Equity required to bet river

It's probably under the assumption that other player can't raise and defends MDF.

As Haizemberg93 said, this seems right if we assume villain can't raise and defends MDF.

We obviously only want to bet when EV(bet) > EV(check).

When IP we are guaranteed to realize 100% of our equity by checking,
so the EV of checking with 70% equity is: 70% * 1 - 30% * 0 = 0.7x pot

If we bet 0.3x pot, villain will fold: 0.3 / 1.3 = 23%

Villain has 70 combos we beat and 30 combos that beat us.
He folds 23 of the combos we beat, leaving him with 47 combos we beat.
Our equity when called is then: 47 / (47 + 30) = 61%

EV(bet 0.3x pot): (23% fold * 1x pot) + (77% call * 61% win * 1.3x pot) - (77% call * 39% lose * 0.3x pot) = 0.75x pot EV = higher than EV(check).

If we were to plot it on a graph we would see that ~0.3x pot is indeed the highest EV size, and we would blunder if we bet more than 0.67x pot
because then EV(check) > EV(bet), since our equity when called is then <50%.

But again, in this scenario villain is not allowed to raise, which would lower our EV.

I guess this is why solvers don't like to bet small on the river in position, because the EV of checking is high, and betting small
doesn't add much EV even if villain is not allowed to raise. Allow villain to raise and the EV of betting small for thin value can easily become lower than checking.

Here's a graphing calculator you can play around with (hopefully my formulas are right)

https://www.desmos.com/calculator/nzhcebeehe

Desmos

Desmos | Graphing Calculator

x = bet-size

f = fold%

c = call%

b = equity before call

a = equity after call

y = EV

Another issue with the graph is that it doesn't consider the composition of villain's range.

Let's say if villain's range was:
1% nuts
1% bluffcatchers
99% air

Our hand would have 99% equity, but it would clearly be a big mistake to overbet our hand, since more than half of the resulting calling range would be the nuts.

It's more accurate to say that betsizing depends on the *ratio* of nuts and bluffcatchers in villain's range. The more hands he has that beat your bluffs, the more it allows you to size up - even if your hand is not that nutted.

In that case vilain will defend mdf of the range that is not air. So your effective equity will be 50% not 99%.

Formula is fairly good if you take equity vs range that beats bluffs. Something to keep in mind:

1.You should bet smaller(or not at all) if opponent will often raise with range of stronger value hands and bluffs
2.Sometimes you go bigger if that prevents opponent from raising too much. For example when you go all in he cant raise anymore
3.Sometimes you also want to go smaller if that induces a value raise form weaker hands
4.Blockers-If hand blocks a lot of calling range vs big bet, you might go smaller to get called more often.

In this "rare" scenario, what is the bet size to choose ? Tiny bet (maybe 1bb) to fold all air at a ridiculous price ?

You would only bet with the nuts and air yourself, probably between block and half pot.
You can essentially remove the air from the equation and ignore it

I'm not exactly sure what you mean.

You can clearly see the formula in the link:
y = 1 * f + c * a * (1+x) - c * (1 - a) * x

Okay. On my screen, the column was narrowed so the graph covered the loss part of the equation. I knew there had to be an explanation.