Coalition formation over graphs is a well studied class of games whose players are vertices and feasible coalitions must be connected subgraphs. In this setting, the existence and computation of equilibria, under various notions of stability, has attracted a lot of attention. However, the natural process by which players, starting from any feasible state, strive to reach an equilibrium after a series of unilateral improving deviations, has been less studied. We investigate the convergence of dynamics towards individually stable outcomes under the following perspective: what are the most general classes of preferences and graph topologies guaranteeing convergence? To this aim, on the one hand, we cover a hierarchy of preferences, ranging from the most general to a subcase of additively separable preferences, including individually rational and monotone cases. On the other hand, given that convergence may fail in graphs admitting a cycle even in our most restrictive preference class, we analyze acyclic graph topologies such as trees, paths, and stars.
{"title":"Individually Stable Dynamics in Coalition Formation over Graphs","authors":"Angelo FanelliLAMSADE, Laurent GourvèsLAMSADE, Ayumi IgarashiUTokyo, Luca MoscardelliUd'A","doi":"arxiv-2408.11488","DOIUrl":"https://doi.org/arxiv-2408.11488","url":null,"abstract":"Coalition formation over graphs is a well studied class of games whose\u0000players are vertices and feasible coalitions must be connected subgraphs. In\u0000this setting, the existence and computation of equilibria, under various\u0000notions of stability, has attracted a lot of attention. However, the natural\u0000process by which players, starting from any feasible state, strive to reach an\u0000equilibrium after a series of unilateral improving deviations, has been less\u0000studied. We investigate the convergence of dynamics towards individually stable\u0000outcomes under the following perspective: what are the most general classes of\u0000preferences and graph topologies guaranteeing convergence? To this aim, on the\u0000one hand, we cover a hierarchy of preferences, ranging from the most general to\u0000a subcase of additively separable preferences, including individually rational\u0000and monotone cases. On the other hand, given that convergence may fail in\u0000graphs admitting a cycle even in our most restrictive preference class, we\u0000analyze acyclic graph topologies such as trees, paths, and stars.","PeriodicalId":501316,"journal":{"name":"arXiv - CS - Computer Science and Game Theory","volume":"4 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142197547","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Issues of inequity in U.S. high schools' course scheduling did not previously exist. However, in recent years, with the increase in student population and course variety, students perceive that the course scheduling method is unfair. Current integer programming (IP) methods to the high school scheduling problem (HSSP) fall short in addressing these fairness concerns. The purpose of this research is to develop a solution methodology that generates feasible and fair course schedules using student preferences. Utilizing principles of fairness, which have been well studied in market design, we define the fair high school scheduling problem (FHSSP), a novel extension to the HSSP, and devise a corresponding algorithm based on integer programming to solve the FHSSP. We test our approach on a real course request dataset from a high school in California, USA. Results show that our algorithm can generate schedules that are both feasible and fair. In this paper, we demonstrate that our IP algorithm not only solves the HSSP and FHSSP in the United States but has the potential to be applied to various real-world scheduling problems. Additionally, we show the feasibility of integrating human emotions into mathematical modeling.
美国高中课程安排中的不公平问题以前并不存在。然而，近年来，随着学生人数的增加和课程种类的增多，学生们认为课程安排方法有失公平。本研究的目的是开发一种求解方法，利用学生的偏好生成可行且公平的课程安排。利用在市场设计中得到充分研究的公平原则，我们定义了公平高中课程安排问题（FHSSP）--HSSP 的一种新扩展，并设计了一种基于整数编程的相应算法来解决 FHSSP。我们在美国加利福尼亚州一所高中的真实课程请求数据集上测试了我们的方法。结果表明，我们的算法可以生成既可行又公平的课程表。在本文中，我们证明了我们的 IP 算法不仅能解决美国的 HSSP 和 FHSSP 问题，而且有潜力应用于现实世界中的各种排课问题。此外，我们还展示了将人类情感融入数学建模的可行性。
{"title":"A Constraint Programming Approach to Fair High School Course Scheduling","authors":"Mitsuka Kiyohara, Masakazu Ishihata","doi":"arxiv-2408.12032","DOIUrl":"https://doi.org/arxiv-2408.12032","url":null,"abstract":"Issues of inequity in U.S. high schools' course scheduling did not previously\u0000exist. However, in recent years, with the increase in student population and\u0000course variety, students perceive that the course scheduling method is unfair.\u0000Current integer programming (IP) methods to the high school scheduling problem\u0000(HSSP) fall short in addressing these fairness concerns. The purpose of this\u0000research is to develop a solution methodology that generates feasible and fair\u0000course schedules using student preferences. Utilizing principles of fairness,\u0000which have been well studied in market design, we define the fair high school\u0000scheduling problem (FHSSP), a novel extension to the HSSP, and devise a\u0000corresponding algorithm based on integer programming to solve the FHSSP. We\u0000test our approach on a real course request dataset from a high school in\u0000California, USA. Results show that our algorithm can generate schedules that\u0000are both feasible and fair. In this paper, we demonstrate that our IP algorithm\u0000not only solves the HSSP and FHSSP in the United States but has the potential\u0000to be applied to various real-world scheduling problems. Additionally, we show\u0000the feasibility of integrating human emotions into mathematical modeling.","PeriodicalId":501316,"journal":{"name":"arXiv - CS - Computer Science and Game Theory","volume":"43 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142197545","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In cross-silo federated learning (FL), companies collaboratively train a shared global model without sharing heterogeneous data. Prior related work focused on algorithm development to tackle data heterogeneity. However, the dual problem of coopetition, i.e., FL collaboration and market competition, remains under-explored. This paper studies the FL coopetition using a dynamic two-period game model. In period 1, an incumbent company trains a local model and provides model-based services at a chosen price to users. In period 2, an entrant company enters, and both companies decide whether to engage in FL collaboration and then compete in selling model-based services at different prices to users. Analyzing the two-period game is challenging due to data heterogeneity, and that the incumbent's period one pricing has a temporal impact on coopetition in period 2, resulting in a non-concave problem. To address this issue, we decompose the problem into several concave sub-problems and develop an algorithm that achieves a global optimum. Numerical results on three public datasets show two interesting insights. First, FL training brings model performance gain as well as competition loss, and collaboration occurs only when the performance gain outweighs the loss. Second, data heterogeneity can incentivize the incumbent to limit market penetration in period 1 and promote price competition in period 2.
{"title":"Technical Report: Coopetition in Heterogeneous Cross-Silo Federated Learning","authors":"Chao Huang, Justin Dachille, Xin Liu","doi":"arxiv-2408.11355","DOIUrl":"https://doi.org/arxiv-2408.11355","url":null,"abstract":"In cross-silo federated learning (FL), companies collaboratively train a\u0000shared global model without sharing heterogeneous data. Prior related work\u0000focused on algorithm development to tackle data heterogeneity. However, the\u0000dual problem of coopetition, i.e., FL collaboration and market competition,\u0000remains under-explored. This paper studies the FL coopetition using a dynamic\u0000two-period game model. In period 1, an incumbent company trains a local model\u0000and provides model-based services at a chosen price to users. In period 2, an\u0000entrant company enters, and both companies decide whether to engage in FL\u0000collaboration and then compete in selling model-based services at different\u0000prices to users. Analyzing the two-period game is challenging due to data\u0000heterogeneity, and that the incumbent's period one pricing has a temporal\u0000impact on coopetition in period 2, resulting in a non-concave problem. To\u0000address this issue, we decompose the problem into several concave sub-problems\u0000and develop an algorithm that achieves a global optimum. Numerical results on\u0000three public datasets show two interesting insights. First, FL training brings\u0000model performance gain as well as competition loss, and collaboration occurs\u0000only when the performance gain outweighs the loss. Second, data heterogeneity\u0000can incentivize the incumbent to limit market penetration in period 1 and\u0000promote price competition in period 2.","PeriodicalId":501316,"journal":{"name":"arXiv - CS - Computer Science and Game Theory","volume":"171 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142197548","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Congestion games are attractive because they can model many concrete situations where some competing entities interact through the use of some shared resources, and also because they always admit pure Nash equilibria which correspond to the local minima of a potential function. We explore the problem of computing a state of minimum potential in this setting. Using the maximum number of resources that a player can use at a time, and the possible symmetry in the players' strategy spaces, we settle the complexity of the problem for instances having monotone (i.e., either non-decreasing or non-increasing) latency functions on their resources. The picture, delineating polynomial and NP-hard cases, is complemented with tight approximation algorithms.
{"title":"Minimizing Rosenthal's Potential in Monotone Congestion Games","authors":"Vittorio BilòLAMSADE, Angelo FanelliLAMSADE, Laurent GourvèsLAMSADE, Christos TsoufisLAMSADE, Cosimo Vinci","doi":"arxiv-2408.11489","DOIUrl":"https://doi.org/arxiv-2408.11489","url":null,"abstract":"Congestion games are attractive because they can model many concrete\u0000situations where some competing entities interact through the use of some\u0000shared resources, and also because they always admit pure Nash equilibria which\u0000correspond to the local minima of a potential function. We explore the problem\u0000of computing a state of minimum potential in this setting. Using the maximum\u0000number of resources that a player can use at a time, and the possible symmetry\u0000in the players' strategy spaces, we settle the complexity of the problem for\u0000instances having monotone (i.e., either non-decreasing or non-increasing)\u0000latency functions on their resources. The picture, delineating polynomial and\u0000NP-hard cases, is complemented with tight approximation algorithms.","PeriodicalId":501316,"journal":{"name":"arXiv - CS - Computer Science and Game Theory","volume":"11 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142197550","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Valentin Zech, Niclas Boehmer, Edith Elkind, Nicholas Teh
We study two-stage committee elections where voters have dynamic preferences over candidates; at each stage, a committee is chosen under a given voting rule. We are interested in identifying a winning committee for the second stage that overlaps as much as possible with the first-stage committee. We show a full complexity dichotomy for the class of Thiele rules: this problem is tractable for Approval Voting (AV) and hard for all other Thiele rules (including, in particular, Proportional Approval Voting and the Chamberlin-Courant rule). We extend this dichotomy to the greedy variants of Thiele rules. We also explore this problem from a parameterized complexity perspective for several natural parameters. We complement the theory with experimental analysis: e.g., we investigate the average number of changes in the committee as a function of changes in voters' preferences and the role of ties.
{"title":"Multiwinner Temporal Voting with Aversion to Change","authors":"Valentin Zech, Niclas Boehmer, Edith Elkind, Nicholas Teh","doi":"arxiv-2408.11017","DOIUrl":"https://doi.org/arxiv-2408.11017","url":null,"abstract":"We study two-stage committee elections where voters have dynamic preferences\u0000over candidates; at each stage, a committee is chosen under a given voting\u0000rule. We are interested in identifying a winning committee for the second stage\u0000that overlaps as much as possible with the first-stage committee. We show a\u0000full complexity dichotomy for the class of Thiele rules: this problem is\u0000tractable for Approval Voting (AV) and hard for all other Thiele rules\u0000(including, in particular, Proportional Approval Voting and the\u0000Chamberlin-Courant rule). We extend this dichotomy to the greedy variants of\u0000Thiele rules. We also explore this problem from a parameterized complexity\u0000perspective for several natural parameters. We complement the theory with\u0000experimental analysis: e.g., we investigate the average number of changes in\u0000the committee as a function of changes in voters' preferences and the role of\u0000ties.","PeriodicalId":501316,"journal":{"name":"arXiv - CS - Computer Science and Game Theory","volume":"204 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142197580","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The sorting and filtering capabilities offered by modern e-commerce platforms significantly impact customers' purchase decisions, as well as the resulting prices set by competing sellers on these platforms. Motivated by this practical reality, we study price competition under a flexible choice model: Consider-then-Choose with Lexicographic Choice (CLC). In this model, a customer first forms a consideration set of sellers based on (i) her willingness-to-pay and (ii) an arbitrary set of criteria on items' non-price attributes; she then chooses the highest-ranked item according to a lexicographic ranking in which items with better performance on more important attributes are ranked higher. We provide a structural characterization of equilibria in the resulting game of price competition, and derive an economically interpretable condition, which we call gradient dominance, under which equilibria can be computed efficiently. For this subclass of CLC models, we prove that distributed gradient-based pricing dynamics converge to the set of equilibria. Extensive numerical experiments show robustness of our theoretical findings when gradient dominance does not hold.
{"title":"Price Competition Under A Consider-Then-Choose Model With Lexicographic Choice","authors":"Siddhartha Banerjee, Chamsi Hssaine, Vijay Kamble","doi":"arxiv-2408.10429","DOIUrl":"https://doi.org/arxiv-2408.10429","url":null,"abstract":"The sorting and filtering capabilities offered by modern e-commerce platforms\u0000significantly impact customers' purchase decisions, as well as the resulting\u0000prices set by competing sellers on these platforms. Motivated by this practical\u0000reality, we study price competition under a flexible choice model:\u0000Consider-then-Choose with Lexicographic Choice (CLC). In this model, a customer\u0000first forms a consideration set of sellers based on (i) her willingness-to-pay\u0000and (ii) an arbitrary set of criteria on items' non-price attributes; she then\u0000chooses the highest-ranked item according to a lexicographic ranking in which\u0000items with better performance on more important attributes are ranked higher.\u0000We provide a structural characterization of equilibria in the resulting game of\u0000price competition, and derive an economically interpretable condition, which we\u0000call gradient dominance, under which equilibria can be computed efficiently.\u0000For this subclass of CLC models, we prove that distributed gradient-based\u0000pricing dynamics converge to the set of equilibria. Extensive numerical\u0000experiments show robustness of our theoretical findings when gradient dominance\u0000does not hold.","PeriodicalId":501316,"journal":{"name":"arXiv - CS - Computer Science and Game Theory","volume":"69 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142197552","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Online advertising is a primary source of income for e-commerce platforms. In the current advertising pattern, the oriented targets are the online store owners who are willing to pay extra fees to enhance the position of their stores. On the other hand, brand suppliers are also desirable to advertise their products in stores to boost brand sales. However, the currently used advertising mode cannot satisfy the demand of both stores and brand suppliers simultaneously. To address this, we innovatively propose a joint advertising model termed Joint Auction, allowing brand suppliers and stores to collaboratively bid for advertising slots, catering to both their needs. However, conventional advertising auction mechanisms are not suitable for this novel scenario. In this paper, we propose JRegNet, a neural network architecture for the optimal joint auction design, to generate mechanisms that can achieve the optimal revenue and guarantee near dominant strategy incentive compatibility and individual rationality. Finally, multiple experiments are conducted on synthetic and real data to demonstrate that our proposed joint auction significantly improves platform revenue compared to the known baselines.
{"title":"Joint Auction in the Online Advertising Market","authors":"Zhen Zhang, Weian Li, Yahui Lei, Bingzhe Wang, Zhicheng Zhang, Qi Qi, Qiang Liu, Xingxing Wang","doi":"arxiv-2408.09885","DOIUrl":"https://doi.org/arxiv-2408.09885","url":null,"abstract":"Online advertising is a primary source of income for e-commerce platforms. In\u0000the current advertising pattern, the oriented targets are the online store\u0000owners who are willing to pay extra fees to enhance the position of their\u0000stores. On the other hand, brand suppliers are also desirable to advertise\u0000their products in stores to boost brand sales. However, the currently used\u0000advertising mode cannot satisfy the demand of both stores and brand suppliers\u0000simultaneously. To address this, we innovatively propose a joint advertising\u0000model termed Joint Auction, allowing brand suppliers and stores to\u0000collaboratively bid for advertising slots, catering to both their needs.\u0000However, conventional advertising auction mechanisms are not suitable for this\u0000novel scenario. In this paper, we propose JRegNet, a neural network\u0000architecture for the optimal joint auction design, to generate mechanisms that\u0000can achieve the optimal revenue and guarantee near dominant strategy incentive\u0000compatibility and individual rationality. Finally, multiple experiments are\u0000conducted on synthetic and real data to demonstrate that our proposed joint\u0000auction significantly improves platform revenue compared to the known\u0000baselines.","PeriodicalId":501316,"journal":{"name":"arXiv - CS - Computer Science and Game Theory","volume":"27 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142197555","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Weighted voting games are a well-known and useful class of succinctly representable simple games that have many real-world applications, e.g., to model collective decision-making in legislative bodies or shareholder voting. Among the structural control types being analyzing, one is control by adding players to weighted voting games, so as to either change or to maintain a player's power in the sense of the (probabilistic) Penrose-Banzhaf power index or the Shapley-Shubik power index. For the problems related to this control, the best known lower bound is PP-hardness, where PP is "probabilistic polynomial time," and the best known upper bound is the class NP^PP, i.e., the class NP with a PP oracle. We optimally raise this lower bound by showing NP^PP-hardness of all these problems for the Penrose-Banzhaf and the Shapley-Shubik indices, thus establishing completeness for them in that class. Our proof technique may turn out to be useful for solving other open problems related to weighted voting games with such a complexity gap as well.
{"title":"Control by Adding Players to Change or Maintain the Shapley-Shubik or the Penrose-Banzhaf Power Index in Weighted Voting Games Is Complete for NP^PP","authors":"Joanna Kaczmarek, Jörg Rothe","doi":"arxiv-2408.09953","DOIUrl":"https://doi.org/arxiv-2408.09953","url":null,"abstract":"Weighted voting games are a well-known and useful class of succinctly\u0000representable simple games that have many real-world applications, e.g., to\u0000model collective decision-making in legislative bodies or shareholder voting.\u0000Among the structural control types being analyzing, one is control by adding\u0000players to weighted voting games, so as to either change or to maintain a\u0000player's power in the sense of the (probabilistic) Penrose-Banzhaf power index\u0000or the Shapley-Shubik power index. For the problems related to this control,\u0000the best known lower bound is PP-hardness, where PP is \"probabilistic\u0000polynomial time,\" and the best known upper bound is the class NP^PP, i.e., the\u0000class NP with a PP oracle. We optimally raise this lower bound by showing\u0000NP^PP-hardness of all these problems for the Penrose-Banzhaf and the\u0000Shapley-Shubik indices, thus establishing completeness for them in that class.\u0000Our proof technique may turn out to be useful for solving other open problems\u0000related to weighted voting games with such a complexity gap as well.","PeriodicalId":501316,"journal":{"name":"arXiv - CS - Computer Science and Game Theory","volume":"30 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142197554","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
A tournament on $n$ agents is a complete oriented graph with the agents as vertices and edges that describe the win-loss outcomes of the $binom{n}{2}$ matches played between each pair of agents. The winner of a tournament is determined by a tournament rule that maps tournaments to probability distributions over the agents. We want these rules to be fair (choose a high-quality agent) and robust to strategic manipulation. Prior work has shown that under minimally fair rules, manipulations between two agents can be prevented when utility is nontransferable but not when utility is completely transferable. We introduce a partially transferable utility model that interpolates between these two extremes using a selfishness parameter $lambda$. Our model is that an agent may be willing to lose on purpose, sacrificing some of her own chance of winning, but only if the colluding pair's joint gain is more than $lambda$ times the individual's sacrifice. We show that no fair tournament rule can prevent manipulations when $lambda < 1$. We computationally solve for fair and manipulation-resistant tournament rules for $lambda = 1$ for up to 6 agents. We conjecture and leave as a major open problem that such a tournament rule exists for all $n$. We analyze the trade-offs between ``relative'' and ``absolute'' approximate strategyproofness for previously studied rules and derive as a corollary that all of these rules require $lambda geq Omega(n)$ to be robust to manipulation. We show that for stronger notions of fairness, non-manipulable tournament rules are closely related to tournament rules that witness decreasing gains from manipulation as the number of agents increases.
{"title":"Toward Fair and Strategyproof Tournament Rules for Tournaments with Partially Transferable Utilities","authors":"David Pennock, Ariel Schvartzman, Eric Xue","doi":"arxiv-2408.10346","DOIUrl":"https://doi.org/arxiv-2408.10346","url":null,"abstract":"A tournament on $n$ agents is a complete oriented graph with the agents as\u0000vertices and edges that describe the win-loss outcomes of the $binom{n}{2}$\u0000matches played between each pair of agents. The winner of a tournament is\u0000determined by a tournament rule that maps tournaments to probability\u0000distributions over the agents. We want these rules to be fair (choose a\u0000high-quality agent) and robust to strategic manipulation. Prior work has shown\u0000that under minimally fair rules, manipulations between two agents can be\u0000prevented when utility is nontransferable but not when utility is completely\u0000transferable. We introduce a partially transferable utility model that\u0000interpolates between these two extremes using a selfishness parameter\u0000$lambda$. Our model is that an agent may be willing to lose on purpose,\u0000sacrificing some of her own chance of winning, but only if the colluding pair's\u0000joint gain is more than $lambda$ times the individual's sacrifice. We show that no fair tournament rule can prevent manipulations when $lambda\u0000< 1$. We computationally solve for fair and manipulation-resistant tournament\u0000rules for $lambda = 1$ for up to 6 agents. We conjecture and leave as a major\u0000open problem that such a tournament rule exists for all $n$. We analyze the\u0000trade-offs between ``relative'' and ``absolute'' approximate strategyproofness\u0000for previously studied rules and derive as a corollary that all of these rules\u0000require $lambda geq Omega(n)$ to be robust to manipulation. We show that for\u0000stronger notions of fairness, non-manipulable tournament rules are closely\u0000related to tournament rules that witness decreasing gains from manipulation as\u0000the number of agents increases.","PeriodicalId":501316,"journal":{"name":"arXiv - CS - Computer Science and Game Theory","volume":"11 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142197553","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Focusing on the bipartite Stable Marriage problem, we investigate different robustness measures related to stable matchings. We analyze the computational complexity of computing them and analyze their behavior in extensive experiments on synthetic instances. For instance, we examine whether a stable matching is guaranteed to remain stable if a given number of adversarial swaps in the agent's preferences are performed and the probability of stability when applying swaps uniformly at random. Our results reveal that stable matchings in our synthetic data are highly unrobust to adversarial swaps, whereas the average-case view presents a more nuanced and informative picture.
{"title":"Worst- and Average-Case Robustness of Stable Matchings: (Counting) Complexity and Experiments","authors":"Kimon Boehmer, Niclas Boehmer","doi":"arxiv-2408.09160","DOIUrl":"https://doi.org/arxiv-2408.09160","url":null,"abstract":"Focusing on the bipartite Stable Marriage problem, we investigate different\u0000robustness measures related to stable matchings. We analyze the computational\u0000complexity of computing them and analyze their behavior in extensive\u0000experiments on synthetic instances. For instance, we examine whether a stable\u0000matching is guaranteed to remain stable if a given number of adversarial swaps\u0000in the agent's preferences are performed and the probability of stability when\u0000applying swaps uniformly at random. Our results reveal that stable matchings in\u0000our synthetic data are highly unrobust to adversarial swaps, whereas the\u0000average-case view presents a more nuanced and informative picture.","PeriodicalId":501316,"journal":{"name":"arXiv - CS - Computer Science and Game Theory","volume":"29 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142197556","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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