Extreme-case risk measures provide an approach for quantifying the upper and lower bounds of risk in situations where limited information is available regarding the underlying distributions. Previous research has demonstrated that for popular risk measures, such as value-at-risk and conditional value-at-risk, the worst-case counterparts can be evaluated in closed form when only the first two moments of the underlying distributions are known. In this study, we extend these findings by presenting closed-form solutions for a general class of distortion risk measures, which consists of various popular risk measures as special cases when the first and certain higher-order (i.e., second or more) absolute center moments, alongside the symmetry properties of the underlying distributions, are known. Moreover, we characterize the extreme-case distributions with convex or concave envelopes of the corresponding distributions. By providing closed-form solutions for extreme-case distortion risk measures and characterizations for the corresponding distributions, our research contributes to the understanding and application of risk quantification methodologies.Funding: H. Shao acknowledges support from the Yangtze River Delta Science and Technology Innovation Community Joint Research Program [Grant 2022CSJGG0800]. Z. G. Zhang acknowledges support from the Canadian Network for Research and Innovation in Machining Technology, Natural Sciences and Engineering Research Council of Canada [Grant RGPIN-2019-06364].Supplemental Material: The online appendix is available at https://doi.org/10.1287/moor.2022.0156 .
{"title":"Extreme-Case Distortion Risk Measures: A Unification and Generalization of Closed-Form Solutions","authors":"Hui Shao, Zhe George Zhang","doi":"10.1287/moor.2022.0156","DOIUrl":"https://doi.org/10.1287/moor.2022.0156","url":null,"abstract":"Extreme-case risk measures provide an approach for quantifying the upper and lower bounds of risk in situations where limited information is available regarding the underlying distributions. Previous research has demonstrated that for popular risk measures, such as value-at-risk and conditional value-at-risk, the worst-case counterparts can be evaluated in closed form when only the first two moments of the underlying distributions are known. In this study, we extend these findings by presenting closed-form solutions for a general class of distortion risk measures, which consists of various popular risk measures as special cases when the first and certain higher-order (i.e., second or more) absolute center moments, alongside the symmetry properties of the underlying distributions, are known. Moreover, we characterize the extreme-case distributions with convex or concave envelopes of the corresponding distributions. By providing closed-form solutions for extreme-case distortion risk measures and characterizations for the corresponding distributions, our research contributes to the understanding and application of risk quantification methodologies.Funding: H. Shao acknowledges support from the Yangtze River Delta Science and Technology Innovation Community Joint Research Program [Grant 2022CSJGG0800]. Z. G. Zhang acknowledges support from the Canadian Network for Research and Innovation in Machining Technology, Natural Sciences and Engineering Research Council of Canada [Grant RGPIN-2019-06364].Supplemental Material: The online appendix is available at https://doi.org/10.1287/moor.2022.0156 .","PeriodicalId":49852,"journal":{"name":"Mathematics of Operations Research","volume":"229 4","pages":""},"PeriodicalIF":1.7,"publicationDate":"2023-11-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138508627","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Bhaskar Ray Chaudhury, Jugal Garg, Kurt Mehlhorn, Ruta Mehta, Pranabendu Misra
We study the problem of fairly allocating a set of indivisible goods among n agents with additive valuations. Envy freeness up to any good (EFX) is arguably the most compelling fairness notion in this context. However, the existence of an EFX allocation has not been settled and is one of the most important problems in fair division. Toward resolving this question, many impressive results show the existence of its relaxations. In particular, it is known that 0.618-EFX allocations exist and that EFX allocation exists if we do not allocate at most (n-1) goods. Reducing the number of unallocated goods has emerged as a systematic way to tackle the main question. For example, follow-up works on three- and four-agents cases, respectively, allocated two more unallocated goods through an involved procedure. In this paper, we study the general case and achieve sublinear numbers of unallocated goods. Through a new approach, we show that for every [Formula: see text], there always exists a [Formula: see text]-EFX allocation with sublinear number of unallocated goods and high Nash welfare. For this, we reduce the EFX problem to a novel problem in extremal graph theory. We define the notion of rainbow cycle number [Formula: see text] in directed graphs. For all [Formula: see text] is the largest k such that there exists a k-partite graph [Formula: see text], in which each part has at most d vertices (i.e., [Formula: see text] for all [Formula: see text]); for any two parts Vi and Vj, each vertex in Vi has an incoming edge from some vertex in Vj and vice versa; and there exists no cycle in G that contains at most one vertex from each part. We show that any upper bound on [Formula: see text] directly translates to a sublinear bound on the number of unallocated goods. We establish a polynomial upper bound on [Formula: see text], yielding our main result. Furthermore, our approach is constructive, which also gives a polynomial-time algorithm for finding such an allocation.Funding: J. Garg was supported by the Directorate for Computer and Information Science and Engineering [Grant CCF-1942321]. R. Mehta was supported by the Directorate for Computer and Information Science and Engineering [Grant CCF-1750436].
{"title":"Improving Envy Freeness up to Any Good Guarantees Through Rainbow Cycle Number","authors":"Bhaskar Ray Chaudhury, Jugal Garg, Kurt Mehlhorn, Ruta Mehta, Pranabendu Misra","doi":"10.1287/moor.2021.0252","DOIUrl":"https://doi.org/10.1287/moor.2021.0252","url":null,"abstract":"We study the problem of fairly allocating a set of indivisible goods among n agents with additive valuations. Envy freeness up to any good (EFX) is arguably the most compelling fairness notion in this context. However, the existence of an EFX allocation has not been settled and is one of the most important problems in fair division. Toward resolving this question, many impressive results show the existence of its relaxations. In particular, it is known that 0.618-EFX allocations exist and that EFX allocation exists if we do not allocate at most (n-1) goods. Reducing the number of unallocated goods has emerged as a systematic way to tackle the main question. For example, follow-up works on three- and four-agents cases, respectively, allocated two more unallocated goods through an involved procedure. In this paper, we study the general case and achieve sublinear numbers of unallocated goods. Through a new approach, we show that for every [Formula: see text], there always exists a [Formula: see text]-EFX allocation with sublinear number of unallocated goods and high Nash welfare. For this, we reduce the EFX problem to a novel problem in extremal graph theory. We define the notion of rainbow cycle number [Formula: see text] in directed graphs. For all [Formula: see text] is the largest k such that there exists a k-partite graph [Formula: see text], in which each part has at most d vertices (i.e., [Formula: see text] for all [Formula: see text]); for any two parts V<jats:sub>i</jats:sub> and V<jats:sub>j</jats:sub>, each vertex in V<jats:sub>i</jats:sub> has an incoming edge from some vertex in V<jats:sub>j</jats:sub> and vice versa; and there exists no cycle in G that contains at most one vertex from each part. We show that any upper bound on [Formula: see text] directly translates to a sublinear bound on the number of unallocated goods. We establish a polynomial upper bound on [Formula: see text], yielding our main result. Furthermore, our approach is constructive, which also gives a polynomial-time algorithm for finding such an allocation.Funding: J. Garg was supported by the Directorate for Computer and Information Science and Engineering [Grant CCF-1942321]. R. Mehta was supported by the Directorate for Computer and Information Science and Engineering [Grant CCF-1750436].","PeriodicalId":49852,"journal":{"name":"Mathematics of Operations Research","volume":"232 2","pages":""},"PeriodicalIF":1.7,"publicationDate":"2023-11-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138508634","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
A major challenge of multiagent reinforcement learning (MARL) is the curse of multiagents, where the size of the joint action space scales exponentially with the number of agents. This remains to be a bottleneck for designing efficient MARL algorithms, even in a basic scenario with finitely many states and actions. This paper resolves this challenge for the model of episodic Markov games. We design a new class of fully decentralized algorithms—V-learning, which provably learns Nash equilibria (in the two-player zero-sum setting), correlated equilibria, and coarse correlated equilibria (in the multiplayer general-sum setting) in a number of samples that only scales with [Formula: see text], where Ai is the number of actions for the ith player. This is in sharp contrast to the size of the joint action space, which is [Formula: see text]. V-learning (in its basic form) is a new class of single-agent reinforcement learning (RL) algorithms that convert any adversarial bandit algorithm with suitable regret guarantees into an RL algorithm. Similar to the classical Q-learning algorithm, it performs incremental updates to the value functions. Different from Q-learning, it only maintains the estimates of V-values instead of Q-values. This key difference allows V-learning to achieve the claimed guarantees in the MARL setting by simply letting all agents run V-learning independently.Funding: This work was partially supported by Office of Naval Research Grant N00014-22-1-2253.
{"title":"V-Learning—A Simple, Efficient, Decentralized Algorithm for Multiagent Reinforcement Learning","authors":"Chi Jin, Qinghua Liu, Yuanhao Wang, Tiancheng Yu","doi":"10.1287/moor.2021.0317","DOIUrl":"https://doi.org/10.1287/moor.2021.0317","url":null,"abstract":"A major challenge of multiagent reinforcement learning (MARL) is the curse of multiagents, where the size of the joint action space scales exponentially with the number of agents. This remains to be a bottleneck for designing efficient MARL algorithms, even in a basic scenario with finitely many states and actions. This paper resolves this challenge for the model of episodic Markov games. We design a new class of fully decentralized algorithms—V-learning, which provably learns Nash equilibria (in the two-player zero-sum setting), correlated equilibria, and coarse correlated equilibria (in the multiplayer general-sum setting) in a number of samples that only scales with [Formula: see text], where A<jats:sub>i</jats:sub> is the number of actions for the ith player. This is in sharp contrast to the size of the joint action space, which is [Formula: see text]. V-learning (in its basic form) is a new class of single-agent reinforcement learning (RL) algorithms that convert any adversarial bandit algorithm with suitable regret guarantees into an RL algorithm. Similar to the classical Q-learning algorithm, it performs incremental updates to the value functions. Different from Q-learning, it only maintains the estimates of V-values instead of Q-values. This key difference allows V-learning to achieve the claimed guarantees in the MARL setting by simply letting all agents run V-learning independently.Funding: This work was partially supported by Office of Naval Research Grant N00014-22-1-2253.","PeriodicalId":49852,"journal":{"name":"Mathematics of Operations Research","volume":"35 1","pages":""},"PeriodicalIF":1.7,"publicationDate":"2023-11-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138543166","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
A single-server queue with renewal arrivals and generally distributed independent and identically distributed service times is considered. Customers are served using the longest remaining time first scheduling algorithm. In case of a tie, processor sharing is utilized. We introduce a fluid model for the evolution of a measure-valued state descriptor of this queue, and we investigate its properties. We also prove a fluid limit theorem justifying our fluid model as the first-order approximation of the queueing system under consideration.
{"title":"Fluid Limits for Longest Remaining Time First Queues","authors":"Łukasz Kruk","doi":"10.1287/moor.2023.0090","DOIUrl":"https://doi.org/10.1287/moor.2023.0090","url":null,"abstract":"A single-server queue with renewal arrivals and generally distributed independent and identically distributed service times is considered. Customers are served using the longest remaining time first scheduling algorithm. In case of a tie, processor sharing is utilized. We introduce a fluid model for the evolution of a measure-valued state descriptor of this queue, and we investigate its properties. We also prove a fluid limit theorem justifying our fluid model as the first-order approximation of the queueing system under consideration.","PeriodicalId":49852,"journal":{"name":"Mathematics of Operations Research","volume":"1 4","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-11-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135934325","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We propose a class of semimetrics for acyclic preference relations, any one of which is an alternative to the classical Kemeny-Snell-Bogart metric. These semimetrics are based solely on the implications of preferences for choice behavior and thus appear more suitable in economic contexts and choice experiments. We obtain a fairly simple axiomatic characterization for the class we propose. The apparently most important member of this class, which we dub the “top-difference semimetric,” is characterized separately. We also obtain alternative formulae for it and, relative to this particular metric, compute the diameter of the space of complete and transitive preferences, as well as the best transitive extension of a given acyclic preference relation. Supplemental Material: The e-companion is available at https://doi.org/10.1287/moor.2022.0351 .
{"title":"A Class of Dissimilarity Semimetrics for Preference Relations","authors":"Hiroki Nishimura, Efe A. Ok","doi":"10.1287/moor.2022.0351","DOIUrl":"https://doi.org/10.1287/moor.2022.0351","url":null,"abstract":"We propose a class of semimetrics for acyclic preference relations, any one of which is an alternative to the classical Kemeny-Snell-Bogart metric. These semimetrics are based solely on the implications of preferences for choice behavior and thus appear more suitable in economic contexts and choice experiments. We obtain a fairly simple axiomatic characterization for the class we propose. The apparently most important member of this class, which we dub the “top-difference semimetric,” is characterized separately. We also obtain alternative formulae for it and, relative to this particular metric, compute the diameter of the space of complete and transitive preferences, as well as the best transitive extension of a given acyclic preference relation. Supplemental Material: The e-companion is available at https://doi.org/10.1287/moor.2022.0351 .","PeriodicalId":49852,"journal":{"name":"Mathematics of Operations Research","volume":"23 20","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135863725","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Albert S. Berahas, Frank E. Curtis, Michael J. O'Neill, Daniel P. Robinson
A sequential quadratic optimization algorithm is proposed for solving smooth nonlinear-equality-constrained optimization problems in which the objective function is defined by an expectation. The algorithmic structure of the proposed method is based on a step decomposition strategy that is known in the literature to be widely effective in practice, wherein each search direction is computed as the sum of a normal step (toward linearized feasibility) and a tangential step (toward objective decrease in the null space of the constraint Jacobian). However, the proposed method is unique from others in the literature in that it both allows the use of stochastic objective gradient estimates and possesses convergence guarantees even in the setting in which the constraint Jacobians may be rank-deficient. The results of numerical experiments demonstrate that the algorithm offers superior performance when compared with popular alternatives. Funding: This material is based upon work supported by the U.S. National Science Foundation’s Division of Computing and Communication Foundations under award [CF-1740796], by the Office of Naval Research under award [N00014-21-1-2532], and by the National Science Foundation under award [2030859] to the Computing Research Association for the CIFellows Project.
{"title":"A Stochastic Sequential Quadratic Optimization Algorithm for Nonlinear-Equality-Constrained Optimization with Rank-Deficient Jacobians","authors":"Albert S. Berahas, Frank E. Curtis, Michael J. O'Neill, Daniel P. Robinson","doi":"10.1287/moor.2021.0154","DOIUrl":"https://doi.org/10.1287/moor.2021.0154","url":null,"abstract":"A sequential quadratic optimization algorithm is proposed for solving smooth nonlinear-equality-constrained optimization problems in which the objective function is defined by an expectation. The algorithmic structure of the proposed method is based on a step decomposition strategy that is known in the literature to be widely effective in practice, wherein each search direction is computed as the sum of a normal step (toward linearized feasibility) and a tangential step (toward objective decrease in the null space of the constraint Jacobian). However, the proposed method is unique from others in the literature in that it both allows the use of stochastic objective gradient estimates and possesses convergence guarantees even in the setting in which the constraint Jacobians may be rank-deficient. The results of numerical experiments demonstrate that the algorithm offers superior performance when compared with popular alternatives. Funding: This material is based upon work supported by the U.S. National Science Foundation’s Division of Computing and Communication Foundations under award [CF-1740796], by the Office of Naval Research under award [N00014-21-1-2532], and by the National Science Foundation under award [2030859] to the Computing Research Association for the CIFellows Project.","PeriodicalId":49852,"journal":{"name":"Mathematics of Operations Research","volume":"20 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136018095","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We consider the problem of fair allocation of indivisible goods to n agents with no transfers. When agents have equal entitlements, the well-established notion of the maximin share (MMS) serves as an attractive fairness criterion for which, to qualify as fair, an allocation needs to give every agent at least a substantial fraction of the agent’s MMS. In this paper, we consider the case of arbitrary (unequal) entitlements. We explain shortcomings in previous attempts that extend the MMS to unequal entitlements. Our conceptual contribution is the introduction of a new notion of a share, the AnyPrice share (APS), that is appropriate for settings with arbitrary entitlements. Even for the equal entitlements case, this notion is new and satisfies [Formula: see text], for which the inequality is sometimes strict. We present two equivalent definitions for the APS (one as a minimization problem, the other as a maximization problem) and provide comparisons between the APS and previous notions of fairness. Our main result concerns additive valuations and arbitrary entitlements, for which we provide a polynomial-time algorithm that gives every agent at least a [Formula: see text] - fraction of the agent’s APS. This algorithm can also be viewed as providing strategies in a certain natural bidding game, and these strategies secure each agent at least a [Formula: see text] - fraction of the agent’s APS. Funding: T. Ezra’s research is partially supported by the European Research Council Advanced [Grant 788893] AMDROMA “Algorithmic and Mechanism Design Research in Online Markets” and MIUR PRIN project ALGADIMAR “Algorithms, Games, and Digital Markets.” U. Feige’s research is supported in part by the Israel Science Foundation [Grant 1122/22].
{"title":"Fair-Share Allocations for Agents with Arbitrary Entitlements","authors":"Moshe Babaioff, Tomer Ezra, Uriel Feige","doi":"10.1287/moor.2021.0199","DOIUrl":"https://doi.org/10.1287/moor.2021.0199","url":null,"abstract":"We consider the problem of fair allocation of indivisible goods to n agents with no transfers. When agents have equal entitlements, the well-established notion of the maximin share (MMS) serves as an attractive fairness criterion for which, to qualify as fair, an allocation needs to give every agent at least a substantial fraction of the agent’s MMS. In this paper, we consider the case of arbitrary (unequal) entitlements. We explain shortcomings in previous attempts that extend the MMS to unequal entitlements. Our conceptual contribution is the introduction of a new notion of a share, the AnyPrice share (APS), that is appropriate for settings with arbitrary entitlements. Even for the equal entitlements case, this notion is new and satisfies [Formula: see text], for which the inequality is sometimes strict. We present two equivalent definitions for the APS (one as a minimization problem, the other as a maximization problem) and provide comparisons between the APS and previous notions of fairness. Our main result concerns additive valuations and arbitrary entitlements, for which we provide a polynomial-time algorithm that gives every agent at least a [Formula: see text] - fraction of the agent’s APS. This algorithm can also be viewed as providing strategies in a certain natural bidding game, and these strategies secure each agent at least a [Formula: see text] - fraction of the agent’s APS. Funding: T. Ezra’s research is partially supported by the European Research Council Advanced [Grant 788893] AMDROMA “Algorithmic and Mechanism Design Research in Online Markets” and MIUR PRIN project ALGADIMAR “Algorithms, Games, and Digital Markets.” U. Feige’s research is supported in part by the Israel Science Foundation [Grant 1122/22].","PeriodicalId":49852,"journal":{"name":"Mathematics of Operations Research","volume":"196 4","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136377186","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
This paper concerns discrete-time infinite-horizon stochastic control systems with Borel state and action spaces and universally measurable policies. We study optimization problems on strategic measures induced by the policies in these systems. The results are then applied to risk-neutral and risk-sensitive Markov decision processes to establish the measurability of the optimal value functions and the existence of universally measurable, randomized or nonrandomized, ϵ-optimal policies, for a variety of average cost criteria and risk criteria. We also extend our analysis to a class of minimax control problems and establish similar optimality results under the axiom of analytic determinacy. Funding: This work was supported by grants from DeepMind, the Alberta Machine Intelligence Institute (AMII), and Alberta Innovates-Technology Futures (AITF).
{"title":"On Strategic Measures and Optimality Properties in Discrete-Time Stochastic Control with Universally Measurable Policies","authors":"Huizhen Yu","doi":"10.1287/moor.2022.0188","DOIUrl":"https://doi.org/10.1287/moor.2022.0188","url":null,"abstract":"This paper concerns discrete-time infinite-horizon stochastic control systems with Borel state and action spaces and universally measurable policies. We study optimization problems on strategic measures induced by the policies in these systems. The results are then applied to risk-neutral and risk-sensitive Markov decision processes to establish the measurability of the optimal value functions and the existence of universally measurable, randomized or nonrandomized, ϵ-optimal policies, for a variety of average cost criteria and risk criteria. We also extend our analysis to a class of minimax control problems and establish similar optimality results under the axiom of analytic determinacy. Funding: This work was supported by grants from DeepMind, the Alberta Machine Intelligence Institute (AMII), and Alberta Innovates-Technology Futures (AITF).","PeriodicalId":49852,"journal":{"name":"Mathematics of Operations Research","volume":"48 3","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135273257","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Johannes Brustle, José Correa, Paul Duetting, Victor Verdugo
We study the competition complexity of dynamic pricing relative to the optimal auction in the fundamental single-item setting. In prophet inequality terminology, we compare the expected reward [Formula: see text] achievable by the optimal online policy on m independent and identically distributed (i.i.d.) random variables distributed according to F to the expected maximum [Formula: see text] of n i.i.d. draws from F. We ask how big m has to be to ensure that [Formula: see text] for all F. We resolve this question and characterize the competition complexity as a function of ε. When [Formula: see text], the competition complexity is unbounded. That is, for any n and m there is a distribution F such that [Formula: see text]. In contrast, for any [Formula: see text], it is sufficient and necessary to have [Formula: see text], where [Formula: see text]. Therefore, the competition complexity not only drops from unbounded to linear, it is actually linear with a very small constant. The technical core of our analysis is a lossless reduction to an infinite dimensional and nonlinear optimization problem that we solve optimally. A corollary of this reduction is a novel proof of the factor [Formula: see text] i.i.d. prophet inequality, which simultaneously establishes matching upper and lower bounds. Funding: This work was supported by ANID (Anillo ICMD) [Grant ACT210005] and the Center for Mathematical Modeling [Grant FB210005].
{"title":"The Competition Complexity of Dynamic Pricing","authors":"Johannes Brustle, José Correa, Paul Duetting, Victor Verdugo","doi":"10.1287/moor.2022.0230","DOIUrl":"https://doi.org/10.1287/moor.2022.0230","url":null,"abstract":"We study the competition complexity of dynamic pricing relative to the optimal auction in the fundamental single-item setting. In prophet inequality terminology, we compare the expected reward [Formula: see text] achievable by the optimal online policy on m independent and identically distributed (i.i.d.) random variables distributed according to F to the expected maximum [Formula: see text] of n i.i.d. draws from F. We ask how big m has to be to ensure that [Formula: see text] for all F. We resolve this question and characterize the competition complexity as a function of ε. When [Formula: see text], the competition complexity is unbounded. That is, for any n and m there is a distribution F such that [Formula: see text]. In contrast, for any [Formula: see text], it is sufficient and necessary to have [Formula: see text], where [Formula: see text]. Therefore, the competition complexity not only drops from unbounded to linear, it is actually linear with a very small constant. The technical core of our analysis is a lossless reduction to an infinite dimensional and nonlinear optimization problem that we solve optimally. A corollary of this reduction is a novel proof of the factor [Formula: see text] i.i.d. prophet inequality, which simultaneously establishes matching upper and lower bounds. Funding: This work was supported by ANID (Anillo ICMD) [Grant ACT210005] and the Center for Mathematical Modeling [Grant FB210005].","PeriodicalId":49852,"journal":{"name":"Mathematics of Operations Research","volume":"10 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135617278","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Given a nonnegative matrix [Formula: see text], the matrix scaling problem asks whether A can be scaled to a doubly stochastic matrix [Formula: see text] for some positive diagonal matrices D 1 , D 2 . The Sinkhorn algorithm is a simple iterative algorithm, which repeats row-normalization [Formula: see text] and column-normalization [Formula: see text] alternatively. By this algorithm, A converges to a doubly stochastic matrix in limit if and only if the bipartite graph associated with A has a perfect matching. This property can decide the existence of a perfect matching in a given bipartite graph G, which is identified with the 0, 1-matrix A G . Linial et al. (2000) showed that [Formula: see text] iterations for A G decide whether G has a perfect matching. Here, n is the number of vertices in one of the color classes of G. In this paper, we show an extension of this result. If G has no perfect matching, then a polynomial number of the Sinkhorn iterations identifies a Hall blocker—a vertex subset X having neighbors [Formula: see text] with [Formula: see text], which is a certificate of the nonexistence of a perfect matching. Specifically, we show that [Formula: see text] iterations can identify one Hall blocker and that further polynomial iterations can also identify all parametric Hall blockers X of maximizing [Formula: see text] for [Formula: see text]. The former result is based on an interpretation of the Sinkhorn algorithm as alternating minimization for geometric programming. The latter is on an interpretation as alternating minimization for Kullback–Leibler (KL) divergence and on its limiting behavior for a nonscalable matrix. We also relate the Sinkhorn limit with parametric network flow, principal partition of polymatroids, and the Dulmage–Mendelsohn decomposition of a bipartite graph. Funding: K. Hayashi was supported by the Japan Society for the Promotion of Science [Grant JP19J22605]. H. Hirai was supported by Precursory Research for Embryonic Science and Technology [Grant JPMJPR192A].
{"title":"Finding Hall Blockers by Matrix Scaling","authors":"Koyo Hayashi, Hiroshi Hirai, Keiya Sakabe","doi":"10.1287/moor.2022.0198","DOIUrl":"https://doi.org/10.1287/moor.2022.0198","url":null,"abstract":"Given a nonnegative matrix [Formula: see text], the matrix scaling problem asks whether A can be scaled to a doubly stochastic matrix [Formula: see text] for some positive diagonal matrices D 1 , D 2 . The Sinkhorn algorithm is a simple iterative algorithm, which repeats row-normalization [Formula: see text] and column-normalization [Formula: see text] alternatively. By this algorithm, A converges to a doubly stochastic matrix in limit if and only if the bipartite graph associated with A has a perfect matching. This property can decide the existence of a perfect matching in a given bipartite graph G, which is identified with the 0, 1-matrix A G . Linial et al. (2000) showed that [Formula: see text] iterations for A G decide whether G has a perfect matching. Here, n is the number of vertices in one of the color classes of G. In this paper, we show an extension of this result. If G has no perfect matching, then a polynomial number of the Sinkhorn iterations identifies a Hall blocker—a vertex subset X having neighbors [Formula: see text] with [Formula: see text], which is a certificate of the nonexistence of a perfect matching. Specifically, we show that [Formula: see text] iterations can identify one Hall blocker and that further polynomial iterations can also identify all parametric Hall blockers X of maximizing [Formula: see text] for [Formula: see text]. The former result is based on an interpretation of the Sinkhorn algorithm as alternating minimization for geometric programming. The latter is on an interpretation as alternating minimization for Kullback–Leibler (KL) divergence and on its limiting behavior for a nonscalable matrix. We also relate the Sinkhorn limit with parametric network flow, principal partition of polymatroids, and the Dulmage–Mendelsohn decomposition of a bipartite graph. Funding: K. Hayashi was supported by the Japan Society for the Promotion of Science [Grant JP19J22605]. H. Hirai was supported by Precursory Research for Embryonic Science and Technology [Grant JPMJPR192A].","PeriodicalId":49852,"journal":{"name":"Mathematics of Operations Research","volume":"19 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135883272","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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