Dynamics of Multi-Player Games
E. Ben-Naim, B. Kahng, and J.S. Kim
We analyze 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
$m$th ranked player wins. We solve for the long time limit of the
distribution of the number of wins for all $n$ and $m$ 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. When the worst player
wins, the standings are least competitive as there is one tier in
which all of the players are equal. This behavior is understood via
scaling analysis of the nonlinear evolution equations.
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