In adversarial resource allocation settings, forming alliances can enhance performance, but the benefits may diminish if alliance formation becomes costly. In this work, we explore this issue using the framework of the coalitional Blotto game, in which two players compete separately against a common adversary across valued contests. Each player aims to win contests by allocating more of their limited budget than their opponent. Previous work on alliance formation in coalitional Blotto games has shown that if one player transfers a portion of their budget to the other, then both players can perform better; however, it is also known that it is never in either players' interest to simply concede a portion of their budget. In this work, we study the setting in which transfers are costly, meaning that if one player donates a portion of their budget, the recipient only receives a fraction of the transferred amount. We show that mutually beneficial costly transfers exist, and we provide necessary and sufficient conditions for the existence of such a costly transfer. Then, we consider the setting in which players can transfer budgets and contests at a cost, and we show that this alliance strategy is mutually beneficial in almost all game instances.
{"title":"Inefficient Alliance Formation in Coalitional Blotto Games","authors":"Vade Shah, Keith Paarporn, Jason R. Marden","doi":"arxiv-2409.06899","DOIUrl":"https://doi.org/arxiv-2409.06899","url":null,"abstract":"In adversarial resource allocation settings, forming alliances can enhance\u0000performance, but the benefits may diminish if alliance formation becomes\u0000costly. In this work, we explore this issue using the framework of the\u0000coalitional Blotto game, in which two players compete separately against a\u0000common adversary across valued contests. Each player aims to win contests by\u0000allocating more of their limited budget than their opponent. Previous work on\u0000alliance formation in coalitional Blotto games has shown that if one player\u0000transfers a portion of their budget to the other, then both players can perform\u0000better; however, it is also known that it is never in either players' interest\u0000to simply concede a portion of their budget. In this work, we study the setting\u0000in which transfers are costly, meaning that if one player donates a portion of\u0000their budget, the recipient only receives a fraction of the transferred amount.\u0000We show that mutually beneficial costly transfers exist, and we provide\u0000necessary and sufficient conditions for the existence of such a costly\u0000transfer. Then, we consider the setting in which players can transfer budgets\u0000and contests at a cost, and we show that this alliance strategy is mutually\u0000beneficial in almost all game instances.","PeriodicalId":501316,"journal":{"name":"arXiv - CS - Computer Science and Game Theory","volume":"5 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142197668","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}
Laura Georgescu, James Fox, Anna Gautier, Michael Wooldridge
Quadratic Voting (QV) is a social choice mechanism that addresses the "tyranny of the majority" of one-person-one-vote mechanisms. Agents express not only their preference ordering but also their preference intensity by purchasing $x$ votes at a cost of $x^2$. Although this pricing rule maximizes utilitarian social welfare and is robust against strategic manipulation, it has not yet found many real-life applications. One key reason is that the original QV mechanism does not limit voter budgets. Two variations have since been proposed: a (no-budget) multiple-issue generalization and a fixed-budget version that allocates a constant number of credits to agents for use in multiple binary elections. While some analysis has been undertaken with respect to the multiple-issue variation, the fixed-budget version has not yet been rigorously studied. In this work, we formally propose a novel fixed-budget multiple-issue QV mechanism. This integrates the advantages of both the aforementioned variations, laying the theoretical foundations for practical use cases of QV, such as multi-agent resource allocation. We analyse our fixed-budget multiple-issue QV by comparing it with traditional voting systems, exploring potential collusion strategies, and showing that checking whether strategy profiles form a Nash equilibrium is tractable.
{"title":"Fixed-budget and Multiple-issue Quadratic Voting","authors":"Laura Georgescu, James Fox, Anna Gautier, Michael Wooldridge","doi":"arxiv-2409.06614","DOIUrl":"https://doi.org/arxiv-2409.06614","url":null,"abstract":"Quadratic Voting (QV) is a social choice mechanism that addresses the\u0000\"tyranny of the majority\" of one-person-one-vote mechanisms. Agents express not\u0000only their preference ordering but also their preference intensity by\u0000purchasing $x$ votes at a cost of $x^2$. Although this pricing rule maximizes\u0000utilitarian social welfare and is robust against strategic manipulation, it has\u0000not yet found many real-life applications. One key reason is that the original\u0000QV mechanism does not limit voter budgets. Two variations have since been\u0000proposed: a (no-budget) multiple-issue generalization and a fixed-budget\u0000version that allocates a constant number of credits to agents for use in\u0000multiple binary elections. While some analysis has been undertaken with respect\u0000to the multiple-issue variation, the fixed-budget version has not yet been\u0000rigorously studied. In this work, we formally propose a novel fixed-budget\u0000multiple-issue QV mechanism. This integrates the advantages of both the\u0000aforementioned variations, laying the theoretical foundations for practical use\u0000cases of QV, such as multi-agent resource allocation. We analyse our\u0000fixed-budget multiple-issue QV by comparing it with traditional voting systems,\u0000exploring potential collusion strategies, and showing that checking whether\u0000strategy profiles form a Nash equilibrium is tractable.","PeriodicalId":501316,"journal":{"name":"arXiv - CS - Computer Science and Game Theory","volume":"27 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142197670","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 games in which a leader makes a single commitment, and then multiple followers (each with a different utility function) respond. In particular, we study ambiguous commitment strategies in these games, in which the leader may commit to a set of mixed strategies, and ambiguity-averse followers respond to maximize their worst-case utility over the set of leader strategies. Special cases of this setting have previously been studied when there is a single follower: in these cases, it is known that the leader can increase her utility by making an ambiguous commitment if the follower is restricted to playing a pure strategy, but that no gain can be had from ambiguity if the follower may mix. We confirm that this result continues to hold in the setting of general Stackelberg games. We then develop a theory of ambiguous commitment in games with multiple followers. We begin by considering the case where the leader must make the same commitment against each follower. We establish that -- unlike the case of a single follower -- ambiguous commitment can improve the leader's utility by an unboundedly large factor, even when followers are permitted to respond with mixed strategies and even. We go on to show an advantage for the leader coupling the same commitment across all followers, even when she has the ability to make a separate commitment to each follower. In particular, there exist general sum games in which the leader can enjoy an unboundedly large advantage by coupling her ambiguous commitment across multiple followers rather than committing against each individually. In zero-sum games we show there can be no such coupling advantage. Finally, we give a polynomial time algorithm for computing the optimal leader commitment strategy in the special case in which the leader has 2 actions (and k followers may have m actions), and prove that in the general case, the problem is NP-hard.
我们研究了领导者做出单一承诺，然后多个追随者（每个追随者都有不同的效用函数）做出回应的博弈。特别是，我们研究了这些博弈中的模糊承诺策略，其中领导者可能承诺采取一组混合策略，而模糊规避型追随者的回应是在领导者策略集上最大化他们的最坏情况效用。在这种情况下，众所周知，如果追随者只能采取纯策略，那么领导者可以通过做出模棱两可的承诺来增加自己的效用；但如果追随者可以采取混合策略，那么领导者就无法从模棱两可中获得收益。我们证实，这一结果在一般斯塔克尔伯格博弈中依然成立。然后，我们建立了一个多追随者博弈中的模糊承诺理论。我们首先考虑领导者必须对每个追随者做出相同承诺的情况。我们发现，与单个追随者的情况不同，模棱两可的承诺能以无限大的系数提高领导者的效用，即使在允许追随者以混合策略做出回应的情况下也是如此。我们进而证明，即使领导者有能力对每个追随者做出单独的承诺，对所有追随者做出相同的承诺也是有优势的。特别是，在一般和博弈中，领导者可以通过在多个追随者之间做出模糊承诺，而不是对每个追随者单独做出承诺，从而获得无限大的优势。在零和博弈中，我们证明不可能存在这种耦合优势。最后，在领导者有 2 个行动（k 个追随者可能有 m 个行动）的特殊情况下，我们给出了计算最优领导者承诺策略的多项式时间算法，并证明在一般情况下，这个问题是 NP 难的。
{"title":"The Value of Ambiguous Commitments in Multi-Follower Games","authors":"Natalie Collina, Rabanus Derr, Aaron Roth","doi":"arxiv-2409.05608","DOIUrl":"https://doi.org/arxiv-2409.05608","url":null,"abstract":"We study games in which a leader makes a single commitment, and then multiple\u0000followers (each with a different utility function) respond. In particular, we\u0000study ambiguous commitment strategies in these games, in which the leader may\u0000commit to a set of mixed strategies, and ambiguity-averse followers respond to\u0000maximize their worst-case utility over the set of leader strategies. Special\u0000cases of this setting have previously been studied when there is a single\u0000follower: in these cases, it is known that the leader can increase her utility\u0000by making an ambiguous commitment if the follower is restricted to playing a\u0000pure strategy, but that no gain can be had from ambiguity if the follower may\u0000mix. We confirm that this result continues to hold in the setting of general\u0000Stackelberg games. We then develop a theory of ambiguous commitment in games\u0000with multiple followers. We begin by considering the case where the leader must\u0000make the same commitment against each follower. We establish that -- unlike the\u0000case of a single follower -- ambiguous commitment can improve the leader's\u0000utility by an unboundedly large factor, even when followers are permitted to\u0000respond with mixed strategies and even. We go on to show an advantage for the\u0000leader coupling the same commitment across all followers, even when she has the\u0000ability to make a separate commitment to each follower. In particular, there\u0000exist general sum games in which the leader can enjoy an unboundedly large\u0000advantage by coupling her ambiguous commitment across multiple followers rather\u0000than committing against each individually. In zero-sum games we show there can\u0000be no such coupling advantage. Finally, we give a polynomial time algorithm for\u0000computing the optimal leader commitment strategy in the special case in which\u0000the leader has 2 actions (and k followers may have m actions), and prove that\u0000in the general case, the problem is NP-hard.","PeriodicalId":501316,"journal":{"name":"arXiv - CS - Computer Science and Game Theory","volume":"204 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142197673","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}
ChatGPT has established Generative AI (GenAI) as a significant technological advancement. However, GenAI's intricate relationship with competing platforms and its downstream impact on users remains under-explored. This paper initiates the study of GenAI's long-term social impact resulting from the weakening network effect of human-based platforms like Stack Overflow. First, we study GenAI's revenue-maximization optimization problem. We develop an approximately optimal solution and show that the optimal solution has a non-cyclic structure. Then, we analyze the social impact, showing that GenAI could be socially harmful. Specifically, we present an analog to Braess's paradox in which all users would be better off without GenAI. Finally, we develop necessary and sufficient conditions for a regulator with incomplete information to ensure that GenAI is socially beneficial.
{"title":"Braess's Paradox of Generative AI","authors":"Boaz Taitler, Omer Ben-Porat","doi":"arxiv-2409.05506","DOIUrl":"https://doi.org/arxiv-2409.05506","url":null,"abstract":"ChatGPT has established Generative AI (GenAI) as a significant technological\u0000advancement. However, GenAI's intricate relationship with competing platforms\u0000and its downstream impact on users remains under-explored. This paper initiates\u0000the study of GenAI's long-term social impact resulting from the weakening\u0000network effect of human-based platforms like Stack Overflow. First, we study\u0000GenAI's revenue-maximization optimization problem. We develop an approximately\u0000optimal solution and show that the optimal solution has a non-cyclic structure.\u0000Then, we analyze the social impact, showing that GenAI could be socially\u0000harmful. Specifically, we present an analog to Braess's paradox in which all\u0000users would be better off without GenAI. Finally, we develop necessary and\u0000sufficient conditions for a regulator with incomplete information to ensure\u0000that GenAI is socially beneficial.","PeriodicalId":501316,"journal":{"name":"arXiv - CS - Computer Science and Game Theory","volume":"132 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142197509","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 explores concurrent FL processes within a three-tier system, with edge servers between edge devices and FL servers. A challenge in this setup is the limited bandwidth from edge devices to edge servers. Thus, allocating the bandwidth efficiently and fairly to support simultaneous FL processes becomes crucial. We propose a game-theoretic approach to model the bandwidth allocation problem and develop distributed and centralized heuristic schemes to find an approximate Nash Equilibrium of the game. We proposed the approach mentioned above using centralized and entirely distributed assumptions. Through rigorous analysis and experimentation, we demonstrate that our schemes efficiently and fairly assign the bandwidth to the FL processes for centralized and distributed solutions and outperform a baseline scheme where each edge server assigns bandwidth proportionally to the FL servers' requests that it receives. The proposed distributed and centralized schemes have comptetive performance.
{"title":"Fair Allocation of Bandwidth At Edge Servers For Concurrent Hierarchical Federated Learning","authors":"Md Anwar Hossen, Fatema Siddika, Wensheng Zhang","doi":"arxiv-2409.04921","DOIUrl":"https://doi.org/arxiv-2409.04921","url":null,"abstract":"This paper explores concurrent FL processes within a three-tier system, with\u0000edge servers between edge devices and FL servers. A challenge in this setup is\u0000the limited bandwidth from edge devices to edge servers. Thus, allocating the\u0000bandwidth efficiently and fairly to support simultaneous FL processes becomes\u0000crucial. We propose a game-theoretic approach to model the bandwidth allocation\u0000problem and develop distributed and centralized heuristic schemes to find an\u0000approximate Nash Equilibrium of the game. We proposed the approach mentioned\u0000above using centralized and entirely distributed assumptions. Through rigorous\u0000analysis and experimentation, we demonstrate that our schemes efficiently and\u0000fairly assign the bandwidth to the FL processes for centralized and distributed\u0000solutions and outperform a baseline scheme where each edge server assigns\u0000bandwidth proportionally to the FL servers' requests that it receives. The\u0000proposed distributed and centralized schemes have comptetive performance.","PeriodicalId":501316,"journal":{"name":"arXiv - CS - Computer Science and Game Theory","volume":"26 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142197508","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 envy-free up to any item (EFX) allocations on graphs where vertices and edges represent agents and items respectively. An agent is only interested in items that are incident to her and all other items have zero marginal values to her. Christodoulou et al. [EC, 2023] first proposed this setting and studied the case of goods. We extend this setting to the case of mixed manna where an item may be liked or disliked by its endpoint agents. In our problem, an agent has an arbitrary valuation over her incident items such that the items she likes have non-negative marginal values to her and those she dislikes have non-positive marginal values. We provide a complete study of the four notions of EFX for mixed manna in the literature, which differ by whether the removed item can have zero marginal value. We prove that an allocation that satisfies the notion of EFX where the virtually-removed item could always have zero marginal value may not exist and determining its existence is NP-complete, while one that satisfies any of the other three notions always exists and can be computed in polynomial time. We also prove that an orientation (i.e., a special allocation where each edge must be allocated to one of its endpoint agents) that satisfies any of the four notions may not exist, and determining its existence is NP-complete.
{"title":"A Complete Landscape of EFX Allocations of Mixed Manna on Graphs","authors":"Yu Zhou, Tianze Wei, Minming Li, Bo Li","doi":"arxiv-2409.03594","DOIUrl":"https://doi.org/arxiv-2409.03594","url":null,"abstract":"We study envy-free up to any item (EFX) allocations on graphs where vertices\u0000and edges represent agents and items respectively. An agent is only interested\u0000in items that are incident to her and all other items have zero marginal values\u0000to her. Christodoulou et al. [EC, 2023] first proposed this setting and studied\u0000the case of goods. We extend this setting to the case of mixed manna where an\u0000item may be liked or disliked by its endpoint agents. In our problem, an agent\u0000has an arbitrary valuation over her incident items such that the items she\u0000likes have non-negative marginal values to her and those she dislikes have\u0000non-positive marginal values. We provide a complete study of the four notions\u0000of EFX for mixed manna in the literature, which differ by whether the removed\u0000item can have zero marginal value. We prove that an allocation that satisfies\u0000the notion of EFX where the virtually-removed item could always have zero\u0000marginal value may not exist and determining its existence is NP-complete,\u0000while one that satisfies any of the other three notions always exists and can\u0000be computed in polynomial time. We also prove that an orientation (i.e., a\u0000special allocation where each edge must be allocated to one of its endpoint\u0000agents) that satisfies any of the four notions may not exist, and determining\u0000its existence is NP-complete.","PeriodicalId":501316,"journal":{"name":"arXiv - CS - Computer Science and Game Theory","volume":"10 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142197505","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}
When learning to play an imperfect information game, it is often easier to first start with the basic mechanics of the game rules. For example, one can play several example rounds with private cards revealed to all players to better understand the basic actions and their effects. Building on this intuition, this paper introduces {it progressive hiding}, an algorithm that learns to play imperfect information games by first learning the basic mechanics and then progressively adding information constraints over time. Progressive hiding is inspired by methods from stochastic multistage optimization such as scenario decomposition and progressive hedging. We prove that it enables the adaptation of counterfactual regret minimization to games where perfect recall is not satisfied. Numerical experiments illustrate that progressive hiding can achieve optimal payoff in a benchmark of emergent communication trading game.
{"title":"Learning in Games with progressive hiding","authors":"Heymann Benjamin, Lanctot Marc","doi":"arxiv-2409.03875","DOIUrl":"https://doi.org/arxiv-2409.03875","url":null,"abstract":"When learning to play an imperfect information game, it is often easier to\u0000first start with the basic mechanics of the game rules. For example, one can\u0000play several example rounds with private cards revealed to all players to\u0000better understand the basic actions and their effects. Building on this\u0000intuition, this paper introduces {it progressive hiding}, an algorithm that\u0000learns to play imperfect information games by first learning the basic\u0000mechanics and then progressively adding information constraints over time.\u0000Progressive hiding is inspired by methods from stochastic multistage\u0000optimization such as scenario decomposition and progressive hedging. We prove\u0000that it enables the adaptation of counterfactual regret minimization to games\u0000where perfect recall is not satisfied. Numerical experiments illustrate that\u0000progressive hiding can achieve optimal payoff in a benchmark of emergent\u0000communication trading game.","PeriodicalId":501316,"journal":{"name":"arXiv - CS - Computer Science and Game Theory","volume":"30 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142197672","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 display advertising, advertisers want to achieve a marketing objective with constraints on budget and cost-per-outcome. This is usually formulated as an optimization problem that maximizes the total utility under constraints. The optimization is carried out in an online fashion in the dual space - for an incoming Ad auction, a bid is placed using an optimal bidding formula, assuming optimal values for the dual variables; based on the outcome of the previous auctions, the dual variables are updated in an online fashion. While this approach is theoretically sound, in practice, the dual variables are not optimal from the beginning, but rather converge over time. Specifically, for the cost-constraint, the convergence is asymptotic. As a result, we find that cost-control is ineffective. In this work, we analyse the shortcomings of the optimal bidding formula and propose a modification that deviates from the theoretical derivation. We simulate various practical scenarios and study the cost-control behaviors of the two algorithms. Through a large-scale evaluation on the real-word data, we show that the proposed modification reduces the cost violations by 50%, thereby achieving a better cost-control than the theoretical bidding formula.
{"title":"Cost-Control in Display Advertising: Theory vs Practice","authors":"Anoop R Katti, Rui C. Gonçalves, Rinchin Iakovlev","doi":"arxiv-2409.03874","DOIUrl":"https://doi.org/arxiv-2409.03874","url":null,"abstract":"In display advertising, advertisers want to achieve a marketing objective\u0000with constraints on budget and cost-per-outcome. This is usually formulated as\u0000an optimization problem that maximizes the total utility under constraints. The\u0000optimization is carried out in an online fashion in the dual space - for an\u0000incoming Ad auction, a bid is placed using an optimal bidding formula, assuming\u0000optimal values for the dual variables; based on the outcome of the previous\u0000auctions, the dual variables are updated in an online fashion. While this\u0000approach is theoretically sound, in practice, the dual variables are not\u0000optimal from the beginning, but rather converge over time. Specifically, for\u0000the cost-constraint, the convergence is asymptotic. As a result, we find that\u0000cost-control is ineffective. In this work, we analyse the shortcomings of the\u0000optimal bidding formula and propose a modification that deviates from the\u0000theoretical derivation. We simulate various practical scenarios and study the\u0000cost-control behaviors of the two algorithms. Through a large-scale evaluation\u0000on the real-word data, we show that the proposed modification reduces the cost\u0000violations by 50%, thereby achieving a better cost-control than the theoretical\u0000bidding formula.","PeriodicalId":501316,"journal":{"name":"arXiv - CS - Computer Science and Game Theory","volume":"43 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142197674","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 Knaster-Tarski theorem, also known as Tarski's theorem, guarantees that every monotone function defined on a complete lattice has a fixed point. We analyze the query complexity of finding such a fixed point on the $k$-dimensional grid of side length $n$ under the $leq$ relation. Specifically, there is an unknown monotone function $f: {0,1,ldots, n-1}^k to {0,1,ldots, n-1}^k$ and an algorithm must query a vertex $v$ to learn $f(v)$. Our main result is a randomized lower bound of $Omegaleft( k + frac{k cdot log{n}}{log{k}} right)$ for the $k$-dimensional grid of side length $n$, which is nearly optimal in high dimensions when $k$ is large relative to $n$. As a corollary, we characterize the randomized and deterministic query complexity on the Boolean hypercube ${0,1}^k$ as $Theta(k)$.
{"title":"Randomized Lower Bounds for Tarski Fixed Points in High Dimensions","authors":"Simina Brânzei, Reed Phillips, Nicholas Recker","doi":"arxiv-2409.03751","DOIUrl":"https://doi.org/arxiv-2409.03751","url":null,"abstract":"The Knaster-Tarski theorem, also known as Tarski's theorem, guarantees that\u0000every monotone function defined on a complete lattice has a fixed point. We\u0000analyze the query complexity of finding such a fixed point on the\u0000$k$-dimensional grid of side length $n$ under the $leq$ relation.\u0000Specifically, there is an unknown monotone function $f: {0,1,ldots, n-1}^k\u0000to {0,1,ldots, n-1}^k$ and an algorithm must query a vertex $v$ to learn\u0000$f(v)$. Our main result is a randomized lower bound of $Omegaleft( k + frac{k\u0000cdot log{n}}{log{k}} right)$ for the $k$-dimensional grid of side length\u0000$n$, which is nearly optimal in high dimensions when $k$ is large relative to\u0000$n$. As a corollary, we characterize the randomized and deterministic query\u0000complexity on the Boolean hypercube ${0,1}^k$ as $Theta(k)$.","PeriodicalId":501316,"journal":{"name":"arXiv - CS - Computer Science and Game Theory","volume":"11 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142197506","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 the domain of machine learning and game theory, the quest for Nash Equilibrium (NE) in extensive-form games with incomplete information is challenging yet crucial for enhancing AI's decision-making support under varied scenarios. Traditional Counterfactual Regret Minimization (CFR) techniques excel in navigating towards NE, focusing on scenarios where opponents deploy optimal strategies. However, the essence of machine learning in strategic game play extends beyond reacting to optimal moves; it encompasses aiding human decision-making in all circumstances. This includes not only crafting responses to optimal strategies but also recovering from suboptimal decisions and capitalizing on opponents' errors. Herein lies the significance of transitioning from NE to Bayesian Perfect Equilibrium (BPE), which accounts for every possible condition, including the irrationality of opponents. To bridge this gap, we propose Belief Update Fictitious Play (BUFP), which innovatively blends fictitious play with belief to target BPE, a more comprehensive equilibrium concept than NE. Specifically, through adjusting iteration stepsizes, BUFP allows for strategic convergence to both NE and BPE. For instance, in our experiments, BUFP(EF) leverages the stepsize of Extensive Form Fictitious Play (EFFP) to achieve BPE, outperforming traditional CFR by securing a 48.53% increase in benefits in scenarios characterized by dominated strategies.
在机器学习和博弈论领域，在信息不完全的广式博弈中寻求纳什均衡（NashEquilibrium，NE）是一项挑战，但对于增强人工智能在各种场景下的决策支持至关重要。传统的 "反事实遗憾最小化"（CFR）技术能很好地实现 NE 导航，重点关注对手部署最佳策略的场景。然而，战略游戏中机器学习的本质不仅仅是对最优策略做出反应，它还包括在所有情况下帮助人类做出决策。这不仅包括对最优策略做出反应，还包括从次优决策中恢复，以及利用对手的失误。从 NE 过渡到贝叶斯完美均衡（BPE）的意义就在于此，后者考虑了所有可能的情况，包括对手的非理性。为了弥补这一差距，我们提出了 "信念更新虚构对局"（BUFP），它创新性地将虚构对局与信念相结合，以贝叶斯完美均衡（BPE）为目标，这是一个比NE更全面的均衡概念。例如，在我们的实验中，BUFP（EF）利用扩展形式虚构博弈（ExtensiveForm Fictitious Play，EFFP）的步长来实现 BPE，其表现优于传统的 CFR，在以主导战略为特征的场景中确保了 48.53% 的收益增长。
{"title":"Beyond Nash Equilibrium: Achieving Bayesian Perfect Equilibrium with Belief Update Fictitious Play","authors":"Qi Ju, Zhemei Fang, Yunfeng Luo","doi":"arxiv-2409.02706","DOIUrl":"https://doi.org/arxiv-2409.02706","url":null,"abstract":"In the domain of machine learning and game theory, the quest for Nash\u0000Equilibrium (NE) in extensive-form games with incomplete information is\u0000challenging yet crucial for enhancing AI's decision-making support under varied\u0000scenarios. Traditional Counterfactual Regret Minimization (CFR) techniques\u0000excel in navigating towards NE, focusing on scenarios where opponents deploy\u0000optimal strategies. However, the essence of machine learning in strategic game\u0000play extends beyond reacting to optimal moves; it encompasses aiding human\u0000decision-making in all circumstances. This includes not only crafting responses\u0000to optimal strategies but also recovering from suboptimal decisions and\u0000capitalizing on opponents' errors. Herein lies the significance of\u0000transitioning from NE to Bayesian Perfect Equilibrium (BPE), which accounts for\u0000every possible condition, including the irrationality of opponents. To bridge this gap, we propose Belief Update Fictitious Play (BUFP), which\u0000innovatively blends fictitious play with belief to target BPE, a more\u0000comprehensive equilibrium concept than NE. Specifically, through adjusting\u0000iteration stepsizes, BUFP allows for strategic convergence to both NE and BPE.\u0000For instance, in our experiments, BUFP(EF) leverages the stepsize of Extensive\u0000Form Fictitious Play (EFFP) to achieve BPE, outperforming traditional CFR by\u0000securing a 48.53% increase in benefits in scenarios characterized by dominated\u0000strategies.","PeriodicalId":501316,"journal":{"name":"arXiv - CS - Computer Science and Game Theory","volume":"88 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142197596","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}
京公网安备 11010802042870号
京ICP备2023020795号-1
扫码关注我们
求助内容：
应助结果提醒方式：