Pairwise Comparison Model with a Diverging Number of Nuisance Parameters

Abstract

We consider a ranking problem of players in a game where the number of players diverges but the number of games per player is ¯xed. The standard maximum likelihood estimator based on Thurstone model is inconsistent since the number of nuisance parameters (i.e. the ability of each player) diverges. We propose a new estimator based on the conditional likelihood. We propose a conditional pairwise comparison likelihood and prove that the maximum conditional likelihood estimator is consistent. Also, we develop an e±cient computational algorithm to calculate the maximum conditional likelihood. We illustrate the proposed method on data generated from a game called the Skid Rush that is a web based multi-player car racing game.