2

Suppose there is a single, indivisible resource that can be used or enjoyed by just one of two parties at a time. You can imagine a field of crops that can be plowed by just one of two farmers in any year. My question concerns the equitable division of that resource over time. (I feel confident the following is a classic problem in economics or game theory related to, yet different from, basic compound interest.)

Suppose Farmer A uses the field for each of ten consecutive years. Next it is Farmer B's turn. Suppose Farmer B uses the field for each of the next ten consecutive years. After those 20 years, each farmer has had the field for the same total time. Seems that they're "even"... a "fair" apportionment.

But think again.

After the first year A has enjoyed more farming time than B. Likewise after year 2... Likewise after year 11. And after year 12... it is only the full 20 years do we come to a situation where A hasn't had more time than B. And B NEVER experiences the condition that he has had more time with the resource than has A. SURELY A has had a better deal, overall. Right? It seems that B should be given a few "extra" years to experience what A has experienced for 20 years straight: a greater total usage time than his counterpart farmer.

Now I'm well familiar with compound interest, discounting, the "time value of money," and such, but in this case it isn't clear what discount rate or other time penalty is appropriate.

Hence my questions:

  • Is there a term (presumably from mathematical economics or game theory) to describe precisely this situation?
  • Is the above logic correct (or at least justified) that Farmer B should have more than the nominal 10 years in order to achieve a fair apportionment between A and B?
  • In the absence of an external "interest rate", is there a principled method for computing how many years B should have to achieve parity?
Arctic Char
  • 16,007
David G. Stork
  • 29,774
  • 1
    I have had Questions like this myself. Even here, the situation is more complicated than you made it by saying that B never had more than A. I make more complicated by saying that Initially A used the field to grow something while B was totally idle. Later when B was using the filed, A was utilizing what he grew. { A might have stored wood to make furniture , stored fruits to make pickles , stored money to go on vacation } Alternatively, given the time spans, we can say A used his youth to enjoy his old age , whereas B wasted his youth & has to toil in old age. [ Here, A has a better deal ! ] – Prem Sep 07 '22 at 06:51
  • Yep: In that case I want to make the simplifying assumption that there is some compound interest rate (that we keep the same.... for instance that the farmer sells the crop and puts the funds in an interest-bearing account). In that case, I can slog through and numerically find when the two "final values" are the same. But I'm uncomfortable with this, as I always seem to need to impose an interest rate (or effective interest rate), and I see no principled way to do that. Perhaps one can only solve this problem with some effective interest rate. Let's see if other solvers have insights. – David G. Stork Sep 07 '22 at 06:56
  • Everything lies in the "fair" word ... what do you mean by "fair" ? Here is my thougts about your question : If you can describe your problem in terms of utility functions and constraints that are supposed to make the deal "fair", all that remains will be mathematics .. – MrSmithGoesToWashington Sep 07 '22 at 08:05
  • 1
    Perhaps relevant: https://en.wikipedia.org/wiki/Thue%E2%80%93Morse_sequence#Equitable_sequencing – Qiaochu Yuan Sep 07 '22 at 08:56
  • @QiaochuYuan: Very interesting... but not quite relevant. Thanks, though! – David G. Stork Sep 07 '22 at 09:27
  • It sounds like you want a fairness criterion. Have you looked at https://en.wikipedia.org/wiki/Efficient_envy-free_division keeping in mind that a bundle may be an infitie sequence? – Trurl Sep 18 '22 at 23:20

0 Answers0