Edge-wise rating difference bound over polynomial time

Prove that, for the Elo process with match-up probabilities q on edges E of the comparison graph and step-size η, the edge-wise rating differences satisfy max_{i,j : q_{i,j} > 0} |X^t_i − X^t_j| < M_E + 1 for polynomially many time steps, where M_E = max_{i,j : q_{i,j} > 0} |ρ_i − ρ_j|.

Background

The paper compares its results with recent advances on regularized maximum likelihood estimation in the BTL model. The authors note that a version of their analysis could use M_E (the maximum true-rating difference across edges of the comparison graph) in place of the global cap M to potentially tighten dependencies.

However, adopting M_E requires showing that the Elo edge-wise rating differences remain bounded over polynomially long times. The authors explicitly state they are currently unable to prove this, and it limits replacing global bounds by edge-local ones in their analysis.