Abstract

Many methods of dominance rank ordination were
recently reviewed by de Vries (1998). Overall, two
types of method for finding a dominance rank order can
be distinguished. In one group of methods some numerical
criterion, calculated for the dominance matrix as a
whole, is minimized (or maximized) resulting in a reorganized
matrix for which this criterion
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is smallest (or
largest). The result produced by each of these methods is
a rank order of the individuals, that is, the most plausible
one relative to the specific criterion used, and given the
dominance encounters observed. This group includes
methods developed by Slater (1961), de Vries (1998),
McMahan & Morris (1984), Brown (1975), Bossuyt
(1990), Crow (1990) and Boyd & Silk (1983). The second
class of methods aims to provide a suitable measure of
individual overall success in the group, from which a rank
order can be directly derived. Measures that have been
put forward for this purpose include: number of individuals
dominated; proportion of total encounters won;
Clutton-Brock et al. s (1979) index of fighting success;
David s (1987, 1988) score; and Jameson et al. s (1999)
score. As yet, however, none of these success measures
appears to be generally accepted (see also de Vries &
Appleby 2000).
All these methods start by observing behaviour for a
certain period of time after which the outcomes of the
dominance encounters are arranged in a matrix. When
sufficient interactions between the contestants have been
observed, a rank ordination method is used that yields a
dominance order that is presumed to have existed during
the whole observation period. Basically this means that
it is assumed that specific interactions between two
individuals reflect the dominance order rather than
influence it.
In this paper we present the Elo-rating method which
provides sequential estimations of individual dominance
strengths based on the actual sequence of dominance
interactions. From the values of the individual Elo-ratings
an estimated rank order can be derived at any moment in
time. Elo-rating was developed and subsequently named after Arpad Elo (1961, 1978). It is intended and still used
as a fair method for ranking chess players. The Elo-rating
calculation procedure is based on the assumption that the
chance of A winning from B is a function of the difference
in current ratings of the two contestants. After each
contest the Elo-ratings of the two contestants are updated
in proportion to the deviation of the actual outcome
(win, loss or tie) from the expected outcome for each of
the two contestants. The expected outcome for a contestant
is based on the rating difference of the two contestants
at the moment of the contest. The winner s rating
increases (and the loser s rating decreases) in proportion
to the deviation from the expected outcome. As outstanding
features of the Elo-rating method in comparison to
other methods of estimating dominance strength, we
mention that it is independent of the number of contestants
(which may vary over time), it takes the sequence of
interactions into account, and gives a continuous update
so that the process of dominance strength acquisition can
be followed from interaction to interaction. Our main
aim is to present Elo-rating as a method for the sequential
estimation of dominance strengths. However, from a
different perspective it is also possible to consider the
Elo-rating updating process as a model of the way in
which dominance is generated within a group. The
underlying model here is based on the positive (negative)
reinforcement of some internal variable when an individual
wins (loses) a dominance interaction. We also
briefly discuss the application of Elo-rating in a
simulation modelling context.
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