It is generally challenging to characterize the optimal selling mechanism even when the seller knows the buyer's valuation distributions in multi-item screening. An insightful and significant result in robust mechanism design literature is that if the seller knows only marginal distributions of the buyer's valuation, then separable mechanisms, in which all items are sold independently, are robustly optimal under the maximin revenue objectives. While the separable mechanism is simple to implement, the literature also indicates that separate selling can not guarantee any substantial fraction of the potential optimal revenue for given distributions. To design a simple mechanism with a good performance guarantee, we introduce a novel class of mechanisms, termed "semi-separable mechanism". In these mechanisms, the allocation and payment rule of each item is a function solely of the corresponding item's valuation, which retains the separable mechanism's practical simplicity. However, the design of the allocation and payment function is enhanced by leveraging the joint distributional information, thereby improving the performance guarantee against the hindsight optimal revenue. We establish that a semi-separable mechanism achieves the optimal performance ratio among all incentive-compatible and individually rational mechanisms when only marginal support information is known. This result demonstrates that the semi-separable mechanisms ensure both the interpretation and implementation simplicity, and performance superiority. Our framework is also applicable to scenarios where the seller possesses information about the aggregate valuations of product bundles within any given partition of the product set. Furthermore, our results also provide guidelines for the multi-item screening problem with non-standard ambiguity sets.
{"title":"Semi-Separable Mechanisms in Multi-Item Robust Screening","authors":"Shixin Wang","doi":"arxiv-2408.13580","DOIUrl":"https://doi.org/arxiv-2408.13580","url":null,"abstract":"It is generally challenging to characterize the optimal selling mechanism\u0000even when the seller knows the buyer's valuation distributions in multi-item\u0000screening. An insightful and significant result in robust mechanism design\u0000literature is that if the seller knows only marginal distributions of the\u0000buyer's valuation, then separable mechanisms, in which all items are sold\u0000independently, are robustly optimal under the maximin revenue objectives. While\u0000the separable mechanism is simple to implement, the literature also indicates\u0000that separate selling can not guarantee any substantial fraction of the\u0000potential optimal revenue for given distributions. To design a simple mechanism\u0000with a good performance guarantee, we introduce a novel class of mechanisms,\u0000termed \"semi-separable mechanism\". In these mechanisms, the allocation and\u0000payment rule of each item is a function solely of the corresponding item's\u0000valuation, which retains the separable mechanism's practical simplicity.\u0000However, the design of the allocation and payment function is enhanced by\u0000leveraging the joint distributional information, thereby improving the\u0000performance guarantee against the hindsight optimal revenue. We establish that\u0000a semi-separable mechanism achieves the optimal performance ratio among all\u0000incentive-compatible and individually rational mechanisms when only marginal\u0000support information is known. This result demonstrates that the semi-separable\u0000mechanisms ensure both the interpretation and implementation simplicity, and\u0000performance superiority. Our framework is also applicable to scenarios where\u0000the seller possesses information about the aggregate valuations of product\u0000bundles within any given partition of the product set. Furthermore, our results\u0000also provide guidelines for the multi-item screening problem with non-standard\u0000ambiguity sets.","PeriodicalId":501188,"journal":{"name":"arXiv - ECON - Theoretical Economics","volume":"27 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142197113","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}
Blockchain trade intent auctions currently intermediate approximately USD 5 billion monthly. Due to production complementarities, the auction is combinatorial: when multiple trade intents from different traders are auctioned off simultaneously, a bidder (here called solver) can generate additional efficiencies by winning a batch of multiple trade intents. However, unlike other combinatorial auctions studied in the literature, the auction has no numeraire. Fairness is a concern as the efficiencies from batching cannot be easily shared between traders. We formalize this problem and study the most commonly used auction formats: batch auctions and multiple simultaneous auctions. We also propose a novel fair combinatorial auction that combines batch auction and multiple simultaneous auctions: solvers submit individual-trade bids and batched bids, but batched bids are considered only if they are better for all traders relative to the outcome of multiple simultaneous auctions (constructed using the individual-trade bids). We find a trade-off between the fairness guarantees provided by the auction (i.e., the minimum each trader can expect to receive) and the expected value of the assets returned to the traders. Also, the amount that each trader receives in the equilibrium of the fair combinatorial auction may be higher or lower than what they receive in the equilibrium of the simultaneous auctions used as a benchmark for fairness.
{"title":"Combinatorial Auctions without a Numeraire: The Case of Blockchain Trade-Intent Auctions","authors":"Andrea Canidio, Felix Henneke","doi":"arxiv-2408.12225","DOIUrl":"https://doi.org/arxiv-2408.12225","url":null,"abstract":"Blockchain trade intent auctions currently intermediate approximately USD 5\u0000billion monthly. Due to production complementarities, the auction is\u0000combinatorial: when multiple trade intents from different traders are auctioned\u0000off simultaneously, a bidder (here called solver) can generate additional\u0000efficiencies by winning a batch of multiple trade intents. However, unlike\u0000other combinatorial auctions studied in the literature, the auction has no\u0000numeraire. Fairness is a concern as the efficiencies from batching cannot be\u0000easily shared between traders. We formalize this problem and study the most\u0000commonly used auction formats: batch auctions and multiple simultaneous\u0000auctions. We also propose a novel fair combinatorial auction that combines\u0000batch auction and multiple simultaneous auctions: solvers submit\u0000individual-trade bids and batched bids, but batched bids are considered only if\u0000they are better for all traders relative to the outcome of multiple\u0000simultaneous auctions (constructed using the individual-trade bids). We find a\u0000trade-off between the fairness guarantees provided by the auction (i.e., the\u0000minimum each trader can expect to receive) and the expected value of the assets\u0000returned to the traders. Also, the amount that each trader receives in the\u0000equilibrium of the fair combinatorial auction may be higher or lower than what\u0000they receive in the equilibrium of the simultaneous auctions used as a\u0000benchmark for fairness.","PeriodicalId":501188,"journal":{"name":"arXiv - ECON - Theoretical Economics","volume":"81 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142197116","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}
Lev Razumovskiy, Mariya Gerasimova, Nikolay Karenin
We study a mathematical model for the optimization of the price of real estate (RE). This model can be characterised by a limited amount of goods, fixed sales horizon and presence of intermediate sales and revenue goals. We develop it as an enhancement and upgrade of the model presented by Besbes and Maglaras now also taking into account variable demand, time value of money, and growth of the objective value of Real Estate with the development stage.
{"title":"Dynamic Pricing for Real Estate","authors":"Lev Razumovskiy, Mariya Gerasimova, Nikolay Karenin","doi":"arxiv-2408.12553","DOIUrl":"https://doi.org/arxiv-2408.12553","url":null,"abstract":"We study a mathematical model for the optimization of the price of real\u0000estate (RE). This model can be characterised by a limited amount of goods,\u0000fixed sales horizon and presence of intermediate sales and revenue goals. We\u0000develop it as an enhancement and upgrade of the model presented by Besbes and\u0000Maglaras now also taking into account variable demand, time value of money, and\u0000growth of the objective value of Real Estate with the development stage.","PeriodicalId":501188,"journal":{"name":"arXiv - ECON - Theoretical Economics","volume":"197 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142197117","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}
Fourier-Motzkin elimination, a standard method for solving systems of linear inequalities, leads to an elementary, short, and self-contained proof of von Neumann's minimax theorem.
{"title":"Von Neumann's minimax theorem through Fourier-Motzkin elimination","authors":"Mark Voorneveld","doi":"arxiv-2408.11504","DOIUrl":"https://doi.org/arxiv-2408.11504","url":null,"abstract":"Fourier-Motzkin elimination, a standard method for solving systems of linear\u0000inequalities, leads to an elementary, short, and self-contained proof of von\u0000Neumann's minimax theorem.","PeriodicalId":501188,"journal":{"name":"arXiv - ECON - Theoretical Economics","volume":"17 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142197147","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 practice, most auction mechanisms are not strategy-proof, so equilibrium analysis is required to predict bidding behavior. In many auctions, though, an exact equilibrium is not known and one would like to understand whether -- manually or computationally generated -- bidding strategies constitute an approximate equilibrium. We develop a framework and methods for estimating the distance of a strategy profile from equilibrium, based on samples from the prior and either bidding strategies or sample bids. We estimate an agent's utility gain from deviating to strategies from a constructed finite subset of the strategy space. We use PAC-learning to give error bounds, both for independent and interdependent prior distributions. The primary challenge is that one may miss large utility gains by considering only a finite subset of the strategy space. Our work differs from prior research in two critical ways. First, we explore the impact of bidding strategies on altering opponents' perceived prior distributions -- instead of assuming the other agents to bid truthfully. Second, we delve into reasoning with interdependent priors, where the type of one agent may imply a distinct distribution for other agents. Our main contribution lies in establishing sufficient conditions for strategy profiles and a closeness criterion for conditional distributions to ensure that utility gains estimated through our finite subset closely approximate the maximum gains. To our knowledge, ours is the first method to verify approximate equilibrium in any auctions beyond single-item ones. Also, ours is the first sample-based method for approximate equilibrium verification.
{"title":"Verifying Approximate Equilibrium in Auctions","authors":"Fabian R. Pieroth, Tuomas Sandholm","doi":"arxiv-2408.11445","DOIUrl":"https://doi.org/arxiv-2408.11445","url":null,"abstract":"In practice, most auction mechanisms are not strategy-proof, so equilibrium\u0000analysis is required to predict bidding behavior. In many auctions, though, an\u0000exact equilibrium is not known and one would like to understand whether --\u0000manually or computationally generated -- bidding strategies constitute an\u0000approximate equilibrium. We develop a framework and methods for estimating the\u0000distance of a strategy profile from equilibrium, based on samples from the\u0000prior and either bidding strategies or sample bids. We estimate an agent's\u0000utility gain from deviating to strategies from a constructed finite subset of\u0000the strategy space. We use PAC-learning to give error bounds, both for\u0000independent and interdependent prior distributions. The primary challenge is\u0000that one may miss large utility gains by considering only a finite subset of\u0000the strategy space. Our work differs from prior research in two critical ways.\u0000First, we explore the impact of bidding strategies on altering opponents'\u0000perceived prior distributions -- instead of assuming the other agents to bid\u0000truthfully. Second, we delve into reasoning with interdependent priors, where\u0000the type of one agent may imply a distinct distribution for other agents. Our\u0000main contribution lies in establishing sufficient conditions for strategy\u0000profiles and a closeness criterion for conditional distributions to ensure that\u0000utility gains estimated through our finite subset closely approximate the\u0000maximum gains. To our knowledge, ours is the first method to verify approximate\u0000equilibrium in any auctions beyond single-item ones. Also, ours is the first\u0000sample-based method for approximate equilibrium verification.","PeriodicalId":501188,"journal":{"name":"arXiv - ECON - Theoretical Economics","volume":"81 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142197127","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 explore the nonlinear dynamics of a macroeconomic model with resource constraints. The dynamics is derived from a production function that considers capital and a generalized form of energy as inputs. Energy, the new variable, is depleted during the production process and has to be renewed, whereas capital grows with production and decreases from depreciation. Dependent on time scales and energy related control parameters, we obtain steady states of high or low production, but also sustained oscillations that show properties of business cycles. We also find conditions for the coexistence of stable fixed points and limit cycles. Our model allows to specify investment and saving functions for Kaldor's model of business cycles. We provide evidence for an endogenous origin of business cycles if depleting resources are taken into account.
{"title":"The nonlinear economy (I): How resource constrains lead to business cycles","authors":"Frank Schweitzer, Giona Casiraghi","doi":"arxiv-2408.16015","DOIUrl":"https://doi.org/arxiv-2408.16015","url":null,"abstract":"We explore the nonlinear dynamics of a macroeconomic model with resource\u0000constraints. The dynamics is derived from a production function that considers\u0000capital and a generalized form of energy as inputs. Energy, the new variable,\u0000is depleted during the production process and has to be renewed, whereas\u0000capital grows with production and decreases from depreciation. Dependent on\u0000time scales and energy related control parameters, we obtain steady states of\u0000high or low production, but also sustained oscillations that show properties of\u0000business cycles. We also find conditions for the coexistence of stable fixed\u0000points and limit cycles. Our model allows to specify investment and saving\u0000functions for Kaldor's model of business cycles. We provide evidence for an\u0000endogenous origin of business cycles if depleting resources are taken into\u0000account.","PeriodicalId":501188,"journal":{"name":"arXiv - ECON - Theoretical Economics","volume":"27 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142197140","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 propose a combinatorial ascending auction that is "approximately" optimal, requiring minimal rationality to achieve this level of optimality, and is robust to strategic and distributional uncertainties. Specifically, the auction is rank-guaranteed, meaning that for any menu M and any valuation profile, the ex-post revenue is guaranteed to be at least as high as the highest revenue achievable from feasible allocations, taking the (|M|+ 1)th-highest valuation for each bundle as the price. Our analysis highlights a crucial aspect of combinatorial auction design, namely, the design of menus. We provide simple and approximately optimal menus in various settings.
我们提出了一种 "近似 "最优的组合式升序拍卖，这种拍卖只需要最低限度的理性就能达到最优，而且对战略和分配的不确定性也是稳健的。具体来说，该拍卖是有等级保证的，这意味着对于任何菜单 M 和任何估值情况，事后收益都能保证至少与可行分配的最高收益一样高，并以每个捆绑包的 (|M|+ 1) 最高估值作为价格。我们的分析强调了组合拍卖设计的一个重要方面，即菜单的设计。我们提供了各种情况下的简单近似最优菜单。
{"title":"Rank-Guaranteed Auctions","authors":"Wei He, Jiangtao Li, Weijie Zhong","doi":"arxiv-2408.12001","DOIUrl":"https://doi.org/arxiv-2408.12001","url":null,"abstract":"We propose a combinatorial ascending auction that is \"approximately\" optimal,\u0000requiring minimal rationality to achieve this level of optimality, and is\u0000robust to strategic and distributional uncertainties. Specifically, the auction\u0000is rank-guaranteed, meaning that for any menu M and any valuation profile, the\u0000ex-post revenue is guaranteed to be at least as high as the highest revenue\u0000achievable from feasible allocations, taking the (|M|+ 1)th-highest valuation\u0000for each bundle as the price. Our analysis highlights a crucial aspect of\u0000combinatorial auction design, namely, the design of menus. We provide simple\u0000and approximately optimal menus in various settings.","PeriodicalId":501188,"journal":{"name":"arXiv - ECON - Theoretical Economics","volume":"2013 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142197119","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}
Indirect reciprocity is a plausible mechanism for sustaining cooperation: people cooperate with those who have a good reputation, which can be acquired by helping others. However, this mechanism requires the population to agree on who has good or bad moral standing. Consensus can be provided by a central institution that monitors and broadcasts reputations. But how might such an institution be maintained, and how can a population ensure that it is effective and incorruptible? Here we explore a simple mechanism to sustain an institution of reputational judgment: a compulsory contribution from each member of the population, i.e., a tax. We analyze the maximum possible tax rate that individuals will rationally pay to sustain an institution of judgment, which provides a public good in the form of information, and we derive necessary conditions for individuals to resist the temptation to evade their tax payment. We also consider the possibility that institution members may be corrupt and subject to bribery, and we analyze how often an institution must be audited to prevent bribery. Our analysis has implications for the establishment of robust public institutions that provide social information to support cooperation in large populations -- and the potential negative consequences associated with wealth or income inequality.
{"title":"Establishing incorruptible institutions by social contract and taxation","authors":"Taylor A. Kessinger, Joshua B. Plotkin","doi":"arxiv-2408.11199","DOIUrl":"https://doi.org/arxiv-2408.11199","url":null,"abstract":"Indirect reciprocity is a plausible mechanism for sustaining cooperation:\u0000people cooperate with those who have a good reputation, which can be acquired\u0000by helping others. However, this mechanism requires the population to agree on\u0000who has good or bad moral standing. Consensus can be provided by a central\u0000institution that monitors and broadcasts reputations. But how might such an\u0000institution be maintained, and how can a population ensure that it is effective\u0000and incorruptible? Here we explore a simple mechanism to sustain an institution\u0000of reputational judgment: a compulsory contribution from each member of the\u0000population, i.e., a tax. We analyze the maximum possible tax rate that\u0000individuals will rationally pay to sustain an institution of judgment, which\u0000provides a public good in the form of information, and we derive necessary\u0000conditions for individuals to resist the temptation to evade their tax payment.\u0000We also consider the possibility that institution members may be corrupt and\u0000subject to bribery, and we analyze how often an institution must be audited to\u0000prevent bribery. Our analysis has implications for the establishment of robust\u0000public institutions that provide social information to support cooperation in\u0000large populations -- and the potential negative consequences associated with\u0000wealth or income inequality.","PeriodicalId":501188,"journal":{"name":"arXiv - ECON - Theoretical Economics","volume":"43 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142197118","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}
Rashida Hakim, Jason Milionis, Christos Papadimitriou, Georgios Piliouras
During 2023, two interesting results were proven about the limit behavior of game dynamics: First, it was shown that there is a game for which no dynamics converges to the Nash equilibria. Second, it was shown that the sink equilibria of a game adequately capture the limit behavior of natural game dynamics. These two results have created a need and opportunity to articulate a principled computational theory of the meaning of the game that is based on game dynamics. Given any game in normal form, and any prior distribution of play, we study the problem of computing the asymptotic behavior of a class of natural dynamics called the noisy replicator dynamics as a limit distribution over the sink equilibria of the game. When the prior distribution has pure strategy support, we prove this distribution can be computed efficiently, in near-linear time to the size of the best-response graph. When the distribution can be sampled -- for example, if it is the uniform distribution over all mixed strategy profiles -- we show through experiments that the limit distribution of reasonably large games can be estimated quite accurately through sampling and simulation.
{"title":"Swim till You Sink: Computing the Limit of a Game","authors":"Rashida Hakim, Jason Milionis, Christos Papadimitriou, Georgios Piliouras","doi":"arxiv-2408.11146","DOIUrl":"https://doi.org/arxiv-2408.11146","url":null,"abstract":"During 2023, two interesting results were proven about the limit behavior of\u0000game dynamics: First, it was shown that there is a game for which no dynamics\u0000converges to the Nash equilibria. Second, it was shown that the sink equilibria\u0000of a game adequately capture the limit behavior of natural game dynamics. These\u0000two results have created a need and opportunity to articulate a principled\u0000computational theory of the meaning of the game that is based on game dynamics.\u0000Given any game in normal form, and any prior distribution of play, we study the\u0000problem of computing the asymptotic behavior of a class of natural dynamics\u0000called the noisy replicator dynamics as a limit distribution over the sink\u0000equilibria of the game. When the prior distribution has pure strategy support,\u0000we prove this distribution can be computed efficiently, in near-linear time to\u0000the size of the best-response graph. When the distribution can be sampled --\u0000for example, if it is the uniform distribution over all mixed strategy profiles\u0000-- we show through experiments that the limit distribution of reasonably large\u0000games can be estimated quite accurately through sampling and simulation.","PeriodicalId":501188,"journal":{"name":"arXiv - ECON - Theoretical Economics","volume":"27 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142197133","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}
Prophet inequalities are a cornerstone in optimal stopping and online decision-making. Traditionally, they involve the sequential observation of $n$ non-negative independent random variables and face irrevocable accept-or-reject choices. The goal is to provide policies that provide a good approximation ratio against the optimal offline solution that can access all the values upfront -- the so-called prophet value. In the prophet inequality over time problem (POT), the decision-maker can commit to an accepted value for $tau$ units of time, during which no new values can be accepted. This creates a trade-off between the duration of commitment and the opportunity to capture potentially higher future values. In this work, we provide best possible worst-case approximation ratios in the IID setting of POT for single-threshold algorithms and the optimal dynamic programming policy. We show a single-threshold algorithm that achieves an approximation ratio of $(1+e^{-2})/2approx 0.567$, and we prove that no single-threshold algorithm can surpass this guarantee. With our techniques, we can analyze simple algorithms using $k$ thresholds and show that with $k=3$ it is possible to get an approximation ratio larger than $approx 0.602$. Then, for each $n$, we prove it is possible to compute the tight worst-case approximation ratio of the optimal dynamic programming policy for instances with $n$ values by solving a convex optimization program. A limit analysis of the first-order optimality conditions yields a nonlinear differential equation showing that the optimal dynamic programming policy's asymptotic worst-case approximation ratio is $approx 0.618$. Finally, we extend the discussion to adversarial settings and show an optimal worst-case approximation ratio of $approx 0.162$ when the values are streamed in random order.
{"title":"Optimal Guarantees for Online Selection Over Time","authors":"Sebastian Perez-Salazar, Victor Verdugo","doi":"arxiv-2408.11224","DOIUrl":"https://doi.org/arxiv-2408.11224","url":null,"abstract":"Prophet inequalities are a cornerstone in optimal stopping and online\u0000decision-making. Traditionally, they involve the sequential observation of $n$\u0000non-negative independent random variables and face irrevocable accept-or-reject\u0000choices. The goal is to provide policies that provide a good approximation\u0000ratio against the optimal offline solution that can access all the values\u0000upfront -- the so-called prophet value. In the prophet inequality over time\u0000problem (POT), the decision-maker can commit to an accepted value for $tau$\u0000units of time, during which no new values can be accepted. This creates a\u0000trade-off between the duration of commitment and the opportunity to capture\u0000potentially higher future values. In this work, we provide best possible worst-case approximation ratios in the\u0000IID setting of POT for single-threshold algorithms and the optimal dynamic\u0000programming policy. We show a single-threshold algorithm that achieves an\u0000approximation ratio of $(1+e^{-2})/2approx 0.567$, and we prove that no\u0000single-threshold algorithm can surpass this guarantee. With our techniques, we\u0000can analyze simple algorithms using $k$ thresholds and show that with $k=3$ it\u0000is possible to get an approximation ratio larger than $approx 0.602$. Then,\u0000for each $n$, we prove it is possible to compute the tight worst-case\u0000approximation ratio of the optimal dynamic programming policy for instances\u0000with $n$ values by solving a convex optimization program. A limit analysis of\u0000the first-order optimality conditions yields a nonlinear differential equation\u0000showing that the optimal dynamic programming policy's asymptotic worst-case\u0000approximation ratio is $approx 0.618$. Finally, we extend the discussion to\u0000adversarial settings and show an optimal worst-case approximation ratio of\u0000$approx 0.162$ when the values are streamed in random order.","PeriodicalId":501188,"journal":{"name":"arXiv - ECON - Theoretical Economics","volume":"26 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142197192","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：