[Santa Claus Trouble 1 Game Free Download
Everardo Laboy
Egalitarian item allocation, also called max-min item allocation is a fair item allocation problem, in which the fairness criterion follows the egalitarian rule. The goal is to maximize the minimum value of an agent. That is, among all possible allocations, the goal is to find an allocation in which the smallest value of an agent is as large as possible. In case there are two or more allocations with the same smallest value, then the goal is to select, from among these allocations, the one in which the second-smallest value is as large as possible, and so on (by the leximin order). Therefore, an egalitarian item allocation is sometimes called a leximin item allocation.
The special case in which the value of each item j to each agent is either 0 or some constant vj is called the santa claus problem: santa claus has a fixed set of gifts, and wants to allocate them among children such that the least-happy child is as happy as possible.
The two rules are equivalent when the agents' valuations are already normalized, that is, all agents assign the same value to the set of all items. However, they may differ with non-normalized valuations. For example, if there are four items, Alice values them at 1,1,1,1 and George values them at 3,3,3,3, then the absolute-leximin rule would give three items to Alice and one item to George, since the utility profile in this case is (3,3), which is optimal. In contrast, the relative-leximin rule would give two items to each agent, since the normalized utility profile in this case, when the total value of both agents is normalized to 1, is (0.5,0.5), which is optimal.
The standard egalitarian rule requires that each agent assigns a numeric value to each object. Often, the agents only have ordinal utilities. There are two generalizations of the egalitarian rule to ordinal settings.
Whenever a proportional allocation exists, the relative-leximin allocation is proportional. This is because, in a proportional allocation, the smallest relative value of an agent is at least 1/n, so the same must hold when we maximize the smallest relative value. However, the absolute-leximin allocation might not be proportional, as shown in the example above.
3. When all agents have valuations that are matroid rank functions (i.e., submodular with binary marginals), the set of absolute-leximin allocations is equivalent to the set of max-product allocations; all such allocations are max-sum and EF1.[21]
4. In the context of indivisible allocation of chores (items with negative utilities), with 3 or 4 agents with additive valuations, any leximin-optimal allocation is PROP1 and PO; with n agents with general identical valuations, any leximin-optimal allocation is EFX.[23]
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