ESS patterns: Adding pairs to an ESS

The notion of a pattern of evolutionarily stable strategies was introduced by Cannings and Vickers in 1988 (J. Theor. Biol. 132:387–420). In this paper a specific class of patterns is considered. Suppose that there is an evolutionarily stable strategy (ESS) on some set of n strategies {1, 2, …, n} and that new strategies {n + 1, n + 2, …, n + k} are added. Supposing that for this new enlarged conflict there is still an ESS on {1, 2, …, n} and also that there are ESSs on {n + i, j} for 1 ≤ i ≤ k and j ∈ Si ⊂ {1, 2, …, n}, the authors investigate the restrictions on the Si. These restrictions are related to certain properties of strong tournaments introduced by Reid and Beineke. We also specify, given the Si, what ESSs of the form {n + i, n + j} can be added.