1020 IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS, VOL. 24, NO. 5, MAY 2006 Market Sharing Games Applied to Content Distribution in Ad Hoc Networks Michel X. Goemans, Li (Erran) Li, Vahab S. Mirrokni, and Marina Thottan, Member, IEEE Abstract—In third-generation (3G) wireless data networks, re- I. INTRODUCTION peated requests for popular data items can exacerbate the already scarce wireless spectrum. In this paper, we propose an architec- HIRD-GENERATION (3G) wide-area wireless networks tural and protocol framework that allows 3G service providers to T have recently experienced tremendous growth, with the host efficient content distribution services. We offload the spec- number of subscribers reaching more than 70 million world- trum intensive task of content distribution to an ad hoc network. wide [1]. Many 3G service providers have started offering con- Less mobile users (resident subscribers) are provided incentives to tent-rich services such as sports replays, news headlines, music cache popular data items, while mobile users (transit subscribers) access this data from resident subscribers through the ad hoc net- videos, and movie trailers [1]. work. Since the participants of this data distribution network act The 3G subscriber market can be categorized into groups as selfish agents, they may collude to maximize their individual with shared interest in location-based services, e.g., the pre- payoff. Our proposed protocol discourages potential collusion sce- view of movies in a theater or the scene of the beach nearby. narios. In this architecture, the goal (social function) of the 3G ser- Since the 3G radio resources are limited, it is expensive to re- vice provider is to have the selfishly motivated resident subscribers service as many data requests as possible. However, the choice of peatedly transmit large quantities of data over the air interface which set of items to cache is left to the individual user. The caching from the base station (BS). It is more economical for the ser- activity among the different users can be modeled as a market vice provider to offload such repeated requests on to the ad hoc sharing game. network comprised of its subscribers, where some of them may In this work, we study the Nash equilibria of market sharing have recently acquired a copy of the data. In this scenario, the games and the performance of such equilibria in terms of a social function. These games are a special case of congestion games that goal for the service provider is to give incentives for peer sub- have been studied in the economics literature. In particular, pure scribers in the system to cache and forward the data to the re- strategy Nash equilibria for this set of games exist. We give a poly- questing subscribers. Since each data item is large in size and nomial-time algorithm to find a pure strategy Nash equilibrium for transit subscribers are mobile, we assume that the data transfer a special case, while it is NP-hard to do so in the general case. As for occurs in within a range of a few hops. the performance of Nash equilibria, we show that the price of an- archy—the worst case ratio between the social function at any Nash We envision a system consisting of two groups of subscribers: equilibrium and at the social optimum—can be upper bounded resident and transit subscribers. Resident subscribers are less by a factor of 2. When the popularity follows a Zipf distribution, mobile and mostly confined to a certain geographical area. Res- the price of anarchy is bounded by 1.45 in the special case where ident subscribers have incentives to cache data items that are caching any item has a positive reward for all players. We prove specific to this geographical region since the service provider that the selfish behavior of computationally bounded agents con- verges to an approximate Nash equilibrium in a finite number of gives monetary rewards for satisfying the queries of transit sub- improvements. Furthermore, we prove that, after each agent com- scribers. Transit subscribers request their favorite data items putes its response function once using a constant factor approxi- when they visit a particular region. Since the service provider mation algorithm, the outcome of the game is within a factor of does not have knowledge of the spatial and temporal distribu- @log A of the optimal social value, where is the number of tion of requests, it is difficult if not impossible for the provider agents. Our simulation scenarios show that the price of anarchy is 30% better than that of the worst case analysis and that the system to stipulate which subscriber should cache which set of data quickly (1 or 2 steps) converges to a Nash equilibrium. items. Therefore, the decision of what to cache is left to each in- Index Terms—Mobile ad hoc networks, Nash equilibrium, dividual subscriber. The realization of this content distribution price of anarchy, third-generation (3G) wireless networks, unified system depends on two main issues. First, since subscribers are architecture. selfish agents, they may collude to increase their individual pay- offs. Collusion can result in other subscribers being cheated of Manuscript received February 15, 2005; revised January 15, 2006. This their rewards. In this work, we address the problem of colluding work was supported in part by the Office of Naval Research under Grant subscribers by providing a protocol framework that discourages N00014-05-1-0148. The work of L. Li was supported in part by the National or prevents collusions. The second issue is that the payoff of Science Foundation (NSF) under Grant ANI-0335244. This paper was pre- sented at the ACM Mobile Ad Hoc Networking and Computing Symposium, each item for each agent must be a function of the number of 2004. cache requests it services. This in turn depends on the number M. X. Goemans and V. S. Mirrokni are with the Department of Mathe- of agents who cache a given item since each agent has limited matics and the Computer Science and Artificial Intelligence Laboratories, Massachusetts Institute of Technology, Cambridge, MA 02139 USA (e-mail: storage space (due to form factor, current 3G devices only have [email protected]; [email protected]). a couple of MB flash memory [1]). Therefore, each selfish agent L. Li and M. Thottan are with the Center for Networking Research, Bell Lab- may change the set of items it caches in response to the set of oratories, Lucent Technologies, Murray Hill, NJ 07974 USA (e-mail: erranlli@ research.bell-labs.com; [email protected]). items cached by others. This leads to a noncooperative caching Digital Object Identifier 10.1109/JSAC.2006.872884 scenario which we model as a market sharing game. 0733-8716/$20.00 © 2006 IEEE GOEMANS et al.: MARKET SHARING GAMES APPLIED TO CONTENT DISTRIBUTION IN AD HOC NETWORKS 1021 In the market sharing game, the primary questions are game and in Section VI, we show that pure strategy Nash equi- whether the system converges, i.e., results in a pure strategy librium exists, but it is NP-hard to find such an equilibrium. We Nash equilibrium and how long it will take to converge. The also provide a polynomial-time algorithm to find a pure strategy goal of the service provider is to offload as many cache requests Nash equilibrium for the special case when all the items have the as possible to the ad hoc network. We refer to this goal as same size. In Section VII, we model computationally bounded the social optimum. However, given the selfish behavior of agents by approximation algorithms and analyze the outcome the agents, it is unlikely that the system will result in a social of their selfish behavior. We observe that such agents converge optimum. Therefore, we would like to bound the ratio between to an approximate Nash equilibrium. We also show that, after the optimal social value and the outcome of the selfish behavior one step of improvements the outcome of the game is within an of players. We refer to this ratio as the price of anarchy. Fur- factor of the optimum. In Section VIII, we investigate thermore, when computing the selfish behavior of individual the price of anarchy for a set of sample instances and the con- players, it is essential to consider the computational constraint vergence rate to exact and approximate Nash equilibria. Related on the individual subscribers. We model computationally work is provided in Section IX. A discussion on our results and bounded agents using approximation algorithms and evaluate other relevant issues are described in Section X. We conclude in how fast the selfish behavior of such agents will converge to Section XI. an approximate Nash equilibrium or arrive at an approximate solution to the social function. II. SYSTEM ARCHITECTURE,TRUST MODEL, AND PROTOCOLS The main contributions of this paper are as follows. First, we study the applicability of noncooperative caching in wire- In this section, we briefly discuss the system architecture and less networks and propose a detailed protocol that provides in- the trust model for distributed noncooperative caching. We then centives for selfish agents to service other agents, while dis- outline the protocols required for offloading popular data items couraging collusion among participating agents. We model the from 3G networks to the multihop ad hoc network. Our archi- caching game among the different mobile users as a market tecture is largely the same as UCAN [14]. In our system archi- sharing game. We show that this is a special case of conges- tecture, we assume the following. tion games. It is known that pure strategy Nash equilibria exist • A service provider operates a 3G data network, e.g., for these games [21].