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.

翻译：在竞技游戏中，通常为每位玩家分配一个表示其技能水平的实数评分。评分系统是一套程序，玩家每次获胜时其评分上调，每次失败时则下调。许多匹配系统允许玩家对其对手的评分施加一定程度的控制；例如，玩家可能有选择地发起与评分公开可见的对手的比赛，或者在比赛开始前瞥见对手评分后无惩罚地中止比赛。一个自然的问题是：能否设计一种评分系统，使得追求评分最大化的玩家没有动机采取策略性行为，即不会刻意寻求与某一评分而非另一评分的对手进行比赛？我们证明了以下结论：- 这种“对手无关性”属性的完整版本要求过强，实际上不可行。尽管某些评分系统满足该属性，但这些系统缺乏一些理想的表达能力，表明它们不适合捕捉大多数有意义的游戏场景。- 然而，存在一种自然的松弛条件，大致要求玩家在与自己“实力相当”的任意两个对手之间保持无差异。我们证明这种松弛的对手无关性变体——我们称之为 $P$ 对手无关性——是可行的。事实上，$P$ 对手无关性的某个强版本恰好刻画了评分系统 Sonas，该系统最初因其在高级别国际象棋比赛结果预测上的经验准确性而被提出。