n class we analyzed an ultimatum $($take$-$it$-$or$-$leave$-$it$)$ bargaining game in which two

players are bargaining over how to divide a $$pie$$ of size $1$ between them. Player $1$

can make any proposal $0$ x $1,$ where x is the proposed share for player $1$ and the

remainder, $1$ x$,$ is the proposed share for player $2.$ Player $2$ can accept or reject this

proposal. If player $2$ accepts then each side gets its proposed share. If player $2$ rejects

then each side gets $0.$

Consider a variant in which if player $2$ rejects, then player $1$ gets $0$ and player $2$ gets

a payoff of b$,$ where $0<$ b $<1.($In bargaining terminology, this amount is player

$2$s $$outside option$$ if no agreement is reached.$)$

$($a$)(10$ points$)$ Draw the game$-$tree.

$($b$)(20$ points$)$ Use backwards induction to find the SPE. $($The SPE includes player

$2$s entire acceptance rule, as well as the specific proposal that player $1$ ends up

making.$)$

$($c$)(10$ points$)$ As b becomes larger, what happens to player $1$s share of the pie in

the SPE outcome? Explain.

$2$