This paper develops a framework for the design of scoring rules to optimally incentivize an agent to exert a multi-dimensional effort. This framework is a generalization to strategic agents of the classical knapsack problem (cf. Briest, Krysta, and V\"ocking, 2005, Singer, 2010) and it is foundational to applying algorithmic mechanism design to the classroom. The paper identifies two simple families of scoring rules that guarantee constant approximations to the optimal scoring rule. The truncated separate scoring rule is the sum of single dimensional scoring rules that is truncated to the bounded range of feasible scores. The threshold scoring rule gives the maximum score if reports exceed a threshold and zero otherwise. Approximate optimality of one or the other of these rules is similar to the bundling or selling separately result of Babaioff, Immorlica, Lucier, and Weinberg (2014). Finally, we show that the approximate optimality of the best of those two simple scoring rules is robust when the agent's choice of effort is made sequentially.

翻译：本文构建了一个评分规则设计框架，以最优激励代理人投入多维努力。该框架是对经典背包问题（参见Briest、Krysta与Vöcking，2005；Singer，2010）在策略代理人情境下的泛化，构成了将算法机制设计应用于课堂场景的基础。本文确定了两种保证与最优评分规则保持恒定近似比的简单评分规则族：截断分离评分规则是各单维评分规则之和经可行得分有界范围截断得到的规则；阈值评分规则在报告超过阈值时赋予最高分，否则赋予零分。这两类规则之一的近似最优性类似于Babaioff、Immorlica、Lucier与Weinberg（2014）提出的捆绑销售与单独销售结论。最后，我们证明当代理人序贯选择努力时，这两种简单评分规则中较优者的近似最优性具有鲁棒性。