A characterization of combinatorial demand C. Chambers F. Echenique - - PowerPoint PPT Presentation

A characterization of combinatorial demand

• C. Chambers
• F. Echenique

UC San Diego Caltech

Montreal Nov 19, 2016

This paper

Literature on matching (e.g Kelso-Crawford) and combinatorial auctions (e.g Milgrom): D(p) = argmax{v(A) −

• a∈A

pa : A ⊆ X} (*) When is * true? What is the behavioral content of the combined assumptions of rationality and quasilinearity?

Chambers-Echenique Combinatorial demand

Notation

◮ Let X be a finite set (of items). ◮ Let S be the set of all nonempty subsets of 2X ◮ (so the empty set is not in S, but {∅} is). ◮ Identify A ⊆ X with 1A ∈ RX. ◮ If p ∈ RX then p, A = x∈A px.

Chambers-Echenique Combinatorial demand

Demand

A demand function is D : RX

++ → S

s.t. ∃¯ p ∈ RX

++ with D(p) = {∅} for all p ≥ ¯

p. (¯ p a choke price)

Chambers-Echenique Combinatorial demand

Demand

D is quasilinear rationalizable if ∃ v : 2X → R s.t D(p) = argmaxA⊆Xv(A) − p, A

Chambers-Echenique Combinatorial demand

Suppose D is QL-rationalizable

Let A ∈ D(p) and B ∈ D(q). v(A) − p, A ≥ v(B) − p, B v(B) − q, B ≥ v(A) − q, A. Thus: p − q, A − B ≤ 0.

Chambers-Echenique Combinatorial demand

Suppose D is QL-rationalizable

Let A ∈ D(p) and B ∈ D(q). v(A) − p, A ≥ v(B) − p, B v(B) − q, B ≥ v(A) − q, A. Thus: p − q, A − B ≤ 0. The law of demand!

Chambers-Echenique Combinatorial demand

Demand

A demand function D

◮ satisfies the law of demand if for all p, q ∈ RX ++, and all

A ∈ D(p) and B ∈ D(q), p − q, A − B ≤ 0;

◮ is upper hemicontinuous if, ∀p ∈ RX ++, ∃ nbd V of p s.t.

D(q) ⊆ D(p) when q ∈ V .

Chambers-Echenique Combinatorial demand

Main result

Theorem

A demand function is quasilinear rationalizable iff it is upper hemicontinuous and satisfies the law of demand.

Chambers-Echenique Combinatorial demand

Identification

Theorem

For any quasilinear rationalizable D, there is a unique monotone v : 2X → R for which v(∅) = 0 which rationalizes D. Utility is identified up to an additive constant.

Chambers-Echenique Combinatorial demand

Monotone rationalization

D is monotone, concave, quasilinear rationalizable (MCQ-rationalizable) if ∃ a monotone, concave g : RX

+ → R s.t

v(A) = g(1A), and D(p) = argmax{v(A) − p, A : A ⊆ X}.

Corollary

If a demand function is quasilinear rationalizable, then it is MCQ-rationalizable.

Chambers-Echenique Combinatorial demand

Proof ideas

D(p) = argmaxA⊆Xv(A) − p, A If A ∈ D(p) then we want p to be the “gradient of v at A.” Can recover v by “integrating” over p.

Chambers-Echenique Combinatorial demand

Chambers-Echenique Combinatorial demand

Cyclic monotonicity

D satisfies cyclic monotonicity if, for all n (using summation mod n),

n

• i=1

pi, Ai − Ai+1 ≤ 0, where Ai ∈ D(pi), for all sequences {pi}n

i=1.

Chambers-Echenique Combinatorial demand

Cyclic monotonicity

Define: v(A) = infp1, A − A1 + . . . + ¯ p, Ak − ∅, inf is taken over all finite seq. (pi, Ai)k

i=1 with Ai ∈ D(pi).

Chambers-Echenique Combinatorial demand

Cyclic monotonicity

Define: v(A) = infp1, A − A1 + . . . + ¯ p, Ak − ∅, inf is taken over all finite seq. (pi, Ai)k

i=1 with Ai ∈ D(pi).

Observe, by CM, − {p1, A − A1 + . . . + ¯ p, Ak − ∅} + p, A − ∅ ≤ 0. So v(A) is well defined (and ≥ 0).

Chambers-Echenique Combinatorial demand

Let A ∈ D(p) and B ⊆ X (B ∈ D(RX

++) need a different arg.

• therwise).

By defn. of v, v(B) ≤ p, B − A + v(A). Thus v(A) − p, A ≥ v(B) − p, B.

Chambers-Echenique Combinatorial demand

Let A ∈ D(p) and B ⊆ X (B ∈ D(RX

++) need a different arg.

• therwise).

By defn. of v, v(B) ≤ p, B − A + v(A). Thus v(A) − p, A ≥ v(B) − p, B. Proof that if A ∈ D(p) and B / ∈ D(p) then v(A) − p, A > v(B) − p, B requires more.

Chambers-Echenique Combinatorial demand

D satisfies condition ♠ if ∀p and B / ∈ D(p) ∃A ∈ D(p) and p′ s.t A ∈ D(p′) and p′, A − B > p, A − B.

Lemma

If D is upper hemicontinuous, then it satisfies condition ♠.

Chambers-Echenique Combinatorial demand

Cyclic monotonicity

Lemma

If D satisfies cyclic monotonicity, and condition ♠, then it is quasilinear rationalizable. Based on ideas in Rochet/Rockafellar (but ♠ plays a technical role).

Chambers-Echenique Combinatorial demand

Lemma

A demand function satisfies cyclic monotonicity if it satisfies the law of demand. Follows from recent results in mech. design (Lavi, Mu’alem, and Nisan; Saks and Yu; and Ashlagi, Braverman, Hassidim, and Monderer).

Chambers-Echenique Combinatorial demand

Related literature

◮ Rochet/Rockafeller ◮ Brown and Calsamiglia ◮ Sher and Kim ◮ Lavi, Mu’alem, and Nisan; ◮ Saks and Yu; ◮ Ashlagi, Braverman, Hassidim, and Monderer

Chambers-Echenique Combinatorial demand

Conclusions

◮ Quasilinear rational demand is a ubiquitous assumption. ◮ Our result is the first characterization in terms of observable

behavior.

◮ Identification enables welfare analysis. ◮ New use for recent results in mech. design.

Chambers-Echenique Combinatorial demand