0
$\begingroup$

In traditional rating systems (such as Elo), a player's strength is represented by a single scalar value, which is assumed to be consistent across different opponents. However, in some games, the interaction between players can introduce complexities that aren't captured by a single number. This leads to potential scenarios where $p_1>p_2$​, $p_2>p_3$, and $p_3>p_1$​, suggesting that player strengths might be better represented in an $n$-dimensional space (ex. $\text{style}_1$ beats $\text{style}_2$ and $\text{style}_2$ beats $\text{style}_3$ but $\text{style}_3$ beats $\text{style}_1$).

  1. Are there established methods or models in the mathematical literature that deal with such multi-dimensional interactions in the context of rating or ranking systems?
  2. How might one approach the problem of clustering players based on their multi-dimensional strengths or styles?

Any references or insights would be appreciated.

Improve this question
$\endgroup$
5
  • 1
    $\begingroup$ Munshi, Jamal, The Relative Playing Strength of Chess Players: A Note (August 8, 2014). Available at SSRN: ssrn.com/abstract=2477868 or dx.doi.org/10.2139/ssrn.2477868 "The conventional method of rating chess players uses a method called "scoring" to reduce a trinomial process to a binomial making it possible to rate chess playing strength with a single scalar measure. ... Two dimensional measures of playing strength, though more cumbersome to use, may have greater validity because they contain more information." $\endgroup$
    – Gerry Myerson
    Commented Oct 18, 2023 at 22:41
  • $\begingroup$ Section 2.4 of Pelanek R., Applications of the Elo Rating System in Adaptive Educational Systems, Computers & Education (2016), doi:10.1016/j.compedu.2016.03.017. available at fi.muni.cz/~xpelanek/publications/CAE-elo.pdf is entitled, Multivariate Extensions. $\endgroup$
    – Gerry Myerson
    Commented Oct 18, 2023 at 22:46
  • $\begingroup$ @GerryMyerson - Pelanek (2016) assumes testability of each skill individually. "The multi- variate extension is used as follows. Based on available data we compute correlations $c_{ij}$ between knowledge components. For each knowledge component $i$ we have a student skill parameter $\theta_{si}$. After a student $s$ answers a question belonging to a knowledge component $i$ we update estimates of all skills $j$ ..." $\endgroup$
    – mb1
    Commented Oct 18, 2023 at 23:35
  • $\begingroup$ @GerryMyerson - Munshi (2014) seems to only talk about "percent of win" as a second metric. This does not necessarily generalize well to style in most games $\endgroup$
    – mb1
    Commented Oct 18, 2023 at 23:40
  • $\begingroup$ I think the Blade-Chest model is exactly what you're looking for: cs.cornell.edu/people/tj/publications/chen_joachims_16a.pdf. Every player is represented using two vectors (their "blade" and "chest") representing their offensive and defensive abilities. You can fit this model to data and represent intransitivity in matchups. $\endgroup$
    – Kiran Tomlinson
    Commented Oct 19, 2023 at 13:26

0

Reset to default

You must log in to answer this question.