{"id":47,"date":"2020-01-21T16:43:27","date_gmt":"2020-01-21T16:43:27","guid":{"rendered":"http:\/\/www.lancaster.ac.uk\/stor-i-student-sites\/kim-ward\/?p=47"},"modified":"2020-01-21T16:43:29","modified_gmt":"2020-01-21T16:43:29","slug":"mathematical-fairness","status":"publish","type":"post","link":"https:\/\/www.lancaster.ac.uk\/stor-i-student-sites\/kim-ward\/2020\/01\/21\/mathematical-fairness\/","title":{"rendered":"Mathematical Fairness"},"content":{"rendered":"\n

A while ago, when I was in the “MOOCs are great!” phase of my life, I took an online course on negotiation<\/a>. I never finished the practical part of it, but I remember the theory being surprisingly mathematical. It centred around constructing theoretically sound mathematical concepts of “fairness” and then applying them to a variety of different scenarios to construct an “optimally fair to all parties” solution – accompanied, of course, by an explanation of the various ways people attempt to obscure this to skew the negotiation situation in their favour.<\/p>\n\n\n\n

A simple example of mathematical fairness is known as the “Principle of the Divided Cloth”, and it goes all the way back to the Talmud. If there is a roll of cloth that person A claims to own all of and person B claims to own half of, how should it be divided? Well, assuming the claims are both reasonable, one would be tempted to split the cloth in proportion to the amount that was claimed by each side, in a 2:1 ratio. However, the Principle of the Divided Cloth instead proposes a 3:1 split, following the logic that only half of the cloth is in dispute – nobody is saying that A doesn’t own the first half – so the disputed part should be split equally.<\/p>\n\n\n\n

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The Principle fo the Divided Cloth, taken from a video in the Negotiation course<\/figcaption><\/figure>\n\n\n\n

This principle can be applied to a wide variety of problems where two (or more) parties can come together to create something more valuable than they could working apart, and must decide how to divide that extra value.<\/p>\n\n\n\n

Fairness in Optimisation<\/h2>\n\n\n\n

In an optimisation problem, you’re trying to locate the “best” solution. “Best” how and for whom, you ask? Good question. The usual answer is “best for whomever is paying me and however they define ‘best'”. However, in applying optimisation methods to complex problems when multiple parties have stakes in the outcome, you can find yourself unwillingly appointed to the position of arbitrator.<\/p>\n\n\n\n

The example we’ve looked at in STOR-i relates to the OR-MASTER<\/a> project about scheduling flight capacity on airport runways. The airport itself would like as much capacity as possible to be used (making it extra money), without going over capacity and causing delays as planes are held up for lack of runways. Each airline wants to be able to schedule its flights to take off and land whenever it wants, which may include during “peak periods” when the runways are too busy to accommodate everyone’s requests. The airlines as a collective want a solution they understand the mechanics of that means they don’t have to play convoluted guessing games with to get the flight slots they want.<\/p>\n\n\n\n

There are two conflicts at play here:<\/p>\n\n\n\n