[Nick C]

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by "bluff more than 30% of the time" I mean more than 30% of his river bets being bluffs.

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Yeah, and if 70 percent of his "showdown" hands are winners and he bets all of them, then he can't bluff more than 30 percent on the river unless he's also betting some non-showdown hands.

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if we expect villain to bluff less than 12.5% of the time we should c/f.

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If I'm understanding your numbers correctly, your cutoff point is about 6 percentage points too high here.

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pot is 7BB after villain bets the river. the EV calculation for the river is

7BB*x - 1BB*(1-x) > 0
8BB*x - 1BB > 0
8BB*x > 1BB
x > 1/8
x > 0.125

am I missing something?

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I tried to explain in my edit. Anyway, though, the thing is, Villain is not betting the entire 100 percent of the "showdown" hands you've assigned to him. He's only betting 70 out of those 100 hands, plus a few of the losers. So if he bets 12.5 of the losers, we'll win 12.5 out of 82.5 times we call (about 15.15 percent). This is better than the 12.5 percent we need to break even, so we have a call, and Villain is bluffing too often from a theoretical standpoint. (Edit: Nevertheless, bet-folding would still be better than check-calling given our assumptions, because Villain will call with more losing hands than he'll bet himself.)

So that's why his optimal bluffing frequency should be 10 percent of the showdown hands you've assigned to Villain (assuming he's never betting any non-showdown hands, that is). Basically, whatever hands he's betting, theoretically 1/8 of them should be losers. If he were only betting trip T's but was checking aces up, for instance, then he'd need to start betting a much smaller percentage of his losers to remain theoretically optimal with his bluffs.

Meanwhile, just because Villain starts only bluffing 9.8 losers versus 70 winners doesn't mean check-folding instantly becomes better than bet-folding. He does still gain on those bluffs, and bet-folding was actually quite a bit better than check-folding at that 10-bluffs-per-70-value-bets cutoff point (basically because by bet-folding we still get value from all of Villain's losing showdown hands and meanwhile never sacrifice the pot to one of his bluffs).

Okay, and I do want to emphasize one thing: If Villain bluff-raises sometimes, then that changes everything. The reason I'm emphasizing this is that in our scenario we've made bet-folding a completely safe play, in that it never costs us any more than that one last bet we put in (and we've probably done this more or less appropriately, since Villain is very loose/passive). However, versus a river LAG (instead of our loose-passive in this hand), bet-folding a good hand in a big pot is incredibly risky.