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 \(\mathsf {ATL}\).
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Notes
- 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.
Roughly corresponding to ‘first-person deliberation’ vs. ‘third-person assessment of strategic action in games’ in van Benthem [16].
- 3.
We do, however, discuss briefly in Sect. 5.2 how some concepts of rationality can be expressed in logical languages considered here.
- 4.
We have no strong reason for this finiteness assumption, other than common sense and technical convenience.
- 5.
We assume that there is an ordering on \(\mathbb {A}\mathrm {gt} \) which is respected in the definition of tuples etc.
- 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.
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.
- 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.
- 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.
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.
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.
This corresponds to the notion of synchronous perfect recall according to [41].
- 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.
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.
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.
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.
- 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.
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\(\mathsf {STIT}\).
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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.
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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
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