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PDL As a Multi-Agent Strategy Logic

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PDL As a Multi-Agent Strategy Logic PDL as a Multi-Agent Strategy Logic ∗ Extended Abstract Jan van Eijck CWI and ILLC Science Park 123 1098 XG Amsterdam, The Netherlands [email protected] ABSTRACT The logic we propose follows a suggestion made in Van Propositional Dynamic Logic or PDL was invented as a logic Benthem [4] (in [11]) to apply the general perspective of ac- for reasoning about regular programming constructs. We tion logic to reasoning about strategies in games, and links propose a new perspective on PDL as a multi-agent strategic up to propositional dynamic logic (PDL), viewed as a gen- logic (MASL). This logic for strategic reasoning has group eral logic of action [29, 19]. Van Benthem takes individual strategies as first class citizens, and brings game logic closer strategies as basic actions and proposes to view group strate- to standard modal logic. We demonstrate that MASL can gies as intersections of individual strategies (compare also express key notions of game theory, social choice theory and [1] for this perspective). We will turn this around: we take voting theory in a natural way, we give a sound and complete the full group strategies (or: full strategy profiles) as basic, proof system for MASL, and we show that MASL encodes and construct individual strategies from these by means of coalition logic. Next, we extend the language to epistemic strategy union. multi-agent strategic logic (EMASL), we give examples of A fragment of the logic we analyze in this paper was pro- what it can express, we propose to use it for posing new posed in [10] as a logic for strategic reasoning in voting (the questions in epistemic social choice theory, and we give a cal- system in [10] does not have current strategies). culus for reasoning about a natural class of epistemic game The plan of the paper is as follows. In Section 2 we re- models. We end by listing avenues for future research and by view key concepts from strategic game theory, and hint at tracing connections to a number of other logics for reasoning how these will show up in our logic. Section 3 does the about strategies. same for voting theory. Section 4 gives a motivating ex- ample about coalition formation and strategic reasoning in voting. Section 5 presents the language of MASL, and gives Categories and Subject Descriptors the semantics. Next we show, in Section 6, that the key F.4.1 [Mathematical Logic]: Modal Logic; F.4.1 [Mathe- concepts of strategic game theory and voting theory are matical Logic]: Proof Theory; I.2.3 [Artificial Intelli- expressible in MASL. Section 7 extends the proof system gence]: Deduction and Theorem Proving for PDL to a sound and complete proof system for MASL. Section 8 gives an embedding of coalition logic into MASL. Section 9 extends MASL to an epistemic logic for reason- Keywords ing about knowledge in games, Section 10 gives examples of Strategies, Strategic Games, Coalition Logic, Modal Logic, what EMASL can express, and Section 11 sketches a calcu- Dynamic Logic, Voting Theory lus for EMASL. Section 12 concludes. Key contributions of the paper are a demonstration of how 1. INTRODUCTION PDL can be turned into a game logic for strategic games, and how this game logic can be extended to an epistemic game In this paper we propose a simple and natural multi-agent logic with PDL style modalities for game strategies and for strategy logic, with explicit representations for individual epistemic operators. This makes all the logical and model and group strategies. The logic can be viewed as an ex- checking tools for PDL available for analyzing properties of tension of the well-known propositional logic of programs strategic games and epistemic strategic games. PDL. We show that the logic can express key notions of game theory and voting theory, such as Nash equilibrium, and the properties of voting rules that are used to prove the 2. GAME TERMINOLOGY Gibbard-Satterthwaite theorem. A strategic game form is a pair Unlike most other game logics, our logic uses explicit rep- resentations of group strategies in N-player games, with (n; fSigi2f1;:::;ng) N ≥ 2, and treats coalitions as a derived notion. where f1; : : : ; ng with n > 1 is the set of players, and each ∗ A full version of this paper is available at www.cwi.nl/ Si is a non-empty set of strategies (the available actions for ~jve/papers/13 player i). Below we will impose the restriction that the game forms are finite: each Si is a finite non-empty set. TARK 2013, Chennai, India. We use N for the set f1; : : : ; ng, and S for S1×· · ·×Sn, and Copyright 2013 by the author. we call a member of S a strategy profile. Thus, a strategy profile s is an n-tuple of strategies, one for each player. If s It should be noted that payoff functions are a special case is a strategy profile, we use si or s[i] for its i-th component. of output functions. In the example of PD with payoffs, Strategy profiles are in one-to-one correspondence to game we can view the payoff function as an output function with outcomes, and in fact we can view s 2 S also as a game range f0; 1; 2; 3g2. outcome [22]. Below, we will assume that output functions are of type Consider the prisoner's dilemma game PD for two players o : S ! P , and we will introduce proposition letters to as an example. Both players have two strategies: c for co- range over P . This allows us to view the game forms as operate, d for defect. The possible game outcomes are the modal frames, and the games including the output functions four strategy profiles (c; c); (c; d); (d; c); (d; d). as models, with the output function fixing the valuation by means of \the valuation makes p true in a state s iff s 2 c d o−1(p)." c c; c c; d A special case of this is the case where the P are payoff d d; c d; d vectors. Valuations that are payoff vectors allow us to ex- press preferences of the players for an outcome as boolean It is useful to be able to classify game outcomes. A P - formulas (see below). 0 outcome function for game form (N; S) is a function o : Let (si; s−i) be the strategy profile that is like s for all 0 S ! P . players except i, but has si replaced by si. A strategy si is For the example of the PD game, o could be a function a best response in s if 2 with range fx; y; z; ug , as follows: 0 0 8si 2 Si ui(s) ≥ ui(si; s−i): c d A strategy profile s is a (pure) Nash equilibrium if each c x; x y; z si is a best response in s: d z; y u; u 0 0 8i 2 N 8si 2 Si ui(s) ≥ ui(si; s−i): If C ⊆ N, we let S = Q S be the set of group C i2C i A game G is Nash if G has a (pure) Nash equilibrium. strategies for C. If s 2 S and t 2 S we use (s; t) for C N−C These key notions of game theory will reappear below the strategy profile that results from combining s and t, i.e., when we discuss the expressiveness of MASL. for the strategy profile u given by u[i] = s[i] if i 2 C; u[i] = t[i] otherwise. 3. VOTING AS A MULTI-AGENT GAME The group strategies for the PD game coincide with the Voting can be seen as a form of multi-agent decision mak- strategy profiles. ing, with the voters as agents [14]. Voting is the process An abstract game G is a tuple of selecting an item or a set of items from a finite set A of alternatives, on the basis of the stated preferences of a set (N; S; {≥igi2N ); of voters. See [7] for a detailed account. We assume that the preferences of a voter are represented where (N; S) is a game structure, and each ≥i is a prefer- by a ballot, where a ballot is a linear ordering of A. Let ence relation on S1 × · · · × Sn. These preference relations are assumed to be transitive, reflexive, and complete, where ord(A) be the set of all ballots on A. completeness means that for all different s; t 2 S, one of If there are three alternatives a; b; c, and a voter prefers a s ≥ t, t ≥ s holds. over b and b over c, then her ballot is abc. i i 0 In the PD game example, with the output function as Assume the set of voters is N = f1; : : : ; ng. If we use b; b above, the preferences could be fixed by adding the infor- to range over ballots, then a profile P is a vector (b1;:::; bn) mation that z > x > u > y. of ballots, one for each voter. If P is a profile, we use Pi for The preference relations may also be encoded as numerical the ballot of voter i in P. utilities. A payoff function or utility function for a player i The following represents the profile P where the first voter has ballot abc, the second voter has ballot abc, the third is a function ui from strategy profiles to real numbers. A voter has ballot bca, and so on: payoff function ui represents the preference ordering ≥i of player i if s ≥i t iff ui(s) ≥ ui(t), for all strategy profiles (abc; abc; bca; abc; cab; acb): s; t.
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