I'm starting up a summoner character in a Pathfinder campaign, but we're a little unsure of the rules for eidolons. While I know the summoner is the one who tells her eidolon what to do, I'm wondering whether the player or the GM decides what the eidolon actually does.
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There's no fixed rule for this, so it will be up to you and your GM to work this out.
An Eidolon has its own feelings, and Intelligence score, and the same alignment as you.
I would say that 90% of the time, the Eidolon should be controlled by the player. If the GM needs to have a RP moment between you and the Eidolon, then that is a situation where it might be valid. (Or if the GM feels that you are not playing the Eidolon's alignment or whatnot).
An important thing to remember, is the Eidolon is a crucial part of your class skills and features, and I feel it would be unsatisfying for you to not get to play your Eidolon.
Much like summoned creatures, animal companions, bonded mounts, familiars, etc., this is totally up to your DM. All the rules say is that eidolons are treated as summoned creatures, which similarly have ambiguous autonomy.
Typically, DMs will just have PCs control the eidolon directly. The vast majority of gaming groups do this. As with animal companions and the like, there's always the possibility that the DM may feel like the creature would act differently - they have to heed the summoner's commands but in terms of individual initiative and cleverness when say, the summoner is indisposed, it's a grey area.