SABANCI UNIVERSITY
ECON 201 – A
GAMES AND STRATEGY
FINAL
EXAMINATION ANSWERS
June 13, 2007
1. (
25 points
). An individual privately knows whether he has good health (type-G) or bad health
(type-B). Type-G individuals become ill with probability 0.1, whereas type-Bs become ill with
probability 0.5. When the individual becomes ill, his medical expenses are 1900TLs. The
individual’s initial wealth is 10.000TLs and his final utility U depends on his wealth as follows
U(W) = W
0.5
.
(i)
[5p]
Compute the expected utility of the type-G and the type-B (without any
insurance).
(ii)
[10p] What premium will the insurance company ask if it knows that the individual is
type-G? Same question, if it knows that the individual is type-B? (Assume that the
insurance company makes a take-it-or-leave-it premium offer, which the individual
rejects or accepts, and the game ends. If the offer is rejected, the insurance company
gets zero, whereas the individual gets the utility you computed in (i)).
(iii)
[10p] Suppose that the insurance company does not know the type of the individual.
It is common knowledge, as the insurance company also knows, that this individual is
type-G with probability 0.5 and type-B with probability 0.5. Draw the game tree, and
find the premium that the insurance company will offer in the sequential equilibrium.
ANSWER. (i) EU of type-G is 0.1(90) + 0.9(100) = 99, EU of type-B is 0.5(90) + 0.5(100) = 95.
(ii) The maximum that type-G is willing to pay, PG, satisfies the condition
99 = (10000 – PG)
0.5
, thus, PG = 199TL, Similarly, PB = 975TL.
(iii) The company has two choices. If it charges PL, only type-B will pay, thus, the company will
get 0.5(975 – 0.5(1900)) = 12.5TL. If it charges PM, both types will accept and the company will
get 199 – 0.5(0.1(1900)) – 0.5(0.5(1900)) = 199 – 560 < 0. Therefore it will charge the price PB.
2. (
35 points
) Consider the following games
GAME A
GAME B
Ayse
Ali
Football
Movie
Football
4 , 2
1 , 1
Movie
0 , 0
X , 4
(i)
[10 p] Find the Nash equilibria of both games (in Game A, indicate what equilibria
may arise according the value that X may take).
(ii)
[10p] One of these games will be played. The type of Ayse determines the game that
will be played, which Ayse knows but Ali does not. It is common knowledge that
Ayse is of type-A (hence Game A will be played) with probability 0.4. Draw the
game tree. Suppose X = 0 and find ALL pure strategy Bayesian Nash equilibria of
this game.
(iii)