Lexicographic dominance

Lexicographic dominance is a total order between random variables. It is a form of stochastic ordering. It is defined as follows.^[1]^: 8 Random variable A has lexicographic dominance over random variable B (denoted $A\succ _{ld}B$ ) if one of the following holds:

A has a higher probability than B of receiving the best outcome.
A and B have an equal probability of receiving the best outcome, but A has a higher probability of receiving the 2nd-best outcome.
A and B have an equal probability of receiving the best and 2nd-best outcomes, but A has a higher probability of receiving the 3rd-best outcome.

In other words: let k be the first index for which the probability of receiving the k-th best outcome is different for A and B. Then this probability should be higher for A.

Variants

Upward lexicographic dominance is defined as follows.^[2] Random variable A has upward lexicographic dominance over random variable B (denoted $A\succ _{ul}B$ ) if one of the following holds:

A has a lower probability than B of receiving the worst outcome.
A and B have an equal probability of receiving the worst outcome, but A has a lower probability of receiving the 2nd-worst outcome.
A and B have an equal probability of receiving the worst and 2nd-worst outcomes, but A has a lower probability of receiving the 3rd-worst outcomes.

To distinguish between the two notions, the standard lexicographic dominance notion is sometimes called downward lexicographic dominance and denoted $A\succ _{dl}B$ .

Relation to other dominance notions

First-order stochastic dominance implies both downward-lexicographic and upward-lexicographic dominance.^[3] The opposite is not true. For example, suppose there are four outcomes ranked z > y > x > w. Consider the two lotteries that assign to z, y, x, w the following probabilities:

A: .2, .4, .2, .2
B: .2, .3, .4, .1

Then the following holds:

$A\succ _{dl}B$ , since they assign the same probability to z but A assigns more probability to y.
$B\succ _{ul}A$ , since B assigns less probability to the worst outcome w.
$A\not \succ _{sd}B$ , since B assigns more probability to the three best outcomes {z,y,x}. If, for example, the value of z,y,x is very near 1, and the value of w is 0, then the expected value of B is near 0.9 while the expected value of A is near 0.8.
$B\not \succ _{sd}A$ , since A assigns more probability to the two best outcomes {z,y}. If, for example, the value of z,y is very near 1, and the value of x,w is 0, then the expected value of B is near 0.5 while the expected value of A is near 0.6.

Applications

Lexicographic dominance relations are used in social choice theory to define notions of strategyproofness,^[2] incentives for participation,^[4] ordinal efficiency^[3] and envy-freeness.^[5]

Hosseini and Larson^[6] analyse the properties of rules for fair random assignment based on lexicographic dominance.

References

^ Chakrabarty, Deeparnab; Swamy, Chaitanya (2014-01-12). "Welfare maximization and truthfulness in mechanism design with ordinal preferences". Proceedings of the 5th conference on Innovations in theoretical computer science. ITCS '14. New York, NY, USA: Association for Computing Machinery. pp. 105–120. doi:10.1145/2554797.2554810. ISBN 978-1-4503-2698-8. S2CID 2428592.
^ ^a ^b Cho, Wonki Jo (2016-01-01). "Incentive properties for ordinal mechanisms". Games and Economic Behavior. 95: 168–177. doi:10.1016/j.geb.2015.12.003. ISSN 0899-8256.
^ ^a ^b Cho, Wonki Jo; Doğan, Battal (2016-09-01). "Equivalence of efficiency notions for ordinal assignment problems". Economics Letters. 146: 8–12. doi:10.1016/j.econlet.2016.07.007. ISSN 0165-1765.
^ Aziz, Haris (2016-11-08). "Participation Incentives in Randomized Social Choice". arXiv:1602.02174 [cs.GT].
^ Cho, Wonki Jo (2018-06-01). "Probabilistic assignment: an extension approach". Social Choice and Welfare. 51 (1): 137–162. doi:10.1007/s00355-018-1110-z. ISSN 1432-217X. S2CID 19700606.
^ Hadi Hosseini, Kate Larson (2015-07-24). Strategyproof Quota Mechanisms for Multiple Assignment Problems. OCLC 1106222190.

[:02-1] Chakrabarty, Deeparnab; Swamy, Chaitanya (2014-01-12). "Welfare maximization and truthfulness in mechanism design with ordinal preferences". Proceedings of the 5th conference on Innovations in theoretical computer science. ITCS '14. New York, NY, USA: Association for Computing Machinery. pp. 105–120. doi:10.1145/2554797.2554810. ISBN 978-1-4503-2698-8. S2CID 2428592.

[:0-2] Cho, Wonki Jo (2016-01-01). "Incentive properties for ordinal mechanisms". Games and Economic Behavior. 95: 168–177. doi:10.1016/j.geb.2015.12.003. ISSN 0899-8256.

[:1-3] Cho, Wonki Jo; Doğan, Battal (2016-09-01). "Equivalence of efficiency notions for ordinal assignment problems". Economics Letters. 146: 8–12. doi:10.1016/j.econlet.2016.07.007. ISSN 0165-1765.

[4] Aziz, Haris (2016-11-08). "Participation Incentives in Randomized Social Choice". arXiv:1602.02174 [cs.GT].

[5] Cho, Wonki Jo (2018-06-01). "Probabilistic assignment: an extension approach". Social Choice and Welfare. 51 (1): 137–162. doi:10.1007/s00355-018-1110-z. ISSN 1432-217X. S2CID 19700606.

[:12-6] Hadi Hosseini, Kate Larson (2015-07-24). Strategyproof Quota Mechanisms for Multiple Assignment Problems. OCLC 1106222190.

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