Probabilistic Agent-Dependent Oughts Conclusion References Thijs - PowerPoint PPT Presentation

Stit Theory The Dominance Approach The Language The Probabilistic Approach Probabilistic Oughts Probabilistic Agent-Dependent Oughts Conclusion References Thijs De Coninck, Nathan Wood Centre for Logic and Philosophy of Science Ghent University PhDs in Logic, Bern, 26.4.2019 1 [1 2 14]

Table of Contents Stit Theory The Dominance Approach Stit Theory The Language The Probabilistic The Dominance Approach Approach Probabilistic Oughts The Language Conclusion References The Probabilistic Approach Probabilistic Oughts Conclusion 2 [2 2 14]

Stit Theory Stit Theory The Dominance Approach Stit Theory: theory of “seeing to it that” something is the The Language The Probabilistic case. Approach ◮ Modal logic of agency Probabilistic Oughts ◮ Cast within a theory of indeterministic branching time Conclusion References ◮ A brief history of Stit theory: Stit theory – Belnap et al. (2001) Deontic Stit – Horty (2001) Indexed Deontic Stit – Kooi and Tamminga (2008) Probabilistic Stit – Broersen (2013) 3 [3 3 14]

The Dominance Approach Stit Theory The Dominance Dominance Approach: An action X dominates an action Approach Y for an agent i if, given all possible combined actions of The Language all other agents, X has better consequences for i than the The Probabilistic Approach action Y Probabilistic Oughts j Conclusion Y Y ′ References X ( 1 , 1 ) ( 3 , 0 ) i X ′ ( 0 , 3 ) ( 2 , 2 ) Table: The two player prisoner’s dilemma. Each agent has a real-valued utility function U i over worlds that represents their preferences. 4 [4 5 14]

The Problem – Traffic Lights Stit Theory The Dominance Approach The Language The Probabilistic Approach Probabilistic Oughts Conclusion References Figure: Traffic Lights 5 [5 5 14]

The Language Stit Theory The Dominance Approach The Language The Probabilistic Approach Probabilistic Oughts ϕ, ψ ::= p | ¬ ϕ | ϕ ∨ ψ | ♦ ϕ | α i | [ α i ] ϕ | O i α i | P i α i Conclusion References 6 [6 6 14]

The Probabilistic Approach Stit Theory The Dominance Approach Probabilistic Approach: An action X is better than Y if the The Language expected utility of X is greater than the expected utility of The Probabilistic Approach Y . Probabilistic Oughts j Conclusion Y Y ′ References X ( 0 , 0 ) ( 1 , 1 ) i X ′ ( 1 , 1 ) ( 0 , 0 ) Table: Coordination game (without probabilities) 7 [7 9 14]

The Probabilistic Approach Stit Theory The Dominance Approach Probabilistic Approach: An action X is better than Y if the The Language expected utility of X is greater than the expected utility of The Probabilistic Y . Approach Probabilistic Oughts j Conclusion 0 . 9 0 . 1 References Y ′ Y X ( 0 , 0 ) ( 1 , 1 ) i X ′ ( 1 , 1 ) ( 0 , 0 ) Table: Coordination game (with probabilities) 8 [8 9 14]

Stit Theory The Dominance Approach The Language Definition The Probabilistic A belief function is a function B i : N × 2 W → [ 0 , 1 ] such Approach that Probabilistic Oughts C1.1 B i ( j , X ) = 0 if X / ∈ Choice j ( F ) Conclusion C1.2 B i ( j , X ) > 0 if X ∈ Choice j ( F ) References C1.3 � X ∈ Choice j ( F ) B i ( j , X ) = 1 provided that i � = j C1.4 B i ( i , X ) = 1 if X ∈ Choice i ( F ) 9 [9 9 14]

Degree of belief that w is realized Stit Theory B ∗ i : W → [ 0 , 1 ] : The Dominance Approach The Language � B ∗ i ( w ) = B i ( j , Choice j ( w )) The Probabilistic Approach j ∈ N Probabilistic Oughts Conclusion j References 0 . 05 0 . 9 0 . 05 Y ′ Y ′′ Y X 2 1 4 i X ′ 0 7 4 Table: The Potluck 10 [10 12 14]

Probabilistic Oughts Stit Theory δ i : 2 W → R : The Dominance Approach The Language � B ∗ δ i ( X ) = i ( w ) · U i ( w ) The Probabilistic Approach w ∈ X Probabilistic Oughts Conclusion j References 0 . 05 0 . 9 0 . 05 Y ′ Y ′′ Y X 2 1 4 i X ′ 0 7 4 Table: The Potluck 11 [11 12 14]

Probabilistic Oughts Stit Theory δ i : 2 W → R : The Dominance Approach The Language � B ∗ δ i ( X ) = i ( w ) · U i ( w ) The Probabilistic Approach w ∈ X Probabilistic Oughts Conclusion j References > 0 . 6 < 0 . 2 0 . 2 Y ′ Y ′′ Y X 2 1 4 i X ′ 0 7 4 Table: The Potluck 12 [12 12 14]

Further Work Stit Theory The Dominance Approach The Language The Probabilistic Approach Probabilistic ◮ Conditional Oughts Oughts Conclusion ◮ Group Oughts References ◮ Connections to Epistemic Game Theory 13 [13 14 14]

References Stit Theory The Dominance Approach Belnap, N., Perloff, M., and Xu, M. (2001). Facing the The Language Future: Agents and Choices in Our Indeterminist The Probabilistic Approach World . Oxford University Press. Probabilistic Broersen, J. (2013). Probabilistic stit logic and its Oughts Conclusion decomposition. International Journal of Approximate References Reasoning , 54(4):467–477. Horty, J. F . (2001). Agency and Deontic Logic . Oxford University Press. Kooi, B. and Tamminga, A. (2008). Moral conflicts between groups of agents. Journal of Philosophical Logic , 37(1):1–21. 14 [14 14 14]