Stit Logics, Games, Knowledge, and Freedom

Abstract

This paper has two main goals: highlighting the connections between Stit logics and game theory and comparing Stit logics with Matrix Game Logic, a Dynamic Logic introduced by van Benthem order to model some interesting epistemic notions from game theory. Achieving the first goal will prove the flexibility of Stit logics and their applicability in the logical foundations of game theory, and will lay the groundwork for accomplishing the second. A comparison between Stit logics and Matrix Game Logic is already offered in a recent work by van Benthem and Pacuit. Here, we push the comparison further by embedding Matrix Game Logic into a fragment of group Stit logic, and using the embedding to derive some properties of Matrix Game Logic-in particular, undecidability and the lack of finite axiomatizability. In addition, the embedding sheds light on some open issues about the so-called "freedom operator" of Matrix Game Logic.

Doing Without Nature

“…used in [8] to study the complexity of STIT logic for groups. They closely resemble the choice structures from [10], the STIT choice scenarios from [16], and the choice Kripke models from [5].…”

“…Horty's [9] work in deontic logic, however, was already strongly inspired by the link between game theory and STIT theory. Publications that are directly relevant to this paper are [5,13,14,16].…”

Choice-Driven Counterfactuals

“…In particular, logics of this sort seem to be necessary to clarify complex moral and legal ideas, such as the concept of responsibility [2,11,12,20,32] and "could have done otherwise" [5]. In addition, the discussion in Section 4 and Remark 2 suggests that a stit logic with counterfactuals may be fruitfully used to incorporate strategic reasoning in stit, thus advancing recent research connecting stit and game-theory, see, e.g., [19,29,48,51]). We conjecture that the latter application may call for a framework combining our approach to the semantics of counterfactuals with extensions of stit logics with epistemic operators [23,27,50] and probabilistic belief operators [13].…”

“…David betting). 19 To account for the Assume Independence intuition, we supplement Definition 9 with a further requirement on unconstrained agents. Recall that an agent i is unconstrained at a moment m when none of the actions available to her at m is deviant (cf.…”

“…Then the number of independent events for any histories h 1 and h 2 is defined as: 19 To account for the reasoning in (2) in the context of branching time, Thomason and Gupta [49] impose constraints of "causal coherence" on their models. Yet, they acknowledge that this move adds a substantial layer of complexity to their theory.…”