Opponent Indifference in Rating Systems: A Theoretical Case for Sonas

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.