Elo-rating as a tool in the sequential estimation of dominance strengths

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 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.