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Opponent Indifference in Rating Systems: A Theoretical Case for Sonas

Authors Greg Bodwin, Forest Zhang

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  • 21 pages

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Author Details

Greg Bodwin
  • University of Michigan, Ann Arbor, MI, United States
Forest Zhang
  • University of Michigan, Ann Arbor, MI, United States


We are grateful to Jeff Sonas for a helpful discussion on the history and relationship between rating systems, and we are grateful to an anonymous reviewer for thorough and helpful feedback.

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Greg Bodwin and Forest Zhang. Opponent Indifference in Rating Systems: A Theoretical Case for Sonas. In 14th Innovations in Theoretical Computer Science Conference (ITCS 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 251, pp. 21:1-21:21, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2023)


In competitive games, it is common to assign each player a real number rating signifying their skill level. A rating system is a procedure by which player ratings are adjusted upwards each time they win, or downwards each time they lose. Many matchmaking systems give players some control over their opponent’s rating; for example, a player might be able to selectively initiate games against opponents whose ratings are publicly visible, or abort a game without penalty before it begins but after glimpsing their opponent’s rating. It is natural to ask whether one can design a rating system that does not incentivize a rating-maximizing player to act strategically, seeking games against opponents of one rating over another. We show the following: - The full version of this "opponent indifference" property is unfortunately too strong to be feasible. Although it is satisfied by some rating systems, these systems lack certain desirable expressiveness properties, suggesting that they are not suitable to capture most games of interest. - However, there is a natural relaxation, roughly requiring indifference between any two opponents who are both "reasonably evenly matched" with the choosing player. We prove that this relaxed variant of opponent indifference, which we call P opponent indifference, is viable. In fact, a certain strong version of P opponent indifference precisely characterizes the rating system Sonas, which was originally proposed for its empirical predictive accuracy on the outcomes of high-level chess games.

Subject Classification

ACM Subject Classification
  • Theory of computation → Algorithmic game theory and mechanism design
  • Rating systems
  • opponent indifference
  • incentive compatibility
  • mechanism design
  • game theory


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