Markets with asymmetric information can be model information, resulting in Bayes
ID: 370180 • Letter: M
Question
Markets with asymmetric information can be model information, resulting in Bayesian Nash equilibrium outcomes trade outcomes. with inefficien Harsanyi's purification theorem suggests that mixed-strategy Bayesian Nash equilibria of games with heterogeneous players,trae equilibria pure-stra . nformation can be thought of as representing Exercises Chicken Revisited: Consider the game of chicken in Section 12 parameters R 8, H = 16, and L = 0 as described there. A knows some game theory, decides to use this model to claim that movin a society in which all parents are lenient will have detrimental effects behavior of teenagers. Does equilibrium analysis support this claim? Wha R=8, H =0, and L = 16? 12.2.1 with the preacher 12.1 , who Cournot Revisited: Consider the Cournot duopoly model in which two firm I and 2, simultaneously choose the quantities they supply, qi and price each will face is determined by the market demand function p(ah.g a-b(qi + q2). Each firm has a probability of having a marginal unit cost of c, and a probability 1- of having a marginal unit cost of cH. These probabilities are common knowledge, but the true type is revealed only each firm individually. Solve for the Bayesian Nash equilibrium. 12.2 2. The 12.3 Armed Conflict: Consider the following strategic situation: Two rival armies plan to seize a disputed territory. Each army's general can choose either to attack (A) or to not attack (N). In addition, each army is either strong (S) or weak (W) with equal probability, and the realizations for each army are independent. Furthermore the type of each army is known only to that armys general. An army can capture the territory if either (i) it attacks and its rival does not or (ii) it and its rival attack, but it is strong and the rival is we ak. If As both attack and are of equal strength then neither captures the tern tory for payoffs, the territory is worth m if captured and each army has a cost Ou fighting equal to s if it is strong and w if it is weak, where sExplanation / Answer
Solution:
The Action Spaces are ----- A1 = A2= (A,N)
The Type spaces are T1 = T2= (S,W)
The probabilities are
P(S) = P(W) = 0.5
The strategy sets------------ S1 = S2 = (AA,AN,NA,NN)
{AA= Attack in any case, AN= Attack if strong, NA= not attack is strong, NN= Never attack}
Player 2
Player 1
(1)/(2)
AA
AN
NA
NN
AA
(-3/4), (-3/4)
(3/4),(1/4)
(3/4), -1
3,0
AN
(1/4),(3/4)
(1/2), (1/2)
(5/4),(1/4)
(3/2),0
NA
(-1) , (3/4)
(1/4) , (5/4)
(1/4) , (1/4)
(3/2), 0
NN
0,3
0,(3/2)
0,(3/2)
0,0
2. The payoff matrix, when m=3, w=4, s=2
Player 2
Player 1
(1)/(2)
AA
AN
NA
NN
AA
(-9/4) , (-9/4)
0, (-1/4)
(-3/4), -2
3 , 0
AN
(-1/4), 0
(1/4), (1/4)
1 , (-1/4)
(3/2) , 0
NA
(-2) , (-3/4)
(-1/4), (1)
(-1/4), (-1/4)
(3/2), 0
NN
0,3
0, (3/2)
0, (3/2)
0,0
nash equilibrium for this game theory = {(AA, NN);(NN,AA)}
Player 2 Player 1 (1)/(2) AA AN NA NN AA M/4 - s+w/2 ; M /4 - s+w/2 M/2-s+w/4;M/4-s/2 3M /4 - s+w/ 2 ; w/2 m,0 AN M/ 4- s/2 ; M/2-s+w/2 M-s/ 4 ; M-s /4 M/ 2- s/ 4 ; M-w/4 m/2,0 NA (-w/2); 3M/ 4 - s+w/2 m-w/4; m/2 - s/4 m-w/4,m-w/4 m/2,0 NN 0,m 0, m/2 0, m/2 0,0