A state-independent preference representation in he continuous case David R´ ıos, Enrique Miranda Rey Juan Carlos University, University of Oviedo COST meeting, October 2008 A state-independent preference representation in he continuous case – p. 1/39
The setting Take a set of alternatives A , a set of states S and a set of consequences C . We consider an order � between the alternatives, so: • a � b means ‘alternative a is preferred to alternative b ’. • a ≻ b means ‘alternative a is strictly preferred to alternative b ’. • a ∼ b means ‘alternative a is indifferent to alternative b ’. The idea of an axiomatisation is to provide necessary and sufficient conditions on � to be able to represent it by means of an expected utility model . A state-independent preference representation in he continuous case – p. 2/39
Some axiomatisations • L. Savage, The foundations of statistics . Wiley, 1954. • F. Anscombe and R. Aumann, A definition of subjective probability . Annals of Mathematical Statistics, 34, 199-205, 1963. • M. de Groot, Optimal Statistical Decisions . McGraw Hill, 1970. A state-independent preference representation in he continuous case – p. 3/39
The completeness axiom The axiomatisations above all require that � is weak order, i.e., complete and transitive: this means in particular that we can express our preferences between any pair of alternatives. Then we obtain a unique utility function u over C and a unique probability p over s such that � � a � b ⇔ u ( c ( a, s )) p ( s ) dcds S C � � ≥ u ( c ( b, s )) p ( s ) dcds. S C A state-independent preference representation in he continuous case – p. 4/39
Dealing with incomplete information If we do not have enough information, it is more reasonable that the order between the alternatives is only a quasi-order (reflexive and transitive): there will be alternatives for which we cannot express a preference with guarantees. ֒ → But then there will not be a unique probability and/or utility representing our information! A state-independent preference representation in he continuous case – p. 5/39
Generalisations to imprecise utilities We consider a unique probability distribution over S and a set U of utility functions over C. • R. Aumann, Utility theory without the completeness axiom . Econometrica 30, 445-462, 1962. • J. Dubra, F. Maccheroni, E. Ok, Expected utility theory without the completeness axiom . Journal of Economic Theory, 115, 118-133, 2004. A state-independent preference representation in he continuous case – p. 6/39
Generalisations to imprecise beliefs We consider a convex set P of probability distributions over S and a unique utility function u . • D. Ríos Insua, F. Ruggeri, Robust Bayesian Analysis . Lecture Notes in Statistics 152. Springer, 2000. • P. Walley, Statistical Reasoning with Imprecise Probabilities . Chapman and Hall, 1991. • R. Rigotti, C. Shannon, Uncertainty and risk in financial markets . Econometrica, 73, 203–243, 2005. A state-independent preference representation in he continuous case – p. 7/39
Imprecise utilities and beliefs Our goal is to give an axiomatisation for the case where both probabilities and utilities are imprecise, so we have a set P of probabilities and a set U of utilities which are paired up arbitrarily. Some early work in this direction can be found in • D. Ríos Insua, Sensitivity analysis in multiobjective decision making . Springer, 1990. • D. Ríos Insua, On the foundations of decision making under partial information . Theory and Decision, 33, 83-100, 1992. A state-independent preference representation in he continuous case – p. 8/39
State dependence and independence In general the axiomatisations for imprecise beliefs and utilities are made for so-called state-dependent utilities, i.e., functions v : S × C → R , such that � � a � b ⇔ v ( s, c ( a, s )) dcds S C � � ≥ v ( s, c ( b, s )) dcds ∀ v ∈ V. S C v is called state-independent or a probability-utility pair when it can be expressed as a product of a probability p over S and a utility U over C: v ( s, c ) = p ( s ) u ( c ) ∀ s, c. A state-independent preference representation in he continuous case – p. 9/39
Some state independent representa- tions • R. Nau, The shape of incomplete preferences . Annals of Statistics, 34(5), 2430-2448, 2006. • T. Seidenfeld, M. Schervisch, J. Kadane, A representation of partially ordered preferences . Annals of Statistics, 23(6), 2168-2217, 1995. • A. García del Amo and D. Ríos Insua, A note on an open problem in the foundations of statstics . RACSAM, 96(1), 55-61, 2002. A state-independent preference representation in he continuous case – p. 10/39
Nau’s framework • A finite set of states S and a finite set of consequences C. • The set B of horse lotteries f : S → P ( C ) . • H c denotes the lottery such that H c ( s )( c ) = 1 ∀ s ∈ S . • 1 denotes the best consequence in C , and 0 the worst. • For any E ⊆ S and any horse lotteries f, g , Ef + E c g is the horse lottery equal to f ( s ) if s ∈ E and to g ( s ) is s / ∈ E . A state-independent preference representation in he continuous case – p. 11/39
The axioms (A1) � is transitive and reflexive. (A2) f � g ⇔ αf + (1 − α ) h � αg + (1 − α ) h ∀ α ∈ (0 , 1) , h . (A3) f n � g n ∀ n, f n → f, g n → g ⇒ f � g . (A4) H 1 � H c � H 0 ∀ c . (A5) H 1 ≻ H 0 . A state-independent preference representation in he continuous case – p. 12/39
A state-dependent representation � satisfies A1–A5 ⇔ it is represented by a closed convex set of state-dependent utility functions V , in the sense that f � g ⇔ U v ( f ) ≥ U v ( g ) ∀ v ∈ V , where � U v ( f ) = f ( s, c ) v ( s, c ) . s ∈ S,c ∈ C A state-independent preference representation in he continuous case – p. 13/39
A state-independent representation (A6) If f, g are constant, f ′ � g ′ , H E � H p , H F � H q with p > 0 , then αEf + (1 − α ) f ′ � αEg + ( a − α ) g ′ ⇒ βFf + (1 − β ) f ′ � βFg + (1 − β ) g ′ β p α for β = 1 if α = 1 and for β s.t. 1 − β ≤ q . 1 − α � satisfies (A1)–(A6) if and only if it is represented by a set V ′ of state-independent utilities, f � g ⇔ U v ( f ) ≥ U v ( g ) ∀ v ∈ V ′ , where U v ( f ) = � s ∈ S,c ∈ C f ( s, c ) p ( s ) u ( c ) . A state-independent preference representation in he continuous case – p. 14/39
Seidenfeld, Schervisch, Kadane • A countable set of consequences C . • A finite set of states S . • Horse lotteries f : S → P ( C ) , and in particular simple horse lotteries, i.e., horse lotteries for which f ( s ) is a simple probability distribution for all s . • A strict preference relationship ≻ over horse lotteries. A state-independent preference representation in he continuous case – p. 15/39
The axioms (A1) ≻ is transitive and irreflexive. (A2) For any f, g, h , and any α ∈ (0 , 1) , αf + (1 − α ) h ≻ αg + (1 − α ) h ⇔ f ≻ g . (A3) Let ( f n ) n → f, ( g n ) n → g . Then: • f n ≻ g n ∀ n and g ≻ h ⇒ f ≻ h . • f n ≻ g n ∀ n and h ≻ f ⇒ h ≻ g . If ≻ satisfies axioms (A1)–(A3), then: • It can be extended to a weak order � satisfying (A2), (A3). • ≻ is uniquely represented by a (bounded) utility v that agrees with ≻ on simple horse lotteries. A state-independent preference representation in he continuous case – p. 16/39
The representation theorem above is made in terms of state-dependent utilities: any v has associated a probability p and utility functions u 1 , . . . , u n , so that for every horse lottery f , n � v ( f ) = p ( s j ) u j ( f ( s )) . j =1 The goal would be to have u 1 = . . . , u n , i.e., state-independent utilities. A state-independent preference representation in he continuous case – p. 17/39
Almost state-independent utilities ≻ admits almost state-independent utilities when for any finite set of rewards { r 1 , . . . , r n } , ǫ > 0 , there is a pair ( p, u j ) s.t. for any { s 1 , . . . , s k } s.t. � k i =1 p ( s i ) > 1 − ǫ , 1 ≤ i ≤ n, 1 ≤ j � = j ′ ≤ k | u j ( r i ) − u j ′ ( r i ) | < ǫ. max A state-independent preference representation in he continuous case – p. 18/39
Some definitions A state s is ≻ - potentially null when for any horse lotteries f, g with f ( s ′ ) = g ( s ′ ) ∀ s ′ � = s , f ∼ g . We denote f L the horse lottery which is constant on the probability distribution L over C . Given a constant horse lottery f L α , � (1 − 2 − m ) f 0 + 2 − m f L α if s � = s j f α j,m := f L α if s = s j A state-independent preference representation in he continuous case – p. 19/39
An (almost) state-independent repre- sentation • (A4) If s j is not ≻ potentially null, then for each acts f L 1 , f L 2 , f 1 , f 2 , f L 1 ≻ f L 2 ⇔ f 1 ≻ f 2 , where f i ( s ) = f i if s = s j , f 1 ( s ) = f 2 ( s ) otherwise. • (A5) For any two constant horse lotteries f L α , f L β , it holds that j,m ≻ f β f L α ≻ f L β ⇔ f α j,m ∀ m ∈ N , ∀ j . If ≻ satisfies (A1)–(A5), then it admits almost state-independent utilites. A state-independent preference representation in he continuous case – p. 20/39
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