We study the sensitivity of optimal solutions of convex separable optimization problems over an integral polymatroid base polytope with respect to parameters determining both the cost of each element and the polytope. Under convexity and a regularity assumption on the functional dependency of the cost function with respect to the parameters, we show that reoptimization after a change in parameters can be done by elementary local operations. Applying this result, we derive that starting from any optimal solution there is a new optimal solution to new parameters such that the L1-norm of the difference of the two solutions is at most two times the L1 norm of the difference of the parameters. We apply these sensitivity results to a class of non-cooperative polymatroid games and derive the existence of pure Nash equilibria. We complement our results by showing that polymatroids are the maximal combinatorial structure enabling these results. For any non-polymatroid region, there is a corresponding optimization problem for which the sensitivity results do not hold. In addition, there is a game where the players strategies are isomorphic to the non-polymatroid region and that does not admit a pure Nash equilibrium.
Parts of the results of this paper have been presented by the authors in less general and preliminary form in the Proceedings of the 10th Conference on Web and Internet Economics @cite_35 .
@cite_35
2145397675 2165707497 1992658674 1971105953
Abstract We envision a future in which the global economy and the Internet will merge, evolving into an information economy bustling with billions of economically motivated software agents that exchange information goods and services with humans and other agents. Economic software agents will differ in important ways from their human counterparts, and these differences may have significant beneficial or harmful effects upon the global economy. It is therefore important to consider the economic incentives and behaviors of economic software agents, and to use every available means to anticipate their collective interactions. We survey research conducted by the Information Economies group at IBM Research aimed at understanding collective interactions among agents that dynamically price information goods or services. In particular, we study the potential impact of widespread shopbot usage on prices, the price dynamics that may ensue from various mixtures of automated pricing agents (or “pricebots”), the potential use of machine-learning algorithms to improve profits, and more generally the interplay among learning, optimization, and dynamics in agent-based information economies. These studies illustrate both beneficial and harmful collective behaviors that can arise in such systems, suggest possible cures for some of the undesired phenomena, and raise fundamental theoretical issues, particularly in the realms of multi-agent learning and dynamic optimization. The current economic crisis requires fast information to predict economic behavior early, which is difficult at times of structural changes. This paper suggests an innovative new method of using data on internet activity for that purpose. It demonstrates strong correlations between keyword searches and unemployment rates using monthly German data and exhibits a strong potential for the method used. The Internet has been a key enabling technology for many new distributed applications and services. However, the deployment of new protocols and services in the Internet infrastructure itself has been sluggish, especially where economic incentives for network providers are unclear. In our work, we seek to develop an "economy plane" for the Internet that enables network providers to offer new network-based services (QoS, storage, etc.) for sale to customers. The explicit connection between economic relationships and network services across various time scales enables users to select among service alternatives. The resulting competition among network service providers will lead to overall better technological solutions and more competitive prices. In this paper, we present the architectural aspects of our ChoiceNet economy plane as well as some of the technological problems that need to be addressed in a practical deployment. We live in a computerized and networked society where many of our actions leave a digital trace and affect other people’s actions. This has lead to the emergence of a new data-driven research field: mathematical methods of computer science, statistical physics and sociometry provide insights on a wide range of disciplines ranging from social science to human mobility. A recent important discovery is that search engine traffic (i.e., the number of requests submitted by users to search engines on the www) can be used to track and, in some cases, to anticipate the dynamics of social phenomena. Successful examples include unemployment levels, car and home sales, and epidemics spreading. Few recent works applied this approach to stock prices and market sentiment. However, it remains unclear if trends in financial markets can be anticipated by the collective wisdom of on-line users on the web. Here we show that daily trading volumes of stocks traded in NASDAQ-100 are correlated with daily volumes of queries related to the same stocks. In particular, query volumes anticipate in many cases peaks of trading by one day or more. Our analysis is carried out on a unique dataset of queries, submitted to an important web search engine, which enable us to investigate also the user behavior. We show that the query volume dynamics emerges from the collective but seemingly uncoordinated activity of many users. These findings contribute to the debate on the identification of early warnings of financial systemic risk, based on the activity of users of the www.