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 ${\displaystyle A\succ _{ld}B}$) 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.

Upward lexicographic dominance is defined as follows.^[2] Random variable A has upward lexicographic dominance over random variable B (denoted ${\displaystyle A\succ _{ul}B}$) 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 ${\displaystyle A\succ _{dl}B}$.

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:

• ${\displaystyle A\succ _{dl}B}$, since they assign the same probability to z but A assigns more probability to y.
• ${\displaystyle B\succ _{ul}A}$, since B assigns less probability to the worst outcome w.
• ${\displaystyle A\not \succ _{sd}B}$, 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.
• ${\displaystyle B\not \succ _{sd}A}$, 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.

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.