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 ) − p a : A ⊆ X } (*) a ∈ A 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 2 X ◮ (so the empty set is not in S , but { ∅ } is). ◮ Identify A ⊆ X with 1 A ∈ R X . ◮ If p ∈ R X then � p , A � = � x ∈ A p x . Chambers-Echenique Combinatorial demand

Demand A demand function is D : R X ++ → S p ∈ R X s.t. ∃ ¯ ++ with D ( p ) = { ∅ } for all p ≥ ¯ p . (¯ p a choke price) Chambers-Echenique Combinatorial demand

Demand D is quasilinear rationalizable if ∃ v : 2 X → R s.t D ( p ) = argmax A ⊆ X v ( 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 ∈ R X ++ , and all A ∈ D ( p ) and B ∈ D ( q ), � p − q , A − B � ≤ 0; ◮ is upper hemicontinuous if, ∀ p ∈ R X ++ , ∃ 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 : 2 X → 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 : R X + → R s.t v ( A ) = g (1 A ), 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 ) = argmax A ⊆ X v ( 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 � � p i , A i − A i +1 � ≤ 0 , i =1 where A i ∈ D ( p i ), for all sequences { p i } n i =1 . Chambers-Echenique Combinatorial demand

Cyclic monotonicity Define: v ( A ) = inf � p 1 , A − A 1 � + . . . + � ¯ p , A k − ∅ � , inf is taken over all finite seq. ( p i , A i ) k i =1 with A i ∈ D ( p i ). Chambers-Echenique Combinatorial demand

Cyclic monotonicity Define: v ( A ) = inf � p 1 , A − A 1 � + . . . + � ¯ p , A k − ∅ � , inf is taken over all finite seq. ( p i , A i ) k i =1 with A i ∈ D ( p i ). Observe, by CM, − {� p 1 , A − A 1 � + . . . + � ¯ p , A k − ∅ �} + � p , A − ∅ � ≤ 0 . So v ( A ) is well defined (and ≥ 0). Chambers-Echenique Combinatorial demand

Let A ∈ D ( p ) and B ⊆ X ( B ∈ D ( R X ++ ) need a different arg. otherwise). 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 ( R X ++ ) need a different arg. otherwise). 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 ∈ D ( p ) ∃ A ∈ D ( p ) and p ′ s.t ∀ p and B / 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