a$)$ Explain why in a Nash equilibrium, it must be that p $1=$ p $2=0.$

$($Hint: compare player i$$s payoff of choosing $($p i $=$ x$,$ r i $=$ y$)$ with the payoff of

choosing $($p i $=0,$ r i $=$ x $+$ y$)).$

$($b$)$ Given that p $1=$ p $2=0$ in any Nash Equilibrium, each council member$$s payoff

function can be simplified,

u $1($r $1,$ r $2)=2$r $1$ r $2$

$1$ r $1+$ r $2$

$2=3$

$2$ r $1$ r $2$

$1$ r $2$

$2$

u $2($r $1,$ r $2)=2$r $2$ r $2$

$2$ r $1+$ r $2$

$2=3$

$2$ r $2$ r $2$

$2$ r $1$

$2$

Find council member $1$s best response to level r $2.$

$($Hint: Use the math fact$)$

$($c$)$ Find the Nash equilibrium of the game.

$($Hint: each player has a dominant strategy$).$