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

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PHIL309P Methods in Philosophy, Politics and Economics

Eric Pacuit University of Maryland

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Cardinal Utility Theory

u : X → R Which comparisons are meaningful?

1. u(x) and u(y)? (ordinal utility)

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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)?

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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)?

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Ordinal vs. Cardinal Utility

Ordinal scale: Qualitative comparisons of objects allowed, no information about differences or ratios.

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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).

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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 (= 4.53592kg) is twice as much as 5lb (= 2.26796kg).

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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.

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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.

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A Choice

R B W S Take or Gamble? B R S Take Gamble 0.5 0.5 [B : 1] ∼ [R : p, S : 1 − p]

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A Choice

R B W S Take or Gamble? B R S Take Gamble 0.5 0.5 [B : 1] ∼ [R : p, S : 1 − p]

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A Choice

R B W S Take or Gamble? B R S Take Gamble 0.5 0.5 [B : 1] ∼ [R : p, S : 1 − p]

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A Choice

R B W S Take or Gamble? B R S Take Gamble p 1 − p [B : 1] ∼ [R : p, S : 1 − p]

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A Choice

R B W S Take or Gamble? B R S Take Gamble p 1 − p [B : 1] ∼ [R : p, S : 1 − p]

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A Choice

R B W S Take or Gamble? B R S Take Gamble p 1 − p 1 ∗ u(B) = p ∗ u(R) + (1 − p) ∗ u(S)

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A Choice

R B W S Take or Gamble? B R S Take Gamble p 1 − p u(B) = p ∗ 1 + (1 − p) ∗ 0 = p

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Lotteries

Suppose that X = {x1, . . . , xn} is a set of outcomes. A lottery over X is a tuple [x1 : p1, x2 : p2, . . . , xn : pn] where

i pi = 1.

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Lotteries

Suppose that X = {x1, . . . , xn} is a set of outcomes. A lottery over X is a tuple [x1 : p1, x2 : p2, . . . , xn : pn] where

i pi = 1.

x1 x2 · · · xn−1 xn p1 p2 pn−1 pn We identify an element x ∈ X with the lottery [x : 1] ∈ L

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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?

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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′)

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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 = [x1 : p1, . . . , xn : pn], we would like to show that U(L) = n

k=1 pkU(xk).

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Independence Axiom

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<

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<

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=

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=

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=

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>

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>

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L = [A1 : p1, A2 : p2, A3 : p3]

u(A1)

A1 p1 A2 p2 A3 p3

p1 p1 + p2 p1 + p2 + p3 10 / 16

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L = [A1 : p1, A2 : p2, A3 : p3]

u(A1)

A1 p1 A2 p2 A3 p3

p1 p1 + p2 p1 + p2 + p3 = 1 10 / 16

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L = [A1 : p1, A2 : p2, A3 : p3]

u(A1)

A1 p1 A2 p2 A3 p3

p1 p1 + p2 p1 + p2 + p3 = 1 10 / 16

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L = [A1 : p1, A2 : p2, A3 : p3]

u(A1)

A1 A2 p2 A3 p3

p1 p1 + p2 p1 + p2 + p3 = 1 10 / 16

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L = [A1 : p1, A2 : p2, A3 : p3]

u(A1)

A1 A2 A3 p3

p1 p1 + p2 p1 + p2 + p3 = 1 10 / 16

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L = [A1 : p1, A2 : p2, A3 : p3]

u(A1)

A1 A2 A3

p1 p1 + p2 p1 + p2 + p3 = 1 10 / 16

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L = [A1 : p1, A2 : p2, A3 : p3]

p1 ∗ u(A1) u(A1) p2 ∗ u(A2) u(A2) p3 ∗ u(A3) u(A3) p1 p1 + p2 p1 + p2 + p3 = 1 10 / 16

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L = [A1 : p1, A2 : p2, A3 : p3]

p1 ∗ u(A1) u(A1) p2 ∗ u(A2) u(A2) p3 ∗ u(A3) u(A3) p1 p1 + p2 p1 + p2 + p3 = 1 1 10 / 16

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L = [A1 : p1, A2 : p2, A3 : p3]

p2 ∗ u(A2) u(A2) p3 ∗ u(A3) u(A3) p1 p1 + p2 p1 + p2 + p3 = 1 10 / 16

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