What do the indices of reproductive skew measure?

Several indices of reproductive skew that quantify the degree of unequal partitioning of reproductive output among individuals have been proposed without consensus on their merits and defects. We believe that the major reason for the disagreement is the lack of discussion on what the population parameter of skew (population skew or true skew) should measure. In our view, the skew index should be an unbiased estimate of a population skew, and the estimated skew needs to satisfy the following two conditions. First, if the group size is equal and the distribution of potential of reproductive output (pi), which is scaled by the proportion of the individual's to the total group reproductive output, is also fixed, skew remains constant even when the total number of offspring in the group changes. Second, if the group size is different, skew should have intuitive biological meaning. Our analyses revealed that, among various indices so far proposed, only the skews estimated by Kokko and Lindstrom's lambda and Morisita's I-delta satisfy the first condition. However, the two indices estimate different population parameters, thus implying different biological meanings. Morisita's I-delta is a linear function of CV2 (squared coefficient of variation) of pi, and lambda is a positive function of Sigma pi (2)(i) when offspring number follows a multinomial distribution. In the special cases where a group consists of discrete classes of breeders and nonbreeders, lambda behaves roughly inversely parallel to the absolute number of breeders, while I-delta moves almost parallel to the proportion of nonbreeders. Furthermore, lambda is sensitive to the total proportion of reproductive output possessed by the dominants but is relatively less sensitive to the number of subordinates. We discussed the possible situations where either of the two indices will be useful.