Alright, back to California...
From the NamedTownie variant, we have a lower bound - if the Cop has no actual investigation (or doesn't use it, or is known to be Naive/Paranoid, etc.), the EV is 163/630 (~25.87%).
For an upper bound, consider the case where the scum can't fake claim or counterclaim the Cop at all (even in an endgame). Now the problem is simply one of figuring out when the Cop should claim and reveal his results. For a larger game, this might actually be a tricky question (though I don't doubt we could throw together a recursive spreadsheet or a program to solve it). However, for this setup, it turns out that the Cop should almost never claim until it's down to three, unless he is the lynch target.
If the (randomly selected) lynch target is the Cop, the Cop claims. Pick a different lynch target, Cop will be nightkilled, game becomes vanilla.
If a scum is lynched Day 1, obviously the Cop should reveal if he gets a guilty after that. If he gets an innocent Night 1, however, he should stay hidden - revealing gives an EV of 2/3, but if instead we select a lynch target at random with the Cop only claiming if the chosen lynch target is the Cop or the investigation choice, the EV jumps to 7/9 (because either the scum is the lynch target, or if an innocent is lynched there is still a good chance the Cop will survive and win the game). In fact, it doesn't matter whether the Cop steps in to save the investigated innocent if that player is chosen as the lynch target (2/3 either way).
If a townie is lynched Day 1, the Cop should follow a similar plan with an innocent investigation - protect himself or the confirmed innocent, but otherwise stay hidden. The benefits of staying hidden to endgame outweigh the risk of getting killed before confirming the single investigation.
The less obvious case is that if the Cop gets a guilty, he should *still* stay hidden, at least temporarily. The instinct is "I have a guilty, it's lylo, better claim", but in fact the correct play in this case is to pick a target at random, and have the Cop come out only if the lynch target is not the player he got a guilty on. It's likely the Cop will be forced to come out, but if the lynch randomly lands on the investigated scum, the EV doubles.
Anyway, putting it all together I get an EV of 6859/15750 (~43.55%).
I suspect we can push the upper bound down some by adding the following rule: If the Day 1 lynch is on scum, the scum are told who the Cop is (this is equivalent to allowing the scum to fake claim, forcing the Cop to counter). I suppose it's possible that the Cop should stay hidden, but I think it's unlikely. Anyway, if we add in that rule just for Day 1, it pushes the EV down to ~35.78%. Adding it in across the board would push the EV lower still (may go back through and calculate this, have other things to do at the moment).
There are more complicated things going on in the actual California setup. As was pointed out earlier in the thread, the scum shouldn't claim Cop 100% of the time, or a Townie claim is confirmed innocent - rather, they should claim Cop some large fraction of the time, in order to draw out the Cop most of the time but ensure that the town must lynch Townie claims. The Cop may be better off staying hidden if scum claim Cop - if the Cop is NKed, the scum gets lynched the next day, and the real Cop gets an investigation, but on the other hand the odds of a successful D1 lynch go way down and when the Cop does come out D2 he won't necessarily be believed (since if this strategy was viable, it may also be viable for scum to leave the Cop alive N1 and counterclaim him D2). Then you've got the whole "which claim is more likely to be true" problem, in all cases (both claim D1, both claim D2, one claims D1 and the other D2). It's a mess.
I think I'll tackle 2 Mafia, 1 Doctor, 4 Townies next. Tired of dealing with Cops.