We study the problem of agent selection in causal strategic learning under multiple decision makers and address two key challenges that come with it. Firstly, while much of prior work focuses on studying a fixed pool of agents that remains static regardless of their evaluations, we consider the impact of selection procedure by which agents are not only evaluated, but also selected. When each decision maker unilaterally selects agents by maximising their own utility, we show that the optimal selection rule is a trade-off between selecting the best agents and providing incentives to maximise the agents' improvement. Furthermore, this optimal selection rule relies on incorrect predictions of agents' outcomes. Hence, we study the conditions under which a decision maker's optimal selection rule will not lead to deterioration of agents' outcome nor cause unjust reduction in agents' selection chance. To that end, we provide an analytical form of the optimal selection rule and a mechanism to retrieve the causal parameters from observational data, under certain assumptions on agents' behaviour. Secondly, when there are multiple decision makers, the interference between selection rules introduces another source of biases in estimating the underlying causal parameters. To address this problem, we provide a cooperative protocol which all decision makers must collectively adopt to recover the true causal parameters. Lastly, we complement our theoretical results with simulation studies. Our results highlight not only the importance of causal modeling as a strategy to mitigate the effect of gaming, as suggested by previous work, but also the need of a benevolent regulator to enable it.

翻译：我们研究了在多个决策者参与的因果战略学习中代理选择问题，并应对随之而来的两个关键挑战。首先，尽管先前大量工作聚焦于静态代理池（其评估结果不影响代理池组成），但我们考虑了选择程序的影响——代理不仅被评估，还会被选中。当每个决策者通过最大化自身效用单方面选择代理时，我们证明最优选择规则是在选择最佳代理与提供激励以最大化代理改进之间的权衡。此外，该最优选择规则依赖于对代理结果的不准确预测。因此，我们研究了决策者最优选择规则不会导致代理结果恶化或造成代理选择机会不公正减少的条件。为此，我们在代理行为的特定假设下，给出了最优选择规则的解析形式以及从观测数据中恢复因果参数的机制。其次，当存在多个决策者时，选择规则之间的相互干扰会为底层因果参数估计引入新的偏差源。为解决此问题，我们提出了一种合作协议，所有决策者必须集体采用该协议以恢复真实因果参数。最后，我们通过仿真研究补充了理论结果。研究结果不仅强调了因果建模作为减轻博弈效应策略的重要性（如先前研究所示），还揭示了仁慈监管机构在推动其实施中的必要性。