We propose two solution concepts for matchings under preferences: robustness and near stability. The former strengthens while the latter relaxes the classical definition of stability by Gale and Shapley (1962). Informally speaking, robustness requires that a matching must be stable in the classical sense, even if the agents slightly change their preferences. Near stability, however, imposes that a matching must become stable (again, in the classical sense) provided the agents are willing to adjust their preferences a bit. Both of our concepts are quantitative; together they provide means for a fine-grained analysis of the stability of matchings. Moreover, our concepts allow the exploration of tradeoffs between stability and other criteria of social optimality, such as the egalitarian cost and the number of unmatched agents. We investigate the computational complexity of finding matchings that implement certain predefined tradeoffs. We provide a polynomial-time algorithm that, given agent preferences, returns a socially optimal robust matching (if it exists), and we prove that finding a socially optimal and nearly stable matching is computationally hard.
{"title":"Matchings under Preferences: Strength of Stability and Tradeoffs","authors":"Jiehua Chen, P. Skowron, Manuel Sorge","doi":"10.1145/3485000","DOIUrl":"https://doi.org/10.1145/3485000","url":null,"abstract":"We propose two solution concepts for matchings under preferences: robustness and near stability. The former strengthens while the latter relaxes the classical definition of stability by Gale and Shapley (1962). Informally speaking, robustness requires that a matching must be stable in the classical sense, even if the agents slightly change their preferences. Near stability, however, imposes that a matching must become stable (again, in the classical sense) provided the agents are willing to adjust their preferences a bit. Both of our concepts are quantitative; together they provide means for a fine-grained analysis of the stability of matchings. Moreover, our concepts allow the exploration of tradeoffs between stability and other criteria of social optimality, such as the egalitarian cost and the number of unmatched agents. We investigate the computational complexity of finding matchings that implement certain predefined tradeoffs. We provide a polynomial-time algorithm that, given agent preferences, returns a socially optimal robust matching (if it exists), and we prove that finding a socially optimal and nearly stable matching is computationally hard.","PeriodicalId":42216,"journal":{"name":"ACM Transactions on Economics and Computation","volume":"9 1","pages":"1 - 55"},"PeriodicalIF":1.2,"publicationDate":"2021-10-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43455540","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}
Stable matching in a community consisting of N men and N women is a classical combinatorial problem that has been the subject of intense theoretical and empirical study since its introduction in 1962 in a seminal work by Gale and Shapley. When the input preference profile is generated from a distribution, we study the output distribution of two stable matching procedures: women-proposing-deferred-acceptance and men-proposing-deferred-acceptance. We show that the two procedures are ex-ante equivalent—that is, under certain conditions on the input distribution, their output distributions are identical. In terms of technical contributions, we generalize (to the non-uniform case) an integral formula, due to Knuth and Pittel, which gives the probability that a fixed matching is stable. Using an inclusion-exclusion principle on the set of rotations, we give a new formula that gives the probability that a fixed matching is the women/men-optimal stable matching.
{"title":"Two-Sided Random Matching Markets: Ex-Ante Equivalence of the Deferred Acceptance Procedures","authors":"Simon Mauras","doi":"10.1145/3485010","DOIUrl":"https://doi.org/10.1145/3485010","url":null,"abstract":"Stable matching in a community consisting of N men and N women is a classical combinatorial problem that has been the subject of intense theoretical and empirical study since its introduction in 1962 in a seminal work by Gale and Shapley. When the input preference profile is generated from a distribution, we study the output distribution of two stable matching procedures: women-proposing-deferred-acceptance and men-proposing-deferred-acceptance. We show that the two procedures are ex-ante equivalent—that is, under certain conditions on the input distribution, their output distributions are identical. In terms of technical contributions, we generalize (to the non-uniform case) an integral formula, due to Knuth and Pittel, which gives the probability that a fixed matching is stable. Using an inclusion-exclusion principle on the set of rotations, we give a new formula that gives the probability that a fixed matching is the women/men-optimal stable matching.","PeriodicalId":42216,"journal":{"name":"ACM Transactions on Economics and Computation","volume":"9 1","pages":"1 - 14"},"PeriodicalIF":1.2,"publicationDate":"2021-10-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46836525","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 matching markets (OMMs) are commonly used in today’s world to pair agents from two parties (whom we will call offline and online agents) for mutual benefit. However, studies have shown that the algorithms making decisions in these OMMs often leave disparities in matching rates, especially for offline agents. In this article, we propose online matching algorithms that optimize for either individual or group-level fairness among offline agents in OMMs. We present two linear-programming (LP) based sampling algorithms, which achieve competitive ratios at least 0.725 for individual fairness maximization and 0.719 for group fairness maximization. We derive further bounds based on fairness parameters, demonstrating conditions under which the competitive ratio can increase to 100%. There are two key ideas helping us break the barrier of 1-1/𝖾~ 63.2% for competitive ratio in online matching. One is boosting, which is to adaptively re-distribute all sampling probabilities among only the available neighbors for every arriving online agent. The other is attenuation, which aims to balance the matching probabilities among offline agents with different mass allocated by the benchmark LP. We conduct extensive numerical experiments and results show that our boosted version of sampling algorithms are not only conceptually easy to implement but also highly effective in practical instances of OMMs where fairness is a concern.
{"title":"Fairness Maximization among Offline Agents in Online-Matching Markets","authors":"Will Ma, Pan Xu, Yifan Xu","doi":"10.1145/3569705","DOIUrl":"https://doi.org/10.1145/3569705","url":null,"abstract":"Online matching markets (OMMs) are commonly used in today’s world to pair agents from two parties (whom we will call offline and online agents) for mutual benefit. However, studies have shown that the algorithms making decisions in these OMMs often leave disparities in matching rates, especially for offline agents. In this article, we propose online matching algorithms that optimize for either individual or group-level fairness among offline agents in OMMs. We present two linear-programming (LP) based sampling algorithms, which achieve competitive ratios at least 0.725 for individual fairness maximization and 0.719 for group fairness maximization. We derive further bounds based on fairness parameters, demonstrating conditions under which the competitive ratio can increase to 100%. There are two key ideas helping us break the barrier of 1-1/𝖾~ 63.2% for competitive ratio in online matching. One is boosting, which is to adaptively re-distribute all sampling probabilities among only the available neighbors for every arriving online agent. The other is attenuation, which aims to balance the matching probabilities among offline agents with different mass allocated by the benchmark LP. We conduct extensive numerical experiments and results show that our boosted version of sampling algorithms are not only conceptually easy to implement but also highly effective in practical instances of OMMs where fairness is a concern.","PeriodicalId":42216,"journal":{"name":"ACM Transactions on Economics and Computation","volume":"10 1","pages":"1 - 27"},"PeriodicalIF":1.2,"publicationDate":"2021-09-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42880422","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}
Pub Date : 2021-05-18DOI: 10.1609/aaai.v35i6.16658
F. Brandt, Martin Bullinger, A. Wilczynski
The formal study of coalition formation in multi-agent systems is typically realized in the framework of hedonic games, which originate from economic theory. The main focus of this branch of research has been on the existence and the computational complexity of deciding the existence of coalition structures that satisfy various stability criteria. The actual process of forming coalitions based on individual behavior has received little attention. In this article, we study the convergence of simple dynamics leading to stable partitions in a variety of established classes of hedonic games, including anonymous, dichotomous, fractional, and hedonic diversity games. The dynamics we consider is based on individual stability: an agent will join another coalition if she is better off and no member of the welcoming coalition is worse off. Our results are threefold. First, we identify conditions for the (fast) convergence of our dynamics. To this end, we develop new techniques based on the simultaneous usage of multiple intertwined potential functions and establish a reduction uncovering a close relationship between anonymous hedonic games and hedonic diversity games. Second, we provide elaborate counterexamples determining tight boundaries for the existence of individually stable partitions. Third, we study the computational complexity of problems related to the coalition formation dynamics. In particular, we settle open problems suggested by Bogomolnaia and Jackson, Brandl et al., and Boehmer and Elkind.
{"title":"Reaching Individually Stable Coalition Structures","authors":"F. Brandt, Martin Bullinger, A. Wilczynski","doi":"10.1609/aaai.v35i6.16658","DOIUrl":"https://doi.org/10.1609/aaai.v35i6.16658","url":null,"abstract":"The formal study of coalition formation in multi-agent systems is typically realized in the framework of hedonic games, which originate from economic theory. The main focus of this branch of research has been on the existence and the computational complexity of deciding the existence of coalition structures that satisfy various stability criteria. The actual process of forming coalitions based on individual behavior has received little attention. In this article, we study the convergence of simple dynamics leading to stable partitions in a variety of established classes of hedonic games, including anonymous, dichotomous, fractional, and hedonic diversity games. The dynamics we consider is based on individual stability: an agent will join another coalition if she is better off and no member of the welcoming coalition is worse off. Our results are threefold. First, we identify conditions for the (fast) convergence of our dynamics. To this end, we develop new techniques based on the simultaneous usage of multiple intertwined potential functions and establish a reduction uncovering a close relationship between anonymous hedonic games and hedonic diversity games. Second, we provide elaborate counterexamples determining tight boundaries for the existence of individually stable partitions. Third, we study the computational complexity of problems related to the coalition formation dynamics. In particular, we settle open problems suggested by Bogomolnaia and Jackson, Brandl et al., and Boehmer and Elkind.","PeriodicalId":42216,"journal":{"name":"ACM Transactions on Economics and Computation","volume":"11 1","pages":"1 - 65"},"PeriodicalIF":1.2,"publicationDate":"2021-05-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44649010","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 NP-hard Hospital Residents problem with lower and upper quotas (HR-QLU), the goal is to find a stable matching of residents to hospitals where the number of residents matched to a hospital is either between its lower and upper quota or zero. We analyze this problem from a parameterized complexity perspective using several natural parameters such as the number of hospitals and the number of residents. Moreover, answering an open question of Biró et al. [TCS 2010], we present an involved polynomial-time algorithm that finds a stable matching (if it exists) on instances with maximum lower quota two. Alongside HR-QLU, we also consider two closely related models of independent interest, namely, the special case of HR-QLU where each hospital has only a lower quota but no upper quota and the variation of HR-QLU where hospitals do not have preferences over residents, which is also known as the House Allocation problem with lower and upper quotas. Last, we investigate the parameterized complexity of these three NP-hard problems when preferences may contain ties.
{"title":"A Fine-grained View on Stable Many-to-one Matching Problems with Lower and Upper Quotas","authors":"Niclas Boehmer, Klaus Heeger","doi":"10.1145/3546605","DOIUrl":"https://doi.org/10.1145/3546605","url":null,"abstract":"In the NP-hard Hospital Residents problem with lower and upper quotas (HR-QLU), the goal is to find a stable matching of residents to hospitals where the number of residents matched to a hospital is either between its lower and upper quota or zero. We analyze this problem from a parameterized complexity perspective using several natural parameters such as the number of hospitals and the number of residents. Moreover, answering an open question of Biró et al. [TCS 2010], we present an involved polynomial-time algorithm that finds a stable matching (if it exists) on instances with maximum lower quota two. Alongside HR-QLU, we also consider two closely related models of independent interest, namely, the special case of HR-QLU where each hospital has only a lower quota but no upper quota and the variation of HR-QLU where hospitals do not have preferences over residents, which is also known as the House Allocation problem with lower and upper quotas. Last, we investigate the parameterized complexity of these three NP-hard problems when preferences may contain ties.","PeriodicalId":42216,"journal":{"name":"ACM Transactions on Economics and Computation","volume":"10 1","pages":"1 - 53"},"PeriodicalIF":1.2,"publicationDate":"2020-09-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44194019","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 the efficiency of first-/second-price sequential multiunit auctions with two buyers and complete information. Extending the primal-dual framework for obtaining efficiency bounds to this sequential setting, we obtain tight price of anarchy bounds. For general valuation functions, we show that the price of anarchy is exactly 1/T for auctions with T items for sale. For concave valuation functions, we show that the price of anarchy is bounded below by 1-1/e≃ 0.632. This bound is asymptotically tight as the number of items sold tends to infinity.
{"title":"The Price of Anarchy of Two-Buyer Sequential Multiunit Auctions","authors":"Mete cSeref Ahunbay, A. Vetta","doi":"10.1145/3584864","DOIUrl":"https://doi.org/10.1145/3584864","url":null,"abstract":"We study the efficiency of first-/second-price sequential multiunit auctions with two buyers and complete information. Extending the primal-dual framework for obtaining efficiency bounds to this sequential setting, we obtain tight price of anarchy bounds. For general valuation functions, we show that the price of anarchy is exactly 1/T for auctions with T items for sale. For concave valuation functions, we show that the price of anarchy is bounded below by 1-1/e≃ 0.632. This bound is asymptotically tight as the number of items sold tends to infinity.","PeriodicalId":42216,"journal":{"name":"ACM Transactions on Economics and Computation","volume":"11 1","pages":"1 - 26"},"PeriodicalIF":1.2,"publicationDate":"2020-07-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44662271","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}
Vittorio Bilò, G. Monaco, L. Moscardelli, Cosimo Vinci
In load-balancing problems there is a set of clients, each wishing to select a resource from a set of permissible ones to execute a certain task. Each resource has a latency function, which depends on its workload, and a client’s cost is the completion time of her chosen resource. Two fundamental variants of load-balancing problems are selfish load balancing (a.k.a. load-balancing games), where clients are non-cooperative selfish players aimed at minimizing their own cost solely, and online load balancing, where clients appear online and have to be irrevocably assigned to a resource without any knowledge about future requests. We revisit both problems under the objective of minimizing the Nash Social Welfare, i.e., the geometric mean of the clients’ costs. To the best of our knowledge, despite being a celebrated welfare estimator in many social contexts, the Nash Social Welfare has not been considered so far as a benchmarking quality measure in load-balancing problems. We provide tight bounds on the price of anarchy of pure Nash equilibria and on the competitive ratio of the greedy algorithm under very general latency functions, including polynomial ones. For this particular class, we also prove that the greedy strategy is optimal, as it matches the performance of any possible online algorithm.
{"title":"Nash Social Welfare in Selfish and Online Load Balancing","authors":"Vittorio Bilò, G. Monaco, L. Moscardelli, Cosimo Vinci","doi":"10.1145/3544978","DOIUrl":"https://doi.org/10.1145/3544978","url":null,"abstract":"In load-balancing problems there is a set of clients, each wishing to select a resource from a set of permissible ones to execute a certain task. Each resource has a latency function, which depends on its workload, and a client’s cost is the completion time of her chosen resource. Two fundamental variants of load-balancing problems are selfish load balancing (a.k.a. load-balancing games), where clients are non-cooperative selfish players aimed at minimizing their own cost solely, and online load balancing, where clients appear online and have to be irrevocably assigned to a resource without any knowledge about future requests. We revisit both problems under the objective of minimizing the Nash Social Welfare, i.e., the geometric mean of the clients’ costs. To the best of our knowledge, despite being a celebrated welfare estimator in many social contexts, the Nash Social Welfare has not been considered so far as a benchmarking quality measure in load-balancing problems. We provide tight bounds on the price of anarchy of pure Nash equilibria and on the competitive ratio of the greedy algorithm under very general latency functions, including polynomial ones. For this particular class, we also prove that the greedy strategy is optimal, as it matches the performance of any possible online algorithm.","PeriodicalId":42216,"journal":{"name":"ACM Transactions on Economics and Computation","volume":"10 1","pages":"1 - 41"},"PeriodicalIF":1.2,"publicationDate":"2020-07-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46639980","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}
P. Goldberg, Edwin Lock, Francisco Javier Marmolejo-Cossío
This article addresses the computational challenges of learning strong substitutes demand when given access to a demand (or valuation) oracle. Strong substitutes demand generalises the well-studied gross substitutes demand to a multi-unit setting. Recent work by Baldwin and Klemperer shows that any such demand can be expressed in a natural way as a finite list of weighted bid vectors. A simplified version of this bidding language has been used by the Bank of England. Assuming access to a demand oracle, we provide an algorithm that computes the unique list of weighted bid vectors corresponding to a bidder’s demand preferences. In the special case where their demand can be expressed using positive bids only, we have an efficient algorithm that learns this list in linear time. We also show super-polynomial lower bounds on the query complexity of computing the list of bids in the general case where bids may be positive and negative. Our algorithms constitute the first systematic approach for bidders to construct a bid list corresponding to non-trivial demand, allowing them to participate in “product-mix” auctions.
{"title":"Learning Strong Substitutes Demand via Queries","authors":"P. Goldberg, Edwin Lock, Francisco Javier Marmolejo-Cossío","doi":"10.1145/3546604","DOIUrl":"https://doi.org/10.1145/3546604","url":null,"abstract":"This article addresses the computational challenges of learning strong substitutes demand when given access to a demand (or valuation) oracle. Strong substitutes demand generalises the well-studied gross substitutes demand to a multi-unit setting. Recent work by Baldwin and Klemperer shows that any such demand can be expressed in a natural way as a finite list of weighted bid vectors. A simplified version of this bidding language has been used by the Bank of England. Assuming access to a demand oracle, we provide an algorithm that computes the unique list of weighted bid vectors corresponding to a bidder’s demand preferences. In the special case where their demand can be expressed using positive bids only, we have an efficient algorithm that learns this list in linear time. We also show super-polynomial lower bounds on the query complexity of computing the list of bids in the general case where bids may be positive and negative. Our algorithms constitute the first systematic approach for bidders to construct a bid list corresponding to non-trivial demand, allowing them to participate in “product-mix” auctions.","PeriodicalId":42216,"journal":{"name":"ACM Transactions on Economics and Computation","volume":"10 1","pages":"1 - 22"},"PeriodicalIF":1.2,"publicationDate":"2020-05-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47102245","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}
Brouwer’s fixed point theorem states that any continuous function from a compact convex space to itself has a fixed point. Roughgarden and Weinstein (FOCS 2016) initiated the study of fixed point computation in the two-player communication model, where each player gets a function from [0,1]^n to [0,1]^n, and their goal is to find an approximate fixed point of the composition of the two functions. They left it as an open question to show a lower bound of 2^{Omega (n)} for the (randomized) communication complexity of this problem, in the range of parameters which make it a total search problem. We answer this question affirmatively. Additionally, we introduce two natural fixed point problems in the two-player communication model. Each player is given a function from [0,1]^n to [0,1]^{n/2}, and their goal is to find an approximate fixed point of the concatenation of the functions. Each player is given a function from [0,1]^n to [0,1]^{n}, and their goal is to find an approximate fixed point of the mean of the functions. We show a randomized communication complexity lower bound of 2^{Omega (n)} for these problems (for some constant approximation factor). Finally, we initiate the study of finding a panchromatic simplex in a Sperner-coloring of a triangulation (guaranteed by Sperner’s lemma) in the two-player communication model: A triangulation T of the d-simplex is publicly known and one player is given a set S_Asubset T and a coloring function from S_A to lbrace 0,ldots ,d/2rbrace, and the other player is given a set S_Bsubset T and a coloring function from S_B to lbrace d/2+1,ldots ,drbrace, such that S_Adot{cup }S_B=T, and their goal is to find a panchromatic simplex. We show a randomized communication complexity lower bound of |T|^{Omega (1)} for the aforementioned problem as well (when d is large). On the positive side, we show that if dle 4 then there is a deterministic protocol for the Sperner problem with O((log |T|)^2) bits of communication.
{"title":"On Communication Complexity of Fixed Point Computation","authors":"Anat Ganor, S. KarthikC., Dömötör Pálvölgyi","doi":"10.1145/3485004","DOIUrl":"https://doi.org/10.1145/3485004","url":null,"abstract":"Brouwer’s fixed point theorem states that any continuous function from a compact convex space to itself has a fixed point. Roughgarden and Weinstein (FOCS 2016) initiated the study of fixed point computation in the two-player communication model, where each player gets a function from [0,1]^n to [0,1]^n, and their goal is to find an approximate fixed point of the composition of the two functions. They left it as an open question to show a lower bound of 2^{Omega (n)} for the (randomized) communication complexity of this problem, in the range of parameters which make it a total search problem. We answer this question affirmatively. Additionally, we introduce two natural fixed point problems in the two-player communication model. Each player is given a function from [0,1]^n to [0,1]^{n/2}, and their goal is to find an approximate fixed point of the concatenation of the functions. Each player is given a function from [0,1]^n to [0,1]^{n}, and their goal is to find an approximate fixed point of the mean of the functions. We show a randomized communication complexity lower bound of 2^{Omega (n)} for these problems (for some constant approximation factor). Finally, we initiate the study of finding a panchromatic simplex in a Sperner-coloring of a triangulation (guaranteed by Sperner’s lemma) in the two-player communication model: A triangulation T of the d-simplex is publicly known and one player is given a set S_Asubset T and a coloring function from S_A to lbrace 0,ldots ,d/2rbrace, and the other player is given a set S_Bsubset T and a coloring function from S_B to lbrace d/2+1,ldots ,drbrace, such that S_Adot{cup }S_B=T, and their goal is to find a panchromatic simplex. We show a randomized communication complexity lower bound of |T|^{Omega (1)} for the aforementioned problem as well (when d is large). On the positive side, we show that if dle 4 then there is a deterministic protocol for the Sperner problem with O((log |T|)^2) bits of communication.","PeriodicalId":42216,"journal":{"name":"ACM Transactions on Economics and Computation","volume":"9 1","pages":"1 - 27"},"PeriodicalIF":1.2,"publicationDate":"2019-09-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49648536","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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