Evolutionary Games with Sequential Decisions and Dollar Auctions

Conflict occurs throughout the animal world. Such conflicts are often modelled by evolutionary games, where individual animals make a single decision each within the game. These decisions can be sequential, in either order, or simultaneous, and the outcome of the game can depend strongly upon which case is assumed to occur. Real conflicts are generally more complex, however. A fight over a territory, for instance, can involve a succession of different stages and, therefore, choices to be made by the protagonists. In this paper we thus introduce a method of modelling a more complex class of interactions, where each individual can make a sequence of decisions. We show that despite the inherent complexity, under certain assumptions, the resulting game often leads to the case where both animals fight to the fullest extent or where one concedes immediately, thus mirroring the outcomes of simpler single decision games. However, for other cases we see that the outcome is not so simple, and intermediate level contests can occur. This happens principally in cases where the duration of contests is uncertain, and partially governed by external factors which can bring the contest to a sudden end, such as the weather or the appearance of a predator. We thus develop a theory grounded in simple evolutionary models, but extending them in various important ways.