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

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.

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

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

[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].

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} }

Document

**Published in:** Dagstuhl Seminar Proceedings, Volume 5361, Algorithmic Aspects of Large and Complex Networks (2006)

We propose a simple and intuitive cost mechanism which assigns costs for the competitive usage of $m$ resources by $n$ selfish agents. Each agent has an individual demand; demands are drawn according to some probability distribution. The cost paid by an agent for a resource she chooses is the total demand put on the resource divided by the number of agents who chose that same resource. So, resources charge costs in an equitable, fair way, while each resource makes no profit out of the agents.
We call our model the Fair Pricing model. Its fair cost mechanism induces a non-cooperative game among the agents. To evaluate the Nash equilibria of this game, we introduce the Diffuse Price of Anarchy, as an extension of the Price of Anarchy that takes into account the probability distribution on the demands. We prove:
(1) Pure Nash equilibria may not exist, unless all chosen demands are identical; in contrast, a fully mixed Nash equilibrium exists for all possible choices of the demands. Further on, the fully mixed Nash equilibrium is the unique Nash equilibrium in case there are only two agents.
(2) In the worst-case choice of demands, the Price of Anarchy is $Theta (n)$;
for the special case of two agents, the Price of Anarchy is less than $2 - frac{1}{m}$.
(3) Assume now that demands are drawn from a bounded, independent probability distribution, where all demands are identically distributed and each is at most a (universal for the class) constant times its expectation. Then, the Diffuse Price of Anarchy is at most that same constant, which is just $2$ when each demand is distributed symmetrically around its expectation.

Marios Mavronicolas, Panagiota Panagopoulou, and Paul G. Spirakis. A Cost Mechanism for Fair Pricing of Resource Usage. In Algorithmic Aspects of Large and Complex Networks. Dagstuhl Seminar Proceedings, Volume 5361, pp. 1-15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2006)

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@InProceedings{mavronicolas_et_al:DagSemProc.05361.2, author = {Mavronicolas, Marios and Panagopoulou, Panagiota and Spirakis, Paul G.}, title = {{A Cost Mechanism for Fair Pricing of Resource Usage}}, booktitle = {Algorithmic Aspects of Large and Complex Networks}, pages = {1--15}, series = {Dagstuhl Seminar Proceedings (DagSemProc)}, ISSN = {1862-4405}, year = {2006}, volume = {5361}, editor = {Stefano Leonardi and Friedhelm Meyer auf der Heide and Dorothea Wagner}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/DagSemProc.05361.2}, URN = {urn:nbn:de:0030-drops-5646}, doi = {10.4230/DagSemProc.05361.2}, annote = {Keywords: Cost Sharing, Diffuse Price of Anarchy, Fair Pricing, Resources} }

Document

**Published in:** Dagstuhl Seminar Proceedings, Volume 5361, Algorithmic Aspects of Large and Complex Networks (2006)

We propose a simple and intuitive cost mechanism which assigns costs for the competitive usage of $m$ resources by $n$ selfish agents. Each agent has an individual demand; demands are drawn according to some probability distribution. The cost paid by an agent for a resource she chooses is the total demand put on the resource divided by the number of agents who chose that same resource. So, resources charge costs in an equitable, fair way, while each resource makes no profit out of the agents.
We call our model the Fair Pricing model. Its fair cost mechanism induces a non-cooperative game among the agents. To evaluate the Nash equilibria of this game, we introduce the Diffuse Price of Anarchy, as an extension of the Price of Anarchy that takes into account the probability distribution on the demands. We prove:
(1) Pure Nash equilibria may not exist, unless all chosen demands are identical. In contrast, we have been able to prove that pure Nash equilibria do exist for two closely related cost sharing models, namely the Average Cost Pricing and the Serial Cost Sharing models.
(2) A fully mixed Nash equilibrium exists for all possible choices of the demands. Further on, the fully mixed Nash equilibrium is the unique Nash equilibrium in case there are only two agents.
(3) In the worst-case choice of demands, the Price of Anarchy is $Theta (n)$; for the special case of two agents, the Price of Anarchy is less than $2 - frac{1}{m}$.
(4) Assume now that demands are drawn from a bounded, independent probability distribution, where all demands are identically distributed and each is at most a (universal for the class) constant times its expectation. Then, the Diffuse Price of Anarchy is at most that same constant, which is just 2 when each demand is distributed symmetrically around its expectation.

Marios Mavronicolas, Panagiota Panagopoulou, and Paul G. Spirakis. Cost Sharing Mechanisms for Fair Pricing of Resources Usage. In Algorithmic Aspects of Large and Complex Networks. Dagstuhl Seminar Proceedings, Volume 5361, pp. 1-18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2006)

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@InProceedings{mavronicolas_et_al:DagSemProc.05361.5, author = {Mavronicolas, Marios and Panagopoulou, Panagiota and Spirakis, Paul G.}, title = {{Cost Sharing Mechanisms for Fair Pricing of Resources Usage}}, booktitle = {Algorithmic Aspects of Large and Complex Networks}, pages = {1--18}, series = {Dagstuhl Seminar Proceedings (DagSemProc)}, ISSN = {1862-4405}, year = {2006}, volume = {5361}, editor = {Stefano Leonardi and Friedhelm Meyer auf der Heide and Dorothea Wagner}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/DagSemProc.05361.5}, URN = {urn:nbn:de:0030-drops-5665}, doi = {10.4230/DagSemProc.05361.5}, annote = {Keywords: Cost Sharing, Diffuse Price of Anarchy, Fair Pricing, Resources} }

Document

**Published in:** Dagstuhl Seminar Proceedings, Volume 5011, Computing and Markets (2005)

In this work, we consider an interesting variant
of the well-studied KP model [KP99] for selfish
routing that reflects some influence from the much
older Wardrop [War52]. In the new model, user
traffics are still unsplittable, while social cost
is now the expectation of the sum, over all links,
of a certain polynomial evaluated at the total
latency incurred by all users choosing the link;
we call it polynomial social cost. The polynomials
that we consider have non-negative coefficients.
We are interested in evaluating Nash equilibria in
this model, and we use the Price of Anarchy as our
evaluation measure. We prove the Fully Mixed Nash
Equilibrium Conjecture for identical users and two
links, and establish an approximate version of the
conjecture for arbitrary many links. Moreover, we
give upper bounds on the Price of Anarchy.

Martin Gairing, Thomas Lücking, Marios Mavronicolas, and Burkhard Monien. The Price of Anarchy for Polynomial Social Cost. In Computing and Markets. Dagstuhl Seminar Proceedings, Volume 5011, pp. 1-12, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2005)

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@InProceedings{gairing_et_al:DagSemProc.05011.19, author = {Gairing, Martin and L\"{u}cking, Thomas and Mavronicolas, Marios and Monien, Burkhard}, title = {{The Price of Anarchy for Polynomial Social Cost}}, booktitle = {Computing and Markets}, pages = {1--12}, series = {Dagstuhl Seminar Proceedings (DagSemProc)}, ISSN = {1862-4405}, year = {2005}, volume = {5011}, editor = {Daniel Lehmann and Rudolf M\"{u}ller and Tuomas Sandholm}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/DagSemProc.05011.19}, URN = {urn:nbn:de:0030-drops-2005}, doi = {10.4230/DagSemProc.05011.19}, annote = {Keywords: selfish routing , KP-model , price of anarchy , fully mixed Nash Equilibrium} }

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