As we all know, in the polarized river toy game, our EV increases the more we increase the bet-size.

I watched a RIO video by Steven Paul and he explained it like this:








This gave me the impression that our value bets gains its increased EV from when it gets called,
but when I broke it down further that doesn't seem to be the case.

Let's look at three different bet-sizes.

EV of Bluffs:
0.5x = 33.3% * 1 - 66.7% * 0.5 = 0
1x = 50% * 1 - 50% * 1 = 0
2x = 66.7% * 1 - 33.3% * 2 = 0

EV of value getting called:
0.5x = 66.7% * 1.5 = 1
1x = 50% * 2 = 1
2x = 33.3% * 3 = 1

EV of value getting folds:
0.5x = 33.3% * 1 = 0.33
1x = 50% * 1 = 0.5
2x = 66.7% * 1 = 0.667

So the EV of our value getting called remains the same, while the EV of our value getting folds increases.
I found this surprising for some reason...

EV of bluffs stays the same
EV of value increases

EV of getting folds doesn't increase, your equation is just showing that villain folds more vs bigger sizings.

The reason that bigger bets make more money is not that you can bluff more, this is almost an afterthought. The reason is that bigger bets need to get called disproportionately in order to make you indifferent to bluffing.

4x pot is 8 times half pot, but you don't get 8 times as many folds

Of course bluffs stays EV0. That is exactly what both Steven Paul and I said.

But what I showed (I think) is that the EV of value getting called doesn't actually increase.

EV of bluffs:
0.5x = 33.3% * 1 - 66.7% * 0.5 = 0
1x = 50% * 1 - 50% * 1 = 0
2x = 66.7% * 1 - 33.3% * 2 = 0

EV of value getting called:
0.5x = 66.7% * 1.5 = 1
1x = 50% * 2 = 1
2x = 33.3% * 3 = 1

EV of value getting folds:
0.5x = 33.3% * 1 = 0.33
1x = 50% * 1 = 0.5
2x = 66.7% * 1 = 0.667

No, what you showed is that you get called less and you get more calls, which is self evident.
The EV of getting a fold is always the size of the pot.
The EV of getting a call (if you have the nuts) is the size of the pot+bet, which increases with betsize

Your EV when an event happens doesn't depend on the frequency of that event

Please show me a mathematical example of how the EV of our value increases with bet-size. I want to understand.

Because this is what I am confused about right now.

Yes, (Pot + Bet) obviously increases with bet-size, but at the same time calling frequency decreases.

So Call% * (Pot + Bet) doesn't actually increase with bet-size. It seems to stay at exactly 1x pot regardless of bet-size.

One way to think about is this:
With nuts you already won the pot EV of cheking is 1, so you just trying to maximize the addition money that goes into a pot.
If you bet pot you get 0.5, if you bet 2x the pot you get 0.67 ect.

This is a clever observation.

You're looking at half the equation. The EV of betting = Call%(EV when called) + Fold%(EV when they fold). You need to add these together to find the total EV of betting the nuts:



When you add those together you can clearly see the nuts makes more money when you bet bigger:

It is interesting, however, that Call% * (Pot + Bet) remains constant. I've never noticed that before. It appears that in order to make your bluffs indifferent, it passes the EV of fold equity from your bluffs to your nuts.

Yeah, I know. In the first post I calculated both parts, but I wanted to point out the call part of the equation, because I expected it to increase.

The explanation I've received before (and also Aner0 seems to believe) is that the EV of our value increases because "villain is forced to put more money into the
pot to make our bluffs indifferent".


Or as Steven Paul put it: "B/(B+1) increases as B increases".

That's why I was surprised to see the that EV of the call side of the equation doesn't increase, but the fold side!

Anyway, thanks for clarifying!

This is how I’ve always thought about it too.