Cut and choose

Two children have one cake and both want a fair share. Here is the oldest fair rule there is: one cuts, the other chooses.

Two children, one cake: $1$ cake in all. One cuts, the other chooses.

Cake $K=1$; cutter $C$, chooser $H$; pieces $a$ and $1-a$.

The cutter would love the biggest piece, but she cannot know which piece the other will take. So the only safe plan is to cut two pieces that are exactly equal. If one piece were bigger, the chooser would take it and the cutter would be left with less than half.

The cutter makes pieces $a$ and $1-a$. The chooser takes the bigger one, so the cutter gets $\min (a,1-a)$. That is at most $\frac{1}{2}$, and it reaches $\frac{1}{2}$ only when $a=\frac{1}{2}$. So she must cut in half.

$C$ gets $\min (a,1-a)\le \frac{1}{2}$, equal only for $a=\frac{1}{2}$.

The chooser is safe as well. Whichever piece he takes, at least half of the cake is his, and if the cutter was careless, he gets even more.

The chooser gets $\max (a,1-a)\ge \frac{1}{2}$, however the cake is cut.

$H$ gets $\max (a,1-a)\ge \frac{1}{2}$.

So nobody can complain: each child is sure of at least half, and the only way to be sure of half is to cut honestly. This kind of fairness needs no judge, only a good rule.

Honest cutting means $a=\frac{1}{2}$, so each child is sure of at least $\frac{1}{2}$.

$a=\frac{1}{2}$, so both are sure of $\frac{1}{2}$ or more.