Abstract of Proposal ?Collaborative Research: The Nonatomic-game Approach to Revenue Management under Competition?
The grant provides funding for the characterization of nonatomic-game equilibria of revenue management problems. Due to its innate difficulty, revenue management problems in competitive settings have not been adequately examined. This project will avoid complexities posed by multi-firm interactions by going to the infinite-firm extreme, considering a continuum of firms among whom none has any discernible impact on the overall evolution of the market. The focus of the project will be the establishment and characterization of equilibrium market processes and pricing policies that mutually sustain one another. Two types of problems involving stochastic and price-sensitive demand arrivals will be tackled, with each posing its unique challenges. In one type, firms can freely choose from a menu of prices, while in another type, firms cannot revisit prices they have themselves charged. Deterministic counterparts will be studied as benchmarks for heavy-volume limits, and computational studies will be conducted to assess the appropriateness of the infinite-firm approximation. If successful, results of this research will lead to a deeper understanding of competitive revenue management problems. General methods, specific tools, and intermediate results of this research will likely be applicable to theoretical endeavors elsewhere. In particular, novel uses of the Knaster-Taski and Kakutani-Glicksberg-Fan fixed point theorems may inspire further research. Equilibrium pricing policies identified by the project will serve as useful guidelines to real firms practicing dynamic price adjustments in response to competitors? price movements. The proposed computational study will address the problem of how accurate the infinite-firm approximation of the real finite-firm setting can be. Research outcomes will prove to be more practicable if it turns out that fewer firms are needed in order to realize strong incentives for real firms to adopt nonatomic-game pricing equilibria.
This research focuses on helping companies decide the best prices to charge in the face of intense competition and changing inventory levels. The traditional revenue management problem concerning one single firm is already quite difficult when demand reacts to prices in a random fashion and the concerned firm has to keep track of its own inventory levels. In the presence of multiple competitors whose pricing actions change unpredictably and whose inventory levels are unobservable, a competing firm faces an even daunting task. We treat this problem approximately by going to the infinite-firm limit. The benefit provided by this limit is that no single firm holds any sway over the overall market condition. This way, we can again talk about an optimal policy, one that is a best response to a given market trajectory which is the aggregate of all firms’ actions. This is in contrast to competitive situations where we have to deal with equilibria, a game-theoretic concept. In addition, the randomness of all firms cancels out each other with the resultant market trajectory being deterministic in nature. Hence, the infinite-firm approach simplifies rather than complicates the problem. Using this nonatomic-game (no player constitutes any undividable atom) approach, we study two versions of the competitive dynamic pricing problem. In one version, any firm can freely charge a given menu of prices, while in another version, prices charged earlier cannot be revisited. For both situations, we identify best prices as functions of both the time and the current firm’s own inventory level. All firms behaving thusly would help lead the market trajectory to one that induces these pricing policies. A nice feature about these pricing policies is that they are simple in not dependent on competing firms’ status. A relevant question is of course how good our infinite-firm approximation can be for real finite-firm situations. To this, we have tried to answer from both theoretical and computational aspects. Theoretically, we established that policies identified in infinite-firm settings would offer nearly best actions to finite-firm situations when the number of firms increases indefinitely. Our computational tests verified that in certain applications, having 30 to 50 firms would suffice for the infinite-firm approach to work reasonably well. Besides the above main thrust of research, we have investigated a slew of closely related problems. For general infinite-player competitive situations exhibiting complementary relationships, we provided general theoretical results. For single-firm dynamic pricing situation involving time-dependent demands, we gave shapes of optimal pricing policies. For problems involving not only pricing but also production or ordering, we showed how joint pricing-production policies could be formed. Our research outcomes have been recorded in eight papers. Among them, one has been published, two have been accepted, and the other five are at various stages of the peer review process. The following is a summary of these papers’ status: Yang, J. 2011. Asymptotic Interpretations for Equilibria of Nonatomic Games. Journal of Mathematical Economics, 47, pp. 491-499. Chen, F.Y., W. Xue, and J. Yang. 2012. Note: Optimal Inventory Policy in the Presence of a Long-term Supplier and a Spot Market. Operations Research, forthcoming. Yang, J. and Y. Xia. 2011. A Nonatomic-game Approach to Dynamic Pricing under Competition. Production and Operations Management, forthcoming. Yang, J., Y. Xia, X. Qi, and Y. Liu. 2011. A Nonatomic-game Model for Clearance Sales under Competition. Under 2nd review with Naval Research Logistics. Liu, Y. and J. Yang. 2011. Joint Pricing-procurement Control under Fluctuating Raw Material Prices. Under 2nd review with Naval Research Logistics Yang, J. and X. Qi. 2011. The Nonatomic Supermodular Game. Under 2nd review with Games and Economic Behavior. Liu, Y. and J. Yang. 2012. A Revisit to the Markup Practice of Dynamic Pricing, Under 1st review with Naval Research Logistics. Yang, J. 2012. Analysis of Markovian Competitive Situations using NonatomicGames. Under 1st review with Games and Economic Behavior. The PI's page at http://andromeda.rutgers.edu/~jy380 contains links to these papers. Research findings were disseminated in academic conferences as well. These include the third annual OCSAMSE conference in July 2010 in Beijing, China, the 2011 annual INFORMS meeting in Charlotte, NC, the 2012 annual POMS meeting in Chicago, as well as the CMMI grantees' conferences held in January 2011 in Atlanta, GA and in July 2012 in Boston, MA. This award benefited the Ph.D. education of two students at NJIT. Mojisola Otegbeye received one-year’s support from this project and finished her thesis on the topic of "Addressing the Procurement and Inventory Decision in a Volatile Commodity Price Environment". While working extensively on two research projects, Yifeng Liu gained valuable experience that would benefit him in his future career. He was able to expand his knowledge base in stochastic processes, optimization, dynamic programming, and game theory while dealing with technical problems involved in these projects. The computational requirements of the projects helped to hone Yifeng's programming skills.
