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Ordinal Pareto efficiency
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<p>Ordinal Pareto efficiency refers to several adaptations of the concept of <a href="/facts/Pareto-efficiency/nMTdVPqy">Pareto-efficiency</a> to settings in which the agents only express <a href="/facts/Ordinal_utility/pFbpa2dg">ordinal utilities</a> over items, but not over bundles. That is, agents rank the items from best to worst, but they do not rank the subsets of items. In particular, they do not specify a numeric value for each item. This may cause an ambiguity regarding whether certain allocations are Pareto-efficient or not. As an example, consider an economy with three items and two agents, with the following rankings: </p> <ul><li>Alice: x > y > z.</li> <li>George: x > z > y.</li></ul> <p>Consider the allocation [Alice: x, George: y,z]. Whether or not this allocation is Pareto-efficient depends on the agents' numeric valuations. For example: </p> <ul><li>It is possible that Alice prefers {y,z} to {x} and George prefers {x} to {y,z} (for example: Alice's valuations for x,y,z are 8,7,6 and George's valuations are 7,1,2, so the utility profile is 8,3). Then the allocation is not Pareto-efficient, since both Alice and George would be better-off by exchanging their bundles (the utility profile would be 13,7).</li> <li>In contrast, it is possible that Alice prefers {x} to {y,z} and George prefers {y,z} to {x} (for example: Alice's valuations are 12,4,2 and George's valuations are 6,3,4). Then the allocation is Pareto-efficient: in any other allocation, if Alice still gets x, then George's utility is lower; if Alice does not get x, then Alice's utility is lower. Moreover, the allocation is Pareto-efficient even if the items are divisible (that is, it is <a href="/facts/Fractionally_Pareto_efficient/GEmD4okd">fractionally Pareto efficient</a>): if Alice yields any amount <i>r</i> of x to George, then George would have to give her at least 3<i>r</i> of y or 6<i>r</i> of z in order to keep her utility at the same level. But then George's utility would change by 6<i>r</i>-9<i>r</i> or 6<i>r</i>-24<i>r</i>, which is negative.</li></ul> <p>Since the Pareto-efficiency of an allocation depends on the rankings of bundles, it is a-priori not clear how to determine the efficiency of an allocation when only rankings of items are given. </p>