G.T. Vickers

Multi-player matrix games

Game theory has had remarkable success as a framework for the discussion of animal behaviour and evolution. It suggested new interpretations and prompted new observational studies. Most of this work has been done with 2-player games. That is the individuals of a population compete in pairwise interactions. While this is often the case in nature, it is not exclusively so. Here we introduce a class of models for situations in which more than two (possibly very many) individuals compete simultaneously. It is shown that the solutions (i.e. the behaviour which may be expected to be observable for long periods) are more complex than for 2-player games. The concluding section lists some of the new phenomena which can occur.

On the definition of an evolutionarily stable strategy

An ESS must be able to withstand invasion by a small group. It is shown that there are (at least) two possible mathematical interpretations of this statement. In some important applications the two definitions of an ESS are equivalent, but this is not generally the case and a simple example is given to illustrate this. In the weaker form, not favoured here, an ESS may not withstand some infinitely small invasions, but in the stronger form the ESS is certain to survive all invasions up to a fixed fraction of the population. Also, when a pay-off matrix is used to define a dynamic, the stronger definition is the more convenient.

Patterns of ESS's. II

A matrix may have several evolutionarily stable strategies (ESSs). It is thus possible for different populations of a species to adopt a different ESS even when the pay-offs for the populations are the same. The occurrence of different strategies does not imply different circumstances. However, there are constraints upon the collection of supports of the ESSs (i.e. pattern) that any matrix can have. The best-known of these is that the support of one ESS cannot be contained in that of another and this gives bounds on the number of different patterns possible for n x n matrices. Other general constraints are presented here. The enumeration of the patterns for 3 x 3 and 4 x 4 matrices is completed and considerable progress made on 5 x 5 matrices where the number of (permutationally distinct, maximal) patterns exceeds 16.

On the number of local maxima of a constrained quadratic form

We consider the problem of determining the greatest number of local maxima that a quadratic form can have when the vector is constrained to lie within the unit simplex. Specifically, we investigate the local maxima of V = pTAp, where p = (p1, p2, . . . , pn)∊ ∆n = {X ∊ Rn: xi ≽ 0, ∑ixi = 1} and A = (aij) is a real, symmetric n x n matrix. Considering the central role played by quadratic forms in the history of mathematics in general and algebra in particular, it is perhaps surprising that this problem does not appear to have received any attention. It is a rather awkward problem because the constraint cannot be readily incorporated. A complete solution to the problem is lacking, but we show that the greatest number of maxima that any n x n matrix can have increases geometrically with n and also present some results on the lengths (i. e. the number of non-zero elements) of the maximizing vectors.

A sequential-arrivals model of territory acquisition II

Birds arrive sequentially at their breeding ground where the nest sites vary in value (measured by reproductive success). Each bird must either choose a vacant site or challenge an occupier for its site. In the latter case we assume the occupier to be the more likely winner. The loser of the contest incurs a cost and must go to a vacant site. The rational strategy for such a contest is found. There is a threshold phenomenon; early arrivals occupy vacant sites, late arrivals fight. This result is intuitively reasonable, but the sequence of sites chosen is complex. A recursive method for specifying the solution is described and applied explicitly to some illustrative cases.Copyright 1997 Academic Press Limited Copyright 1997 Academic Press Limited

On the number of stable equilibria in a one-locus, multi-allelic system

It is known that in the evolution of a one-locus, multi-allelic system, the fitness increases and the system always approaches a stable equilibrium point (except for degenerate cases). However, there may be several such stable points and various results, assertions and conjectures exist in the literature concerning the number of these points. Here we present counter examples to some of these claims and give a proof of a result of some importance when new alleles are being introduced.