• Market valuation. One possibility would be to sell the cake (at fair market value) and divide the cash proceeds. Then Jane and Dick could each buy a smaller cake with lots of frosting and Dick a fruitcake without frosting, or even something else. But market transactions are not free. Even selling through E-bay has its risks and costs.
• Authority valuation. The two could submit their distribution problem to an authority figure - perhaps Mom or even a judge. But even an impartial authority may not know about their individual preferences (frosting versus quantity, smooth versus crunchy) and make mistakes.
• Fair division. The two could use a method to divide the cake so that each considers the result to be fair. One age-old method is for Jane to cut and Dick to choose -- or the other way around and, of course, under Mom's watchful eye. Their method becomes a "market" for their expression of their individual preferences.

Game theory has developed procedures (algorithms) for the division among n-players of a divisible object (a cake) or a set of objects (property in divorce) so that each player considers the division "fair" according to his or her value system. For example, how might neighbors solve the problem that one consider the other to be a nuisance. See Meurer, "Fair Division" 47 Buff. L. Rev. 937 (1999).

Here are some algorithms for solving Mom's cake problem:

Cut and choose method. This method works for two players, as in the case of Jane and Dick. (Although Mom often takes credit for the method, it was first described in Hesiod's Theogeny where Prometheus placed meat in two piles and Zeus got to choose.)

Moving knife procedure. Suppose there are three kids who are to split the cake. Can this be done fairly? One strategy is for Mom to place a knife over one corner of the cake and begin to move it slowly across the cake. When any of the kids says "stop," that kid (K1 let's say) gets the piece. Presumably K1 thinks she got at least 1/3, and K2 and K3 (who did not speak up) believe the remainder is at least 2/3. Mom keeps on moving the knife, until K2 says "stop." K2 is happy, because he thinks he got at least 1/3 -- and the same for K3! All thought their piece was "fair" at the time they chose. But there's a problem. K1 might be envious. For example, K1 might think K3 got more than 1/3 when K2 said "stop" too early. Although each was happy with her or his decision when made, there might not be a perception of equal sharing -- in our hierarchical world, a condition of complete fairness. [Saul Levmore and Elizabeth Cook - continuous envy-free method / Theodore Hill - n-countries divided contiguous disputed territory, all perceive get 1/n th].

Trimming method. Suppose there are three players who want to divide indivisible goods, such as heirs of an estate entitled to an equal share of cash and goods. Is there a procedure that will recognize their individual assessments of the goods? A procedure known since the 1940s, called Knaster's Inheritance Procedure, purports to fairly divide a set of indivisible goods. Here's how it works:

• Each player values the various items according to how much she would pay for each one -- the players must be honest because they'll have to "purchase" items with a high valuation and will reduce their "cash settlement" with a low valuation.
• Using these valuations, each player's own "fair share" of the estate is computed -- that is, how much the player believes she should receive in goods and cash.
• The players then must "purchase" the goods for which they had the highest bids -- this cash can either come from the cash portion of the estate or their own pockets.
• The players then are subject to a "cash adjustment" -- a positive amount if their "purchases" are less than their "fair share" and a negative amount if their "purchases" exceed their "fair share."
• Each player then receives a "cash bonus" equal to the total cash adjustments divided equally among all the players -- this cash bonus, if the players disagree on valuation, is stated as a positive number.
• In a final settlement the players receive their goods after paying (or being paid) their cash adjustment plus cash bonus.

For additional valuation methods that use mathematical models and game theory, see "Merger Simulations."