  In order to understand if and how strategic resource allocation can constrain
the structure of pair-wise competition outcomes in competitive human
competitions we introduce a new multiplayer resource allocation game, the
Population Lotto Game. This new game allows agents to allocate their resources
across a continuum of possible specializations. While this game allows
non-transitive cycles between players, 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/or
non-transitive cycles inside individual leagues. 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.
