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.

我们研究了多个决策者下因果战略学习中的代理选择问题，并解决了相应的两个关键挑战。首先，我们考虑了代理评估和选择过程的影响，发现最佳选择规则在选择最佳代理和提供激励以促进代理改善之间存在权衡。此外，最佳选择规则依赖于代理结果的错误预测。然后，我们研究了决策者的最佳选择规则不会导致代理结果恶化或不公正减少代理选择机会的条件。其次，当存在多个决策者时，选择规则之间的干扰引入了估计潜在因果参数的另一个偏倚来源。为解决这个问题，我们提供了一个合作协议，所有决策者必须共同采用以恢复真实的因果参数。最后，我们通过模拟研究补充了理论结果。我们的结果不仅凸显了因果建模作为减轻操纵效应的策略的重要性，正如以前的工作所建议的，还凸显了需要一位慈善的监管者来使其成为可能。