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Any strong Pareto-improvement is also a weak Pareto-improvement. The opposite is not true; for example, consider a resource allocation problem with two resources, which Alice values at 10, 0 and George values at 5, 5. Consider the allocation giving all resources to Alice, where the utility profile is (10,0).

Any strong Pareto-improvement is also a weak Pareto-improvement. The opposite is not true; for example, consider a resource allocation problem with two resources, which Alice values at 10, 0 and George values at 5, 5. Consider the allocation giving all resources to Alice, where the utility profile is (10,0).

任何强帕累托改进也是弱帕累托改进。反之则不然; 例如，考虑一个包含两个资源的资源分配问题，Alice 的分配值为10,0，George 的分配值为5,5。考虑将所有资源分配给 Alice 的分配，它的配置方案为(10,0)。

任何强帕累托改进也是弱帕累托改进。反之则不然; 例如，考虑一个包含两个资源的资源分配问题，Alice值为10,0，George值为5,5。考虑将所有资源分配给 Alice 的分配，它的配置方案为(10,0)。

Fractional Pareto optimality is a strengthening of Pareto-optimality in the context of fair item allocation. An allocation of indivisible items is fractionally Pareto-optimal (fPO) if it is not Pareto-dominated even by an allocation in which some items are split between agents. This is in contrast to standard Pareto-optimality, which only considers domination by feasible (discrete) allocations.

Fractional Pareto optimality is a strengthening of Pareto-optimality in the context of fair item allocation. An allocation of indivisible items is fractionally Pareto-optimal (fPO) if it is not Pareto-dominated even by an allocation in which some items are split between agents. This is in contrast to standard Pareto-optimality, which only considers domination by feasible (discrete) allocations.

分数帕累托最优是在项目公平分配的背景下加强帕累托最优的一种方法。如果一个不可分割的项目的分配不是帕累托优化(fPO) ，即使是在一个分配过程中，一些项目在代理人之间被分配。这与标准的帕累托最优相反，帕累托最优只考虑可行(离散)分配的控制。

部分帕累托最优是在物品公平分配的背景下对帕累托最优的一个加强。 即使是在一个分配过程中，一些物品在主体之间被分配，如果一个不可分割的物品的分配不是受帕累托支配的，那么它不是部分帕累托最优(fPO)。这与标准的帕累托最优相反，因为它只考虑可行(离散)分配的控制。

As an example, consider an item allocation problem with two items, which Alice values at 3, 2 and George values at 4, 1. Consider the allocation giving the first item to Alice and the second to George, where the utility profile is (3,1).

As an example, consider an item allocation problem with two items, which Alice values at 3, 2 and George values at 4, 1. Consider the allocation giving the first item to Alice and the second to George, where the utility profile is (3,1).

作为一个示例，考虑一个有两个项的项分配问题，Alice 值为3,2，George 值为4,1。考虑将第一个项目分配给 Alice，第二个项目分配给 George，其中实用程序配置文件为(3,1)。

作为一个示例，考虑一个有两个项的项分配问题，Alice 值为3,2，George 值为4,1。考虑将第一个项目分配给 Alice，第二个项目分配给 George，其中配置方案为(3,1)。