This webpage is aimed at introducing something known by many names such as the Fair Share problem or Cake Sharing problem. We will first look at some History and Background around this famous problem and go through some of the mathematics and explanation of where it comes from, and then the website will detail the significance and applications of this problem. For a more visual representation go to this Numberphile explanation.

Historical Development and Background:

Fair Cake cutting has been a documented problem for at least the last seven decades. This has been a problem that was extensively studied in areas of computer science and othe social sciences as the idea of fair shairing has implications beyond just cake. There are many different methods for solving the fair share problem, the latest being discovered and proved in 2017. This method was a step above the rest because it offered a truly bounded number of cuts to solve the problem. The rest of the methods while genios ways to solve the problem could theoretically have an unbounded number of cuts and are therefore unfeasible in some senarios, especially with large numbers of people sharing the cake¹.

One of the first documented methods was published in 1948, and is the most interesting method to me. It's called the "Last Deminisher" method. This method was published in Econometrica by Hugo Steinhaus who was a Polish-Jewish professor at the University of Warsaw who was actually in hiding duriing WWII when he first became interested in this problem. With some help from a couple of his students, Stefan Banach and Bronislaw Knaster, Steinhaus was able publish this so called "Last Deminisher" method that is still widely used today. ².

Explanation of Mathematical Content:

While there are many ways to solve the Fair Division problem, I will focus on Steinhaus' method. Many could argue that the mathematics is better for the Bounded method found in 2017, but that math is much more convoluted and complicated to be used in real life situations in my opinion. If you would like to learn more about this method ^{click here}. I believe that Steinhaus' "Last Diminisher" method is the perfect amount of mathematically complex such that it can be explained to regular people. I think it also makes the most intuitive sense and therefore is why I will focus on explaining that method. The one caveat is that this method while being practical for a lot of cases, ironically would probably not be practical for cutting cakes.

First, let's talk about the "rules" of the Last Diminisher method. To start, everyone is numbered as a way to give people order for when their turn is. The first person cuts the cake into two pieces however they would like. At this point, there should always be a piece of the cake that is being selected and a piece of the cake not yet up for selection. The second person gets the choice whether to cut what I will call the selection piece. If they do not, then the same choice goes to the next person. If the second person chooses to cut, they must cut off a portion from the selection piece. Each time a portion is cut off the selection piece, it joins the non-selection piece off to the side. This pattern continues in a cycle where the last person to cut the cake, gets the selection piece, that is if everyone else has been given the chance to cut the cake and has chosen not to. This continues until everyone has gotten a piece of cake. This is why it is called the Last Diminisher method.

Next, let's talk about how this satisfies the fair division. The integral part of this method is that each person has a choice to cut the selection piece. This means that if the person believes the piece is already "unfair" or would be with part cut off, they will not get it. This also means that if they do cut the piece they are accepting or stating that they think it would be fair. The logical part of this method says that at this point, the selection piece would continue to be cut until it reached 1/n proportion of the whole cake, at which time nobody else would cut a piece off afterwards. This means that every person would eventually get 1/n of the original cake.

Applications:

The applications basically boil down to the importance of resource allocation between large numbers of people or departments. For example, we commonly need to allocate school supplies to children course slots in universities to students machine processing time to users kidneys to kidney transplant patients etc.
"Understanding who gets what, and how and why, is still very much a work in progress."
-- Alvin E. Roth, 2012 Nobel Laureate in Economics

Resources: *Some of these resources were not used in the above webpage.* If you would like to learn more about Fair Division, these Resources will expand your understanding.
Aziz, H. (2017). A Discrete and Bounded Envy-free Cake Cutting Protocol for Any Number of Agents. Carnegie Melon University.¹
Steinhaus, H. (1948) Report of the Washington Meeting, September 6-18, 1947. Econometrica, 16(1), 33-111. ².
Arzi, O., Aumann, Y., & Dombb, Y. (2015). Toss one’s cake, and eat it too: partial divisions can improve social welfare in cake cutting. Social Choice and Welfare, 46(4), 933–954.
Brams, S. J., & Taylor, A. D. (1996). Fair Division: From Cake-Cutting to Dispute Resolution. Fair Division.
Chen, A. links open overlay panelY. (2013). Truth, justice, and cake cutting. Games and Economic Behavoir, 77(1).
Chung, F. (2000). Fete of Combinatorics and Computer Science. Mathematical Studies, 63–93.
Detemple, D. (1987). Applications of Euler’s formula to partition problems. The Mathematical Gazette, 71(456), 104-107.
Robertson, J. (1998). Cake-Cutting Algorithms.
Procaccia, A. D. (2013). Cake cutting. Communications of the ACM, 56(7), 78.
Segal-Halevi, E., Nitzan, S., Hassidim, A., & Aumann, Y. (2017). Fair and square: Cake-cutting in two dimensions. Journal of Mathematical Economics, 70, 1–28.
Shin , J. (2018). CONCEPTUAL ANALYSIS OF STUDENTS’ SOLVING EQUAL SHARING PROBLEMS. Early Algebra, Algebra, and Number Concepts.
Yuen, G. (2008). Problem solving strategies students use when solving combinatorial problems. University of British Columbia.