![]() In theory, we would need to do this an infinite number of times, because there are an infinite number of points on the edge of a pie. Once we’ve gone all the way around the pie, you’d tell me which positions of the knife gave you the piece with the most value, and I’d cut there. You would again note your perceived value of each piece. I would again adjust the second knife to suggest two pieces that I think are of equal value. Next, I would rotate the first knife a bit, say, to one o’clock. Then, you would note your perceived value of each piece. Then I would put the second knife in a position going from the center to another point on the edge, so as to produce two pieces I perceive as equal in value. First, I would position the end of one knife at the center of the pie and point the knife toward 12 o’clock. If you and I were using “I cut, you choose” to divide a pie, we could use a method that is like using knives as clock hands. But if there is a special piece of chocolate beside the 2 on the clock face, I might point the hands at, say, the 1 and the 3 if I thought a small piece of pie with the chocolate would have the same value as a far larger piece with no chocolate. If the pie had a clock face and the clock’s two hands were knives, pointing the hands to 12 and 6, for example, or to 10 and 4 would produce two equal-size pieces. However, since you get to choose between the two pieces, you may perceive your piece as greater in value.Īpplying the method to pies is more complicated, because there are infinitely many ways to cut a pie into two pieces that I perceive as equal in value. But it probably won’t be equitable: From my perspective, I will get half the value of the cake. Neither of us will envy the other, and there will be no way of increasing one person’s share without decreasing the other’s share. The method is like the classical “I cut, you choose” approach to cutting a cake: I cut the cake into two pieces that I perceive to be equal in value if not in size. They know how to cut a pie into two pieces in a way that is envy-free and efficient, but not necessarily equitable. Since we each think we got 60 percent, it’s “equitable.”Īlthough researchers know that equitable division exists, they don’t know how to produce it. Splitting a decorated pie between two people is not so tough, but creating fair shares for more than two people may be impossible.įor two people, the researchers found that it is possible to cut slices that are not only envy-free and efficient, but also “equitable.” For example, you might get a larger piece than I get, but I may think that I got 60 percent of the value of the pie because I got the side with all the coconut, while you think you got 60 percent of the value of the pie because you got the side with the cherry on it. ![]() But for pie, the situation is more difficult, the researchers found. They also wanted to make sure the division is “efficient,” so that no other way of dividing the pie would be better for one person without becoming worse for others.įor cake, there are procedures for finding, or at least approximating, envy-free and efficient cuts for any number of people. That complicates the challenges of cutting fair sizes of cake and makes pie cutting even more complex, if not impossible.īarbanel and Brams wanted to find a way of cutting a pie that is “envy-free,” meaning that each person is at least as happy with his own piece as he would be with anyone else’s. That simple difference leads to a world of mathematical trouble.įurthermore, mathematical analysis of cutting decorated cakes and pies assumes that the portions are not necessarily of equal size. Following that rule, there is only one way to split a rectangular cake into two equal-size pieces.įor a pie, however, there are an infinite number of ways, because there an infinite number of lines going through the center of a circle. Mathematicians simplify cake cutting by assuming that all slices are perpendicular to one particular side of a rectangular cake, with no crosswise cuts. We slice pies into wedges, but we cut sheet cakes into rectangles. ![]()
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