there exists an open set of types for i that results
in a being chosen. With our assumptions, we show that if all
2-cycles in the corresponding valuation graph have nonnegative
length, then in fact all k-cycles in this graph have zero length for
any positive integer k. An implication of this result is that a
k-cycle condition that is necessary for DISC is binding for all
positive integer k.
In proving our main result, we identify and exploit some geometric
properties of this problem.
Notation and Basic Definitions
N: finite set of individuals
Ω: finite set of outcomes
T i : i’s characteristic type space for i ∈ N
Q
T −i = j∈N\{i} T j : the characteristic type space of individuals
other than i
T i × T −i : the characteristic type space
(t i , t −i ) ∈ T i × T −i : the characteristic type profile, which is
private information
v i : Ω × T i → R: i’s valuation function
A direct mechanism consists of
an allocation function G : T i × T −i → Ω and
a payment function P ≡ (P 1 , . . . P n ) : T i × T −i → Rn ,
where P i : T i × T −i → Rn is the payment function for
individual i.
Given the other individuals’ reported types t −i ∈ T −i , the utility
of individual i with characteristic type t i ∈ T i and reported type
s i ∈ T i is
v i (G (s i , t −i )|t i ) − P i (s i , t −i ).
An allocation function G is dominant strategy incentive
compatible (DSIC) if there exists a payment function P such that
for all i ∈ N and all t −i ∈ T −i ,
v i (G (t i , t −i )|t i ) − P i (t i , t −i ) ≥ v i (G (s i , t −i )|t i ) − P i (s i , t −i ),
∀s i , t i ∈ T i .
By fixing the individual i ∈ N and the characteristic types t −i of
the other individuals and by letting t = t i , and T = T i , we can
define a single person mechanism with allocation function
g : T → A and payment function p : T → R by setting
g (t) = G (t, t −i ) and p(t) = P i (t, t −i ) for all t ∈ T ,
where
A = {a1 , . . . , am } is the finite set of attainable outcomes given
t −i .
That is, A = {a ∈ Ω|g (t) = a for some t ∈ T }.
Note that g is onto.
Let Ra = {t ∈ T |g (t) = a} be the set of characteristic types for i
that induce outcome a with the allocation function g .
By construction, Ra is nonempty for all a ∈ A.
The Rochet–Rockafellar Theorem
For (g , p), by letting v = v i , the DISC condition is
v (g (t)|t) − p(t) ≥ v (g (s)|t) − p(s) ∀s, t ∈ T .
(1)
For the mechanism g , the characteristic network Tg is the
complete directed graph with nodes T and arc length
d(s, t) = v (g (t)|t) − v (g (s)|t)
for the directed arc (s, t) from s to t.
Note that d(s, t) is the increase in the valuation if the true
characteristic type t is reported instead of the characteristic type s.
This increase in valuation is not the increase in the utility because
the payments have not been taken into account.
For any positive integer k, a k-cycle in the characteristic network
Tg is a sequence of arcs (t1 , t2 ), . . . , (tk−1 , tk ), (tk , t1 ) whose
length is defined to be the sum of the lengths of the arcs in the
cycle, i.e., d(t1 , t2 ) + · · · + d(tk−1 , tk ) + d(tk , t1 ).
Theorem 1 [Rockafellar (1970) – Rochet (1987)]
An allocation function g : T → A satisfies DISC if and only if the
corresponding characteristic network Tg does not contain a k-cycle
with negative length for any finite positive integer k.
It is a straightforward implication of DISC that the k-cycle
condition
d(t1 , t2 ) + · · · + d(tk−1 , tk ) + d(tk , t1 ) ≥ 0
holds for all finite positive integer k.
If g satisfies DISC and g (s) = g (t), we must have p(s) = p(t) as
well.
As a consequence, if DISC is satisfied, then there exists a payment
function p̃ : A → R that assigns payments to outcomes for which
g (t) ∈ arg max v (a|t) − p̃(a),
a∈A
∀t ∈ T .
(2)
The existence of a payment function p̃ satisfying (2) is known as
the taxation principle.
Allocation Networks
Given the allocation function g , the corresponding allocation
network Γg is the complete directed graph which has A as the set
of nodes and `(a, b) as the length of the directed arc from node a
to node b, where
`(a, b) = inf [v (b|t) − v (a|t)] = inf [v (g (t)|t) − v (a|t)] ,
t∈Rb
t∈Rb
∀a, b ∈ A.
For any positive integer k, a k-cycle in the allocation network Γg
is a sequence of arcs (a1 , a2 ), . . . , (ak−1 , ak ), (ak , a1 ) whose length
is defined to be the sum of the lengths of its arcs in the cycle, i.e.,
`(a1 , a2 ) + · · · + `(ak−1 , ak ) + `(ak , a1 ).
Restated in terms of allocation networks, the Rockafellar–Rochet
Theorem says:
Theorem 2
An allocation function g : T → A satisfies DISC if and only if the
corresponding allocation network Γg does not contain a k-cycle
with negative length for any finite positive integer k.
The 2-Cycle Condition
An allocation function g satisfies the 2-cycle condition if in the
characteristic network Tg ,
d(s, t) + d(t, s) ≥ 0,
∀s, t ∈ T .
(3)
Note that (3) is equivalent to:
v (g (t)|t) − v (g (s)|t) ≥ v (g (t)|s) − v (g (s)|s),
∀s, t ∈ T .
That is, the increase in valuation obtained by replacing g (s) with
g (t) is at least as large for t as for s. For this reason, the 2-cycle
condition is also known as weak monotonicity.
The allocation network Γg satisfies the 2-cycle condition if
`(a, b) + `(b, a) ≥ 0,
∀a, b ∈ A.
(4)
Proposition 1
An allocation function g satisfies the 2-cycle condition (3) if and
only if the corresponding allocation network Γg satisfies the 2-cycle
condition (4).
The Saks–Yu Theorem
It follows straightforwardly from the incentive constraints that the
2-cycle condition is a necessary condition for an allocation function
g to satisfy DISC.
Bikhchandani, Chatterji, Lavi, Mu’alem, Nisan, and Sen (2006)
and Saks and Yu (2005) have identified restrictions on v under
which the 2-cycle condition is sufficient for DISC.
The BCLMNS condition is a rather abstract domain richness
condition.
V = {v ∈ Rm |v = (v (a1 |t), . . . , v (am |t)) for some t ∈ T }.
V is i’s valuation type space (given t −i ).
Each characteristic type t ∈ T has associated with it a
corresponding valuation type v t = (vat1 , . . . , vatm ) ∈ V, where
vat = v (a|t) for all a ∈ A.
Note that if characteristic types s and t have the same associated
valuation type v , then there is no loss of generality in identifying
them. Henceforth, we assume that if s 6= t, then v s 6= v t . With
this assumption, there is a unique t ∈ T associated with each
v ∈ V. Let t v denote the characteristic type associated with v .
Theorem 3 [Saks–Yu (2005)]
If V ⊆ Rm is convex, then an allocation function g : T → A
satisfies DISC if and only if the corresponding allocation network
Γg satisfies the 2-cycle condition.
A version of the Saks-Yu Theorem is implicit in Jehiel, Moldovanu,
and Stacchetti (1999).
Generalizations of this theorem that allow for infinite sets of
alternatives appear in Archer and Kleinberg (2008) and Monderer
(2008).
Partitioning the Valuation Type Space
Recall that Ra is the set of characteristic types that the allocation
function g maps into outcome a. The sets Ra for a ∈ A induce a
partition of the valuation type space V. Our results are obtained
by investigating the geometry of this partition.
For all a, b ∈ A with a 6= b, let
Qab = {v ∈ Rm |va − vb ≥ `(b, a)}.
Qab is a closed halfspace in Rm .
For all a ∈ A, let
Qa =
\
b∈A\{a}
Qab .
Qa is a closed convex polyhedron in Rm .
Proposition 2
For any allocation function g and any outcome a ∈ A, (i) for any
characteristic type t ∈ Ra , the corresponding valuation type v t
must be in Qa ∩ V and (ii) if the allocation function g satisfies the
2-cycle condition, then for any value type v in the interior of
Qa ∩ V, the corresponding characteristic type t v must be in Ra .
Proposition 3
If an allocation function g satisfies the 2-cycle condition, then for
any characteristic type t ∈ Ra and any valuation type v 0 ∈ V with
0
v 0 ≥ v t for which va0 > vat and vb0 = vbt , the characteristic type t v
corresponding to v 0 is not in Rb .
Proposition 3 is a monotonicity result. Suppose that outcome a is
chosen. If the valuation type increases in the value of outcome a
and does not decrease in the valuation of any other outcome, then
with the new valuation type, no outcome can be chosen whose
valuation has not changed.
The Main Theorem
Theorem 4
If (i) the allocation function g satisfies the 2-cycle condition, (ii)
the value type space V is a convex product subset of Rm , and (iii)
Qa ∩ V has a nonempty interior for all a ∈ A, then for all positive
integer k, any k-cycle in the corresponding allocation network Γg
must have zero length.
Thus, if assumptions (ii) and (iii) are satisfied, then not only is the
2-cycle condition sufficient for an allocation function to satisfy
DISC, which was known from the Saks–Yu Theorem, it is also the
case that all the k-cycle conditions bind.
Theorem 4 is established by a series of lemmas.
Lemma 1
Under the assumptions of Theorem 4, any 2-cycle in the
corresponding allocation network Γg must have zero length.
For the special case in which V is all of Rm , Lemma 1 has been
established in Lavi, Mu’alem, and Nisan (2009).
If there are only two outcomes (i.e., if V ⊆ R2 ), then the
conclusion of Lemma 1 holds if the allocation rule g satisfies the
2-cycle condition and the value type space V is a convex.
When there are three or more outcomes, the conclusion of Lemma
1 need not hold if the assumption that Qa ∩ V has a nonempty
interior for all a ∈ A is not satisfied.
Lemma 2
If all 2-cycles in the allocation network Γg have zero length and all
3-cycles in Γg have nonnegative length, then for any finite positive
integer k, any k-cycle in Γg must have zero length.
Consider any 3-cycle (a1 , a2 ), (a2 , a3 ), (a3 , a1 ). Because all 3-cycles
have nonnegative length,
`(a1 , a2 ) + `(a2 , a3 ) + `(a3 , a1 ) ≥ 0.
(5)
Because all 2-cycles have zero length, (5) is equivalent to
−`(a2 , a1 ) − `(a3 , a2 ) − `(a1 , a3 ) ≥ 0,
or, equivalently,
`(a1 , a3 ) + `(a3 , a2 ) + `(a2 , a1 ) ≤ 0.
Because all 3-cycles have nonnegative length, the last inequality
implies that the 3-cycle (a1 , a3 ), (a3 , a2 ), (a2 , a1 ) must have zero
length, which implies that the original 3-cycle
(a1 , a2 ), (a2 , a3 ), (a3 , a1 ) must have zero length.
Consider any 4-cycle (a1 , a2 ), (a2 , a3 ), (a3 , a4 ), (a4 , a1 ). Because all
2-cycles have zero length, the length of this 4-cycle is equal to the
sum of the lengths of the following 3-cycles:
(a1 , a2 ), (a2 , a4 ), (a4 , a1 )
(a2 , a3 ), (a3 , a4 ), (a4 , a2 )
both of which have length zero.
Induction is used to prove the lemma for larger values of k.
Lemma 3
Under the assumptions of Theorem 4, any 3-cycle in the
corresponding allocation network Γg must have nonnegative length.