? E XERCISE 222.2 (Variant of ultimatum game with equity-conscious players) Con- sider a variant…

?  EXERCISE 222.2 (Variant of ultimatum game with equity-conscious players)  Con-

sider a variant of the game in Exercise 181.1 in which β1 = 0, and the person 2 whom person 1 faces is drawn randomly from a population in which the fraction

p have β2  = 0 and the remaining fraction 1 − p have β2  = 1.  When making    her

 

1

 

 

 

 

 

 

 

0, 1                   1, 0

 

Figure 223.1 An extensive game with perfect information and chance moves. The label c denotes chance; the number beside each action of chance is the probability with which that action is chosen.

 

 

offer, person 1 knows only that her opponent’s characteristic is β2 = 0 with prob- ability p and β2  = 1 with probability 1 − p.  Model this situation as an  extensive

3   game with perfect information and chance moves in which person 1 makes an of- fer, then chance determines the type of person 2, and finally person 2 accepts or rejects person 1’s offer. opponent Find the subgame perfect equilibria of this game. (Use the fact that if β2 = 0, then in any subgame perfect equilibrium of the game in Exercise 181.1 person 2 accepts all offers x > 0, rejects all offers x 0, and may accept or reject the offer 0, and if β2 = 1 then she accepts all offers x > 1 , may accept or reject the offer 1 , and rejects all offers x 1 .) Are there any values of p

3                                                     3

for which an offer is rejected in equilibrium?