I'm an old geezer, long time rec player with terrible math skills.
Don't know if this question maybe belongs in beginner section but here goes.

If I bluff pot on the river, what percentage does villain need to fold, in order for it to be a profitable play?

Let's say for example, pot is 100 and I bluff bet 100. The size of the pot.

To break even with an airball bluff, you need villain to have a folding frequency of bet/(pot+bet)

So if you bet 100 into a pot of 100, you need a folding frequency of 100/(100+100) = 1/2 = 50%.

Since it is rare that a hand has 0 equity (airball bluff), I copied the following table from the draft book "Hold ‘em Poker by the Numbers" to show the required fold equity for various hero hand equities ranging from 0% to 40%. Blank entries signify that the situation has positive EV without the need for fold equity.

I’ve never seen that. Thanks.

Keep in mind that your hand's equity only matters if your opponent responds by either folding or calling. If they continue by raising, then your hand's equity doesn't matter and you can just assume you're bluffing with complete air.

I think this is very useful for 15 to 45bb poker, particularly tourney poker because of the frequency of overbetting on the flop and turn.

With more than 45bb in play, I would think that the bluffs would come from the inherently profitable triangle of blanks at the bottom left. These hands can bet fractions of the pot for profit and the stronger blanks can stand up to a raise. This is an important range construction crossroads:
.................../.................................\
bluff with strong draws.............bluff with weak draws

I have a loose theory that the (weak draw bluffs) portion of my bluffing range are the only hands in my range that would fold to a minimum raise. This is why I think that low equity semibluffs are, or should be, low frequency actions in big bet games.* This strengthens your strategy vs a minraise.

This is why I think total polarization is a mistake for preflop 3+ betting. What are you going to do with a range that is (33% bluffs) when facing a minraise that's offering you at least 3:1?

*In limit games, there are many more available strong hands, which allows you to bluff more (combos) but less (frequently). In many instances its correct to bet 100%; this is usually because of either a scary board, or a low opponent raising frequency, or both.

I am very interested in your book. Do you get into things like Minimum Defense Frequency? Most of what is out there is not useful at a table. They generally ignore that MDF must take into account the preflop ranges, not just simple math. It also must include made hands and strong draws.

I am working on software that simulates play using only external files for decisions. Info at peakholdem.com. Might be useful to you.

What I am working on now is board analysis. Will Tipton has suggested that there are 103 boards that represent all combinations. I am struggling with how to reduce board analysis to something truly representative but is simple enough to be useful at the table. Have you done anything on that?

Maybe I'm confused, but isn't it better EV to be checking down with any hand less than 50% equity when 0% hands are getting folded. How does a half pot bet with 30% equity make more profit than checking?

The chart doesn't say whether it is more profitable to check or bet. It just says when it is profitable to bet.

Kelvis
"The chart doesn't say whether it is more profitable to check or bet. It just says when it is profitable to bet"
+1

The phrase “when 0% hands are getting folded” implies that a no-fold is given, which is not the case. The chart shows the required fold equity to make a bet profitable given card equity against villain’s calling range.

This range will normally be narrower than villain’s check-check range. So, for example, if hero checks and villain also checks with a range for which hero has 40% equity (>30%), then you can show that if villain folds 1/3 of the time facing a half-pot bet , hero EV = 0.40, same as the check-check option. This assumes no raise and the hand is subsequently checked down if not a river bet.

EVchk = 0.40*1 = 0.4

EVbet = 0.33*1 + 0.67*(0.30*(1+2*0.5)-0.5) = 0.4

Okay, that makes sense. I guess, in a way, it sort of compares itself to folding since that is 0 EV.

Is there a problem with using this logic instead:
If I make a pot sized bet that is getting called with 25% equity then it can be considered risking only 3/4 of the pot since 25% of what I'm betting will be won back. Is this too much of a simplification?

According to that logic if you have 50% equity you risk 50% of the bet. In reality you're not risking any of the bet because you have 50% over the bet + call.

No, it would be risking 25% of the bet.