I'm studying the different variations of the ultimatum games. I've spent some time on this following game:
Assume now that each player does not only care about the amount of money she receives, but also about the equity of the allocation. Specifically, suppose that player i’s preferences are represented by the following utility function $$u_{i}(s_{1},s_{2}) = s_{i} - \beta_{i}|s_{1}-s_{2}|$$ with $\beta_{i} > \frac{1}{2}$ and $|s_{1}-s_{2}|$ denoting the absolute value og $s_{1} - s_{2}$.
Find the SPNE.
First, we can see that a term is "missing" in the Fehr-Schmidt utility function, which implies that $\alpha_{i} = 0$. Thus, we only have the "own payoff" which is the first term in the utility function and the term which measures the loss from advantageous inequality.
My approach is to use backward induction because we have a dynamic game of complete (and perfect) information. So, logically we should start by solving the second stage and then move back to the first stage.
It is not written in the what payoffs players get if the responder declines the offer. But I guess we can assume that each get nothing i.e. 0. Apart from declining the offer the responder can also accept the offer in which he will receive $$u_{2}(s_{1},s_{2}) = s_{2} - \beta_{2}|s_{1}-s_{2}|.$$ So, to solve for the second stage player 2(the responder) gets $s_{2}$ and player 1 (the proposer) gets $1 - s_{2}$. Thus, we can insert $1 - s_{2}$ in the utility equation for player 2 $$u_{2}(s_{1},s_{2}) = s_{2} - \beta_{2}|1 - s_{2} - s_{2}| = s_{2} - \beta_{2}|1-2s_{2}|.$$ Thus, the responder will reject the offer if $$s_{2} - \beta_{2}|1-2s_{2}|<0 \quad \Rightarrow \quad s_{2} < \frac{b_{2}}{2b_{2}+1}, \frac{b_{2}}{2b_{2}-1} < s_{2}.$$
When $s_{2} - \beta_{2}|1-2s_{2}|=0$ player 2 is indifferent. In case of indifference we are allowed to assume that player 2 chooses to accept.
Now that we know the threshold for acceptance of player 2, we can turn to player 1. Player 1 has the same Fehr-Schmidt utility function as player 2. Since the Fehr-Schmidt utility function is strictly increasing in own payoff, player 1 wants to keep as much of the share as he can. Anticipating player 2's acceptance threshold, player 1 will offer $s_{2} = \frac{b_{2}}{2b_{2}+1}$ and $\frac{b_{2}}{2b_{2}-1} = s_{2}$. In which he will keep $$s_{1} = 1 - \frac{b_{2}}{2b_{2}+1} \quad \text{and} \quad s_{1} = 1 -\frac{b_{2}}{2b_{2}-1}.$$ In that case the SPNE of the game is either of the two following $$E^{SPNE}=\left(1 - \frac{b_{2}}{2b_{2}+1},\frac{b_{2}}{2b_{2}+1}\right) \quad \text{or} \quad E^{SPNE}=\left(1 - \frac{b_{2}}{2b_{2}-1},\frac{b_{2}}{2b_{2}-1}\right).$$
So, my concern is if this is done correctly or if there is another method which is easier. Any comment, constructive criticism, and in generel help is much appriciated.
Note I have looked at this post for some help:
Fehr & Schmidt, ultimatum game, inequaltiy aversion, perfect subgame Nash equilibrium
Thanks in advance.
