JOURNAL OF COMBINATORIAL THEORY, Series A 42, 300-301 (1986)

Note

A Note on the Cake-Division Problem

D. R. WOODALL

Department sf Mathematics, University of Nottingham, NG7 ZRD, England

Communicated by the Managing Editors

Received May 17, 1985

Let C, hereafter referred to as the cake, be a compact convex set in some Euclidean space. Suppose that each of n people defines a non-atomic probability measure on the Bore1 sets of C. The problem of dividing C among the n people in such a way that each receives at least l/n of it (in his own measure) is solved and well known [l-8]. The authors of [l, 43 prove the interesting and possibly surprising extension that, if the n measures are not all identical, then the division can be effected in such a way that every person receives more than l/n of the cake (in his own measure). Their proof uses Lyapunov’s theorem and so is non-constructive. The purpose of this note is to give an algorithmic proof of this result, assuming that a piece of cake C, can be identified to which some two of the people, say A and B, assign different measures c1> fl. The algorithm is based on one of Fink [2], which I incorrectly attributed in [S] to Saaty [S].

We start by dividing the cake between A and B in such a way that each thinks he has more than half the cake. To do this, choose a rational num- ber p/q with CL >p/q > 8. Denoting A’s and B’s measures by pLa and pLg respectively, let A divide CO into p PA-equal pieces X1,..., X, and B divide C\C, into q-p pL,-equal pieces Yr,..., YqPP, so that pA(Xi)> l/q and P~( Yi) > l/q for each i and j. (By the definition of a probability measure, pA(C) =pLB(C) = 1.) Since C,“=, pB(Xi) = fi<p/q, there is an i such that pue(Xi) < l/q. Similarly, there is a j such that ,u~( Y,) < l/q. Thus, with this choice of i and j, pA(Xi) > pA( Y,) and pLg( Y,) > pa(Xi). Now let A and B divide C\(X,u Y,) between them in the standard way: A divides it into two ,u,-equal portions, A’ and Y, and B chooses the pe-larger, say Y. Then ~A(XuXi)>~a(Yu Y,) and pLB(Yu Yj)>pg(XuXi), so that pA(Xu Xi) > + and pLg( Yu Y,) > f.

Now let A, = A and A2 = B, and suppose that at some later stage we have divided the cake among k people Al ,..., Ak (2 6 k < n - 1) in such a

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ON THE CAKE-DIVISION PROBLEM 301

way that each thinks he has more than l/k of the cake. For each i (i= l,..., k), let Ai and Ak+l divide Ais current portion, say Zi, between them as follows. Denoting Als measure by pi, so that pi(Zi) > l/k, choose a positive integer q such that (qk - l)[p,(Z,) - l/k] > l/k(k + l), so that (qk - 1) pi(Zi) > q - l/(k + 1) and

Let Ai divide Zi into q(k + 1) - 1 pi-equal pieces, and let Ak+ I choose the q pLk+ ,-largest of these. The portion remaining to Ai then has measure (pi) equal to the LHS of (1). And the portion taken by Ak+ 1 has measure bk+ l) at least

A k+l receives such a portion for each i, and Uf= 1 Zi = C, so Ak + , receives more than l/(k + 1) of the whole cake.

Continuing this process until k + 1 = n, the required division is com- pleted.

In conclusion, note that there is still no known algorithm for the problem, introduced by Gamow and Stern [3] and considered in [6,8], of dividing the cake among n people in such a way that each person thinks he has at least as much cake as anyone else. J. L. Selfridge has discovered an algorithm (reproduced in [S] ) in the case n = 3, but it has not been extended to n > 3.

REFERENCES

1. L. E. DUBINS AND E. H. SPANIER, How to cut a cake fairly, Amer. Math. Monthly 68 (1961), l-17.

2. A. M. FINK, A note on the fair division problem, Math. Mug. 37 (1964), 341-342. 3. G. GAMOW AND M. STERN, “Puzzle-Math”, Viking, New York, 1958. 4. K. REBMAN, How to get (at least) a fair share of the cake, in “Mathematical Plums”

(R. Honsberger, Ed.), pp. 22-37, Math. Assoc. Amer., Washington, D. C., 1979. 5. T. L. SAATY. “Optimization in Integers and Related Extremal Problems,” McGraw-Hill,

New York, 1970. 6. W. STROMQUIST, How to cut a cake fairly, Amer. Math. Monthly 87 (1980), 64G644, 88

(1981), 613-614. 7. W. STROMQLJIST AND D. R. WOODALL, Sets on which several measures agree, J. Math. Anal.

A@., 108 (1985), 241-248. 8. D. R. WOODALL, Dividing a cake fairly, J. Mafh. Appl. 78 (1980). 233-247.

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