question 6) The centipede game illustrated in Figure 3.8 can be solved by using rollback without drawing the complete game tree. In the version of the game discussed in the text, players A and B alternately had the opportunity to claim or pass a growing pile of dimes (to a maximum of 10) placed on the table.

Suppose the rules of the games were changed so that:

(b) Two rounds of the game are played with the same two players, A and B. In the first round, A may not keep more than five dimes, and B may not keep more than nine. Find the rollback equilibrium strategies for each player.

(c) Two rounds of the game are played with the same two players, A and B. In the first round, A may not keep more than five dimes, and B may not keep more than four. Find the rollback equilibrium strategies for each player

question 10 ) consider the survivor game tree illustrated in Figure 3.9. Suppose that, unlike in Figure 3.9, you want to use only general values for the various probabilities

In particular, suppose that the probability of winning the immunity challenge when Rich chooses continue is x for Rich,y for Kelly, and 1-x-y

for Rudy; similarly, the probability of winning when Rich chooses Give Up is z for Kelly and 1- z for Rudy. Further, suppose that Rich's chance of being picked by the jury is p if he has won immunity and has voted off Rudy;his chance of being picked is 4 if Kelly has won immunity and has voted off Rudy. continue to assume that, if Rudy wins immunity, he keeps Rich with probability 1, and that Rudy wins the game with probability I if he ends up in the final two.

(a) What is the algebraic formula for the probability, in terms of p, q, x, and y that Rich wins the million dollars if he chooses Continue? What is the probability that he wins if he chooses Give up? can you determine Rich'soptimal strategy with only this level of information?

(b) The discussionin Section3 .7 suggestst hat Give up is optimal for Rich as long as (i) Kelly is very likely to win the immunity challenge once Rich gives up and (ii) Rich wins the jury's final vote more often when Kelly has voted out Rudy than when Rich has done so. write out expressions entailing the general probabilities (P,4,x,y) that summarize these two conditions.

(c) Suppose that the two conditions from part b hold. Prove that Rich's optimal strategyis Give UP.