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**Published in:** LIPIcs, Volume 289, 41st International Symposium on Theoretical Aspects of Computer Science (STACS 2024)

The recent banking crisis has again emphasized the importance of understanding and mitigating systemic risk in financial networks. In this paper, we study a market-driven approach to rescue a bank in distress based on the idea of claims trading, a notion defined in Chapter 11 of the U.S. Bankruptcy Code. We formalize the idea in the context of the seminal model of financial networks by Eisenberg and Noe [Eisenberg and Noe, 2001]. For two given banks v and w, we consider the operation that w takes over some claims of v and in return gives liquidity to v (or creditors of v) to ultimately rescue v (or mitigate contagion effects). We study the structural properties and computational complexity of decision and optimization problems for several variants of claims trading.
When trading incoming edges of v (i.e., claims for which v is the creditor), we show that there is no trade in which both banks v and w strictly improve their assets. We therefore consider creditor-positive trades, in which v profits strictly and w remains indifferent. For a given set C of incoming edges of v, we provide an efficient algorithm to compute payments by w that result in a creditor-positive trade and maximal assets of v. When the set C must also be chosen, the problem becomes weakly NP-hard. Our main result here is a bicriteria FPTAS to compute an approximate trade, which allows for slightly increased payments by w. The approximate trade results in nearly the optimal amount of assets of v in any exact trade. Our results extend to the case in which banks use general monotone payment functions to settle their debt and the emerging clearing state can be computed efficiently.
In contrast, for trading outgoing edges of v (i.e., claims for which v is the debtor), the goal is to maximize the increase in assets for the creditors of v. Notably, for these results the characteristics of the payment functions of the banks are essential. For payments ranking creditors one by one, we show NP-hardness of approximation within a factor polynomial in the network size, in both problem variants when the set of claims C is part of the input or not. Instead, for payments proportional to the value of each debt, our results indicate more favorable conditions.

Martin Hoefer, Carmine Ventre, and Lisa Wilhelmi. Algorithms for Claims Trading. In 41st International Symposium on Theoretical Aspects of Computer Science (STACS 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 289, pp. 42:1-42:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{hoefer_et_al:LIPIcs.STACS.2024.42, author = {Hoefer, Martin and Ventre, Carmine and Wilhelmi, Lisa}, title = {{Algorithms for Claims Trading}}, booktitle = {41st International Symposium on Theoretical Aspects of Computer Science (STACS 2024)}, pages = {42:1--42:17}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-311-9}, ISSN = {1868-8969}, year = {2024}, volume = {289}, editor = {Beyersdorff, Olaf and Kant\'{e}, Mamadou Moustapha and Kupferman, Orna and Lokshtanov, Daniel}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2024.42}, URN = {urn:nbn:de:0030-drops-197523}, doi = {10.4230/LIPIcs.STACS.2024.42}, annote = {Keywords: Financial Networks, Claims Trade, Systemic Risk} }

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**Published in:** LIPIcs, Volume 274, 31st Annual European Symposium on Algorithms (ESA 2023)

We study threshold testing, an elementary probing model with the goal to choose a large value out of n i.i.d. random variables. An algorithm can test each variable X_i once for some threshold t_i, and the test returns binary feedback whether X_i ≥ t_i or not. Thresholds can be chosen adaptively or non-adaptively by the algorithm. Given the results for the tests of each variable, we then select the variable with highest conditional expectation. We compare the expected value obtained by the testing algorithm with expected maximum of the variables.
Threshold testing is a semi-online variant of the gambler’s problem and prophet inequalities. Indeed, the optimal performance of non-adaptive algorithms for threshold testing is governed by the standard i.i.d. prophet inequality of approximately 0.745 + o(1) as n → ∞. We show how adaptive algorithms can significantly improve upon this ratio. Our adaptive testing strategy guarantees a competitive ratio of at least 0.869 - o(1). Moreover, we show that there are distributions that admit only a constant ratio c < 1, even when n → ∞. Finally, when each box can be tested multiple times (with n tests in total), we design an algorithm that achieves a ratio of 1 - o(1).

Martin Hoefer and Kevin Schewior. Threshold Testing and Semi-Online Prophet Inequalities. In 31st Annual European Symposium on Algorithms (ESA 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 274, pp. 62:1-62:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{hoefer_et_al:LIPIcs.ESA.2023.62, author = {Hoefer, Martin and Schewior, Kevin}, title = {{Threshold Testing and Semi-Online Prophet Inequalities}}, booktitle = {31st Annual European Symposium on Algorithms (ESA 2023)}, pages = {62:1--62:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-295-2}, ISSN = {1868-8969}, year = {2023}, volume = {274}, editor = {G{\o}rtz, Inge Li and Farach-Colton, Martin and Puglisi, Simon J. and Herman, Grzegorz}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2023.62}, URN = {urn:nbn:de:0030-drops-187159}, doi = {10.4230/LIPIcs.ESA.2023.62}, annote = {Keywords: Prophet Inequalities, Testing, Stochastic Probing} }

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**Published in:** Dagstuhl Reports, Volume 12, Issue 11 (2023)

This report documents the program and outcomes of Dagstuhl Seminar 22452 "Computational Social Dynamics". The seminar addressed social and dynamic problems in the field of algorithmic game theory, and their implications in numerous applications, such as fair division, financial networks, or behavioral game theory. We summarize organizational aspects of the seminar, the talk abstracts, and the problems that were discussed in the open problem sessions.

Martin Hoefer, Sigal Oren, Roger Wattenhofer, and Giovanna Varricchio. Computational Social Dynamics (Dagstuhl Seminar 22452). In Dagstuhl Reports, Volume 12, Issue 11, pp. 28-44, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@Article{hoefer_et_al:DagRep.12.11.28, author = {Hoefer, Martin and Oren, Sigal and Wattenhofer, Roger and Varricchio, Giovanna}, title = {{Computational Social Dynamics (Dagstuhl Seminar 22452)}}, pages = {28--44}, journal = {Dagstuhl Reports}, ISSN = {2192-5283}, year = {2023}, volume = {12}, number = {11}, editor = {Hoefer, Martin and Oren, Sigal and Wattenhofer, Roger and Varricchio, Giovanna}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/DagRep.12.11.28}, URN = {urn:nbn:de:0030-drops-178346}, doi = {10.4230/DagRep.12.11.28}, annote = {Keywords: algorithmic game theory, behavioral economics, fair division, financial networks, social networks} }

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**Published in:** LIPIcs, Volume 185, 12th Innovations in Theoretical Computer Science Conference (ITCS 2021)

We consider a game of persuasion with evidence between a sender and a receiver. The sender has private information. By presenting evidence on the information, the sender wishes to persuade the receiver to take a single action (e.g., hire a job candidate, or convict a defendant). The sender’s utility depends solely on whether or not the receiver takes the action. The receiver’s utility depends on both the action as well as the sender’s private information. We study three natural variations. First, we consider sequential equilibria of the game without commitment power. Second, we consider a persuasion variant, where the sender commits to a signaling scheme and then the receiver, after seeing the evidence, takes the action or not. Third, we study a delegation variant, where the receiver first commits to taking the action if being presented certain evidence, and then the sender presents evidence to maximize the probability the action is taken. We study these variants through the computational lens, and give hardness results, optimal approximation algorithms, as well as polynomial-time algorithms for special cases. Among our results is an approximation algorithm that rounds a semidefinite program that might be of independent interest, since, to the best of our knowledge, it is the first such approximation algorithm for a natural problem in algorithmic economics.

Martin Hoefer, Pasin Manurangsi, and Alexandros Psomas. Algorithmic Persuasion with Evidence. In 12th Innovations in Theoretical Computer Science Conference (ITCS 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 185, pp. 3:1-3:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{hoefer_et_al:LIPIcs.ITCS.2021.3, author = {Hoefer, Martin and Manurangsi, Pasin and Psomas, Alexandros}, title = {{Algorithmic Persuasion with Evidence}}, booktitle = {12th Innovations in Theoretical Computer Science Conference (ITCS 2021)}, pages = {3:1--3:20}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-177-1}, ISSN = {1868-8969}, year = {2021}, volume = {185}, editor = {Lee, James R.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2021.3}, URN = {urn:nbn:de:0030-drops-135420}, doi = {10.4230/LIPIcs.ITCS.2021.3}, annote = {Keywords: Bayesian Persuasion, Semidefinite Programming, Approximation Algorithms} }

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**Published in:** LIPIcs, Volume 151, 11th Innovations in Theoretical Computer Science Conference (ITCS 2020)

In their seminal work on systemic risk in financial markets, Eisenberg and Noe [Larry Eisenberg and Thomas Noe, 2001] proposed and studied a model with n firms embedded into a network of debt relations. We analyze this model from a game-theoretic point of view. Every firm is a rational agent in a directed graph that has an incentive to allocate payments in order to clear as much of its debt as possible. Each edge is weighted and describes a liability between the firms. We consider several variants of the game that differ in the permissible payment strategies. We study the existence and computational complexity of pure Nash and strong equilibria, and we provide bounds on the (strong) prices of anarchy and stability for a natural notion of social welfare. Our results highlight the power of financial regulation - if payments of insolvent firms can be centrally assigned, a socially optimal strong equilibrium can be found in polynomial time. In contrast, worst-case strong equilibria can be a factor of Ω(n) away from optimal, and, in general, computing a best response is an NP-hard problem. For less permissible sets of strategies, we show that pure equilibria might not exist, and deciding their existence as well as computing them if they exist constitute NP-hard problems.

Nils Bertschinger, Martin Hoefer, and Daniel Schmand. Strategic Payments in Financial Networks. In 11th Innovations in Theoretical Computer Science Conference (ITCS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 151, pp. 46:1-46:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{bertschinger_et_al:LIPIcs.ITCS.2020.46, author = {Bertschinger, Nils and Hoefer, Martin and Schmand, Daniel}, title = {{Strategic Payments in Financial Networks}}, booktitle = {11th Innovations in Theoretical Computer Science Conference (ITCS 2020)}, pages = {46:1--46:16}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-134-4}, ISSN = {1868-8969}, year = {2020}, volume = {151}, editor = {Vidick, Thomas}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2020.46}, URN = {urn:nbn:de:0030-drops-117316}, doi = {10.4230/LIPIcs.ITCS.2020.46}, annote = {Keywords: Nash Equilibrium, Financial Network, Systemic Risk, Price of Anarchy, Equilibrium Computation} }

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**Published in:** LIPIcs, Volume 123, 29th International Symposium on Algorithms and Computation (ISAAC 2018)

We study online secretary problems with returns in combinatorial packing domains with n candidates that arrive sequentially over time in random order. The goal is to accept a feasible packing of candidates of maximum total value. In the first variant, each candidate arrives exactly twice. All 2n arrivals occur in random order. We propose a simple 0.5-competitive algorithm that can be combined with arbitrary approximation algorithms for the packing domain, even when the total value of candidates is a subadditive function. For bipartite matching, we obtain an algorithm with competitive ratio at least 0.5721 - o(1) for growing n, and an algorithm with ratio at least 0.5459 for all n >= 1. We extend all algorithms and ratios to k >= 2 arrivals per candidate.
In the second variant, there is a pool of undecided candidates. In each round, a random candidate from the pool arrives. Upon arrival a candidate can be either decided (accept/reject) or postponed (returned into the pool). We mainly focus on minimizing the expected number of postponements when computing an optimal solution. An expected number of Theta(n log n) is always sufficient. For matroids, we show that the expected number can be reduced to O(r log (n/r)), where r <=n/2 is the minimum of the ranks of matroid and dual matroid. For bipartite matching, we show a bound of O(r log n), where r is the size of the optimum matching. For general packing, we show a lower bound of Omega(n log log n), even when the size of the optimum is r = Theta(log n).

Martin Hoefer and Lisa Wilhelmi. Packing Returning Secretaries. In 29th International Symposium on Algorithms and Computation (ISAAC 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 123, pp. 65:1-65:12, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{hoefer_et_al:LIPIcs.ISAAC.2018.65, author = {Hoefer, Martin and Wilhelmi, Lisa}, title = {{Packing Returning Secretaries}}, booktitle = {29th International Symposium on Algorithms and Computation (ISAAC 2018)}, pages = {65:1--65:12}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-094-1}, ISSN = {1868-8969}, year = {2018}, volume = {123}, editor = {Hsu, Wen-Lian and Lee, Der-Tsai and Liao, Chung-Shou}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ISAAC.2018.65}, URN = {urn:nbn:de:0030-drops-100133}, doi = {10.4230/LIPIcs.ISAAC.2018.65}, annote = {Keywords: Secretary Problem, Coupon Collector Problem, Matroids} }

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**Published in:** LIPIcs, Volume 122, 38th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2018)

We consider the task of assigning indivisible goods to a set of agents in a fair manner. Our notion of fairness is Nash social welfare, i.e., the goal is to maximize the geometric mean of the utilities of the agents. Each good comes in multiple items or copies, and the utility of an agent diminishes as it receives more items of the same good. The utility of a bundle of items for an agent is the sum of the utilities of the items in the bundle. Each agent has a utility cap beyond which he does not value additional items. We give a polynomial time approximation algorithm that maximizes Nash social welfare up to a factor of e^{1/{e}} ~~ 1.445. The computed allocation is Pareto-optimal and approximates envy-freeness up to one item up to a factor of 2 + epsilon.

Bhaskar Ray Chaudhury, Yun Kuen Cheung, Jugal Garg, Naveen Garg, Martin Hoefer, and Kurt Mehlhorn. On Fair Division for Indivisible Items. In 38th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 122, pp. 25:1-25:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{chaudhury_et_al:LIPIcs.FSTTCS.2018.25, author = {Chaudhury, Bhaskar Ray and Cheung, Yun Kuen and Garg, Jugal and Garg, Naveen and Hoefer, Martin and Mehlhorn, Kurt}, title = {{On Fair Division for Indivisible Items}}, booktitle = {38th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2018)}, pages = {25:1--25:17}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-093-4}, ISSN = {1868-8969}, year = {2018}, volume = {122}, editor = {Ganguly, Sumit and Pandya, Paritosh}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2018.25}, URN = {urn:nbn:de:0030-drops-99242}, doi = {10.4230/LIPIcs.FSTTCS.2018.25}, annote = {Keywords: Fair Division, Indivisible Goods, Envy-Free} }

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**Published in:** LIPIcs, Volume 107, 45th International Colloquium on Automata, Languages, and Programming (ICALP 2018)

In cost sharing games with delays, a set of agents jointly uses a finite subset of resources. Each resource has a fixed cost that has to be shared by the players, and each agent has a non-shareable player-specific delay for each resource. A prominent example is uncapacitated facility location (UFL), where facilities need to be opened (at a shareable cost) and clients want to connect to opened facilities. Each client pays a cost share and his non-shareable physical connection cost. Given any profile of subsets used by the agents, a separable cost sharing protocol determines cost shares that satisfy budget balance on every resource and separability over the resources. Moreover, a separable protocol guarantees existence of pure Nash equilibria in the induced strategic game for the agents.
In this paper, we study separable cost sharing protocols in several general combinatorial domains. We provide black-box reductions to reduce the design of a separable cost sharing protocol to the design of an approximation algorithm for the underlying cost minimization problem. In this way, we obtain new separable cost sharing protocols in games based on arbitrary player-specific matroids, single-source connection games without delays, and connection games on n-series-parallel graphs with delays. All these reductions are efficiently computable - given an initial allocation profile, we obtain a profile of no larger cost and separable cost shares turning the profile into a pure Nash equilibrium. Hence, in these domains any approximation algorithm can be used to obtain a separable cost sharing protocol with a price of stability bounded by the approximation factor.

Tobias Harks, Martin Hoefer, Anja Huber, and Manuel Surek. Efficient Black-Box Reductions for Separable Cost Sharing. In 45th International Colloquium on Automata, Languages, and Programming (ICALP 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 107, pp. 154:1-154:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{harks_et_al:LIPIcs.ICALP.2018.154, author = {Harks, Tobias and Hoefer, Martin and Huber, Anja and Surek, Manuel}, title = {{Efficient Black-Box Reductions for Separable Cost Sharing}}, booktitle = {45th International Colloquium on Automata, Languages, and Programming (ICALP 2018)}, pages = {154:1--154:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-076-7}, ISSN = {1868-8969}, year = {2018}, volume = {107}, editor = {Chatzigiannakis, Ioannis and Kaklamanis, Christos and Marx, D\'{a}niel and Sannella, Donald}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2018.154}, URN = {urn:nbn:de:0030-drops-91587}, doi = {10.4230/LIPIcs.ICALP.2018.154}, annote = {Keywords: Cost Sharing, Price of Stability, Matroids, Connection Games} }

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**Published in:** LIPIcs, Volume 80, 44th International Colloquium on Automata, Languages, and Programming (ICALP 2017)

The secretary problem is a classic model for online decision making. Recently, combinatorial extensions such as matroid or matching secretary problems have become an important tool to study algorithmic problems in dynamic markets. Here the decision maker must know the numerical value of each arriving element, which can be a demanding informational assumption. In this paper, we initiate the study of combinatorial secretary problems with ordinal information, in which the decision maker only needs to be aware of a preference order consistent with the values of arrived elements. The goal is to design online algorithms with small competitive ratios.
For a variety of combinatorial problems, such as bipartite matching, general packing LPs, and independent set with bounded local independence number, we design new algorithms that obtain constant competitive ratios.
For the matroid secretary problem, we observe that many existing algorithms for special matroid structures maintain their competitive ratios even in the ordinal model. In these cases, the restriction to ordinal information does not represent any additional obstacle. Moreover, we show that ordinal variants of the submodular matroid secretary problems can be solved using algorithms for the linear versions by extending [Feldman and Zenklusen, 2015]. In contrast, we provide a lower bound of Omega(sqrt(n)/log(n)) for algorithms that are oblivious to the matroid structure, where n is the total number of elements. This contrasts an upper bound of O(log n) in the cardinal model, and it shows that the technique of thresholding is not sufficient for good algorithms in the ordinal model.

Martin Hoefer and Bojana Kodric. Combinatorial Secretary Problems with Ordinal Information. In 44th International Colloquium on Automata, Languages, and Programming (ICALP 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 80, pp. 133:1-133:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@InProceedings{hoefer_et_al:LIPIcs.ICALP.2017.133, author = {Hoefer, Martin and Kodric, Bojana}, title = {{Combinatorial Secretary Problems with Ordinal Information}}, booktitle = {44th International Colloquium on Automata, Languages, and Programming (ICALP 2017)}, pages = {133:1--133:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-041-5}, ISSN = {1868-8969}, year = {2017}, volume = {80}, editor = {Chatzigiannakis, Ioannis and Indyk, Piotr and Kuhn, Fabian and Muscholl, Anca}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2017.133}, URN = {urn:nbn:de:0030-drops-74594}, doi = {10.4230/LIPIcs.ICALP.2017.133}, annote = {Keywords: Secretary Problem, Matroid Secretary, Ordinal Information, Online Algorithms} }

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**Published in:** LIPIcs, Volume 57, 24th Annual European Symposium on Algorithms (ESA 2016)

We present the first analysis of Fisher markets with buyers that have budget-additive utility functions. Budget-additive utilities are elementary concave functions with numerous applications in online adword markets and revenue optimization problems. They extend the standard case of linear utilities and have been studied in a variety of other market models. In contrast to the frequently studied CES utilities, they have a global satiation point which can imply multiple market equilibria with quite different characteristics. Our main result is an efficient combinatorial algorithm to compute a market equilibrium with a Pareto-optimal allocation of goods. It relies on a new descending-price approach and, as a special case, also implies a novel combinatorial algorithm for computing a market equilibrium in linear Fisher markets. We complement this positive result with a number of hardness results for related computational questions. We prove that it isNP-hard to compute a market equilibrium that maximizes social welfare, and it is PPAD-hard to find any market equilibrium with utility functions with separate satiation points for each buyer and each good.

Xiaohui Bei, Jugal Garg, Martin Hoefer, and Kurt Mehlhorn. Computing Equilibria in Markets with Budget-Additive Utilities. In 24th Annual European Symposium on Algorithms (ESA 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 57, pp. 8:1-8:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)

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@InProceedings{bei_et_al:LIPIcs.ESA.2016.8, author = {Bei, Xiaohui and Garg, Jugal and Hoefer, Martin and Mehlhorn, Kurt}, title = {{Computing Equilibria in Markets with Budget-Additive Utilities}}, booktitle = {24th Annual European Symposium on Algorithms (ESA 2016)}, pages = {8:1--8:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-015-6}, ISSN = {1868-8969}, year = {2016}, volume = {57}, editor = {Sankowski, Piotr and Zaroliagis, Christos}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2016.8}, URN = {urn:nbn:de:0030-drops-63504}, doi = {10.4230/LIPIcs.ESA.2016.8}, annote = {Keywords: Budget-Additive Utility, Market Equilibrium, Equilibrium Computation} }

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**Published in:** LIPIcs, Volume 1, 25th International Symposium on Theoretical Aspects of Computer Science (2008)

We study a multi-player one-round game termed Stackelberg Network
Pricing Game, in which a leader can set prices for a subset of $m$
priceable edges in a graph. The other edges have a fixed cost.
Based on the leader's decision one or more followers optimize a
polynomial-time solvable combinatorial minimization problem and
choose a minimum cost solution satisfying their requirements based
on the fixed costs and the leader's prices. The leader receives as
revenue the total amount of prices paid by the followers for
priceable edges in their solutions, and the problem is to find
revenue maximizing prices. Our model extends several known pricing
problems, including single-minded and unit-demand pricing, as well
as Stackelberg pricing for certain follower problems like shortest
path or minimum spanning tree. Our first main result is a tight
analysis of a single-price algorithm for the single follower game,
which provides a $(1+varepsilon) log m$-approximation for any
$varepsilon >0$. This can be extended to provide a
$(1+varepsilon )(log k + log m)$-approximation for the general
problem and $k$ followers. The latter result is essentially best
possible, as the problem is shown to be hard to approximate within
$mathcal{O(log^varepsilon k + log^varepsilon m)$. If
followers have demands, the single-price algorithm provides a
$(1+varepsilon )m^2$-approximation, and the problem is hard to
approximate within $mathcal{O(m^varepsilon)$ for some
$varepsilon >0$. Our second main result is a polynomial time
algorithm for revenue maximization in the special case of
Stackelberg bipartite vertex cover, which is based on non-trivial
max-flow and LP-duality techniques. Our results can be extended to
provide constant-factor approximations for any constant number of
followers.

Patrick Briest, Martin Hoefer, and Piotr Krysta. Stackelberg Network Pricing Games. In 25th International Symposium on Theoretical Aspects of Computer Science. Leibniz International Proceedings in Informatics (LIPIcs), Volume 1, pp. 133-142, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2008)

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@InProceedings{briest_et_al:LIPIcs.STACS.2008.1340, author = {Briest, Patrick and Hoefer, Martin and Krysta, Piotr}, title = {{Stackelberg Network Pricing Games}}, booktitle = {25th International Symposium on Theoretical Aspects of Computer Science}, pages = {133--142}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-06-4}, ISSN = {1868-8969}, year = {2008}, volume = {1}, editor = {Albers, Susanne and Weil, Pascal}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2008.1340}, URN = {urn:nbn:de:0030-drops-13406}, doi = {10.4230/LIPIcs.STACS.2008.1340}, annote = {Keywords: Stackelberg Games, Algorithmic Pricing, Approximation Algorithms, Inapproximability.} }

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