Nikhil R. Devanur, Nima Haghpanah, Alexandros Psomas
We study a pricing problem that is motivated by the following examples. A cloud computing platform such as Amazon EC2 sells virtual machines to clients, each of who needs a different number of virtual machine hours. Similarly, cloud storage providers such as Dropbox have customers that require different amounts of storage. Software companies such as Microsoft sell software subscriptions that can have different levels of service. The levels could be the number of different documents you are allowed to create, or the number of hours you are allowed to use the software. Companies like Google and Microsoft sell API calls to artificial intelligence software such as face recognition, to other software developers. Video and mobile games are increasingly designed in such a way that one can pay for better access to certain features. Spotify and iTunes sell music subscription, and different people listen to different number of songs in a month. Cellphone service providers like AT&T and Verizon offer cellular phone call minutes and data. People have widely varying amounts of data consumption.
{"title":"Optimal Multi-Unit Mechanisms with Private Demands","authors":"Nikhil R. Devanur, Nima Haghpanah, Alexandros Psomas","doi":"10.1145/3033274.3085122","DOIUrl":"https://doi.org/10.1145/3033274.3085122","url":null,"abstract":"We study a pricing problem that is motivated by the following examples. A cloud computing platform such as Amazon EC2 sells virtual machines to clients, each of who needs a different number of virtual machine hours. Similarly, cloud storage providers such as Dropbox have customers that require different amounts of storage. Software companies such as Microsoft sell software subscriptions that can have different levels of service. The levels could be the number of different documents you are allowed to create, or the number of hours you are allowed to use the software. Companies like Google and Microsoft sell API calls to artificial intelligence software such as face recognition, to other software developers. Video and mobile games are increasingly designed in such a way that one can pay for better access to certain features. Spotify and iTunes sell music subscription, and different people listen to different number of songs in a month. Cellphone service providers like AT&T and Verizon offer cellular phone call minutes and data. People have widely varying amounts of data consumption.","PeriodicalId":287551,"journal":{"name":"Proceedings of the 2017 ACM Conference on Economics and Computation","volume":"174 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2017-04-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"132291747","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}
Market equilibria of matching markets offer an intuitive and fair solution for matching problems without money with agents who have preferences over the items. Such a matching market can be viewed as a variation of Fisher market, albeit with rather peculiar preferences of agents. These preferences can be described by piece-wise linear concave (PLC) functions, which however, are not separable (due to each agent only asking for one item), are not monotone, and do not satisfy the gross substitute property-- increase in price of an item can result in increased demand for the item. Devanur and Kannan in FOCS 08 showed that market clearing prices can be found in polynomial time in markets with fixed number of items and general PLC preferences. They also consider Fischer markets with fixed number of agents (instead of fixed number of items), and give a polynomial time algorithm for this case if preferences are separable functions of the items, in addition to being PLC functions. Our main result is a polynomial time algorithm for finding market clearing prices in matching markets with fixed number of different agent preferences, despite that the utility corresponding to matching markets is not separable. We also give a simpler algorithm for the case of matching markets with fixed number of different items.
{"title":"Computing Equilibrium in Matching Markets","authors":"S. Alaei, Pooya Jalaly, É. Tardos","doi":"10.1145/3033274.3085150","DOIUrl":"https://doi.org/10.1145/3033274.3085150","url":null,"abstract":"Market equilibria of matching markets offer an intuitive and fair solution for matching problems without money with agents who have preferences over the items. Such a matching market can be viewed as a variation of Fisher market, albeit with rather peculiar preferences of agents. These preferences can be described by piece-wise linear concave (PLC) functions, which however, are not separable (due to each agent only asking for one item), are not monotone, and do not satisfy the gross substitute property-- increase in price of an item can result in increased demand for the item. Devanur and Kannan in FOCS 08 showed that market clearing prices can be found in polynomial time in markets with fixed number of items and general PLC preferences. They also consider Fischer markets with fixed number of agents (instead of fixed number of items), and give a polynomial time algorithm for this case if preferences are separable functions of the items, in addition to being PLC functions. Our main result is a polynomial time algorithm for finding market clearing prices in matching markets with fixed number of different agent preferences, despite that the utility corresponding to matching markets is not separable. We also give a simpler algorithm for the case of matching markets with fixed number of different items.","PeriodicalId":287551,"journal":{"name":"Proceedings of the 2017 ACM Conference on Economics and Computation","volume":"76 1-2","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2017-03-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"132812281","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}
Bayesian experts with a common prior that are exposed to different evidence possibly make contradicting probabilistic forecasts. A policy maker who receives the forecasts must aggregate them in the best way possible. This is a challenge whenever the policy maker is not familiar with the prior nor the model and evidence available to the experts. We propose a model of non-Bayesian forecast aggregation and adapt the notion of regret as a means for evaluating the policy maker's performance. Whenever experts are Blackwell ordered taking a weighted average of the two forecasts, the weight of which is proportional to its precision (the reciprocal of the variance), is optimal. The resulting regret is equal 1/8(5√ 5-11) approx 0.0225425, which is 3 to 4 times better than naive approaches such as choosing one expert at random or taking the non-weighted average.
{"title":"Forecast Aggregation","authors":"Itai Arieli, Y. Babichenko, Rann Smorodinsky","doi":"10.2139/ssrn.2934104","DOIUrl":"https://doi.org/10.2139/ssrn.2934104","url":null,"abstract":"Bayesian experts with a common prior that are exposed to different evidence possibly make contradicting probabilistic forecasts. A policy maker who receives the forecasts must aggregate them in the best way possible. This is a challenge whenever the policy maker is not familiar with the prior nor the model and evidence available to the experts. We propose a model of non-Bayesian forecast aggregation and adapt the notion of regret as a means for evaluating the policy maker's performance. Whenever experts are Blackwell ordered taking a weighted average of the two forecasts, the weight of which is proportional to its precision (the reciprocal of the variance), is optimal. The resulting regret is equal 1/8(5√ 5-11) approx 0.0225425, which is 3 to 4 times better than naive approaches such as choosing one expert at random or taking the non-weighted average.","PeriodicalId":287551,"journal":{"name":"Proceedings of the 2017 ACM Conference on Economics and Computation","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2017-03-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"115374288","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 consider the problem of dividing indivisible goods fairly among n agents who have additive and submodular valuations for the goods. Our fairness guarantees are in terms of the maximin share, that is defined to be the maximum value that an agent can ensure for herself, if she were to partition the goods into n bundles, and then receive a minimum valued bundle. Since maximin fair allocations (i.e., allocations in which each agent gets at least her maximin share) do not always exist, prior work has focussed on approximation results that aim to find allocations in which the value of the bundle allocated to each agent is (multiplicatively) as close to her maximin share as possible. In particular, Procaccia and Wang (2014) along with Amanatidis et al. (2015) have shown that under additive valuations a 2/3-approximate maximin fair allocation always exists and can be found in polynomial time. We complement these results by developing a simple and efficient algorithm that achieves the same approximation guarantee. Furthermore, we initiate the study of approximate maximin fair division under submodular valuations. Specifically, we show that when the valuations of the agents are nonnegative, monotone, and submodular, then a 1/10-approximate maximin fair allocation is guaranteed to exist. In fact, we show that such an allocation can be efficiently found by using a simple round-robin algorithm. A technical contribution of the paper is to analyze the performance of this combinatorial algorithm by employing the concept of multilinear extensions.
{"title":"Approximation Algorithms for Maximin Fair Division","authors":"Siddharth Barman, S. K. Murthy","doi":"10.1145/3033274.3085136","DOIUrl":"https://doi.org/10.1145/3033274.3085136","url":null,"abstract":"We consider the problem of dividing indivisible goods fairly among n agents who have additive and submodular valuations for the goods. Our fairness guarantees are in terms of the maximin share, that is defined to be the maximum value that an agent can ensure for herself, if she were to partition the goods into n bundles, and then receive a minimum valued bundle. Since maximin fair allocations (i.e., allocations in which each agent gets at least her maximin share) do not always exist, prior work has focussed on approximation results that aim to find allocations in which the value of the bundle allocated to each agent is (multiplicatively) as close to her maximin share as possible. In particular, Procaccia and Wang (2014) along with Amanatidis et al. (2015) have shown that under additive valuations a 2/3-approximate maximin fair allocation always exists and can be found in polynomial time. We complement these results by developing a simple and efficient algorithm that achieves the same approximation guarantee. Furthermore, we initiate the study of approximate maximin fair division under submodular valuations. Specifically, we show that when the valuations of the agents are nonnegative, monotone, and submodular, then a 1/10-approximate maximin fair allocation is guaranteed to exist. In fact, we show that such an allocation can be efficiently found by using a simple round-robin algorithm. A technical contribution of the paper is to analyze the performance of this combinatorial algorithm by employing the concept of multilinear extensions.","PeriodicalId":287551,"journal":{"name":"Proceedings of the 2017 ACM Conference on Economics and Computation","volume":"42 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2017-03-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"122703122","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}
Shuchi Chawla, Nikhil R. Devanur, Janardhan Kulkarni, Rad Niazadeh
We consider a scheduling problem where a cloud service provider has multiple units of a resource available over time. Selfish clients submit jobs, each with an arrival time, deadline, length, and value. The service provider's goal is to implement a truthful online mechanism for scheduling jobs so as to maximize the social welfare of the schedule. Recent work shows that under a stochastic assumption on job arrivals, there is a single-parameter family of mechanisms that achieves near-optimal social welfare. We show that given any such family of near-optimal online mechanisms, there exists an online mechanism that in the worst case performs nearly as well as the best of the given mechanisms. Our mechanism is truthful whenever the mechanisms in the given family are truthful and prompt, and achieves optimal (within constant factors) regret. We model the problem of competing against a family of online scheduling mechanisms as one of learning from expert advice. A primary challenge is that any scheduling decisions we make affect not only the payoff at the current step, but also the resource availability and payoffs in future steps. Furthermore, switching from one algorithm (a.k.a. expert) to another in an online fashion is challenging both because it requires synchronization with the state of the latter algorithm as well as because it affects the incentive structure of the algorithms. We further show how to adapt our algorithm to a non-clairvoyant setting where job lengths are unknown until jobs are run to completion. Once again, in this setting, we obtain truthfulness along with asymptotically optimal regret (within polylogarithmic factors).
{"title":"Truth and Regret in Online Scheduling","authors":"Shuchi Chawla, Nikhil R. Devanur, Janardhan Kulkarni, Rad Niazadeh","doi":"10.1145/3033274.3085119","DOIUrl":"https://doi.org/10.1145/3033274.3085119","url":null,"abstract":"We consider a scheduling problem where a cloud service provider has multiple units of a resource available over time. Selfish clients submit jobs, each with an arrival time, deadline, length, and value. The service provider's goal is to implement a truthful online mechanism for scheduling jobs so as to maximize the social welfare of the schedule. Recent work shows that under a stochastic assumption on job arrivals, there is a single-parameter family of mechanisms that achieves near-optimal social welfare. We show that given any such family of near-optimal online mechanisms, there exists an online mechanism that in the worst case performs nearly as well as the best of the given mechanisms. Our mechanism is truthful whenever the mechanisms in the given family are truthful and prompt, and achieves optimal (within constant factors) regret. We model the problem of competing against a family of online scheduling mechanisms as one of learning from expert advice. A primary challenge is that any scheduling decisions we make affect not only the payoff at the current step, but also the resource availability and payoffs in future steps. Furthermore, switching from one algorithm (a.k.a. expert) to another in an online fashion is challenging both because it requires synchronization with the state of the latter algorithm as well as because it affects the incentive structure of the algorithms. We further show how to adapt our algorithm to a non-clairvoyant setting where job lengths are unknown until jobs are run to completion. Once again, in this setting, we obtain truthfulness along with asymptotically optimal regret (within polylogarithmic factors).","PeriodicalId":287551,"journal":{"name":"Proceedings of the 2017 ACM Conference on Economics and Computation","volume":"2015 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2017-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"130112859","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 classic sequential screening problem under ex-post participation constraints. Thus the seller is required to satisfy buyers' ex-post participation constraints. A leading example is the online display advertising market, in which publishers frequently cannot use up-front fees and instead use transaction-contingent fees. We establish when the optimal selling mechanism is static (buyers are not screened) or dynamic (buyers are screened), and obtain a full characterization of such contracts. We begin by analyzing our model within the leading case of exponential distributions with two types. We provide a necessary and sufficient condition for the optimality of the static contract. If the means of the two types are sufficiently close, then no screening is optimal. If they are sufficiently apart, then a dynamic contract becomes optimal. Importantly, the latter contract randomizes the low type buyer while giving a deterministic allocation to the high type. It also makes the low type worse-off and the high type better-off compared to the contract the seller would offer if he knew the buyer's type. Our main result establishes a necessary and sufficient condition under which the static contract is optimal for general distributions. We show that when this condition fails, a dynamic contract that randomizes the low type buyer is optimal.
{"title":"The Scope of Sequential Screening with Ex Post Participation Constraints","authors":"D. Bergemann, Francisco Castro, G. Weintraub","doi":"10.2139/ssrn.3569697","DOIUrl":"https://doi.org/10.2139/ssrn.3569697","url":null,"abstract":"We study the classic sequential screening problem under ex-post participation constraints. Thus the seller is required to satisfy buyers' ex-post participation constraints. A leading example is the online display advertising market, in which publishers frequently cannot use up-front fees and instead use transaction-contingent fees. We establish when the optimal selling mechanism is static (buyers are not screened) or dynamic (buyers are screened), and obtain a full characterization of such contracts. We begin by analyzing our model within the leading case of exponential distributions with two types. We provide a necessary and sufficient condition for the optimality of the static contract. If the means of the two types are sufficiently close, then no screening is optimal. If they are sufficiently apart, then a dynamic contract becomes optimal. Importantly, the latter contract randomizes the low type buyer while giving a deterministic allocation to the high type. It also makes the low type worse-off and the high type better-off compared to the contract the seller would offer if he knew the buyer's type. Our main result establishes a necessary and sufficient condition under which the static contract is optimal for general distributions. We show that when this condition fails, a dynamic contract that randomizes the low type buyer is optimal.","PeriodicalId":287551,"journal":{"name":"Proceedings of the 2017 ACM Conference on Economics and Computation","volume":"50 17 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2017-02-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"122215809","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}
Christian Kroer, K. Waugh, F. Kılınç-Karzan, T. Sandholm
Sparse iterative methods, in particular first-order methods, are known to be among the most effective in solving large-scale two-player zero-sum extensive-form games. The convergence rates of these methods depend heavily on the properties of the distance-generating function that they are based on. We investigate the acceleration of first-order methods for solving extensive-form games through better design of the dilated entropy function---a class of distance-generating functions related to the domains associated with the extensive-form games. By introducing a new weighting scheme for the dilated entropy function, we develop the first distance-generating function for the strategy spaces of sequential games that only a logarithmic dependence on the branching factor of the player. This result improves the convergence rate of several first-order methods by a factor of Ω(bdd), where b is the branching factor of the player, and d is the depth of the game tree. Thus far, counterfactual regret minimization methods have been faster in practice, and more popular, than first-order methods despite their theoretically inferior convergence rates. Using our new weighting scheme and practical tuning we show that, for the first time, the excessive gap technique can be made faster than the fastest counterfactual regret minimization algorithm, CFRP, in practice.
{"title":"Theoretical and Practical Advances on Smoothing for Extensive-Form Games","authors":"Christian Kroer, K. Waugh, F. Kılınç-Karzan, T. Sandholm","doi":"10.1145/3033274.3085131","DOIUrl":"https://doi.org/10.1145/3033274.3085131","url":null,"abstract":"Sparse iterative methods, in particular first-order methods, are known to be among the most effective in solving large-scale two-player zero-sum extensive-form games. The convergence rates of these methods depend heavily on the properties of the distance-generating function that they are based on. We investigate the acceleration of first-order methods for solving extensive-form games through better design of the dilated entropy function---a class of distance-generating functions related to the domains associated with the extensive-form games. By introducing a new weighting scheme for the dilated entropy function, we develop the first distance-generating function for the strategy spaces of sequential games that only a logarithmic dependence on the branching factor of the player. This result improves the convergence rate of several first-order methods by a factor of Ω(bdd), where b is the branching factor of the player, and d is the depth of the game tree. Thus far, counterfactual regret minimization methods have been faster in practice, and more popular, than first-order methods despite their theoretically inferior convergence rates. Using our new weighting scheme and practical tuning we show that, for the first time, the excessive gap technique can be made faster than the fastest counterfactual regret minimization algorithm, CFRP, in practice.","PeriodicalId":287551,"journal":{"name":"Proceedings of the 2017 ACM Conference on Economics and Computation","volume":"27 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2017-02-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"125089971","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 suggest a general method for inferring players' values from their actions in repeated games. The method extends and improves upon the recent suggestion of (Nekipelov et al., EC 2015) and is based on the assumption that players are more likely to exhibit sequences of actions that have lower regret. We evaluate this "quantal-regret" method on two different datasets from experiments of repeated games with controlled player values: those of (Selten and Chmura, AER 2008) on a variety of two-player 2x2 games and our own experiment on ad-auctions (Noti et al., WWW 2014). We find that the quantal-regret method is consistently and significantly more precise than either "classic" econometric methods that are based on Nash equilibria, or the "min-regret" method of (Nekipelov et al., EC 2015).
我们建议一种从玩家在重复游戏中的行为推断其价值的通用方法。该方法扩展并改进了Nekipelov等人(EC 2015)的最新建议，并基于玩家更有可能表现出具有较低后悔的行动序列的假设。我们在两个不同的数据集上评估了这种“量子后悔”方法，这些数据集来自控制玩家价值的重复游戏实验:一个是(Selten和Chmura, AER 2008)关于各种双人2x2游戏的数据集，另一个是我们自己的广告拍卖实验(Noti et al.， WWW 2014)。我们发现，与基于纳什均衡的“经典”计量经济学方法或(Nekipelov et al.， EC 2015)的“最小后悔”方法相比，量子后悔方法始终且明显更精确。
{"title":"A \"Quantal Regret\" Method for Structural Econometrics in Repeated Games","authors":"N. Nisan, Gali Noti","doi":"10.1145/3033274.3085111","DOIUrl":"https://doi.org/10.1145/3033274.3085111","url":null,"abstract":"We suggest a general method for inferring players' values from their actions in repeated games. The method extends and improves upon the recent suggestion of (Nekipelov et al., EC 2015) and is based on the assumption that players are more likely to exhibit sequences of actions that have lower regret. We evaluate this \"quantal-regret\" method on two different datasets from experiments of repeated games with controlled player values: those of (Selten and Chmura, AER 2008) on a variety of two-player 2x2 games and our own experiment on ad-auctions (Noti et al., WWW 2014). We find that the quantal-regret method is consistently and significantly more precise than either \"classic\" econometric methods that are based on Nash equilibria, or the \"min-regret\" method of (Nekipelov et al., EC 2015).","PeriodicalId":287551,"journal":{"name":"Proceedings of the 2017 ACM Conference on Economics and Computation","volume":"31 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2017-02-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"129351863","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}
Users of social, economic, or medical networks share personal information in exchange for tangible benefits, but may be harmed by leakage and misuse of the shared information. I analyze the effect of enhancing privacy in the presence of two opposing forces: network effects and informational interdependencies. I show that two privacy enhancements---reducing the likelihood of leakage and decreasing the level of informational interdependence---have opposite effects on the volume of information sharing, and that although they always seem beneficial to non-strategic users, both privacy enhancements may backfire when users are strategic.
{"title":"Information Sharing and Privacy in Networks","authors":"R. Gradwohl","doi":"10.1145/3033274.3085095","DOIUrl":"https://doi.org/10.1145/3033274.3085095","url":null,"abstract":"Users of social, economic, or medical networks share personal information in exchange for tangible benefits, but may be harmed by leakage and misuse of the shared information. I analyze the effect of enhancing privacy in the presence of two opposing forces: network effects and informational interdependencies. I show that two privacy enhancements---reducing the likelihood of leakage and decreasing the level of informational interdependence---have opposite effects on the volume of information sharing, and that although they always seem beneficial to non-strategic users, both privacy enhancements may backfire when users are strategic.","PeriodicalId":287551,"journal":{"name":"Proceedings of the 2017 ACM Conference on Economics and Computation","volume":"20 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2017-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"128595615","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 consider ε-equilibria notions for a constant value of ε in n-player m-action games, where m is a constant. We focus on the following question: What is the largest grid size over the mixed strategies such that ε-equilibrium is guaranteed to exist over this grid. For Nash equilibrium, we prove that constant grid size (that depends on ε and m, but not on n) is sufficient to guarantee the existence of a weak approximate equilibrium. This result implies a polynomial (in the input) algorithm for a weak approximate equilibrium. For approximate Nash equilibrium we introduce a closely related question and prove its equivalence to the well-known Beck-Fiala conjecture from discrepancy theory. To the best of our knowledge, this is the first result that introduces a connection between game theory and discrepancy theory. For a correlated equilibrium, we prove a O(1 over log n) lower-bound on the grid size, which matches the known upper bound of Ω(1 over log n). Our result implies an Ω(log n) lower bound on the rate of convergence of any dynamic to approximate correlated (and coarse correlated) equilibrium. Again, this lower bound matches the O(log n) upper bound that is achieved by regret minimizing algorithms.
{"title":"Simple Approximate Equilibria in Games with Many Players","authors":"Itai Arieli, Y. Babichenko","doi":"10.1145/3033274.3085110","DOIUrl":"https://doi.org/10.1145/3033274.3085110","url":null,"abstract":"We consider ε-equilibria notions for a constant value of ε in n-player m-action games, where m is a constant. We focus on the following question: What is the largest grid size over the mixed strategies such that ε-equilibrium is guaranteed to exist over this grid. For Nash equilibrium, we prove that constant grid size (that depends on ε and m, but not on n) is sufficient to guarantee the existence of a weak approximate equilibrium. This result implies a polynomial (in the input) algorithm for a weak approximate equilibrium. For approximate Nash equilibrium we introduce a closely related question and prove its equivalence to the well-known Beck-Fiala conjecture from discrepancy theory. To the best of our knowledge, this is the first result that introduces a connection between game theory and discrepancy theory. For a correlated equilibrium, we prove a O(1 over log n) lower-bound on the grid size, which matches the known upper bound of Ω(1 over log n). Our result implies an Ω(log n) lower bound on the rate of convergence of any dynamic to approximate correlated (and coarse correlated) equilibrium. Again, this lower bound matches the O(log n) upper bound that is achieved by regret minimizing algorithms.","PeriodicalId":287551,"journal":{"name":"Proceedings of the 2017 ACM Conference on Economics and Computation","volume":"58 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2017-01-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"122282180","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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