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

PHIL309P Methods in Philosophy, Politics and Economics

Eric Pacuit University of Maryland

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Utility Function

A utility function on a set X is a function u : X → R

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Utility Function

A utility function on a set X is a function u : X → R A preference ordering is represented by a utility function iff x is (weakly) preferred to y provided u(x) ≥ u(y)

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Utility Function

A utility function on a set X is a function u : X → R A preference ordering is represented by a utility function iff x is (weakly) preferred to y provided u(x) ≥ u(y) What properties does such a preference ordering have?

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Ordinal Utility Theory

• Fact. Suppose that X is finite and is a complete and transitive ordering over

X, then there is a utility function u : X → R that represents (i.e., x y iff u(x) ≥ u(y))

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Ordinal Utility Theory

• Fact. Suppose that X is finite and is a complete and transitive ordering over

X, then there is a utility function u : X → R that represents (i.e., x y iff u(x) ≥ u(y)) Utility is defined in terms of preference (so it is an error to say that the agent prefers x to y because she assigns a higher utility to x than to y).

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Important

All three of the utility functions represent the preference x ≻ y ≻ z Item u1 u2 u3 x 3 10 1000 y 2 5 99 z 1 1 x ≻ y ≻ z is represented by both (3, 2, 1) and (1000, 999, 1), so one cannot say that y is “closer” to x than to z.

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What is utility?

◮ usefulness ◮ from Principle of Utility: an object’s “tendency to produce benefit, advantage, pleasure, good, or happiness” (Broome, p19) for all people ◮ a person’s personal, subjective good ◮ “the value of a function that represents a person’s preferences” (Reiss, p21)

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Economists primarily use the last sense of utility (as will we), which is not problematic, however, “[i]f...you use ‘utility’ to stand for a representation of a person’s preferences, and at the same time for the person’s good, you cannot even express the question [of whether or not persons always act so as to maximize their utility]. You will say: by definition, what a person prefers has more utility for her, so how can it fail to have more utility for her? The ambiguity is intolerable.” (Reiss, p. 21)

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Types of Choices

◮ certainty: highly confident about the relationship between actions and

• utcomes

◮ risk: clear sense of possibilities and their likelihoods ◮ (Knightian) uncertainty: the relationship between actions and outcomes is so imprecise that it is not possible to assign likelihoods

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Lotteries

Suppose that X is a set of outcomes. A (simple) lottery over X is denoted [x1 : p1, x2 : p2, . . . , xn : pn] where for i = 1, . . . , n, xi ∈ X and pi ∈ [0, 1], and

i pi = 1.

Let L be the set of (simple) lotteries over X. We identify elements x ∈ X with the lottery [x : 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, . . . , 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, . . . , xn : pn] where

i pi = 1.

x1 x2 · · · xn−1 xn p1 p2 pn−1 pn Let L be the set of lotteries.

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Expected Value of a Lottery

Suppose that the outcomes of a lottery are monetary values. So, L = [x1 : p1, x2 : p2, . . . , xn : pn], where each xi is an amount of money. Then, EV(L) =

• i

pi × xi

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Expected Value of a Lottery

Suppose that the outcomes of a lottery are monetary values. So, L = [x1 : p1, x2 : p2, . . . , xn : pn], where each xi is an amount of money. Then, EV(L) =

• i

pi × xi E.g., if L = [$100 : 0.55, $50 : 0.25, $0 : 0.20], then EV(L) = 0.55 ∗ 100 + 0.25 ∗ 50 + 0.2 ∗ 0 = 80

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You are given a choice between two lotteries L1 and L2. The outcome of the lotteries is determined by flipping a fair coin. The payoff for the two lotteries are given in the following table: Heads Tails L1 $1M $1M L2 $3M $0 Which of the two lotteries would you choose?

• 1. L1
• 2. L2
• 3. I am indifferent between the two lotteries

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Expected monetary value

Suppose that the outcomes of a lottery are monetary values. So, L = [x1 : p1, x2 : p2, . . . , xn : pn], where each xi is an amount of money. Then, EV(L) =

• i

pi × xi

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Expected monetary value

Suppose that the outcomes of a lottery are monetary values. So, L = [x1 : p1, x2 : p2, . . . , xn : pn], where each xi is an amount of money. Then, EV(L) =

• i

pi × xi E.g., if L = [$100 : 0.55, $50 : 0.25, $0 : 0.20], then EV(L) = 0.55 ∗ 100 + 0.25 ∗ 50 + 0.2 ∗ 0 = 80

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Problems with using monetary payoffs

◮ Overly Restrictive: We care about more things than money.

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Problems with using monetary payoffs

◮ Overly Restrictive: We care about more things than money. ◮ The St. Petersburg Paradox: Consider the following wager: I will flip a fair coin until it comes up heads; if the first time it comes up heads is the nth toss, then I will pay you 2n. What’s the most you’d be willing to pay for this wager? What is its expected monetary value?

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Problems with using monetary payoffs

◮ Overly Restrictive: We care about more things than money. ◮ The St. Petersburg Paradox: Consider the following wager: I will flip a fair coin until it comes up heads; if the first time it comes up heads is the nth toss, then I will pay you 2n. What’s the most you’d be willing to pay for this wager? What is its expected monetary value? ◮ Valuing Money: Doesn’t the value of a wager depend on more than merely how much it’s expected to pay out? (I.e., your total fortune, how much you personally care about money, etc.)

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Problems with using monetary payoffs

◮ Overly Restrictive: We care about more things than money. ◮ The St. Petersburg Paradox: Consider the following wager: I will flip a fair coin until it comes up heads; if the first time it comes up heads is the nth toss, then I will pay you 2n. What’s the most you’d be willing to pay for this wager? What is its expected monetary value? ◮ Valuing Money: Doesn’t the value of a wager depend on more than merely how much it’s expected to pay out? (I.e., your total fortune, how much you personally care about money, etc.) ◮ Risk-aversion: Is it irrational to prefer a sure-thing $x to a wager whose expected payout is $x?

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We should move away from “monetary payouts” to “utility”.

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Comments on Expected Utility

Options 1/2 1/2 L1 1M 1M L2 3M 0M

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Comments on Expected Utility

Options 1/2 1/2 L1 1M 1M L2 3M 0M EVM(L1) = 1/2 · 1 + 1/2 · 1 = 1 EVM(L1) = 1/2 · 3 + 1/2 · 0 = 1.5

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Comments on Expected Utility

Options 1/2 1/2 L1 1M 1M L2 3M 0M EVM(L1) = 1/2 · 1 + 1/2 · 1 = 1 EVM(L1) = 1/2 · 3 + 1/2 · 0 = 1.5 What numbers should we use in place of monetary value?

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Comments on Expected Utility

Options 1/2 1/2 L1 1M 1M L2 3M 0M EVM(L1) = 1/2 · 1 + 1/2 · 1 = 1 EVM(L1) = 1/2 · 3 + 1/2 · 0 = 1.5 What numbers should we use in place of monetary value? (moral) value? personal utility?

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Expected Utility

Suppose that X = {x1, . . . , xn} and u : X → R is a utility function on X. This can be extended to an expected utility function EU : L(X) → R where EU([x1 : p1, . . . , xn : pn]) = p1 × u(x1) + · · · + pn × u(xn) = n

i=1 pi × u(xi)

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Suppose that Ann is faced with the choice between lotteries L1 and L2 where: L1 = [o1 : 0.4, o2 : 0.6] L2 = [o3 : 1.0] Can expected utility theory tell us how Ann should rank L1 and L2?

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Suppose that Ann is faced with the choice between lotteries L1 and L2 where: L1 = [o1 : 0.4, o2 : 0.6] L2 = [o3 : 1.0] Can expected utility theory tell us how Ann should rank L1 and L2? No! Suppose that Ann is also faced with the choice between lotteries L3 and L4 where: L3 = [o1 : 0.2, o2 : 0.8] L4 = [o3 : 0.5, o2 : 0.5]

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Suppose that Ann is faced with the choice between lotteries L1 and L2 where: L1 = [o1 : 0.4, o2 : 0.6] L2 = [o3 : 1.0] Can expected utility theory tell us how Ann should rank L1 and L2? No! Suppose that Ann is also faced with the choice between lotteries L3 and L4 where: L3 = [o1 : 0.2, o2 : 0.8] L4 = [o3 : 0.5, o2 : 0.5] If we know that Ann ranks L2 over L1 (e.g., L2 ≻ L1), can we conclude anything about how Ann ranks L3 and L4?

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Suppose that Ann is faced with the choice between lotteries L1 and L2 where: L1 = [o1 : 0.4, o2 : 0.6] L2 = [o3 : 1.0] Can expected utility theory tell us how Ann should rank L1 and L2? No! Suppose that Ann is also faced with the choice between lotteries L3 and L4 where: L3 = [o1 : 0.2, o2 : 0.8] L4 = [o3 : 0.5, o2 : 0.5] If we know that Ann ranks L2 over L1 (e.g., L2 ≻ L1), can we conclude anything about how Ann ranks L3 and L4? Yes: Ann must rank L4 over L3 (e.g., L4 ≻ L3).

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x [x : p, y : (1 − p)] L y u(x) pu(x) + (1 − p)u(y) u(y) u(EU(L)) u(EU(L)) Risk neutral Risk seeking Risk averse

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x [x : p, y : (1 − p)] L y u(x) pu(x) + (1 − p)u(y) u(y) u(EU(L)) u(EU(L)) Risk neutral Risk seeking Risk averse

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x [x : p, y : (1 − p)] L y u(x) pu(x) + (1 − p)u(y) u(y) u(L) u(L) Risk neutral Risk seeking Risk averse

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x [x : p, y : (1 − p)] L y u(x) pu(x) + (1 − p)u(y) u(y) u(L) u(L) Risk neutral Risk seeking Risk averse

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x [x : p, y : (1 − p)] L y u(x) pu(x) + (1 − p)u(y) u(y) u(L) u(L) Risk neutral Risk seeking Risk averse

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