On modal logic interpretations of games

2013

Propositional Dynamic Logic or PDL was invented as a logic for reasoning about regular programming constructs. We propose a new perspective on PDL as a multi-agent strategic logic (MASL). This logic for strategic reasoning has group strategies as rst class citizens, and brings game logic closer to standard modal logic. We demonstrate that MASL can express key notions of game theory, social choice theory and voting theory in a natural way, we give a sound and complete proof system for MASL, and we show that MASL encodes coalition logic. Next, we extend the language to epistemic multi-agent strategic logic (EMASL), we give examples of what it can express, we propose to use it for posing new questions in epistemic social choice theory, and we give a calculus for reasoning about a natural class of epistemic game models. We end by listing avenues for future research and by tracing connections to a number of other logics for reasoning about strategies.

2009

Abstract Strategic reasoning representation is a key issue in the theoretical study and the implementation of Multi Agent System. To this purpose a language is developed that can explicitly describe the dynamics of strategies, the preferences and their interaction. The relation with Pauly's Coalition Logic is studied and a full reduction is shown, together with the characterization of gametheoretical concept such as undominated choice.

2018

We revisit the discussion on reasoning about games in dynamic-epistemic logic and present a language for describing reasoning in possibly infinite games from the perspective of the players. We argue that even though a plethora of sophisticated logics of strategic reasoning in games are available, it is still worthwhile to consider the game structures themselves from the perspective of logic. In the process, we provide complete axiom systems for these games satisfying characteristic properties from the gametheoretic literature. Decidability of the satisfiability problem is also taken up to consider the existence of games following certain rules that can be expressed in the logical language.

2006

Preference is a basic notion in human behaviour, underlying such varied phenomena as individual rationality in the philosophy of action and game theory, obligations in deontic logic (we should aim for the best of all possible worlds), or collective decisions in social choice theory. Also, in a more abstract sense, preference orderings are used in conditional logic or non-monotonic reasoning as a way of arranging worlds into more or less plausible ones. The field of preference logic (cf. Hansson ) studies formal systems that can express and analyze notions of preference between various sorts of entities: worlds, actions, or propositions. The art is of course to design a language that combines perspicuity and low complexity with reasonable expressive power. In this paper, we take a particularly simple approach. As preferences are binary relations between worlds, they naturally support standard unary modalities. In particular, our key modality ♦ϕ will just say that is ϕ true in some world which is at least as good as the current one. Of course, this notion can also be indexed to separate agents. The essence of this language is already in [4], but our semantics is more general, and so are our applications and later language extensions. Our modal language can express a variety of preference notions between propositions. Moreover, as already suggested in [9], it can "deconstruct" standard conditionals, providing an embedding of conditional logic into more standard modal logics. Next, we take the language to the analysis of games, where some sort of preference logic is evidently needed ([23] has a binary modal analysis different from ours). We show how a qualitative unary preference modality suffices for defining Nash Equilibrium in strategic games, and also the Backward Induction solution for finite extensive games. Finally, from a technical perspective, our treatment adds a new twist. Each application considered in this paper suggests the need for some additional access to worlds before the preference modality can unfold its true power. For this purpose, we use various extras from the modern literature: the global modality, further hybrid logic operators, action modalities from propositional dynamic logic, and modalities of individual and distributed knowledge from epistemic logic. The total package is still modal, but we can now capture a large variety of new notions. Finally, our emphasis in this paper is wholly on expressive power. Axiomatic completeness results for our languages can be found in the follow-up paper [27].

2012

Propositional Dynamic Logic or PDL was invented as a logic for reasoning about regular programming constructs. We propose a new perspective on PDL as a multi-agent strategic logic (MASL). This logic for strategic reasoning has group strategies as first class citizens, and brings game logic closer to standard modal logic. We demonstrate that MASL can express key notions of game theory, social choice theory and voting theory in a natural way, we give a sound and complete proof system for MASL, and we show that MASL encodes coalition logic. Next, we extend the language to epistemic multi-agent strategic logic (EMASL), we give examples of what it can express, we propose to use it for posing new questions in epistemic social choice theory, and we give a calculus for reasoning about a natural class of epistemic game models. We end by listing avenues for future research and by tracing connections to a number of other logics for reasoning about strategies.

Risk Decision and Policy, 2002

Two views of game theory are discussed: (1) game theory as a description of the behavior of rational individuals who recognize each other's reationality and reasoning abilities, and (2) game theory as an internally consistent recommendation to individuals on how to act in interactive situations. It is shown that the same mathematical tool, namely modal logic, can be used to explicitly model both views.

Studia Logica, 2014

Modal logics have proven to be a very successful tool for reasoning about games. However, until now, although logics have been put forward for games in both normal form and games in extensive form, and for games with complete and incomplete information, the focus in the logic community has hitherto been on games with pure strategies. This paper is a first to widen the scope to logics for games that allow mixed strategies. We present a modal logic for games in normal form with mixed strategies, and demonstrate its soundness and strong completeness. Characteristic for our logic is a number of infinite rules.

2008

We show that Pauly's Coalition Logic can be embedded into a richer normal modal logic that we call Normal Simulation of Coalition Logic (NCL). We establish that the latter is strictly more expressive than the former by proving that it is NEXPTIME complete in the case of at least two agents.

New types of knowledge constructions in epistemic logic are defined, whose semantics is given by a combination of game-theoretic notions and modal semantics. A new notion of knowledge introduced in this framework is focussed knowledge, which arises from the game-theoretic phenomenon of imperfect information in quantified versions of epistemic logics. This notion is shown to be useful in knowledge representation of multi-agent systems with uncertainty. In general, imperfect information gives rise to partialised versions of these logics.

Log. Methods Comput. Sci., 2017

Strategy Logic (SL, for short) has been introduced by Mogavero, Murano, and Vardi as a useful formalism for reasoning explicitly about strategies, as first-order objects, in multi-agent concurrent games. This logic turns out to be very powerful, subsuming all major previously studied modal logics for strategic reasoning, including ATL, ATL*, and the like. Unfortunately, due to its high expressiveness, SL has a non-elementarily decidable model-checking problem and the satisfiability question is undecidable, specifically Sigma_1^1. In order to obtain a decidable sublogic, we introduce and study here One-Goal Strategy Logic (SL[1G], for short). This is a syntactic fragment of SL, strictly subsuming ATL*, which encompasses formulas in prenex normal form having a single temporal goal at a time, for every strategy quantification of agents. We prove that, unlike SL, SL[1G] has the bounded tree-model property and its satisfiability problem is decidable in 2ExpTime, thus not harder than the ...