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In general, a lexmaxmin optimization problem may have more than one optimal solution. If and are two optimal solutions, then their ordered value vector must be the same, that is, [] = [] for all [], [5]: Thm.2 that is, the smallest value is the same, the second-smallest value is the same, and so on.
The leximin allocation yields the utility profile (3, 2ε, 2ε): the first item must go to agent 1 - otherwise the smallest utility is 0. Then, the second and third items must go to agent 2 - otherwise the next-smallest utility is ε or less; so agent 3 gets only the last item. The maximin-share values of the agents are 0, ε, 1.
The first element of the sequence (0) has no previous value. The nearest (only) smaller value previous to 8 and to 4 is 0. All three values previous to 12 are smaller, but the nearest one is 4. Continuing in the same way, the nearest previous smaller values for this sequence (indicating the nonexistence of a previous smaller value by a dash) are
For example, a state with utility profile (0,100,100) has the same minimum value as a state with utility profile (0,0,0). In this case, the egalitarian rule often uses the leximin order, that is: subject to maximizing the smallest utility, it aims to maximize the next-smallest utility; subject to that, maximize the next-smallest utility, and so on.
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As a baseline algorithm, selection of the th smallest value in a collection of values can be performed by the following two steps: . Sort the collection; If the output of the sorting algorithm is an array, retrieve its th element; otherwise, scan the sorted sequence to find the th element.
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