Ordinal Pareto efficiency explained

Ordinal Pareto efficiency refers to several adaptations of the concept of Pareto-efficiency to settings in which the agents only express ordinal utilities 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:

• Alice: x > y > z.
• George: x > z > y.

Consider the allocation [Alice: x, George: y,z]. Whether or not this allocation is Pareto-efficient depends on the agents' numeric valuations. For example:

• It is possible that Alice prefers to and George prefers to (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).
• In contrast, it is possible that Alice prefers to and George prefers to (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 fractionally Pareto efficient): if Alice yields any amount r of x to George, then George would have to give her at least 3r of y or 6r of z in order to keep her utility at the same level. But then George's utility would change by 6r-9r or 6r-24r, which is negative.

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.

Definitions

An allocation X = (X₁,...,X_n) Pareto-dominates another allocation Y = (Y₁,...,Y_n), if every agent i weakly prefers the bundle X_i to the bundle Y_i, and at least one agent j strictly prefers X_j to Y_j. An allocation X is Pareto-efficient if no other allocation Pareto-dominates it. Sometimes, a distinction is made between discrete-Pareto-efficiency, which means that an allocation is not dominated by a discrete allocation, and the stronger concept of Fractional Pareto efficiency, which means that an allocation is not dominated even by a fractional allocation.

The above definitions depend on the agents' ranking of bundles (sets of items). In our setting, agents report only their rankings of items. A bundle ranking is called consistent with an item ranking if it ranks the singleton bundles in the same order as the items they contain. For example, if Alice's ranking is, then any consistent bundle ranking must have < < <