Given the poisonous atmosphere at the Trump White House, carrying out witch hunts won’t stop the leaking. The only way to address the problem is to change the culture, foster a sense of team spirit, and reward people for being loyal. In terms of game theory, you need to alter the rewards and punishments that individual staffers perceive to be attached to their actions, so that coöperation, rather than backstabbing, emerges as an equilibrium strategy.
That is the sort of environment that previous Presidents created. But, of course, teamwork is anathema to Trump, who is himself addicted to leaking. In conversations with his cronies, which take place on a regular basis, he routinely bad-mouths certain officials, and the cronies then tell journalists—on a non-attributable basis, of course—that Person X or Person Y has fallen out of favor. That helps explain why there are so many stories about who’s up and who’s down, so many personnel changes, and so many leaks. The prisoners’ dilemma illustrates how the process works. But, in this case, it could be renamed the Trump dilemma. He created it.