Fehr & Schmidt, ultimatum game, inequaltiy aversion, perfect subgame Nash equilibrium
There are a couple of things here. First of all, envy is when your own payoff is smaller than the other player's payoff, which is the case when $z_i&lt;z_j$. So the coefficients here are $0.8$ for envy and $0.4$ for compassion (envy looms larger).

Solving the game using backward induction, we need to first find the minimum value that player 2 is willing to accept. A purely self-regarding player will accept any positive offer from player 1. But with fairness preferences, player 2 will actually decline low positive offers as unfair. To find the threshold, we first note that player 2 gets a utility of $0$ if she declines the offer (in which case both receive 0 payoff). Suppose player 1 offers player 2 an amount $z_2$. Player 1 would keep the remainder, $80-z_2$. The utility is piece-wise linear, with a kink at 40 (equal split), like this:



We need to find the value for $z_2$ where player 2's utility crosses the x-axis (red dot). Since this happens below 40, the relevant part of the utility function is the envy case. Thus, we solve $z_2-0.8(80-z_2-z_2)=0$, which turns out to be $z_2^0=\frac{320}{13}\approx 24.6$.

Knowing the acceptance threshold we can now turn to player 1. Player 1 has the same Fehr-Schmidt preferences as player 2. Since the utility function is strictly increasing in own payoff, player 1 wants to keep as much of the pie as possible. Anticipating player 2's acceptance threshold, player 1 will offer $z_2^0=\frac{320}{13}$, maybe adding a small $\varepsilon$ to resolve player 2's indifference.