Lexicographic dominance is a total order between random variables.
) if one of the following holds: 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.
) if one of the following holds: To distinguish between the two notions, the standard lexicographic dominance notion is sometimes called downward lexicographic dominance and denoted
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: Then the following holds: 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.