An Algorithm For Super Envy-Free Cake Division

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1 Journal of Mathematical Analysis and Applications 239, (1999) Article ID jrnaa , available online at on I DE hl3 An Algorithm For Super Envy-Free Cake Division William A. Webb Department of Pure and Applied Mathematics, Washington State Unir ersily, Pullman, Washington I.? Submitted by RichardA. Duke Received December 11, 1997 The standard mathematical setting for fair division problems begins with a cake X which is a compact subset of some Euclidean space and n players P,,...,el each with an additive nonatomic probability measure p,,.., p,, on a a-algebra of measureable subsets of X, and asks for a partition {X,,..., X,} of X such that P, is satisfied to receive X, under some definition of fairness. The original problem introduced by Steinhaus in 1946 asked for a simple fair assignment where pi(xj) 2 l/n whenever 1 I i I n [23]. Other common criteria for fairness include strongly fair assignment where pj(xi) > l/n whenever 1 I i I n, envy-free where pi(x,) 2 pi(xj) whenever 1 I i, j I n and strongly envy-free where p,(x,) > p,(xj) whenever 1 I i, j I n and i # j. Recently, Barbanel in- troduced the condition of super envy-free where pi(x,) > l/n and p,(xj) < l/n whenever 1 5 i, j 5 n and i f j 131. Barbanel proved that a super envy-free partition exists if and only if the measures are linearly independent, that is c1 p c, p,, = 0 only if c1 =... = c, = 0. He raised the question of whether an algorithm can be found which produces such a super envy-free partition. We will answer this question in the affirmative. Algorithms for fair division are of two main types; moving knife and finite, with the latter usually being more difficult since the rules defining a cut are more restrictive. We will show here that there exists a finite algorithm to produce a super envy-free partition. The proof uses a theorem on near-exact division from [22]. The algorithm is finite but does not have a bound on the number of steps. There are a variety of algorithms, both moving knife and finite, which produce simple fair, strongly fair, envyfree, and strongly envy-free partitions. The algorithms for simple fair and envy-free generally work for any X/99 $30.00 Copyrigtit by Academic Piess All rigtits of reproduction in any form reserved.

2 176 WILLIAM A. WEBB set of measures. However, consider what happens for strongly fair. If all of the measures happen to be identical then strongly fair is impossible. Even if the measures are different, under the rules we are using there is no guarantee that a piece on which players disagree will ever be cut, although such pieces exist. Thus for example, Woodall s algorithm for strongly fair division begins with a given piece A on which at least two players disagree This piece A serves as a witness to the fact that the measures are not all identical. What is required for a witness for linear independence of the measures? This is a much stronger condition and at least n pieces are needed for such a witness. Hence, we suppose we are given n pieces A,,..., A, with p,(a,) = all such that the matrix M = [a,,] is nonsingular. Such a collection of pieces is easily seen to be a witness for the linear independence of the measures. We begin by showing that we may assume that the A, are a partition of X. First, if two of the sets, say A, and A, are not disjoint let A, n A, = B and b = the column vector [ p,(b)]. Then at least one of the collections A, - B, A,,..., A, or A,, A, - B,..., A,, or A, - B, A, - B,..., A, is also a witness. Let m, be the ith column of M. Since det M # 0, it cannot be the case that all three of the determinants det[m, - b, m,,..., m,,] = det M - det[b, m,,..., m,,] det[m,, m, - b,..., m,,] = det M - det[m,, b,..., m,,] det[m, - b,m,, -b,..., m,,] = det M - det[b,m,,..., m,,] - det[m,, b,..., m,] are equal to zero. Using this observation repeatedly, we may assume that all of the Ai are disjoint. If A,,..., A,, is not a partition of X, then let 2 be the solution of MZ = [l,..., 1IT = 1. At least one element of 2, say z,, is not zero. Hence, det[l - m, -..* -m,,,m,,...,ma] = det[l,m,,...,ma]# 0, and X -A, -A, An, A,,..., A,, is also a witness. There is another type of fair division which will be a key part of the algorithm for super envy-free division. DEFINITION. Given an E > 0, the partition (X,,..., X,) of X is &-near exact in the ratio r1 : r, : 1.. : r, provided whenever 1 < i, k 4 n.

3 AN ALGORITHM FOR SUPER ENVY-FREE CAKE DIVISION 177 The following theorem, proved in [ZZI, guarantees an algorithm of the desired kind [221. THEOREM. Gicen any E > 0 there is a finite algorithm which produces &-near exact division in the ratio r, : rg :... : r,. Beginning with the witness A,,..., A, which is a partition of X, the matrix M = [aij] is a stochastic matrix so its inverse M-' also has row sums of one, but may have negative entries. Let t = minimum element of M-'. Since MM-' =I it is clear that t I 0. Choose 6 such that 0 < 6 < (n- l)/n(l - tn). Define the matrix N = [aij] by nii = (l/n) + 6 and nij = (l/n) - (S/(n- 1)) for i # j. This 6 is sufficiently small that all elements of MplN = R = [rij] are positive and also all row sums are one. Super envy-free division is now accomplished by partitioning each piece Aj into (X,,,..., X,,,) which is &-near exact in the ratio rj, :...: rj,, with Pi receiving Xjj for 1 I j I n, where F = 6/n2. Then pi(x. ) = ajjrjk + qjk where lqjkl < F. Then Pi evaluates the!k entire share given to Pk as worth + &ilk +... fainrnk + &irlk = nik + &ilk +... feirlk which is strictly greater than l/n when k = i, and strictly less than l/n when k # i. More generally, we can begin with any partition A,,..., A,, of X however generated, such that the corresponding matrix M is nonsingular, and any matrix N sufficiently close to the matrix all of whose elements are l/n. Essentially the same algorithm produces a partition X,,..., X, of X for which pj(xj) is arbitrarily close to the entry njj of N. The algorithm described above begins with a witness consisting of a partition of a subset of X for which the corresponding matrix M is nonsingular. The referee has suggested a way in which such a witness can be found. For simplicity take the cake X to be a compact subset of two-dimensional Euclidean space, suppose that the measures are all continuous and assume that a witness consisting of disjoint open sets A,, A,,..., A,, exists. (These are not the weakest assumptions one could make, but a thorough discussion of this is beyond the scope of this paper.) Enclose X in a square S, normalized to have a side of one. At step j subdivide S into squares having sides (i)j. Consider all possible partitions of all subcollections of these squares into n subsets and calculate the corresponding matrix for each such partition. Whenever a nonsingular matrix occurs we have found a suitable witness. Since det M # 0 for the sets A,,..., A,,, by the continuity of the measures we will eventually approximate the A; as unions of squares sufficiently close to guarantee

4 178 WILLIAM A. WEBB such a nonsingular matrix in our search (although there is no a priori bound on the number of steps required). In practice we may well be lucky enough to find a suitable partition long before we closely approximate the Ai. REFERENCES 1. A. K. Austin, Sharing a cake, Mathematical Gazette 66 (1982), J. Barbanel, Game-theoretic algorithms for fair and strongly fair cake division with entitlements, Colloq. Math. 69 (1995), J. Barbanel, Super envy-free cake division and independence of measures, J. Math. Anal. Appl. 197 (1996) J. Barbanel, 011 the possibilities for partitioning a cake, Proc. Am. Math. SOC. 124 (1996), J. Barbanel and A. Taylor, Preference relations and measures in the context of fair division, Proc. Am. Math. Soc. 123 (19951, S. J. Brarris and A. D. Taylor, An envy-free cake division protocol, Am. Math. Monthly 102 (1995) S. J. Brarris arid A. D. Taylor, A note on erivy-free cake division, J. Comhinatorial Theory Ser. A 70 (1995), S. J. Brams and A. D. Taylor, Fair Division: From Cake Cutting to Dispute Resolution, Cambridge Uriiv. Press, Cambridge, UK, S. J. Brams, arid A. D. Taylor, and W. S. Zwicker, Old arid new moving-knife schemes, Math. lntelligencer 17 (1995) S. J. Brams, A. D. Taylor, and W. S. Zwicker, A moving-knife solution to the four-person erivy-free cake division problem, Proc. Am. Math. Soc. 125 (1997), L. E. Dubins and E. H. Spanier, How to cut a cake fairly, Amer. Math. Monthly 68 (1961), S. Even arid A. Paz, A note on cake cutting, Discrete Appl. Math. 7 (1984), A. M. Fink, A note on the fair division problem, Math. Mag. (1964), D. Gale, Mathematical entertainments, Math lntelligencer 15 (1993) M. Gardrier, aha! Insight, Sci. Am., pp , W.H. Freeman arid Company, New York, B. Knaster, Sur le probleme du partage pragmatique de H. Steinhaus, Ann. Polon. Math. 19 (19461, D. Olivastro, Preferred shared, The Sciences, Mar/April 1992, K. Rebman, How to get (at least) a fair share of the cake, in Mathematical Plums (Ross Horisberged, Ed.), pp , Math. Assoc. of America, Washington, DC, J. M. Robertson arid W. A. Webb, Minimal number of cuts for fair division, Ars Comhin. 31 (19911, J. M. Robertson and W. A. Webb, Approximating fair division with a limited number of cuts, J. Comhin. Theory Ser. A 72 (1995), J. M. Robertson and W. A. Webb, Extensions of cut and choose fair division, Elem. Math. 52 (19971, J. M. Robertson and W. A. Webb, Near exact and envy-free cake division, Ars Comhin. 45 (19971,

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