This paper considers a problem where multiple users make repeated decisions based on their own observed events. The events and decisions at each time step determine the values of a utility function and a collection of penalty functions. The goal is to make distributed decisions over time to maximize time average utility subject to time average constraints on the penalties. An example is a collection of power constrained sensor nodes that repeatedly report their own observations to a fusion center. Maximum time average utility is fundamentally reduced because users do not know the events observed by others. Optimality is characterized for this distributed context. It is shown that optimality is achieved by correlating user decisions through a commonly known pseudorandom sequence. An optimal algorithm is developed that chooses pure strategies at each time step based on a set of time-varying weights.
The framework of partial knowledge at each user is similar in spirit to a @cite_9 @cite_14 . There, the goal is to design competitive strategies that lead to a Nash equilibrium. This is significantly different from the goal of the current paper. The current paper is not concerned with competition or equilibrium. Rather, there is a single utility function that all users desire to maximize. Distributed algorithms are developed to maximize time average utility subject to time average penalty constraints.
@cite_9 @cite_14
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We study in this paper a noncooperative approach for sharing resources of a common pool among users, wherein each user strives to maximize its own utility. The optimality notion is then a Nash equilibrium. First, we present a general framework of systems wherein a Nash equilibrium is Pareto inefficient, which are similar to the 'tragedy of the commons' in economics. As examples that fit in the above framework, we consider noncooperative flow-control problems in communication networks where each user decides its throughput to optimize its own utility. As such a utility, we first consider the power which is defined as the throughput divided by the expected end-to-end packet delay, and then consider another utility of additive costs. For both utilities, we establish the non-efficiency of the Nash equilibria. We address the problem of spectrum pricing in a cognitive radio network where multiple primary service providers compete with each other to offer spectrum access opportunities to the secondary users. By using an equilibrium pricing scheme, each of the primary service providers aims to maximize its profit under quality of service (QoS) constraint for primary users. We formulate this situation as an oligopoly market consisting of a few firms and a consumer. The QoS degradation of the primary services is considered as the cost in offering spectrum access to the secondary users. For the secondary users, we adopt a utility function to obtain the demand function. With a Bertrand game model, we analyze the impacts of several system parameters such as spectrum substitutability and channel quality on the Nash equilibrium (i.e., equilibrium pricing adopted by the primary services). We present distributed algorithms to obtain the solution for this dynamic game. The stability of the proposed dynamic game algorithms in terms of convergence to the Nash equilibrium is studied. However, the Nash equilibrium is not efficient in the sense that the total profit of the primary service providers is not maximized. An optimal solution to gain the highest total profit can be obtained. A collusion can be established among the primary services so that they gain higher profit than that for the Nash equilibrium. However, since one or more of the primary service providers may deviate from the optimal solution, a punishment mechanism may be applied to the deviating primary service provider. A repeated game among primary service providers is formulated to show that the collusion can be maintained if all of the primary service providers are aware of this punishment mechanism, and therefore, properly weight their profits to be obtained in the future. In the traffic assignment problem, first proposed by Wardrop in 1952, commuters select the shortest available path to travel from their origins to their destinations. We study a generalization of this problem in which competitors, who may control a nonnegligible fraction of the total flow, ship goods across a network. This type of games, usually referred to as atomic games, readily applies to situations in which the competing freight companies have market power. Other applications include intelligent transportation systems, competition among telecommunication network service providers, and scheduling with flexible machines. Our goal is to determine to what extent these systems can benefit from some form of coordination or regulation. We measure the quality of the outcome of the game without centralized control by computing the worst-case inefficiency of Nash equilibria. The main conclusion is that although self-interested competitors will not achieve a fully efficient solution from the system's point of view, the loss is not too severe. We show how to compute several bounds for the worst-case inefficiency that depend on the characteristics of cost functions and on the market structure in the game. In addition, building upon the work of Catoni and Pallotino, we show examples in which market aggregation (or collusion) adversely impacts the aggregated competitors, even though their market power increases. For example, Nash equilibria of atomic network games may be less efficient than the corresponding Wardrop equilibria. When competitors are completely symmetric, we provide a characterization of the Nash equilibrium using a potential function, and prove that this counterintuitive phenomenon does not arise. Finally, we study a pricing mechanism that elicits more coordination from the players by reducing the worst-case inefficiency of Nash equilibria. The Generalized Second Price Auction has been the main mechanism used by search companies to auction positions for advertisements on search pages. In this paper we study the social welfare of the Nash equilibria of this game in various models. In the full information setting, socially optimal Nash equilibria are known to exist (i.e., the Price of Stability is 1). This paper is the first to prove bounds on the price of anarchy, and to give any bounds in the Bayesian setting. Our main result is to show that the price of anarchy is small assuming that all bidders play un-dominated strategies. In the full information setting we prove a bound of 1.618 for the price of anarchy for pure Nash equilibria, and a bound of 4 for mixed Nash equilibria. We also prove a bound of 8 for the price of anarchy in the Bayesian setting, when valuations are drawn independently, and the valuation is known only to the bidder and only the distributions used are common knowledge. Our proof exhibits a combinatorial structure of Nash equilibria and uses this structure to bound the price of anarchy. While establishing the structure is simple in the case of pure and mixed Nash equilibria, the extension to the Bayesian setting requires the use of novel combinatorial techniques that can be of independent interest. We study the following Bayesian setting: mitems are sold to nselfish bidders in mindependent second-price auctions. Each bidder has a privatevaluation function that expresses complex preferences over allsubsets of items. Bidders only have beliefsabout the valuation functions of the other bidders, in the form of probability distributions. The objective is to allocate the items to the bidders in a way that provides a good approximation to the optimal social welfare value. We show that if bidders have submodular valuation functions, then every Bayesian Nash equilibrium of the resulting game provides a 2-approximation to the optimal social welfare. Moreover, we show that in the full-information game a pure Nash always exists and can be found in time that is polynomial in both mand n. This paper develops a strategy to coordinate the charging of autonomous plug-in electric vehicles (PEVs) using concepts from non-cooperative games. The foundation of the paper is a model that assumes PEVs are cost-minimizing and weakly coupled via a common electricity price. At a Nash equilibrium, each PEV reacts optimally with respect to a commonly observed charging trajectory that is the average of all PEV strategies. This average is given by the solution of a fixed point problem in the limit of infinite population size. The ideal solution minimizes electricity generation costs by scheduling PEV demand to fill the overnight non-PEV demand “valley”. The paper's central theoretical result is a proof of the existence of a unique Nash equilibrium that almost satisfies that ideal. This result is accompanied by a decentralized computational algorithm and a proof that the algorithm converges to the Nash equilibrium in the infinite system limit. Several numerical examples are used to illustrate the performance of the solution strategy for finite populations. The examples demonstrate that convergence to the Nash equilibrium occurs very quickly over a broad range of parameters, and suggest this method could be useful in situations where frequent communication with PEVs is not possible. The method is useful in applications where fully centralized control is not possible, but where optimal or near-optimal charging patterns are essential to system operation. In this paper we consider online auction mechanisms for the allocation of M items that are identical to each other except for the fact that they have different expiration times, and each item must be allocated before it expires. Players arrive at different times, and wish to buy one item before their deadline. The main difficulty is that players act "selfishly" and may mis-report their values, deadlines, or arrival times. We begin by showing that the usual notion of truthfulness (where players follow a single dominant strategy) cannot be used in this case, since any (deterministic) truthful auction cannot obtain better than an M-approximation of the social welfare. Therefore, instead of designing auctions in which players should follow a single strategy, we design two auctions that perform well under a wide class of selfish, "semi-myopic", strategies. For every combination of such strategies, the auction is associated with a different algorithm, and so we have a family of "semi-myopic" algorithms. We show that any algorithm in this family obtains a 3-approximation, and by this conclude that our auctions will perform well under any choice of such semi-myopic behaviors. We next turn to provide a game-theoretic justification for acting in such a semi-myopic way. We suggest a new notion of "Set-Nash" equilibrium, where we cannot pin-point a single best-response strategy, but rather only a set of possible best-response strategies. We show that our auctions have a Set-Nash equilibrium which is all semi-myopic, hence guarantees a 3-approximation. We believe that this notion is of independent interest. This paper addresses the behavior of the selfish service providers in the form of IP sinks providing high-speed IP access. Service providers compete for mobile users by adjusting the price they charge for their services. Their aim is to maximize the total collected profit. Mobile users are also selfish choosing the service provider offering the best quality of service and price combination. As the service providers come closer to each other, we show the existence of three critical phase transitions in their behavior. Depending on the separation between them, there may exists a unique Nash equilibrium, or a continuum of Nash equilibria, or no Nash equilibrium. We completely characterize the pricing strategies of service providers at Nash equilibria. We also prove that the total social welfare in the presence of selfish providers is close to the maximum social welfare that can reached through non- selfish optimization. We study combinatorial auctions where each item is sold separately but simultaneously via a second price auction. We ask whether it is possible to efficiently compute in this game a pure Nash equilibrium with social welfare close to the optimal one. We show that when the valuations of the bidders are submodular, in many interesting settings (e.g., constant number of bidders, budget additive bidders) computing an equilibrium with good welfare is essentially as easy as computing, completely ignoring incentives issues, an allocation with good welfare. On the other hand, for subadditive valuations, we show that computing an equilibrium requires exponential communication. Finally, for XOS (a.k.a. fractionally subadditive) valuations, we show that if there exists an efficient algorithm that finds an equilibrium, it must use techniques that are very different from our current ones. We study competition between wireless devices with incomplete information about their opponents. We model such interactions as Bayesian interference games. Each wireless device selects a power profile over the entire available bandwidth to maximize its data rate. Such competitive models represent situations in which several wireless devices share spectrum without any central authority or coordinated protocol. In contrast to games where devices have complete information about their opponents, we consider scenarios where the devices are unaware of the interference they cause to other devices. Such games, which are modeled as Bayesian games, can exhibit significantly different equilibria. We first consider a simple scenario of simultaneous move games, where we show that the unique Bayes-Nash equilibrium is where both devices spread their power equally across the entire bandwidth. We then extend this model to a two-tiered spectrum sharing case where users act sequentially. Here one of the devices, called the primary user, is the owner of the spectrum and it selects its power profile first. The second device (called the secondary user) then responds by choosing a power profile to maximize its Shannon capacity. In such sequential move games, we show that there exist equilibria in which the primary user obtains a higher data rate by using only a part of the bandwidth. In a repeated Bayesian interference game, we show the existence of reputation effects: an informed primary user can bluff to prevent spectrum usage by a secondary user who suffers from lack of information about the channel gains. The resulting equilibrium can be highly inefficient, suggesting that competitive spectrum sharing is highly suboptimal. This paper addresses the control of a team of autonomous agents pursuing a smart evader in a non-accurately mapped terrain. By describing the problem as a partial information Markov game, we are able to integrate map-learning and pursuit. We propose receding horizon control policies, in which the pursuers and the evader try to respectively maximize and minimize the probability of capture at the next time instant. Since this probability is conditioned to distinct observations for each team, the resulting game is nonzero-sum. When the evader has access to the pursuers' information, we show that a Nash solution to the one-step nonzero-sum game always exists. Moreover, we propose a method to compute the Nash equilibrium policies by solving an equivalent zero-sum matrix game. A simulation example shows the feasibility of the proposed approach. This paper is concerned with the game theoretical approach in designing the multiuser downlink beamformers in multicell systems. Sharing the same physical resource, the base-station of each cell wishes to minimize its transmit power subject to a set of target signal-to-interference-plus-noise ratios (SINRs) at the multiple users in the cell. In this context, at first, the paper considers a strategic noncooperative game (SNG) where each base-station greedily determines its optimal downlink beamformer strategy in a distributed manner, without any coordination between the cells. Via the game theory framework, it is shown that this game belongs to the framework of standard functions. The conditions guaranteeing the existence and uniqueness of a Nash Equilibrium (NE) in this competitive design are subsequently examined. The paper then makes a revisit to the fully coordinated design in multicell downlink beamforming, where the optimal beamformers are jointly designed between the base-stations. A comparison between the competitive and coordinated designs shows the benefits of applying the former over the latter in terms of each design's distributed implementation. Finally, in order to improve the efficiency of the NE in the competitive design, the paper considers a more cooperative game through a pricing mechanism. The pricing consideration enables a base-station to steer its beamformers in a more cooperative manner, which ultimately limits the interference induced to other cells. The study on the existence and uniqueness of the new game's NE is then given. The paper also presents a condition on the pricing factors that allow the new NE point to approach the performance established by the coordinated design, while retaining the distributed nature of the multicell game. This paper tackles the problem of joint users’ uplink transmission power and data rate allocation in multi-service two-tier femtocell networks. Each user—either macrocell (MUE) or femtocell user equipment (FUE)—is associated with a two-variable utility function that represents his perceived satisfaction with respect to his allocated resources (i.e., power and rate). User's utility function is differentiated based both on the tier that the user belongs to and the service he requests. The joint resource allocation problem is directly confronted as a two-variable optimization problem and formulated as a non-cooperative game. The theory of supermodular games is utilized towards treating the two-variable optimization problem and the inherent multidimensional competition that arises among the users. The existence of proposed game's Nash Equilibrium (NE) point is analytically shown, while game's convergence to its NE point is proven. A distributed and iterative algorithm for computing the desired NE is introduced, where the optimal values of each user's uplink transmission power and data rate are simultaneously updated at the same step. The performance of the proposed approach is evaluated via modeling and simulation and its superiority compared to other state of the art approaches is illustrated. We consider the problem of designing (perhaps massively distributed) collectives of computational processes to maximize a provided "world utility" function. We consider this problem when the behavior of each process in the collective can be cast as striving to maximize its own payoff utility function. For such cases the central design issue is how to initialize update those payoff utility functions of the individual processes so as to induce behavior of the entire collective having good values of the world utility. Traditional "team game" approaches to this problem simply assign to each process the world utility as its payoff utility function. In previous work we used the "Collective Intelligence" (COIN) framework to derive a better choice of payoff utility functions, one that results in world utility performance up to orders of magnitude superior to that ensuing from the use of the team game utility. In this paper, we extend these results using a novel mathematical framework. Under that new framework we review the derivation of the general class of payoff utility functions that both (i) are easy for the individual processes to try to maximize, and (ii) have the property that if good values of them are achieved, then we are assured a high value of world utility. These are the "Aristocrat Utility" and a new variant of the "Wonderful Life Utility" that was introduced in the previous COIN work. We demonstrate experimentally that using these new utility functions can result in significantly improved performance over that of previously investigated COIN payoff utilities, over and above those previous utilities' superiority to the conventional team game utility. These results also illustrate the substantial superiority of these payoff functions to perhaps the most natural version of the economics technique of "endogenizing externalities." We introduce a simple network design game that models how independent selfish agents can build or maintain a large network. In our game every agent has a specific connectivity requirement, i.e. each agent has a set of terminals and wants to build a network in which his terminals are connected. Possible edges in the network have costs and each agent's goal is to pay as little as possible. Determining whether or not a Nash equilibrium exists in this game is NP-complete. However, when the goal of each player is to connect a terminal to a common source, we prove that there is a Nash equilibrium as cheap as the optimal network, and give a polynomial time algorithm to find a (1+e)-approximate Nash equilibrium that does not cost much more. For the general connection game we prove that there is a 3-approximate Nash equilibrium that is as cheap as the optimal network, and give an algorithm to find a (4.65+e)-approximate Nash equilibrium that does not cost much more. In this paper, we investigate a distributed learning scheme for a broad class of stochastic optimization problems and games that arise in signal processing and wireless communications. The proposed algorithm relies on the method of matrix exponential learning (MXL) and only requires locally computable gradient observations that are possibly imperfect. To analyze it, we introduce the notion of a stable Nash equilibrium and we show that the algorithm is globally convergent to such equilibria—or locally convergent when an equilibrium is only locally stable. To complement our convergence analysis, we also derive explicit bounds for the algorithm's convergence speed and we test it in realistic multicarrier multiple-antenna wireless scenarios where several users seek to maximize their energy efficiency. Our results show that learning allows users to attain a net increase between @math and @math in energy efficiency, even under very high uncertainty. In this paper, we consider the demand response problem in smart grid consisting of a retailer and multiple residential consumers, where the retailer determines consumers' payments based on their power consumption profile. Our aim is to propose a fully distributed algorithm that is able to optimize the aggregate cost, utility, and retailer's profit simultaneously. To this end, we first formulate the consumer-side trend as a constrained convex optimization problem and propose a fully distributed adaptive diffusion algorithm to solve it. In addition, we design a one-leader N-follower Stackelberg game to model interactions among the retailer and consumers. The proposed framework is able to continuously track the drifts resulting from the changes in the real-time pricing or the consumer preferences. Moreover, it is scalable and does not require network-wide information or rely on central controller. We provide comprehensive simulation results to show the effectiveness of the proposed framework. This paper considers the problem of pricing and transmission scheduling for an access point (AP) in a wireless network, where the AP provides service to a set of mobile users. The goal of the AP is to maximize its own time-average profit. We first obtain the optimum time-average profit of the AP and prove the "Optimality of Two Prices" theorem. We then develop an online scheme that jointly solves the pricing and transmission scheduling problem in a dynamic environment. The scheme uses an admission price and a business decision as tools to regulate the incoming traffic and to maximize revenue. We show the scheme can achieve any average profit that is arbitrarily close to the optimum, with a tradeoff in average delay. This holds for general Markovian dynamics for channel and user state variation, and does not require a priori knowledge of the Markov model. The model and methodology developed in this paper are general and apply to other stochastic settings where a single party tries to maximize its time-average profit. Why might markets tend toward and remain near equilibrium prices? In an effort to shed light on this question from an algorithmic perspective, this paper formalizes the setting of Ongoing Markets, by contrast with the classic market scenario, which we term One-Time Markets. The Ongoing Market allows trade at non-equilibrium prices, and, as its name suggests, continues over time. As such, it appears to be a more plausible model of actual markets. For both market settings, this paper defines and analyzes variants of a simple tatonnement algorithm that differs from previous algorithms that have been subject to asymptotic analysis in three significant respects: the price update for a good depends only on the price, demand, and supply for that good, and on no other information; the price update for each good occurs distributively and asynchronously; the algorithms work (and the analyses hold) from an arbitrary starting point. Our algorithm introduces a new and natural update rule. We show that this update rule leads to fast convergence toward equilibrium prices in a broad class of markets that satisfy the weak gross substitutes property. These are the first analyses for computationally and informationally distributed algorithms that demonstrate polynomial convergence. Our analysis identifies three parameters characterizing the markets, which govern the rate of convergence of our protocols. These parameters are, broadly speaking: 1. A bound on the fractional rate of change of demand for each good with respect to fractional changes in its price. 2. A bound on the fractional rate of change of demand for each good with respect to fractional changes in wealth. 3. The closeness of the market to a Fisher market (a market with buyers starting with money alone). We give two types of protocols. The first type assumes global knowledge of only (an upper bound on) the first parameter. For this protocol, we also provide a matching lower bound in terms of these parameters for the One-Time Market. Our second protocol, which is analyzed for the One-Time Market alone, assumes no global knowledge whatsoever. We study the following simple Bayesian auction setting: m items are sold to n selfish bidders in m independent second-price auctions. Each bidder has a private valuation function that specifies his or her complex preferences over all subsets of items. Bidders only have beliefs about the valuation functions of the other bidders, in the form of probability distributions. The objective is to allocate the items to the bidders in a way that provides a good approximation to the optimal social welfare value. We show that if bidders have submodular or, more generally, fractionally subadditive (aka XOS) valuation functions, every Bayes-Nash equilibrium of the resulting game provides a 2-approximation to the optimal social welfare. Moreover, we show that in the full-information game, a pure Nash always exists and can be found in time that is polynomial in both m and n.