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Stit Theory The Dominance Approach The Language The Probabilistic Approach Probabilistic Oughts Conclusion References

Probabilistic Agent-Dependent Oughts

Thijs De Coninck, Nathan Wood

Centre for Logic and Philosophy of Science Ghent University

PhDs in Logic, Bern, 26.4.2019

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Stit Theory The Dominance Approach The Language The Probabilistic Approach Probabilistic Oughts Conclusion References

Table of Contents

Stit Theory The Dominance Approach The Language The Probabilistic Approach Probabilistic Oughts Conclusion

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Stit Theory The Dominance Approach The Language The Probabilistic Approach Probabilistic Oughts Conclusion References

Stit Theory

Stit Theory: theory of “seeing to it that” something is the case.

◮ Modal logic of agency ◮ Cast within a theory of indeterministic branching time ◮ 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)

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Stit Theory The Dominance Approach The Language The Probabilistic Approach Probabilistic Oughts Conclusion References

The Dominance Approach

Dominance Approach: An action X dominates an action Y for an agent i if, given all possible combined actions of all other agents, X has better consequences for i than the action Y j Y Y ′ i X (1, 1) (3, 0) X ′ (0, 3) (2, 2)

Table: The two player prisoner’s dilemma.

Each agent has a real-valued utility function Ui over worlds that represents their preferences.

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The Problem – Traffic Lights

Figure: Traffic Lights

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The Language

ϕ, ψ ::= p | ¬ϕ | ϕ ∨ ψ | ♦ϕ | αi | [αi]ϕ | Oiαi | Piαi

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The Probabilistic Approach

Probabilistic Approach: An action X is better than Y if the expected utility of X is greater than the expected utility of Y. j Y Y ′ i X (0, 0) (1, 1) X ′ (1, 1) (0, 0)

Table: Coordination game (without probabilities)

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The Probabilistic Approach

Probabilistic Approach: An action X is better than Y if the expected utility of X is greater than the expected utility of Y. j 0.9 0.1 Y Y ′ i X (0, 0) (1, 1) X ′ (1, 1) (0, 0)

Table: Coordination game (with probabilities)

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Definition

A belief function is a function Bi : N × 2W → [0, 1] such that C1.1 Bi(j, X) = 0 if X / ∈ Choicej(F) C1.2 Bi(j, X) > 0 if X ∈ Choicej(F) C1.3

X∈Choicej(F) Bi(j, X) = 1 provided that i = j

C1.4 Bi(i, X) = 1 if X ∈ Choicei(F)

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Stit Theory The Dominance Approach The Language The Probabilistic Approach Probabilistic Oughts Conclusion References

Degree of belief that w is realized

B∗

i : W → [0, 1] :

B∗

i (w) =

• j∈N

Bi(j, Choicej(w)) j 0.05 0.9 0.05 Y Y ′ Y ′′ i X 2 1 4 X ′ 7 4

Table: The Potluck

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Stit Theory The Dominance Approach The Language The Probabilistic Approach Probabilistic Oughts Conclusion References

Probabilistic Oughts

δi : 2W → R : δi(X) =

• w∈X

B∗

i (w) · Ui(w)

j 0.05 0.9 0.05 Y Y ′ Y ′′ i X 2 1 4 X ′ 7 4

Table: The Potluck

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Stit Theory The Dominance Approach The Language The Probabilistic Approach Probabilistic Oughts Conclusion References

Probabilistic Oughts

δi : 2W → R : δi(X) =

• w∈X

B∗

i (w) · Ui(w)

j > 0.6 < 0.2 0.2 Y Y ′ Y ′′ i X 2 1 4 X ′ 7 4

Table: The Potluck

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Further Work

◮ Conditional Oughts ◮ Group Oughts ◮ Connections to Epistemic Game Theory

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References

Belnap, N., Perloff, M., and Xu, M. (2001). Facing the Future: Agents and Choices in Our Indeterminist

• World. Oxford University Press.

Broersen, J. (2013). Probabilistic stit logic and its

• decomposition. International Journal of Approximate

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.

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