Abhay Shivgounda Patil

About what matters.

Mathematics for justice!

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Can we divide something between people such that everyone is guaranteed to be satisfied?  Yes, for sure – and the logic behind it is not as difficult to understand as you may think.

I attended a wonderful Fun-DO-Math session by Prof. M. Prakash today.  He mixed puzzle, mathematics and psychology to debate ways to divide something desirable (a cake, or a plot of land) between 2 or more people where everyone would be guaranteed to be happy at the end!  Here the astonishing part is one can actually prove that everyone would be happy – isn’t that amazing?

Now – some mathematically literate folks may shrug their shoulders at my naivete – but I shall still proceed to describe this method.  I was aware of the first part, where we have two people fighting for the share.  But I had not thought about the case when there are more than two.  It’s quite simple – so even if you are not a math person – read on.  Here we go. (This is a quite well known problem of fair share, but my little google search did not yield an elegantly stated solution. It may still be there – I just didn’t find it.)

Let’s say Baba and Dada want to share a cake. Both want at least half of it. If we appoint a third person who will cut the cake and give a piece to each – it is very much possible that one of them may cry foul. So – what do we do?  So, here is the procedure that you should follow.  You ask Baba to cut the cake in two parts and let Dada go first to choose a piece for himself.  BTW – note that you actually don’t even need to tell Baba that he has to cut it in two equal pieces!  They are both smart people with steady hands and a good sense of proportion.  The moment they understand the procedure – it is guaranteed that both will have no reason to feel unhappy at the end!  Baba is the last one to pick the piece, but he gets to cut it in two.  Dada goes first – so has no reason to complain.

Now, imagine Bhau joins the fray and we need to make three equal parts – and we also want to ensure that it is guaranteed that all of them will be satisfied with what they get.

We have already solved the problem for two people.  We need to build the solution for three people.  There is one added complexity.  We need to ensure two of them don’t gang up against the third!  Let’s say we ask Baba to cut the cake in three, and let him go last.  Then we ask Dada to pick a piece first, followed by Bhau; and so Baba gets the last remaining one.  Knowing the procedure, smart that he is, Baba may enter in a pact with Dada.  He may cut one big piece and two small ones.  Dada will then pick a big one.  Bhau will have to choose one of two smaller ones.  Now it may appear that Baba too gets a small one – but Dada will actually share his spoils with Baba and keep Bhau fuming!  So this procedure won’t cut it. (Pun unintended 😉 )

There are actually two (or even more?) solutions – let’s look at the one that’s easy to understand.  We will first ask Baba to cut the cake in two as if he is sharing it with only one other (and not two other) person.  We then ask Dada to choose one of the pieces.  This is exactly like the first case.  Then we ask Baba to cut his piece in three parts – where it is made clear to him that Bhau will be asked to pick any one of those three pieces.  We tell the same thing to Dada.  After Bhau picks one piece each from Baba’s and Dada’s sets, we are done!  Each one will have two pieces with them – and each of them will be guaranteed to be satisfied that they got at least 1/3rd of the original cake.

Check for yourself why no one will have a reason to complain.  Post it as a comment here.  If you think someone could get unhappy, post that argument too.


Written by Abhay Shivgounda Patil

December 23, 2012 at 9:30 pm

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