Computational aspects of combinatorial pricing problems

Combinatorial pricing encompasses a wide range of natural optimization problems that arise in the computation of revenue maximizing pricing schemes for a given set of goods, as well as the design of profit maximizing auctions in strategic settings. We consider the computational side of several different multi-product and network pricing problems and, as most of the problems in this area are NP-hard, we focus on the design of approximation algorithms and corresponding inapproximability results. In the unit-demand multi-product pricing problem it is assumed that each consumer has different budgets for the products she is interested in and purchases a single product out of her set of alternatives. Depending on how consumers choose their products once prices are fixed we distinguish the min-buying, max-buying and rank-buying models, in which consumers select the affordable product with smallest price, highest price or highest rank according to some predefined preference list, respectively. We prove that the max-buying model allows for constant approximation guarantees and this is true even in the case of limited product supply. For the min-buying model we prove inapproximability beyond the known logarithmic guarantees under standard complexity theoretic assumptions. Surprisingly, this result even extends to the case of pricing with a price ladder constraint, i.e., a predefined relative order on the product prices. Furthermore, similar results can be shown for the uniform-budget version of the problem, which corresponds to a special case of the unit-demand envy-free pricing problem, under an assumption about the average case hardness of refuting random 3SAT-instances. Introducing the notion of stochastic selection rules we show that among a large class of selection rules based on the order of product prices the maxbuying model is in fact the only one allowing for sub-logarithmic approximation guarantees. In the single-minded pricing problem each consumer is interested in a single set of products, which she purchases if the sum of prices does not exceed her budget. It turns out that our results on envyfree unit-demand pricing can be extended to this scenario and yield inapproximability results for ratios expressed in terms of the number of distinct products, thereby complementing existing hardness results. On the algorithmic side, we present an algorithm with approximation guarantee that depends only on the maximum size of the sets and the number of requests per product. Our algorithm’s ratio matches previously known results in the worst case but has significantly better provable performance guarantees on sparse problem instances. Viewing single-minded as a network pricing problem in which we assign prices to edges and consumers want to purchase paths in the network, it is proven that the problem remains APX-hard even on extremely sparse instances. For the special case of pricing on a line with paths that are nested, we design an FPTAS and prove NP-hardness. In a Stackelberg network pricing game a so-called leader sets the prices on a subset of the edges of a network, the remaining edges have associated fixed costs. Once prices are fixed, one or more followers purchase min-cost subnetworks according to their requirements and pay the leader for all pricable edges contained in their networks. We extend the analysis of the known single-price algorithm, which assigns the same price to all pricable edges, from cases in which the feasible subnetworks of a follower form the basis of a matroid to the general case, thus, obtaining logarithmic approximation guarantees for general Stackelberg games. We then consider a special 2-player game in which the follower buys a min-cost vertex cover in a bipartite graph and the leader sets prices on a subset of the vertices. We prove that this problem is polynomial time solvable in some cases and allows for constant approximation guarantees in general. Finally, we point out that results on unit-demand and single-minded pricing yield several strong inapproximability results for Stackelberg pricing games with multiple followers

Stackelberg Network Pricing Games

We study a multi-player one-round game termed Stackelberg Network Pricing Game, in which a leader can set prices for a subset of $m$ priceable edges in a graph. The other edges have a fixed cost. Based on the leader's decision one or more followers optimize a polynomial-time solvable combinatorial minimization problem and choose a minimum cost solution satisfying their requirements based on the fixed costs and the leader's prices. The leader receives as revenue the total amount of prices paid by the followers for priceable edges in their solutions, and the problem is to find revenue maximizing prices. Our model extends several known pricing problems, including single-minded and unit-demand pricing, as well as Stackelberg pricing for certain follower problems like shortest path or minimum spanning tree. Our first main result is a tight analysis of a single-price algorithm for the single follower game, which provides a $(1+\epsilon) \log m$-approximation for any $\epsilon >0$. This can be extended to provide a $(1+\epsilon)(\log k + \log m)$-approximation for the general problem and $k$ followers. The latter result is essentially best possible, as the problem is shown to be hard to approximate within \mathcal{O(\log^\epsilon k + \log^\epsilon m). If followers have demands, the single-price algorithm provides a $(1+\epsilon)m^2$-approximation, and the problem is hard to approximate within \mathcal{O(m^\epsilon) for some $\epsilon >0$. Our second main result is a polynomial time algorithm for revenue maximization in the special case of Stackelberg bipartite vertex cover, which is based on non-trivial max-flow and LP-duality techniques. Our results can be extended to provide constant-factor approximations for any constant number of followers

Российская этика бизнеса в контексте национальных особенностей

Анализируются основные элементы бизнес этики, позволяющие утверждать, что этические воззрения не могут носить универсальный характер, поскольку в основе лежат принципиальные культурные различия. Установлены элементы, указывающие на отличия в бизнес этике России, США и европейских государств. Сделаны выводы о культурных предпосылках, которые позволяют увидеть причину, по которой бизнесмены России и США будут по-разному реагировать на аналогичную этическую дилемму

Approximate Equilibria in Games with Few Players

We study the problem of computing approximate Nash equilibria (epsilon-Nash equilibria) in normal form games, where the number of players is a small constant. We consider the approach of looking for solutions with constant support size. It is known from recent work that in the 2-player case, a 1/2-Nash equilibrium can be easily found, but in general one cannot achieve a smaller value of epsilon than 1/2. In this paper we extend those results to the k-player case, and find that epsilon = 1-1/k is feasible, but cannot be improved upon. We show how stronger results for the 2-player case may be used in order to slightly improve upon the epsilon = 1-1/k obtained in the k-player case