I have a question related to the "pizza splitting" problem (split the pizza in a fair way), and I am looking for an answer of hopefully the same elegance (first splits, second chooses).
Not sure if this is the right Stack Exchange site to ask this, as it has some economics traits to it.
Problem statement: Two participants A and B buy an indivisible item to use together, for an initial price of P'. Some time passes (the "price" is now only P < P') and the two of them need to split the item: A keeps the item and pays B some money for it, or the other way around.
I am looking for a fair protocol (a bidding system perhaps?), input of which would be "what value does the item have for A and B" and the output would be "who gets the item and how much money he pays to the other".
Ideas: I thought that the item has 2 "values" for each of the participants: the maximum amount of money he would pay to keep it P_keep(), and the minimum amount he would sell it for P_sell(). Not sure whether it is always true that one of these is always larger than the other one. I am guessing this could at least be a validation of some sort: if P_keep(A) < P_sell(B) and P_keep(B) < P_sell(A) then there is no way to split this.
What should I be looking for here? Wikipedia has some articles about splitting and fair protocols, but it always seems to be the case that the value is the same for each of the participants...
A great help to me would be some insight into how P, P', P_keep() and P_sell() work, how they are e.g. ordered.