[W. Deranged]

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Nice post, I think given the read F>R. I won't start counting combos and estimating what hands Button will raise, but that's my feeling.

I would be interested in some thoughts about the risk of giving the 4th player in this pot a freecard. There's also a risk of him having our hand beat. Will this effect your calcs our do you consider his presence negligible as well?

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Basically, in some sense we have to discount the presence of the two intermediate players at least partially or else the problem becomes so difficult that we can't really create any reasonable model like this to understand it.
We have no reason to believe that the cold-caller has us beat anyway; he's at least somewhat likely to dump on the turn, and, if he has a draw, it may well be the same draw as villain anyway and in that sense giving him a free card is the same as giving villain a free card anyway. We should not that, since betting may have some extra value behind it since that player may call with a worse hand, betting becomes more appealing. I think it is pretty hard to speculate how big that effect will be, though.

Casting the problem in terms of a heads-up problem with villain does help to clarify the issue. If we think it's close between checking and calling on the turn, with the third player also in there, probably holding a hand we can beat, and maybe having outs against us, I think that would certainly help to "break the tie" so to speak and push us toward betting.

I'm really, really interested by this question of how the "check the turn with outs" and the "don't give free cards" principles work in coordination with each other on the turn in situations like this. I'm thinking all probably write up something pretty lengthy about it along the lines of my response to this post and post it here in a little bit. I've played a couple of hands that got me thinking of similar principles and I think they'd be worthy to see compared.