It works for uniform cake. The failure mode is when the cake is non-uniform, and the children want different things. In this scenario the top of the cake has two pieces of fruit, a slice of kiwi and a slice of strawberry. It's a soft cake, so the fruit can't be split or cut. Child A cuts the cake into two slices, each with one piece of fruit. Which slice does Child B choose? If you're lucky, A likes strawberry and B likes kiwi, and B chooses what they want. If you're not, A likes strawberry and B is an asshole that doesn't care about fruit, so B chooses the strawberry piece knowing that A likes strawberry, and picks that piece just to ruin A's day. That's over the top, mean and spiteful you say? Have you met children!?
> Game theory rises to the level of Solomon. Has it ever failed?
There’s a reason we study the Prisoner’s dilemma and its failure modes.
With regards to the cake example more specifically, what makes it somewhat trivial is that the state of the world is fully observable, utilities are easily computable (volume of cake), etc.