Fair Division

Assume you have 10 candies, how do you divide them "fairly" among 5 children?

At face value this question might look simple and not fit for college class discussion. However, let's waite. First let's see at some questions that need answering .

Are the candies identical?

Are the children entitled to equal shares?


If the candies are identical and the children are entitled to equal shares then the problem is indeed a Grade 2 level division problem. Each child should get 10/5 = 2 candies.


On the other hand, if the answer to either one of the above questions is no, then we are in for a big surprise.


Should we, regardless of the type of the candies involved, insist on giving the children two candies each? If we do so, what will happen? If we did that, can we, with assurance, say we are dividing the candies "fairly"?


Before we go any further, what does "fairly" mean in this context?


Is this really a problem? After all, what is the big deal of dividing candies? Who cares whether we divide the candies fairly or not? Is this a question that only mathematicians, in their crazy mind, ask?


Well, what if the candies are replaced by:


And instead of children we have players who lay claim to these items? These could be:


What about if the items are:


We can easily see that the candy talk is only a metaphor to what can come in the heart of many court battles of divorce settlement and inheritance fights.


In short, regardless of what we are dividing, be it candies or jewelry, art work or land, we are talking of a set of "items" that are to be be divided and a set of players which could be a set of:

that are entitled to share the "items".


What is a fair share?

A fair share, in the case we are talking here is any share that in the opinion of the person getting it is at least "one Nth" of the item(s) to be divided. Here N is the number of players. Let's note here that this "fair share" is relative to the one receiving it. The opinion of the others doesn't matter. In otherwords, we are implying only proportional share not necessarily an "envy-free" share. Proportional shares are more easy to come by than envy-free shares. Here we discuss the proportional shares.

Types of fair division:

Continuous fair division

  1. Two players

  2. More than two players

Discrete Fair-Division