Expeted Utility Problems
1. Suppose the outome spae onsists of two points {Good, Bad}. Desribe
a lottery over outomes by a single number p ∈ [0, 1] interpreted as the
probability with whih the good outome ours. Player 1's preferenes
over lotteries are given by p º1 p′ i uG p+(1 − p) uB ≥ uG p′ +(1 − p′ ) uB ,
while player 2's preferenes are given by p º2 p′ if vG p + (1 − p) vB ≥
vG p′ + (1 − p′ ) vB . The number uG > 0. Show that player 1 and 2 have
the same preferene if and only if there is a positive number λ suh that
λ (uG , uB ) = (vG , vB ).
2. Suppose there are three states in S , and two outomes (Good, Bad) in χ.
A plan is a vetor p = {p1 , p2 , p3 } for whih p1 is the probability the good
outome is hosen in state 1, p2 is the probabilty of the good outome in
state 2, and so on. Suppose that preferenes over plans are given by
U (p) =
b1g p1 + b1b (1 − p1 ) + b2g p2 + b2b (1 − p2 ) + b3g p3 + b3b (1 − p3 )
for some set of onstants for whih b2g > b2b > 0. Now suppose that
U (p1 , p2 , p3 ) ≥ U (p1 , p′2 , p3 ) =⇒ U (p2 , p1 , p3 ) ≥ U (p′2 , p1 , p3 ). Then
prove that (b2g , b2b ) = λ (b1g , b1b ) for some positive onstant λ.
3. Suppose there are four outomes denoted χ = {tl, tr, bl, br} and 2 states.
is attahed to eah of these
Suppose that a state ontingent probablity psi P
outomes having the property that for eah s, χ psx = 1. What property
must the probabilities psx have in order that the lottery on χ an be interpreted as as the lottery generated by a pair of mixed strategies. Give an
example of a lottery of these ations that an't be interpreted this way.
4. The game of mathing pennies is given by
H
T
H 1,-1 -1,1
T -1,1 1,-1
Interpret the payos in eah ell as utility values. The olumn player has
two belief types, th and tt . The olumn player of type th plays H while the
olumn player of type tt plays T. If the row player of type th plays H and
T with equal probability, what must his subjetive prior belief about the
type of olumn player be if he is an expeted utility maximizing player?
Now suppose the olumn player of type th plays H with probability 34
while the olumn player of type tt plays H and T with equal probability.
If the row player plays H and T with equal probability, what must his
subjetive prior belief about the type of the olumn player be? Using
the last assumption about the olumn player and your answer to the last
question, explain why eah player or eah belief type is using a strategy
that is a best reply to a strategy that is a best reply,...., et.
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5. In a simple version of the Ellsberg experiment, there are 100 blak and
white balls in urn A. An objetive lottery draws a ball at random from
the urn, and this ball turns out to be white with probability q . A deision maker ranks gambles that pay $1 when a white ball is drawn
by alulating the expeted utility as follows: q º q ′ if and only if
ln (qu (1) + (1 − q) u (0)) ≥ ln (q ′ u (1) + (1 − q ′ ) u (0)). Normalize u (0) =
0 and u (1) = 1. Then betting on a white ball yeilds payo q while betting
on a blak ball yeilds payo (1 − q). A seond urn B has 100 blak and
white balls but no information is given about the odds of drawing a white
ball.
Suppose the deision maker values ompound lotteries over bets on a white
ball as follow: if p is a ompound lottery in whih simple lottery qi is drawn
with probability λi , then p º p′ if and only if
X
λi ln (qi u (1) + (1 − qi ) u (0)) ≥
i
X
λ′i ln (qi′ u (1) + (1 − qi′ ) u (0)) .
i
Now assume the deision maker believess that it is equally likely that
either all the balls in urn B are white, or all of them are blak - i.e., he
treats the unknown urn as if it were a ompound lottery. Find the interval
of probabilities q of drawing a white ball from urn A for whih the deision
maker will prefer to draw a ball from urn A instead of urn B whihever
olor he is betting on. Desribe the interval of probabilities of drawing a
white ball from urn A for whih he will prefer to draw from urn B when
he is betting on a blak ball.
6. Two balls are to be drawn from two separate urns ontaining blak and
white balls. The proportion of white balls in eah urn is unknown. A
deision maker has expeted utility preferenes, and redues ompound
lotteries. However he has a prior belief that the proportion of white balls in
eah urn is the same. He doesn't know the exat proportion, but believes
that there is a probability distribution F with the probability that the
probability with whih this proportion is less than or equal to q is F (q).
He values all bets using expeted subjetive utility. He is oered two bets:
bet 1 - the balls with have the same olor; bet 2 - the balls will have
dierent olors. What is his ranking of these two bets.
7. Prospet Theory: Suppose the state spae is S = [0, 1]. Suppose the set
of events is equal to the set of all onvex intervals [a, b] with a ≤ b. The
probability of an event [a, b] is given by b − a. The apaity used by a
deision maker to evaluate bets on dierent events is given by
F ([a, b]) =
(
1
2
1
4
(b − a)
¡
¢
+ 32 b − a − 21
(b − a) ≤ 21
(b − a) > 21 .
Assume that the utility for wealth is v (w) = w. In experiment A, there
are two prospets: {{[0, .1] , $1000} , {[.1, .99] , $500} , {[.99, 1] , $0}}, and
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{[01] , 500}. In experiment B there are two distint prospets {{[0, .1] , $1000} , {[.1, 1] , 0}}
and {{[0.11] , $500} , {[.11, 1] , 0}}. Calulate the value of eah of these
prospets using the apaity F and the value funtion v . Whih prospet
with be hosen in eah of the two experiments.
8. Loss Aversion: Using events as in the previous question, let F + ([a, b]) =
g (b − a) for some stritly onvex funtion g satisfying g (0) = 0 and g (1) =
1 (while F − ([a, b]) = b − a. Utility for wealth is given by v (w) = w − 500.
Using prospet theory evalute the prospets {{[0.5] , 650} , {[.5, 1] , 550}}
ompared to {{[0.5] , 600} , {[.5, 1] , 600}}, then {{[0.5] , 750} , {[.5, 1] , 650}}
ompared to {{[0.5] , 700} , {[.5, 1] , 700}}, then {{[0.5] , 550} , {[.5, 1] , 450}}
ompared to {{[0.5] , 500} , {[.5, 1] , 500}}. Is this deision maker risk averse?
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