In many competitive settings, from education to politics, rules do not reward effort evenly, and thresholds (e.g., grade cutoffs or electoral majorities) make some moments disproportionately important. Success thus depends on efficiently allocating limited resources. However, empirical demonstration has been difficult because effort allocation is rarely observable and feedback is often delayed, limiting our understanding of expertise. Professional tennis provides an ideal natural experiment. Because each game resets after a player wins four points and points in a lost game are wasted, the value of a point varies sharply across scores. Efficient allocation should therefore win games without wasting points, conserving resources for future games. Such allocation manifests in score-dependent point-winning probabilities, from which we derive each player's Pareto frontier-the theoretical limit of the trade-off between game-winning probability and the expected points per game. Here, we show that top players operate closer to this frontier, converting points to game wins more efficiently. Optimal strategies reduce the probability of winning points when the player is far behind (e.g.,0-2, 0-3). This behavior is psychologically difficult-letting go of the current game-but represents a rational energy conservation strategy. Top players exhibit this pattern especially in return games, where winning points is harder than in service games, requiring them to drastically vary their efforts, consistent with game-theoretic predictions. These findings suggest that elite performance reflects efficient adaptation to rule-created value structures; knowing when to give up may be as fundamental to expertise as knowing when to compete.

翻译：在许多竞争环境中，从教育到政治，规则并不均等地回报努力，而阈值（如成绩分界线或选举多数）使得某些时刻具有不成比例的重要性。因此，成功取决于对有限资源的高效分配。然而，由于努力分配很少能被直接观察且反馈常常延迟，实证证明一直很困难，这限制了我们对于专业知识的理解。职业网球提供了一个理想的自然实验。由于每局比赛在球员赢得四分后重置，且输掉局中的分数作废，不同比分下每一分的价值差异显著。因此，高效的分配应能在不浪费分数的情况下赢得局分，从而为后续比赛保存资源。这种分配体现在依赖于比分的得分概率中，我们由此推导出每位球员的帕累托前沿——即赢得局分的概率与每局预期得分之间权衡的理论极限。本文表明，顶尖球员更接近这一前沿，从而更高效地将分数转化为局分胜利。最优策略会降低球员在比分大幅落后时（例如0-2、0-3）的得分概率。这种行为在心理上是困难的——即放弃当前局——但代表了一种理性的能量保存策略。顶尖球员在接发球局中尤其表现出这种模式，因为在接发球局中得分比在发球局中更为困难，这要求他们根据比分大幅调整努力程度，这与博弈论的预测一致。这些发现表明，精英表现反映了对规则创造的价值结构的高效适应；知道何时放弃可能与知道何时竞争一样，是专业素养的根本所在。