Characterization of dominant strategy mechanisms in rich
domains
Juan Carlos Carbajal
School of Economics
University of Queensland
[email protected]
Rabee Tourky
School of Economics
University of Queensland
[email protected]
September 2008
Extended Abstract
Our aim in this paper is to provide a complete characterization of dominant strategy
mechanisms in terms of generalized Groves mechanisms in social choice settings quasilinear preferences, private values, large choice sets and rich preference domains. This
characterization is composed of two elements: (i) we show that a social choice rule (SCR)
is dominant strategy implementable if and only if it is the solution to a well defined social
problem; (ii) we show that the transfer scheme that implements a corresponding SCR is
unique, up to affine transformations.
Not every choice rule is dominant strategy implementable, at least not in interesting social choice frameworks. Thus, once such framework has been specified in terms
of the set of social alternatives, the set of agents, the domain of preferences and the
informational structure, one can ask the question of what are the properties of a SCR
that characterize implementation. On the other hand, if a given SCR is known to be
dominant strategy implementable, one can ask what properties, if any, have in common the possible transfer schemes that implement this SCR. The first question has been
addressed by Roberts [8], and more recently by Bikhchandani et al [1], Lavi et al [6],
and Saks and Yu [9]. All of these papers deal with a finite set of social alternatives
(e.g., auction-like environments). The second question, known in the mechanism design
literature as Revenue Equivalence (or Payoff Equivalence) has been addressed by Green
and Laffont [4], and more recently by Milgrom and Segal [7], Chung and Olszewski [3],
Heydenreich et al [5] and Carbajal [2].
Previous work in the literature addresses these two fundamental issues of mechanism design separately. A remarkable exception is Roberts [8], who showed that if the
set of social alternatives is finite and the domain of preferences is unrestricted, then
any SCR satisfying a weak monotonicity condition called PAD (see below) is implementable, and uniquely so, by generalized Groves transfers. We follow Roberts and
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address both questions simultaneously, providing a characterization for dominant strategy direct mechanisms in terms of generalized Groves mechanisms. The main difference
between Roberts [8] and our paper is that we deal with large choice sets and rich (but
not unrestricted) preference domains. Our motivation is rooted in the fact that, while
the assumption of a finite choice set covers a great deal of interesting economic problems
(auction settings being a prominent example), it leaves out many equally interesting
problems, e.g., the provision of public goods, the design of pollution permits, the allocation of water or other divisible natural resources among different populations, etc. With
that in mind, we deal with choice sets that are uncountably infinite.
The formal details of our social choice setting are as follows: there is a set X = [0, 1]
of social choices or alternatives, which we refer to as the choice set, and a finite group of
agents N = {1, . . . , n}, each of whom has quasi-linear preferences over social alternatives
and money. Given (x, ti ) ∈ X × <, the utility of agent i ∈ N is represented by:
ui (x, ti ) = vi (x) + ti ,
where the individual valuation function vi : X → < is private information. Agent i’s type
space (valuation space) is denoted by Vi ; we let V = ×Vi and V−i = ×j6=i Vj . A social
choice environment E is then the tuple {N , X , (Vi )i∈N }. This will constitute the basic
framework of our analysis.
A direct mechanism is a pair Γ = (ψ, τ ), where the social choice rule (SCR) ψ maps
V into X and the transfer scheme τ maps V into <n . We focus on dominant strategy
implementation.
Definition 1. A direct mechanism Γ = (ψ, τ ) is said to be dominant strategy (incentive
compatible) if for every agent truth-telling is a dominant strategy; i.e., for all i ∈ N , all
v−i ∈ V−i and all vi , vi0 ∈ Vi :
vi (ψ(vi , v−i )) + τi (vi , v−i ) ≥ vi (ψ(vi0 , v−i )) + τi (vi0 , v−i ).
Throughout this paper we consider choice rules that are onto; i.e., given ψ : V → X ,
we assume that for every choice x ∈ X there exists a type profile v = (v1 , . . . , vn ) ∈ V
such that x = ψ(v). One shall not consider a priori that every SCR is implementable
in dominant strategies. In fact, as we shall see, if the domain of individual preferences
is sufficiently rich, the only implementable choice rules are solutions to a well-defined
social problem.
Definition 2. A SCR ψ : V → X is said to be an affine maximizer with respect to
(k, F ) if there exist a vector of individual weights k = (k1 , . . . , kn ) ∈ <n+ , k 6= 0, and an
adjustment function F : X → R such that for all v ∈ V :
nXn
o
ψ(v) ∈ arg max
ki vi (x) + F (x) .
i=1
x∈X
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If ψ is an affine maximizer choice rule then it can be implemented by means of a
generalized Groves transfer scheme. Together, these two objects define what we refer to
as generalized Groves mechanisms.
Definition 3. A mechanism Γ = (ψ, τ ) is called generalized Groves if ψ is an affine
maximizer with respect to (k, F ), and if for each agent i ∈ N with ki > 0, transfers are
given by:
τi (v) =
o
1 nX
kj vj (ψ(v)) + F (ψ(v)) + h̄i (v−i ) ,
j6=i
ki
∀ v ∈ V.
Observe that, in the above definition, there is no need to specify payments to an
agent j with kj = 0, since the choices made under ψ will not depend on j’s reports.
Let BC[0, 1] denote the space of bounded, upper semi-continuous measurable functions
defined on [0, 1]. We shall use the following definition of rich domain for individual
preferences over social alternatives.
Definition 4. The domain of preferences in a social choice setting E = {N , X , (Vi )i∈N }
is said to be rich if Vi = BC[0, 1] for every agent i ∈ N .
Thus, under the Rich Domain (RD) assumption, V = (BC[0, 1])n . We can now
state our main result, which puts certain limits not only on the class of transfers that
implement SCRs (generalized Groves transfers), but also on the class of SCRs that can
be implemented in dominant strategies (affine maximizers), in social choice environments
with rich domains and uncountable choice sets.
Theorem 5. Let E = {N , X , (Vi )i∈N } be a social choice environment satisfying the RD
assumption. Let Γ = (ψ, τ ) be a direct mechanism with an onto choice rule. Then Γ is
dominant strategy if and only if it is a generalized Groves mechanism.
We divide the proof of Theorem 5 in several propositions. Following Roberts [8],
we start by showing that dominant strategy implementation implies a monotonicity
condition (PAD) on the choice rules (Proposition 7). We then show that if a choice
rule satisfies PAD then it must be an affine maximizer (Proposition 12). To do so,
we associate to every pair of social alternatives (x, y) a half-space in <n , the space
where the value difference v(x) − v(y) lies. We then employ PAD to show that, in the
case of rich domains, these half spaces are equivalent after a certain normalization has
been imposed on them. This normalization gives us the weights (k1 , . . . , kn ) and the
adjustment function F needed to construct the affine maximization problem.1 The next
step is to show that an affine maximizer SCR is dominant strategy implementable by
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This clever technique was first presented by Roberts [8] for the case of finite choice sets and unrestricted domains, and later refined by Lavi et al [6] (still with finite choice sets and unrestricted domains).
The structure of our proof is closer to this last work.
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generalized Groves transfers (Proposition 13). Finally, we show that any transfer scheme
that implements an affine maximizer SCR must be a generalized Groves transfer scheme
(Proposition 14), and thus any dominant strategy mechanism must be a generalized
Groves mechanism. This implies as a corollary the important Revenue Equivalence
principle: in rich domains, any two dominant strategy mechanisms sharing the same
SCR generate transfers that are equal, up to an affine transformation. For this last step
of our characterization result, we adapt an argument originally presented by Green and
Laffont [4]. PAD is defined next.
Definition 6. The SCR ψ : V → X satisifies the Positive Association of Difference
condition (PAD) if for any pair of type profiles v, v 0 in V : v 0 (x) − v(x) v 0 (y) − v(y)
for all y ∈ X , y 6= x, and x = ψ(v) imply that x = ψ(v 0 ).
In the above definition, v(x) = (v1 (x), . . . , vn (x)), v(y) = (v1 (y), . . . , vn (y)), and we
write v(x) v(y) if vi (x) > vi (y) for all i ∈ N .
Proposition 7. If ψ is dominant strategy implementable then ψ satisfies PAD.
It shall be noted that this result holds for any social choice setting E, not just environments that satisfy the rich domain assumption. On the other hand Proposition 12,
which shows that if ψ satisfies PAD then ψ is an affine maximizer, makes extensive use
of the RD assumption. We arrive at this conclusion using Lemma 8 through 11, all of
which assume that ψ satisfies PAD.
Lemma 8. Let v, v 0 ∈ V be two distinct type profiles. If x = ψ(v) and v 0 (x) − v(x) v 0 (y) − v(y) for some y ∈ X , y 6= x, then it must be that y 6= ψ(v 0 ).
We now construct the appropriate half-spaces in <n for each pair of social alternatives
x 6= y. Recall that ψ is onto, hence for any alternative x ∈ X , there exists a type profile
v ∈ V such that x = ψ(v). Take any other y ∈ X , and write v(x) − v(y) = α. We define
the set H(x, y) ⊆ <n by
H(x, y) := {α ∈ <n : ∃ v ∈ V such that v(x) − v(y) = α, x = ψ(v)}
Let Pα = {α + <n++ } denote the strictly positive orthant translated by α. We shall
see that if α belongs to H(x, y) then Pα is a subset of H(x, y). This allows one to derive
important properties of the sets H(x, y), which we do in the next lemma.
Lemma 9. For any x, y ∈ X , x 6= y, and any α ∈ <n , the following holds:
9.1. H(x, y) is a nonempty subset of <n .
9.2. If α ∈ H(x, y) then Pα ∩ H(x, y) = Pα and −Pα ∩ H(y, x) = ∅.
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9.3. If α 6∈ H(x, y) then −α ∈ H(y, x).
9.4. If α and α0 belong to H(x, y), then co(Pα ∪ Pα0 ) ⊆ H(x, y).
In addition, for any z ∈ X and z 6= y, then the following holds:
9.5. If α ∈ H(x, y) and α0 ∈ H(y, z) then Pα+α0 ⊆ H(x, z).
Lemma 9 provides the basic structure of the sets H(x, y). We proceed to look for a
common structure for the sets H(x, y) and H(w, z). We employ the following notation:
for any real number q, let q = (q, . . . , q) ∈ <n . For any x, y ∈ X , x 6= y, define
q(x, y) := inf{q ∈ < : q ∈ H(x, y)}. These lower bounds are then used to normalize the
different sets H(x, y), H(w, z).
Lemma 10. Let x, y, z be different social alternatives in X . Then the following holds:
10.1. −∞ < q(x, y) < +∞.
10.2. q(x, y) = −q(y, x).
10.3. q(x, z) = q(x, y) + q(y, z).
We proceed to shift the set H(x, y) using q(x, y) = (q(x, y), . . . , q(x, y)) ∈ <n . For
any x 6= y, define I(x, y) := H(x, y) − q(x, y), and denote I ◦ (x, y) the interior of I(x, y).
Lemma 11. For any x, y, w, z in X , with x 6= y and w 6= z, the following holds:
1. Given any 0, ∈ I ◦ (x, y) and 0 6∈ I ◦ (x, y).
2. I ◦ (x, y) = I ◦ (w, z) =: I ◦ .
3. I ◦ is a convex subset of <n .
Note that we can conclude, from the preceding lemma, that the closure of I ◦ is a
convex set in <n containing the origin. Applying the Separating Hyperplane Theorem,
there exists a vector k = (k1 , . . . , kn ) ∈ <n+ , k 6= 0, such that if α ∈ cl(I ◦ ) then k · α ≥ 0.
Now fix an arbitrary x0 ∈ X and define the function F : X → < by
F (x) = k · q(x0 , x),
where we set q(x0 , x0 ) = 0. We use the pair (k, F ) to arrive at our desired result.
Proposition 12. If ψ satisfy PAD then ψ is an affine maximizer with respect to (k, F );
i.e., for every v ∈ V :
k · v(ψ(v)) + F (ψ(v)) ≥ k · v(y) + F (y),
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∀ y ∈ X.
It is now immediate to show that if the SCR ψ is an affine maximizer, then it is
dominant strategy implementable by means of generalized Groves transfers.
Proposition 13. Let ψ be an affine maximizer SCR with respect to (k, F ), where k =
(k1 , . . . , kn ) ∈ <n+ , k 6= 0, and F : X → <. Let τ : V → <n be a generalized Groves
transfer scheme. Then the direct mechanism Γ = (ψ, τ ) is dominant strategy.
Taken together, Propositions 7, 12, and 13 show that with rich domains and large
choice sets, a SCR is dominant strategy implementable iff it satisfies PAD iff it is an
affine maximizer. Moreover, Proposition 13 tells us that we can use generalized Groves
transfers to implement any affine maximizer SCR. To complete the characterization of
dominant strategy mechanisms in terms of generalized Groves mechanisms, it remains
to show that any transfer scheme implementing a SCR must be generalized Groves.
Proposition 14. Let ψ be a dominant strategy implementable SCR. If τ : V → <n
implements ψ, then τ must be a generalized Groves transfer scheme.
The proof of Theorem 5 follows readily from Propositions 7, 12, 13, and 14. Note, as
we mentioned earlier, that as a corollary of our characterization result we have that any
dominant strategy implementable choice rule satisfies the Revenue Equivalence principle. More specifically, if E is a social choice environment satisfying the RD assumption,
then given any two dominant strategy mechanisms with a common SCR, transfers implementing this choice rule are equal up to an affine transformation, since they must be
generalized Groves transfers.
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