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Documents authored by Bilò, Vittorio

Document
DOI: 10.4230/LIPIcs.ESA.2024.26
Achieving Envy-Freeness Through Items Sale

Authors: Vittorio Bilò, Evangelos Markakis, and Cosimo Vinci

Published in: LIPIcs, Volume 308, 32nd Annual European Symposium on Algorithms (ESA 2024)

Abstract
We consider a fair division setting of allocating indivisible items to a set of agents. In order to cope with the well-known impossibility results related to the non-existence of envy-free allocations, we allow the option of selling some of the items so as to compensate envious agents with monetary rewards. In fact, this approach is not new in practice, as it is applied in some countries in inheritance or divorce cases. A drawback of this approach is that it may create a value loss, since the market value derived by selling an item can be less than the value perceived by the agents. Therefore, given the market values of all items, a natural goal is to identify which items to sell so as to arrive at an envy-free allocation, while at the same time maximizing the overall social welfare. Our work is focused on the algorithmic study of this problem, and we provide both positive and negative results on its approximability. When the agents have a commonly accepted value for each item, our results show a sharp separation between the cases of two or more agents. In particular, we establish a PTAS for two agents, and we complement this with a hardness result, that for three or more agents, the best approximation guarantee is provided by essentially selling all items. This hardness barrier, however, is relieved when the number of distinct item values is constant, as we provide an efficient algorithm for any number of agents. We also explore the generalization to heterogeneous valuations, where the hardness result continues to hold, and where we provide positive results for certain special cases.
Cite as

Vittorio Bilò, Evangelos Markakis, and Cosimo Vinci. Achieving Envy-Freeness Through Items Sale. In 32nd Annual European Symposium on Algorithms (ESA 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 308, pp. 26:1-26:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{bilo_et_al:LIPIcs.ESA.2024.26,
  author =	{Bil\`{o}, Vittorio and Markakis, Evangelos and Vinci, Cosimo},
  title =	{{Achieving Envy-Freeness Through Items Sale}},
  booktitle =	{32nd Annual European Symposium on Algorithms (ESA 2024)},
  pages =	{26:1--26:16},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-338-6},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{308},
  editor =	{Chan, Timothy and Fischer, Johannes and Iacono, John and Herman, Grzegorz},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2024.26},
  URN =		{urn:nbn:de:0030-drops-210977},
  doi =		{10.4230/LIPIcs.ESA.2024.26},
  annote =	{Keywords: Fair Item Allocation, Approximation Algorithms, Envy-freeness, Markets}
}
Document
DOI: 10.4230/LIPIcs.MFCS.2020.15
Topological Influence and Locality in Swap Schelling Games

Authors: Davide Bilò, Vittorio Bilò, Pascal Lenzner, and Louise Molitor

Published in: LIPIcs, Volume 170, 45th International Symposium on Mathematical Foundations of Computer Science (MFCS 2020)

Abstract
Residential segregation is a wide-spread phenomenon that can be observed in almost every major city. In these urban areas residents with different racial or socioeconomic background tend to form homogeneous clusters. Schelling’s famous agent-based model for residential segregation explains how such clusters can form even if all agents are tolerant, i.e., if they agree to live in mixed neighborhoods. For segregation to occur, all it needs is a slight bias towards agents preferring similar neighbors. Very recently, Schelling’s model has been investigated from a game-theoretic point of view with selfish agents that strategically select their residential location. In these games, agents can improve on their current location by performing a location swap with another agent who is willing to swap. We significantly deepen these investigations by studying the influence of the underlying topology modeling the residential area on the existence of equilibria, the Price of Anarchy and on the dynamic properties of the resulting strategic multi-agent system. Moreover, as a new conceptual contribution, we also consider the influence of locality, i.e., if the location swaps are restricted to swaps of neighboring agents. We give improved almost tight bounds on the Price of Anarchy for arbitrary underlying graphs and we present (almost) tight bounds for regular graphs, paths and cycles. Moreover, we give almost tight bounds for grids, which are commonly used in empirical studies. For grids we also show that locality has a severe impact on the game dynamics.
Cite as

Davide Bilò, Vittorio Bilò, Pascal Lenzner, and Louise Molitor. Topological Influence and Locality in Swap Schelling Games. In 45th International Symposium on Mathematical Foundations of Computer Science (MFCS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 170, pp. 15:1-15:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{bilo_et_al:LIPIcs.MFCS.2020.15,
  author =	{Bil\`{o}, Davide and Bil\`{o}, Vittorio and Lenzner, Pascal and Molitor, Louise},
  title =	{{Topological Influence and Locality in Swap Schelling Games}},
  booktitle =	{45th International Symposium on Mathematical Foundations of Computer Science (MFCS 2020)},
  pages =	{15:1--15:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-159-7},
  ISSN =	{1868-8969},
  year =	{2020},
  volume =	{170},
  editor =	{Esparza, Javier and Kr\'{a}l', Daniel},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2020.15},
  URN =		{urn:nbn:de:0030-drops-126841},
  doi =		{10.4230/LIPIcs.MFCS.2020.15},
  annote =	{Keywords: Residential Segregation, Schelling’s Segregation Model, Non-cooperative Games, Price of Anarchy, Game Dynamics}
}
Document
DOI: 10.4230/LIPIcs.ITCS.2019.14
Almost Envy-Free Allocations with Connected Bundles

Authors: Vittorio Bilò, Ioannis Caragiannis, Michele Flammini, Ayumi Igarashi, Gianpiero Monaco, Dominik Peters, Cosimo Vinci, and William S. Zwicker

Published in: LIPIcs, Volume 124, 10th Innovations in Theoretical Computer Science Conference (ITCS 2019)

Abstract
We study the existence of allocations of indivisible goods that are envy-free up to one good (EF1), under the additional constraint that each bundle needs to be connected in an underlying item graph G. When the items are arranged in a path, we show that EF1 allocations are guaranteed to exist for arbitrary monotonic utility functions over bundles, provided that either there are at most four agents, or there are any number of agents but they all have identical utility functions. Our existence proofs are based on classical arguments from the divisible cake-cutting setting, and involve discrete analogues of cut-and-choose, of Stromquist's moving-knife protocol, and of the Su-Simmons argument based on Sperner's lemma. Sperner's lemma can also be used to show that on a path, an EF2 allocation exists for any number of agents. Except for the results using Sperner's lemma, all of our procedures can be implemented by efficient algorithms. Our positive results for paths imply the existence of connected EF1 or EF2 allocations whenever G is traceable, i.e., contains a Hamiltonian path. For the case of two agents, we completely characterize the class of graphs G that guarantee the existence of EF1 allocations as the class of graphs whose biconnected components are arranged in a path. This class is strictly larger than the class of traceable graphs; one can check in linear time whether a graph belongs to this class, and if so return an EF1 allocation.
Cite as

Vittorio Bilò, Ioannis Caragiannis, Michele Flammini, Ayumi Igarashi, Gianpiero Monaco, Dominik Peters, Cosimo Vinci, and William S. Zwicker. Almost Envy-Free Allocations with Connected Bundles. In 10th Innovations in Theoretical Computer Science Conference (ITCS 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 124, pp. 14:1-14:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{bilo_et_al:LIPIcs.ITCS.2019.14,
  author =	{Bil\`{o}, Vittorio and Caragiannis, Ioannis and Flammini, Michele and Igarashi, Ayumi and Monaco, Gianpiero and Peters, Dominik and Vinci, Cosimo and Zwicker, William S.},
  title =	{{Almost Envy-Free Allocations with Connected Bundles}},
  booktitle =	{10th Innovations in Theoretical Computer Science Conference (ITCS 2019)},
  pages =	{14:1--14:21},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-095-8},
  ISSN =	{1868-8969},
  year =	{2019},
  volume =	{124},
  editor =	{Blum, Avrim},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2019.14},
  URN =		{urn:nbn:de:0030-drops-101078},
  doi =		{10.4230/LIPIcs.ITCS.2019.14},
  annote =	{Keywords: Envy-free Division, Cake-cutting, Resource Allocation, Algorithmic Game Theory}
}
Document
DOI: 10.4230/LIPIcs.MFCS.2018.47
Pricing Problems with Buyer Preselection

Authors: Vittorio Bilò, Michele Flammini, Gianpiero Monaco, and Luca Moscardelli

Published in: LIPIcs, Volume 117, 43rd International Symposium on Mathematical Foundations of Computer Science (MFCS 2018)

Abstract
We investigate the problem of preselecting a subset of buyers participating in a market so as to optimize the performance of stable outcomes. We consider four scenarios arising from the combination of two stability notions, item and bundle envy-freeness, with the two classical objective functions, i.e., the social welfare and the seller's revenue. When adopting the notion of item envy-freeness, we prove that, for both the two objective functions, the problem cannot be approximated within n^(1-epsilon) for any epsilon >0, and provide tight or nearly tight approximation algorithms. We also prove that maximizing the seller's revenue is NP-hard even for a single buyer, thus closing an open question. Under bundle envy-freeness, instead, we show how to transform in polynomial time any stable outcome for a market involving only a subset of buyers to a stable one for the whole market without worsening its performance, both for the social welfare and the seller's revenue. Finally, we consider multi-unit markets, where all items are of the same type and are assigned the same price. For this specific case, we show that buyer preselection can improve the performance of stable outcomes in all of the four considered scenarios, and we design corresponding approximation algorithms.
Cite as

Vittorio Bilò, Michele Flammini, Gianpiero Monaco, and Luca Moscardelli. Pricing Problems with Buyer Preselection. In 43rd International Symposium on Mathematical Foundations of Computer Science (MFCS 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 117, pp. 47:1-47:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{bilo_et_al:LIPIcs.MFCS.2018.47,
  author =	{Bil\`{o}, Vittorio and Flammini, Michele and Monaco, Gianpiero and Moscardelli, Luca},
  title =	{{Pricing Problems with Buyer Preselection}},
  booktitle =	{43rd International Symposium on Mathematical Foundations of Computer Science (MFCS 2018)},
  pages =	{47:1--47:16},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-086-6},
  ISSN =	{1868-8969},
  year =	{2018},
  volume =	{117},
  editor =	{Potapov, Igor and Spirakis, Paul and Worrell, James},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2018.47},
  URN =		{urn:nbn:de:0030-drops-96292},
  doi =		{10.4230/LIPIcs.MFCS.2018.47},
  annote =	{Keywords: Pricing problems, Envy-freeness, Revenue maximization, Social Welfare maximization}
}
Document
DOI: 10.4230/LIPIcs.ICALP.2018.146
Uniform Mixed Equilibria in Network Congestion Games with Link Failures

Authors: Vittorio Bilò, Luca Moscardelli, and Cosimo Vinci

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

Abstract
Motivated by possible applications in fault-tolerant routing, we introduce the notion of uniform mixed equilibria in network congestion games with adversarial link failures, where players need to route traffic from a source to a destination node. Given an integer rho >= 1, a rho-uniform mixed strategy is a mixed strategy in which a player plays exactly rho edge disjoint paths with uniform probabilities, so that a rho-uniform mixed equilibrium is a tuple of rho-uniform mixed strategies, one for each player, in which no player can lower her cost by deviating to another rho-uniform mixed strategy. For games with weighted players and affine latency functions, we show existence of rho-uniform mixed equilibria and provide a tight characterization of their price of anarchy. For games with unweighted players, instead, we extend the existential guarantee to any class of latency functions and, restricted to games with affine latencies, we derive a tight characterization of both the prices of anarchy and stability.
Cite as

Vittorio Bilò, Luca Moscardelli, and Cosimo Vinci. Uniform Mixed Equilibria in Network Congestion Games with Link Failures. In 45th International Colloquium on Automata, Languages, and Programming (ICALP 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 107, pp. 146:1-146:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{bilo_et_al:LIPIcs.ICALP.2018.146,
  author =	{Bil\`{o}, Vittorio and Moscardelli, Luca and Vinci, Cosimo},
  title =	{{Uniform Mixed Equilibria in Network Congestion Games with Link Failures}},
  booktitle =	{45th International Colloquium on Automata, Languages, and Programming (ICALP 2018)},
  pages =	{146:1--146:14},
  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.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2018.146},
  URN =		{urn:nbn:de:0030-drops-91508},
  doi =		{10.4230/LIPIcs.ICALP.2018.146},
  annote =	{Keywords: Network Congestion Games, Fault-Tolerant Routing, Nash Equilibria, Price of Anarchy, Price of Stability}
}
Document
DOI: 10.4230/LIPIcs.ESA.2017.17
On the Impact of Singleton Strategies in Congestion Games

Authors: Vittorio Bilò and Cosimo Vinci

Published in: LIPIcs, Volume 87, 25th Annual European Symposium on Algorithms (ESA 2017)

Abstract
To what extent does the structure of the players' strategy space influence the efficiency of decentralized solutions in congestion games? In this work, we investigate whether better performance is possible when restricting to load balancing games in which players can only choose among single resources. We consider three different solutions concepts, namely, approximate pure Nash equilibria, approximate one-round walks generated by selfish players aiming at minimizing their personal cost and approximate one-round walks generated by cooperative players aiming at minimizing the marginal increase in the sum of the players' personal costs. The last two concepts can also be interpreted as solutions of simple greedy online algorithms for the related resource selection problem. Under fairly general latency functions on the resources, we show that, for all three types of solutions, better bounds cannot be achieved if players are either weighted or asymmetric. On the positive side, we prove that, under mild assumptions on the latency functions, improvements on the performance of approximate pure Nash equilibria are possible for load balancing games with weighted and symmetric players in the case of identical resources. We also design lower bounds on the performance of one-round walks in load balancing games with unweighted players and identical resources (in this case, solutions generated by selfish and cooperative players coincide).
Cite as

Vittorio Bilò and Cosimo Vinci. On the Impact of Singleton Strategies in Congestion Games. In 25th Annual European Symposium on Algorithms (ESA 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 87, pp. 17:1-17:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@InProceedings{bilo_et_al:LIPIcs.ESA.2017.17,
  author =	{Bil\`{o}, Vittorio and Vinci, Cosimo},
  title =	{{On the Impact of Singleton Strategies in Congestion Games}},
  booktitle =	{25th Annual European Symposium on Algorithms (ESA 2017)},
  pages =	{17:1--17:14},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-049-1},
  ISSN =	{1868-8969},
  year =	{2017},
  volume =	{87},
  editor =	{Pruhs, Kirk and Sohler, Christian},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2017.17},
  URN =		{urn:nbn:de:0030-drops-78576},
  doi =		{10.4230/LIPIcs.ESA.2017.17},
  annote =	{Keywords: Congestion games, Nash equilibrium, price of anarchy, online load balancing, greedy algorithms}
}
Document
DOI: 10.4230/LIPIcs.ICALP.2017.125
Simple Greedy Algorithms for Fundamental Multidimensional Graph Problems

Authors: Vittorio Bilò, Ioannis Caragiannis, Angelo Fanelli, Michele Flammini, and Gianpiero Monaco

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

Abstract
We revisit fundamental problems in undirected and directed graphs, such as the problems of computing spanning trees, shortest paths, steiner trees, and spanning arborescences of minimum cost. We assume that there are d different cost functions associated with the edges of the input graph and seek for solutions to the resulting multidimensional graph problems so that the p-norm of the different costs of the solution is minimized. We present combinatorial algorithms that achieve very good approximations for this objective. The main advantage of our algorithms is their simplicity: they are as simple as classical combinatorial graph algorithms of Dijkstra and Kruskal, or the greedy algorithm for matroids.
Cite as

Vittorio Bilò, Ioannis Caragiannis, Angelo Fanelli, Michele Flammini, and Gianpiero Monaco. Simple Greedy Algorithms for Fundamental Multidimensional Graph Problems. In 44th International Colloquium on Automata, Languages, and Programming (ICALP 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 80, pp. 125:1-125:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@InProceedings{bilo_et_al:LIPIcs.ICALP.2017.125,
  author =	{Bil\`{o}, Vittorio and Caragiannis, Ioannis and Fanelli, Angelo and Flammini, Michele and Monaco, Gianpiero},
  title =	{{Simple Greedy Algorithms for Fundamental Multidimensional Graph Problems}},
  booktitle =	{44th International Colloquium on Automata, Languages, and Programming (ICALP 2017)},
  pages =	{125:1--125:13},
  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.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2017.125},
  URN =		{urn:nbn:de:0030-drops-74669},
  doi =		{10.4230/LIPIcs.ICALP.2017.125},
  annote =	{Keywords: multidimensional graph problems, matroids, shortest paths, Steiner trees, arborescences}
}
Document
DOI: 10.4230/LIPIcs.STACS.2017.13
Existential-R-Complete Decision Problems about Symmetric Nash Equilibria in Symmetric Multi-Player Games

Authors: Vittorio Bilò and Marios Mavronicolas

Published in: LIPIcs, Volume 66, 34th Symposium on Theoretical Aspects of Computer Science (STACS 2017)

Abstract
We study the complexity of decision problems about symmetric Nash equilibria for symmetric multi-player games. These decision problems concern the existence of a symmetric Nash equilibrium with certain natural properties. We show that a handful of such decision problems are Existential-R-complete; that is, they are exactly as hard as deciding the Existential Theory of the Reals.
Cite as

Vittorio Bilò and Marios Mavronicolas. Existential-R-Complete Decision Problems about Symmetric Nash Equilibria in Symmetric Multi-Player Games. In 34th Symposium on Theoretical Aspects of Computer Science (STACS 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 66, pp. 13:1-13:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@InProceedings{bilo_et_al:LIPIcs.STACS.2017.13,
  author =	{Bil\`{o}, Vittorio and Mavronicolas, Marios},
  title =	{{Existential-R-Complete Decision Problems about Symmetric Nash Equilibria in Symmetric Multi-Player Games}},
  booktitle =	{34th Symposium on Theoretical Aspects of Computer Science (STACS 2017)},
  pages =	{13:1--13:14},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-028-6},
  ISSN =	{1868-8969},
  year =	{2017},
  volume =	{66},
  editor =	{Vollmer, Heribert and Vall\'{e}e, Brigitte},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2017.13},
  URN =		{urn:nbn:de:0030-drops-70200},
  doi =		{10.4230/LIPIcs.STACS.2017.13},
  annote =	{Keywords: Nash equilibrium, complexity of equilibria, ExistentialR-completeness}
}
Document
DOI: 10.4230/LIPIcs.STACS.2016.17
A Catalog of EXISTS-R-Complete Decision Problems About Nash Equilibria in Multi-Player Games

Authors: Vittorio Bilò and Marios Mavronicolas

Published in: LIPIcs, Volume 47, 33rd Symposium on Theoretical Aspects of Computer Science (STACS 2016)

Abstract
[Schaefer and Stefankovic, Theory of Computing Systems, 2015] provided an explicit formulation of EXISTS-R as the class capturing the complexity of deciding the Existential Theory of the Reals, and established that deciding, given a 3-player game, whether or not it has a Nash equilibrium with no probability exceeding a given rational is EXISTS-R-complete. Four more decision problems about Nash equilibria for 3-player games were very recently shown EXISTS-R-complete via a chain of individual, problem-specific reductions in [Garg et al., Proceedings of ICALP 2015]; determining more such EXISTS-R-complete problems was posed there as an open problem. In this work, we deliver an extensive catalog of EXISTS-R-complete decision problems about Nash equilibria in 3-player games, thus resolving completely the open problem from [Garg et al., Proceedings of ICALP 2015]. Towards this end, we present a single and very simple, unifying reduction from the EXISTS-R-complete decision problem from [Schaefer and Stefankovic, Theory of Computing Systems, 2015] to (almost) all the decision problems about Nash equilibria that were before shown NP-complete for 2-player games in [Bilo and Mavronicolas, Proceedings of SAGT 2012; Conitzer and Sandholm, Games and Economic Behavior, 2008; Gilboa and Zemel, Games and Economic Behavior, 1989]. Encompassed in the catalog are the four decision problems shown EXISTS-R-complete in [Garg et al., Proceedings of ICALP 2015].
Cite as

Vittorio Bilò and Marios Mavronicolas. A Catalog of EXISTS-R-Complete Decision Problems About Nash Equilibria in Multi-Player Games. In 33rd Symposium on Theoretical Aspects of Computer Science (STACS 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 47, pp. 17:1-17:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)

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@InProceedings{bilo_et_al:LIPIcs.STACS.2016.17,
  author =	{Bil\`{o}, Vittorio and Mavronicolas, Marios},
  title =	{{A Catalog of EXISTS-R-Complete Decision Problems About Nash Equilibria in Multi-Player Games}},
  booktitle =	{33rd Symposium on Theoretical Aspects of Computer Science (STACS 2016)},
  pages =	{17:1--17:13},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-001-9},
  ISSN =	{1868-8969},
  year =	{2016},
  volume =	{47},
  editor =	{Ollinger, Nicolas and Vollmer, Heribert},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2016.17},
  URN =		{urn:nbn:de:0030-drops-57189},
  doi =		{10.4230/LIPIcs.STACS.2016.17},
  annote =	{Keywords: Nash equilibrium, complexity of equilibria, EXISTS-R-completeness}
}
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