Ordinal Pareto efficiency


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 . 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 . Then the allocation is not Pareto-efficient, since both Alice and George would be better-off by exchanging their bundles.
  • In contrast, it is possible that Alice prefers to and George prefers to . 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 : 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 = Pareto-dominates another allocation Y =, if every agent i weakly prefers the bundle Xi to the bundle Yi, and at least one agent j strictly prefers Xj to Yj. 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. 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 < < <

Necessary Pareto-efficiency

Brams, Edelman and Fishburn call an allocation Pareto-ensuring if it is Pareto-efficient for all bundle rankings that are consistent with the agents' item rankings. For example:
  • If agents' valuations are assumed to be positive, then every allocation giving all items to a single agent is Pareto-ensuring.
  • If Alice's ranking is x>y and George's ranking is y>x, then the allocation is Pareto-ensuring.
  • If Alice's ranking is x>y>z and George's ranking is x>z>y and the allocations must be discrete, then the allocation is Pareto-ensuring.
  • With the above rankings, the allocation is not Pareto-ensuring. As explained in the introduction, it is not Pareto-efficient e.g. when Alice's valuations for x,y,z are 8,7,6 and George's valuations are 7,1,2. Note that both these valuations are consistent with the agents' rankings.
Bouveret, Endriss and Lang. use an equivalent definition. They say that an allocation X possibly Pareto-dominates an allocation Y if there exists some bundle rankings consistent with the agents' item rankings, for which X Pareto-dominates Y. An allocation is called Necessarily-Pareto-efficient if no other allocation possibly-Pareto-dominates it.
The two definitions are logically equivalent:
  • "X is Pareto-ensuring" is equivalent to "For every consistent bundle ranking, for every other allocation Y, Y does not Pareto-dominate X".
  • "X is NecPE" is equivalent to "For every other allocation Y, for every consistent bundle ranking, Y does not Pareto-dominate X". Exchanging the order of "for all" quantifiers does not change the logical meaning.
The NecPE condition remains the same whether we allow all additive bundle rankings, or we allow only rankings that are based on additive valuations with diminishing differences.

Existence

NecPE is a very strong requirement, which often cannot be satisfied. For example, suppose two agents have the same item ranking. One of them, say Alice, necessarily receives the lowest-ranked item. There are consistent additive bundle-rankings in which Alices values this item at 0 while George values it at 1. Hence, giving it to Alice is not Pareto-efficient.
If we require that all items have a strictly positive value, then giving all items to a single agent is trivially NecPE, but it very unfair. If fractional allocations are allowed, then there may be no NecPE allocation which gives both agents a positive value. For example, suppose Alice and George both have the ranking x>y. If both get a positive value, then either Alice gets some x and George gets some y, or vice-versa. In the former case, it is possible that Alice's valuations are e.g. 4,2 and George's valuations are 8,1, so Alice can exchange a small amount r of x for a small amount 3r of y. Alice gains 6r-4r and George gains 8r-3r, so both gains are positive. In the latter case, an analogous argument holds.

Possible Pareto-efficiency

Brams, Edelman and Fishburn call an allocation Pareto-possible if it is Pareto-efficient for some bundle rankings that are consistent with the agents' item rankings. Obviously, every Pareto-ensuring allocation is Pareto-possible. In addition:
  • If Alice's ranking is x>y>z and George's ranking is x>z>y, then the allocation is Pareto-possible. As explained in the introduction, it is Pareto-efficient e.g. when Alice's valuations for x,y,z are 12,4,2 and George's valuations are 6,3,4. Note that both these valuations are consistent with the agents' rankings.
  • If Alice's ranking is x>y and George's ranking is y>x, then the allocation is not Pareto-possible, since it is always Pareto-dominated by the allocation .
Bouveret, Endriss and Lang. use a different definition. They say that an allocation X necessarily Pareto-dominates an allocation Y if for all bundle rankings consistent with the agents' item rankings, X Pareto-dominates Y. An allocation is called Possibly-Pareto-efficient if no other allocation necessarily-Pareto-dominates it.
The two definitions are not logically equivalent:
  • "X is Pareto-possible" is equivalent to "There exist a consistent bundle ranking for which, for every other allocation Y, Y does not dominate X". It must be the same bundle ranking for all other allocations Y.
  • "X is PosPE" is equivalent to "For every other allocation Y, there exists a consistent bundle ranking, for which Y does not dominate X". There can be a different bundle ranking for every other allocation Y.
If X is Pareto-possible then it is PosPE, but the other implication is not true.
The Pareto-possible condition remains the same whether we allow all additive bundle rankings, or we allow only rankings that are based on additive valuations with diminishing differences.

Stochastic-dominance Pareto-efficiency

Bogomolnaia and Moulin' present an efficiency notion for the setting of fair random assignment. It is based on the notion of stochastic dominance.
For each agent i, A bundle Xi weakly-stochastically dominates ' a bundle Yi if for every item z, the total fraction of items better than z in Xi is at least as large as in Yi. The sd relation has several equivalent definitions; see responsive set extension. In particular, Xi sd Yi if-and-only-if, for every bundle ranking consistent with the item ranking, Xi is at least as good as Yi. A bundle Xi strictly-stochastically dominates ' a bundle Yi if Xi wsd Yi and Xi ≠ Yi. Equivalently, for at least one item z, the "at least as large as in Yi" becomes "strictly larger than in Yi". In the ssd relation is written as "Xi ''>> Yi".
An allocation X = stochastically dominates another allocation Y =, if for every agent i'': Xi wsd Yi, and Y≠X. In the stochastic domination relation between allocations is also written as "X ''>> Y". This is equivalent to necessary Pareto-domination.
An allocation is called sd-efficient ' if there no allocation that stochastically dominates it. This is similar to PosPE, but emphasizes that the bundle rankings must be based on additive utility functions, and the allocations may be fractional''.

Equivalences

As noted above, Pareto-possible implies PosPE, but the other direction is not logically true. McLennan proves that they are equivalent in the fair random assignment problem. Particularly, he proves that the following are equivalent:
  • X is sd-efficient ;
  • there exists additive bundle-rankings consistent with the agents' item-rankings for which X is fractionally-Pareto-efficient ;
  • there exists additive bundle-rankings consistent with the agents' item-rankings for which X maximizes the sum of agents' utilities.
The implications → → are easy; the challenging part is to prove that →. McLennan proves it using the polyhedral separating hyperplane theorem.
Bogomolnaia and Moulin prove another useful characterization of sd-efficiency, for the same fair random assignment setting but with strict item rankings. Define the exchange graph of a given fractional allocation as a directed graph in which the nodes are the items, and there is an arc x→y iff there exists an agent i that prefers x and receives a positive fraction of y. Define an allocation as acyclic if its exchange graph has no directed cycles. Then, an allocation sd-efficient iff it is acyclic.
Fishburn proved the following equivalence on dominance relations of discrete bundles, with responsive bundle rankings:
  • If Xi ''>> Yi , then for every responsive bundle-ranking consistent with the item-ranking, Xi >Yi.
  • If not Xi >> Yi, then there exists at least one responsive bundle-ranking consistent with the item-ranking, for which Xi i.
Therefore, the following holds for dominance relations of discrete allocations: X'' >> Y iff X necessarily Pareto-dominates Y.