Expected Utility Theory Economics 302 - Microeconomic Theory II: - - PowerPoint PPT Presentation

Expected Utility Theory

Economics 302 - Microeconomic Theory II: Strategic Behavior Shih En Lu

Simon Fraser University (with thanks to Anke Kessler)

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Topics

1

Introduction to choice under uncertainty

2

Expected utility theory

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Most Important Things to Learn

1

What is a lottery?

2

What is expected utility?

3

When can one’s preferences over lotteries be represented by an expected utility function?

4

Problems with expected utility theory

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Choice Under Uncertainty

You have learned about preferences and utility functions over sure

• utcomes.

Recall: means "is preferred to", means "is strictly preferred to", and ∼ means "is as good as". Definition: a utility function u represents preferences if for any

• utcomes A and B, u(A) ≥ u(B) exactly when A B.

Recall: This is only possible when preferences are rational (complete and transitive). For example, not possible if A B C A. But life is full of uncertainty! You often have to decide between choices that each lead to an uncertain outcome. Today’s goal: represent preferences over uncertain outcomes. This is important for modeling strategic behaviour: how do you react to others when their actions are uncertain?

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Example

You don’t know if it’s going to rain, and you have to decide whether to carry an umbrella. If you carry an umbrella: 10% chance you lose it, 60% chance you carry it around needlessly, 30% chance you use it If you leave umbrella at home: 0.1% chance you lose it, 66.6% chance you don’t need it, 33.3% chance you get wet What factors matter in your decision? Your decision depends on the probability of each outcome, and on how much you like/hate each outcome. We will assume that these are the only relevant factors. But see "Ellsberg Paradox" (slide 13) for an example where this is not true for some people.

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Notation and Terminology

Suppose a situation has n possible and mutually exclusive outcomes, labeled 1, 2, ..., n.

Example: Lose umbrella (1), carry it needlessly (2), use umbrella (3), don’t carry and don’t need it (4), get wet (5)

A lottery [p1, p2, ..., pn] is a list of probabilities, where pi is the probability that outcome i occurs.

Example: Carrying the umbrella leads to lottery [0.1, 0.6, 0.3, 0, 0], not carrying it leads to lottery [0.001, 0, 0, 0.666, 0.333].

Note: because p1, p2, ..., pn are the probabilities of all possible and mutually exclusive outcomes, we must have p1 + p2 + ... + pn = 1.

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Expected Utility

Suppose outcome 1 gives you utility u1, outcome 2 u2, and so on. What is your utility from lottery L = [p1, p2, ..., pn]? Natural answer: p1u1 + p2u2 + ... + pnun, which is the L’s (von Neumann-Morgenstern) expected utility. But even if the ui’s represent your preferences over outcomes, expected utility may not represent your preferences over lotteries. Example: Fido prefers chicken (outcome 1) over pears (outcome 2)

• ver apples (outcome 3). Assigning u1 = 2, u2 = 1 and u3 = 0 would

represent its preferences over these sure outcomes. But suppose Fido prefers [0.4, 0, 0.6] over [0, 1, 0]. With the above utilities, does expected utility represent Fido’s preferences over lotteries? So it’s important to assign the right utility to each outcome - not just the order matters (ordinal utility), but size matters too (cardinal utility).

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Axioms for Expected Utility (I)

Given preferences over lotteries, it’s not always possible to find utilities over outcomes u1, u2, ..., un such that expected utility represents the said preferences over lotteries. Just as you needed assumptions on preferences over outcomes to build a utility function representing them, you need assumptions on preferences over lotteries to build an expected utility function representing them. There are four required axioms. The first two are the same as the axioms needed on preferences over outcomes, but now applied to lotteries:

1

Completeness: For any lotteries L and L, either L L, L ∼ L, or L ≺ L.

2

Transitivity: If L L and L L, then L L.

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Compound Lottery

To state the next two axioms, we need more terminology. Suppose there are two lotteries, L = [p1, p2, ..., pn] and L = [p

1, p 2, ..., p n].

Suppose L occurs with probability α, and L occurs with probability 1 − α.

Example: You ask your mom to put the umbrella in your bag, but she might forget.

We effectively have a "lottery of lotteries," or a compound lottery, and we denote it αL + (1 − α)L. The probabilities of outcomes 1, 2, ..., n associated with αL + (1 − α)L are: [αp1 + (1 − α)p

1, αp2 + (1 − α)p 2, ..., αpn + (1 − α)p n].

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Axioms for Expected Utility (II)

We can now state the third axiom of expected utility theory:

• 3. Continuity: If L L L, then there exists α ∈ [0, 1] such that

L ∼ αL + (1 − α)L.

This seems like a sensible axiom, but its applicability is in doubt in some extreme cases. Suppose L is winning $100.01, L is winning $100, and L is dying... Most microeconomic theorists find the last axiom most problematic:

• 4. Independence: For any α ∈ (0, 1] and any lotteries L, L and L,

L L if and only if αL + (1 − α)L αL + (1 − α)L.

In words: your preference over two lotteries isn’t affected by mixing in a third. See "Allais Paradox" (slides 14 and 15) for a situation where this might not hold for some people.

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Expected Utility Theorem

Theorem

If preferences satisfy Axioms 1-4, then it is possible to assign a real number (utility) ui to each outcome i = 1, 2, ..., n such that L L if and only if U(L) ≥ U(L), where U([p1, p2, ..., pn]) = p1u1 + p2u2 + ... + pnun. Due to John von Neumann and Oskar Morgenstern. The theorem tells us that these ui’s exist, but it doesn’t tell us what they are. The proof of the theorem is beyond the scope of this course. Slides 16-18 sketch the argument. The converse of this theorem (that if an expected utility representation exists, Axioms 1-4 must hold) is not hard to show.

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Take Away Message

For the rest of this course, we will assume that the four axioms hold so that agents’ preferences admit an expected utility representation. This is an intuitive assumption, but there are cases where it does not match the real world. Much of economics builds on expected utility theory, but economists also study alternative hypotheses that may shed light on some phenomena - search for prospect theory. We will specifically study lotteries over money later in the semester. Preview: if your utility over monetary outcomes is concave in the amount of money, then you are risk averse.

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Supplementary Material: Ellsberg Paradox

There’s an urn with 3 balls, one of which is red. The other two may be both blue, both yellow, or one blue and one yellow. You choose a colour, and a ball is drawn. If the ball has the colour you chose, you win $10. What colour do you choose? If you strictly prefer red to both blue and yellow, then, assuming that you only care about the probability of winning, you think red is more likely than blue, and also more likely than yellow. But then both Pr(BLUE) and Pr(YELLOW ) are less than Pr(RED) = 1

3.

This can’t be the case since probabilities must add up to 1! This is the Ellsberg Paradox. Lesson: uncertainty about the uncertainty may matter, so people might care about more than the ultimate probabilities and outcomes.

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Supplementary Material: Allais Paradox (I)

Outcomes: [win nothing, win $1,000,000, win $5,000,000]. Consider two choices:

1

A = [0.9, 0, 0.1] or B = [0.89, 0.11, 0]

2

A = [0.01, 0.89, 0.1] or B = [0, 1, 0]

What would you choose in each case? Some people would choose:

1

A over B (winning under A is only slightly less likely, but gives much more money), and

2

B over A ($1,000,000 is guaranteed by B; don’t want to "get greedy" and lose it).

These people’s preferences violate the independence axiom! The next slide shows why.

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Supplementary Material: Allais Paradox (II)

By the formula for compound lotteries, we can write:

A = 0.11[ 1

11, 0, 10 11 ] + 0.89[1, 0, 0]

B = 0.11[0, 1, 0] + 0.89[1, 0, 0] A = 0.11[ 1

11, 0, 10 11 ] + 0.89[0, 1, 0]

B = 0.11[0, 1, 0] + 0.89[0, 1, 0]

If A B, then 0.11[ 1

11, 0, 10 11] + 0.89[1, 0, 0] 0.11[0, 1, 0] + 0.89[1, 0, 0].

If the independence axiom applies, this means [ 1

11, 0, 10 11 ] [0, 1, 0]

If A ≺ B, then 0.11[ 1

11, 0, 10 11] + 0.89[0, 1, 0] ≺ 0.11[0, 1, 0] + 0.89[0, 1, 0].

If the independence axiom applies, this means [ 1

11, 0, 10 11 ] ≺ [0, 1, 0]

We have a contradiction! So the independence axiom cannot apply. This is the Allais paradox.

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Supplementary Material: Sketch of the Expected Utility Theorem’s Proof

This section is only for those curious about the proof. We only need to show that U(βL + (1 − β)L) = βU(L) + (1 − β)U(L) Once we have the above, we get an expected utility representation, since for any lottery L = [p1, p2, ..., pn], we can write: U(L) = U(p1[1, 0, ..., 0] + p2[0, 1, 0, ..., 0] + ... + pn[0, ..., 0, 1]) = p1U([1, 0, ..., 0]) + p2U([0, 1, 0, ..., 0]) + ... + pn([0, ..., 0, 1]) = p1u1 + p2u2 + ... + pnun. By the continuity axiom, given any lottery L, the agent is indifferent between L and some mixing of the best lottery L and the worst lottery L (which can be shown to exist by completeness, transitivity and independence).

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Supplementary Material: Sketch of the Expected Utility Theorem’s Proof (II)

Suppose L ∼ αLL + (1 − αL)L, and L ∼ αLL + (1 − αL)L. Then by the independence axiom (applied twice), βL + (1 − β)L ∼ β(αLL + (1 − αL)L) + (1 − β)L ∼ β(αLL + (1 − αL)L) + (1 − β)(αLL + (1 − αL)L) ∼ (βαL + (1 − β)αL)L +(β(1 − αL) + (1 − β)(1 − αL))L. This is the crucial step: we have now related all three lotteries that we’re interested in (L, L and βL + (1 − β)L) to the best and worst lotteries.

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Supplementary Material: Sketch of the Expected Utility Theorem’s Proof (III)

Now let’s assign U(L) = 1 and U(L) = 0. (This just linearly scales the utility function and simplifies the algebra.) If U is to have expected utility form, then we must have U(L) = U(αLL + (1 − αL)L) = αLU(L) + (1 − αL)U(L) = αL. Similarly, U(L) = αL, and U(βL + (1 − β)L) = βαL + (1 − β)αL. You can check that U(βL + (1 − β)L) = βU(L) + (1 − β)U(L) indeed holds, so we are done. In a nutshell: Continuity lets us associate every lottery to a mixture of the best and the worst lottery, and independence ensures that these associations are consistent with expected utility - without independence, βL + (1 − β)L might correspond to some other mix of L and L, in which case we wouldn’t get U(βL + (1 − β)L) = βU(L) + (1 − β)U(L).

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