  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.
