This is about the fairness and reciprocity function of Rabin and Charness. Considering the 2 Player game there are the following utility functions:

u1 (a1, b2, c1) = (a1, a2) + f1 (a1, b2) * f2 (b2,c1)

u2 (a2, b1, c2) = (a1, a2) + f1 (a2, b1) * f2 (b1,c2)

u1 (a1, b2, c1) = utiliy function of player one

u2 (a1, b2, c1) = utiliy function of player two

(a1, a2) = Material Benefits

f2 (b2,c1) * f1 (a1, b2) = Fairness Reciprocity

f1 (a1, b2) = Kindness function

f2 (b2,c1) = Perceived kindness function

a1 = action by player one

a2 = action by player two

b1 = Represents player two's beliefs about strategy that player 1 will employ

b2 = Represents player one's beliefs about strategy that player 2 will employ

c1= represents what player one believes player two believes player one's strategy is

c2= represents what player two believes player one believes player two's strategy is

After repeated games we reach the following nash equilibrium: a1 = b1 = c1 for player one or a2 = b2 = c2 for player 2

In order to reflect reality more, I wanted to integrate a variable reflecting in-group bias, meaning that first, individuals see themselves as parts of groups and care about the treatment toward the members of their groups. And second, it is easier for individuals to cooperate if they belong to the same group, for example if they are relatives. There is evidence that individuals tend to treat better those individuals that belong to their own group, even if the group was formed randomly. Does it make sense to include the in-group bias by just including the variable x? u1 (a1, b2, c1) = (a1, a2) + f1 (a1, b2) * f2 (b2,c1) * x u2 (a2, b1, c2) = (a1, a2) + f1 (a2, b1) * f2 (b1,c2) * x And if it makes sense is there a way to calculate the new nash equilibrium? Like how does a1 = b1 = c1 get affected by x?