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Optimal sharing

An equation for dividing up something (like pizza) fairly between two people

Ever made a joint decision with your friend, partner, or family member that you debated hours and hours over? If that decision involved splitting up something between the two of you (e.g., pizza), then it turns out there’s a way to accomplish the sharing that’s mathematically proven to be fair to both parties.

In Section 6.2 of The Calculus of Happiness I discuss the solution. I also talk about its curious origin: John Nash’s (the mathematician the movie A Beautiful Mind centered on). Nash was a pioneer of a field now known as game theory, and one of his research articles focused on a collaborative “game” (roughly defined as an interaction between two or more people involving decisions that affected the people involved). Using some simplifying assumptions (see the Limitations section below), I apply the math in Nash’s research article to the two-person “game” where there’s a decision that needs to be made about splitting up a divisible thing (e.g., pizza). The resulting share each person gets can be quantified into two equations; see equations (6.7a) and (6.7b) in The Calculus of Happiness.

The calculator below uses those equations to help you get a sense of how to divide up the “thing.” It requires the following inputs:

  • The total amount of the “thing” to be divided up (for example, “1” if we’re talking about a pizza, or $100 if we’re talking about a holiday gift from a family member). This is called in the calculator below.
  • How happy you’d be if you received all of the thing, on a scale from 0 to 10, 0 being “unhappy” and 10 being “happy.” This is called in the calculator below.
  • How happy the other person would be (on the same scale) if they received all of the thing. This is called  in the calculator below.
  • Your happiness level (on the same scale) if the two of you couldn’t agree on how to divide up the thing. This is called Yd in the calculator below.
  • The other person’s happiness level (on the same scale) if the two of you couldn’t agree on how to divide up the thing. This is called Pin the calculator below.

There are two natural constraints:

Y_d \leq M, \quad \quad P_d \leq N,

which express the fact that the happiness levels in the event of a disagreement cannot be larger than the maximum happiness levels in the event an agreement is reached. There is also one technical constraint:

\frac{Y_d}{M}+\frac{P_d}{N} \leq 1,

which is required for a solution (the optimal splitting) to exist. This constraint effectively says that a solution exists only when Yand Paren’t too large (relative to M and N).

Limitations

The formulas the calculator above is based on quantifying happiness using linear functions. This isn’t always an accurate description of happiness. These formulas, and the calculator above, also assume that the inputs (the green cells in the calculator) do not change over the course of the debate as to how much of the thing each person should have. Nonetheless, as I discuss in the book, the formulas (6.7a) and (6.7b) the calculator above is based on yield many useful insights into how to make more productive and fair certain joint decisions between two people.