Consider the incomplete$-$information version of the two$-$round bargaining game, where there is uncertainty on the discount factors.

In the first round, player $1$offers shares $($x$1$; $(1-$x$1)),$where x$1$is a number between $0$and $1.$If player $2$accepts this first offer, then the game ends with payoffs x$1$for player $1$and $(1-$x$1)$for player $2.$If player $2$rejects this offer, then the game proceeds to the second round.

In the second round, player $2$offers shares $($x$2$; $(1-$x$2)),$where x$2$is a number between $0$and $1.$If player $1$accepts this second offer, then the game ends with payoffs delta$1*$x$2$for player $1$and delta$2*(1-$x$2)$for player $2.$If player $1$rejects this offer, then the game ends with $0$payoff for both players.

Player $1\text{'}$s discount factor delta$1$belonging to the interval $(0$; $1)$is common knowledge. But player $2\text{'}$s discount factor delta$2$is private information her : Either player $2$is impatient and her discount factor is delta$2=0,4($with probability p$),$or she is patient and her discount factor is delta$2=0,8($with probability $1-$p$).2$

We will solve for the subgame$-$perfect Bayesian Nash equilibrium outcome of this game by proceeding backwards.

a$)$Construct the subgame$-$perfect Bayesian Nash equilibrium of the second round of the game: Which offers $($which values of x$2)$will be accepted by player $1?$Which offers will be made by the two types of player $2?$

b$)$Going backwards to the first round, given what you found in part a$,$which offers $($which values of x$1)$will be accepted by the impatient type of player $2,$which offers will be accepted by the patient type of player $2?$

c$)$Given what you found in part b$,$find the optimal offer that player $1$will make in the first round as a function of probability p $($the probability that player $2$is impatient$)?$

d$)$In light of what you found above, what would be a reason for bargaining to last longer than predicted by the equilibrium of the bargaining game that we discussed in class?