PHIL309P Methods in Philosophy, Politics and Economics Eric Pacuit - PowerPoint PPT Presentation

PHIL309P Methods in Philosophy, Politics and Economics Eric Pacuit University of Maryland 1 / 16

Cardinal Utility Theory u : X → R Which comparisons are meaningful? 1. u ( x ) and u ( y ) ? (ordinal utility) 2 / 16

Cardinal Utility Theory u : X → R Which comparisons are meaningful? 1. u ( x ) and u ( y ) ? (ordinal utility) 2. u ( x ) − u ( y ) and u ( a ) − u ( b ) ? 2 / 16

Cardinal Utility Theory u : X → R Which comparisons are meaningful? 1. u ( x ) and u ( y ) ? (ordinal utility) 2. u ( x ) − u ( y ) and u ( a ) − u ( b ) ? 3. u ( x ) and 2 ∗ u ( z ) ? 2 / 16

Ordinal vs. Cardinal Utility Ordinal scale : Qualitative comparisons of objects allowed, no information about differences or ratios. 3 / 16

Ordinal vs. Cardinal Utility Ordinal scale : Qualitative comparisons of objects allowed, no information about differences or ratios. Cardinal scales : Interval scale : Quantitative comparisons of objects, accurately reflects differences between objects. E.g., the difference between 75 ◦ F and 70 ◦ F is the same as the difference between 30 ◦ F and 25 ◦ F However, 70 ◦ F ( = 21 . 11 ◦ C) is not twice as hot as 35 ◦ F ( = 1 . 67 ◦ C). 3 / 16

Ordinal vs. Cardinal Utility Ordinal scale : Qualitative comparisons of objects allowed, no information about differences or ratios. Cardinal scales : Interval scale : Quantitative comparisons of objects, accurately reflects differences between objects. E.g., the difference between 75 ◦ F and 70 ◦ F is the same as the difference between 30 ◦ F and 25 ◦ F However, 70 ◦ F ( = 21 . 11 ◦ C) is not twice as hot as 35 ◦ F ( = 1 . 67 ◦ C). Ratio scale : Quantitative comparisons of objects, accurately reflects ratios between objects. E.g., 10lb ( = 4 . 53592 kg ) is twice as much as 5lb ( = 2 . 26796 kg ). 3 / 16

Cardinal Utility Theory x ≻ y ≻ z is represented by both ( 3 , 2 , 1 ) and ( 1000 , 999 , 1 ) , so we cannot say y whether is “closer” to x than to z . 4 / 16

Cardinal Utility Theory x ≻ y ≻ z is represented by both ( 3 , 2 , 1 ) and ( 1000 , 999 , 1 ) , so we cannot say y whether is “closer” to x than to z . Key idea: Ordinal preferences over lotteries allows us to infer a cardinal (interval) scale (with some additional axioms). John von Neumann and Oskar Morgenstern. The Theory of Games and Economic Behavior . Princeton University Press, 1944. 4 / 16

A Choice Take or Gamble ? R Take Gamble B B W S 0 . 5 0 . 5 R S [ B : 1 ] ∼ [ R : p , S : 1 − p ] 5 / 16

A Choice Take or Gamble ? R Take Gamble B B W S 0 . 5 0 . 5 R S [ B : 1 ] ∼ [ R : p , S : 1 − p ] 5 / 16

A Choice Take or Gamble ? R Take Gamble B B W S 0 . 5 0 . 5 R S [ B : 1 ] ∼ [ R : p , S : 1 − p ] 5 / 16

A Choice Take or Gamble ? R Take Gamble B B W p 1 − p S R S [ B : 1 ] ∼ [ R : p , S : 1 − p ] 5 / 16

A Choice Take or Gamble ? R Take Gamble B B W p 1 − p S R S [ B : 1 ] ∼ [ R : p , S : 1 − p ] 5 / 16

A Choice Take or Gamble ? R Take Gamble B B W p 1 − p S R S 1 ∗ u ( B ) = p ∗ u ( R ) + ( 1 − p ) ∗ u ( S ) 5 / 16

A Choice Take or Gamble ? R Take Gamble B B W p 1 − p S R S u ( B ) = p ∗ 1 + ( 1 − p ) ∗ 0 = p 5 / 16

Lotteries Suppose that X = { x 1 , . . . , x n } is a set of outcomes. A lottery over X is a tuple [ x 1 : p 1 , x 2 : p 2 , . . . , x n : p n ] where � i p i = 1. 6 / 16

Lotteries Suppose that X = { x 1 , . . . , x n } is a set of outcomes. A lottery over X is a tuple [ x 1 : p 1 , x 2 : p 2 , . . . , x n : p n ] where � i p i = 1. p 1 p 2 p n − 1 p n · · · x 1 x 2 x n − 1 x n We identify an element x ∈ X with the lottery [ x : 1 ] ∈ L 6 / 16

Let L be the set of lotteries. Suppose that �⊆ L × L is a represents a decision maker’s (rational) preferences on L . What properties should � satisfy? 7 / 16

Let L be the set of lotteries. Suppose that �⊆ L × L is a represents a decision maker’s (rational) preferences on L . What properties should � satisfy? Fact . If � is complete and transitive (plus another condition since L is infinite), then there is a U : L → R such that L � L ′ iff U ( L ) ≥ U ( L ′ ) 7 / 16

Let L be the set of lotteries. Suppose that �⊆ L × L is a represents a decision maker’s (rational) preferences on L . What properties should � satisfy? Fact . If � is complete and transitive (plus another condition since L is infinite), then there is a U : L → R such that L � L ′ iff U ( L ) ≥ U ( L ′ ) For any lottery L = [ x 1 : p 1 , . . . , x n : p n ] , we would like to show that U ( L ) = � n k = 1 p k U ( x k ) . 7 / 16

Independence Axiom 8 / 16

< 9 / 16

< 9 / 16

= 9 / 16

= 9 / 16

= 9 / 16

> 9 / 16

> 9 / 16

L = [ A 1 : p 1 , A 2 : p 2 , A 3 : p 3 ] A 1 A 2 A 3 u ( A 1 ) p 1 p 2 p 3 p 1 p 1 + p 2 p 1 + p 2 + p 3 0 10 / 16

L = [ A 1 : p 1 , A 2 : p 2 , A 3 : p 3 ] A 1 A 2 A 3 u ( A 1 ) p 1 p 2 p 3 p 1 p 1 + p 2 p 1 + p 2 + p 3 = 1 0 10 / 16

L = [ A 1 : p 1 , A 2 : p 2 , A 3 : p 3 ] A 1 A 2 A 3 u ( A 1 ) p 1 p 2 p 3 p 1 p 1 + p 2 p 1 + p 2 + p 3 = 1 0 10 / 16

L = [ A 1 : p 1 , A 2 : p 2 , A 3 : p 3 ] A 1 A 2 A 3 u ( A 1 ) p 2 p 3 p 1 p 1 + p 2 p 1 + p 2 + p 3 = 1 0 10 / 16

L = [ A 1 : p 1 , A 2 : p 2 , A 3 : p 3 ] A 1 A 3 u ( A 1 ) A 2 p 3 p 1 p 1 + p 2 p 1 + p 2 + p 3 = 1 0 10 / 16

L = [ A 1 : p 1 , A 2 : p 2 , A 3 : p 3 ] A 1 u ( A 1 ) A 2 A 3 p 1 p 1 + p 2 p 1 + p 2 + p 3 = 1 0 10 / 16

L = [ A 1 : p 1 , A 2 : p 2 , A 3 : p 3 ] u ( A 1 ) u ( A 2 ) p 1 ∗ u ( A 1 ) p 2 ∗ u ( A 2 ) u ( A 3 ) p 3 ∗ u ( A 3 ) p 1 p 1 + p 2 p 1 + p 2 + p 3 = 1 0 10 / 16

L = [ A 1 : p 1 , A 2 : p 2 , A 3 : p 3 ] u ( A 1 ) u ( A 2 ) p 1 ∗ u ( A 1 ) p 2 ∗ u ( A 2 ) u ( A 3 ) p 3 ∗ u ( A 3 ) p 1 p 1 + p 2 p 1 + p 2 + p 3 = 1 0 0 1 10 / 16

L = [ A 1 : p 1 , A 2 : p 2 , A 3 : p 3 ] u ( A 2 ) p 2 ∗ u ( A 2 ) u ( A 3 ) p 3 ∗ u ( A 3 ) p 1 p 1 + p 2 p 1 + p 2 + p 3 = 1 0 10 / 16

L = [ A 1 : p 1 , A 2 : p 2 , A 3 : p 3 ] u ( A 2 ) p 1 p 1 + p 2 p 1 + p 2 + p 3 = 1 0 10 / 16

[ A 1 : p 1 , A 2 : p 2 , A 3 : p 3 ] [ A 1 : p 1 , B 2 : p 2 , B 3 : p 3 ] u ( A 2 ) u ( A 2 ) p 1 p 1 p 1 + p 2 p 1 + p 2 + p 3 = 1 p 1 + p 2 p 1 + p 2 + p 3 = 1 0 0 [ A 1 : p 1 , B 2 : p 2 , B 3 : p 3 ] ≻ [ A 1 : p 1 , B 2 : p 2 , B 3 : p 3 ] iff [ B 2 : p 2 , B 3 : p 3 ] ≻ [ B 2 : p 2 , B 3 : p 3 ] 10 / 16

[ A 1 : p 1 , A 2 : p 2 , A 3 : p 3 ] [ A 1 : p 1 , B 2 : p 2 , B 3 : p 3 ] u ( A 2 ) u ( A 2 ) p 1 p 1 p 1 + p 2 p 1 + p 2 + p 3 = 1 p 1 + p 2 p 1 + p 2 + p 3 = 1 0 0 [ A 1 : p 1 , A 2 : p 2 , A 3 : p 3 ] ≻ [ A 1 : p 1 , B 2 : p 2 , B 3 : p 3 ] iff [ B 2 : p 2 , B 3 : p 3 ] ≻ [ B 2 : p 2 , B 3 : p 3 ] 10 / 16

[ A 1 : p 1 , A 2 : p 2 , A 3 : p 3 ] [ A 1 : p 1 , B 2 : p 2 , B 3 : p 3 ] u ( A 2 ) u ( A 2 ) p 1 p 1 p 1 + p 2 p 1 + p 2 + p 3 = 1 p 1 + p 2 p 1 + p 2 + p 3 = 1 0 0 [ A 1 : p 1 , A 2 : p 2 , A 3 : p 3 ] ≻ [ A 1 : p 1 , B 2 : p 2 , B 3 : p 3 ] iff [ A 2 : p ′ 2 , A 3 : p ′ 3 ] ≻ [ B 2 : p 2 , B 3 : p 3 ] 10 / 16

[ A 1 : p 1 , A 2 : p 2 , A 3 : p 3 ] [ A 1 : p 1 , B 2 : p 2 , B 3 : p 3 ] u ( A 2 ) u ( A 2 ) p 1 p 1 p 1 + p 2 p 1 + p 2 + p 3 = 1 p 1 + p 2 p 1 + p 2 + p 3 = 1 0 0 [ A 1 : p 1 , A 2 : p 2 , A 3 : p 3 ] ∼ [ A 1 : p 1 , B 2 : p 2 , B 3 : p 3 ] iff [ A 2 : p ′ 2 , A 3 : p ′ 3 ] ∼ [ B 2 : p 2 , B 3 : p 3 ] 10 / 16

[ A 1 : p 1 , A 2 : p 2 , A 3 : p 3 ] [ A 1 : p 1 , B 2 : p 2 , B 3 : p 3 ] u ( A 2 ) u ( A 2 ) p 1 p 1 p 1 + p 2 p 1 + p 2 + p 3 = 1 p 1 + p 2 p 1 + p 2 + p 3 = 1 0 0 [ A 1 : p 1 , A 2 : p 2 , A 3 : p 3 ] ≺ [ A 1 : p 1 , B 2 : p 2 , B 3 : p 3 ] iff [ A 2 : p ′ 2 , A 3 : p ′ 3 ] ≺ [ B 2 : p 2 , B 3 : p 3 ] 10 / 16

[ A 1 : p 1 , A 2 : p 2 , A 3 : p 3 ] [ A 1 : p 1 , B 2 : p 2 , B 3 : p 3 ] u ( A 2 ) u ( A 2 ) p 1 p 1 p 1 + p 2 p 1 + p 2 + p 3 = 1 p 1 + p 2 p 1 + p 2 + p 3 = 1 0 0 [ A 1 : p 1 , A 2 : p 2 , A 3 : p 3 ] ≻ / ∼ / ≺ [ A 1 : p 1 , B 2 : q 2 , B 3 : q 3 ] iff [ A 2 : p ′ 2 , A 3 : p ′ 3 ] ≻ / ∼ / ≺ [ B 2 : q ′ 2 , B 3 : q ′ 3 ] 10 / 16

[ A 1 : p 1 , A 2 : p 2 , A 3 : p 3 ] [ A 1 : p 1 , B 2 : p 2 , B 3 : p 3 ] u ( A 2 ) u ( A 2 ) p 1 p 1 p 1 + p 2 p 1 + p 2 + p 3 = 1 p 1 + p 2 p 1 + p 2 + p 3 = 1 0 0 [ A 1 : p 1 , A 2 : p 2 , A 3 : p 3 ] ≻ / ∼ / ≺ [ A 1 : p 1 , B 2 : q 2 , B 3 : q 3 ] iff [ A 2 : p ′ 2 , A 3 : p ′ 3 ] ≻ / ∼ / ≺ [ B 2 : q ′ 2 , B 3 : q ′ 3 ] 10 / 16

Independence For all L 1 , L 2 , L 3 ∈ L and a ∈ ( 0 , 1 ] , L 1 ≻ L 2 if, and only if, [ L 1 : a , L 3 : ( 1 − a )] ≻ [ L 2 : a , L 3 : ( 1 − a )] . L 1 ∼ L 2 if, and only if, [ L 1 : a , L 3 : ( 1 − a )] ∼ [ L 2 : a , L 3 : ( 1 − a )] . 11 / 16

Better Prizes Better prizes : When two lotteries are the same except for one outcome, then the decision maker prefers the lottery with the better outcome. p p ≻ 1 − p 1 − p a c c b ≻ 12 / 16

Better Chances Better Chances : A decision maker prefers a better chance for a better prize a ≻ b p > q p 1 − p ≻ q 1 − q a a b b 13 / 16

Better Chances a ≻ b p > q , so p = q + r and ( 1 − q ) = ( 1 − p ) + r for some r p ≻ q 1 − p 1 − q a a b b 13 / 16