Preference Logic

Preference Logic

Pramod Parajuli 2011

Preference Logic Pramod Parajuli, 2011

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Sorite's Paradox: C0~C1~C2~…~C997~C998~C999 Solution: use of ‘Just Noticeable Di!erence’ ( JND)

Ci�Cj i!, u(Ci)-u(Cj) � " " > 0

C0~C1~C2 C997~C998~C999 Preferences are states of mind whereas choices are actions.

Pramod Parajuli, 2011 Preference Logic

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The problem domain

!  Optimal solutions are di"cult to obtain.

!  Finding sub-optimal solutions •  Relaxing the objective function or •  Relaxing the constraints


4 Agenda

!  To explore preference logic


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Preference types

!  Label preference, e.g. #nite set and alternatives

!  Object preferences, e.g. orange vs. apple

!  Action-object preference, e.g. drinking tea or drinking co!ee

!  Monadic preferences, e.g. good, bad

!  and many other types: intrinsic, extrinsic, conditional preferences etc.


6 Preference operators

!  Preference operator: � e.g. A�B •  transitive, if A�B and B�C, then A�C

!  Indi!erence: ~ e.g. A~B •  re$exive, if A~B, then B~A

!  At least as good as: � e.g. AB •  transitive, re$exive


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Basic preference operator

!  At least as good as: � AB

!  Preference: � A�B i! AB and ¬(BA)

!  Indi!erence: ~ A~B i! AB and BA


8 Choice

!  Choice is revealed preference (action).

!  Choice function Let ‘C’ be a choice function. If ‘C’ is applied for set of alternatives ‘B’ then

for all B�A: C(B)�B, if B#Ø, then C(B)#Ø


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A

B

Choice properties (social)

!  % – property (Cherno!) If B�A then B�C(A) �C(B)

Problem: C({A,B,C}) = {A} C({A,B}) = {B} B = Ø

!  & – property If B�A and X,Y�C(B), then X�C(A) i! Y�C(A)

Problem: One must be Australia champion to become world champion.

!  ! - property (expansion) C(A1)�…�C(An) �C(A1�…�An)


10 Choice properties (economic)

!  Weak axiom of revealed preference (WARP) If X,Y�A and X�C(A), then for all B, if X�B, and Y�C(B), then X�C(B)

Problem: o!ensive choices

!  Strong axiom of revealed preference (SARP) Recursive closure of WARP In words: From a set of alternatives A1, if X is chosen while Y is available, and if in some other sets alternatives A2, Y is chosen while Z is available, then there can be no set of alternatives containing alternatives X and Z for which Z is chosen but X is not.


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Monadic predicates

!  Good: better than its negation A � ¬A

!  Bad: worse than its negation ¬A � A

!  Goodness �B. A � B~¬B

!  Badness �B. B~¬B � A


12 Preference metrics

!  Completeness (incompleteness)

!  Transitivity

!  Order Ai�Xk�Aj or Aj�Xk�Ai holds for each pair of labels (Ai, Aj), i#j


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Constructing preferences from choice

'ree di!erent methods:

i.  At least as good as XY i! for some B, X�C(B) and Y�B X�Y i! XY and ¬(YX) X�Y i! XY and YX

ii.  At least as good as in a binary set XY i! X�C({X,Y}) X�Y i! XY and ¬(YX) X�Y i! XY and YX


14 Constructing preferences from choice

iii.  Strictly preferred to X�Y i! for some B, X�C(B) and Y�[B\C(B)] XY i! ¬(X�Y) X�Y i! XY and YX


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Learning preferences

!  Let’s consider an agent is able to choose among worlds

!  Let’s consider two propositions de#ne possible set of worlds: p – 'e agent mostly visits Bondi beach. q – 'e agent plays skate on the way. Now, four possible set of worlds can be de#ned: W1: p.q W2: p.¬q W3: ¬p.q W4: ¬p. ¬q


16 Learning preferences

!  Now, let’s consider, through interaction with many agents, we found probability and desirability of the possible set of worlds World Probability Desirability W1: p.q 1/6 -2 W2: p.¬q 2/6 1 W3: ¬p.q 2/6 -1 W4: ¬p. ¬q 1/6 3

'e value of a proposition can be evaluated as: value = for all true occurrence/s of proposition, sum(probability $ desirability)


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Learning preferences

!  value of p, #(p) =

!  value of q, #(q) =

!  Similarly, #(¬p) = , #(¬q) =

!  Since #(p)>#(q) and #(¬q)>#(¬p), we conclude that p�q. 'e agent prefers going Bondi beach than playing skate.


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Implementation !  Implementation of preferences has been taken as relaxation in

optimization procedure ( Jayaraman & Govindrajan & Mantha, 1998).

!  It is achieved through constraint relaxation. Let’s consider, we have a function ‘shortest-path’ de#ned as; shortest-path(X, Y, C, P) ⟶ path(X, Y, C, P). – path P with distance C from

X to Y. shortest-path(X,Y,C1,P1) shortest-path(X,Y,C2,P2) ⟵ C1 < C2.


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Challenges

!  Expressivity and completeness

!  Transitivity

!  Di!erent world problem, preference change, temporal preferences

!  Belief

!  Commitments

!  Privacy

!  Criticisms (“People do and should act as problem solvers, not maximizers, because they have many di!erent and incommensurable… goals to achieve” Steven G. Kranz)


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'ank you for your kind attention.

(uestions and suggestions are welcome.


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Bibliography

'e primary source of the concepts presented here is the article ‘Preferences’ in Stanford Encyclopedia of Philosophy, http://plato.stanford.edu/entries/preferences/ Accessed on August 12, 2011.


23 Annex-1: constructing choice from preferences

'ree cases:

i.  'e best choice connection CB(B) = {X � B | �Y � B: (XY)}

ii.  'e non-dominance choice connection CL(B) = {X � B | �Y � B: ¬(Y�X)}

iii.  'e optimization choice connection When cyclic preferences exist, A�B�C�A, CB(A,B,C) = CL(A,B,C) = Ø


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Annex-1: constructing choice from preferences

iii.  'e optimization choice connection continued… Let S be the set of these sets. Now, B is in S i!: a)  B � �(A) b)  for all X,Y: if X�A\B and Y�B then ¬(X �Y) c)  for all F�B there is a Y�F such that for some X�A\F: X�Y Now, the choice function is de#ned as the union of S: CO(A) = � S


25 Annex-2: details of SARP

If

X1,X2, ..., Xn�A1, X2, ..., Xn�A2, ..., Xn-1,Xn�An%1, Xn�An, and X1�C(A1), X2�C(A2), ..., Xn�C(An),

then,

for all B with X1,X2,...,Xn�B, if Xi�C(B), i�{1,...,n},

then X1,X2,...,Xi%1�C(B) (SARP)


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Annex-3: questions raised at the end

Any formal model for desirability?

How would we capture desirability? (ualitative desirability?

How preferences would #t in automated planning? How would the cost model developed?

Can preferences contribute to risk-modeling and mitigation?

How can the machine-learning methods be used for capturing preferences?

How preference-based model di!ers from Bayesian decision making theory?


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Preference Logic