Dynamics of multi-player games

We analyse the dynamics of competitions with a large number of players. In our model, n players compete against each other and the winner is decided based on the standings: in each competition, the mth ranked player wins. We solve for the long time limit of the distribution of the number of wins for all n and m using scaling analysis of the nonlinear evolution equations, and find three different scenarios. When the best player wins, the standings are most competitive as there is one tier with a clear differentiation between strong and weak players. When an intermediate player wins, the standings are two-tier with equally strong players in the top tier and clearly-separated players in the lower tier. Interestingly, the size and the strength of the upper tier are nontrivial. When the worst player wins, the standings are least competitive as there is one tier in which all of the players are equal. We conclude that controlling the rank of the winner provides a way of controlling social inequalities.