$(19$ points$)$ Consider the following game based $on$ the Trust Game $(whichis$ often used $to$

File Preview the level $of$ trust $in$ a society$).An$ "investor" $(player1)$ can decide whether $to$ send

$$20LOan$ "entrepreneur" $(player2).If$ she keeps the money, the game ends with the investor

getting $$20$ and the entrepreneur getting $$0.If$ the investor sends the money, $it$ gets tripled $to$

$$60,$ and the entrepreneur decides whether $to$ keep all the $$60to$ herself, $orto$ split $it$ evenly

with the investor.

$(a)(3pts)$ Draw a game tree $(similarto$ one that $we$ have seen $in$ class$)$ that represents the

possible outcomes and choices $in$ this game. Make sure $to$ label which player makes $a$

decision $at$ each node.

$(b)(2pts)$ First, let's suppose both players are purely self$-$interested, and they both know

that the other $is$ purely self$-$interested. What will Player $2do,if$ Player $1$ chooses $to$

send the money $to$ her?

$(c)(2pts)$ What will Player $1do,$ knowing that Player $2$ will make the choice found above?

$(d)(4pts)$ Now, suppose the two players have distributional preferences $as$ introduced $in$

class: $if{x}_1$ and $x_2$ are the amounts $of$ money players $1$ and $2$ end $up$ with, respectively,

then player $1\text{'}s$ utility $is$

$$$$60$$$20$$$20(19$ points$)$ Consider the following game based $on$ the Trust Game $(whichis$ often used $to$

File Preview the level $of$ trust $in$ a society$).An$ "investor" $(player1)$ can decide whether $to$ send

$$20LOan$ "entrepreneur" $(player2).If$ she keeps the money, the game ends with the investor

getting $$20$ and the entrepreneur getting $$0.If$ the investor sends the money, $it$ gets tripled $to$

$$60,$ and the entrepreneur decides whether $to$ keep all the $$60to$ herself, $orto$ split $it$ evenly

with the investor.

$(a)(3pts)$ Draw a game tree $(similarto$ one that $we$ have seen $in$ class$)$ that represents the

possible outcomes and choices $in$ this game. Make sure $to$ label which player makes $a$

decision $at$ each node.

$(b)(2pts)$ First, let's suppose both players are purely self$-$interested, and they both know

that the other $is$ purely self$-$interested. What will Player $2do,if$ Player $1$ chooses $to$

send the money $to$ her?

$(c)(2pts)$ What will Player $1do,$ knowing that Player $2$ will make the choice found above?

$(d)(4pts)$ Now, suppose the two players have distributional preferences $as$ introduced $in$

class: $if{x}_1$ and $x_2$ are the amounts $of$ money players $1$ and $2$ end $up$ with, respectively,

then player $1\text{'}s$ utility $is$