DECISIONS, ACTIONS, AND GAMES: A LOGICAL PERSPECTIVE Johan van - PDF document

1 DECISIONS, ACTIONS, AND GAMES: A LOGICAL PERSPECTIVE Johan van Benthem, Amsterdam & Stanford, http://staff.science.uva.nl/~johan/ ICLA Third Indian Logic Conference, Chennai January 2009 Logic, rational agency, and intelligent interaction To be rational is to reason intelligently . All available sources of information. Logic as the study of explicit informational processes (inference, observation, communication) 知 问 说 亲 “Zhi: Wen, Shuo, Qin” F. Liu & J. Zhang, 2008, A Note on Mohist Logic , “There are three ways to get knowledge: learning from others, reasoning from what one knows already, and consulting one’s own experience”. To be rational is to act intelligently . Add goals, preferences, decisions, actions. To be rational is to interact intelligently . Argumentation, communication, games. Purpose today Discuss what happens when analyzing agency, mixture computational and philosophical logic, formal results and their significance as a theory of agency. Games as a microcosm for logics of rational agency Just a minimal social scenario: what is involved in explaining/predicting behaviour? A 1, 0 E 0, 100 99, 99 Single actions & strategies (modal logic, dynamic logic, µ -calculus, conditional logic), preference (preference logic), belief & knowledge (doxastic and epistemic logic) – and beyond: intentions, probabilities, iterated games, etc. Logical strategy: ‘deconstruct’.

2 Backward Induction procedure for solving games Suppose E is to move at a node, and all values for daughters are known. E -value = maximum of all E -values on daughters, A -value = minimal A -value at E -best daughters. Dually for A . Extends Zermelo’s algorithm, obvious’ numerical procedure, benchmark for analysis. Statics: defining the BI outcome in modal preference logic Modal preference language <pref i > φ : i prefers some node with φ to the current one. Fact The BI strategy is definable as the unique relation σ satisfying the following axiom for all propositions P – viewed as sets of nodes –, for all players i : (turn i & < σ * >( end & P)) → [move-i]< σ * >( end & <pref i >P). Explanation: (recursively) avoid strictly dominated moves. Dynamics I: Defining the BI procedure in logic of public announcements Dynamic logics of public announcement: Information update as model change. Card examples: learning P eliminates the worlds where P is false. In a picture: from M s to M |P s P ¬P Questions/answers: iterated and common knowledge. Epistemic logic Language p |¬ φ | φ∨ψ | K i φ | C G φ , models M = (W, {~ i | i ∈ G}, V), worlds W , accessibility relations ~ i , valuation V . Truth clauses: M , s |= K i φ iff for all t with s ~ i t : M , t |= φ , M , s |= C G φ iff for all t reachable from s by a finite sequence of ~ i steps ( i ∈ G ): M , t |= φ . Dynamic logic of public announcement PAL : action expressions: !P for all formulas P , and modal operators describing their effects (one simultaneous recursion): M , s |= [!P] φ iff if M , s |=P, then M |P, s |= φ Theorem PAL axiomatized completely by epistemic logic plus recursion axioms : [!P]q ↔ P → q for atomic facts q [!P]¬ φ P → ¬[!P] φ ↔ [!P] φ∧ψ ↔ [!P] φ ∧ [!P] ψ [!P]Ki φ ↔ P → Ki(P → [!P] φ )

3 Backward Induction as repeated public announcement (cf. Muddy Children!): Theorem The Backward Induction solution for extensive games is obtained through repeated announcement of the assertion "No player chooses a move all of whose further histories end worse than all histories after some other available move". Further uses of changing games : improvement. E ’s promises she will not go left – new equilibrium (99, 99) results: both players better off by restricting the freedom of one! A A 1, 0 E 1. 0 E 0, 100 99, 99 99, 99 Theorem Modal logic of games plus announcement axiomatized by basic modal game logic, PAL recursion axioms of PAL for atoms and Booleans, plus a law for moves: <!P><a> ϕ ↔ (P ∧ <a>(P ∧ <!P> ϕ ). (Can be extended to preference logic.) Adding Strategies Using PDL for strategies in games, use relativization closure to get Theorem PDL+ PAL is axiomatized by merging their separate laws while adding the following reduction axiom: [!P]{ σ } φ ↔ (P → { σ |P}[!P] φ ). Dynamics II: The BI procedure as iterated belief revision Conditional logic of relative plausibility: convenient base logic for ‘soft information’: M , s |= B i φ iff M , t |= φ for all worlds t minimal in the ordering λ xy. ≤ i, s xy . P [!P] φ Belief change under hard information: [!P] B i φ ↔ P → B i Theorem The logic of conditional belief under public announcements is axiomatized ψ φ , (b) PAL reduction axioms, plus completely by (a) complete base logic of B i P ∧ [!P] ψ [!P] φ ψ φ ↔ P → B i (c) reduction axiom for conditional beliefs: [!P] B i Allows for interesting scenarios : Misleading true information. New notion: Safe belief . ‘Soft information’ merely changes the plausibility ordering of the existing worlds. Lexicographic upgrade ⇑ P changes the current model M to M ⇑ P : P -worlds now better than all ¬P -worlds; within zones, old order remains .

4 Belief change under soft information ( <> is the epistemic existential modality:) Theorem The dynamic logic of lexicographic upgrade is axiomatized completely by logic of conditional belief + compositional analysis of effects of model change: [ ⇑ P] B ψ φ ↔ (<>( P ∧ [ ⇑ P] ψ ) ∧ B P ∧ [ ⇑ P] ψ [ ⇑ P] φ ) ∨ ( ¬ <>( P ∧ [ ⇑ P] ψ ) ∧ B [ ⇑ P] ψ [ ⇑ P] φ Rethink BI procedure as creating expectations , changing plausibility among branches of game tree. Start with the empty plausibility relation. At a turn for player i, successor node x ‘strictly dominates’ node y for i if all currently most plausible end nodes following x are worse for i than all currently most plausible end nodes following y . Theorem The BI procedure consists in iterated soft updates using the assertion that “No player plays a strictly dominated move at her turns”. New technical issues: (a) Monotonicity: the plausibility links just grow, they do not ‘reverse’. (b) Soft updates need binary ⇑ (P, Q): “make all end-nodes after x more plausible than those after y ”. (c) Formal language stronger fragment of FOL than basic modal one. (d) Final fixed-point relation definable in which logic? (e) Complete dynamic logic for common belief of players? Discussion: what is at stake more generally in our single case study? * Further generalized game scenarios: dynamic logics of preference change. * Not knowing the preferences: conversation, social choice. * New logical issues: combining logics and complexity , varying ‘bridge principles’ (alternatives to Rationality), observational versus theoretical vocabulary ( BI as ‘mechanics’). * On top of standard game theory, DEL gives fine-structure of deliberation and moves. * More general theory of rational agency. References (a) Proceedings ICLA-III 2009 . (b) 'Research' at http://staff.science.uva.nl/~johan/. (c) http://staff.science.uva.nl/~johan/seminar2006.html. (d) http://www.illc.uva.nl/lgc/seminar/