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

Money as a mechanism in a Bewley economy

We study what features an economic environment might possess, such that it would be Pareto efficient for the exchange of goods in that environment to be conducted on spot markets where those goods trade for money. We prove a conjecture that is essentially due to Bewley [1980,1983]. Monetary spot trading is nearly efficient when there is only a single perishable good (or a composite commodity) at each date and state of the world; random shocks are idiosyncratic, privately observed, and temporary; markets are competitive; and the agents are very patient. This result is a fairly close analogue, for trade using outside, fiat money, of a recent characterization by Levine and Zame [2002] of environments in which spot trade using inside money, in the form of one-period debt payable in a commodity, is nearly Pareto efficient. We also study a example where expansionary monetary mechanism Pareto dominates laissez-faire or contractionary monetary mechanism in an environment with impatient agents.Money ; Monetary theory

Computing Equilibria in Markets with Budget-Additive Utilities

We present the first analysis of Fisher markets with buyers that have budget-additive utility functions. Budget-additive utilities are elementary concave functions with numerous applications in online adword markets and revenue optimization problems. They extend the standard case of linear utilities and have been studied in a variety of other market models. In contrast to the frequently studied CES utilities, they have a global satiation point which can imply multiple market equilibria with quite different characteristics. Our main result is an efficient combinatorial algorithm to compute a market equilibrium with a Pareto-optimal allocation of goods. It relies on a new descending-price approach and, as a special case, also implies a novel combinatorial algorithm for computing a market equilibrium in linear Fisher markets. We complement these positive results with a number of hardness results for related computational questions. We prove that it is NP-hard to compute a market equilibrium that maximizes social welfare, and it is PPAD-hard to find any market equilibrium with utility functions with separate satiation points for each buyer and each good.Comment: 21 page

Network Cournot Competition

Cournot competition is a fundamental economic model that represents firms competing in a single market of a homogeneous good. Each firm tries to maximize its utility---a function of the production cost as well as market price of the product---by deciding on the amount of production. In today's dynamic and diverse economy, many firms often compete in more than one market simultaneously, i.e., each market might be shared among a subset of these firms. In this situation, a bipartite graph models the access restriction where firms are on one side, markets are on the other side, and edges demonstrate whether a firm has access to a market or not. We call this game \emph{Network Cournot Competition} (NCC). In this paper, we propose algorithms for finding pure Nash equilibria of NCC games in different situations. First, we carefully design a potential function for NCC, when the price functions for markets are linear functions of the production in that market. However, for nonlinear price functions, this approach is not feasible. We model the problem as a nonlinear complementarity problem in this case, and design a polynomial-time algorithm that finds an equilibrium of the game for strongly convex cost functions and strongly monotone revenue functions. We also explore the class of price functions that ensures strong monotonicity of the revenue function, and show it consists of a broad class of functions. Moreover, we discuss the uniqueness of equilibria in both of these cases which means our algorithms find the unique equilibria of the games. Last but not least, when the cost of production in one market is independent from the cost of production in other markets for all firms, the problem can be separated into several independent classical \emph{Cournot Oligopoly} problems. We give the first combinatorial algorithm for this widely studied problem

Market Equilibrium in Exchange Economies with Some Families of Concave Utility Functions

We present explicit convex programs which characterize the equilibrium for certain additively separable utility functions and CES functions. These include some CES utility functions that do not satisfy weak gross substitutability.Exchange economy, computation of equilibria, convex feasibility problem

Computing Equilibrium in Matching Markets

Market equilibria of matching markets offer an intuitive and fair solution for matching problems without money with agents who have preferences over the items. Such a matching market can be viewed as a variation of Fisher market, albeit with rather peculiar preferences of agents. These preferences can be described by piece-wise linear concave (PLC) functions, which however, are not separable (due to each agent only asking for one item), are not monotone, and do not satisfy the gross substitute property-- increase in price of an item can result in increased demand for the item. Devanur and Kannan in FOCS 08 showed that market clearing prices can be found in polynomial time in markets with fixed number of items and general PLC preferences. They also consider Fischer markets with fixed number of agents (instead of fixed number of items), and give a polynomial time algorithm for this case if preferences are separable functions of the items, in addition to being PLC functions. Our main result is a polynomial time algorithm for finding market clearing prices in matching markets with fixed number of different agent preferences, despite that the utility corresponding to matching markets is not separable. We also give a simpler algorithm for the case of matching markets with fixed number of different items