Suppose player 1 and 2 each has a glass of beverage A and B, respectively. Player 1 is indifferent between the two beverages, while player 2 likes beverage A twice as his own beverage. Can they reach an agreement on exchanging some amount of their beverages?
I'm interested in finding the Nash bargaining solution for the given situation. I can model their utilities by $U_1(a,b)=a+b$ and $U_2(a,b)=2a+b$, but I have no idea how I should represent the feasibility set as a closed subset of $\mathbb{R}^2$ and solve the problem. All of the examples I have seen before, such as splitting a dollar, have one quantity that is being negotiated, while here, there are two.
The feasible set will just be the possible allocations of the two beverages. Picture the utility of player 1 on the first axis and the utility of player 2 on the second axis. If player 2 gets both beverages his utility is $U_2(1,1)=3$ and player 1's utility is $0$. If player 1 gets both beverages his utility is $U_1(1,1)=2$. Since the utility functions are linear you can just connect the two points $(3,0),(2,0)$ in $\mathbb{R}^2$ which gives you the feasible set of possible payoffs. Something like this.