Showing papers on "Stackelberg competition published in 2009"

Distributed Relay Selection and Power Control for Multiuser Cooperative Communication Networks Using Stackelberg Game

Abstract: The performance in cooperative communication depends on careful resource allocation such as relay selection and power control, but the traditional centralized resource allocation requires precise measurements of channel state information (CSI). In this paper, we propose a distributed game-theoretical framework over multiuser cooperative communication networks to achieve optimal relay selection and power allocation without knowledge of CSI. A two-level Stackelberg game is employed to jointly consider the benefits of the source node and the relay nodes in which the source node is modeled as a buyer and the relay nodes are modeled as sellers, respectively. The proposed approach not only helps the source find the relays at relatively better locations and "buyrdquo an optimal amount of power from the relays, but also helps the competing relays maximize their own utilities by asking the optimal prices. The game is proved to converge to a unique optimal equilibrium. Moreover, the proposed resource allocation scheme with the distributed game can achieve comparable performance to that employing centralized schemes.

Stackelberg game for utility-based cooperative cognitiveradio networks

Abstract: With the development of cognitive radio technologies, dynamic spectrum access becomes a promising approach to increase the efficiency of spectrum utilization and solve spectrum scarcity problem. Under dynamic spectrum access, unlicensed wireless users (secondary users) can dynamically access the licensed bands from legacy spectrum holders (primary users) on an opportunistic basis. While most primary users in existing works assume secondary transmissions as negative interference and don't actively involve them into the primary transmission, in this paper, motivated by the idea of cooperative communication, we propose a cooperative cognitive radio framework, where primary users, aware of the existence of secondary users, may select some of them to be the cooperative relay, and in return lease portion of the channel access time to them for their own data transmission. Secondary users cooperating with primary transmissions have the right to decide their payment made for primary user in order to achieve a proportional access time to the wireless media. Both primary and secondary users target at maximizing their utilities in terms of their transmission rate and revenue/payment. This model is formulated as a Stackelberg game and a unique Nash Equilibrium point is achieved in analytical format. Based on the analysis we discuss the condition under which cooperation will increase the performance of the whole system. Both analytical result and numerical result show that the cooperative cognitive radio framework is a promising framework under which the utility of both primary and secondary system are maximized.

Cooperative Advertising and Pricing in a Dynamic Stochastic Supply Chain: Feedback Stackelberg Strategies

Abstract: Cooperative (co-op) advertising is an important instrument for aligning manufacturer and retailer decisions in supply chains. In this, the manufacturer announces a co-op advertising policy, i.e., a participation rate that specifies the percentage of the retailer's advertising expenditure that it will provide. In addition, it also announces the wholesale price. In response, the retailer chooses its optimal advertising and pricing policies. We model this supply chain problem as a stochastic Stackelberg differential game whose dynamics follows Sethi's stochastic sales-advertising model. We obtain the condition when offering co-op advertising is optimal for the manufacturer. We provide in feedback form the optimal advertising and pricing policies for the manufacturer and the retailer. We contrast the results with the advertising and price decisions of the vertically integrated channel, and suggest a method for coordinating the channel.

A Stochastic Multiple-Leader Stackelberg Model: Analysis, Computation, and Application

Abstract: We study an oligopoly consisting of M leaders and N followers that supply a homogeneous product (or service) noncooperatively. Leaders choose their supply levels first, knowing the demand function only in distribution. Followers make their decisions after observing the leader supply levels and the realized demand function. We term the resulting equilibrium a stochastic multiple-leader Stackelberg-Nash-Cournot (SMS) equilibrium. We show the existence and uniqueness of SMS equilibrium under mild assumptions. We also propose a computational approach to find the equilibrium based on the sample average approximation method and analyze its rate of convergence. Finally, we apply this framework to model competition in the telecommunication industry.

Learning and Approximating the Optimal Strategy to Commit To

Abstract: Computing optimal Stackelberg strategies in general two-player Bayesian games (not to be confused with Stackelberg strategies in routing games) is a topic that has recently been gaining attention, due to their application in various security and law enforcement scenarios. Earlier results consider the computation of optimal Stackelberg strategies, given that all the payoffs and the prior distribution over types are known. We extend these results in two different ways. First, we consider learning optimal Stackelberg strategies. Our results here are mostly positive. Second, we consider computing approximately optimal Stackelberg strategies. Our results here are mostly negative.

Introducing hierarchy in energy games

Abstract: In this work, we introduce hierarchy in wireless networks that can be modeled by a decentralized multiple access channel and for which energy-efficiency is the main performance index. In these networks users are free to choose their power control strategy to selfishly maximize their energy-efficiency. Specifically, we introduce hierarchy in two different ways: 1. Assuming single-user decoding at the receiver, we investigate a Stackelberg formulation of the game where one user is the leader whereas the other users are assumed to be able to react to the leader's decisions; 2. Assuming neither leader nor followers among the users, we introduce hierarchy by assuming successive interference cancellation at the receiver. It is shown that introducing a certain degree of hierarchy in non-cooperative power control games not only improves the individual energy efficiency of all the users but can also be a way of insuring the existence of a non-saturated equilibrium and reaching a desired trade-off between the global network performance at the equilibrium and the requested amount of signaling. In this respect, the way of measuring the global performance of an energy-efficient network is shown to be a critical issue.

Coalition Stability in Assembly Models

Abstract: In this paper, we study dynamic supplier alliances in a decentralized assembly system. We examine a supply chain in which n suppliers sell complementary components to a downstream assembler, who faces a price-sensitive deterministic demand. We analyze alliance/coalition formation between suppliers, using a two-stage approach. In Stage 1, suppliers form coalitions that each agree to sell a kit of components to the assembler. In Stage 2, coalitions make wholesale price decisions, whereas the assembler buys the components (kits) from the coalitions and sets the selling price of the product. Stage 2 is modeled as a competitive game, in which the primary competition is vertical (i.e., supplier coalitions compete against the downstream assembler), and the secondary competition is horizontal, in that coalitions compete against each other. Here, we consider three modes of competition---Supplier Stackelberg, Vertical Nash, and Assembler Stackelberg models---that correspond to different power structures in the market. In Stage 1, we analyze the stability of coalition structures. We assume that suppliers are farsighted, that is, each coalition considers the possibility that once it acts, another coalition may react, and a third coalition might in turn react, and so on. Using this framework, we predict the structure of possible supplier alliances as a function of the power structure in the market, the number of suppliers, and the structure of the demand.

Cognitive Mobile Virtual Network Operator: Investment and Pricing with Supply Uncertainty

Abstract: This paper studies the optimal investment and pricing decisions of a cognitive mobile virtual network operator (C-MVNO) under spectrum supply uncertainty. Compared with a traditional MVNO who often leases spectrum via long-term contracts, a C-MVNO can acquire spectrum dynamically in short-term by both sensing the empty "spectrum holes" of licensed bands and dynamically leasing from the spectrum owner. As a result, a C-MVNO can make flexible investment and pricing decisions to match the current demands of the secondary unlicensed users. Compared to dynamic spectrum leasing, spectrum sensing is typically cheaper, but the obtained useful spectrum amount is random due to primary licensed users' stochastic traffic. The C-MVNO needs to determine the optimal amounts of spectrum sensing and leasing by evaluating the trade off between cost and uncertainty. The C-MVNO also needs to determine the optimal price to sell the spectrum to the secondary unlicensed users, taking into account wireless heterogeneity of users such as different maximum transmission power levels and channel gains. We model and analyze the interactions between the C-MVNO and secondary unlicensed users as a Stackelberg game. We show several interesting properties of the network equilibrium, including threshold structures of the optimal investment and pricing decisions, the independence of the optimal price on users' wireless characteristics, and guaranteed fair and predictable QoS among users. We prove that these properties hold for general SNR regime and general continuous distributions of sensing uncertainty. We show that spectrum sensing can significantly improve the C-MVNO's expected profit and users' payoffs.

Strategic inventory deployment for retail and e-tail stores

Abstract: In this paper, we study a supply chain comprising one manufacturer and one retailer. Customers can make purchases either from the retailer or directly from the manufacturer via an e-tail channel. From the manufacturer's perspective of managing the two channels, we study three different inventory strategies, namely centralized inventory strategy, a Stackelberg inventory strategy, and a strategy where the e-tail operation is outsourced to a third party logistics provider (3PL). For each strategy, we obtain the optimal inventory levels in retail and e-tail stores and the respective expected profits. We compare the performance of various strategies. Managerial insights are provided through analytical and numerical analyses to illustrate the applicability of different strategies.

Effective solutions for real-world Stackelberg games: when agents must deal with human uncertainties

Abstract: How do we build multiagent algorithms for agent interactions with human adversaries? Stackelberg games are natural models for many important applications that involve human interaction, such as oligopolistic markets and security domains. In Stackelberg games, one player, the leader, commits to a strategy and the follower makes their decision with knowledge of the leader's commitment. Existing algorithms for Stackelberg games efficiently find optimal solutions (leader strategy), but they critically assume that the follower plays optimally. Unfortunately, in real-world applications, agents face human followers (adversaries) who --- because of their bounded rationality and limited observation of the leader strategy --- may deviate from their expected optimal response. Not taking into account these likely deviations when dealing with human adversaries can cause an unacceptable degradation in the leader's reward, particularly in security applications where these algorithms have seen real-world deployment. To address this crucial problem, this paper introduces three new mixed-integer linear programs (MILPs) for Stackelberg games to consider human adversaries, incorporating: (i) novel anchoring theories on human perception of probability distributions and (ii) robustness approaches for MILPs to address human imprecision. Since these new approaches consider human adversaries, traditional proofs of correctness or optimality are insufficient; instead, it is necessary to rely on empirical validation. To that end, this paper considers two settings based on real deployed security systems, and compares 6 different approaches (three new with three previous approaches), in 4 different observability conditions, involving 98 human subjects playing 1360 games in total. The final conclusion was that a model which incorporates both the ideas of robustness and anchoring achieves statistically significant better rewards and also maintains equivalent or faster solution speeds compared to existing approaches.

Optimal Order Quantities with Remanufacturing Across New Product Generations

Abstract: We address the problem of determining the optimal retailer order quantities from a manufacturer who makes new products in conjunction with ordering remanufactured products from a remanufacturer using used and unsold products from the previous product generation. Specifically, we determine the optimal order quantity by the retailer for four systems of decision-making: (a) the three firms make their decisions in a coordinated fashion, (b) the retailer acts independently while the manufacturer and remanufacturer coordinate their decisions, (c) the remanufacturer acts independently while the retailer and manufacturer coordinate their decisions, and (d) all three firms act independently. We model the four options described above as centralized or decentralized decision-making systems with the manufacturer being the Stackelberg leader and provide insights into the optimal order quantities. Coordination mechanisms are then provided which enable the different players to achieve jointly the equivalent profits in a coordinated channel.

A Game Theoretical Model for Adversarial Learning

Abstract: It is now widely accepted that in many situations where classifiers are deployed, adversaries deliberately manipulate data in order to reduce the classifier’s accuracy. The most prominent example is email spam, where spammers routinely modify emails to get past classifier-based spam filters. In this paper we model the interaction between the adversary and the data miner as a two-person sequential noncooperative Stackelberg game and analyze the outcomes when there is a natural leader and a follower. We then proceed to model the interaction (both discrete and continuous) as an optimization problem and note that even solving linear Stackelberg game is NP-Hard. Finally we use a real spam email data set and evaluate the performance of local search algorithm under different strategy spaces.

A new perspective on multi-user power control games in interference channels

Abstract: This paper considers the problem of how to allocate power among competing users sharing a frequency-selective interference channel. We model the interaction between selfish users as a non-cooperative game. As opposed to the existing iterative water-filling algorithm that studies the myopic users, this paper studies how a foresighted user, who knows the channel state information and response strategies of its competing users, should optimize its transmission strategy. To characterize this multi-user interaction, the Stackelberg equilibrium is introduced, and the existence of this equilibrium for the investigated noncooperative game is shown. We analyze this interaction in more detail using a simple two-user example, where the foresighted user determines its transmission strategy by solving as a bi-level program which allows him to account for the myopic user's response. It is analytically shown that a foresighted user can improve its performance, if it has the necessary information about its competitors. Since the optimal solution of Stackelberg equilibrium is computationally prohibitive, we propose a practical low-complexity approach based on Lagrangian duality theory. Numerical simulations verify the performance improvements. Possible ways to acquire the required information and to extend the formulation to more than two users are also discussed.

Spatial SINR Games Combining Base Station Placement and Mobile Association

Abstract: We study in this paper the question of determining locations of base stations (BSs) that may belong to the same or to competing service providers, taking into account the impact of these decisions on the behavior of intelligent mobile terminals who can connect to the base station that offers the best utility. We first study the SINR association-game: we determine the cells corresponding to each base stations, i.e. the locations at which mobile terminals prefer to connect to a given base station than to other. The signal to interference and noise ratio (SINR) is used as the quantity that determines the association. We make some surprising observations: (i) displacing a base station a little in one direction may result in a displacement of the boundary of the corresponding cell to the opposite direction; (ii) A cell corresponding to a BS may be the union of disconnected sub-cells. We then study the Stackelberg equilibrium in the combined BS location and mobile association problem: we determine where to locate the BSs so as to maximize the revenues obtained at the induced SINR mobile association game. We consider the cases of single frequency band and two frequency bands of operation. Finally, we also consider Stackelberg equilibria in two frequency systems with successive interference cancellation.

Stackelberg Strategies for Selfish Routing in General Multicommodity Networks

Abstract: A natural generalization of the selfish routing setting arises when some of the users obey a central coordinating authority, while the rest act selfishly. Such behavior can be modeled by dividing the users into an α fraction that are routed according to the central coordinator’s routing strategy (Stackelberg strategy), and the remaining 1−α that determine their strategy selfishly, given the routing of the coordinated users. One seeks to quantify the resulting price of anarchy, i.e., the ratio of the cost of the worst traffic equilibrium to the system optimum, as a function of α. It is well-known that for α=0 and linear latency functions the price of anarchy is at most 4/3 (J. ACM 49, 236–259, 2002). If α tends to 1, the price of anarchy should also tend to 1 for any reasonable algorithm used by the coordinator. We analyze two such algorithms for Stackelberg routing, LLF and SCALE. For general topology networks, multicommodity users, and linear latency functions, we show a price of anarchy bound for SCALE which decreases from 4/3 to 1 as α increases from 0 to 1, and depends only on α. Up to this work, such a tradeoff was known only for the case of two nodes connected with parallel links (SIAM J. Comput. 33, 332–350, 2004), while for general networks it was not clear whether such a result could be achieved, even in the single-commodity case. We show a weaker bound for LLF and also some extensions to general latency functions. The existence of a central coordinator is a rather strong requirement for a network. We show that we can do away with such a coordinator, as long as we are allowed to impose taxes (tolls) on the edges in order to steer the selfish users towards an improved system cost. As long as there is at least a fraction α of users that pay their taxes, we show the existence of taxes that lead to the simulation of SCALE by the tax-payers. The extension of the results mentioned above quantifies the improvement on the system cost as the number of tax-evaders decreases.

Co-op Advertising in Dynamic Retail Oligopolies

Abstract: We study a supply chain in which a consumer goods manufacturer sells its product through a retailer The retailer undertakes promotional expenditures, uch as advertising, to increase sales and to compete against other retailer(s) The manufacturer supports the retailer’s promotional expenditure through a cooperative advertising program by reimbursing a portion (called the subsidy rate) of the retailer’s promotional expenditure To determine the subsidy rate, we formulate a Stackelberg differential game between the manufacturer and the retailer, and a Nash differential subgame between the retailer and the competing retailer(s) We derive the optimal feedback promotional expenditures of the retailers and the optimal feedback subsidy rate of the manufacturer, and show how they are influenced by market parameters An important finding is that the manufacturer should support its retailer only when a subsidy threshold is crossed The impact of competition on this threshold is non-monotone Specifically, the manufacturer offers more support when its retailer competes with one other retailer but its support starts decreasing with the presence of additional retailers In the case where the manufacturer sells through all retailers, we show under certain assumptions that it should support only one dominant retailer Also, we describe how we can incorporate retail price competition into the model

Stackelberg contention games in multiuser networks

Abstract: Interactions among selfish users sharing a common transmission channel can be modeled as a noncooperative game using the game theory framework. When selfish users choose their transmission probabilities independently without any coordination mechanism, Nash equilibria usually result in a network collapse. We propose a methodology that transforms the noncooperative game into a Stackelberg game. Stackelberg equilibria of the Stackelberg game can overcome the deficiency of the Nash equilibria of the original game. A particular type of Stackelberg intervention is constructed to show that any positive payoff profile feasible with independent transmission probabilities can be achieved as a Stackelberg equilibrium payoff profile.We discuss criteria to select an operating point of the network and informational requirements for the Stackelberg game. We relax the requirements and examine the effects of relaxation on performance.

Mixed oligopoly, sequential entry, and spatial price discrimination

Abstract: This paper is the first to examine the welfare consequences of a public firm in a traditional model of spatial price discrimination. It demonstrates that when a private firm acts as a Stackelberg location leader, the presence of a public firm always improves welfare. Moreover, when three firms locate sequentially, the presence of a public firm improves social welfare unless it locates last. Thus, despite examining a variety of location timings, including simultaneous location, privatization never improves welfare and usually harms welfare. This conclusion differs from several currently in the literature in which privatization often improves welfare.

Phenomena in Inverse Stackelberg Games, Part 1: Static Problems

Abstract: Dynamic two-person games are considered, in which the roles of the players are hierarchical. One player behaves as a leader, the other one as a follower. Such games are named after Stackelberg. In the current paper, a special type of such games is considered, known in the literature as inverse Stackelberg games. In such games, the leader announces his strategy as a mapping from the follower’s decision space into his own decision space. Arguments for studying such problems are given. This paper specifically studies dynamic games, i.e. the underlying model is described by an ordinary differential equation. The decisions of both players have a time component. As in the static case, the routine way of analysis, leading to a study of composed functions, is not very fruitful. Other approaches are given, mainly by studying specific examples.

Profit Raising Entry

Abstract: Common wisdom suggests that entry reduces profits of incumbent firms. On the contrary, we demonstrate that if the incumbents differ in marginal costs and the entrants behave like Stackelberg followers, then entry may benefit the cost efficient incumbents while hurting the cost inefficient ones. And the total outputs of all incumbents may be higher under entry.

On a branch-and-bound approach for a Huff-like Stackelberg location problem

Abstract: Modelling the location decision of two competing firms that intend to build a new facility in a planar market can be done by a Huff-like Stackelberg location problem. In a Huff-like model, the market share captured by a firm is given by a gravity model determined by distance calculations to facilities. In a Stackelberg model, the leader is the firm that locates first and takes into account the actions of the competing chain (follower) locating a new facility after the leader. The follower problem is known to be a hard global optimisation problem. The leader problem is even harder, since the leader has to decide on location given the optimal action of the follower. So far, in literature only heuristic approaches have been tested to solve the leader problem. Our research question is to solve the leader problem rigorously in the sense of having a guarantee on the reached accuracy. To answer this question, we develop a branch-and-bound approach. Essentially, the bounding is based on the zero sum concept: what is gain for one chain is loss for the other. We also discuss several ways of creating bounds for the underlying (follower) sub-problems, and show their performance for numerical cases.