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0. Introduction. The cake division problem is the problem of dividing a cake among a certain number of individuals in such a way that each individual is “satisfied”, where we assume that each such individual has some means of comparing the relative worth of different pieces of cake, but where we do not necessarily assume that different individuals agree with each other in these evaluations. Of course, it is not yet clear what we mean by “satisfied”. To make this precise, let us assume that there are n individuals, and n associated finitely additive, non-atomic measures m1, . . . ,mn, each defined on some algebra of subsets of our cake C, such that for each such mi, mi(C) = 1. Various notions of what it means for each of the n individuals to be satisfied with a partition 〈P1, . . . , Pn〉 of C have been considered. See, for example, Brams and Taylor [2], or Barbanel [1]. We shall consider, and then generalize, two of these notions: The partition 〈P1, . . . , Pn〉 of C is fair with respect to the sequence of measures 〈m1, . . . ,mn〉 if and only if for each i = 1, . . . , n, mi(Pi) ≥ 1/n, and is strongly fair with respect to the sequence of measures 〈m1, . . . ,mn〉 if and only if for each i = 1, . . . , n, mi(Pi) > 1/n. Banach and Knaster (see Steinhaus [6] or [7]) presented an algorithm for obtaining a partition of a cake C among n individuals that is fair, and Woodall [8] presented an algorithm for obtaining a partition of a cake C among n individuals that is strongly fair, assuming that the measures are not identical and that we are given a witness of the disagreement of some two of the measures. We note that it is clearly impossible to obtain a strongly fair partition of C if all of the involved measures are identical. In this paper, we study a generalization of these notions. Let 〈α1, . . . , αn〉 be a sequence of positive numbers satisfying α1 + . . . + αn = 1. We shall call such a sequence an entitlement sequence. Then, we say that the partition 〈P1, . . . , Pn〉 of C is fair with respect to the sequence of measures 〈m1, . . . ,mn〉 and the entitlement sequence 〈α1, . . . , αn〉 if and only if for

Game-theoretic algorithms for fair and strongly fair cake division with entitlements