Except Vinny wasn't in on this call. only Ansom can see it. Think about it like a telephone. You only get half the conversation. For instance, when Stanley calls Maggie, Parson only hears what Maggie says.

Speaking of which...

How exactly did Stanley call Maggie???

Unless he can order her to call him even when she is out of ear-shot, and she (via natural thinkamancy) knows, and is compelled to obey? Ansom needs the hats, but that could be because he was communicating with a non-thinkamancer.

That kind of 'spooky action at a distance' is essentially what happens when you play a TBS game, so it is thematically appropriate, but it also does add a huge level of significance that needs to be accounted for when speculating. For instance, Stanley could disband parson et al regardless of whether or not he is at GK...