o Ec no m ic s 6 21 H and o ut # 5
Extensive Form Games: Uncertainty and belief structures
Two examples:
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A system of beliefs for an extensive form game is a mapping : X [0, 1] such that for all x in
X,
x* H(x)(x*) = 1.
[ A probability assessment at x by (H(x)). ]
E[ui H, , i, i ] is player i's expected utility starting at information set H if i's beliefs of
conditional probabilities of being at the various nodes of H are given by , if i follows strategy i
and everyone else follows i.
A strategy profile = (1, 2, ..., I) in game is sequentially rational at information set H given
a system of beliefs if for individual (H) we have
E[u(H) H, , (H), (H) ] E[u(H) H, , (H), (H) ]
for all (H) (S(H)). is sequentially rational given a system of beliefs if it is sequentially
rational at information set H given a system of beliefs for all information sets H.
A strategy-beliefs pair (, ) is a weak perfect Bayesian equilibrium (weak PBE) in game if:
(i) is sequentially rational given a system of beliefs ;
(ii) For every H such that p(H ) > 0, and for all x in H,
We say is a weak PBE if there exists at least one system of beliefs such that (, ) is a
weak PBE.
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Bierman and Fernandez
Exercise 15.4. Show that the car buying game is a Lemons Market. That is, the game has a
WPBE in which the car is sold iff it is a lemon.
Exercise 15.5. The following strategies and beliefs constitute another WPBE of the car buying
game.
Beverlys strategy: If the car is good, always offer to sell the car for $5000; if the car is a
lemon, offer to sell it for $5000 with probability p and offer to sell it for $2000 with probability
1p.
Jims strategy: Always reject a price above $5000; accept a price of v$5000 with
probability q; always reject a price between $2000 and $5000; always accept a price at or below
$2000.
Jims beliefs: If the price is below $5000, then the car is a lemon for sure; if the price is
$5000 or higher, then the car is a lemon with probability .
Find the values of p, q, and as follows:
(1) Given Beverlys strtategy, use Bayes theorem to find an expression for in terms of
p, denoted (p).
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(2) If Jim rejects a price of $5000, his expected payoff is 0. If accepts the price, his
expected payoff equals ($6000 $5000)(1 ) + ($2000 $5000). Since his equilibrium
strategy calls for him to randomize when Beverly offers to sell for $5000, these two expected
payoffs must be equal. this Use fact to solve for .
(3) Now use the value of and your expression for (p) to solve for p.
(4) If Beverlys car is a lemon and she offers a price of $2000, then her expected payoff is
$2000 $1000. If she offers a price of $5000, then her expected payoff equals $0 (1 q) +
($5000 $1000)q. Since her equilibrium strategy calls for her to randomize when her car is a
lemon, these two expected payoff must be equal. Use this fact to solve for q.
What is the probability that the car is sold if it is a lemon? What is the probability that the car is
sold if it is good?
Exercise. Revisit Exercises 15.4 and 15.5 from Bierman and Fernandez but change the
distribution of car types in the population so that 1/4 (not ) of cars are good and 3/4 (not ) are
lemons.
(A) Show that there is a perfect Bayesian equilibrium in which a car is sold if and only if it is a
lemon. What is Beverlys expected value from discovering a car in her driveway?
Expected value = 1/4 [ 2000P(car sells for 2000 / car is good) + 4500P(car does not sell / car
is good)] + 3/4 [2000P(car sells for 2000 / car is lemon) + 1000P(car does not sell / car is
lemon)]
(B) Determine p, q, and so that the proposal in Exercise 15.5 (with a mixed strategy involving
a price of $5000) is a perfect Bayesian equilibrium, what is Beverlys expected value
from discovering a car in her driveway. [Again assume the probability is .75 the car is a
lemon and .25 the car is good.]
Expected value = 1/4 [5000P(car sells for 5000 / car is good) + 2000P(car sells for 2000 / car
is good) + 4500P(car does not sell / car is good)] + 3/4 [5000P(car sells for 5000 / car is
lemon) + 2000P(car sells for 2000 / car is lemon) + 1000P(car does not sell / car is lemon)]
(C) Is there another solution, similar in form to the one in Part (B), but at a price V different from
$5,000 that gives her a higher expected revenue? Let V stand for the higher price
(Assume 4500 < V < 6000), and go through the calculations suggested in Bierman and
Fernandez, using V instead of 5000 throughout. Show that 5000 is NOT the value of V
that maximizes Beverlys expected value. Explain your answer carefully.
Expected value = 1/4 [VP(car sells for V / car is good) + 2000P(car sells for 2000 / car is
good) + 4500P(car does not sell / car is good)] + 3/4 [VP(car sells for V / car is lemon) +
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2000P(car sells for 2000 / car is lemon) + 1000P(car does not sell / car is lemon)]
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