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Consider the problem of fairly sharing a homogeneous cake between two people. It is well-known that a fair division can be achieved through the divide and choose procedure: player 1 cuts the cake into two pieces and player 2 chooses a piece.

This problem can be generalized to non-homogeneous cake, more than two players, etc.

My question is: are there any concrete examples of people using the fair division procedures (such as divide and choose) that emerge as the solution to such problems in practical applications?

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The easiest generalization, of envy free sharing of a heterogeneous cake between two cake eaters is quite common. My family growing up frequently used the you divide and I choose method for sharing a lone piece of dessert. Depending on what you'd accept for "concrete example", Abraham and Lot use this method to divide the land of Canaan. A two-stage fair division problem was used in the partitioning of Germany after World War II. The Talmud has examples of fair division rules, which though are only thought experiments, are thought to have been applied in the Jewish diaspora over inheritance matters.

I would also argue that the marriage / matching problems of Lloyd Shapley and Alvin Roth are a form of fair division, and one that scales to very large numbers of players and applies only to multiple, indivisible "cakes. Their methods have been used extensively in real world problems of school, job, and kidney assignment.

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  • $\begingroup$ This hits the nail on the head. I was just thinking of writing about Gale-Shapley marriage stability actually. Neat stuff. $\endgroup$ – Kitsune Cavalry Oct 3 '15 at 18:20
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Most division problems are over land/property or belongings of the deceased or business acquisitions, things that aren't infinitely divisible, are not homogeneous, and often that involve more than two players. Cut and choose collapses with more than two players. There is a lot more literature about specific procedures such as:

  • Surplus Procedure: referee game, cannot guarantee both envy-free and equitable cut, may or may not be maxmin strategy proof, given which of those properties you go for
  • Banach-Knaster Procedure: generalizable to n players, proportional but not envy free
  • Dubins-Spanier Moving Knife Procedure, Steinhaus Procedure: 3 players, proportional but not envy free
  • Stromquist Procedure: envy-free
  • and in particular, Adjusted Winner for two people

Try a quick search of these with Google Scholar or www.fairoutcomes.com Fair division is a fascinating branch of economics and looking at efficiency vs welfare.

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  • $\begingroup$ I am aware that other procedures are needed for more general games. But the question is: are there practical examples of people using (any of) these procedures? $\endgroup$ – Ubiquitous Oct 2 '15 at 16:00
  • $\begingroup$ Huh, apparently my link is broken now. It used to give some practical examples of people using it. That is odd, sorry. $\endgroup$ – Kitsune Cavalry Oct 2 '15 at 16:47

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