Scalable Inversion of Contests with Correlated Performances, Including Softmax and Multinomial Probit

Multinomial probit choice probabilities over n alternatives are Gaussian orthant integrals, computed by simulation for thirty years, one expensive integral per alternative. Inversion, which is to say determining item attractiveness consistent with a prescribed choice probability vector, is even more difficult and has been considered impractical for correlated contests when n is large. Yet here, for families lying within a grammar including factor, block and hierarchical covariance structures, we exhibit a calibration tested at n = 1,000,000 reproducing probabilities to very high accuracy, even in the extreme tail. We must return to much smaller problems for any performance comparison to be possible due to limitations of the prior art. The Geweke-Hajivassiliou-Keane simulator is the standard (and still appropriate for high rank) but is two hundred times slower already at n = 200, and its measured cost grows roughly as n^2.8 while ours is linear. Furthermore our approach applies to any continuous performance distributions within reason: the Thurstone-Mosteller model families thereby become a practical alternative to logit at modern scale.