Auction Design with Fairness Concerns: Subsidies vs. Set-Asides (joint with Mallesh Pai)

Government procurement and allocation programs often use subsidies and set-asides favoring small businesses and other target groups to address fairness concerns. These concerns are in addition to standard objectives such as efficiency and revenue. We study the design of the optimal mechanism for a seller concerned with efficiency, subject to a constraint to favor a target group. In our model, buyers' private values are determined by costly pre-auction investment. If the constraint is distributional, i.e. to guarantee that the target group wins `sufficiently often', then the constrained efficient mechanism is a flat subsidy. This is consistent with findings in the empirical literature. In contrast, if the constraint is to ensure a certain investment level by the target group, the optimal mechanism is a type dependent subsidy. In this case a set aside may be better than a flat or percentage subsidy.

Price Discrimination Through Communication (joint with Itai Sher)

We study a seller's optimal mechanism for maximizing revenue when the buyer may present evidence relevant to the buyer's value, or when different types of buyer have a differential ability to communicate. We introduce a dynamic bargaining protocol in which the buyer first makes a sequence of concessions in a cheap talk phase, and then at a time determined by the seller, the buyer presents evidence to support his previous assertions, and then the seller makes a take-it-or-leave-it offer. Our main result is that the optimal mechanism can be implemented as a sequential equilibrium of our dynamic bargaining protocol. Unlike the optimal mechanism to which the seller can commit, the equilibrium of the bargaining protocol also provides incentives for the seller to behave as required. We thereby provide a natural procedure whereby the seller can optimally price discriminate on the basis of the buyer's evidence

Optimal Dynamic Auctions, 2008, with Mallesh Pai.

We consider a dynamic auction problem motivated by the traditional single-leg, multi-period revenue management problem. A seller with C units to sell faces potential buyers with unit demand who arrive and depart over the course of T time periods. The time at which a buyer arrives, her value for a unit as well as the time by which she must make the purchase are private information. In this environment,
we derive the revenue maximizing Bayesian incentive compatible selling mechanism.

Axiomatic characterization of the absolute median on cube-free median networks ,
2006, with H. M. Mulder.

In Vohra (1996) a characterization of the absolute median of a tree network using three simple axioms is presented. This note extends that result from tree networks to cube-free median networks. A special case of such networks is the grid structure of roads found in cities equipped with the Manhattan metric.

Bounds on the Inefficiency of Sequential Auctions, 2006, with Junjik Bae, Eyal Beigman, Randall Berry and Micahel Honig.

We study the sequential second price auction of multiple units of a homogeneous commodity. It is well known that such auctions can have inefficient equilibria. For the case of two bidders, we show that the value of the allocation obtained in a the unique subgame perfect equilibrium is at least $1-e^{-1}$ of the value of the efficient allocation. Furthermore, we show that this bound is asymptotically tight.

Non Parametric Learnability of Income-Lipschitz Demand Functions, 2006, with Eyal Beigman.

A sequence of prices and demands are rationalizable if there exists a concave, continuous and monotone utility function such that the demands are the maximizers of the utility function over the budget set corresponding to the price. Afriat(1967) presented necessary and sufficient conditions for a finite sequence to be rationalizable. Varian(1982) and later Blundell et al. (2003, 2005) continued this line of work studying nonparametric methods to forecasts demand. Their methods do not implement any probabilistic model and therefore fall short of giving a general degree of confidence in the forecast. The present paper complements this line of research by introducing a statistical model and a measure of complexity through which we are able to study the learnability of classes of demand functions and derive a degree of confidence in the forecasts.

In this paper we develop a framework to study the learnability of real vector valued demand functions through observations on prices and demand. Our results give lower and upper bounds on the sample complexity of PAC learnability and show that the sample complexity of learning a class of vector valued functions with finite fat shattering dimension increases by a linear factor of the dimension. We show that classes of income-Lipschitz demand functions with global bounds on the Lipschitz constant have finite fat shattering dimension.

ADVANCED MATHEMATICAL ECONOMICS

This is the cover of a book based on lecture notes for a Ph.D. class I have taught. The book is published by Routledge .

The blurb for the book reads as follows:

This concise textbook presents students with all they need for advancing in mathematical economics. Higher level undergrads as well as postgrad students in mathematical economics will find this book to be extremely useful in the course of their development as economists.

The table of contents appears below. To my chagrin I have learned why books have mistakes. The errors uncovered so far can be found here. Debasis Mishra has been kind enough to compile a more extensive list and this can be found here.

Table of Contents
1 Things To Know

1.1 Sets
1.2 The Space We Work In
1.3 Facts from Real Analysis
1.4 Facts from Linear Algebra
1.5 Facts from Graph Theory
1.5.1 Directed Graphs

2 Feasibility

2.1 Fundamental Theorem of Linear Algebra
2.2 Linear Inequalities
2.3 Non-negative Solutions
2.4 The General Case
2.5 Application: Arbitrage
2.5.1 Black-Scholes Formula
2.6 Application: Co-operative Games
2.7 Application: Auctions

3 Convex Sets

3.1 Separating Hyperplane Theorem
3.2 Polyhedrons and Polytopes
3.3 Dimension of a Set
3.4 Properties of Convex Sets
3.5 Application: Linear Production Model

4 Linear Programming

4.1 Basic Solutions
4.2 Duality
4.3 Writing Down the Dual
4.4 Interpreting the Dual
4.5 Marginal Value Theorem
4.6 Application: Zero-Sum Games
4.7 Application: Afriat's Theorem
4.8 Integer Programming
4.9 Application: Efficient Assignment
4.10 Application: Arrow's Theorem

5 Non-linear Programming

5.1 Necessary Conditions for Local Optimality
5.2 Sufficient Conditions for Optimality
5.2.1 Concave Functions
5.2.2 Concave Programming
5.2.3 Constraint Qualifications
5.3 Envelope Theorem
5.4 An Aside on Utility Functions
5.5 Application: Market Games
5.6 Application: Principal-Agent Problem

6 Fixed Points

6.1 Banach Fixed Point Theorem
6.2 Brouwer Fixed Point Theorem
6.2.1 Sperner's Lemma
6.2.2 Application: Cake Cutting
6.2.3 Proof of Brouwer's Theorem
6.3 Application: Nash Equilibrium
6.4 Application: Equilibrium in Exchange Economies
6.5 Application: Hex
6.6 Kakutani's Fixed Point Theorem

7 Lattices and Supermodularity

7.1 Abstract Lattices
7.2 Application: Supermodular Games
7.3 Application: Transportation Problem
7.4 Application: Efficient Assignment and the Core
7.5 Application: Stable Matchings

8 Matroids

8.1 Introduction
8.2 Matroid Optimization
8.3 Rank Functions
8.4 Deletion and Contraction
8.5 Matroid Intersection and Partitioning
8.5.1 Application: Hall Marriage Theorem
8.6 Polymatroids
8.7 Application: Efficient Allocation with Indivisibilities
8.8 Application: Shannon Switching Game