Active learning enhances the performance of machine learning methods, particularly in semi-supervised cases, by judiciously selecting a limited number of unlabeled data points for labeling, with the goal of improving the performance of an underlying classifier. In this work, we introduce the Multiclass Active Learning with Auction Dynamics on Graphs (MALADY) framework which leverages the auction dynamics algorithm on similarity graphs for efficient active learning. In particular, we generalize the auction dynamics algorithm on similarity graphs for semi-supervised learning in [24] to incorporate a more general optimization functional. Moreover, we introduce a novel active learning acquisition function that uses the dual variable of the auction algorithm to measure the uncertainty in the classifier to prioritize queries near the decision boundaries between different classes. Lastly, using experiments on classification tasks, we evaluate the performance of our proposed method and show that it exceeds that of comparison algorithms.
主动学习可以提高机器学习方法的性能，尤其是在半监督情况下，它可以明智地选择数量有限的未标记数据点进行标记，从而提高底层分类器的性能。在这项工作中，我们介绍了图形拍卖动态多类主动学习（Multiclass Active Learning with Auction Dynamics on Graphs，MALADY）框架，该框架利用相似性图形上的拍卖动态算法实现高效的主动学习。特别是，我们对 [24] 中用于半监督学习的相似性图上拍卖动态算法进行了概括，纳入了一个更通用的优化函数。此外，我们还引入了一种新的主动学习获取函数，它使用拍卖算法的对偶变量来衡量分类器的不确定性，从而优先处理不同类别之间决策边界附近的查询。最后，通过分类任务的实验，我们评估了我们提出的方法的性能，结果表明它超过了比较算法。
{"title":"MALADY: Multiclass Active Learning with Auction Dynamics on Graphs","authors":"Gokul Bhusal, Kevin Miller, Ekaterina Merkurjev","doi":"arxiv-2409.09475","DOIUrl":"https://doi.org/arxiv-2409.09475","url":null,"abstract":"Active learning enhances the performance of machine learning methods,\u0000particularly in semi-supervised cases, by judiciously selecting a limited\u0000number of unlabeled data points for labeling, with the goal of improving the\u0000performance of an underlying classifier. In this work, we introduce the\u0000Multiclass Active Learning with Auction Dynamics on Graphs (MALADY) framework\u0000which leverages the auction dynamics algorithm on similarity graphs for\u0000efficient active learning. In particular, we generalize the auction dynamics\u0000algorithm on similarity graphs for semi-supervised learning in [24] to\u0000incorporate a more general optimization functional. Moreover, we introduce a\u0000novel active learning acquisition function that uses the dual variable of the\u0000auction algorithm to measure the uncertainty in the classifier to prioritize\u0000queries near the decision boundaries between different classes. Lastly, using\u0000experiments on classification tasks, we evaluate the performance of our\u0000proposed method and show that it exceeds that of comparison algorithms.","PeriodicalId":501316,"journal":{"name":"arXiv - CS - Computer Science and Game Theory","volume":"12 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142265125","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}
This paper reexamines the classic problem of revenue maximization in single-item auctions with $n$ buyers under the lens of the robust optimization framework. The celebrated Myerson's mechanism is the format that maximizes the seller's revenue under the prior distribution, which is mutually independent across all $n$ buyers. As argued in a recent line of work (Caragiannis et al. 22), (Dughmi et al. 24), mutual independence is a strong assumption that is extremely hard to verify statistically, thus it is important to relax the assumption. While optimal under mutual independent prior, we find that Myerson's mechanism may lose almost all of its revenue when the independence assumption is relaxed to pairwise independence, i.e., Myerson's mechanism is not pairwise-robust. The mechanism regains robustness when the prior is assumed to be 3-wise independent. In contrast, we show that second-price auctions with anonymous reserve, including optimal auctions under i.i.d. priors, lose at most a constant fraction of their revenues on any regular pairwise independent prior. Our findings draw a comprehensive picture of robustness to $k$-wise independence in single-item auction settings.
本文在稳健优化框架的视角下重新审视了有 n 个买家的单品拍卖中收益最大化的经典问题。著名的迈尔森机制是在先验分布下使卖方收益最大化的形式，而先验分布在所有 $n$ 买方中是相互独立的。正如最近的一些研究（Caragiannis et al.22）和（Dughmi et al.24）所指出的，相互独立是一个很强的假设，在统计上极难验证，因此放宽这一假设非常重要。我们发现，虽然迈尔森机制在相互独立的先验条件下是最优的，但当独立性假设放宽到成对独立性时，迈尔森机制可能会失去几乎所有的收益，也就是说，迈尔森机制并不是成对稳健的。当先验假定为三向独立时，该机制就会恢复稳健性。与此相反，我们证明了带有匿名储备金的二次定价拍卖（包括在 i.i.d. 先验下的最优拍卖）在任何常规的成对独立先验下最多损失其收入的固定部分。我们的发现全面描绘了在单项拍卖中$k$智独立的稳健性。
{"title":"On Robustness to $k$-wise Independence of Optimal Bayesian Mechanisms","authors":"Nick Gravin, Zhiqi Wang","doi":"arxiv-2409.08547","DOIUrl":"https://doi.org/arxiv-2409.08547","url":null,"abstract":"This paper reexamines the classic problem of revenue maximization in\u0000single-item auctions with $n$ buyers under the lens of the robust optimization\u0000framework. The celebrated Myerson's mechanism is the format that maximizes the\u0000seller's revenue under the prior distribution, which is mutually independent\u0000across all $n$ buyers. As argued in a recent line of work (Caragiannis et al.\u000022), (Dughmi et al. 24), mutual independence is a strong assumption that is\u0000extremely hard to verify statistically, thus it is important to relax the\u0000assumption. While optimal under mutual independent prior, we find that Myerson's\u0000mechanism may lose almost all of its revenue when the independence assumption\u0000is relaxed to pairwise independence, i.e., Myerson's mechanism is not\u0000pairwise-robust. The mechanism regains robustness when the prior is assumed to\u0000be 3-wise independent. In contrast, we show that second-price auctions with\u0000anonymous reserve, including optimal auctions under i.i.d. priors, lose at most\u0000a constant fraction of their revenues on any regular pairwise independent\u0000prior. Our findings draw a comprehensive picture of robustness to $k$-wise\u0000independence in single-item auction settings.","PeriodicalId":501316,"journal":{"name":"arXiv - CS - Computer Science and Game Theory","volume":"30 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142269510","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}
How to design a fair and reasonable allocation plan for the common revenue of the alliance is considered in this paper. We regard the common revenue to be allocated as an exogenous variable which will not participate in the subsequent production process. The production organizations can cooperate with each other and form alliances. As the DEA cross-efficiency combines self- and peer-evaluation mechanisms, and the cooperative game allows fair negotiation among participants, we combine the cross-efficiency with the cooperative game theory and construct the modified Shapley value to reflect the contribution of the evaluated participant to the alliance. In addition, for each participant, both the optimistic and the pessimistic modified Shapley values are considered, and thus the upper and lower bounds of the allocation revenue are obtained, correspondingly. A numerical example is presented to illustrate the operation procedure. Finally, we apply the approach to an empirical application concerning a city commercial bank with 18 branches in China.
本文将探讨如何为联盟的共同收益制定公平合理的分配方案。我们将待分配的共同收益视为一个外生变量，它不参与后续的生产过程。生产组织可以相互合作，形成联盟。由于 DEA 交叉效率结合了自评和互评机制，而合作博弈允许参与者之间进行公平协商，因此我们将交叉效率与合作博弈理论相结合，构建修正的夏普利值来反映被评价者对联盟的贡献。此外，对于每个参与者，我们都会考虑乐观和悲观的修正夏普利值，从而相应地得到分配收益的上限和下限。我们举了一个数字例子来说明操作过程。最后，我们将该方法应用于一个有关中国一家拥有 18 家分行的城市商业银行的实证应用中。
{"title":"The common revenue allocation based on modified Shapley value and DEA cross-efficiency","authors":"Xinyu Wanga, Qianwei Zhanga, Binwei Guib, Yingdi Zhaoa","doi":"arxiv-2409.08491","DOIUrl":"https://doi.org/arxiv-2409.08491","url":null,"abstract":"How to design a fair and reasonable allocation plan for the common revenue of\u0000the alliance is considered in this paper. We regard the common revenue to be\u0000allocated as an exogenous variable which will not participate in the subsequent\u0000production process. The production organizations can cooperate with each other\u0000and form alliances. As the DEA cross-efficiency combines self- and\u0000peer-evaluation mechanisms, and the cooperative game allows fair negotiation\u0000among participants, we combine the cross-efficiency with the cooperative game\u0000theory and construct the modified Shapley value to reflect the contribution of\u0000the evaluated participant to the alliance. In addition, for each participant,\u0000both the optimistic and the pessimistic modified Shapley values are considered,\u0000and thus the upper and lower bounds of the allocation revenue are obtained,\u0000correspondingly. A numerical example is presented to illustrate the operation\u0000procedure. Finally, we apply the approach to an empirical application\u0000concerning a city commercial bank with 18 branches in China.","PeriodicalId":501316,"journal":{"name":"arXiv - CS - Computer Science and Game Theory","volume":"56 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142265130","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}
We study a variation of facility location problems (FLPs) that aims to improve the accessibility of agents to the facility within the context of mechanism design without money. In such a variation, agents have preferences on the ideal locations of the facility on a real line, and the facility's location is fixed in advance where (re)locating the facility is not possible due to various constraints (e.g., limited space and construction costs). To improve the accessibility of agents to facilities, existing mechanism design literature in FLPs has proposed to structurally modify the real line (e.g., by adding a new interval) or provide shuttle services between two points when structural modifications are not possible. In this paper, we focus on the latter approach and propose to construct an accessibility range to extend the accessibility of the facility. In the range, agents can receive accommodations (e.g., school buses, campus shuttles, or pickup services) to help reach the facility. Therefore, the cost of each agent is the distance from their ideal location to the facility (possibility) through the range. We focus on designing strategyproof mechanisms that elicit true ideal locations from the agents and construct accessibility ranges (intervals) to approximately minimize the social cost or the maximum cost of agents. For both social and maximum costs, we design group strategyproof mechanisms with asymptotically tight bounds on the approximation ratios.
{"title":"Mechanism Design for Extending the Accessibility of Facilities","authors":"Hau Chan, Jianan Lin, Chenhao Wang, Yanxi Xie","doi":"arxiv-2409.08993","DOIUrl":"https://doi.org/arxiv-2409.08993","url":null,"abstract":"We study a variation of facility location problems (FLPs) that aims to\u0000improve the accessibility of agents to the facility within the context of\u0000mechanism design without money. In such a variation, agents have preferences on\u0000the ideal locations of the facility on a real line, and the facility's location\u0000is fixed in advance where (re)locating the facility is not possible due to\u0000various constraints (e.g., limited space and construction costs). To improve\u0000the accessibility of agents to facilities, existing mechanism design literature\u0000in FLPs has proposed to structurally modify the real line (e.g., by adding a\u0000new interval) or provide shuttle services between two points when structural\u0000modifications are not possible. In this paper, we focus on the latter approach\u0000and propose to construct an accessibility range to extend the accessibility of\u0000the facility. In the range, agents can receive accommodations (e.g., school\u0000buses, campus shuttles, or pickup services) to help reach the facility.\u0000Therefore, the cost of each agent is the distance from their ideal location to\u0000the facility (possibility) through the range. We focus on designing\u0000strategyproof mechanisms that elicit true ideal locations from the agents and\u0000construct accessibility ranges (intervals) to approximately minimize the social\u0000cost or the maximum cost of agents. For both social and maximum costs, we\u0000design group strategyproof mechanisms with asymptotically tight bounds on the\u0000approximation ratios.","PeriodicalId":501316,"journal":{"name":"arXiv - CS - Computer Science and Game Theory","volume":"5 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142265126","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 this paper, we examine two-stage production organizations as decision-making units (DMUs) that can collaborate to form alliances. We present a novel approach to transform a grand coalition of n DMUs with a two-stage structure into 2n single-stage sub-DMUs by extending the vectors of the initial input, intermediate product, and final output, thus creating a 2n*2n DEA cross-efficiency (CREE) matrix. By combining cooperative game theory with CREE and utilizing three cooperative game solution concepts, namely, the nucleolus, the least core and the Shapley value, a characteristic function is developed to account for two types of allocation, i.e., direct allocation and secondary allocation. Moreover, the super-additivity and the core non-emptiness properties are explored. It is found that the sum of the revenue allocated to all DMUs will remain constant at each stage regardless of the allocation manner and the cooperative solution concept selected. To illustrate the efficiency and practicality of the proposed approach, both a numerical example and an empirical application are provided.
在本文中，我们将两阶段生产组织视为可以合作结成联盟的决策单元（DMU）。我们提出了一种新方法，通过扩展初始投入、中间产品和最终产出的向量，将一个由 n 个具有两阶段结构的 DMU 组成的大联盟转化为 2n 个单一阶段的子 DMU，从而创建一个 2n*2n 的 DEA 交叉效率（CREE）矩阵。通过将合作博弈理论与 CREE 相结合，并利用三个合作博弈解概念，即核、最小核心和沙普利值，建立了一个特征函数，以考虑两种分配方式，即直接分配和二次分配。此外，还探讨了超加性和核不emptiness 特性。研究发现，无论选择哪种分配方式和合作方案概念，分配给所有 DMU 的收入总和在每个阶段都将保持不变。为了说明所提方法的效率和实用性，提供了一个数值示例和一个经验应用。
{"title":"Common revenue allocation in DMUs with two stages based on DEA cross-efficiency and cooperative game","authors":"Xinyu Wang, Qianwei Zhang, Yilun Lu, Yingdi Zhao","doi":"arxiv-2409.08502","DOIUrl":"https://doi.org/arxiv-2409.08502","url":null,"abstract":"In this paper, we examine two-stage production organizations as\u0000decision-making units (DMUs) that can collaborate to form alliances. We present\u0000a novel approach to transform a grand coalition of n DMUs with a two-stage\u0000structure into 2n single-stage sub-DMUs by extending the vectors of the initial\u0000input, intermediate product, and final output, thus creating a 2n*2n DEA\u0000cross-efficiency (CREE) matrix. By combining cooperative game theory with CREE\u0000and utilizing three cooperative game solution concepts, namely, the nucleolus,\u0000the least core and the Shapley value, a characteristic function is developed to\u0000account for two types of allocation, i.e., direct allocation and secondary\u0000allocation. Moreover, the super-additivity and the core non-emptiness\u0000properties are explored. It is found that the sum of the revenue allocated to\u0000all DMUs will remain constant at each stage regardless of the allocation manner\u0000and the cooperative solution concept selected. To illustrate the efficiency and\u0000practicality of the proposed approach, both a numerical example and an\u0000empirical application are provided.","PeriodicalId":501316,"journal":{"name":"arXiv - CS - Computer Science and Game Theory","volume":"16 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142265127","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}
$B$-matching is a special case of matching problems where nodes can join multiple matchings with the degree of each node constrained by an upper bound, the node's $B$-value. The core solution of a bipartite $B$-matching is both a matching between the agents respecting the upper bound constraint and an allocation of the value of the edge among its nodes such that no group of agents can deviate and collectively gain higher allocation. We present two learning dynamics that converge to the core of the bipartite $B$-matching problems. The first dynamics are centralized dynamics in the nature of the Hungarian method, which converge to the core in a polynomial time. The second dynamics are distributed dynamics, which converge to the core with probability one. For the distributed dynamics, a node maintains only a state consisting of (i) its aspiration levels for all of its possible matches and (ii) the matches, if any, to which it belongs. The node does not keep track of its history nor is it aware of the environment state. In each stage, a randomly activated node proposes to form a new match and changes its aspiration based on the success or failure of its proposal. At this stage, the proposing node inquires about the aspiration of the agent it wants to match with to calculate the feasibility of the match. The environment matching structure changes whenever a proposal succeeds. A state is absorbing for the distributed dynamics if and only if it is in the core of the $B$-matching.
{"title":"Distributed Learning Dynamics Converging to the Core of $B$-Matchings","authors":"Aya Hamed, Jeff S. Shamma","doi":"arxiv-2409.07754","DOIUrl":"https://doi.org/arxiv-2409.07754","url":null,"abstract":"$B$-matching is a special case of matching problems where nodes can join\u0000multiple matchings with the degree of each node constrained by an upper bound,\u0000the node's $B$-value. The core solution of a bipartite $B$-matching is both a\u0000matching between the agents respecting the upper bound constraint and an\u0000allocation of the value of the edge among its nodes such that no group of\u0000agents can deviate and collectively gain higher allocation. We present two\u0000learning dynamics that converge to the core of the bipartite $B$-matching\u0000problems. The first dynamics are centralized dynamics in the nature of the\u0000Hungarian method, which converge to the core in a polynomial time. The second\u0000dynamics are distributed dynamics, which converge to the core with probability\u0000one. For the distributed dynamics, a node maintains only a state consisting of\u0000(i) its aspiration levels for all of its possible matches and (ii) the matches,\u0000if any, to which it belongs. The node does not keep track of its history nor is\u0000it aware of the environment state. In each stage, a randomly activated node\u0000proposes to form a new match and changes its aspiration based on the success or\u0000failure of its proposal. At this stage, the proposing node inquires about the\u0000aspiration of the agent it wants to match with to calculate the feasibility of\u0000the match. The environment matching structure changes whenever a proposal\u0000succeeds. A state is absorbing for the distributed dynamics if and only if it\u0000is in the core of the $B$-matching.","PeriodicalId":501316,"journal":{"name":"arXiv - CS - Computer Science and Game Theory","volume":"98 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142197671","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}
Shiri Ron, Clayton Thomas, S. Matthew Weinberg, Qianfan Zhang
We study the communication complexity of truthful combinatorial auctions, and in particular the case where valuations are either subadditive or single-minded, which we denote with $mathsf{SubAdd}cupmathsf{SingleM}$. We show that for three bidders with valuations in $mathsf{SubAdd}cupmathsf{SingleM}$, any deterministic truthful mechanism that achieves at least a $0.366$-approximation requires $exp(m)$ communication. In contrast, a natural extension of [Fei09] yields a non-truthful $mathrm{poly}(m)$-communication protocol that achieves a $frac{1}{2}$-approximation, demonstrating a gap between the power of truthful mechanisms and non-truthful protocols for this problem. Our approach follows the taxation complexity framework laid out in [Dob16b], but applies this framework in a setting not encompassed by the techniques used in past work. In particular, the only successful prior application of this framework uses a reduction to simultaneous protocols which only applies for two bidders [AKSW20], whereas our three-player lower bounds are stronger than what can possibly arise from a two-player construction (since a trivial truthful auction guarantees a $frac{1}{2}$-approximation for two players).
{"title":"Communication Separations for Truthful Auctions: Breaking the Two-Player Barrier","authors":"Shiri Ron, Clayton Thomas, S. Matthew Weinberg, Qianfan Zhang","doi":"arxiv-2409.08241","DOIUrl":"https://doi.org/arxiv-2409.08241","url":null,"abstract":"We study the communication complexity of truthful combinatorial auctions, and\u0000in particular the case where valuations are either subadditive or\u0000single-minded, which we denote with $mathsf{SubAdd}cupmathsf{SingleM}$. We\u0000show that for three bidders with valuations in\u0000$mathsf{SubAdd}cupmathsf{SingleM}$, any deterministic truthful mechanism\u0000that achieves at least a $0.366$-approximation requires $exp(m)$\u0000communication. In contrast, a natural extension of [Fei09] yields a\u0000non-truthful $mathrm{poly}(m)$-communication protocol that achieves a\u0000$frac{1}{2}$-approximation, demonstrating a gap between the power of truthful\u0000mechanisms and non-truthful protocols for this problem. Our approach follows the taxation complexity framework laid out in [Dob16b],\u0000but applies this framework in a setting not encompassed by the techniques used\u0000in past work. In particular, the only successful prior application of this\u0000framework uses a reduction to simultaneous protocols which only applies for two\u0000bidders [AKSW20], whereas our three-player lower bounds are stronger than what\u0000can possibly arise from a two-player construction (since a trivial truthful\u0000auction guarantees a $frac{1}{2}$-approximation for two players).","PeriodicalId":501316,"journal":{"name":"arXiv - CS - Computer Science and Game Theory","volume":"65 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142197666","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}
Gagan Aggarwal, Ashwinkumar Badanidiyuru, Paul Dütting, Federico Fusco
Motivated by online retail, we consider the problem of selling one item (e.g., an ad slot) to two non-excludable buyers (say, a merchant and a brand). This problem captures, for example, situations where a merchant and a brand cooperatively bid in an auction to advertise a product, and both benefit from the ad being shown. A mechanism collects bids from the two and decides whether to allocate and which payments the two parties should make. This gives rise to intricate incentive compatibility constraints, e.g., on how to split payments between the two parties. We approach the problem of finding a revenue-maximizing incentive-compatible mechanism from an online learning perspective; this poses significant technical challenges. First, the action space (the class of all possible mechanisms) is huge; second, the function that maps mechanisms to revenue is highly irregular, ruling out standard discretization-based approaches. In the stochastic setting, we design an efficient learning algorithm achieving a regret bound of $O(T^{3/4})$. Our approach is based on an adaptive discretization scheme of the space of mechanisms, as any non-adaptive discretization fails to achieve sublinear regret. In the adversarial setting, we exploit the non-Lipschitzness of the problem to prove a strong negative result, namely that no learning algorithm can achieve more than half of the revenue of the best fixed mechanism in hindsight. We then consider the $sigma$-smooth adversary; we construct an efficient learning algorithm that achieves a regret bound of $O(T^{2/3})$ and builds on a succinct encoding of exponentially many experts. Finally, we prove that no learning algorithm can achieve less than $Omega(sqrt T)$ regret in both the stochastic and the smooth setting, thus narrowing the range where the minimax regret rates for these two problems lie.
{"title":"Selling Joint Ads: A Regret Minimization Perspective","authors":"Gagan Aggarwal, Ashwinkumar Badanidiyuru, Paul Dütting, Federico Fusco","doi":"arxiv-2409.07819","DOIUrl":"https://doi.org/arxiv-2409.07819","url":null,"abstract":"Motivated by online retail, we consider the problem of selling one item\u0000(e.g., an ad slot) to two non-excludable buyers (say, a merchant and a brand).\u0000This problem captures, for example, situations where a merchant and a brand\u0000cooperatively bid in an auction to advertise a product, and both benefit from\u0000the ad being shown. A mechanism collects bids from the two and decides whether\u0000to allocate and which payments the two parties should make. This gives rise to\u0000intricate incentive compatibility constraints, e.g., on how to split payments\u0000between the two parties. We approach the problem of finding a\u0000revenue-maximizing incentive-compatible mechanism from an online learning\u0000perspective; this poses significant technical challenges. First, the action\u0000space (the class of all possible mechanisms) is huge; second, the function that\u0000maps mechanisms to revenue is highly irregular, ruling out standard\u0000discretization-based approaches. In the stochastic setting, we design an efficient learning algorithm\u0000achieving a regret bound of $O(T^{3/4})$. Our approach is based on an adaptive\u0000discretization scheme of the space of mechanisms, as any non-adaptive\u0000discretization fails to achieve sublinear regret. In the adversarial setting,\u0000we exploit the non-Lipschitzness of the problem to prove a strong negative\u0000result, namely that no learning algorithm can achieve more than half of the\u0000revenue of the best fixed mechanism in hindsight. We then consider the\u0000$sigma$-smooth adversary; we construct an efficient learning algorithm that\u0000achieves a regret bound of $O(T^{2/3})$ and builds on a succinct encoding of\u0000exponentially many experts. Finally, we prove that no learning algorithm can\u0000achieve less than $Omega(sqrt T)$ regret in both the stochastic and the\u0000smooth setting, thus narrowing the range where the minimax regret rates for\u0000these two problems lie.","PeriodicalId":501316,"journal":{"name":"arXiv - CS - Computer Science and Game Theory","volume":"29 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142197669","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}
Alexandros Hollender, Gilbert Maystre, Sai Ganesh Nagarajan
Adversarial multiplayer games are an important object of study in multiagent learning. In particular, polymatrix zero-sum games are a multiplayer setting where Nash equilibria are known to be efficiently computable. Towards understanding the limits of tractability in polymatrix games, we study the computation of Nash equilibria in such games where each pair of players plays either a zero-sum or a coordination game. We are particularly interested in the setting where players can be grouped into a small number of teams of identical interest. While the three-team version of the problem is known to be PPAD-complete, the complexity for two teams has remained open. Our main contribution is to prove that the two-team version remains hard, namely it is CLS-hard. Furthermore, we show that this lower bound is tight for the setting where one of the teams consists of multiple independent adversaries. On the way to obtaining our main result, we prove hardness of finding any stationary point in the simplest type of non-convex-concave min-max constrained optimization problem, namely for a class of bilinear polynomial objective functions.
{"title":"The Complexity of Two-Team Polymatrix Games with Independent Adversaries","authors":"Alexandros Hollender, Gilbert Maystre, Sai Ganesh Nagarajan","doi":"arxiv-2409.07398","DOIUrl":"https://doi.org/arxiv-2409.07398","url":null,"abstract":"Adversarial multiplayer games are an important object of study in multiagent\u0000learning. In particular, polymatrix zero-sum games are a multiplayer setting\u0000where Nash equilibria are known to be efficiently computable. Towards\u0000understanding the limits of tractability in polymatrix games, we study the\u0000computation of Nash equilibria in such games where each pair of players plays\u0000either a zero-sum or a coordination game. We are particularly interested in the\u0000setting where players can be grouped into a small number of teams of identical\u0000interest. While the three-team version of the problem is known to be\u0000PPAD-complete, the complexity for two teams has remained open. Our main\u0000contribution is to prove that the two-team version remains hard, namely it is\u0000CLS-hard. Furthermore, we show that this lower bound is tight for the setting\u0000where one of the teams consists of multiple independent adversaries. On the way\u0000to obtaining our main result, we prove hardness of finding any stationary point\u0000in the simplest type of non-convex-concave min-max constrained optimization\u0000problem, namely for a class of bilinear polynomial objective functions.","PeriodicalId":501316,"journal":{"name":"arXiv - CS - Computer Science and Game Theory","volume":"171 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142197667","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}
Eric Balkanski, Vasilis Gkatzelis, Golnoosh Shahkarami
We revisit the canonical problem of strategic facility location and study the power and limitations of randomization in the design of truthful mechanisms augmented with machine-learned predictions. In the strategic facility location problem, a set of agents are asked to report their locations in some metric space and the goal is to use these reported locations to determine where to open a new facility, aiming to optimize some aggregate measure of distance of the agents from that facility. However, the agents are strategic and can misreport their locations if this may lead to a facility location choice that they prefer. The goal is to design truthful mechanisms, which ensure the agents cannot benefit by misreporting. A lot of prior work has studied this problem from a worst-case perspective, but a recent line of work proposed a framework to refine these results when the designer is provided with some, possible incorrect, predictions regarding the agents' true locations. The goal is to simultaneously provide strong consistency guarantees (i.e., guarantees when the predictions provided to the mechanism are correct) and near-optimal robustness guarantees (i.e., guarantees that hold irrespective of how inaccurate the predictions may be). In this work we focus on the power of randomization in this problem and analyze the best approximation guarantees achievable with respect to the egalitarian social cost measure for one- and two-dimensional Euclidean spaces.
{"title":"Randomized Strategic Facility Location with Predictions","authors":"Eric Balkanski, Vasilis Gkatzelis, Golnoosh Shahkarami","doi":"arxiv-2409.07142","DOIUrl":"https://doi.org/arxiv-2409.07142","url":null,"abstract":"We revisit the canonical problem of strategic facility location and study the\u0000power and limitations of randomization in the design of truthful mechanisms\u0000augmented with machine-learned predictions. In the strategic facility location\u0000problem, a set of agents are asked to report their locations in some metric\u0000space and the goal is to use these reported locations to determine where to\u0000open a new facility, aiming to optimize some aggregate measure of distance of\u0000the agents from that facility. However, the agents are strategic and can\u0000misreport their locations if this may lead to a facility location choice that\u0000they prefer. The goal is to design truthful mechanisms, which ensure the agents\u0000cannot benefit by misreporting. A lot of prior work has studied this problem\u0000from a worst-case perspective, but a recent line of work proposed a framework\u0000to refine these results when the designer is provided with some, possible\u0000incorrect, predictions regarding the agents' true locations. The goal is to\u0000simultaneously provide strong consistency guarantees (i.e., guarantees when the\u0000predictions provided to the mechanism are correct) and near-optimal robustness\u0000guarantees (i.e., guarantees that hold irrespective of how inaccurate the\u0000predictions may be). In this work we focus on the power of randomization in\u0000this problem and analyze the best approximation guarantees achievable with\u0000respect to the egalitarian social cost measure for one- and two-dimensional\u0000Euclidean spaces.","PeriodicalId":501316,"journal":{"name":"arXiv - CS - Computer Science and Game Theory","volume":"12 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142197517","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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