Opponent Indifference in Rating Systems: A Theoretical Case for Sonas
Greg Bodwin, Forest Zhang
TL;DR
The paper addresses incentive-compatibility in rating systems by formalizing a base, one-dimensional, memoryless, zero-sum framework with a skill curve $\sigma$ and an adjustment function $\alpha$. It shows that full opponent indifference is incompatible with full-scale growth, while a relaxed $P$-opponent indifference is viable and precisely characterized by $P$-separability of $\sigma$ and $P$-constancy of $K$, with strong $P$-indifference corresponding to Sonas-like curves and a $P$-constant $K$. This connects robustness to opponent-selection to the empirical Sonas model, offering a principled justification for Sonas in high-level play and clarifying limits of strategyproofness in traditional rating systems. The results illuminate how display of opponent ratings and matchmaking dynamics interact with fundamental properties of rating updates, and they open directions for extensions beyond the current one-dimensional, translation-invariant setting. Overall, the work provides a rigorous foundation for designing and selecting rating systems that balance predictive accuracy with resistance to opponent-selection incentives, highlighting Sonas as a particularly favorable option under realistic constraints.
Abstract
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 matches against opponents whose ratings are publicly visible, or abort a match 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 matches 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 "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 matches.