Game Solution, Epistemic Dynamics and Fixed-Point Logics

Abstract

Current methods for solving games embody a form of "procedural rationality" that invites logical analysis in its own right. This paper is a brief case study of Backward Induction for extensive games, replacing earlier static logical definitions by stepwise dynamic ones. We consider a number of analysis from recent years that look different conceptually, and find that they are all mathematically equivalent. This shows how an abstract logical perspective can bring out basic invariant structure in games. We then generalize this to an exploration of fixed-point logics on finite trees that best fit game-theoretic equilibria. We end with some open questions that suggest a broader program for merging current computational logics with notions and results from game theory. This paper is largely a program for opening up an area: an extended version of the technical results will be found in the forthcoming dissertation [26].Logics that describe games In recent years, many logical analyses have been given of both strategic and extensive games, through introducing formal languages that describe game structure while raising logical questions of definability and axiomatization ([17], [4], [19], [29]). A benchmark for logics in this tradition has been the definition of Backward Induction ("BI" for short), the most common method for solving finite extensive games of perfect information ([39], [40]). In this same arena, basic foundational results have been obtained in epistemic game theory, endowing bare games with epistemic assumptions about players. A pilot result was the characterization of the BI outcome in terms of assuming common knowledge, or common true belief, in rationality, meaning that players choose those actions that they believe to be best for themselves ([1]). Recently,[9] has suggested that the main focus here should be shifted: away from a static assumption of known or believed rationality to the underlying "procedural rationality" of plausible procedures that players engage in when analyzing and playing a game, and the way these result in stable limit models where rationality becomes common knowledge. 1 Thus, [4] shows how game-theoretic equilibrium fits with the computational perspective of fixed-point logics, and [11] gives several dynamic procedures that analyze BI. This paper will analyze these proposals further, and find their common mathematical background. This will then be our starting point for suggesting a more general line of investigation. Analyzing solution procedures Basics of extensive gamesWe assume some basic game theory, and we will work with finite extensive games of perfect information, i.e., finite trees with labelled nodes, where each node is either an end node, or an intermediate node that represents the turn of a unique player. 2 We will mostly think of 2player games, though much of what we say generalizes to more players. While game trees with moves are simple computational structures, the essence of rational action arises with the way players evaluate outcomes. Thus, there is also a fur...

Preference at First Sight

“…Recursive solution procedures naturally correspond to definitions in existing fixed-point logics, such as the widely used system LFP(FO). An LFP(FO) formula mirroring the recursive nature of BI is constructed in [4,6] to define the classical BI relation, based on the above property RAT. Now, we have shown that sight-restricted SCBI, too, is a recursive game solution procedure.…”

“…Therefore, we know that M satisfies RATS. P However, compared with the generic logic in [4,3,2], the given definition in our logic is local. It is obvious that CF, the formula defining the property RAT, is insensitive to models -while our CFS M relies on a given model for its ranges of big disjunctions and conjunctions, and in its model-dependent notations like s(h) and h 1 ≥ h 2 .…”

Public Announcement Logic in Geometric Frameworks

“…For a model M and a formula ϕ, we define the announcement limit lim ϕ M as the first model which is reached by successive announcements of ϕ that no longer changes after the last announcement is made. Announcement limits exist in both finite and infinite models (van Benthem & Gheerbrant, 2010). For instance, for any model M , lim p M = M |p for propositional variable p. Therefore, the limit model is the first updated model when the announcement is a ground Boolean formula.…”

A Logic for Strategy Updates

“…The game-theoretical approach to the centipede game was initiated by Rosenthal [21], which was followed by several solution methods [2,13]. More recently, the role of rationality in its solution methods has been discussed [3,6] and extended to some other cases [16].The role knowledge as opposed to beliefs has been discussed in similar contexts by Artemov [1]. Moreover, based on experimental work, Ghosh, Meijering and Verbrugge aim to observe how humans reason strategically [11].…”