I am doing a self study on behavioral economics and I am trying to solve behavioral version of the dictator game with following utilities for person 1 and 2.
$$ u_1( \sigma_1, \sigma_2 ) = \begin{cases} \sigma_1 - 2 \left( \frac{\sigma_1}{\sigma_1+\sigma_2}-\frac{1}{2}\right)^2 \text{ if } \sigma_1 + \sigma_2>0\\ 0 \text{ if } \sigma_1 + \sigma_2 = 0 \end{cases} $$
$$ u_2(\sigma_1, \sigma_2) = \sigma_2 - 2 \max\{\sigma_1 -\sigma_2,0\} - \frac{1}{4} \max\{\sigma_2 - \sigma_1, 0\}$$
So to solve this I first try to derive best response functions for both players with respect to their own control variable $\sigma_i$. For the first person the best response function would be:
$$BR_1 = 1 - 4 \left( \frac{\sigma_1}{\sigma_1+\sigma_2}-\frac{1}{2}\right) \left( \frac{\sigma_2}{(\sigma_1+\sigma_2)^2} \right)= 0 $$
However, I have a bit problem deriving the best response function of the second player because the function is kinky. I know that for example in standard consumer theory when there is kinky utility function like with perfect complements, lets say $\min \{\alpha x_1, \beta x_2\}$ the marginal utility of $x_1$ and $x_2$ would be $\alpha$ and $\beta$ respectively, but I usually dont work with kinky functions so I kinda just learn that part by heart and I am not sure if I am correct by saying that the best response function of the second player is:
$$BR_2 = 1 + 2 -\frac{1}{4} = 2 \frac{3}{4}$$
I got $BR_2$ just by extending the logic from the perfect complements utility function, but it seems to me like its somehow missing something. Just thinking about it intuitively it seems to me that person 2 cares about equality since the expression includes both $\max \{\sigma_1-\sigma_2,0\}$ and $\max \{\sigma_2-\sigma_1,0\}$, but then I dont understand why we end up just with constant best response function? Shouldnt it depend on what actually $\sigma_1$ is or its just that my calculations are wrong?
Also, I know from a standard dictator game that to calculate the minimum acceptable payoff the utility just has to be higher then 0. An extension of that should of course also hold here, but since here the zero utility outcome occurs only if $\sigma_1+\sigma_2=0$ I am bit unsure of how to account for that.
So to sum up my questions are:
- is my derivation of best response functions correct?
- provided that 1 is correct what is the intuition behind constant BR in the case of second player?
- How would I derive minimum acceptable payoff for this dictator game. Would it be just checking for which values of $\sigma_i$ the $u_i>0$ for $i=1,2$?
