Gamble space transformations that preserve coherence Possibility spaces X and Z Transformation Γ from L ( Z ) to L ( X ) Conditions for preserving coherence Positive homogeneity: λ > 0 ⇒ Γ ( λ f ) = λΓ f Additivity: Γ ( f + g ) = Γ f + Γ g Positivity: f > 0 ⇔ Γ f > 0 f < 0 ⇔ Γ f < 0 Negativity: which imply λ ∈ R ⇒ Γ ( λ f + g ) = λΓ f + Γ g Linearity: f > g ⇔ Γ f > Γ g Monotonicity: Coherence Preserving Transformation Proposition A transformation preserves coherence if and only if it is linear and monotone.
Transformation of a set of desirable gambles D Γ := { h ∈ L ( Z ): Γ h ∈ D}
Transformation of a set of desirable gambles D Γ := { h ∈ L ( Z ): Γ h ∈ D} b b b b D Γ D a a d d ◮ Γ : L ( { d , b } ) → L ( { a , b } ) � ( a ) = 1 ◮ � Γ h 2 h ( d ) and � Γ h � ( b ) = h ( b )
Taking a slice of a set of desirable gambles ( − 1 3 , 0 , 4 3 ) I c D 2 I b − 1 3 I a I b ( 4 3 , 0 , − 1 3 ) ( 5 3 , 0 , − 2 3 ) ( 1 6 , 5 4 , − 5 12 ) (1 , 5 9 , − 5 9 )
Taking a slice of a set of desirable gambles ( − 1 3 , 0 , 4 3 ) d d D Γ 1 I c D c c 2 I b − 1 3 I a I b ( 4 3 , 0 , − 1 3 ) Γ 1 ( 5 3 , 0 , − 2 3 ) ( 1 6 , 5 4 , − 5 12 ) (1 , 5 9 , − 5 9 ) ◮ Γ 1 : L ( { c , d } ) → L ( { a , b , c } ) ◮ � Γ 1 h � ( a ) = � Γ 1 h � ( b ) = h ( d ) and � Γ 1 h � ( c ) = h ( c )
Taking a slice of a set of desirable gambles ( − 1 3 , 0 , 4 3 ) d d D Γ 1 I c D c c 2 I b − 1 3 I a d d I b ( 4 3 , 0 , − 1 3 ) Γ 2 Γ 1 D Γ 2 ( 5 3 , 0 , − 2 3 ) ( 1 6 , 5 4 , − 5 12 ) (1 , 5 9 , − 5 c c 9 ) ◮ Γ 1 : L ( { c , d } ) → L ( { a , b , c } ) ◮ � Γ 1 h � ( a ) = � Γ 1 h � ( b ) = h ( d ) and � Γ 1 h � ( c ) = h ( c ) ◮ Γ 2 : L ( { c , d } ) → L ( { a , b , c } ) � ( a ) = 3 � ( b ) = 1 ◮ � Γ 2 h 4 h ( d ) , � Γ 2 h 4 h ( d ) and � Γ 2 h � ( c ) = h ( c )
Conditional sets of desirable gambles Conditioning event B ⊆ X is what the experiment’s outcome is assumed to belong to Contingent gambles are those for which, if B does not occur, status quo is maintained Transformation ∤ B c maps gambles on B to contingent gambles on X : � h ( x ) , x ∈ B , � ∤ B c h � ( x ) = x ∈ B c , 0 ,
Conditional sets of desirable gambles Conditioning event B ⊆ X is what the experiment’s outcome is assumed to belong to Contingent gambles are those for which, if B does not occur, status quo is maintained Transformation ∤ B c maps gambles on B to contingent gambles on X : � h ( x ) , x ∈ B , � ∤ B c h � ( x ) = x ∈ B c , 0 , Conditional set of desirable gambles Given a set of desirable gambles D ⊆ L ( X ) , the set of desirable gambles conditional on B is D| B := D ∤ Bc = { h ∈ L ( B ): ∤ B c h ∈ D} ◮ Other formats: ∤ B c ( D| B ) = { f ∈ D : f = fI B } and � = { f ∈ L ( X ): fI B ∈ D} � L ( B c ) ∤ B c ( D| B ) + ∤ B ◮ Can be used as an updated set of desirable gambles
Conditional sets of desirable gambles: example ( − 1 3 , 0 , 4 3 ) I c D 2 I b − 1 3 I a I b ( 4 3 , 0 , − 1 3 ) ( 5 3 , 0 , − 2 3 ) ( 1 6 , 5 4 , − 5 12 ) (1 , 5 9 , − 5 9 )
Conditional sets of desirable gambles: example ( − 1 3 , 0 , 4 3 ) I c D 2 I b − 1 3 I a I b ( 4 3 , 0 , − 1 3 ) ( 5 3 , 0 , − 2 3 ) ( 1 6 , 5 4 , − 5 12 ) (1 , 5 9 , − 5 9 ) b b ( − 1 3 , 4 3 ) D|{ a , b } a a
Conditional sets of desirable gambles: example ( − 1 3 , 0 , 4 3 ) I c D 2 I b − 1 3 I a I b ( 4 3 , 0 , − 1 3 ) ( 5 3 , 0 , − 2 3 ) ( 1 6 , 5 4 , − 5 12 ) (1 , 5 9 , − 5 9 ) c c b b ( − 1 3 , 4 3 ) D|{ a , b } D|{ b , c } a a b b ( 25 18 , − 7 18 )
Conditional sets of desirable gambles: example ( − 1 3 , 0 , 4 3 ) I c D 2 I b − 1 3 I a I b ( 4 3 , 0 , − 1 3 ) ( 5 3 , 0 , − 2 3 ) ( 1 6 , 5 4 , − 5 12 ) (1 , 5 9 , − 5 9 ) c c a a b b ( − 1 3 , 4 ( − 1 3 , 4 3 ) 3 ) D|{ a , b } D|{ b , c } D|{ c , a } a a c c b b ( 4 3 , − 1 3 ) ( 25 18 , − 7 18 )
Marginal sets of desirable gambles Cartesian product possibility space X = Y × Z , focus on Y -component (ignore Z -component) Cylindrical extension ↑ Z maps gambles from the source gamble space to its cartesian product with L ( Z ) : � ↑ Z h � ( y , z ) = h ( y )
Marginal sets of desirable gambles Cartesian product possibility space X = Y × Z , focus on Y -component (ignore Z -component) Cylindrical extension ↑ Z maps gambles from the source gamble space to its cartesian product with L ( Z ) : � ↑ Z h � ( y , z ) = h ( y ) Marginal set of desirable gambles Given a set of desirable gambles D ⊆ L ( Y × Z ) , its Y - marginal is D ↓ Y := D ↑ Z = { h ∈ L ( Y ): ↑ Z h ∈ D} ( a , c ) ( a , c ) ↑ { b , c } ( a , b ) ( a , b )
Marginals for surjective maps and partitions Essential features of marginalization: Surjective map γ ↓Y from X = Y × Z to Y such that ↑ Z h = h ◦ γ ↓Y : γ ↓Y ( y , z ) = y Partition B γ ↓Y can function as the possibility space of the Y -marginal: � = � γ − 1 B γ ↓Y := ↓Y ( y ): y ∈ Y � { y } × Z : y ∈ Y �
Marginals for surjective maps and partitions Essential features of marginalization: Surjective map γ ↓Y from X = Y × Z to Y such that ↑ Z h = h ◦ γ ↓Y : γ ↓Y ( y , z ) = y Partition B γ ↓Y can function as the possibility space of the Y -marginal: � = � γ − 1 B γ ↓Y := ↓Y ( y ): y ∈ Y � { y } × Z : y ∈ Y � Generalization from the Cartesian product case: Surjective map γ Associated transformation Γ γ h = h ◦ γ � γ − 1 ( y ): y ∈ Y and partition B γ := � ; resulting γ - marginal D γ := D Γ γ . Partition B Analogous; define γ B for all x ∈ X by letting γ B ( x ) equal that B in B for which x ∈ B .
Outline Reasoning about and with sets of desirable gambles Derived coherent sets of desirable gambles Combining sets of desirable gambles Joining compatible individuals Marginal extension Partial preference orders Maximally committal sets of strictly desirable gambles Relationships with other, nonequivalent models
Joining compatible individuals How can we combine individual sets of desirable gambles into a joint ? ◮ View the individual sets as derived from the joint: specify the transformations between the individual gamble spaces and the joint gamble space. ◮ The union of the transformed individual sets is taken as an assessment. ◮ Check whether this the individual sets are compatible ; i.e., if the assessment avoids partial loss ◮ If so, the natural extension of the assessment is the joint; if not, there is no coherent joint
Joining compatible individuals How can we combine individual sets of desirable gambles into a joint ? ◮ View the individual sets as derived from the joint: specify the transformations between the individual gamble spaces and the joint gamble space. ◮ The union of the transformed individual sets is taken as an assessment. ◮ Check whether this the individual sets are compatible ; i.e., if the assessment avoids partial loss ◮ If so, the natural extension of the assessment is the joint; if not, there is no coherent joint Consider the following individually coherent conditional sets of desirable gambles: ◮ E ( { ( − 2 , 1) } ) ⊂ L ( { a , b } ) ; a contingent gamble: ( − 2 , 1 , 0) ◮ E ( { ( − 2 , 1) } ) ⊂ L ( { b , c } ) ; a contingent gamble: (0 , − 2 , 1) ◮ E ( { ( − 2 , 1) } ) ⊂ L ( { c , a } ) ; a contingent gamble: (1 , 0 , − 2) They are incompatible: the sum of the given contingent desirable gambles, ( − 1 , − 1 , − 1) , incurs sure loss.
Combining sets of desirable gambles: example ( − 1 3 , 0 , 4 3 ) I c D 2 I b − 1 3 I a I b ( 4 3 , 0 , − 1 3 ) ( 5 3 , 0 , − 2 3 ) ( 1 6 , 5 4 , − 5 12 ) (1 , 5 9 , − 5 9 )
Combining sets of desirable gambles: example ( − 1 3 , 0 , 4 3 ) I c D 2 I b − 1 3 I a I b ( 4 3 , 0 , − 1 3 ) Γ 2 Γ 1 ( 5 3 , 0 , − 2 3 ) ( 1 6 , 5 4 , − 5 12 ) (1 , 5 9 , − 5 9 ) A := Γ 1 ( D Γ 1 ) ∪ Γ 2 ( D Γ 2 ) ∪ ∤ { c } ( D|{ a , b } ) ∪ ∤ { a } ( D|{ b , c } ) ∪ ∤ { b } ( D|{ c , a } )
Combining sets of desirable gambles: example ( − 1 3 , 0 , 4 3 ) ( − 1 3 , 0 , 4 3 ) I c I c D E ( A ) 2 I b − 1 3 ( − 1 3 , 4 3 , 0) I a I a I b I b ( 4 3 , 0 , − 1 3 ) Γ 2 Γ 1 ( 4 3 , 0 , − 1 3 ) ( 25 18 , 0 , − 7 18 ) ( 5 3 , 0 , − 2 3 ) ( 75 63 , 25 63 , − 37 63 ) ( 25 33 , 25 33 , − 17 ( 1 6 , 5 4 , − 5 33 ) 12 ) (1 , 5 9 , − 5 9 ) A := Γ 1 ( D Γ 1 ) ∪ Γ 2 ( D Γ 2 ) ∪ ∤ { c } ( D|{ a , b } ) ∪ ∤ { a } ( D|{ b , c } ) ∪ ∤ { b } ( D|{ c , a } )
Marginal extension Separately specified conditional sets of desirable gambles have disjunct possibility spaces Separately coherent conditional sets of desirable gambles are separately specified and individually coherent
Marginal extension Separately specified conditional sets of desirable gambles have disjunct possibility spaces Separately coherent conditional sets of desirable gambles are separately specified and individually coherent Marginal Extension Theorem Given a partition B of X , a coherent B -marginal D B ⊂ L ( B ) , and separately coherent conditional sets of desirable gambles D| B ⊂ L ( B ) , B ∈ B , then their combination D : = E ( A ) ⊆ L ( X ) , with A : = Γ B ( D B ) ∪ � B ∈B ∤ B c ( D| B ) , is coherent as well.
Marginal extension Separately specified conditional sets of desirable gambles have disjunct possibility spaces Separately coherent conditional sets of desirable gambles are separately specified and individually coherent Marginal Extension Theorem Given a partition B of X , a coherent B -marginal D B ⊂ L ( B ) , and separately coherent conditional sets of desirable gambles D| B ⊂ L ( B ) , B ∈ B , then their combination D : = E ( A ) ⊆ L ( X ) , with A : = Γ B ( D B ) ∪ � B ∈B ∤ B c ( D| B ) , is coherent as well. I c I c E ( A ) ( − 1 3 , 4 3 , 0) I a I a I b I b Γ {{ a , b } , { c }} 2 I b − 1 3 ( 5 3 , 0 , − 2 ( 25 33 , 25 33 , − 17 3 ) 33 )
Exercises 1. Explicitly show that the transformation Γ γ associated to the surjective map γ : { 0 , 1 } 2 → { 0 , 1 , 2 } : x �→ x 1 + x 2 preserves coherence. 1.1 What slice of L ( { 0 , 1 } 2 ) does Γ γ generate? 1.2 What is the partition associated to γ ? 2. Show that the transformation Γ : L ( { 0 , 1 , 2 } ) → L ([0 , 1]) that maps a gamble g to the parabola g (0)(1 − θ ) 2 + g (1) θ (1 − θ ) + 2 g (2) θ 2 in θ does not preserve coherence, by considering 1 − 4 θ + 4 θ 2 . 2.1 Describe the linear subspace of L ([0 , 1]) generated by Γ . 2.2 Define a vector ordering on this subspace that makes Γ preserve coherence. 3. Take E ( A 7 ) from Exercise 2.2 of the previous series. 3.1 Calculate its conditionals for all nonempty events of { a , b , c } , give the extreme-ray representation in all three formats. 3.2 Calculate its marginals for all partitions of { a , b , c } . 3.3 Calculate the marginal extensions of the appropriate derived conditionals and marginals for all partitions of { a , b , c } . 4. Prove the Marginal Extension Theorem.
Outline Reasoning about and with sets of desirable gambles Derived coherent sets of desirable gambles Combining sets of desirable gambles Partial preference orders Strict preference Nonstrict preference Nonstrict preferences implied by strict ones Strict preferences implied by nonstrict ones Maximally committal sets of strictly desirable gambles Relationships with other, nonequivalent models
Partial strict preference order Strict preference f ≻ g if we are eager to exchange g for f Partial . . . order The order does not have to be complete, f ⊁ g ∧ g ⊁ f is possible Strict desirability is strict preference over status quo, the zero gamble 0 : f ≻ g ⇔ f − g ≻ 0 ⇔ f − g ∈ D
Partial strict preference order Strict preference f ≻ g if we are eager to exchange g for f Partial . . . order The order does not have to be complete, f ⊁ g ∧ g ⊁ f is possible Strict desirability is strict preference over status quo, the zero gamble 0 : f ≻ g ⇔ f − g ≻ 0 ⇔ f − g ∈ D Rationality criteria for strict preference relations ≻ on L ( X ) × L ( X ) : f ⊁ f Irreflexivity: Transitivity: f ≻ g ∧ g ≻ h ⇒ f ≻ h � f ≻ g ⇔ µ f + (1 − µ ) h ≻ µ g + (1 − µ ) h � Mix-indep.: 0 < µ ≤ 1 ⇒ Monotonicity: f > g ⇒ f ≻ g
Partial strict preference order Strict preference f ≻ g if we are eager to exchange g for f Partial . . . order The order does not have to be complete, f ⊁ g ∧ g ⊁ f is possible Strict desirability is strict preference over status quo, the zero gamble 0 : f ≻ g ⇔ f − g ≻ 0 ⇔ f − g ∈ D Rationality criteria for strict preference relations ≻ on L ( X ) × L ( X ) : f ⊁ f Irreflexivity: Transitivity: f ≻ g ∧ g ≻ h ⇒ f ≻ h � f ≻ g ⇔ µ f + (1 − µ ) h ≻ µ g + (1 − µ ) h � Mix-indep.: 0 < µ ≤ 1 ⇒ Monotonicity: f > g ⇒ f ≻ g Strengthening coherence criteria for sets of desirable gambles D : Avoiding nonpositivity: f ≤ 0 ⇒ f / ∈ D
Partial strict preference order Strict preference f ≻ g if we are eager to exchange g for f Partial . . . order The order does not have to be complete, f ⊁ g ∧ g ⊁ f is possible Strict desirability is strict preference over status quo, the zero gamble 0 : f ≻ g ⇔ f − g ≻ 0 ⇔ f − g ∈ D Rationality criteria for strict preference relations ≻ on L ( X ) × L ( X ) : f ⊁ f Irreflexivity: Transitivity: f ≻ g ∧ g ≻ h ⇒ f ≻ h � f ≻ g ⇔ µ f + (1 − µ ) h ≻ µ g + (1 − µ ) h � Mix-indep.: 0 < µ ≤ 1 ⇒ Monotonicity: f > g ⇒ f ≻ g Strengthening coherence criteria for sets of desirable gambles D : D ∩ L − Avoiding nonpositivity: 0 ( X ) = ∅
Partial strict preference order Strict preference f ≻ g if we are eager to exchange g for f Partial . . . order The order does not have to be complete, f ⊁ g ∧ g ⊁ f is possible Strict desirability is strict preference over status quo, the zero gamble 0 : f ≻ g ⇔ f − g ≻ 0 ⇔ f − g ∈ D Rationality criteria for strict preference relations ≻ on L ( X ) × L ( X ) : f ⊁ f Irreflexivity: Transitivity: f ≻ g ∧ g ≻ h ⇒ f ≻ h � f ≻ g ⇔ µ f + (1 − µ ) h ≻ µ g + (1 − µ ) h � Mix-indep.: 0 < µ ≤ 1 ⇒ Monotonicity: f > g ⇒ f ≻ g Strengthening coherence criteria for sets of desirable gambles D : Avoiding nonpositivity: 0 / ∈ D
Partial nonstrict preference order Nonstrict preference f � g if we are willing, i.e., not adverse, to exchange g for f Partial . . . order The order does not have to be complete Nonstrict desirability is nonstrict preference over status quo: f � g ⇔ f − g � 0 ⇔ f − g ∈ D
Partial nonstrict preference order Nonstrict preference f � g if we are willing, i.e., not adverse, to exchange g for f Partial . . . order The order does not have to be complete Nonstrict desirability is nonstrict preference over status quo: f � g ⇔ f − g � 0 ⇔ f − g ∈ D Rationality criteria for nonstrict preference relations � on L ( X ) × L ( X ) : Reflexivity: f � f Transitivity: g � h ∧ f � g ⇒ f � h � f � g ⇔ µ f + (1 − µ ) h � µ g + (1 − µ ) h � Mix-indep.: 0 < µ ≤ 1 ⇒ Monotonicity: f > g ⇒ f � g ∧ g � f
Partial nonstrict preference order Nonstrict preference f � g if we are willing, i.e., not adverse, to exchange g for f Partial . . . order The order does not have to be complete Nonstrict desirability is nonstrict preference over status quo: f � g ⇔ f − g � 0 ⇔ f − g ∈ D Rationality criteria for nonstrict preference relations � on L ( X ) × L ( X ) : Reflexivity: f � f Transitivity: g � h ∧ f � g ⇒ f � h � f � g ⇔ µ f + (1 − µ ) h � µ g + (1 − µ ) h � Mix-indep.: 0 < µ ≤ 1 ⇒ Monotonicity: f > g ⇒ f � g ∧ g � f Strengthening coherence criteria for sets of desirable gambles D : Accepting nonnegativity: f ≥ 0 ⇒ f ∈ D
Partial nonstrict preference order Nonstrict preference f � g if we are willing, i.e., not adverse, to exchange g for f Partial . . . order The order does not have to be complete Nonstrict desirability is nonstrict preference over status quo: f � g ⇔ f − g � 0 ⇔ f − g ∈ D Rationality criteria for nonstrict preference relations � on L ( X ) × L ( X ) : Reflexivity: f � f Transitivity: g � h ∧ f � g ⇒ f � h � f � g ⇔ µ f + (1 − µ ) h � µ g + (1 − µ ) h � Mix-indep.: 0 < µ ≤ 1 ⇒ Monotonicity: f > g ⇒ f � g ∧ g � f Strengthening coherence criteria for sets of desirable gambles D : L + Accepting nonnegativity: 0 ( X ) ⊆ D
Strict vs. nonstrict ◮ Strict preference is more useful for decision making
Strict vs. nonstrict ◮ Strict preference is more useful for decision making ◮ Advantages of nonstrict preference: Indifference is the equivalence relation defined by symmetric nonstrict preference: f ≡ g ⇔ f � g ∧ g � f Incomparability is the irreflexive relation defined by symmetric nonstrict nonpreference: f ⊲ ⊳ g ⇔ f � g ∧ g � f
Strict vs. nonstrict ◮ Strict preference is more useful for decision making ◮ Advantages of nonstrict preference: Indifference is the equivalence relation defined by symmetric nonstrict preference: f ≡ g ⇔ f � g ∧ g � f Incomparability is the irreflexive relation defined by symmetric nonstrict nonpreference: f ⊲ ⊳ g ⇔ f � g ∧ g � f K 1 Example: ◮ ≡ -equivalence classes K 1 , K 2 , K 3 K 2 K 3 ◮ intransitivity of ⊲ ⊳ : K 1 ⊲ ⊳ K 3 and K 3 ⊲ ⊳ K 2 , but K 1 � K 2 0
Nonstrict preferences implied by strict ones Motivation Indifference and incomparability are useful concepts Associate a nonstrict preference relation � to a strict one ≻ ; a set of nonstrictly desirable gambles D � to a set of strictly desirable gambles D ≻
Nonstrict preferences implied by strict ones Motivation Indifference and incomparability are useful concepts Associate a nonstrict preference relation � to a strict one ≻ ; a set of nonstrictly desirable gambles D � to a set of strictly desirable gambles D ≻ Bad proposal Let D ≻ := D � ∪ { 0 } ; it makes the difference between � and ≻ vacuous
Nonstrict preferences implied by strict ones Motivation Indifference and incomparability are useful concepts Associate a nonstrict preference relation � to a strict one ≻ ; a set of nonstrictly desirable gambles D � to a set of strictly desirable gambles D ≻ Bad proposal Let D ≻ := D � ∪ { 0 } ; it makes the difference between � and ≻ vacuous Better proposal ‘Making a sweet deal by sweetening an OK deal’: f � g ⇔ f − g � 0 ⇔ ( f − g ) + D ≻ ⊆ D ≻ Immediate consequence: f ≻ g ⇒ g � � f Incomparability ≍ and indifference ≈
Strict and the associated nonstrict preferences: examples g f − g f − g f g − f f g g − f f g − f f − g g
Strict and the associated nonstrict preferences: examples g f − g f − g f g − f f g g − f f g − f f − g g f ≍ g
Strict and the associated nonstrict preferences: examples g f − g f − g f g − f f g g − f f g − f f − g g f ≍ g f ≻ g
Strict and the associated nonstrict preferences: examples g f − g f − g f g − f f g g − f f g − f f − g g f ≍ g f ≻ g f � g
Strict and the associated nonstrict preferences: examples g f − g f − g f g − f f g g − f f g − f f − g g f ≍ g f ≻ g f � g f − g f − g f − g f g − f g − f g f f g − f g g
Strict and the associated nonstrict preferences: examples g f − g f − g f g − f f g g − f f g − f f − g g f ≍ g f ≻ g f � g f − g f − g f − g f g − f g − f g f f g − f g g f ≈ g
Strict and the associated nonstrict preferences: examples g f − g f − g f g − f f g g − f f g − f f − g g f ≍ g f ≻ g f � g f − g f − g f − g f g − f g − f g f f g − f g g f ≈ g f ≻ g
Strict and the associated nonstrict preferences: examples g f − g f − g f g − f f g g − f f g − f f − g g f ≍ g f ≻ g f � g f − g f − g f − g f g − f g − f g f f g − f g g f ≈ g f ≻ g f ≻ g
Strict preferences implied by nonstrict ones Motivation Strict preferences are useful for decision making Associate a strict preference relation ⊲ to a nonstrict one � ; a set of strictly desirable gambles D ⊲ to a set of nonstrictly desirable gambles D �
Strict preferences implied by nonstrict ones Motivation Strict preferences are useful for decision making Associate a strict preference relation ⊲ to a nonstrict one � ; a set of strictly desirable gambles D ⊲ to a set of nonstrictly desirable gambles D � Reuse deal-sweetening? Does not work in general: some D � cannot be associated to any D ⊲
Strict preferences implied by nonstrict ones Motivation Strict preferences are useful for decision making Associate a strict preference relation ⊲ to a nonstrict one � ; a set of strictly desirable gambles D ⊲ to a set of nonstrictly desirable gambles D � Reuse deal-sweetening? Does not work in general: some D � cannot be associated to any D ⊲ Other options? Not pursued: no proliferation of interpretations We continue with strict desirability as the primitive notion
Exercises 1. Possibility space { a , b } . 1.1 Which of ( − 4 , 3) , ( − 3 , 4) , and (3 , − 3) belong to D ≻ , D � , both, or neither, when (5 , − 2) ≈ ( − 2 , 5) . 1.2 Which, or both, or neither of { ( − 1 , 1) } and { (2 , − 3) } is compatible as an assessment with (5 , − 3) ≍ (4 , − 1) . 2. Prove the equivalence of the rationality criteria for strict preference and strict desirability. 3. Prove that � satisfies the rationality criteria of nonstrict preference (assume they are equivalent to those for nonstrict desirability).
Outline Reasoning about and with sets of desirable gambles Derived coherent sets of desirable gambles Combining sets of desirable gambles Partial preference orders Maximally committal sets of strictly desirable gambles Maximally committal coherent extensions Maximality & transformations Relationships with other, nonequivalent models
Maximally committal sets of strictly desirable gambles Maximal coherent sets of (strictly) desirable gambles . . . ◮ are the maximal elements of D ( X ) ordered by inclusion ◮ are not included in any other coherent set of desirable gambles ◮ result in assessments that incur nonpositivity when any gamble in its complement is added to it
Maximally committal sets of strictly desirable gambles Maximal coherent sets of (strictly) desirable gambles . . . ◮ are the maximal elements of D ( X ) ordered by inclusion ◮ are not included in any other coherent set of desirable gambles ◮ result in assessments that incur nonpositivity when any gamble in its complement is added to it Characterization of Maximal Sets of Desirable Gambles The set D in D ( X ) is maximal if and only if f ∈ D ⇔ − f / ∈ D for all nonzero gambles f on X .
Maximally committal sets of strictly desirable gambles Maximal coherent sets of (strictly) desirable gambles . . . ◮ are the maximal elements of D ( X ) ordered by inclusion ◮ are not included in any other coherent set of desirable gambles ◮ result in assessments that incur nonpositivity when any gamble in its complement is added to it Characterization of Maximal Sets of Desirable Gambles The set D in D ( X ) is maximal if and only if f ∈ D ⇔ − f / ∈ D for all nonzero gambles f on X . ◮ are halfspaces that are neither open nor closed ◮ belong to the set ˆ D ( X )
Maximally committal coherent extensions Maximal coherent extension of an assessment A ⊆ L ( X ) Any encompas- sing maximally committal coherent set of desirable gambles b A D 4 D 5 D 6 a a Set of maximal coherent extensions ˆ D A := {D ∈ ˆ D ( X ): A ⊆ D}
Maximally committal coherent extensions Maximal coherent extension of an assessment A ⊆ L ( X ) Any encompas- sing maximally committal coherent set of desirable gambles b A D 4 D 5 D 6 a a Set of maximal coherent extensions ˆ D A := {D ∈ ˆ D ( X ): A ⊆ D} Maximal Sets and Nonpositivity Avoidance Theorem An assessment A ⊆ L ( X ) avoids nonpositivity if and only if ˆ D A � = ∅ .
Maximally committal coherent extensions Maximal coherent extension of an assessment A ⊆ L ( X ) Any encompas- sing maximally committal coherent set of desirable gambles b A D 4 D 5 D 6 a a Set of maximal coherent extensions ˆ D A := {D ∈ ˆ D ( X ): A ⊆ D} Maximal Sets and Nonpositivity Avoidance Theorem An assessment A ⊆ L ( X ) avoids nonpositivity if and only if ˆ D A � = ∅ . Maximal Sets and Natural Extension Corollary The least committal extension of an assessment A ⊆ L ( X ) that avoids nonpositivity, i.e., its natural extension E ( A ) , is the intersection � ˆ D A of the encompassing maximal sets of desirable gambles.
Maximality & transformations Maximality Preserving Transformations Proposition A coherence preserving transformation preserves maximality.
Exercises 1. Possibility space { a , b , c } ; let f := ( − 1 , 1 , 1) be an extreme ray of a maximal set of desirable gambles. 1.1 Draw the intersection with the sum-one plane of the ones for which respectively f + I b − I a and f + I c − I a are nonstrictly desirable. 1.2 Also draw their intersection with the sum-minus one plane. 2. Prove the Characterization of Maximal Sets of Desirable Gambles 3. Prove the Maximal Sets and Natural Extension Corollary 4. Prove the Maximality Preserving Transformations Proposition
Outline Reasoning about and with sets of desirable gambles Derived coherent sets of desirable gambles Combining sets of desirable gambles Partial preference orders Maximally committal sets of strictly desirable gambles Relationships with other, nonequivalent models Linear previsions Credal sets To lower & upper previsions Simplified variants of desirability From lower previsions Conditional lower previsions
Linear previsions Linear previsions . . . ◮ are positive linear normed expectation operators ◮ provide fair prices for gambles in L ( X ) ◮ are equivalent to (finitely additive) probability measures and, on finite X , to probability mass functions
Linear previsions Linear previsions . . . ◮ are positive linear normed expectation operators ◮ provide fair prices for gambles in L ( X ) ◮ are equivalent to (finitely additive) probability measures and, on finite X , to probability mass functions ◮ belong to the closed convex set P ( X ) which is, for finite X , the unit simplex spanned by the degenerate previsions (or { 0 , 1 } -valued probability mass functions) P c P a P b
Linear previsions Linear previsions . . . ◮ are positive linear normed expectation operators ◮ provide fair prices for gambles in L ( X ) ◮ are equivalent to (finitely additive) probability measures and, on finite X , to probability mass functions ◮ belong to the closed convex set P ( X ) which is, for finite X , the unit simplex spanned by the degenerate previsions (or { 0 , 1 } -valued probability mass functions) P c P a P b ◮ provide probabilities for events, as fair prices for their indicators
From linear previsions to sets of desirable gambles Given a linear prevision P ∈ P ( X ) , gambles with a strictly positive fair price are strictly desirable: D P := E ( A P ) , with A P := � f ∈ L ( X ): P ( f ) > 0 �
From linear previsions to sets of desirable gambles Given a linear prevision P ∈ P ( X ) , gambles with a strictly positive fair price are strictly desirable: D P := E ( A P ) , with A P := � f ∈ L ( X ): P ( f ) > 0 � Observations: � is a linear subspace of L ( X ) ◮ � f ∈ L ( X ): P ( f ) = 0 ◮ So A P is an open halfspace ◮ Except in a few borderline cases, so is D P b b b b a a a a ◮ Except in two nontrivial cases, D P is nonmaximal, so ˆ D P ⊆ D P are nontrivial
From credal sets to sets of desirable gambles A credal set is a set of linear previsions Given a credal set M ⊆ P ( X ) , gambles with a strictly positive fair price for every linear prevision in the credal set are strictly desirable: D M := E ( A M ) , with A M := � f ∈ L ( X ): ( ∀ P ∈ M : P ( f ) > 0) � � = A P P ∈M
From credal sets to sets of desirable gambles A credal set is a set of linear previsions Given a credal set M ⊆ P ( X ) , gambles with a strictly positive fair price for every linear prevision in the credal set are strictly desirable: D M := E ( A M ) , with A M := � f ∈ L ( X ): ( ∀ P ∈ M : P ( f ) > 0) � � = A P P ∈M Observations: ◮ Each prevision gives rise to a linear constraint in gamble space ◮ Constraints from linear previsions strictly in the convex hull of M are redundant ◮ So the border structure of M is uniquely important
From credal sets to sets of desirable gambles: example P c ( 1 6 , 1 6 , 2 3 ) ( 1 3 , 0 , 2 3 ) ( 1 6 , 1 3 , 1 2 ) ( 2 3 , 0 , 1 3 ) ( 1 6 , 1 2 , 1 3 ) M ( 2 3 , 1 6 , 1 ( 1 6 , 2 3 , 1 6 ) 6 ) P a P b ( 2 3 , 1 ( 1 3 , 2 3 , 0) 3 , 0)
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