The Complexities of Cake
February 3, 2013, 11:19 pm
Filed under: Editorializing

There is a classic scenario from game theory that solves the problem of a parent dividing a cake between two siblings: let one cut the cake in two, and let the other choose which piece to take, and the result will invariably be that the pieces are about equal and neither kid can complain. Well, this hypothetical parent should be glad I’m not their kid, ’cause I’ve been thinking …

Cutting a cake by straight slicing is not the optimal solution. If you’re the kid who’s cutting, it’s better to take human foibles into account. If your sibling likes icing, cut the cake horizontally so that a smaller amount of cake has all the icing. Between these two pieces, the icing lover will choose the smaller top piece, which gets you more cake.

Or … cut a smaller circle out of the inside of the cake (assuming it’s a round cake). The smaller circle and remaining outer ring will be of equal sizes if the smaller circle’s diameter is 70.7% of the full cake’s diameter. (Note: the smaller circle need not be concentric with the full cake — it could be, for instance, tangent to the full cake.) If you cut the smaller circle to have a diameter that’s only 65% of the full cake’s diameter, then the smaller circle will still look like a larger piece, and thus get chosen by your sibling, leaving you with the larger outer ring.

With some ingenuity in tool making (or tool finding, if you have a curved grapefruit knife on hand), you could combine these techniques and cut a parabolic shape out of the cake, leaving a smaller section containing all top and side icing for your sibling.

(If you’re the kid who chooses rather than cuts, well, you just have to do your best. You can only choose from the options you’re presented with.)

In conclusion, the invisible hand of economic rationality doesn’t ensure fairness, and game theory doesn’t predict real life because it oversimplifies the options, which in the real world include leveraging trade-offs (e.g. craving icing, or wanting low price goods even if they’re not sustainably produced) and exploiting confusing images (e.g. sizes that look larger than they are, or conflating brand with value).


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