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

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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)

“Zhi: Wen, Shuo, Qin”

知 问 说 亲

• 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’.

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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 <prefi>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: (turni & <σ*>(end & P)) → [move-i]<σ*>(end & <prefi>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 | Kiφ | CGφ, models M = (W, {~i | i∈G}, V), worlds W, accessibility relations ~i, valuation V. Truth clauses: M, s |= Kiφ iff for all t with s ~i t: M, t |= φ, M, s |= CGφ 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]Kiφ ↔ 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 |= Biφ iff M, t |=φ for all worlds t minimal in the ordering λxy. ≤i, s xy. Belief change under hard information: [!P] Biφ ↔ P → Bi

P [!P]φ

Theorem The logic of conditional belief under public announcements is axiomatized completely by (a) complete base logic of Bi

ψ φ, (b) PAL reduction axioms, plus

(c) reduction axiom for conditional beliefs: [!P] Bi

ψ φ ↔ P → Bi P ∧ [!P]ψ [!P]φ

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] Bψφ ↔ (<>(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),

• bservational 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/