Did you get your fair share of the cake?

We claim all envy-free divisions are proportional, but not all proportional divisions are envy-free.

Not all proportional divisions are envy-free: Suppose Alice, Bob, and Carol are dividing a cake and Alice starts by splitting the cake into three pieces, $P_1, P_2, P_3$ each with value $\frac{1}{3}$ . She takes the first piece $P_1,$ and then Bob and Carol together use the cut-and-choose method to divide pieces $P_2$ and $P_3$ . Suppose Bob thinks $P_2$ is slightly larger in value than $P_3$ , so he divides $P_2$ into two parts $P_2^*$ and $P_2^{**}$ and gives Carol the option to choose either $P_2^*$ or $\{ P_2^{**}, P_3 \}.$ Now, since Alice believes $P_3$ has value exactly $\frac{1}{3}$ of the cake, the portion $\{ P_2^{**}, P_3\}$ has value strictly larger than $\frac{1}{3}$ , so Alice does not receive the piece she believes to be largest in this division. This gives an example of a proportional division that is not envy-free.

All envy-free divisions are proportional: Consider an envy-free division among $n$ people and consider one person's (say, Alice's) perspective on the cake division. Since the division is envy-free, Alice believes she receives the largest piece among the $n$ pieces. Let the value of the entire cake be $1$ . Then if Alice values each piece in the division at strictly less than $\frac{1}{n}$ of the entire cake, then the entire cake has value

$\mbox{Value of entire cake } < \frac{1}{n} + \frac{1}{n} + \cdots + \frac{1}{n} = 1.$

This contradicts the assumption that the value of the entire cake is $1$ , so there must be at least one piece that Alice values at $\frac{1}{n}$ of the cake or larger. Since Alice receives the piece she believes to have largest value, her piece has value at least $\frac{1}{n}$ . Since this holds for all $n$ people in the division, this shows any envy-free division is also a proportional division.

This shows all envy-free divisions are proportional, but not all proportional divisions are envy-free

I know this may sound weird or even unreasonable, but how I saw it is; Suppose there were 4 people named A, B, C, and D. They all get exactly 1/4th of the cake, but in A's perspective, C's piece looks more tastier. (maybe it seems it has more chocolate syrup or something) Therefore the person's view of things are quite unpredictable, so, not all proportional divisions are envy-free.