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Documents authored by Brice, Léonard

Document
DOI: 10.4230/LIPIcs.MFCS.2025.30
Finding Equilibria: Simpler for Pessimists, Simplest for Optimists

Authors: Léonard Brice, Thomas A. Henzinger, and K. S. Thejaswini

Published in: LIPIcs, Volume 345, 50th International Symposium on Mathematical Foundations of Computer Science (MFCS 2025)

Abstract
We consider equilibria in multiplayer stochastic graph games with terminal-node rewards. In such games, Nash equilibria are defined assuming that each player seeks to maximise their expected payoff, ignoring their aversion or tolerance to risk. We therefore study risk-sensitive equilibria (RSEs), where the expected payoff is replaced by a risk measure. A classical risk measure in the literature is the entropic risk measure, where each player has a real valued parameter capturing their risk-averseness. We introduce the extreme risk measure, which corresponds to extreme cases of entropic risk measure, where players are either extreme optimists or extreme pessimists. Under extreme risk measure, every player is an extremist: an extreme optimist perceives their reward as the maximum payoff that can be achieved with positive probability, while an extreme pessimist expects the minimum payoff achievable with positive probability. We argue that the extreme risk measure, especially in multi-player graph based settings, is particularly relevant as they can model several real life instances such as interactions between secure systems and potential security threats, or distributed controls for safety critical systems. We prove that RSEs defined with the extreme risk measure are guaranteed to exist when all rewards are non-negative. Furthermore, we prove that the problem of deciding whether a given game contains an RSE that generates risk measures within specified intervals is decidable and NP-complete for our extreme risk measure, and even PTIME-complete when all players are extreme optimists, while that same problem is undecidable using the entropic risk measure or even the classical expected payoff. This establishes, to our knowledge, the first decidable fragment for equilibria in simple stochastic games without restrictions on strategy types or number of players.
Cite as

Léonard Brice, Thomas A. Henzinger, and K. S. Thejaswini. Finding Equilibria: Simpler for Pessimists, Simplest for Optimists. In 50th International Symposium on Mathematical Foundations of Computer Science (MFCS 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 345, pp. 30:1-30:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{brice_et_al:LIPIcs.MFCS.2025.30,
  author =	{Brice, L\'{e}onard and Henzinger, Thomas A. and Thejaswini, K. S.},
  title =	{{Finding Equilibria: Simpler for Pessimists, Simplest for Optimists}},
  booktitle =	{50th International Symposium on Mathematical Foundations of Computer Science (MFCS 2025)},
  pages =	{30:1--30:18},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-388-1},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{345},
  editor =	{Gawrychowski, Pawe{\l} and Mazowiecki, Filip and Skrzypczak, Micha{\l}},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2025.30},
  URN =		{urn:nbn:de:0030-drops-241371},
  doi =		{10.4230/LIPIcs.MFCS.2025.30},
  annote =	{Keywords: Nash equilibria, stochastic games, graph games, risk-sensitive equilibria}
}
Document
DOI: 10.4230/LIPIcs.MFCS.2023.26
Rational Verification for Nash and Subgame-Perfect Equilibria in Graph Games

Authors: Léonard Brice, Jean-François Raskin, and Marie van den Bogaard

Published in: LIPIcs, Volume 272, 48th International Symposium on Mathematical Foundations of Computer Science (MFCS 2023)

Abstract
We study a natural problem about rational behaviors in multiplayer non-zero-sum sequential infinite duration games played on graphs: rational verification, that consists in deciding whether all the rational answers to a given strategy satisfy some specification. We give the complexities of that problem for two major concepts of rationality: Nash equilibria and subgame-perfect equilibria, and for three major classes of payoff functions: energy, discounted-sum, and mean-payoff.
Cite as

Léonard Brice, Jean-François Raskin, and Marie van den Bogaard. Rational Verification for Nash and Subgame-Perfect Equilibria in Graph Games. In 48th International Symposium on Mathematical Foundations of Computer Science (MFCS 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 272, pp. 26:1-26:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{brice_et_al:LIPIcs.MFCS.2023.26,
  author =	{Brice, L\'{e}onard and Raskin, Jean-Fran\c{c}ois and van den Bogaard, Marie},
  title =	{{Rational Verification for Nash and Subgame-Perfect Equilibria in Graph Games}},
  booktitle =	{48th International Symposium on Mathematical Foundations of Computer Science (MFCS 2023)},
  pages =	{26:1--26:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-292-1},
  ISSN =	{1868-8969},
  year =	{2023},
  volume =	{272},
  editor =	{Leroux, J\'{e}r\^{o}me and Lombardy, Sylvain and Peleg, David},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2023.26},
  URN =		{urn:nbn:de:0030-drops-185608},
  doi =		{10.4230/LIPIcs.MFCS.2023.26},
  annote =	{Keywords: Games on graphs, Nash equilibria, subgame-perfect equilibria}
}
Document
Track B: Automata, Logic, Semantics, and Theory of Programming
DOI: 10.4230/LIPIcs.ICALP.2022.116
The Complexity of SPEs in Mean-Payoff Games

Authors: Léonard Brice, Jean-François Raskin, and Marie van den Bogaard

Published in: LIPIcs, Volume 229, 49th International Colloquium on Automata, Languages, and Programming (ICALP 2022)

Abstract
We establish that the subgame perfect equilibrium (SPE) threshold problem for mean-payoff games is NP-complete. While the SPE threshold problem was recently shown to be decidable (in doubly exponential time) and NP-hard, its exact worst case complexity was left open.
Cite as

Léonard Brice, Jean-François Raskin, and Marie van den Bogaard. The Complexity of SPEs in Mean-Payoff Games. In 49th International Colloquium on Automata, Languages, and Programming (ICALP 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 229, pp. 116:1-116:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{brice_et_al:LIPIcs.ICALP.2022.116,
  author =	{Brice, L\'{e}onard and Raskin, Jean-Fran\c{c}ois and van den Bogaard, Marie},
  title =	{{The Complexity of SPEs in Mean-Payoff Games}},
  booktitle =	{49th International Colloquium on Automata, Languages, and Programming (ICALP 2022)},
  pages =	{116:1--116:20},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-235-8},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{229},
  editor =	{Boja\'{n}czyk, Miko{\l}aj and Merelli, Emanuela and Woodruff, David P.},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2022.116},
  URN =		{urn:nbn:de:0030-drops-164574},
  doi =		{10.4230/LIPIcs.ICALP.2022.116},
  annote =	{Keywords: Games on graphs, subgame-perfect equilibria, mean-payoff objectives}
}
Document
DOI: 10.4230/LIPIcs.CSL.2022.10
On the Complexity of SPEs in Parity Games

Authors: Léonard Brice, Jean-François Raskin, and Marie van den Bogaard

Published in: LIPIcs, Volume 216, 30th EACSL Annual Conference on Computer Science Logic (CSL 2022)

Abstract
We study the complexity of problems related to subgame-perfect equilibria (SPEs) in infinite duration non zero-sum multiplayer games played on finite graphs with parity objectives. We present new complexity results that close gaps in the literature. Our techniques are based on a recent characterization of SPEs in prefix-independent games that is grounded on the notions of requirements and negotiation, and according to which the plays supported by SPEs are exactly the plays consistent with the requirement that is the least fixed point of the negotiation function. The new results are as follows. First, checking that a given requirement is a fixed point of the negotiation function is an NP-complete problem. Second, we show that the SPE constrained existence problem is NP-complete, this problem was previously known to be ExpTime-easy and NP-hard. Third, the SPE constrained existence problem is fixed-parameter tractable when the number of players and of colors are parameters. Fourth, deciding whether some requirement is the least fixed point of the negotiation function is complete for the second level of the Boolean hierarchy. Finally, the SPE-verification problem - that is, the problem of deciding whether there exists a play supported by a SPE that satisfies some LTL formula - is PSpace-complete, this problem was known to be ExpTime-easy and PSpace-hard.
Cite as

Léonard Brice, Jean-François Raskin, and Marie van den Bogaard. On the Complexity of SPEs in Parity Games. In 30th EACSL Annual Conference on Computer Science Logic (CSL 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 216, pp. 10:1-10:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{brice_et_al:LIPIcs.CSL.2022.10,
  author =	{Brice, L\'{e}onard and Raskin, Jean-Fran\c{c}ois and van den Bogaard, Marie},
  title =	{{On the Complexity of SPEs in Parity Games}},
  booktitle =	{30th EACSL Annual Conference on Computer Science Logic (CSL 2022)},
  pages =	{10:1--10:17},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-218-1},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{216},
  editor =	{Manea, Florin and Simpson, Alex},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CSL.2022.10},
  URN =		{urn:nbn:de:0030-drops-157306},
  doi =		{10.4230/LIPIcs.CSL.2022.10},
  annote =	{Keywords: Games on graphs, subgame-perfect equilibria, parity objectives}
}
Document
DOI: 10.4230/LIPIcs.CONCUR.2021.8
Subgame-Perfect Equilibria in Mean-Payoff Games

Authors: Léonard Brice, Jean-François Raskin, and Marie van den Bogaard

Published in: LIPIcs, Volume 203, 32nd International Conference on Concurrency Theory (CONCUR 2021)

Abstract
In this paper, we provide an effective characterization of all the subgame-perfect equilibria in infinite duration games played on finite graphs with mean-payoff objectives. To this end, we introduce the notion of requirement, and the notion of negotiation function. We establish that the plays that are supported by SPEs are exactly those that are consistent with the least fixed point of the negotiation function. Finally, we show that the negotiation function is piecewise linear, and can be analyzed using the linear algebraic tool box. As a corollary, we prove the decidability of the SPE constrained existence problem, whose status was left open in the literature.
Cite as

Léonard Brice, Jean-François Raskin, and Marie van den Bogaard. Subgame-Perfect Equilibria in Mean-Payoff Games. In 32nd International Conference on Concurrency Theory (CONCUR 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 203, pp. 8:1-8:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{brice_et_al:LIPIcs.CONCUR.2021.8,
  author =	{Brice, L\'{e}onard and Raskin, Jean-Fran\c{c}ois and van den Bogaard, Marie},
  title =	{{Subgame-Perfect Equilibria in Mean-Payoff Games}},
  booktitle =	{32nd International Conference on Concurrency Theory (CONCUR 2021)},
  pages =	{8:1--8:17},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-203-7},
  ISSN =	{1868-8969},
  year =	{2021},
  volume =	{203},
  editor =	{Haddad, Serge and Varacca, Daniele},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CONCUR.2021.8},
  URN =		{urn:nbn:de:0030-drops-143854},
  doi =		{10.4230/LIPIcs.CONCUR.2021.8},
  annote =	{Keywords: Games on graphs, subgame-perfect equilibria, mean-payoff objectives.}
}
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