Preference and its Dynamics

Transcription

1 Department of Philosophy,Tsinghua University 28 August, 2012, EASLLC

2 Table of contents 1 Introduction 2 Betterness model and dynamics 3 Priorities and dynamics 4 Relating betterness and priority dynamics

3 Closely related references to this lecture : Reasoning about Preference Dynamics, Springer-Verlag, Series: Synthese Library, Vol : A Two-Level Perspective on Preference, Journal of Philosophical Logic, 40(3): , : Von Wright s "The Logic of Preference" Revisited. Synthese, 175(1): 69-88, : Preference Change: A Quantitative Approach, Studies in Logic, 2(3): 12-27, 2009.

4 Closely related references to this lecture Dick de Jongh and : Preference, Priorities and Belief, in T.Grune-Yanoff and S.O. Hansson eds, Preference Change: Approaches from Philosophy, Economics and Psychology, Theory and Decision Library, pp , : Changing for the Better: Preference Dynamics and Agent Diversity, Ph.D dissertation, ILLC, University of Amsterdam, Johan van Benthem and : Dynamic Logic of Preference Upgrade. Journal of Applied Non-Classical Logic, 17(2): 2007.

5 Preference in many research fields Philosophy: preference logic is one of the philosophical logics ([von Wright, 1963], [Hansson, 2001]).

6 Preference in many research fields Philosophy: preference logic is one of the philosophical logics ([von Wright, 1963], [Hansson, 2001]). Decision theory and game theory: preference, choice, utilities are studied together to model how people make decisions.

7 Preference in many research fields Philosophy: preference logic is one of the philosophical logics ([von Wright, 1963], [Hansson, 2001]). Decision theory and game theory: preference, choice, utilities are studied together to model how people make decisions. Computer science and AI: the well-known BDI model ([Rao and Georgeff, 1991]); studies on ceteris paribus preference all else being equal" ([Doyle et al., 1991] and [Doyle and Wellman, 1994])

8 Preference in many research fields Philosophy: preference logic is one of the philosophical logics ([von Wright, 1963], [Hansson, 2001]). Decision theory and game theory: preference, choice, utilities are studied together to model how people make decisions. Computer science and AI: the well-known BDI model ([Rao and Georgeff, 1991]); studies on ceteris paribus preference all else being equal" ([Doyle et al., 1991] and [Doyle and Wellman, 1994]) Till Grune-Yanoff and Sven Ove Hansson (eds.) Preference Change: Approaches from Philosophy, Economics and Psychology, Springer 2009

9 Preference in many research fields Philosophy: preference logic is one of the philosophical logics ([von Wright, 1963], [Hansson, 2001]). Decision theory and game theory: preference, choice, utilities are studied together to model how people make decisions. Computer science and AI: the well-known BDI model ([Rao and Georgeff, 1991]); studies on ceteris paribus preference all else being equal" ([Doyle et al., 1991] and [Doyle and Wellman, 1994]) Till Grune-Yanoff and Sven Ove Hansson (eds.) Preference Change: Approaches from Philosophy, Economics and Psychology, Springer 2009

10 Important aspects of preference The preferences which we shall study are a subject s intrinsic preferences on one occasion only. Thus we exclude both reasons for preferences and the possibility of changes in preferences." ([von Wright, 1963], p.23)

11 Intrinsic preference vs. extrinsic preference... a person says, for example, that he prefers claret to hock, because his doctor has told him or he has found from experience that the first wine is better for his stomach or health in general. In this case a judgement of betterness serves as a ground or reason for a preference. I shall call preferences, which hold this relationship to betterness, extrinsic.... " ([von Wright, 1963], p.14) Extrinsic preference is reason-based, while intrinsic preference is just there.

12 More about reasons for preference I prefer Indian food over Japanese food because Indian food is spicy.

13 More about reasons for preference I prefer Indian food over Japanese food because Indian food is spicy. Reasons may be of many kinds, since the criteria that determine preference can be diverse.

14 More about reasons for preference I prefer Indian food over Japanese food because Indian food is spicy. Reasons may be of many kinds, since the criteria that determine preference can be diverse. For example, our preference is often affected by (or depends on) what others like or dislike, as described in the Chinese classic Record of Music: A ruler has only to be careful of what he likes and dislikes. What the ruler likes, his ministers will practise; and whatever superiors do, their inferiors will follow.

15 Preference change Consider the following twist to our von Wright s scenario: Suppose that before seeing his doctor, he preferred hock to claret. Now the doctor tells him The first wine is better for your health". He then changes his preference, and will now prefer claret to hock! Merely giving information can change preferences!

16 Studies after von Wright Sven Ove Hansson 2001, Preference Logic, in the Handbook of Philosophical Logic.

17 Studies after von Wright Sven Ove Hansson 2001, Preference Logic, in the Handbook of Philosophical Logic. Decision theory and game theory. Computer science and AI.

18 Studies after von Wright Sven Ove Hansson 2001, Preference Logic, in the Handbook of Philosophical Logic. Decision theory and game theory. Computer science and AI. However, little study has been done on the two issues above. [van Benthem et al, 1993] is a first attempt at using dynamic logic for this purpose. Also, influenced by AGM-style belief revision theory, [Hansson,1995] proposed postulates for four basic operations in preference change.

19 Logical models of preference The intrinsic view is reflected in possible worlds models with a betterness ordering.

20 Logical models of preference The intrinsic view is reflected in possible worlds models with a betterness ordering. The extrinsic view that we will adopt was proposed in [de Jongh and Liu, 2009], which introduced priority structures as a reason for preferences.

21 Modal betterness logic Definition A modal betterness model M = (W,, V ) has a non-empty set of worlds W, is a reflexive and transitive relation (the betterness pre-order), and V is a valuation for proposition letters.

22 Modal betterness logic Definition A modal betterness model M = (W,, V ) has a non-empty set of worlds W, is a reflexive and transitive relation (the betterness pre-order), and V is a valuation for proposition letters. If s t but not t s, then t is strictly better than s ( s t ).

23 Modal betterness logic Definition A modal betterness model M = (W,, V ) has a non-empty set of worlds W, is a reflexive and transitive relation (the betterness pre-order), and V is a valuation for proposition letters. If s t but not t s, then t is strictly better than s ( s t ). Definition The modal betterness language over propositional variables Prop is given by the following inductive syntax rule: ϕ := p ϕ ϕ ψ ϕ < ϕ Eϕ.

24 Truth definition Definition Truth conditions for the atomic propositions and Boolean combinations are standard. Modalities work like this: M, s = φ iff for some t wih s t, M, t = φ. M, s = < φ iff for some t with s t, M, t = φ. M, s = Eφ iff for some world t in W, M, t = φ.

25 dynamics in betterness relations Definition Given a modal betterness model (M, s) and formula ϕ, the suggestion upgrade (M ϕ, s) is a model with the same domain, valuation, and actual world as (M, s), the new betterness relation is = {(s, t) M, s = ϕ and M, t = ϕ}.

26 dynamics in betterness relations Definition Given a modal betterness model (M, s) and formula ϕ, the suggestion upgrade (M ϕ, s) is a model with the same domain, valuation, and actual world as (M, s), the new betterness relation is = {(s, t) M, s = ϕ and M, t = ϕ}. In PDL format ϕ(r) := (?ϕ; R;?ϕ) (? ϕ; R;? ϕ) (? ϕ; R;?ϕ). where R is the given input relation of non-strict betterness, while the operation?ϕ tests whether the proposition ϕ holds.

27 Dynamic betterness logics The dynamic betterness language over a set of propositional variables p Prop is given by a mutually recursive syntax rule: ϕ := p ϕ ϕ ψ ϕ < ϕ Eϕ [π]ϕ π := ϕ.

28 Dynamic betterness logics The dynamic betterness language over a set of propositional variables p Prop is given by a mutually recursive syntax rule: ϕ := p ϕ ϕ ψ ϕ < ϕ Eϕ [π]ϕ π := ϕ. Given a betterness model M, the truth definition for formulas is as before, but with one new key clause for the action modality: (M, s) = [ ϕ]ψ iff M ϕ, s = ψ.

29 Key reduction axiom φ ψ ( A φ ψ) (φ φ ψ). ϕ R ψ ( ϕ R ϕ ψ) R (ϕ ϕ ψ).

30 A general view: PDL-programs Definition Betterness change programs are built from tests for modal betterness formulas, weak and strict basic order relations R, R <, and the universal relation, using arbitrary unions and sequential compositions: π :=?ϕ R R < ; These are interpreted as the standard PDL program operations of test?ϕ, sequential composition ; and choice. Many further relation transformers can be defined in PDL format.

31 Example: radical revision Definition Given any modal betterness model (M, s) and formula ϕ, the radical revision (M ϕ, s) is the model with relations defined as follows in PDL-format: ϕ(r) := (?ϕ; R;?ϕ) (? ϕ; R;? ϕ) (? ϕ; ;?ϕ). Here denotes the universal relation. Under this transformation, all ϕ-worlds become better than all ϕ-worlds, whether or not they were better before, and within these two zones, the old ordering remains.

32 Completeness [van Benthem and Liu, 2007] proved that program expressions drive an automatic formulation of dynamic completeness theorems. Theorem Let π be a betterness change program as defined above. There is a complete dynamic betterness logic for π, and its reduction axioms for the weak and strict modalities can be computed automatically.

33 Summary so far By now, we have an abstract model for betterness and intrinsic preference, both weak and strong, and we can deal with dynamic changes to this structure. We now move to extrinsic preference and its logical analysis.

34 Motivating example Example Alice is going to buy a house. For her, there are several things to consider: the cost, the quality and the neighborhood, strictly in that order. All these criteria are clear-cut for her. For instance, the cost is good if it is inside her budget, otherwise it is bad. Her decision is then determined by the fact whether the alternatives have the desirable properties, and also by the given order of importance for the properties. In other words, Alice s preference regarding houses is derived from the priority order of the properties that she considers.

35 Example: dynamic scenario Example (buying with changing priorities) Alice is going to buy a house. For her, there are several things to consider: the cost, the quality and the neighborhood, strictly in that order. One day, Alice luckily wins a lottery prize of ten million dollars. This changes her situation dramatically. Now she considers the quality most important, then neighborhood, then the cost.

36 Main ideas We think of extrinsic preference as given by priority orders of propositions that encode relevant criteria for comparing worlds or objects.

37 Main ideas We think of extrinsic preference as given by priority orders of propositions that encode relevant criteria for comparing worlds or objects. Our format comes from [de Jongh and Liu, 2009], in which betterness order is derived from linearly ordered priorities.

38 Main ideas We think of extrinsic preference as given by priority orders of propositions that encode relevant criteria for comparing worlds or objects. Our format comes from [de Jongh and Liu, 2009], in which betterness order is derived from linearly ordered priorities. We take a more general approach here, using strict partial orders of propositions inducing pre-orders of worlds, following [Andréka et al., 2002].

39 Priorities and dynamics Definition A priority graph G = P, < is a strictly partially ordered set of propositions in a language L.

40 Priorities and dynamics Definition A priority graph G = P, < is a strictly partially ordered set of propositions in a language L. Definition Let G = P, < be a priority graph, and M a model in which the language L defines properties of objects. The induced betterness relation G is defined as follows:y G x := P G((Py Px) P <P(P x P y)).

41 Lexicographic ordering Lexicographic ordering For total orders G, this reduces to lexicographic ordering: y lin G x := P G (Px Py) P G ( P<P (Px Py) (P x P y)). We refer to [Andréka et al., 2002], [Liu, 2008] for mathematical theory of priority graphs.

42 Relating two levels: Representation theorem Theorem Let M = (W,, V ) be any modal model, without constraints on its relation. The following two statements are equivalent: (a) The relation is a reflexive and transitive order, (b) There is a priority graph G = (P, <) such that, for all worlds x, y W, y x iff y G x. If we start with a given betterness order, we can always find a priority graph that derives it.

43 Basic operations on priority graphs The paper [Andréka et al., 2002] has two basic operations on priority graphs G, G : the sequential composition G ;G adds the graph G on top of G in the order: all nodes in the first come before all those in the second, the parallel composition G G is the disjoint union of the graphs G and G, without any order links between them.

44 Basic graph updates and top deletion Definition Let A be the priority graph with one single node A. The set α(g, A) of basic graph updates is defined by: α(g, A) := A G 1 ; G 2 G 1 G 2.

45 Basic graph updates and top deletion Definition Let A be the priority graph with one single node A. The set α(g, A) of basic graph updates is defined by: α(g, A) := A G 1 ; G 2 G 1 G 2. Definition The top deletion of a (non-empty) priority graph G deletes all propositions that are not dominated by another in the graph order, leaving the rest in their old order to produce the new graph del(g ).

46 Valid algebraic equations Fact 1. G ; G G. 2. G G G. 3. G 1 G 2 G 2 G (G 1 G 2 ) < 5. (G 1 ; G 2 ) < (G 1 < G 2) (G 1 G 2 < ). (G 1 < (G 1 G 2 < )).

47 Modal logic of graph-induced relations Definition Consider a set Prop of propositional variables p, and a set Nom of nominals n. Let G be a set of priority graphs G. The modal graph language is defined by the following syntax rule: ϕ := n p ϕ ψ ϕ G ϕ G < ϕ Eϕ. G := G 1 ; G 2 G 1 G 2. [Girard, 2008] axiomatizes this modal graph logic.

48 Relating betterness and priority dynamics Having presented the betterness-based and priority-based preference logic, we now look at the issue how our two styles of representation and their dynamics are related. Definition Let α: (G, A) G, with G, G priority graphs, and A a new proposition which is not in G. Let σ: (, A) be a map with and betterness relations over worlds. We say that α induces σ, if: σ( G, A) = α(g,a) We call the operation α PDL-definable if it induces a relation transformer σ that is PDL-definable in the format afore-mentioned.

49 Cases of harmony Fact Taking a suggestion A given some betterness relation over worlds is induced by the following basic graph update at the priority level: G A. More precisely, the following diagram commutes: G, < A (G A), < W, A W, A( )

50 Cases of harmony Fact Prefixing a new proposition A to a priority graph (G, <) induces the radical upgrade operation A on possible worlds models. More precisely, the following diagram commutes: G, < A;G (A; G ), < W, A W, A( )

51 Basic graph updates Theorem Basic graph updates induce PDL-betterness transformers.

52 Basic graph updates Theorem Basic graph updates induce PDL-betterness transformers. We prove by brute enumeration: Lemma All basic graph updates reduce to a finite set of cases. Up to graph equivalences, all basic graph updates reduce to the five cases A, G, A; G, G ; A, and A G. They are closed under operations ; and. All these operations indeed induce PDL-definable betterness transformers.

53 Obstacles to a complete match Fact The deletion operation del(g ) is not PDL-definable.

54 Obstacles to a complete match Fact The deletion operation del(g ) is not PDL-definable. Fact Not all PDL-definable operations are graph-definable.

55 Obstacles to a complete match Fact The deletion operation del(g ) is not PDL-definable. Fact Not all PDL-definable operations are graph-definable. Proof. Here is a counter-example. Not all betterness transformers preserve the base properties of reflexivity and transitivity. To see this, consider?a; R, that is: keep the old relation only when A is true. This does not preserve reflexivity, as A-worlds have no relations any more. So this relation-transformer cannot be defined using a partial priority graph.

56 Summary We presented a modal model of intrinsic preference.

57 Summary We presented a modal model of intrinsic preference. Priority structure was introduced to represent reasons for preference, thus we get two level model for extrinsic preference.

58 Summary We presented a modal model of intrinsic preference. Priority structure was introduced to represent reasons for preference, thus we get two level model for extrinsic preference. Dynamics of preference has been studied at the two levels.

59 Summary We presented a modal model of intrinsic preference. Priority structure was introduced to represent reasons for preference, thus we get two level model for extrinsic preference. Dynamics of preference has been studied at the two levels. This view will be used to analyze norm change in deontic logic in the next lecture.

60 The End Thanks! Contact me:

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62 Working Notes of the AAAL Symposium on Decision-Theoretic Planning. Girard, P. (2008). Modal Logics for Belief and Preference Change. PhD thesis, Stanford University. Hansson, S. (2001). Preference logic. In Gabbay, D. and Guenthner, F., editors, Handbook of Philosophical Logic, volume 4, chapter 4, pages Dordrecht: Kluwer. Liu, F. (2008). Changing for the Better: Preference Dynamics and Agent Diversity. PhD thesis, ILLC, University of Amsterdam. Rao, A. and Georgeff, M. (1991). Modeling rational agents within a BDI-architecture.

63 In Allen, J., Fikes, R., and Sandewall, E., editors, Proceedings of the 2nd International Conference on Principles of Knowledge Representation and Reasoning, pages Morgan Kaufmann publishers Inc.: San Mateo, CA, USA. van Benthem, J. and Liu, F. (2007). Dynamic logic of preference upgrade. Journal of Applied Non-Classical Logic, 17: von Wright, G. (1963). The Logic of Preference. Edinburgh University Press.