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### Lecture 1: Expected Utility

Yosuke YASUDA

National Graduate Institute for Policy Studies

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### Decision under Uncertainty

We have so far not distinguished between individual’s actions and consequences, but many choices made by agents take place under conditions of uncertainty.

This lecture discusses such a decision under uncertainty, i.e., an environment in which the correspondence between actions and

consequences is not deterministic butstochastic.

◮ To discuss a decision under uncertainty, we extend the domain

of choice functions. The choice of an action is viewed as choosing a “lottery” where the prizes are the consequences.

◮ An implicit assumption is that the decision maker does not

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### Lotteries (1)

We consider preferences and choices over the set of “lotteries.”

◮ Let S be a set of consequences (prizes). We assume thatS is

a finite set and the number of its elements (=|S|) is S.

◮ Alotterypis a function that assigns a nonnegative number to

each prize s, wherePs∈Sp(s) = 1(here p(s) is the objective

probability of obtaining the prize sgiven the lottery p).

◮ Let α◦x⊕(1−α)◦y denote the lottery in which the prizex

is realized with probabilityα and the prize y with 1−α.

◮ Denote by L(S) the (infinite) space containing all lotteries

with prizes in S. That is,{x∈RS+|Pxs= 1}.

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### Lotteries (2)

We impose the following three assumptions on the lotteries.

1. 1◦x⊕(1−1)◦y ∼x: Getting a prize with probability one is

the same as getting the prize for certain.

2. α◦x⊕(1−α)◦y∼(1−α)◦y⊕α◦x: The consumer does

not care about the order in which the lottery is described.

3. β◦(α◦x⊕(1−α)◦y)⊕(1−β)◦y∼(βα)◦x⊕(1−βα)◦y: A consumer’s perception of a lottery depends only on the net probabilities of receiving the various prizes.

The first two assumptions appear to be innocuous.

The third assumption sometimes called “reduction of compound

lotteries” is somewhat suspect.

◮ There is some evidence to suggest that consumers treat

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The most primitive way to evaluate a lottery is to calculate its

mathematical expectation, i.e.,E[p] =PsSp(s)s.

Daniel Bernoulli first doubt this approach in the 18th century when he examined the famous St. Pertersburg paradox.

✞ ✝

☎ ✆

A fair coin is tossed until it shows heads for the first time. If the

first head appears on thek-th trial, a player wins \$2k. How much

are you willing to pay to participate in this lottery?

✝ ☎

Rm The expected value of the lottery is infinite:

2 2 +

22

22 +

23

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The St Petersburg paradox shows that maximizing your dollar expectation may not always be a good idea. It suggests that an agent in risky situation might want to maximize the expectation of some “utility function” with decreasing marginal utility:

E[u(x)] =u(2)1

2+u(4) 1

4+u(8) 1 8+· · ·,

which can be a finite number.

✝ ☎

Q Under what kinds of conditions does a decision maker maximizes the expectation of some “utility function”?

✝ ☎

Rm By utility theory, we know that for any preference relation defined on the space of lotteries that satisfies continuity, there is a

utility representationU: L(S)→R, continuous in the

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### Expected Utility Theory (1)

We will use the following two axioms to isolate a family of preference relations which have a representation by a more structured utility function.

◮ Independence Axiom (I): For anyp, q, r∈L(S)and any

α∈(0,1), p%q ⇔α◦p⊕(1−α)◦r %α◦q⊕(1−α)◦r.

◮ Continuity Axiom (C): Ifp≻q≻r, then there exists

α∈(0,1)such that

q∼[α◦p⊕(1−α)◦r].

Thm Let%be a preference relation overL(S)satisfying the I

andC. There are numbers(v(s))s∈S such that

p%q ⇔U(p) =X

s∈S

p(s)v(s)≥U(q) =X

s∈S

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### Expected Utility Theory (2)

Sketch of the proof LetM andm be a best and a worst certain

lotteries inL(S). When M ∼m, choosingv(s) = 0 for all swe

havePs∈Sp(s)v(s) = 0 for all p∈L(S).

Consider the case thatM ≻m. By I andC, there must be a

single numberv(s)∈[0,1]such that

v(s)◦M⊕(1−v(s))◦m∼[s]

where[s]is a certain lottery with prize s, i.e., [s] = 1◦s.

In particular,v(M) = 1 andv(m) = 0. I implies that

p∼(X

s∈S

p(s)v(s))◦M ⊕(1−X s∈S

p(s)v(s))◦m.

SinceM ≻m, we can show that

p%q ⇔X s∈S

p(s)v(s)≥X s∈S

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### vNM Utility Function (1)

Note the functionU is a utility function representing the

preferences onL(S) whilev is a utility function defined over S,

which is the building block for the construction ofU(p). We refer

tov as a vNM (Von Neumann-Morgenstern) utility function.

✝ ☎

Q How can we construct the vNM utility function?

Letsi(∈S), i= 1, ..., K be a set of consequences ands1, sK be

the best and the worst consequences. That is, for anyi,

[s1]%[si]%[sK].

Then, construct a functionv:S →[0,1]in the following way:

v(s1) = 1 andv(sK) = 0, and

[sj]∼v(sj)◦[s1]⊕(1−v(sj)◦[sK]for all j.

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### vNM Utility Function (2)

✝ ☎

Q To what extent, vNM utility function is unique?

The vNM utilities are unique up to positiveaffine transformation

(multiplication by a positive number and adding any scalar) and

arenot invariant to arbitrary monotonic transformation.

Thm Suppose %is a preference relation defined overL(S) and

letv(s)be the vNM utilities representing the preference relation.

Then, definingw(s) =αv(s) +β for all s(for some α >0 and

someβ), the utility functionW(p) =PsSp(s)w(s) also

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### vNM Utility Function (3)

Proof For any lotteries p, q∈L(S),p%q if and only if

X

s∈S

p(s)v(s)≥X s∈S

q(s)v(s).

Now, the followings hold.

X

s∈S

p(s)w(s) =X

s∈S

p(s)(αv(s) +β) =αX s∈S

p(s)v(s) +β.

X

s∈S

q(s)w(s) =X

s∈S

q(s)(αv(s) +β) =αX s∈S

q(s)v(s) +β.

Thus,Ps∈Sp(s)v(s)≥

P

s∈Sq(s)v(s) holds if and only if

P

s∈Sp(s)w(s)≥

P

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Many experiments reveal systematic deviations from vNM

assumptions. The most famous one is theAllais paradox.

✞ ✝

☎ ✆

Choose first the between

L1 =  andL2 = 0.8◦⊕0.2◦

and then choose between

L3 = 0.5◦⊕0.5◦ andL4 = 0.4◦⊕0.6◦.

Note thatL3= 0.5◦L1⊕0.5◦andL4= 0.5◦L2⊕0.5◦.

AxiomI requires that the preference between L1 and L2 be the

same as that betweenL3 andL4. However, a majority of people

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AssumeL1 ≻L2 butα◦L⊕(1−α)◦L1≺α◦L⊕(1−α)◦L2.

(In our example of Allais paradox,α= 0.5and L= .)

Then, we can perform the following trick on the decision maker:

1. Take α◦L⊕(1−α)◦L1.

2. Take instead α◦L⊕(1−α)◦L2, which you prefer (and you

pay me something...).

3. Let us agree to replace L2 with L1 in case L2 realizes (and

you pay me something now...).

4. Note that you hold α◦L⊕(1−α)◦L1.

5. Let us start from the beginning...

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Allais paradox can be viewed as a violation of independence axiom. The following paradox also shows that many people do not

necessarily follow the expected utility maximization behavior.

✝ ☎

Some bullets are loaded into a revolver with six chambers. The cylinder is then spun and the gun pointed at your head.

Would you be prepared to pay more to get one bullet removed when only one bullet was loaded, or when four bullets were loaded?

✝ ☎

Q People usually say they would pay more in the first case, because they would then be buying their lives for certain. Is this decision reasonable?

✝ ☎

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Suppose\$X (resp. \$Y) is the most that you are willing to pay to

get one bullet removed from a gun containing one (resp. four)

bullet. LetLmean death, and W mean being alive after paying

nothing. LetC mean being alive after paying\$X, and Dmean

being alive after paying\$Y. Note that

u(D)< u(C)⇔D≺C ⇔X < Y.

Letu(L) = 0 andu(W) = 1. Then,

u(C) = 1

6u(L) + 5

6u(W) = 5

6, and

1

2u(L) + 1

2u(D) = 2

3u(L) + 1

3u(W)⇒u(D) = 2 3.

Sinceu(D)< u(C), you must be ready to pay less to get one

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