Lexicographic dominance is a total order between random variables. It is a form of stochastic ordering. It is defined as follows. Random variable A has lexicographic dominance over random variable B (denoted ) 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. Random variable A has upward lexicographic dominance over random variable B (denoted ) 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 .
Relation to other dominance notions First-order stochastic dominance implies both downward-lexicographic and upward-lexicographic dominance. 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:
, since they assign the same probability to z but A assigns more probability to y. , since B assigns less probability to the worst outcome w. , 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. , 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, incentives for participation, ordinal efficiency and envy-freeness.
Hosseini and Larson analyse the properties of rules for fair random assignment based on lexicographic dominance.
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