Solution
Cutting the Cake with Inside Information
Show the problem again
A cake is the interval [0, 1]. A cutter values a piece uniformly, so a piece of length L is worth L to the cutter. A chooser values a piece [a, b] at b² - a² (their value density is 2x, so the whole cake is worth 1 to them). The cutter makes a single cut at a point x of their choosing; the chooser then picks whichever of the two pieces, [0, x] or [x, 1], is worth more to them, and the cutter takes the other piece. The cutter knows the chooser's valuation exactly, and the chooser picks greedily; if the chooser is exactly indifferent, they take the piece the cutter values less. What is the largest value, by the cutter's own measure, that the cutter can guarantee? Give an exact value (a decimal accurate to three places is also accepted).
Worked solution
The chooser values [0, x] at x² and [x, 1] at 1 - x². They take the left piece when x² > 1/2 (that is, x > 1/√2), leaving the cutter 1 - x < 1 - 1/√2 ≈ 0.29. They take the right piece when x < 1/√2, leaving the cutter [0, x] worth x < 1/√2 ≈ 0.707. So the cutter wants x as large as possible without tipping the chooser to the left piece: x = 1/√2, where the chooser is exactly indifferent. By the tie rule the chooser takes the piece the cutter values less, namely [1/√2, 1] (worth 1 - 1/√2 ≈ 0.29 to the cutter), leaving the cutter [0, 1/√2] with value **√2/2 ≈ 0.7071**. Compare the 1/2 the cutter can only guarantee when ignorant of the chooser's tastes. A numerical search over cut points agrees.
Source: Cut-and-choose is the oldest fair-division protocol; the exploitation of the chooser's known preferences is a standard exercise in the cake-cutting literature (see Brams and Taylor, Fair Division, 1996). Numbers chosen for AxiomIQ. Statement written for AxiomIQ.