Problem $1.$ Envy: a capital sin

Consider the following ultimatum bargaining game. There is $1$ unit of a good and player $1$ offers a split $(x,1-x),$ where xin$[0,1]$ is chosen by player $1.$ Player $2$ accepts the offer $(Y)$ or refuses it $(N).$ If player $2$ accepts the offer, then player $1$ gets $x$ and player $2$ gets $1-x.$ If player $2$ refuses the offer, then both players get $0.$ We assume that when player $2$ is indifferent between accepting and refusing, then he accepts.

$($a$)$ Suppose that each player maximizes his payoff. Find the Backward Induction Equilibrium $[$easy; we have done it in class already$].$

Suppose now a more interesting case: players are envious. More precisely, the utility of player $1$ is equal to his payoff minus $$ times the payoff of player $2$ and the utility of player $2$ is equal to his payoff minus $$ times the payoff of player $1,$ with $>0.$ The parameter $$ can therefore be interpreted as a measure of "envy".

$($b$)$ Find the Backward Induction Equilibrium as a function of $.$

$($c$)$ What happens as $$ increases? Interpret the result.