Transitive Outcomes and Population Resource Allocation Games

Competition is critical to our understanding of both ecological and human systems. A surprisingly robust result is that the structure of human competitions is largely transitive. One possible explanation for transitive outcomes is that while the strategy space of human competitions may be complex, strategic competitors allocate their training and preparation in a way that prevents cycles. In order to explore this possible explanation we propose and explore a new multiplayer resource allocation game, the Population Lotto Game, where players design probability distributions subject to 'budget' constraints. We show that the Nash equilibrium of the game also forms a hierarchical structure between discrete `leagues' based on their different resource budgets, with potential sub-league structure and non-transitive cycles remaining possible between players with similar budgets. We provide an algorithm that can find a particular Nash equilibrium for any finite set of discrete sub-population sizes and budgets. Further, our algorithm finds the unique Nash equilibrium that remains stable for the subset of players with budgets below any threshold. Applications and future questions in biology will be discussed at the end