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Documents authored by Bruyere, Veronique

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Bruyère, Véronique

Document
DOI: 10.4230/LIPIcs.CONCUR.2024.14
As Soon as Possible but Rationally

Authors: Véronique Bruyère, Christophe Grandmont, and Jean-François Raskin

Published in: LIPIcs, Volume 311, 35th International Conference on Concurrency Theory (CONCUR 2024)

Abstract
This paper addresses complexity problems in rational verification and synthesis for multi-player games played on weighted graphs, where the objective of each player is to minimize the cost of reaching a specific set of target vertices. In these games, one player, referred to as the system, declares his strategy upfront. The other players, composing the environment, then rationally make their moves according to their objectives. The rational behavior of these responding players is captured through two models: they opt for strategies that either represent a Nash equilibrium or lead to a play with a Pareto-optimal cost tuple.
Cite as

Véronique Bruyère, Christophe Grandmont, and Jean-François Raskin. As Soon as Possible but Rationally. In 35th International Conference on Concurrency Theory (CONCUR 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 311, pp. 14:1-14:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{bruyere_et_al:LIPIcs.CONCUR.2024.14,
  author =	{Bruy\`{e}re, V\'{e}ronique and Grandmont, Christophe and Raskin, Jean-Fran\c{c}ois},
  title =	{{As Soon as Possible but Rationally}},
  booktitle =	{35th International Conference on Concurrency Theory (CONCUR 2024)},
  pages =	{14:1--14:20},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-339-3},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{311},
  editor =	{Majumdar, Rupak and Silva, Alexandra},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CONCUR.2024.14},
  URN =		{urn:nbn:de:0030-drops-207869},
  doi =		{10.4230/LIPIcs.CONCUR.2024.14},
  annote =	{Keywords: Games played on graphs, rational verification, rational synthesis, Nash equilibrium, Pareto-optimality, quantitative reachability objectives}
}
Document
DOI: 10.4230/LIPIcs.CONCUR.2022.33
Pareto-Rational Verification

Authors: Véronique Bruyère, Jean-François Raskin, and Clément Tamines

Published in: LIPIcs, Volume 243, 33rd International Conference on Concurrency Theory (CONCUR 2022)

Abstract
We study the rational verification problem which consists in verifying the correctness of a system executing in an environment that is assumed to behave rationally. We consider the model of rationality in which the environment only executes behaviors that are Pareto-optimal with regard to its set of objectives, given the behavior of the system (which is committed in advance of any interaction). We examine two ways of specifying this behavior, first by means of a deterministic Moore machine, and then by lifting its determinism. In the latter case the machine may embed several different behaviors for the system, and the universal rational verification problem aims at verifying that all of them are correct when the environment is rational. For parity objectives, we prove that the Pareto-rational verification problem is co-NP-complete and that its universal version is in PSPACE and both NP-hard and co-NP-hard. For Boolean Büchi objectives, the former problem is Π₂𝖯-complete and the latter is PSPACE-complete. We also study the case where the objectives are expressed using LTL formulas and show that the first problem is PSPACE-complete, and that the second is 2EXPTIME-complete. Both problems are also shown to be fixed-parameter tractable for parity and Boolean Büchi objectives.
Cite as

Véronique Bruyère, Jean-François Raskin, and Clément Tamines. Pareto-Rational Verification. In 33rd International Conference on Concurrency Theory (CONCUR 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 243, pp. 33:1-33:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{bruyere_et_al:LIPIcs.CONCUR.2022.33,
  author =	{Bruy\`{e}re, V\'{e}ronique and Raskin, Jean-Fran\c{c}ois and Tamines, Cl\'{e}ment},
  title =	{{Pareto-Rational Verification}},
  booktitle =	{33rd International Conference on Concurrency Theory (CONCUR 2022)},
  pages =	{33:1--33:20},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-246-4},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{243},
  editor =	{Klin, Bartek and Lasota, S{\l}awomir 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.CONCUR.2022.33},
  URN =		{urn:nbn:de:0030-drops-170968},
  doi =		{10.4230/LIPIcs.CONCUR.2022.33},
  annote =	{Keywords: Rational verification, Model-checking, Pareto-optimality, \omega-regular objectives}
}
Document
DOI: 10.4230/LIPIcs.CONCUR.2021.27
Stackelberg-Pareto Synthesis

Authors: Véronique Bruyère, Jean-François Raskin, and Clément Tamines

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

Abstract
In this paper, we study the framework of two-player Stackelberg games played on graphs in which Player 0 announces a strategy and Player 1 responds rationally with a strategy that is an optimal response. While it is usually assumed that Player 1 has a single objective, we consider here the new setting where he has several. In this context, after responding with his strategy, Player 1 gets a payoff in the form of a vector of Booleans corresponding to his satisfied objectives. Rationality of Player 1 is encoded by the fact that his response must produce a Pareto-optimal payoff given the strategy of Player 0. We study the Stackelberg-Pareto Synthesis problem which asks whether Player 0 can announce a strategy which satisfies his objective, whatever the rational response of Player 1. For games in which objectives are either all parity or all reachability objectives, we show that this problem is fixed-parameter tractable and NEXPTIME-complete. This problem is already NP-complete in the simple case of reachability objectives and graphs that are trees.
Cite as

Véronique Bruyère, Jean-François Raskin, and Clément Tamines. Stackelberg-Pareto Synthesis. In 32nd International Conference on Concurrency Theory (CONCUR 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 203, pp. 27:1-27:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{bruyere_et_al:LIPIcs.CONCUR.2021.27,
  author =	{Bruy\`{e}re, V\'{e}ronique and Raskin, Jean-Fran\c{c}ois and Tamines, Cl\'{e}ment},
  title =	{{Stackelberg-Pareto Synthesis}},
  booktitle =	{32nd International Conference on Concurrency Theory (CONCUR 2021)},
  pages =	{27:1--27:17},
  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.2021.27},
  URN =		{urn:nbn:de:0030-drops-144040},
  doi =		{10.4230/LIPIcs.CONCUR.2021.27},
  annote =	{Keywords: Stackelberg non-zero sum games played on graphs, synthesis, parity objectives}
}
Document
DOI: 10.4230/LIPIcs.CONCUR.2019.13
The Complexity of Subgame Perfect Equilibria in Quantitative Reachability Games

Authors: Thomas Brihaye, Véronique Bruyère, Aline Goeminne, Jean-François Raskin, and Marie van den Bogaard

Published in: LIPIcs, Volume 140, 30th International Conference on Concurrency Theory (CONCUR 2019)

Abstract
We study multiplayer quantitative reachability games played on a finite directed graph, where the objective of each player is to reach his target set of vertices as quickly as possible. Instead of the well-known notion of Nash equilibrium (NE), we focus on the notion of subgame perfect equilibrium (SPE), a refinement of NE well-suited in the framework of games played on graphs. It is known that there always exists an SPE in quantitative reachability games and that the constrained existence problem is decidable. We here prove that this problem is PSPACE-complete. To obtain this result, we propose a new algorithm that iteratively builds a set of constraints characterizing the set of SPE outcomes in quantitative reachability games. This set of constraints is obtained by iterating an operator that reinforces the constraints up to obtaining a fixpoint. With this fixpoint, the set of SPE outcomes can be represented by a finite graph of size at most exponential. A careful inspection of the computation allows us to establish PSPACE membership.
Cite as

Thomas Brihaye, Véronique Bruyère, Aline Goeminne, Jean-François Raskin, and Marie van den Bogaard. The Complexity of Subgame Perfect Equilibria in Quantitative Reachability Games. In 30th International Conference on Concurrency Theory (CONCUR 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 140, pp. 13:1-13:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{brihaye_et_al:LIPIcs.CONCUR.2019.13,
  author =	{Brihaye, Thomas and Bruy\`{e}re, V\'{e}ronique and Goeminne, Aline and Raskin, Jean-Fran\c{c}ois and van den Bogaard, Marie},
  title =	{{The Complexity of Subgame Perfect Equilibria in Quantitative Reachability Games}},
  booktitle =	{30th International Conference on Concurrency Theory (CONCUR 2019)},
  pages =	{13:1--13:16},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-121-4},
  ISSN =	{1868-8969},
  year =	{2019},
  volume =	{140},
  editor =	{Fokkink, Wan and van Glabbeek, Rob},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CONCUR.2019.13},
  URN =		{urn:nbn:de:0030-drops-109153},
  doi =		{10.4230/LIPIcs.CONCUR.2019.13},
  annote =	{Keywords: multiplayer non-zero-sum games played on graphs, quantitative reachability objectives, subgame perfect equilibria, constrained existence problem}
}
Document
DOI: 10.4230/LIPIcs.CONCUR.2019.21
Energy Mean-Payoff Games

Authors: Véronique Bruyère, Quentin Hautem, Mickael Randour, and Jean-François Raskin

Published in: LIPIcs, Volume 140, 30th International Conference on Concurrency Theory (CONCUR 2019)

Abstract
In this paper, we study one-player and two-player energy mean-payoff games. Energy mean-payoff games are games of infinite duration played on a finite graph with edges labeled by 2-dimensional weight vectors. The objective of the first player (the protagonist) is to satisfy an energy objective on the first dimension and a mean-payoff objective on the second dimension. We show that optimal strategies for the first player may require infinite memory while optimal strategies for the second player (the antagonist) do not require memory. In the one-player case (where only the first player has choices), the problem of deciding who is the winner can be solved in polynomial time while for the two-player case we show co-NP membership and we give effective constructions for the infinite-memory optimal strategies of the protagonist.
Cite as

Véronique Bruyère, Quentin Hautem, Mickael Randour, and Jean-François Raskin. Energy Mean-Payoff Games. In 30th International Conference on Concurrency Theory (CONCUR 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 140, pp. 21:1-21:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{bruyere_et_al:LIPIcs.CONCUR.2019.21,
  author =	{Bruy\`{e}re, V\'{e}ronique and Hautem, Quentin and Randour, Mickael and Raskin, Jean-Fran\c{c}ois},
  title =	{{Energy Mean-Payoff Games}},
  booktitle =	{30th International Conference on Concurrency Theory (CONCUR 2019)},
  pages =	{21:1--21:17},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-121-4},
  ISSN =	{1868-8969},
  year =	{2019},
  volume =	{140},
  editor =	{Fokkink, Wan and van Glabbeek, Rob},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CONCUR.2019.21},
  URN =		{urn:nbn:de:0030-drops-109239},
  doi =		{10.4230/LIPIcs.CONCUR.2019.21},
  annote =	{Keywords: two-player zero-sum games played on graphs, energy and mean-payoff objectives, complexity study and construction of optimal strategies}
}
Document
DOI: 10.4230/LIPIcs.CONCUR.2018.29
Parameterized complexity of games with monotonically ordered omega-regular objectives

Authors: Véronique Bruyère, Quentin Hautem, and Jean-François Raskin

Published in: LIPIcs, Volume 118, 29th International Conference on Concurrency Theory (CONCUR 2018)

Abstract
In recent years, two-player zero-sum games with multiple objectives have received a lot of interest as a model for the synthesis of complex reactive systems. In this framework, Player 1 wins if he can ensure that all objectives are satisfied against any behavior of Player 2. When this is not possible to satisfy all the objectives at once, an alternative is to use some preorder on the objectives according to which subset of objectives Player 1 wants to satisfy. For example, it is often natural to provide more significance to one objective over another, a situation that can be modelled with lexicographically ordered objectives for instance. Inspired by recent work on concurrent games with multiple omega-regular objectives by Bouyer et al., we investigate in detail turned-based games with monotonically ordered and omega-regular objectives. We study the threshold problem which asks whether player 1 can ensure a payoff greater than or equal to a given threshold w.r.t. a given monotonic preorder. As the number of objectives is usually much smaller than the size of the game graph, we provide a parametric complexity analysis and we show that our threshold problem is in FPT for all monotonic preorders and all classical types of omega-regular objectives. We also provide polynomial time algorithms for Büchi, coBüchi and explicit Muller objectives for a large subclass of monotonic preorders that includes among others the lexicographic preorder. In the particular case of lexicographic preorder, we also study the complexity of computing the values and the memory requirements of optimal strategies.
Cite as

Véronique Bruyère, Quentin Hautem, and Jean-François Raskin. Parameterized complexity of games with monotonically ordered omega-regular objectives. In 29th International Conference on Concurrency Theory (CONCUR 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 118, pp. 29:1-29:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{bruyere_et_al:LIPIcs.CONCUR.2018.29,
  author =	{Bruy\`{e}re, V\'{e}ronique and Hautem, Quentin and Raskin, Jean-Fran\c{c}ois},
  title =	{{Parameterized complexity of games with monotonically ordered omega-regular objectives}},
  booktitle =	{29th International Conference on Concurrency Theory (CONCUR 2018)},
  pages =	{29:1--29:16},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-087-3},
  ISSN =	{1868-8969},
  year =	{2018},
  volume =	{118},
  editor =	{Schewe, Sven and Zhang, Lijun},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CONCUR.2018.29},
  URN =		{urn:nbn:de:0030-drops-95673},
  doi =		{10.4230/LIPIcs.CONCUR.2018.29},
  annote =	{Keywords: two-player zero-sum games played on graphs, omega-regular objectives, ordered objectives, parameterized complexity}
}
Document
DOI: 10.4230/LIPIcs.CONCUR.2016.11
On the Complexity of Heterogeneous Multidimensional Games

Authors: Veronique Bruyere, Quentin Hautem, and Jean-Francois Raskin

Published in: LIPIcs, Volume 59, 27th International Conference on Concurrency Theory (CONCUR 2016)

Abstract
We study two-player zero-sum turn-based games played on multidimensional weighted graphs with heterogeneous quantitative objectives. Our objectives are defined starting from the measures Inf, Sup, LimInf, and LimSup of the weights seen along the play, as well as on the window mean-payoff (WMP) measure recently introduced in [Krishnendu,Doyen,Randour,Raskin, Inf. Comput., 2015]. Whereas multidimensional games with Boolean combinations of classical mean-payoff objectives are undecidable [Velner, FOSSACS, 2015], we show that CNF/DNF Boolean combinations for heterogeneous measures taken among {WMP, Inf, Sup, LimInf, LimSup} lead to EXPTIME-completeness with exponential memory strategies for both players. We also identify several interesting fragments with better complexities and memory requirements, and show that some of them are solvable in PTIME.
Cite as

Veronique Bruyere, Quentin Hautem, and Jean-Francois Raskin. On the Complexity of Heterogeneous Multidimensional Games. In 27th International Conference on Concurrency Theory (CONCUR 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 59, pp. 11:1-11:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)

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@InProceedings{bruyere_et_al:LIPIcs.CONCUR.2016.11,
  author =	{Bruyere, Veronique and Hautem, Quentin and Raskin, Jean-Francois},
  title =	{{On the Complexity of Heterogeneous Multidimensional Games}},
  booktitle =	{27th International Conference on Concurrency Theory (CONCUR 2016)},
  pages =	{11:1--11:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-017-0},
  ISSN =	{1868-8969},
  year =	{2016},
  volume =	{59},
  editor =	{Desharnais, Jos\'{e}e and Jagadeesan, Radha},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CONCUR.2016.11},
  URN =		{urn:nbn:de:0030-drops-61618},
  doi =		{10.4230/LIPIcs.CONCUR.2016.11},
  annote =	{Keywords: wo-player zero-sum games played on graphs, quantitative objectives, multidimensional heterogeneous objectives}
}
Document
DOI: 10.4230/LIPIcs.CSL.2015.504
Weak Subgame Perfect Equilibria and their Application to Quantitative Reachability

Authors: Thomas Brihaye, Véronique Bruyère, Noémie Meunier, and Jean-Francois Raskin

Published in: LIPIcs, Volume 41, 24th EACSL Annual Conference on Computer Science Logic (CSL 2015)

Abstract
We study n-player turn-based games played on a finite directed graph. For each play, the players have to pay a cost that they want to minimize. Instead of the well-known notion of Nash equilibrium (NE), we focus on the notion of subgame perfect equilibrium (SPE), a refinement of NE well-suited in the framework of games played on graphs. We also study natural variants of SPE, named weak (resp. very weak) SPE, where players who deviate cannot use the full class of strategies but only a subclass with a finite number of (resp. a unique) deviation step(s). Our results are threefold. Firstly, we characterize in the form of a Folk theorem the set of all plays that are the outcome of a weak SPE. Secondly, for the class of quantitative reachability games, we prove the existence of a finite-memory SPE and provide an algorithm for computing it (only existence was known with no information regarding the memory). Moreover, we show that the existence of a constrained SPE, i.e. an SPE such that each player pays a cost less than a given constant, can be decided. The proofs rely on our Folk theorem for weak SPEs (which coincide with SPEs in the case of quantitative reachability games) and on the decidability of MSO logic on infinite words. Finally with similar techniques, we provide a second general class of games for which the existence of a (constrained) weak SPE is decidable.
Cite as

Thomas Brihaye, Véronique Bruyère, Noémie Meunier, and Jean-Francois Raskin. Weak Subgame Perfect Equilibria and their Application to Quantitative Reachability. In 24th EACSL Annual Conference on Computer Science Logic (CSL 2015). Leibniz International Proceedings in Informatics (LIPIcs), Volume 41, pp. 504-518, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2015)

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@InProceedings{brihaye_et_al:LIPIcs.CSL.2015.504,
  author =	{Brihaye, Thomas and Bruy\`{e}re, V\'{e}ronique and Meunier, No\'{e}mie and Raskin, Jean-Francois},
  title =	{{Weak Subgame Perfect Equilibria and their Application to Quantitative Reachability}},
  booktitle =	{24th EACSL Annual Conference on Computer Science Logic (CSL 2015)},
  pages =	{504--518},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-939897-90-3},
  ISSN =	{1868-8969},
  year =	{2015},
  volume =	{41},
  editor =	{Kreutzer, Stephan},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CSL.2015.504},
  URN =		{urn:nbn:de:0030-drops-54345},
  doi =		{10.4230/LIPIcs.CSL.2015.504},
  annote =	{Keywords: multi-player games on graphs, quantitative objectives, Nash equilibrium, subgame perfect equilibrium, quantitative reachability}
}
Document
DOI: 10.4230/LIPIcs.STACS.2014.199
Meet Your Expectations With Guarantees: Beyond Worst-Case Synthesis in Quantitative Games

Authors: Véronique Bruyère, Emmanuel Filiot, Mickael Randour, and Jean-François Raskin

Published in: LIPIcs, Volume 25, 31st International Symposium on Theoretical Aspects of Computer Science (STACS 2014)

Abstract
Classical analysis of two-player quantitative games involves an adversary (modeling the environment of the system) which is purely antagonistic and asks for strict guarantees while Markov decision processes model systems facing a purely randomized environment: the aim is then to optimize the expected payoff, with no guarantee on individual outcomes. We introduce the beyond worst-case synthesis problem, which is to construct strategies that guarantee some quantitative requirement in the worst-case while providing an higher expected value against a particular stochastic model of the environment given as input. We consider both the mean-payoff value problem and the shortest path problem. In both cases, we show how to decide the existence of finite-memory strategies satisfying the problem and how to synthesize one if one exists. We establish algorithms and we study complexity bounds and memory requirements.
Cite as

Véronique Bruyère, Emmanuel Filiot, Mickael Randour, and Jean-François Raskin. Meet Your Expectations With Guarantees: Beyond Worst-Case Synthesis in Quantitative Games. In 31st International Symposium on Theoretical Aspects of Computer Science (STACS 2014). Leibniz International Proceedings in Informatics (LIPIcs), Volume 25, pp. 199-213, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2014)

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@InProceedings{bruyere_et_al:LIPIcs.STACS.2014.199,
  author =	{Bruy\`{e}re, V\'{e}ronique and Filiot, Emmanuel and Randour, Mickael and Raskin, Jean-Fran\c{c}ois},
  title =	{{Meet Your Expectations With Guarantees: Beyond Worst-Case Synthesis in Quantitative Games}},
  booktitle =	{31st International Symposium on Theoretical Aspects of Computer Science (STACS 2014)},
  pages =	{199--213},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-939897-65-1},
  ISSN =	{1868-8969},
  year =	{2014},
  volume =	{25},
  editor =	{Mayr, Ernst W. and Portier, Natacha},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2014.199},
  URN =		{urn:nbn:de:0030-drops-44589},
  doi =		{10.4230/LIPIcs.STACS.2014.199},
  annote =	{Keywords: two-player games on graphs, Markov decision processes, quantitative objectives, synthesis, worst-case and expected value, mean-payoff, shortest path}
}

Bruyere, Veronique

Document
DOI: 10.4230/LIPIcs.CONCUR.2024.14
As Soon as Possible but Rationally

Authors: Véronique Bruyère, Christophe Grandmont, and Jean-François Raskin

Published in: LIPIcs, Volume 311, 35th International Conference on Concurrency Theory (CONCUR 2024)

Abstract
This paper addresses complexity problems in rational verification and synthesis for multi-player games played on weighted graphs, where the objective of each player is to minimize the cost of reaching a specific set of target vertices. In these games, one player, referred to as the system, declares his strategy upfront. The other players, composing the environment, then rationally make their moves according to their objectives. The rational behavior of these responding players is captured through two models: they opt for strategies that either represent a Nash equilibrium or lead to a play with a Pareto-optimal cost tuple.
Cite as

Véronique Bruyère, Christophe Grandmont, and Jean-François Raskin. As Soon as Possible but Rationally. In 35th International Conference on Concurrency Theory (CONCUR 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 311, pp. 14:1-14:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{bruyere_et_al:LIPIcs.CONCUR.2024.14,
  author =	{Bruy\`{e}re, V\'{e}ronique and Grandmont, Christophe and Raskin, Jean-Fran\c{c}ois},
  title =	{{As Soon as Possible but Rationally}},
  booktitle =	{35th International Conference on Concurrency Theory (CONCUR 2024)},
  pages =	{14:1--14:20},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-339-3},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{311},
  editor =	{Majumdar, Rupak and Silva, Alexandra},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CONCUR.2024.14},
  URN =		{urn:nbn:de:0030-drops-207869},
  doi =		{10.4230/LIPIcs.CONCUR.2024.14},
  annote =	{Keywords: Games played on graphs, rational verification, rational synthesis, Nash equilibrium, Pareto-optimality, quantitative reachability objectives}
}
Document
DOI: 10.4230/LIPIcs.CONCUR.2022.33
Pareto-Rational Verification

Authors: Véronique Bruyère, Jean-François Raskin, and Clément Tamines

Published in: LIPIcs, Volume 243, 33rd International Conference on Concurrency Theory (CONCUR 2022)

Abstract
We study the rational verification problem which consists in verifying the correctness of a system executing in an environment that is assumed to behave rationally. We consider the model of rationality in which the environment only executes behaviors that are Pareto-optimal with regard to its set of objectives, given the behavior of the system (which is committed in advance of any interaction). We examine two ways of specifying this behavior, first by means of a deterministic Moore machine, and then by lifting its determinism. In the latter case the machine may embed several different behaviors for the system, and the universal rational verification problem aims at verifying that all of them are correct when the environment is rational. For parity objectives, we prove that the Pareto-rational verification problem is co-NP-complete and that its universal version is in PSPACE and both NP-hard and co-NP-hard. For Boolean Büchi objectives, the former problem is Π₂𝖯-complete and the latter is PSPACE-complete. We also study the case where the objectives are expressed using LTL formulas and show that the first problem is PSPACE-complete, and that the second is 2EXPTIME-complete. Both problems are also shown to be fixed-parameter tractable for parity and Boolean Büchi objectives.
Cite as

Véronique Bruyère, Jean-François Raskin, and Clément Tamines. Pareto-Rational Verification. In 33rd International Conference on Concurrency Theory (CONCUR 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 243, pp. 33:1-33:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{bruyere_et_al:LIPIcs.CONCUR.2022.33,
  author =	{Bruy\`{e}re, V\'{e}ronique and Raskin, Jean-Fran\c{c}ois and Tamines, Cl\'{e}ment},
  title =	{{Pareto-Rational Verification}},
  booktitle =	{33rd International Conference on Concurrency Theory (CONCUR 2022)},
  pages =	{33:1--33:20},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-246-4},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{243},
  editor =	{Klin, Bartek and Lasota, S{\l}awomir 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.CONCUR.2022.33},
  URN =		{urn:nbn:de:0030-drops-170968},
  doi =		{10.4230/LIPIcs.CONCUR.2022.33},
  annote =	{Keywords: Rational verification, Model-checking, Pareto-optimality, \omega-regular objectives}
}
Document
DOI: 10.4230/LIPIcs.CONCUR.2021.27
Stackelberg-Pareto Synthesis

Authors: Véronique Bruyère, Jean-François Raskin, and Clément Tamines

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

Abstract
In this paper, we study the framework of two-player Stackelberg games played on graphs in which Player 0 announces a strategy and Player 1 responds rationally with a strategy that is an optimal response. While it is usually assumed that Player 1 has a single objective, we consider here the new setting where he has several. In this context, after responding with his strategy, Player 1 gets a payoff in the form of a vector of Booleans corresponding to his satisfied objectives. Rationality of Player 1 is encoded by the fact that his response must produce a Pareto-optimal payoff given the strategy of Player 0. We study the Stackelberg-Pareto Synthesis problem which asks whether Player 0 can announce a strategy which satisfies his objective, whatever the rational response of Player 1. For games in which objectives are either all parity or all reachability objectives, we show that this problem is fixed-parameter tractable and NEXPTIME-complete. This problem is already NP-complete in the simple case of reachability objectives and graphs that are trees.
Cite as

Véronique Bruyère, Jean-François Raskin, and Clément Tamines. Stackelberg-Pareto Synthesis. In 32nd International Conference on Concurrency Theory (CONCUR 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 203, pp. 27:1-27:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{bruyere_et_al:LIPIcs.CONCUR.2021.27,
  author =	{Bruy\`{e}re, V\'{e}ronique and Raskin, Jean-Fran\c{c}ois and Tamines, Cl\'{e}ment},
  title =	{{Stackelberg-Pareto Synthesis}},
  booktitle =	{32nd International Conference on Concurrency Theory (CONCUR 2021)},
  pages =	{27:1--27:17},
  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.2021.27},
  URN =		{urn:nbn:de:0030-drops-144040},
  doi =		{10.4230/LIPIcs.CONCUR.2021.27},
  annote =	{Keywords: Stackelberg non-zero sum games played on graphs, synthesis, parity objectives}
}
Document
DOI: 10.4230/LIPIcs.CONCUR.2019.13
The Complexity of Subgame Perfect Equilibria in Quantitative Reachability Games

Authors: Thomas Brihaye, Véronique Bruyère, Aline Goeminne, Jean-François Raskin, and Marie van den Bogaard

Published in: LIPIcs, Volume 140, 30th International Conference on Concurrency Theory (CONCUR 2019)

Abstract
We study multiplayer quantitative reachability games played on a finite directed graph, where the objective of each player is to reach his target set of vertices as quickly as possible. Instead of the well-known notion of Nash equilibrium (NE), we focus on the notion of subgame perfect equilibrium (SPE), a refinement of NE well-suited in the framework of games played on graphs. It is known that there always exists an SPE in quantitative reachability games and that the constrained existence problem is decidable. We here prove that this problem is PSPACE-complete. To obtain this result, we propose a new algorithm that iteratively builds a set of constraints characterizing the set of SPE outcomes in quantitative reachability games. This set of constraints is obtained by iterating an operator that reinforces the constraints up to obtaining a fixpoint. With this fixpoint, the set of SPE outcomes can be represented by a finite graph of size at most exponential. A careful inspection of the computation allows us to establish PSPACE membership.
Cite as

Thomas Brihaye, Véronique Bruyère, Aline Goeminne, Jean-François Raskin, and Marie van den Bogaard. The Complexity of Subgame Perfect Equilibria in Quantitative Reachability Games. In 30th International Conference on Concurrency Theory (CONCUR 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 140, pp. 13:1-13:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{brihaye_et_al:LIPIcs.CONCUR.2019.13,
  author =	{Brihaye, Thomas and Bruy\`{e}re, V\'{e}ronique and Goeminne, Aline and Raskin, Jean-Fran\c{c}ois and van den Bogaard, Marie},
  title =	{{The Complexity of Subgame Perfect Equilibria in Quantitative Reachability Games}},
  booktitle =	{30th International Conference on Concurrency Theory (CONCUR 2019)},
  pages =	{13:1--13:16},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-121-4},
  ISSN =	{1868-8969},
  year =	{2019},
  volume =	{140},
  editor =	{Fokkink, Wan and van Glabbeek, Rob},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CONCUR.2019.13},
  URN =		{urn:nbn:de:0030-drops-109153},
  doi =		{10.4230/LIPIcs.CONCUR.2019.13},
  annote =	{Keywords: multiplayer non-zero-sum games played on graphs, quantitative reachability objectives, subgame perfect equilibria, constrained existence problem}
}
Document
DOI: 10.4230/LIPIcs.CONCUR.2019.21
Energy Mean-Payoff Games

Authors: Véronique Bruyère, Quentin Hautem, Mickael Randour, and Jean-François Raskin

Published in: LIPIcs, Volume 140, 30th International Conference on Concurrency Theory (CONCUR 2019)

Abstract
In this paper, we study one-player and two-player energy mean-payoff games. Energy mean-payoff games are games of infinite duration played on a finite graph with edges labeled by 2-dimensional weight vectors. The objective of the first player (the protagonist) is to satisfy an energy objective on the first dimension and a mean-payoff objective on the second dimension. We show that optimal strategies for the first player may require infinite memory while optimal strategies for the second player (the antagonist) do not require memory. In the one-player case (where only the first player has choices), the problem of deciding who is the winner can be solved in polynomial time while for the two-player case we show co-NP membership and we give effective constructions for the infinite-memory optimal strategies of the protagonist.
Cite as

Véronique Bruyère, Quentin Hautem, Mickael Randour, and Jean-François Raskin. Energy Mean-Payoff Games. In 30th International Conference on Concurrency Theory (CONCUR 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 140, pp. 21:1-21:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{bruyere_et_al:LIPIcs.CONCUR.2019.21,
  author =	{Bruy\`{e}re, V\'{e}ronique and Hautem, Quentin and Randour, Mickael and Raskin, Jean-Fran\c{c}ois},
  title =	{{Energy Mean-Payoff Games}},
  booktitle =	{30th International Conference on Concurrency Theory (CONCUR 2019)},
  pages =	{21:1--21:17},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-121-4},
  ISSN =	{1868-8969},
  year =	{2019},
  volume =	{140},
  editor =	{Fokkink, Wan and van Glabbeek, Rob},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CONCUR.2019.21},
  URN =		{urn:nbn:de:0030-drops-109239},
  doi =		{10.4230/LIPIcs.CONCUR.2019.21},
  annote =	{Keywords: two-player zero-sum games played on graphs, energy and mean-payoff objectives, complexity study and construction of optimal strategies}
}
Document
DOI: 10.4230/LIPIcs.CONCUR.2018.29
Parameterized complexity of games with monotonically ordered omega-regular objectives

Authors: Véronique Bruyère, Quentin Hautem, and Jean-François Raskin

Published in: LIPIcs, Volume 118, 29th International Conference on Concurrency Theory (CONCUR 2018)

Abstract
In recent years, two-player zero-sum games with multiple objectives have received a lot of interest as a model for the synthesis of complex reactive systems. In this framework, Player 1 wins if he can ensure that all objectives are satisfied against any behavior of Player 2. When this is not possible to satisfy all the objectives at once, an alternative is to use some preorder on the objectives according to which subset of objectives Player 1 wants to satisfy. For example, it is often natural to provide more significance to one objective over another, a situation that can be modelled with lexicographically ordered objectives for instance. Inspired by recent work on concurrent games with multiple omega-regular objectives by Bouyer et al., we investigate in detail turned-based games with monotonically ordered and omega-regular objectives. We study the threshold problem which asks whether player 1 can ensure a payoff greater than or equal to a given threshold w.r.t. a given monotonic preorder. As the number of objectives is usually much smaller than the size of the game graph, we provide a parametric complexity analysis and we show that our threshold problem is in FPT for all monotonic preorders and all classical types of omega-regular objectives. We also provide polynomial time algorithms for Büchi, coBüchi and explicit Muller objectives for a large subclass of monotonic preorders that includes among others the lexicographic preorder. In the particular case of lexicographic preorder, we also study the complexity of computing the values and the memory requirements of optimal strategies.
Cite as

Véronique Bruyère, Quentin Hautem, and Jean-François Raskin. Parameterized complexity of games with monotonically ordered omega-regular objectives. In 29th International Conference on Concurrency Theory (CONCUR 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 118, pp. 29:1-29:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{bruyere_et_al:LIPIcs.CONCUR.2018.29,
  author =	{Bruy\`{e}re, V\'{e}ronique and Hautem, Quentin and Raskin, Jean-Fran\c{c}ois},
  title =	{{Parameterized complexity of games with monotonically ordered omega-regular objectives}},
  booktitle =	{29th International Conference on Concurrency Theory (CONCUR 2018)},
  pages =	{29:1--29:16},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-087-3},
  ISSN =	{1868-8969},
  year =	{2018},
  volume =	{118},
  editor =	{Schewe, Sven and Zhang, Lijun},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CONCUR.2018.29},
  URN =		{urn:nbn:de:0030-drops-95673},
  doi =		{10.4230/LIPIcs.CONCUR.2018.29},
  annote =	{Keywords: two-player zero-sum games played on graphs, omega-regular objectives, ordered objectives, parameterized complexity}
}
Document
DOI: 10.4230/LIPIcs.CONCUR.2016.11
On the Complexity of Heterogeneous Multidimensional Games

Authors: Veronique Bruyere, Quentin Hautem, and Jean-Francois Raskin

Published in: LIPIcs, Volume 59, 27th International Conference on Concurrency Theory (CONCUR 2016)

Abstract
We study two-player zero-sum turn-based games played on multidimensional weighted graphs with heterogeneous quantitative objectives. Our objectives are defined starting from the measures Inf, Sup, LimInf, and LimSup of the weights seen along the play, as well as on the window mean-payoff (WMP) measure recently introduced in [Krishnendu,Doyen,Randour,Raskin, Inf. Comput., 2015]. Whereas multidimensional games with Boolean combinations of classical mean-payoff objectives are undecidable [Velner, FOSSACS, 2015], we show that CNF/DNF Boolean combinations for heterogeneous measures taken among {WMP, Inf, Sup, LimInf, LimSup} lead to EXPTIME-completeness with exponential memory strategies for both players. We also identify several interesting fragments with better complexities and memory requirements, and show that some of them are solvable in PTIME.
Cite as

Veronique Bruyere, Quentin Hautem, and Jean-Francois Raskin. On the Complexity of Heterogeneous Multidimensional Games. In 27th International Conference on Concurrency Theory (CONCUR 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 59, pp. 11:1-11:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)

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@InProceedings{bruyere_et_al:LIPIcs.CONCUR.2016.11,
  author =	{Bruyere, Veronique and Hautem, Quentin and Raskin, Jean-Francois},
  title =	{{On the Complexity of Heterogeneous Multidimensional Games}},
  booktitle =	{27th International Conference on Concurrency Theory (CONCUR 2016)},
  pages =	{11:1--11:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-017-0},
  ISSN =	{1868-8969},
  year =	{2016},
  volume =	{59},
  editor =	{Desharnais, Jos\'{e}e and Jagadeesan, Radha},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CONCUR.2016.11},
  URN =		{urn:nbn:de:0030-drops-61618},
  doi =		{10.4230/LIPIcs.CONCUR.2016.11},
  annote =	{Keywords: wo-player zero-sum games played on graphs, quantitative objectives, multidimensional heterogeneous objectives}
}
Document
DOI: 10.4230/LIPIcs.CSL.2015.504
Weak Subgame Perfect Equilibria and their Application to Quantitative Reachability

Authors: Thomas Brihaye, Véronique Bruyère, Noémie Meunier, and Jean-Francois Raskin

Published in: LIPIcs, Volume 41, 24th EACSL Annual Conference on Computer Science Logic (CSL 2015)

Abstract
We study n-player turn-based games played on a finite directed graph. For each play, the players have to pay a cost that they want to minimize. Instead of the well-known notion of Nash equilibrium (NE), we focus on the notion of subgame perfect equilibrium (SPE), a refinement of NE well-suited in the framework of games played on graphs. We also study natural variants of SPE, named weak (resp. very weak) SPE, where players who deviate cannot use the full class of strategies but only a subclass with a finite number of (resp. a unique) deviation step(s). Our results are threefold. Firstly, we characterize in the form of a Folk theorem the set of all plays that are the outcome of a weak SPE. Secondly, for the class of quantitative reachability games, we prove the existence of a finite-memory SPE and provide an algorithm for computing it (only existence was known with no information regarding the memory). Moreover, we show that the existence of a constrained SPE, i.e. an SPE such that each player pays a cost less than a given constant, can be decided. The proofs rely on our Folk theorem for weak SPEs (which coincide with SPEs in the case of quantitative reachability games) and on the decidability of MSO logic on infinite words. Finally with similar techniques, we provide a second general class of games for which the existence of a (constrained) weak SPE is decidable.
Cite as

Thomas Brihaye, Véronique Bruyère, Noémie Meunier, and Jean-Francois Raskin. Weak Subgame Perfect Equilibria and their Application to Quantitative Reachability. In 24th EACSL Annual Conference on Computer Science Logic (CSL 2015). Leibniz International Proceedings in Informatics (LIPIcs), Volume 41, pp. 504-518, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2015)

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@InProceedings{brihaye_et_al:LIPIcs.CSL.2015.504,
  author =	{Brihaye, Thomas and Bruy\`{e}re, V\'{e}ronique and Meunier, No\'{e}mie and Raskin, Jean-Francois},
  title =	{{Weak Subgame Perfect Equilibria and their Application to Quantitative Reachability}},
  booktitle =	{24th EACSL Annual Conference on Computer Science Logic (CSL 2015)},
  pages =	{504--518},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-939897-90-3},
  ISSN =	{1868-8969},
  year =	{2015},
  volume =	{41},
  editor =	{Kreutzer, Stephan},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CSL.2015.504},
  URN =		{urn:nbn:de:0030-drops-54345},
  doi =		{10.4230/LIPIcs.CSL.2015.504},
  annote =	{Keywords: multi-player games on graphs, quantitative objectives, Nash equilibrium, subgame perfect equilibrium, quantitative reachability}
}
Document
DOI: 10.4230/LIPIcs.STACS.2014.199
Meet Your Expectations With Guarantees: Beyond Worst-Case Synthesis in Quantitative Games

Authors: Véronique Bruyère, Emmanuel Filiot, Mickael Randour, and Jean-François Raskin

Published in: LIPIcs, Volume 25, 31st International Symposium on Theoretical Aspects of Computer Science (STACS 2014)

Abstract
Classical analysis of two-player quantitative games involves an adversary (modeling the environment of the system) which is purely antagonistic and asks for strict guarantees while Markov decision processes model systems facing a purely randomized environment: the aim is then to optimize the expected payoff, with no guarantee on individual outcomes. We introduce the beyond worst-case synthesis problem, which is to construct strategies that guarantee some quantitative requirement in the worst-case while providing an higher expected value against a particular stochastic model of the environment given as input. We consider both the mean-payoff value problem and the shortest path problem. In both cases, we show how to decide the existence of finite-memory strategies satisfying the problem and how to synthesize one if one exists. We establish algorithms and we study complexity bounds and memory requirements.
Cite as

Véronique Bruyère, Emmanuel Filiot, Mickael Randour, and Jean-François Raskin. Meet Your Expectations With Guarantees: Beyond Worst-Case Synthesis in Quantitative Games. In 31st International Symposium on Theoretical Aspects of Computer Science (STACS 2014). Leibniz International Proceedings in Informatics (LIPIcs), Volume 25, pp. 199-213, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2014)

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@InProceedings{bruyere_et_al:LIPIcs.STACS.2014.199,
  author =	{Bruy\`{e}re, V\'{e}ronique and Filiot, Emmanuel and Randour, Mickael and Raskin, Jean-Fran\c{c}ois},
  title =	{{Meet Your Expectations With Guarantees: Beyond Worst-Case Synthesis in Quantitative Games}},
  booktitle =	{31st International Symposium on Theoretical Aspects of Computer Science (STACS 2014)},
  pages =	{199--213},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-939897-65-1},
  ISSN =	{1868-8969},
  year =	{2014},
  volume =	{25},
  editor =	{Mayr, Ernst W. and Portier, Natacha},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2014.199},
  URN =		{urn:nbn:de:0030-drops-44589},
  doi =		{10.4230/LIPIcs.STACS.2014.199},
  annote =	{Keywords: two-player games on graphs, Markov decision processes, quantitative objectives, synthesis, worst-case and expected value, mean-payoff, shortest path}
}
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