Showing papers on "Stackelberg competition published in 2008"

Optimal Pricing of Seasonal Products in the Presence of Forward-Looking Consumers

Abstract: We study the optimal pricing of a finite quantity of a fashion-like seasonal good in the presence of forward-looking (strategic) customers. We distinguish between two classes of pricing strategies: contingent and announced fixed-discount. In both cases, the seller acts as a Stackelberg leader announcing his pricing strategy, while consumers act as followers taking the seller's strategy as given and determining their purchasing behavior. In each case, we identify a subgame-perfect Nash equilibrium and show that given the seller's strategy, the equilibrium in the consumer subgame is unique and consists of symmetric threshold purchasing policies. For both cases, we develop a benchmark model in which customers are nonstrategic (myopic). We conduct a comprehensive numerical study to explore the impact of strategic consumer behavior on pricing policies and expected revenue performance. We show that strategic customer behavior suppresses the benefits of price segmentation, particularly under medium-to-high values of heterogeneity and modest rates of decline in valuations. However, when the level of consumer heterogeneity is small, the rate of decline is medium-to-high, and the seller can optimally choose the time of discount in advance, segmentation can be used quite effectively even with strategic consumers. We find that the seller cannot avoid the adverse impact of strategic consumer behavior even under low levels of initial inventory. We argue that while the seller expects customers to be more concerned about product availability at discount time, he cannot use high-price “betting” strategies as he would in the case of low inventory and myopic customers. Under certain qualifications, announced fixed-discount strategies perform essentially the same as contingent pricing policies in the case of myopic consumers. However, under strategic consumer behavior, announced pricing policies can be advantageous to the seller, compared to contingent pricing schemes. Interestingly, those cases that announced discount strategies offer a significant advantage compared to contingent pricing policies. They appear to offer only a minimal advantage in comparison to fixed-pricing policies. Finally, when the seller incorrectly assumes that strategic customers are myopic in their purchasing decisions, it can be quite costly, reaching potential revenue losses of about 20%.

Playing games for security: an efficient exact algorithm for solving Bayesian Stackelberg games

Abstract: In a class of games known as Stackelberg games, one agent (the leader) must commit to a strategy that can be observed by the other agent (the follower or adversary) before the adversary chooses its own strategy. We consider Bayesian Stackelberg games, in which the leader is uncertain about the types of adversary it may face. Such games are important in security domains, where, for example, a security agent (leader) must commit to a strategy of patrolling certain areas, and a robber (follower) has a chance to observe this strategy over time before choosing its own strategy of where to attack. This paper presents an efficient exact algorithm for finding the optimal strategy for the leader to commit to in these games. This algorithm, DOBSS, is based on a novel and compact mixed-integer linear programming formulation. Compared to the most efficient algorithm known previously for this problem, DOBSS is not only faster, but also leads to higher quality solutions, and does not suffer from problems of infeasibility that were faced by this previous algorithm. Note that DOBSS is at the heart of the ARMOR system that is currently being tested for security scheduling at the Los Angeles International Airport.

The Water-Filling Game in Fading Multiple-Access Channels

Abstract: A game-theoretic framework is developed to design and analyze the resource allocation algorithms in fading multiple-access channels (MACs), where the users are assumed to be selfish, rational, and limited by average power constraints. The maximum sum-rate point on the boundary of the MAC capacity region is shown to be the unique Nash equilibrium of the corresponding water-filling game. This result sheds a new light on the opportunistic communication principle. The base station is then introduced as a player interested in maximizing a weighted sum of the individual rates. A Stackelberg formulation is proposed in which the base station is the designated game leader. In this setup, the base station announces first its strategy defined as the decoding order of the different users, in the successive cancellation receiver, as a function of the channel state. In the second stage, the users compete conditioned on this particular decoding strategy. This formulation is shown to be able to achieve all the corner points of the capacity region, in addition to the maximum sum-rate point. On the negative side, it is shown that there does not exist a base station strategy in this formulation that achieves the rest of the boundary points. To overcome this limitation, a repeated game approach, which achieves the capacity region of the fading MAC, is presented. Finally, the study is extended to vector channels highlighting interesting differences between this scenario and the scalar channel case.

Structural Properties of Buyback Contracts for Price-Setting Newsvendors

Abstract: This paper studies a buyback contract in the Stackelberg framework of a manufacturer (leader) selling to a price-setting newsvendor retailer (follower). Using an analytical model that focuses on a multiplicative demand form, we generalize previous results and produce new structural insights. A novel transformation technique first enables us to establish the unimodality of the profit functions for both channel partners, under relatively mild assumptions. Further analysis identifies the necessary and sufficient condition under which the optimal contract for the manufacturer (wholesale and buyback prices) is distribution free, i.e., independent of the uncertainty in customer demand. A specific instance of the above condition is also necessary and sufficient for a no-buyback contract to be optimal from the manufacturer's perspective. We then prove that the optimal performance of the decentralized channel for distribution-free buyback contracts depends only on the curvature of the deterministic demand part. In addition, some of the optimal decisions and relevant profit ratios for buyback contracts in our setting are shown to be identical to those for their deterministic price-only counterparts.

Stackelberg Competition with Endogenous Entry

Abstract: I characterise endogenous market structures where leaders have a first-mover advantage and entry is endogenous. Leaders are always more aggressive than the followers, independently from strategic substitutability or complementarity. Under quantity competition, leaders produce more than any follower and I determine the conditions for entry-deterrence (high substitutability and non increasing marginal costs). Under price competition, leaders set lower prices than the followers (the opposite than with an exogenous number of firms). In contests, leaders invest more than each follower. In all these cases a leadership improves the allocation of resources compared to the Nash equilibrium with endogenous entry. This article studies endogenous market structures where a firm has a first mover advantage in the sense of Stackelberg (1934) on the other firms and entry in the market is endogenous. The article characterises the equilibrium of these endo genous market structures and provides welfare analysis for some of the applications. These market structures emerge in many sectors which are substantially competitive (a fringe of firms is ready to enter whenever there is a profitable opportunity) but where some incumbent firms have a competitive advantage over the followers (because of technological, historical or legal reasons or just because entry was not possible at an earlier stage) and choose their strategies before them.1 While symmetric models of strategic interactions have been widely studied in the presence of endogenous entry in a Marshallian tradition,2 traditional models with incumbent firms facing a competitive fringe of entrants have been mostly limited to the analysis of markets with homogeneous goods and entry deterrence, for instance in the theories of limit pricing and of contestable markets (Baumol et al, 1982). More recently, Vives (1988) has analysed games with sequential entry of multiple firms but without endogenising the entry process; see also Anderson and Engers (1992, 1994).

Competitive Pricing in Heterogeneous Wireless Access Networks: Issues and Approaches

Abstract: To provide seamless mobility with high-speed wireless connectivity, future generation wireless networks must support heterogeneous wireless access. Pricing schemes adopted by different service providers is crucial and will impact the decisions of users in selecting a network. In this article, we provide a comprehensive survey of the issues related to pricing in heterogeneous wireless networks and possible approaches to the solution of the pricing problem. First, we review the related work on pricing for homogeneous wireless networks in which a single wireless technology is available to the users. Then, we outline the major issues in designing resource allocation and pricing in heterogeneous wireless access networks. To this end, we propose two oligopolistic models for price competition among service providers in a heterogeneous wireless environment consisting of WiMax and WiFi access networks. A non-cooperative game is formulated to obtain the price for the service providers. Two different equilibria, namely, the Nash and the Stackelberg equilibria are considered as the solutions of the simultaneous-play and leader-follower price competitions, respectively.

A stackelberg game for pricing uplink power in wide-band cognitive radio networks

Abstract: We study the problem of pricing uplink power in wide-band cognitive radio networks under the objective of revenue maximization for the service provider and while ensuring incentive compatibility for the users. User utility is modeled as a concave function of the signal-to-noise ratio (SNR) at the base station, and the problem is formulated as a Stackelberg game. Namely, the service provider imposes differentiated prices per unit of transmitting power and the users consequently update their power levels to maximize their net utilities. We devise a pricing policy and give conditions for its optimality when all the users are to be accommodated in the network. We show that there exist infinitely many Nash equilibrium points that reward the service provider with the same revenue. The pricing policy charges more from users that have better channel conditions and more willingness to pay for the provided service. We then study properties of the optimal revenue with respect to different parameters in the network. We show that for regimes with symmetric users who share the same level of willingness to pay, the optimal revenue is concave and increasing in the number of users in the network. We analytically obtain achievable SNRs for this special case, and finally present a numerical study in support of our results.

Leadership and Competition in Network Supply Chains

Abstract: This paper considers network supply chains with price dependent demand by modelling them as large acyclic networks. Such large networks are common in the automobile and apparel industries. We develop a model to analyze the effect of these large-scale problems involving long sequences of contracts, and show that contract leadership, as well as leader position in the network, affect the performance of the entire supply chain. We generalize Spengler (Spengler, J. 1950. Vertical integration and anti-trust policy. J. Political Econom.58 347--352) to a game on a “contract tree” for a particular supply chain and extend the concept of double marginalization so that it can be applied in the form of a transformation to each contract that is offered by one member to another in the “contract tree.” We construct an algorithm to find the equilibrium solution, and derive the optimal location of the leader (“optimal” being the leader location that maximizes total supply chain profits). Our work formalizes many intuitive insights; for example, member profits are determined by systemwide rather than individual costs. Finally, we model Cournot competition between competing supply chains (both two heterogeneous trees and multiple identical trees) and show the effect of changes in leader position as well as cost structure on the equilibrium.

A game theoretic analysis of service competition and pricing in heterogeneous wireless access networks

Abstract: Next generation wireless systems will provide highspeed wireless connectivity and seamless mobility through the provisioning of heterogeneous wireless access. In such a heterogeneous wireless access environment, mobile users will be able to connect to multiple wireless networks (e.g., IEEE 802.16, cellular, and IEEE 802.11-based networks) operated by different service providers, simultaneously. We address the problem of competitive pricing in such a heterogeneous wireless access network. Each of the wireless access networks is assumed to support two types of connections, namely, premium and best-effort connections. For the premium connections, the price is fixed, while for the best effort connections it is dynamic and depends on the competitive or cooperative behavior of the service providers. A competitive pricing model for best-effort connections is developed based on a noncooperative game formulation. We first consider the case where the prices are offered to the users at the same time (i.e., a simultaneous-play game). Nash equilibrium is considered as the solution of this game. Afterwards, we consider the case where a service provider can offer its price before other providers. This is referred to as a leader-follower game for which Stackelberg equilibrium is considered as the solution. We also consider a cooperative pricing model which maximizes the total revenue of the service providers. A method for revenue sharing is presented for this cooperative pricing model. Numerical studies are carried out to evaluate the performances of the different pricing models.

Optimal Product Design Under Price Competition

Abstract: Engineering optimization methods for new product development model consumer demand as a function of product attributes and price in order to identify designs that maximize expected profit. However, prior approaches have ignored the ability of competitors to react to a new product entrant; thus these methods can overestimate expected profit and select suboptimal designs that perform poorly in a competitive market. We propose an efficient approach to new product design accounting for competitor pricing reactions by imposing Nash and Stackelberg conditions as constraints, and we test the method on three product design case studies from the marketing and engineering design literature. We find that a Stackelberg leader strategy generates higher profit than a Nash strategy. Both strategies are superior to ignoring competitor reactions: In our case studies, ignoring price competition results in overestimation of profits by 12%–79%, and accounting for price competition increases realized profits by up to 3.4%. The efficiency, convergence stability, and ease of implementation of the proposed approach enables practical implementation for new product design problems in competitive markets.Copyright © 2008 by ASME

Bayesian stackelberg games and their application for security at Los Angeles international airport

Abstract: Many multiagent settings are appropriately modeled as Stackelberg games [Fudenberg and Tirole 1991; Paruchuri et al. 2007], where a leader commits to a strategy first, and then a follower selfishly optimizes its own reward, considering the strategy chosen by the leader. Stackelberg games are commonly used to model attacker-defender scenarios in security domains [Brown et al. 2006] as well as in patrolling [Paruchuri et al. 2007; Paruchuri et al. 2008]. For example, security personnel patrolling an infrastructure commit to a patrolling strategy first, before their adversaries act taking this committed strategy into account. Indeed, Stackelberg games are being used at the Los Angeles International Airport to schedule security checkpoints and canine patrols [Murr 2007; Paruchuri et al. 2008; Pita et al. 2008a]. They could potentially be used in network routing, pricing in transportation systems and many other situations [Korilis et al. 1997; Cardinal et al. 2005]. Although the follower in a Stackelberg game is allowed to observe the leader’s strategy before choosing its own strategy, there is often an advantage for the leader over the case where both players must choose their moves simultaneously. To see the advantage of being the leader in a Stackelberg game, consider the game with the payoff as shown in Table I. The leader is the row player and the follower is the column player. The only pure-strategy Nash equilibrium for this game is when the leader plays a and the follower plays c which gives the leader a payoff of 2. However, if the leader commits to a mixed strategy of playing a and b with equal (0.5) probability, then the follower will play d, leading to an expected payoff for the leader of 3.5.

Analysis of Price Competition in a Slotted Resource Allocation Game

Abstract: Pricing telecommunication networks has become a highly regarded topic during the last decade, in order to cope with congestion by controlling demand, or to yield proper incentives for a fair sharing of resources. On the other hand, another important factor has to be brought in: there is a rise of competition between service providers in telecommunication networks such as for instance the Internet, and the impact of this competition has to be carefully analyzed. The present paper pertains to this recent stream of works. We consider a slotted resource allocation game with several providers, each of them having a fixed capacity during each time slot, and a fixed access price. Each provider serves its demand up to its capacity, demand in excess being dropped. Total user demand is therefore split among providers according to Wardrop's principle, depending on price and loss probability. Using the characterization of the resulting equilibrium, we prove, under mild conditions, the existence and uniqueness of a Nash equilibrium in the pricing game between providers. We also show that, remarkably, this equilibrium actually corresponds to the socially optimal situation obtained when both users and providers cooperate to maximize the sum of all utilities, this even if providers have the opportunity to artificially reduce their capacity.

Component Procurement Strategies in Decentralized Assemble-to-Order Systems with Time-Dependent Pricing

Abstract: We consider a contract manufacturer who procures multiple components from independent suppliers to produce an assemble-to-order customized product for a client. The unit price of the product depends on the manufacturer's delivery lead time. We explore how the manufacturer can use a vendor-managed consignment inventory (VMCI) scheme to manage the underlying risk and coordinate independent suppliers' decisions on the production quantities of their components under demand uncertainty. We formulate the problem as a Stackelberg game played by the manufacturer against her component suppliers to determine her pricing policy for suppliers' consignment inventories. We further develop an efficient algorithm for finding the manufacturer's optimal pricing scheme. Our results provide useful insights for managing components in these types of assemble-to-order environments and for understanding how component production cost and procurement lead times affect individual firms' performance in decentralized assembly channels.

Application of game theory for duopoly market analysis

Abstract: The paper provides the analysis of game theory models application to identify duopoly market equilibrium (quantities sold and market prices), to evaluate and compare the results of enterprises in a market. The purpose of the analysis is to determine to what extent theoretical models correspond to real life, that is how reliable they are in supporting and estimating decisions of duopoly companies, fortifying market prices and quantities sold, evaluating company's competing positions and possibilities for decision co‐ordination. To describe discrete strategies equilibrium the “Prisoner's Dilemma” model is applied to a hypothetic market entrance game with possible side payments. Further analysis of the market entrance game incorporates mixed strategies based “Matching Pennies” model in case discrete strategies equilibrium does not exist. Continuous strategies are described analyzing hypothetic duopoly by applying Cournot, Stackelberg and Bertrand models. The first and the second mover advantage issu...

Efficient algorithms to solve Bayesian Stackelberg games for security applications

Abstract: In a class of games known as Stackelberg games, one agent (the leader) must commit to a strategy that can be observed by the other agent (the adversary/follower) before the adversary chooses its own strategy. We consider Bayesian Stackelberg games, in which the leader is uncertain about the type of the adversary it may face. Such games are important in security domains, where, for example, a security agent (leader) must commit to a strategy of patrolling certain areas, and an adversary (follower) can observe this strategy over time before choosing where to attack. We present here two different MIP-formulations, ASAP (providing approximate policies with controlled randomization) and DOBSS (providing optimal policies) for Bayesian Stackelberg games. DOBSS is currently the fastest optimal procedure for Bayesian Stackelberg games and is in use by police at the Los Angeles International Airport(LAX) to schedule their activities.

Commitment, first-mover-, and second-mover advantage

Abstract: We identify circumstances under which a firm with a first-mover advan- tage may get leapfrogged by a follower. At the market stage we assume a Stackelberg structure,i.e.theleadercommitstoaquantityandthefollowerreactstoit.Weallowthe owners of both firms to select the internal organization and the production technology before quantities are set. That is, leader and follower can additionally use two com- mitment strategies alternatively or in combination: investing in R&D and delegating quantity decisions to managers. Despite the symmetry of options for the twofirms, we find that there is a unique equilibrium in which bothfirms invest in process R&D, only thefollower delegates, and thefollower can overcome thefirst-mover advantage ofthe quantityleaderandobtainahigherprofitthantheleader.Ouranalysisrevealsthatthere are some important differences between the two commitment devices "cost-reducing R&D" and "delegation to managers".

Analysis of a revenue-sharing contract in supply chain management

Abstract: We consider a supply chain involving one supplier and one retailer in which a revenue-sharing contract is adopted. Under this contract, the retailer can obtain the product from the supplier at a discounted price. As a compensation, the retailer must share his revenue with the supplier at a certain revenue-sharing rate, say r (0≤r≤1), where r represents the portion of the revenue to be kept by the retailer. Our ultimate objective is to help the whole supply chain be more profitable while upholding the individual components' incentives. We use a two-stage (Stackelberg) game to model the problem, where one player is the game's leader and the other the game's follower. Our analysis reveals that for the supply chain to be more profitable, the party that keeps more than half the revenue should serve as the leader of the Stackelberg game.

Feedback Stackelberg equilibrium strategies when the private label competes with the national brand

Abstract: We consider a noncooperative differential game where a retailer sells her own private label in addition to the manufacturer’s brand. We assume that each brand’s goodwill evolves according to a modified Nerlove-Arrow dynamics, in such a way that the advertising effort of one brand hurts the competitor’s goodwill stock. We characterize Feedback-Stackelberg pricing and advertising strategies and employ simulations to analyze their sensitivity to the main model parameters.

Leading and merging:convex costs, Stackelberg and the merger paradox

Abstract: This paper examines the consequences of a Stackelberg leader merging with followers when costs are convex. Such mergers are always profitable for the participants, and the followers often do better merging than remaining excluded rivals. This resolution of the merger paradox cannot be generated either by Stackelberg leadership without convex costs or by convex costs without leadership. In addition, with convex costs, a merger with the leader can actually harm excluded rivals (suggesting why they might object to the merger) and increase social welfare.

The effectiveness of revenue‐sharing contract to coordinate the price‐setting newsvendor products' supply chain

Abstract: Purpose – The purpose of this paper is to numerically analyze the coordinating mechanism of the revenue‐sharing contract in newsvendor products' supply chain The contract considers stochastic and price‐dependent demandDesign/methodology/approach – The paper presents an analytical model for supply chain contracts and then uses numerical methods with the Stackelberg game to identify the contracts' propertiesFindings – Comparing the revenue‐sharing contract with the price‐only contract, the paper finds that a revenue‐sharing contract does improve supply chain performance However, the benefits earned by the revenue‐sharing contract differ among the supply chain partners under the impact of demand variability and price‐sensitivity factors Specifically, the manufacturer will earn a greater share in channel profit in the revenue‐sharing contract than in case of a price‐only contract Also, the incentive by revenue‐sharing contract could not lead to Pareto improvement In other words, the retailer's profit i

Stackelberg Network Pricing Games

Abstract: 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 $(1+varepsilon) log m$-approximation for any $varepsilon >0$. This can be extended to provide a $(1+varepsilon )(log k + log m)$-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 $mathcal{O(log^varepsilon k + log^varepsilon m)$. If followers have demands, the single-price algorithm provides a $(1+varepsilon )m^2$-approximation, and the problem is hard to approximate within $mathcal{O(m^varepsilon)$ for some $varepsilon >0$. 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.