A Geometric Approach to Dominant Strategy...

A Geometric Approach to Dominant StrategyImplementation

John A. Weymark

Vanderbilt University

References

Katherine Cuff, Sunghoon Hong, Jesse A. Schwartz, Quan Wen,and John A. Weymark, Dominant Strategy Implementation with aConvex Product Space of Valuations, Social Choice and Welfare39 (2012), 567–597.

Rakesh V. Vohra, Mechanism Design: A Linear ProgrammingApproach, Cambridge: Cambridge University Press, 2011.

Notation and Basic Definitions

N: finite set of individuals

Ω: finite set of outcomes

T i : i ’s characteristic type space for i ∈ N

T−i =∏

j∈N\i Tj : 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 isprivate 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 ≡ (P1, . . .Pn) : T i × T−i → Rn,

where P i : T i × T−i → Rn is the payment function forindividual i .

Given the other individuals’ reported types t−i ∈ T−i , the utilityof individual i with characteristic type t i ∈ T i and reported types 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 implementable ifthere exists a payment function P such that for all i ∈ N and allt−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 .

Given the allocation function G , for fixed i ∈ N and t−i ∈ T−i ,the characteristic graph TG (t−i ) is the complete directed graphwith nodes T i and arc length

d(s i , t i |t−i ) = v i (G (t i , t−i )|t i )− v i (G (s i , t−i )|t i )

for the directed arc (s i , t i ) from s i to t i .

Note that d(s i , t i |t−i ) is the increase in the valuation if the truecharacteristic type t is reported instead of the characteristic type s.This increase in valuation is not the increase in the utility becausethe payments have not been taken into account.

The Rochet–Rockafellar Theorem

For every integer k ≥ 2, a k-cycle in the characteristic graphTG (t−i ) is a sequence of arcs (t1, t2), . . . , (tk−1, tk), (tk , t1) whoselength is defined to be the sum of the lengths of the arcs in thecycle, i.e., d(t1, t2|t−i ) + · · ·+ d(tk−1, tk |t−i ) + d(tk , t1|t−i ).

Theorem 1 [Rockafellar (1970) – Rochet (1987)]

The allocation function G : T i × T−i → Ω is dominant strategyimplementable if and only if for every i ∈ N, t−i ∈ T−i , andinteger k ≥ 2, all k-cycles in the characteristic graph TG (t−i ) havenonnegative length.

A = a1, . . . , am is the finite set of attainable outcomes given t−i .

Note that m can depend on t−i .

Let Ra(t−i ) = t i ∈ T i | G (t i , t−i ) = a be the set ofcharacteristic types for i that induce outcome a with the allocationfunction G when the other individuals’ types are given by t−i .

By construction, Ra(t−i ) is nonempty for all a ∈ A(t−i )

Allocation Graphs

For the characteristic graph TG (t−i ), the corresponding allocationgraph ΓG (t−i ) is the complete directed graph that has A(t−i ) asthe set of nodes and `(a, b|t−i ) as the length of the directed arcfrom node a to node b, where for all distinct a, b ∈ A(t−i ),

`(a, b|t−i ) = inft i∈Rb(t−i )

[v i (b|t i )− v i (a|t i )

]= inf

t i∈Rb(t−i )

[v i (G (t i , t−i )|t i )− v(a|t i )

].

For any integer k ≥ 2, a k-cycle in the allocation graph ΓG (t−i ) isa sequence of arcs (a1, a2), . . . , (ak−1, ak), (ak , a1) whose length isdefined to be the sum of the lengths of its arcs in the cycle.

The Rockafellar–Rochet Theorem can be restated using allocationgraphs by simply substituting the allocation graph ΓG (t−i ) for thecharacteristic graph TG (t−i ) in the statement of Theorem 1.

In order to analyze dominant strategy implementability, withoutloss of generality, we can consider a fixed individual i ∈ N andfixed types t−i ∈ T−i of the other individuals. Let v = v i , t = t i ,T = T i , and suppress the dependence of A(t−i ), Ra(t−i ),d(s i , t i |t−i ), and `(a, b|t−i ) on t−i .

By fixing i and t−i , (G ,P) defines a single person mechanism(g , p) with allocation function g : T → A and payment functionp : T → R obtained by setting

g(t) = G (t, t−i ) and p(t) = P i (t, t−i ), ∀t ∈ T .

Note that g is surjective. The corresponding characteristic andallocation graphs are denoted by Tg and Γg , respectively.

For (g , p), the dominant strategy implementability condition is

v(g(t)|t)− p(t) ≥ v(g(s)|t)− p(s) ∀s, t ∈ T .

It follows that if g is dominant strategy implementable andg(s) = g(t), then p(s) = p(t) as well.

Theorem 2The following conditions for the allocation function g : T → A areequivalent:

1. g is dominant strategy implementable;

2. for every integer k ≥ 2, all k-cycles in the characteristic graphTg have nonnegative length;

3. for every integer k ≥ 2, all k-cycles in the allocation graph Γg

have nonnegative length.

The 2-Cycle Nonnegativity Condition

An allocation function g satisfies the characteristic graph2-cycle nonnegativity condition if

d(s, t) + d(t, s) ≥ 0, ∀s, t ∈ T , s 6= t

Note that this is equivalent to:

v(g(t)|t)− v(g(s)|t) ≥ v(g(t)|s)− v(g(s)|s), ∀s, t ∈ T , s 6= t.

That is, the increase in valuation obtained by replacing g(s) withg(t) is at least as large for t as for s. For this reason, the 2-cyclenonnegativity condition is also known as weak monotonicity.

An allocation function g satisfies the allocation graph 2-cyclenonnegativity condition if

`(a, b) + `(b, a) ≥ 0, ∀a, b ∈ A, a 6= b.

Theorem 3An allocation function g : T → A satisfies the characteristic graph2-cycle nonnegativity condition if and only if it satisfies theallocation graph 2-cycle nonnegativity condition.

The Saks–Yu Theorem

It follows straightforwardly from the incentive constraints that the2-cycle nonnegativity condition is a necessary condition for anallocation function g to be dominant strategy implementable.

Bikhchandani, Chatterji, Lavi, Mu’alem, Nisan, and Sen (2006)and Saks and Yu (2005) have identified restrictions on v underwhich the 2-cycle nonnegativity condition is sufficient for dominantstrategy implementability. Our results build on Saks and Yu.

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 acorresponding valuation type v t = (v ta1

, . . . , v tam) ∈ V, wherev ta = v(a|t) for all a ∈ A.

Note that if characteristic types s and t have the same associatedvaluation type v , then there is no loss of generality in identifyingthem. Henceforth, we assume that if s 6= t, then v s 6= v t . Withthis assumption, there is a unique t ∈ T associated with eachv ∈ V. Let tv denote the characteristic type associated with v .

Proposition [Saks–Yu (2005)]

If V is convex, then the allocation function g : T → A is dominantstrategy implementable if the 2-cycle nonnegativity condition issatisfied.

The following is the Saks–Yu Theorem.

Theorem 4 [Saks–Yu (2005)]

If V is convex, then the allocation function g : T → A is dominantstrategy implementable if and only if the 2-cycle nonnegativitycondition is satisfied.

Partitioning the Valuation Type Space

Recall that Ra is the set of characteristic types that the allocationfunction g maps into outcome a. The sets Ra for a ∈ A induce apartition of the valuation type space V. Our results are obtainedby investigating the geometry of this partition.

For all a, b ∈ A with a 6= b, the difference set for (a, b) is

Qab = v ∈ Rm|va − vb ≥ `(b, a).

Qab is a closed halfspace in Rm.

For all a ∈ A, letQa =

⋂b∈A\a

Qab.

Qa is a closed convex polyhedron in Rm.

va2

va1

−`(a2, a1)

Qa1 = Qa1a2

`(a1, a2)

Qa2 = Qa2a1

Qa1a2 = v ∈ R2 | va2 ≤ −`(a2, a1) + va1Qa2a1 = v ∈ R2 | va2 ≥ `(a1, a2) + va1.

2-cycle nonnegativity condition: `(a1, a2) ≥ −`(a2, a1)

va2

va1

`(a3, a1) + va3

@@I

Qa1 ∩Υ

`(a3, a2) + va3

@R

Qa2 ∩Υ

−`(a1, a3) + va3*

−`(a2, a3) + va3Z~

Qa3 ∩Υ

Υ = v ∈ R3 | va3 = va3.

Theorem 5For any allocation function g : T → A and any outcome a ∈ A, (i)for any characteristic type t ∈ Ra, the valuation type v t is inQa ∩ V and (ii) if g satisfies the 2-cycle nonnegativity condition,then for any valuation type v ∈ Qa ∩ V, the characteristic type tv

is in Ra.

Theorem 6If the allocation function g : T → A satisfies the 2-cyclenonnegativity condition, then for any characteristic type t ∈ Ra

and any valuation type v ′ ∈ V with v ′ ≥ v t for which v ′a > v ta andv ′b = v tb , the characteristic type tv

′is not in Rb.

Theorem 6 is a monotonicity result. Suppose that outcome a ischosen. If the valuation type increases in the value of outcome aand does not decrease in the valuation of any other outcome, thenwith the new valuation type, no outcome can be chosen whosevaluation has not changed.

Zero Length Cycles

The valuation type space V is a full-dimensional convex productspace if

V = ×a∈A〈La,Ua〉,

where for all a ∈ A, 〈La,Ua〉 is any type of interval of R withendpoints La and Ua for which La < Ua.

Interiority assumption: Qa ∩ V 6= ∅ for all a ∈ A.

Theorem 7Suppose that |A| ≥ 2. If (i) the allocation function g : T → Asatisfies the 2-cycle nonnegativity condition, (ii) the valuation typespace V is a full-dimensional convex product space, and (iii)Qa ∩ V 6= ∅ for all a ∈ A, then for every integer k ≥ 2, allk-cycles in the allocation graph Γg have zero length.

Theorem 7 is established by a series of lemmas.

Lemma 1Under the assumptions of Theorem 7, any 2-cycle in the allocationgraph Γg has zero length.

For the special case in which V is all of Rm, Lemma 1 has beenestablished in Lavi, Mu’alem, and Nisan (2009).

If there are only two outcomes (i.e., if V ⊆ R2), then theconclusion of Lemma 1 holds if the allocation rule g satisfies the2-cycle nonnegativity condition and the value type space V is aconvex.

When there are three or more outcomes, the conclusion of Lemma1 need not hold if the interiority assumption is not satisfied.

vb

va

−`(b, a)

Qa ∩Υ

∂Qab ∩ΥSSo

`(a, b)

Qb ∩Υ

∂Qba ∩ΥZ~

V ∩Υ

?

r(va, vb)

r(va, vb)

Lemma 2If all 2-cycles in the allocation graph Γg have zero length and all3-cycles in Γg have nonnegative length, then for every integerk ≥ 2, any k-cycle in Γg has zero length.

Consider any 3-cycle (a1, a2), (a2, a3), (a3, a1). Because all 3-cycleshave nonnegative length,

`(a1, a2) + `(a2, a3) + `(a3, a1) ≥ 0.

Because all 2-cycles have zero length, this inequality is equivalentto

−`(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 inequalityimplies that the 3-cycle (a1, a3), (a3, a2), (a2, a1) must have zerolength, 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 all2-cycles have zero length, the length of this 4-cycle is equal to thesum 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.

@@

@@

@@@I@

@@@@@@R

-

r

r

r

r

a1

a2

a3

a4

Lemma 3Under the assumptions of Theorem 7, any 3-cycle in the allocationgraph Γg has nonnegative length.

va2

va1

`(a3, a2) + L3

@R

`(a1, a2) `(a3, a1) + L3

V ∩Υ

?

Qa1 ∩Υ

Qa2 ∩Υ

Qa3 ∩Υ

Υ = v ∈ R3 | v3 = L3.

va2

va1

`(a3, a2) + L3

@R

`(a1, a2)

`(a3, a1) + L3@I

Qa1 ∩Υ

Qa2 ∩Υ

Qa3 ∩Υ

`(a3, a2) + L3 = `(a1, a2) + `(a3, a1) + L3 ↔

`(a1, a2) + `(a2, a3) + `(a3, a1) = 0.

Theorem 8If (a) |A| = 1 or (b) |A| ≥ 2, the valuation type space V is afull-dimensional convex product space, and Qa ∩ V 6= ∅ for alla ∈ A, then the following conditions for the allocation functiong : T → A are equivalent:

1. g is dominant strategy implementable;

2. for every integer k ≥ 2, all k-cycles in the allocation graph Γg

have nonnegative length;

3. for every integer k ≥ 2, all k-cycles in the characteristic graphTg have nonnegative length;

4. all 2-cycles in the allocation graph Γg have nonnegativelength;

5. all 2-cycles in the characteristic graph Tg have nonnegativelength;

6. all 2-cycles in the allocation graph Γg have zero length.

Affine Maximizers

For all i , let `(ai , ai ) = 0.

Theorem 9If all 2-cycles and all 3-cycles in the allocation graph Γg have zerolength, then

g(t) ∈ arg maxai∈A

v tai −1

m

m∑j=1

`(aj , ai )

, ∀t ∈ T .

To illustrate that proof strategy, suppose that m = 3 and that a1 isin the argmax. Then,

v ta1− 1

3[`(a2, a1) + `(a3, a1)] ≥ v ta2

− 1

3[`(a1, a2) + `(a3, a2)].

Because 2-cycles have zero length,

v ta1− v ta2

− 2

3`(a2, a1) ≥ 1

3[`(a3, a1)− `(a3, a2)].

So,

v ta1− v ta2

− `(a2, a1) ≥ 1

3[`(a3, a1) + `(a2, a3) + `(a1, a2)].

Because 3-cycles have zero length,

v ta1− v ta2

≥ `(a2, a1).

For all i , let

ki = − 1

m

m∑j=1

`(aj , ai )

Let x ⊕ y = maxx , y and x y = x + y .

Define the tropical polynomial F : V → R by setting

F (v) = k1 va1 ⊕ k2 va2 ⊕ · · · ⊕ km vam , ∀v ∈ V.

We have shown that g(t) is one of the outcomes inA = a1, . . . , am that maximizes F (v t) when all of the 2-cyclesand 3-cycles have zero length.

The tropical hypersurface of F is ∪ai∈A ∂Qai .

The allocation function G : T i × T−i → Ω is nonimposed ifG (T i × T−i ) = Ω

G is an affine maximizer if there exist n nonnegative numbersw1, . . . ,wn, not all of them equal to zero, and M numbers Ka,a ∈ Ω, such that

G (t i , t−i ) ∈ argmaxa∈Ω

n∑j=1

wjvj(a|t j) + Ka

, ∀(t i , t−i ) ∈ T i×T−i .

The affine maximizer G is unresponsive to irrelevant agents if forall i ∈ N for which wi = 0, G (s i , t−i ) = G (t i , t−i ) for alls i , t i ∈ T i and all t−i ∈ T−i .

Theorem 10(a) If an allocation function G : T i × T−i → Ω is an affinemaximizer that is unresponsive to irrelevant agents, then G isdominant strategy implementable.(b) Suppose that there are at least three outcomes in Ω, V i isunrestricted for all i ∈ N, and G : T i × T−i → Ω is a nonimposedallocation function. If G is dominant strategy implementable, thenG is an affine maximizer.

Part (a) is due to Mishra and Sen (2012). Part (b) is due toRoberts (1979).

Example

There is one unit of an indivisible good to be allocated to one oftwo individuals. Possession of the good creates a negativeexternality for the other individual.

a (resp. b) is the outcome in which individual 1 (resp. 2) gets thegood.

T 1 = R+ × R− and T 2 = R− × R+

v1(a|t1) = t1a and v1(b|t1) = t1

b for all t1 ∈ T 1.

v2(b|t2) = t2b and v2(a|t2) = t2

a for all t2 ∈ T 2.

A Vickrey auction has the following allocation and paymentfunctions:

G (t1, t2) =

a if t1

a − t1b ≥ t2

b − t2a

b if t1a − t1

b < t2b − t2

a

and payment function P : T 1 × T 2 → R2 is

P(t1, t2) =

(t2

b − t2a , 0) if t1

a − t1b ≥ t2

b − t2a

(0, t1a − t1

b) if t1a − t1

b < t2b − t2

a .

Each person has an adjusted value for the good given by t1a − t1

b

for person 1 and t2b − t2

a for person 2. Individuals bid their adjustedvalues and the good is awarded to the highest bidder (with a tiebroken in favour of individual 1) with the winner paying thesecond-highest bid (in this case, the other person’s bid) and theloser paying nothing. Note that G chooses the outcome thatmaximizes the sum of the valuations.

For any t2 ∈ T 2, A(t2) = a, b. We have

`(a, b|t2) = inft1∈Rb(t2)

[v(b|t1)− v(a|t1)]

= inft1b−t1

a>−[t2b−t2

a ][t1b − t1

a ] = −[t2b − t2

a ]

because Rb(t2) = t1 ∈ T 1 | t1a − t1

b < t2b − t2

a and we have

`(b, a|t2) = inft1∈Ra(t2)

[v(a|t1)−v(b|t1)] = inft1a−t1

b≥t2b−t2

a

[t1a−t1

b ] = t2b−t2

a

because Ra(t2) = t1 ∈ T 1 | t1a − t1

b ≥ t2b − t2

a. Thus, the only2-cycle in the allocation graph ΓG (t2) has zero length. Similarly,for any t1 ∈ T 1, A(t1) = a, b and the only 2-cycle in ΓG (t1) haszero length.

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