8.6 a. Using the underlining algorithm or other method, one can verify that
b. Cooperation on silent is best sustained using grim strategies as described
c. The low-cost type of player 1 earns 20.25 in the Bayesian–Nash
8.8 a.
b. In a hybrid equilibrium, at least some type of some player plays a mixed
l1
l2
BRHC(l2)
BR2(l1)
BRLC(l2)
LC
HC
1
Pr(H)
= 6/13
Stay
Fold
2
2
n1
n2
Fold
-50, 50
Stay
Stay
Fold
Fold
100, –100
50, –50
50, –50
-100, 100
strategy. If player 1 sees the low card, she prefers the pure strategy of
and
1
be the probabilities that the high type stays and
folds, respectively. In order for the high type to be willing to randomize, it
In order for player 2 to be willing to randomize, he must be
indifferent between staying and folding. His expected payoff from staying
is
Pr( |stay)(100) [1 Pr( |stay)]( 100),HH
and folding in equilibrium, and earns –50 from folding). Given the prior
probabilities of being a high and low type, player 1’s expected payoff
folding in equilibrium and earns –50 from folding). The game is clearly
tilted toward player 1.
a. Cooperating gives a stream of per-period payoffs of 2, for a present
discounted value of
2 (1 ).
If players use tit-for-tat strategies, the
present discounted value from deviating to fink at the start of the game is
22
1
The deviator earns 3 in the first period, followed by a period in which both
fink and earn 1, followed by a return to cooperating in the third period and
thereafter. For the displayed payoff not to exceed
2 (1 ),
1.
The
If players use two periods of punishment, the present discounted
value from deviating is
23
2
1
For the displayed payoff not to exceed
2 (1 ),
we see, upon multiplying
by
1
and simplifying, the required condition is
3
2 1 0.
Factoring,
32
from cooperating,
)1/(2
, exceed that from deviating,
10 11
(1 ) 2
3.
11
Simplifying,
11
2 1 0.
As the graph below shows, the expression
12 11
crosses the x-axis
very slightly to the left of 0.5. Using numerical methods or a more precise
graph, it can be shown that the condition is
50025.0
. The resulting
condition is very close to the condition for cooperation with infinitely
8.10 Refinements of perfect Bayesian equilibrium
a. The key condition is for the firm to be willing to offer a job to an
uneducated worker. (Regarding the other player, the worker, all worker
types obtain the highest payoffs possible, since they are hired and don’t
have to expend the cost of education.) The firm’s expected payoff from J
is
))](|Pr(1[))(|Pr( wNEHwNEH
and from NJ is 0. The displayed expression exceeds 0 if
Pr( | ) .H NE w
According to Bayes’ rule, along the equilibrium path,
and strategies are consistent with this pooling equilibrium. If
Pr( | ) ,H E w
then the firm would choose J conditional on observing
E. On the other hand, if
Pr( | ) ,H E w
then the firm would choose NJ
conditional on observing E.
-0.03
-0.02
-0.01
0
0.01
0.02
0.03
0.49 0.495 0.5 0.505 0.51
imply
Pr( ) .Hw
A high-skilled worker would deviate to E unless the
firm chooses NJ conditional on E. The firm prefers NJ to J conditional on
posterior belief
Pr( | ) 0.LE
Since
0)|Pr( EL
is inconsistent with the
required condition
Pr( | ) 1 ,L E w
the Cho-Kreps intuitive criterion
Behavioral Problems
8.11 Fairness in the Ultimatum Game
a. Solve using backward induction, starting with the responder. The
responder certainly accepts any offer of
0.r
The remaining question is
how he/she responds to an offer of
0.r
It turns out that the responder
must also accept an offer of 0 in equilibrium. If he/she rejected this, there
b. The outcome in the Dictator Game is the same, also involving equilibrium
offer
*0.r
c. (1) The answer here is a bit technical because of the absolute value
sign in the utility function, requiring the analysis of two cases. We
will avoid this technicality by noting that the proposer would never
offer
1 2.r
The responder obtains utility
1 2 0
from offer
12r
obtains and reducing the fairness of the outcome—skewing things
toward the responder) without increasing the chance of acceptance.
So assume
12r
for the rest of the question.
payoff. In fact, we can say more: for the same reasons as in part
non-negative payoff. The responder’s utility is
2( ,1 ) 1
U r r r a r r
yielding first-order condition:
0))(())(( 1122
LrYULrYU
.
Even though the preceding equation cannot be solved explicitly for
*( ),Lr
we can
*2 2 1 1
21
( ) ( ).
U Y r U Y r
dL
dr U U
*
*
11
11 2 1 2 2 1 1
21
12 12
21
()
[ ( )( ) ( ) ( )]
[ ( ) ( )] 0.
dL
U Y r dr
UY r U U U Y r U Y r
UU
UU Y r Y r
UU
12
their joint incomes.