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 ) 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. Then this probability should be higher for A.
Upward lexicographic dominance is defined as follows. [2] Random variable A has upward lexicographic dominance over random variable B (denoted ) 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 .
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:
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.
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 ) 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. Then this probability should be higher for A.
Upward lexicographic dominance is defined as follows. [2] Random variable A has upward lexicographic dominance over random variable B (denoted ) 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 .
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:
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.