What's a Tier List? A mathematical approach

Hi TonyHole, nice to see someone else looking using Nash equilibria to make tier lists. There was a similar discussion here last year.

Two things I’d like to comment on, having looked through your sheet.

  1. You’re directly using Scymrian’s composite matchup chart. I’d recommend reproducing the composite chart instead, because the values given in the original Sheet have been rounded to the nearest 0.25. In addition to the loss of precision, this means some matchups are displayed as non-symmetric, e.g. Grave vs. Jaina is 5.75, but Jaina vs. Grave is 4.5.

  2. Your iterative method, accounting for some scaling differences, takes a matchup matrix M, and converges to the vector v that solves v = Mv. This is not the Nash equilbrium, it’s the matrix’s principal eigenvector, i.e. the part that doesn’t decay with repeated multiplication. I’m not sure how to do it in Sheets, but for the Nash equilibrium you want to solve the equation here. The iterative method is doable in Sheets, but requires a few more steps. You should find the difference between best and worst characters is rather larger than you show here.

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