Ilya Segal

ENVELOPE THEOREMS FOR ARBITRARY CHOICE SETS

The standard envelope theorems apply to choice sets with convex and topological structure, providing sufficient conditions for the value function to be differentiable in a parameter and characterizing its derivative. This paper studies optimization with arbitrary choice sets and shows that the traditional envelope formula holds at any differentiability point of the value function. We also provide conditions for the value function to be, variously, absolutely continuous, left- and right-differentiable, or fully differentiable. These results are applied to mechanism design, convex programming, continuous optimization problems, saddle-point problems, problems with parameterized constraints, and optimal stopping problems.

The Communication Requirements of Efficient Allocations and Supporting Prices

We show that any communication finding a value-maximizing allocation in a private-information economy must also discover supporting prices (in general personalized and nonlinear). In particular, to allocate L indivisible items between two agents, a price must be revealed for each of the 2L-1 bundles. We prove that all monotonic prices for an agent must be used, hence exponential communication in L is needed. Furthermore, exponential communication is needed just to ensure a higher share of surplus than that realized by auctioning all items as a bundle, or even a higher expected surplus (for some probability distribution over valuations). When the utilities are submodular, efficiency still requires exponential communication (and fully polynomial approximation is impossible). When the items are identical, arbitrarily good approximation is obtained with exponentially less communication than exact efficiency.

Dynamic mechanism design: : a myersonian approach

We study mechanism design in dynamic quasilinear environments where private information arrives over time and decisions are made over multiple periods. We make three contributions. First, we provide a necessary condition for incentive compatibility that takes the form of an envelope formula for the derivative of an agents equilibrium expected payoff with respect to his current type. It combines the familiar marginal effect of types on payoffs with novel marginal effects of the current type on future ones that are captured by “impulse response functions.” The formula yields an expression for dynamic virtual surplus that is instrumental to the design of optimal mechanisms and to the study of distortions under such mechanisms. Second, we characterize the transfers that satisfy the envelope formula and establish a sense in which they are pinned down by the allocation rule (“revenue equivalence”). Third, we characterize perfect Bayesian equilibrium‐implementable allocation rules in Markov environments, which yields tractable sufficient conditions that facilitate novel applications. We illustrate the results by applying them to the design of optimal mechanisms for the sale of experience goods (“bandit auctions”).

Antitrust in Innovative Industries

We study the effects of antitrust policy in industries with continual innovation. A more protective antitrust policy may have conflicting effects on innovation incentives, raising the profits of new entrants, but lowering those of continuing incumbents. We show that the direction of the net effect can be determined by analyzing shifts in innovation benefit and supply holding the innovation rate fixed. We apply this framework to analyze several specific antitrust policies. We show that in some cases, holding the innovation rate fixed, as suggested by our comparative statics results, the tension does not arise and a more protective policy necessarily raises the rate of innovation.

Monopoly and Soft Budget Constraint

A benevolent government may decide to subsidize an unprofitable monopoly whose profits do not capture all the social surplus from its production. Anticipating this, the firm may underinvest in order to become unprofitable and extract state subsidies. The resulting welfare loss may exceed by many times the deadweight cost of monopoly pricing. Committing the firm to a price ceiling may soften its budget constraint and thus reduce welfare. Competition can harden budget constraints in industries in which free entry is socially excessive.

Coordination and discrimination in contracting with externalities: divide and conquer?

Abstract The paper studies bilateral contracting between N agents and one principal, whose trade with each agent generates externalities on other agents. It examines the effects of prohibiting the principal from (i) coordinating agents on her preferred equilibrium, and (ii) making different contracts available to different agents. These effects depend on whether an agent is more or less eager to trade when others trade more. The prohibitions reduce the aggregate trade in the former case, and have little or no effect in the latter case. The inefficiencies under different contracting regimes are linked to the sign of the relevant externalities, and are shown to be typically reduced by both prohibitions.

Collusion, Exclusion, and Inclusion in Random-Order Bargaining

This paper examines three types of contracts in a cooperative game solved by a random-order value (such as Shapley value). An exclusive contract delays the contribution of the excluded player j until the arrival of the excluding player i. It is profitable when j is complementary to other players in the absence of i. An inclusive contract brings the included player j forward to player is arrival. It is profitable when j is substitutable to other players in the presence of i. Finally, a collusive contract between i and j can be modeled as a proxy agreement under which i always brings j with him. The profit from collusion therefore equals to the sum of profits from exclusion and inclusion. It is positive when i reduces the complementarity between j and the other players.

The communication requirements of social choice rules and supporting budget sets

The paper examines the communication requirements of social choice rules when the (sincere) agents privately know their preferences. It shows that for a large class of choice rules, any communication verifying that an alternative is in the rule must reveal supporting budget sets for the agents such that the optimality of the proposed alternative to all agents within their respective budget set in itself verifies the alternative. We characterize the budget equilibria that are the minimally informative messages verifying a given choice rule. This characterization is used to identify the communication burden of choice rules, measured with the number of transmitted bits or real variables. Applications include efficiency in convex economies, exact or approximate surplus maximization in combinatorial auctions, the core in indivisible good economies, and stable many-to-one matchings.

Deferred-acceptance auctions and radio spectrum reallocation

Deferred-acceptance auctions choose allocations by an iterative process of rejecting the least attractive bid. These auctions have distinctive computational and incentive properties that make them suitable for application in some challenging environments, such as the planned US auction to repurchase television broadcast rights. For any set of values, any deferred acceptance auction with “threshold pricing”is weakly group strategy-proof, can be implemented using a clock auction, and leads to the same outcome as the complete-information Nash equilibrium of the corresponding paid-as-bid auction. A paid-asbid auction with a non-bossy bid-selection rule is dominance solvable if and only if it is a deferred acceptance auction. ∗The authors thank Alexey Kushnir as well as Mohammad Akbarpour, Piotr Dworczak, Tim Roughgarden, Inbal Talgam Cohen, Joshua Thurston-Milgrom, and Alex Wolitzky for their comments and suggestions. †Department of Economics, Stanford University, Stanford, CA 94305. The authors are members of the Auctionomics’team advising the US Federal Communications Commission concerning the design of the US “incentive auction.”This research analyzes some properties of proposals to which the authors have contributed. ‡Segal gratefully acknowledges the support of the Toulouse Network for Information Technology.

Auctions with severely bounded communication

We study auctions with severe bounds on the communication allowed: each bidder may only transmit t bits of information to the auctioneer. We consider both welfare- and profit-maximizing auctions under this communication restriction. For both measures, we determine the optimal auction and show that the loss incurred relative to unconstrained auctions is mild. We prove non-surprising properties of these kinds of auctions, e.g., that in optimal mechanisms bidders simply report the interval in which their valuation lies in, as well as some surprising properties, e.g., that asymmetric auctions are better than symmetric ones and that multi-round auctions reduce the communication complexity only by a linear factor.