Simple Mechanisms For Agents With Complements

We study the efficiency of simple auctions in the presence of complements. [DMSW15] introduced the single-bid auction, and showed that it has a price of anarchy (PoA) of $O(\log m)$ for complement-free (i.e., subadditive) valuations. Prior to our work, no non-trivial upper bound on the PoA of single bid auctions was known for valuations exhibiting complements. We introduce a hierarchy over valuations, where levels of the hierarchy correspond to the degree of complementarity, and the PoA of the single bid auction degrades gracefully with the level of the hierarchy. This hierarchy is a refinement of the Maximum over Positive Hypergraphs (MPH) hierarchy [FFIILS15], where the degree of complementarity $d$ is captured by the maximum number of neighbors of a node in the positive hypergraph representation. We show that the price of anarchy of the single bid auction for valuations of level $d$ of the hierarchy is $O(d^2 \log(m/d))$, where $m$ is the number of items. We also establish an improved upper bound of $O(d \log m)$ for a subclass where every hyperedge in the positive hypergraph representation is of size at most 2 (but the degree is still $d$). Finally, we show that randomizing between the single bid auction and the grand bundle auction has a price of anarchy of at most $O(\sqrt{m})$ for general valuations. All of our results are derived via the smoothness framework, thus extend to coarse-correlated equilibria and to Bayes Nash equilibria.Comment: Proceedings of the 2016 ACM Conference on Economics and Computatio

Supermodular mechanism design

This paper introduces a mechanism design approach that allows dealing with the multiple equilibrium problem, using mechanisms that are robust to bounded rationality. This approach is a tool for constructing supermodular mechanisms, i.e. mechanisms that induce games with strategic complementarities. In quasilinear environments, I prove that if a social choice function can be implemented by a mechanism that generates bounded strategic substitutes - as opposed to strategic complementarities - then this mechanism can be converted into a supermodular mechanism that implements the social choice function. If the social choice function also satisfies some efficiency criterion, then it admits a supermodular mechanism that balances the budget. Building on these results, I address the multiple equilibrium problem. I provide sufficient conditions for a social choice function to be implementable with a supermodular mechanism whose equilibria are contained in the smallest interval among all supermodular mechanisms. This is followed by conditions for supermodular implementability in unique equilibrium. Finally, I provide a revelation principle for supermodular implementation in environments with general preferences.Implementation, mechanisms, learning, strategic complementarities, supermodular games

Nash Social Welfare Approximation for Strategic Agents

The fair division of resources is an important age-old problem that has led to a rich body of literature. At the center of this literature lies the question of whether there exist fair mechanisms despite strategic behavior of the agents. A fundamental objective function used for measuring fair outcomes is the Nash social welfare, defined as the geometric mean of the agent utilities. This objective function is maximized by widely known solution concepts such as Nash bargaining and the competitive equilibrium with equal incomes. In this work we focus on the question of (approximately) implementing the Nash social welfare. The starting point of our analysis is the Fisher market, a fundamental model of an economy, whose benchmark is precisely the (weighted) Nash social welfare. We begin by studying two extreme classes of valuations functions, namely perfect substitutes and perfect complements, and find that for perfect substitutes, the Fisher market mechanism has a constant approximation: at most 2 and at least e1e. However, for perfect complements, the Fisher market does not work well, its bound degrading linearly with the number of players. Strikingly, the Trading Post mechanism---an indirect market mechanism also known as the Shapley-Shubik game---has significantly better performance than the Fisher market on its own benchmark. Not only does Trading Post achieve an approximation of 2 for perfect substitutes, but this bound holds for all concave utilities and becomes arbitrarily close to optimal for Leontief utilities (perfect complements), where it reaches $(1+\epsilon)$ for every $\epsilon > 0$. Moreover, all the Nash equilibria of the Trading Post mechanism are pure for all concave utilities and satisfy an important notion of fairness known as proportionality

Mechanism design with private communication

We investigate the consequences of assuming "private" communication between the principal and each of his agents in an otherwise standard mechanism design setting.Doing so simplifies significantly optimal mechanisms and institutions. Moreover, it restores continuity of the principal's payoff and of the optimal mechanism with respect to the information structure while still maintaining the useful role of correlation to better extract the agents' information rent. We first prove a "Revelation Principle with private communication" that characterizes the set of allocations implementable under private communication by means of simple "non-manipulability constraints". We also demonstrate a "Taxation Principle" which helps drawing some links between private communication and limited commitment on the principal's side. Equipped with those tools, we derive optimal non-manipulable mechanisms in various environments (unrelated projects, auctions, team production).MECHANISM DESIGN;PRIVATE COMMUNICATION

Informational Substitutes

We propose definitions of substitutes and complements for pieces of information ("signals") in the context of a decision or optimization problem, with game-theoretic and algorithmic applications. In a game-theoretic context, substitutes capture diminishing marginal value of information to a rational decision maker. We use the definitions to address the question of how and when information is aggregated in prediction markets. Substitutes characterize "best-possible" equilibria with immediate information aggregation, while complements characterize "worst-possible", delayed aggregation. Game-theoretic applications also include settings such as crowdsourcing contests and Q\&A forums. In an algorithmic context, where substitutes capture diminishing marginal improvement of information to an optimization problem, substitutes imply efficient approximation algorithms for a very general class of (adaptive) information acquisition problems. In tandem with these broad applications, we examine the structure and design of informational substitutes and complements. They have equivalent, intuitive definitions from disparate perspectives: submodularity, geometry, and information theory. We also consider the design of scoring rules or optimization problems so as to encourage substitutability or complementarity, with positive and negative results. Taken as a whole, the results give some evidence that, in parallel with substitutable items, informational substitutes play a natural conceptual and formal role in game theory and algorithms.Comment: Full version of FOCS 2016 paper. Single-column, 61 pages (48 main text, 13 references and appendix

Common Agency Equilibria with Discrete Mechanisms and Discrete Types

This paper characterizes the equilibrium sets of an intrinsic common agencygame with discrete types and direct revelation mechanisms. After presentinga general algorithm to find the pure-strategy equilibria of this game, we use itto characterize these equilibria when the two principals control activitieswhich are complements in the agent’s objective function. Some of thoseequilibria may entail allocative inefficiency. For the case of substitutes, wedemonstrate non-existence of such equilibria with direct mechanisms, butexistence may be obtained with indirect mechanisms. Finally, we relax theequilibrium concept and analyze quasi-equilibria. We show that existence isthen guaranteed and characterize the corresponding allocations.

The Strategic Value of Incomplete Contracting in a Competing Hierarchies Environment

We explore the strategic value of quantity forcing contracts in a competing manufacturer-retailer hierarchies environment under both adverse selection and moral hazard. Manufacturers dealing with (exclusive) competing retailers may prefer to leave contracts silent on retail prices, whenever other aspects of the retailers’ activity remain nonveri.able. Two effects are at play once one moves from retail price maintenance to quantity forcing. First, restricting the number of screening instruments available to manufacturers has a detrimental effect on their pro.ts as it leaves more possibilities to retailer for getting information rents. Second, such restriction may provide manufacturers with strategic power in that it affects downstream competition. Under some conditions related to the severity of the adverse selection problem and the nature of externalities across retailers, the latter effect may rationalize the use of quantity forcing contracts in a game of competing hierarchies. RPM may be either detrimental or bene.cial to welfare depending upon the type of non-market externalities that retailers impose on each other.Asymmetric Information, Competing Hierarchies, Incomplete Contracts, Quantity Forcing, Resale Price Maintenance, Vertical Contracting

Factive and nonfactive mental state attribution

Factive mental states, such as knowing or being aware, can only link an agent to the truth; by contrast, nonfactive states, such as believing or thinking, can link an agent to either truths or falsehoods. Researchers of mental state attribution often draw a sharp line between the capacity to attribute accurate states of mind and the capacity to attribute inaccurate or “reality-incongruent” states of mind, such as false belief. This article argues that the contrast that really matters for mental state attribution does not divide accurate from inaccurate states, but factive from nonfactive ones

Optimal Combinatorial Mechanism Design

We consider an optimal mechanism design problem with several heterogeneous objects and interdependent values. We characterize ex post incentives using an appropriate monotonicity condition and reformulate the problem in such a way that the choice of an allocation rule can be separated from the choice of the payment rule. Central to our analysis is the formulation of a regularity condition, which gives a recipe for the optimal mechanism. If the problem is regular, then an optimal mechanism can be obtained by solving a combinatorial allocation problem in which objects are allocated in a way to maximize the sum of "virtual" valuations. We identify conditions that imply regularity for two nonnested environments using the techniques of supermodular optimization.