Skip to main content
SpringerLink
Log in

Logics for Reasoning About Strategic Abilities in Multi-player Games

  • Chapter
  • First Online:
Models of Strategic Reasoning

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 8972))

Abstract

We introduce and discuss basic concepts, ideas, and logical formalisms used for reasoning about strategic abilities in multi-player games. In particular, we present concurrent game models and the alternating time temporal logic \(\mathsf {ATL}^{*}\) and its fragment \(\mathsf {ATL}\). We discuss variations of the language and semantics of \(\mathsf {ATL}^{*}\) that take into account the limitations and complications arising from incomplete information, perfect or imperfect memory of players, reasoning within dynamically changing strategy contexts, or using stronger, constructive concepts of strategy. Finally, we briefly summarize some technical results regarding decision problems for some variants of \(\mathsf {ATL}\).

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 39.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Notes

  1. 1.

    While ‘strategy’ is commonly defined as a complete conditional plan, we cannot resist noting here John Lennon’s famous quote: “Life is what happens while you are busy making other plans”.

  2. 2.

    Roughly corresponding to ‘first-person deliberation’ vs. ‘third-person assessment of strategic action in games’ in van Benthem [16].

  3. 3.

    We do, however, discuss briefly in Sect. 5.2 how some concepts of rationality can be expressed in logical languages considered here.

  4. 4.

    We have no strong reason for this finiteness assumption, other than common sense and technical convenience.

  5. 5.

    We assume that there is an ordering on \(\mathbb {A}\mathrm {gt} \) which is respected in the definition of tuples etc.

  6. 6.

    Coalitional effectivity can be regarded as a concept of cooperative game theory from the internal perspective of the coalition, but from the external perspective of the other players it becomes a concept of non-cooperative game theory. We will not dwell into this apparent duality here.

  7. 7.

    This representation theorem was first proved in Pauly [71] for so called “playable” effectivity functions, without the Determinacy requirement. It has been recently shown in [45] that, for games with infinite outcome spaces, “playability” is not sufficient. The Determinacy condition was identified and added to define “truly playable” effectivity functions and prove a correct version of the representation theorem.

  8. 8.

    The set of states is assumed finite in [8] but that restriction is not necessary for our purposes. In Sect. 6.3 we even rely on the fact that the set of states can be infinite.

  9. 9.

    We use the terms objective and goal of a coalition A as synonyms, to indicate the subformula \(\gamma \) of the formula \(\langle \!\langle {A}\rangle \!\rangle _{_{\! }} \gamma \). In doing so, we ignore the issue of whether agents may have (common) goals, how these goals arise, etc.

  10. 10.

    As proved in [42, 43], under natural assumptions the two semantics are equivalent.

  11. 11.

    Of course, \(\mathcal {G}\) is definable as \(\lnot \mathcal {F}\lnot \), but keeping it as a primitive operator in the language is convenient when defining the sublanguage \({\mathsf {ATL}_{\mathsf {}}}\).

  12. 12.

    This is the famous “have the cake or eat it” dilemma. One can keep being able to eat the cake, but only by never eating the cake.

  13. 13.

    Traditionally in game theory two different terms are used to indicate lack of information: “incomplete” and “imperfect”. Usually, the former refers to uncertainties about the game structure and rules, while the latter refers to uncertainties about the history, current state, etc. of the specific play of the game. Here we will use the latter term in about the same sense, whereas we will use “incomplete information” more loosely, to indicate any possible relevant lack of information.

  14. 14.

    This corresponds to the notion of synchronous perfect recall according to [41].

  15. 15.

    Note that uniformity of a joint strategy is based on individual epistemic relations, rather than any collective epistemic relation (representing, e.g., A’s common, mutual, or distributed knowledge, cf. Sect. 6.4). This is because executability of agent a’s choices within strategy \(s_A\) should only depend on what a can observe and deduce.

    Alternative semantics where uniformity of joint strategies is defined in terms of knowledge of the group as a whole have been discussed in [36, 48].

  16. 16.

    The equivalence between \(\langle \!\langle {A}\rangle \!\rangle _{_{\! }}\mathcal {F}\, \varphi \) and \(\varphi \vee \langle \!\langle {A}\rangle \!\rangle _{_{\! }}\mathcal {X}\, \langle \!\langle {A}\rangle \!\rangle _{_{\! }}\mathcal {F}\, \varphi \) is extremely important since it provides a fixpoint characterization of \(\langle \!\langle {A}\rangle \!\rangle _{_{\! }}\mathcal {F}\, \varphi \). The fact that \(\langle \!\langle {A}\rangle \!\rangle _{_{\! }}\mathcal {F}\, \varphi \leftrightarrow \varphi \vee \langle \!\langle {A}\rangle \!\rangle _{_{\! }}\mathcal {X}\, \langle \!\langle {A}\rangle \!\rangle _{_{\! }}\mathcal {F}\, \varphi \) is not valid under incomplete information is one of the main reasons why constructing verification and satisfiability checking algorithms is so difficult for incomplete information strategies.

  17. 17.

    The formula expresses decomposability of conjuctive goals: being able to achieve \(\varphi _1\wedge \varphi _2\) must be equivalent to having a strategy that achieves first \(\varphi _1\) and \(\varphi _2\), or vice versa. It is easy to see that the requirement holds for agents with perfect memory, but not for ones bound to use memoryless strategies (and hence to play the same action whenever the game comes back to a previously visited state).

  18. 18.

    The formula states that, if A has an opening move and a follow-up strategy to achieve eventually \(\varphi \), then both strategies can be combined into a single strategy enforcing eventually \(\varphi \) already from the initial state.

  19. 19.

    For a formal argument, see [2, 57].

  20. 20.

    We cannot replace \(\varphi \,\mathcal {U}_{{}}\,\varphi \) by \(\varphi \) when the latter is a path formula, as then \(\langle \!\langle {\emptyset }\rangle \!\rangle _{_{\! }}\varphi \) would not be a formula of \(\mathsf {CSL}\).

  21. 21.

    Yet, the \(\mathsf {SSTIT}\) semantic structures relate quite naturally to path effectivity models introduced and characterized in [44], and these could provide a more feasible semantics for S\(\mathsf {STIT}\).

References

  1. Abdou, J., Keiding, H.: Effectivity Functions in Social Choice Theory. Kluwer, Netherlands (1991)

    Book  Google Scholar 

  2. Ågotnes, T.: Action and knowledge in alternating-time temporal logic. Synthese 149(2), 377–409 (2006). Section on Knowledge, Rationality and Action

    Article  Google Scholar 

  3. Ågotnes, T., Goranko, V., Jamroga, W.: Alternating-time temporal logics with irrevocable strategies. In: Samet, D. (ed.) Proceeding of TARK XI, pp. 15–24. ACM, New York (2007)

    Google Scholar 

  4. Ågotnes, T., Goranko, V., Jamroga, W.: Strategic commitment and release in logics for multi-agent systems (extended abstract). Technical report IfI-08-01, Clausthal University of Technology (2008)

    Google Scholar 

  5. Ågotnes, T., Walther, D.: A logic of strategic ability under bounded memory. J. Logic Lang. Inform. 18(1), 55–77 (2009)

    Article  MATH  Google Scholar 

  6. Alur, R., Henzinger, T.A., Kupferman, O.: Alternating-time temporal logic. In: Proceedings of the FOCS 1997, pp. 100–109. IEEE Computer Society Press (1997)

    Google Scholar 

  7. Kupferman, O., Alur, R., Henzinger, T.A.: Alternating-time temporal logic. In: de Roever, W.-P., Pnueli, A., Langmaack, H. (eds.) COMPOS 1997. LNCS, vol. 1536, pp. 23–60. Springer, Heidelberg (1998)

    Google Scholar 

  8. Alur, R., Henzinger, T.A., Kupferman, O.: Alternating-time temporal logic. J. ACM 49, 672–713 (2002)

    Article  MathSciNet  Google Scholar 

  9. Baltag, A.: A logic for suspicious players. Bull. Econ. Res. 54(1), 1–46 (2002)

    Article  MathSciNet  Google Scholar 

  10. Baltag, A., Moss, L.S., Solecki, S.: The logic of public announcements, common knowledge and private suspicions. Technical report SEN-R9922, CWI, Amsterdam (1999)

    Google Scholar 

  11. Baltag, A., Smets, S.: A qualitative theory of dynamic interactive belief revision. In: Bonanno, G., van der Hoek, W., Wooldridge, M. (eds.) Logic and the Foundation of Game and Decision Theory (LOFT7), volume 3 of Texts in Logic and Games, pp. 13–60. Amsterdam University Press, Amsterdam (2008)

    Google Scholar 

  12. Baltag, A., Smets, S., Zvesper, J.A.: Keep ‘hoping’ for rationality: A solution to the backward induction paradox. Synthese 169(2), 301–333 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  13. Belnap, N., Perloff, M.: Seeing to it that: a canonical form for agentives. Theoria 54, 175–199 (1988)

    Article  Google Scholar 

  14. van Benthem, J.: Rational dynamics and epistemic logic in games. In: Vannucci, S. (ed.) Logic, Game Theory and Social Choice III, pp. 19–23. ILLC, Amsterdam (2003)

    Google Scholar 

  15. van Benthem, J.: Logic in Games. MIT Press, Cambridge (2013)

    Google Scholar 

  16. van Benthem, J.: Logic of strategies: What and how? In: van Benthem, J., Ghosh, S., Verbrugge, R. (eds.) Models of Strategic Reasoning. LNCS, vol. 8972, pp. 321–332. Springer, Heidelberg (2015)

    Chapter  Google Scholar 

  17. Bonanno, G.: Modal logic and game theory: two alternative approaches. Risk Decis. Policy 7, 309–324 (2002)

    Article  Google Scholar 

  18. Bonanno, G.: Reasoning about strategies and rational play in dynamic games. In: van Benthem, J., Ghosh, S., Verbrugge, R. (eds.) Models of Strategic Reasoning. LNCS, vol. 8972, pp. 34–62. Springer, Heidelberg (2015)

    Chapter  Google Scholar 

  19. Brihaye, T., Da Costa, A., Laroussinie, F., Markey, N.: ATL with strategy contexts and bounded memory. Technical report LSV-08-14, ENS Cachan (2008)

    Google Scholar 

  20. Broersen, J.: CTL.STIT: enhancing ATL to express important multi-agent system verification properties. In: Proceedings of the AAMAS 2010, pp. 683–690 (2010)

    Google Scholar 

  21. Broersen, J., Herzig, A.: Using STIT theory to talk about strategies. In: van Benthem, J., Ghosh, S., Verbrugge, R. (eds.) Models of Strategic Reasoning. Lecturer Notes in Computer Science, vol. 8972, pp. 137–173. Springer, Heidelberg (2015)

    Chapter  Google Scholar 

  22. Broersen, J., Herzig, A., Troquard, N.: Embedding alternating-time temporal logic in strategic STIT logic of agency. J. Log. Comput. 16(5), 559–578 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  23. Brown, M.A.: On the logic of ability. J. Philos. Logic 17, 1–26 (1988)

    Article  MATH  MathSciNet  Google Scholar 

  24. Bulling, N.: Modelling and Verifying Abilities of Rational Agents. Ph.D. thesis, Clausthal University of Technology (2010)

    Google Scholar 

  25. Bulling, N., Dix, J., Jamroga, W.: Model checking logics of strategic ability: Complexity. In: Dastani, M., Hindriks, K.V., Meyer, J.-J.C. (eds.) Specification and Verification of Multi-Agent Systems, pp. 125–159. Springer, New York (2010)

    Chapter  Google Scholar 

  26. Bulling, N., Jamroga, W.: Rational play and rational beliefs under uncertainty. In: Proceedings of AAMAS 2009, pp. 257–264. ACM Press, Budapest, May 2009

    Google Scholar 

  27. Bulling, N., Jamroga, W.: Verifying agents with memory is harder than it seemed. AI Commun. 23(4), 389–403 (2010)

    MATH  MathSciNet  Google Scholar 

  28. Bulling, N., Jamroga, W.: Comparing variants of strategic ability: How uncertainty and memory influence general properties of games. Auton. Agent. Multi-Agent Syst. 28(3), 474–518 (2014)

    Article  Google Scholar 

  29. Bulling, N., Jamroga, W., Dix, J.: Reasoning about temporal properties of rational play. Ann. Math. Artif. Intell. 53(1–4), 51–114 (2009)

    MathSciNet  Google Scholar 

  30. Bulling, N., Jamroga, W., Popovici, M.: ATL* with truly perfect recall: Expressivity and validities. In: Proceedings of the 21st European Conference on Artificial Intelligence (ECAI 2014), pp. 177-182. IOS Press, Prague, August 2014

    Google Scholar 

  31. Chatterjee, K., Piterman, N., Henzinger, T.A.: Strategy logic. In: Caires, L., Vasconcelos, V.T. (eds.) CONCUR 2007. LNCS, vol. 4703, pp. 59–73. Springer, Heidelberg (2007)

    Chapter  Google Scholar 

  32. Chatterjee, K., Henzinger, T.A., Piterman, N.: Strategy logic. Inf. Comput. 208(6), 677–693 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  33. Chellas, B.: Modal Logic: An Introduction. Cambridge University Press, Cambridge (1980)

    Book  MATH  Google Scholar 

  34. Chellas, B.: On bringing it about. J. Philos. Logic 24(6), 563–571 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  35. Clarke, E.M., Emerson, E.A., Sistla, A.P.: Automatic verification of finite-state concurrent systems using temporal logic specifications. ACM Trans. Program. Lang. Syst. 8(2), 244–263 (1986)

    Article  MATH  Google Scholar 

  36. Diaconu, R., Dima, C.: Model-checking alternating-time temporal logic with strategies based on common knowledge is undecidable. Appl. Artif. Intell. 26(4), 331–348 (2012)

    Article  Google Scholar 

  37. Dima, C., Tiplea, F.L.: Model-checking ATL under imperfect information and perfect recall semantics is undecidable (2011). CoRR, abs/1102.4225

    Google Scholar 

  38. van Ditmarsch, H., van der Hoek, W., Kooi, B.: Dynamic Epistemic Logic. Springer, Dordecht (2008)

    MATH  Google Scholar 

  39. van Drimmelen, G.: Satisfiability in alternating-time temporal logic. In: Proceedings of LICS 2003, pp. 208–217. IEEE Computer Society Press (2003)

    Google Scholar 

  40. van Eijck, J.: Strategies in social software. In: van Benthem, J., Ghosh, S., Verbrugge, R. (eds.) Models of Strategic Reasoning. LNCS, vol. 8972, pp. 292–317. Springer, Heidelberg (2015)

    Chapter  Google Scholar 

  41. Fagin, R., Halpern, J.Y., Moses, Y., Vardi, M.Y.: Reasoning about Knowledge. MIT Press, London (1995)

    MATH  Google Scholar 

  42. Goranko, V.: Coalition games and alternating temporal logics. In: van Benthem, J. (ed.) Proceedings of TARK VIII, pp. 259–272. Morgan Kaufmann, San Francisco (2001)

    Google Scholar 

  43. Goranko, V., Jamroga, W.: Comparing semantics of logics for multi-agent systems. Synthese 139(2), 241–280 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  44. Goranko, V., Jamroga, W.: State and path coalition effectivity models for logics of multi-player games. In: Proceedings of AAMAS 2012 (2012)

    Google Scholar 

  45. Goranko, V., Jamroga, W., Turrini, P.: Strategic games and truly playable effectivity functions. In: Proceedings of AAMAS 2011, pp. 727–734 (2011)

    Google Scholar 

  46. Goranko, V., Shkatov, D.: Tableau-based decision procedures for logics of strategic ability in multiagent systems. ACM Trans. Comput. Log. 11(1), 1–48 (2009)

    Article  MathSciNet  Google Scholar 

  47. Goranko, V., van Drimmelen, G.: Complete axiomatization and decidability of alternating-time temporal logic. Theoret. Comput. Sci. 353(1), 93–117 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  48. Guelev, D.P., Dima, C., Enea, C.: An alternating-time temporal logic with knowledge, perfect recall and past: axiomatisation and model-checking. J. Appl. Non-Class. Logics 21(1), 93–131 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  49. Halpern, J.Y.: Reasoning about knowledge: a survey. In: Gabbay, D.M., Hogger, C.J., Robinson, J.A. (eds.) Handbook of Logic in Artificial Intelligence and Logic Programming, vol. 4: Epistemic and Temporal Reasoning, pp. 1–34. Oxford University Press, Oxford (1995)

    Google Scholar 

  50. Harrenstein, B.P., van der Hoek, W., Meyer, J.-J.C., Witteveen, C.: A modal characterization of nash equilibrium. Fundam. Informaticae 57(2–4), 281–321 (2003)

    MATH  Google Scholar 

  51. Hart, S.: Games in extensive and strategic forms. In: Aumann, R.J., Hart, S. (eds.) Handbook of Game Theory with Economic Applications, vol. 1, pp. 19–40. Elsevier, North-Holland (1992)

    Chapter  Google Scholar 

  52. van der Hoek, W., Jamroga, W., Wooldridge, M.: A logic for strategic reasoning. In: Proceedings of AAMAS 2005, pp. 157–164 (2005)

    Google Scholar 

  53. van der Hoek, W., van Otterloo, S., Wooldridge, M.: Preferences in game logics. In: Proceedings of AAMAS-04 (2004)

    Google Scholar 

  54. van der Hoek, W., Verbrugge, R.: Epistemic logic: A survey. Game Theory Appl. 8, 53–94 (2002)

    Google Scholar 

  55. van der Hoek, W., Wooldridge, M.: Tractable multiagent planning for epistemic goals. In: Castelfranchi, C., Johnson, W.L. (eds.) Proceeding of AAMAS-02, pp. 1167–1174. ACM Press, New York (2002)

    Chapter  Google Scholar 

  56. van der Hoek, W., Wooldridge, M.: Cooperation, knowledge and time: Alternating-time temporal epistemic logic and its applications. Stud. Logica 75(1), 125–157 (2003)

    Article  MATH  Google Scholar 

  57. Jamroga, W., Ågotnes, T.: Constructive knowledge: What agents can achieve under incomplete information. J. Appl. Non-Class. Logics 17(4), 423–475 (2007)

    Article  MATH  Google Scholar 

  58. Jamroga, W., van der Hoek, W.: Agents that know how to play. Fundamenta Informaticae 63(2–3), 185–219 (2004)

    MATH  MathSciNet  Google Scholar 

  59. Jamroga, W., Wooldridge, M.J., van der Hoek, W.: Intentions and strategies in game-like scenarios. In: Bento, C., Cardoso, A., Dias, G. (eds.) EPIA 2005. LNCS (LNAI), vol. 3808, pp. 512–523. Springer, Heidelberg (2005)

    Chapter  Google Scholar 

  60. Jonker, G.: Feasible strategies in Alternating-time Temporal Epistemic Logic. Master thesis, University of Utrecht (2003)

    Google Scholar 

  61. Mogavero, F., Murano, A., Perelli, G., Vardi, M.Y.: What makes ATL* decidable? a decidable fragment of strategy logic. In: Ulidowski, I., Koutny, M. (eds.) CONCUR 2012. LNCS, vol. 7454, pp. 193–208. Springer, Heidelberg (2012)

    Chapter  Google Scholar 

  62. Mogavero, F., Murano, A., Vardi, M.Y.: Reasoning about strategies. In: Lodaya, K., Mahajan, M. (eds.) Proceedings of FSTTCS 2010, LIPIcs, pp. 133–144. Atlantis Press, Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik (2010)

    Google Scholar 

  63. Vardi, M.Y., Murano, A., Mogavero, F.: Relentful strategic reasoning in alternating-time temporal logic. In: Voronkov, A., Clarke, E.M. (eds.) LPAR-16 2010. LNCS, vol. 6355, pp. 371–386. Springer, Heidelberg (2010)

    Chapter  Google Scholar 

  64. Moulin, H., Peleg, B.: Cores of effectivity functions and implementation theory. J. Math. Econ. 10(1), 115–145 (1982)

    Article  MATH  MathSciNet  Google Scholar 

  65. Osborne, M., Rubinstein, A.: A Course in Game Theory. MIT Press, Cambridge (1994)

    MATH  Google Scholar 

  66. van Otterloo, S., Jonker, G.: On epistemic temporal strategic logic. Electron. Notes Theoret. Comput. Sci. 126, 35–45 (2004)

    Google Scholar 

  67. van Otterloo, S., Roy, O.: Verification of voting protocols. Working paper, University of Amsterdam (2005)

    Google Scholar 

  68. Pacuit, E.: Dynamic models of rational deliberation in games. In: van Benthem, J., Ghosh, S., Verbrugge, R. (eds.) Models of Strategic Reasoning. LNCS, vol. 8972, pp. 3–33. Springer, Heidelberg (2015)

    Chapter  Google Scholar 

  69. Parikh, R.: The logic of games and its applications. Ann. Discrete Math. 24, 111–140 (1985)

    MathSciNet  Google Scholar 

  70. Paul, S., Ramanujam, R., Simon, S.: Automata and compositional strategies in extensive form games. In: van Benthem, J., Ghosh, S., Verbrugge, R. (eds.) Models of Strategic Reasoning. LNCS, vol. 8972, pp. 174–201. Springer, Heidelberg (2015)

    Chapter  Google Scholar 

  71. Pauly, M.: Logic for Social Software. Ph.D. thesis, University of Amsterdam (2001)

    Google Scholar 

  72. Pauly, M.: A logical framework for coalitional effectivity in dynamic procedures. Bull. Econ. Res. 53(4), 305–324 (2001)

    Article  MathSciNet  Google Scholar 

  73. Pauly, M.: A modal logic for coalitional power in games. J. Logic Comput. 12(1), 149–166 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  74. Perea, A.: Finite reasoning procedures for dynamic games. In: van Benthem, J., Ghosh, S., Verbrugge, R. (eds.) Models of Strategic Reasoning. LNCS, vol. 8972, pp. 63–90. Springer, Heidelberg (2015)

    Chapter  Google Scholar 

  75. Pinchinat, S.: A generic constructive solution for concurrent games with expressive constraints on strategies. In: Okamura, Y., Yoneda, T., Higashino, T., Namjoshi, K.S. (eds.) ATVA 2007. LNCS, vol. 4762, pp. 253–267. Springer, Heidelberg (2007)

    Chapter  Google Scholar 

  76. Sandholm, T.W.: Distributed rational decision making. In: Weiss, G. (ed.) Multiagent Systems: A Modern Approach to Distributed Artificial Intelligence, pp. 201–258. The MIT Press, Cambridge (1999)

    Google Scholar 

  77. Schewe, S.: ATL* satisfiability is 2EXPTIME-Complete. In: Walukiewicz, I., Goldberg, L.A., Halldórsson, M.M., Damgård, I., Ingólfsdóttir, A., Aceto, L. (eds.) ICALP 2008, Part II. LNCS, vol. 5126, pp. 373–385. Springer, Heidelberg (2008)

    Chapter  Google Scholar 

  78. Schobbens, P.Y.: Alternating-time logic with imperfect recall. Electron. Notes Theoret. Comput. Sci. 85(2), 82–93 (2004)

    Article  MathSciNet  Google Scholar 

  79. Troquard, N., Walther, D.: On satisfiability in ATL with strategy contexts. In: del Cerro, L.F., Herzig, A., Mengin, J. (eds.) JELIA 2012. LNCS, vol. 7519, pp. 398–410. Springer, Heidelberg (2012)

    Chapter  Google Scholar 

  80. Walther, D., Lutz, C., Wolter, F., Wooldridge, M.: ATL satisfiability is indeed EXPTIME-complete. J. Logic Comput. 16(6), 765–787 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  81. Walther, D., van der Hoek, W., Wooldridge, M.: Alternating-time temporal logic with explicit strategies. In: Samet, D. (ed.) Proceedings of the TARK XI, pp. 269–278. Presses Universitaires de Louvain, New York (2007)

    Google Scholar 

Download references

Acknowledgments

We are grateful to the participants in the Workshop on Modelling Strategic Reasoning held in February 2012 in the Lorentz Center, Leiden, and particularly to Nicolas Troquard and Dirk Walther, as well as to the anonymous reviewers, for their valuable comments and suggestions. Wojciech Jamroga acknowledges the support of the National Research Fund (FNR) Luxembourg under the project GALOT (INTER/DFG/12/06), as well as the support of the 7th Framework Programme of the European Union under the Marie Curie IEF project ReVINK (PIEF-GA-2012-626398). The final work of Valentin Goranko on this chapter was done while he was an invited visiting professor at the Centre International de Mathématiques et Informatique de Toulouse.

Author information

Authors and Affiliations

  1. Department of Intelligent Systems, TU Delft, Delft, The Netherlands

    Nils Bulling

  2. Department of Philosophy, Stockholm University, Stockholm, Sweden

    Valentin Goranko

  3. Department of Mathematics, University of Johannesburg, Johannesburg, South Africa

    Valentin Goranko

  4. Institute of Computer Science, Polish Academy of Sciences, Warszawa, Poland

    Wojciech Jamroga

  5. Computer Science and Communication Research Unit and Interdisc. Centre on Security, Reliability and Trust, University of Luxembourg, Luxembourg City, Luxembourg

    Wojciech Jamroga

Authors
  1. Nils Bulling
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Valentin Goranko
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Wojciech Jamroga
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Nils Bulling .

Editor information

Editors and Affiliations

  1. Inst for Logic Language & Computation, University of Amsterdam, Amsterdam, Noord-Holland, The Netherlands

    Johan van Benthem

  2. Indian Statistical Institute, Chennai, Tamil Nadu, India

    Sujata Ghosh

  3. Faculty of Math & Natural Sci, University of Groningen, Groningen, Groningen, The Netherlands

    Rineke Verbrugge

Rights and permissions

Reprints and permissions

Copyright information

© 2015 Springer-Verlag Berlin Heidelberg

About this chapter

Cite this chapter

Bulling, N., Goranko, V., Jamroga, W. (2015). Logics for Reasoning About Strategic Abilities in Multi-player Games. In: van Benthem, J., Ghosh, S., Verbrugge, R. (eds) Models of Strategic Reasoning. Lecture Notes in Computer Science(), vol 8972. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-48540-8_4

Download citation

  • DOI: https://doi.org/10.1007/978-3-662-48540-8_4

  • Published:

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-662-48539-2

  • Online ISBN: 978-3-662-48540-8

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation