Approximating fair division with a limited number of cuts

A large class A of finite algorithms for fairly dividing a cake using k of fewer cuts is described. Assume an algorithm assigns piece Xi to player Pi using associated probability measure μi on measurable subsets of the cake X. If M(n, k) = maxA mini(μi(Xi)) and N(n, k) = maxA(number of i such that μ1(X1⩾ 1n) then for n ⩾ 2, M(n, n − 1) = 1(2n − 2), for n ⩾ 3, M(n, n) ⩾ 1(2n − 3), and for n ⩾ 4, M(n, n + 1) ⩾ 1(2n − 4). Also N(2n − 2, n − 1) = n.