Problem $3$: More Players and Extensive Form $[9$ Points$]$

Suppose that three players $($A$,$ B$,$ and C$)$ are playing a game. These three players are choosing

which of them has to mow the lawn. Player A and Player B will vote for who has to mow the lawn.

The game goes like this: Player A and Player B both play simultaneously. Player A gets to choose

a Player, so Player A$,$ Player B$,$ or Player C$.$ Player B also gets to choose a Player, so Player A$,$

Player B$,$ or Player C$.$ Player C then gets to observe the choices that Player A and Player B chose,

and then gets to choose a Player, so Player A$,$ Player B$,$ or Player C$.$

The player with the most votes has to mow the lawn. Whichever player gets the most votes has

to mow the lawn, and receives a payoff of $-10.$ If you do not have to mow the lawn, then you

receive a payoff of $+1,$ enjoying the smell of fresh$-$cut grass without doing any of the work.

If the vote is a tie, then the lawn is not mowed, and everybody receives a payoff of zero.

$7.$ Draw this game in Extensive Form. Include the payoffs each player receives, the strategies

they might choose, and show every point that is an information node. $[3$ points$]$

$8.$ Shrink this game tree as much as you can through backwards induction. $[1$ point$]$

For Q$9$ and Q$10,$ change the game in $1$ way. Now all three players move simultaneously, so Player

C does not get to observe the choices of Players A and B before making their choice.

$9.$ Draw this full game in Matrix Form. $[3$ points$]$

$10.$ What are the Pure Strategy Nash Equilibria in this game? $[2$ points$]$