There’s no simple simple math to decide what point in the ZOPA we settle on.
Actually, you could model this as a bargaining problem. The only issue that the basic theory models only one resource that is to be distributed among two people, whereas now we have two (coconuts and bananas). This means it requires both of us to be honest about our preferences for how we weight bananas vs coconuts. Given that, bargaining theory requires that we define
A feasibility set F, a closed subset of R2 that is often assumed to be convex, the elements of which are interpreted as agreements.F is often assumed to be convex because, for any two feasible outcomes, a convex combination (a weighted average) of them is typically also feasible.
A disagreement, or threat, point d=(d1,d2) , where d1 and d2 are the respective payoffs to player 1 and player 2, which they are guaranteed to receive if they cannot come to a mutual agreement.
The disagreement point (d1,d2) equals the pair of {coconut plus banana utility for (me,you) according to our honest preferences} we get when we don’t trade. The feasibility set consists of all possible pairs of of {coconut plus banana utility for (me,you) according to our honest preferences} when we trade according to any one exchange rate.[1] So point in the ZOPA corresponds to one point in the space.[2] Given this, there’s a unique mathematical solution that follows from some basic properties.
If both parties accept this, this would solve a part of the problem. Not all, since we are still incentivized to lie about our preferences.
There are other assumptions that get to other bargaining solutions, however. Also, it’s unclear to me whether any of the bargaining solutions are morally acceptable in all important cases.
Actually, you could model this as a bargaining problem. The only issue that the basic theory models only one resource that is to be distributed among two people, whereas now we have two (coconuts and bananas). This means it requires both of us to be honest about our preferences for how we weight bananas vs coconuts. Given that, bargaining theory requires that we define
The disagreement point (d1,d2) equals the pair of {coconut plus banana utility for (me,you) according to our honest preferences} we get when we don’t trade. The feasibility set consists of all possible pairs of of {coconut plus banana utility for (me,you) according to our honest preferences} when we trade according to any one exchange rate.[1] So point in the ZOPA corresponds to one point in the space.[2] Given this, there’s a unique mathematical solution that follows from some basic properties.
If both parties accept this, this would solve a part of the problem. Not all, since we are still incentivized to lie about our preferences.
Well, actually it would be the convex closure of those points plus (d1,d2).
I believe you could also include all of the exchange rates outside of ZOPA and it wouldn’t change the solution.
There are other assumptions that get to other bargaining solutions, however. Also, it’s unclear to me whether any of the bargaining solutions are morally acceptable in all important cases.