Consider the following 3 player game:

• At time t = 0, the quality of a good is determined; it can either be good (in which case
• we say ? = G) or bad (in which case we say ? = B)
• At time t=1, player1 observes a signals 1 ?{g,b},whereP[s1 =g|?=G]=P[s1 =
• b | ? = B] = r > 1/2.
• After observing s1, player 1 decides whether or not to adopt the technology. If ? = G, player 1 obtains a payoff of 1 from adoption, and if ? = B, then player 1 obtains a payoff of -1 from adoption. Not adopting gives payoff 0.
• At time t = 2, player 2 observes a signal s2 ? {g, b}, which is independent from and identical distributed as s1. Player 2 also observes whether player 1 adopted or not, but not s1 .
• Player 2 then decides whether to adopt the good or not. Payoffs from each action are the same as for player 1.
• At time t = 3, player 3 observes a signal s3 ? {g,b}, again independent from and identically distributed as s1 and s2
• ? Player 3 then decides whether to adopt the good or not. Payoffs from each action are the same as for player 1. Player 3 also observes whether players 1 and 2 adopted or not, but not s1 or s2.

1. Suppose player 1 believes the prior distribution over ? is such that P[? = G] = q. Find a condition on q and r which ensures that player 1 will (a) always strictly prefer to adopt the technology, and (b) never adopt the technology.
2. Now suppose P[? = G] = P[? = B] = 1/2 for the rest of this problem. Argue that player1 will always adopt if s1 =g and not if s1 =b.
3. Argue that there exists an equilibrium where player 2 will always adopts if s2 = g and not when s2 = b. Your solution should address the fact that player 2 sees player 1's adoption decision.
4. Now consider player 3. Show that if r ? (1/2, 1), then in the equilibrium from the previous part, player 3 will ignore his signal if observing (a) both players adopt, or (b) both players do not adopt.
5. Argue that the same reasoning implies that, even if we extend this game to have infinitely many players (each moving sequentially as above), the probability that the ex-post optimal adoption decision is made does not approach 1, for any r

2. NEW TECHNOLOGY ADOPTION Consider the following 3 player game: 0 At time t = 0, the quality of a good is determined; it can either be good (in which case we say 0 = G) or bad (in which case we say 0 = B) 0 At time t = 1, player 1 observes a signal 31 E {g, b}, where lP'[31 = g | 0 = G] = IP[31 = b|0=B]=r>1/2. 0 After observing 31, player 1 decides whether or not to adopt the technology. If 0 = G, player 1 obtains a payoff of 1 from adoption, and if 0 = B, then player 1 obtains a payoff of -1 from adoption. Not adopting gives payoff 0. 0 At time t = 2, player 2 observes a signal 32 E {g, b}, which is independent from and identical distributed as 31. Player 2 also observes whether player 1 adopted or not, but not 31. 0 Player 2 then decides whether to adopt the good or not. Payoffs from each action are the same as for player 1. 0 At time t = 3, player 3 observes a signal 33 E {9, b}, again independent from and identically distributed as 31 and .52