A note on the cake-division problem

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<ul><li><p>JOURNAL OF COMBINATORIAL THEORY, Series A 42, 300-301 (1986) </p><p>Note </p><p>A Note on the Cake-Division Problem </p><p>D. R. WOODALL </p><p>Department sf Mathematics, University of Nottingham, NG7 ZRD, England </p><p>Communicated by the Managing Editors </p><p>Received May 17, 1985 </p><p>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 Lyapunovs 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&gt; fl. The algorithm is based on one of Fink [2], which I incorrectly attributed in [S] to Saaty [S]. </p><p>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 &gt;p/q &gt; 8. Denoting As and Bs 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)&gt; l/q and P~( Yi) &gt; l/q for each i and j. (By the definition of a probability measure, pA(C) =pLB(C) = 1.) Since C,=, pB(Xi) = fi pA( Y,) and pLg( Y,) &gt; 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)&gt;~a(Yu Y,) and pLB(Yu Yj)&gt;pg(XuXi), so that pA(Xu Xi) &gt; + and pLg( Yu Y,) &gt; f. </p><p>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 &lt; n - 1) in such a </p><p>300 0097-3165186 $3.00 Copyright XJ 1986 by Academc Press, Inc. All rights of reproductmn ,n any lorm reserved. </p></li><li><p>ON THE CAKE-DIVISION PROBLEM 301 </p><p>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) &gt; l/k, choose a positive integer q such that (qk - l)[p,(Z,) - l/k] &gt; l/k(k + l), so that (qk - 1) pi(Zi) &gt; q - l/(k + 1) and </p><p>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 </p><p>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. </p><p>Continuing this process until k + 1 = n, the required division is com- pleted. </p><p>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 &gt; 3. </p><p>REFERENCES </p><p>1. L. E. DUBINS AND E. H. SPANIER, How to cut a cake fairly, Amer. Math. Monthly 68 (1961), l-17. </p><p>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 </p><p>(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, </p><p>New York, 1970. 6. W. STROMQUIST, How to cut a cake fairly, Amer. Math. Monthly 87 (1980), 64G644, 88 </p><p>(1981), 613-614. 7. W. STROMQLJIST AND D. R. WOODALL, Sets on which several measures agree, J. Math. Anal. </p><p>A@., 108 (1985), 241-248. 8. D. R. WOODALL, Dividing a cake fairly, J. Mafh. Appl. 78 (1980). 233-247. </p><p>582a/42/2-10 </p></li></ul>