Fast Algorithms for Online Stochastic Convex Programming

We introduce the online stochastic Convex Programming (CP) problem, a very general version of stochastic online problems which allows arbitrary concave objectives and convex feasibility constraints. Many well-studied problems like online stochastic packing and covering, online stochastic matching with concave returns, etc. form a special case of online stochastic CP. We present fast algorithms for these problems, which achieve near-optimal regret guarantees for both the i.i.d. and the random permutation models of stochastic inputs. When applied to the special case online packing, our ideas yield a simpler and faster primal-dual algorithm for this well studied problem, which achieves the optimal competitive ratio. Our techniques make explicit the connection of primal-dual paradigm and online learning to online stochastic CP.Comment: To appear in SODA 201

First-Come-First-Served for Online Slot Allocation and Huffman Coding

Can one choose a good Huffman code on the fly, without knowing the underlying distribution? Online Slot Allocation (OSA) models this and similar problems: There are n slots, each with a known cost. There are n items. Requests for items are drawn i.i.d. from a fixed but hidden probability distribution p. After each request, if the item, i, was not previously requested, then the algorithm (knowing the slot costs and the requests so far, but not p) must place the item in some vacant slot j(i). The goal is to minimize the sum, over the items, of the probability of the item times the cost of its assigned slot. The optimal offline algorithm is trivial: put the most probable item in the cheapest slot, the second most probable item in the second cheapest slot, etc. The optimal online algorithm is First Come First Served (FCFS): put the first requested item in the cheapest slot, the second (distinct) requested item in the second cheapest slot, etc. The optimal competitive ratios for any online algorithm are 1+H(n-1) ~ ln n for general costs and 2 for concave costs. For logarithmic costs, the ratio is, asymptotically, 1: FCFS gives cost opt + O(log opt). For Huffman coding, FCFS yields an online algorithm (one that allocates codewords on demand, without knowing the underlying probability distribution) that guarantees asymptotically optimal cost: at most opt + 2 log(1+opt) + 2.Comment: ACM-SIAM Symposium on Discrete Algorithms (SODA) 201

Social welfare and profit maximization from revealed preferences

Consider the seller's problem of finding optimal prices for her $n$ (divisible) goods when faced with a set of $m$ consumers, given that she can only observe their purchased bundles at posted prices, i.e., revealed preferences. We study both social welfare and profit maximization with revealed preferences. Although social welfare maximization is a seemingly non-convex optimization problem in prices, we show that (i) it can be reduced to a dual convex optimization problem in prices, and (ii) the revealed preferences can be interpreted as supergradients of the concave conjugate of valuation, with which subgradients of the dual function can be computed. We thereby obtain a simple subgradient-based algorithm for strongly concave valuations and convex cost, with query complexity $O(m^2/\epsilon^2)$, where $\epsilon$ is the additive difference between the social welfare induced by our algorithm and the optimum social welfare. We also study social welfare maximization under the online setting, specifically the random permutation model, where consumers arrive one-by-one in a random order. For the case where consumer valuations can be arbitrary continuous functions, we propose a price posting mechanism that achieves an expected social welfare up to an additive factor of $O(\sqrt{mn})$ from the maximum social welfare. Finally, for profit maximization (which may be non-convex in simple cases), we give nearly matching upper and lower bounds on the query complexity for separable valuations and cost (i.e., each good can be treated independently)

Individual vs. Collective Bargaining in the Large Firm Search Model

We analyze the welfare and employment effects of different wage bargaining regimes. Within the large firm search model, we show that collective bargaining affects employment via two channels. Collective bargaining exerts opposing effects on job creation and wage setting. Firms have a stronger incentive for strategic employment, while workers benefit from the threat of a strike. We find that the employment increase due to the strategic motive is dominated by the employment decrease due to the increase in workers' threat point. In aggregate equilibrium, employment is ineciently low under collective bargaining. But it is not always true that equilibrium wages exceed those under individual bargaining. If unemployment benefits are sufficiently low, collectively bargained wages are smaller. The theory sheds new light on policies concerned with strategic employment and the relation between replacement rates and the extent of collective wage bargaining

Individual vs. Collective Bargaining in the Large Firm Search Model

We analyze the welfare and employment effects of different wage bargaining regimes. Within the large firm search model, we show that collective bargaining affects employment via two channels. Collective bargaining exerts opposing effects on job creation and wage setting. Firms have a stronger incentive for strategic employment, while workers benefit from the threat of a strike. We find that the employment increase due to the strategic motive is dominated by the employment decrease due to the increase in workers' threat point. In aggregate equilibrium, employment is ineciently low under collective bargaining. But it is not always true that equilibrium wages exceed those under individual bargaining. If unemployment benefits are sufficiently low, collectively bargained wages are smaller. The theory sheds new light on policies concerned with strategic employment and the relation between replacement rates and the extent of collective wage bargaining.search; overemployment; collective wage bargaining; wage determination