Hello, everyone, this is Soda.

I have been watching some videos and reading some books to learn game theory, and I am currently learning the [0,1] toy game and having some questions.

There are maybe couple different names for it, just a quick summary of what the game is in case you don't know.

half street game, or you can think of on the river, IP and OP have the same range, which content all the numbers from 0-1 (0.99, 0.98....0.0001). OP is forced to check, and IP can bet 1 pot size or check.

The result is, IP will bet his top 20% for value, and bottom 10%(which is alpha, 1/(1+1)=50%, so 2 value 1 bluff) for bluff, so it is 30% overall betting range.

At Nash equilibrium, OP needs to defend 1-alpha with the range that beat IP's bluff range, so in this case, 1-alpha=50%, and OP's defending range=(100%-10%) * 50% =45%, and, correct me if im wrong, that 45% must include 100% of that 20% IP value betting range, and the rest 25% does not matter, just any 25% that beat IP's bluff.

here is my question:
1. why IP has to bet top 20% for value and 10% for bluff, why not top 40% for value and bottom 20% for bluff. I know it decrease the EV, but what are the factors that make it so?

2. if OP over fold, lets say 5%, how can IP exploit OP? I know IP needs to bluff more, but by how much and why?

My guess this is an ante game and a limit one as well. So, it’s going to be dependent on pot size and bet size. If both players ante $1, giving us a $2 pot, and IP can bet $1, he will need to bet at a certain frequency and his bluff frequency will be tied to the same. I believe that, with a polarized range, the ratio of bluffs to value bets is bet/bet+pot. So a 1/2 PSB gets 1:3 so 1 bluff to 3 value bets. That doesn’t fully answer your question as far as “top 20%” but it’s a start.


With such an extreme deviation (calling half as frequently as is optimal) I would think that IP could print by bluffing 100% of bluff candidates. It’s been a while since I read a similar problem in Play Optimal Poker, and I do believe that’s the conclusion Brokos comes to. I’ll track it down and double check.


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1. The IP player's hand needs to have 50%+ equity when called to be considered a value bet. Therefore, if the OOP player is defending the top 45% of hands vs the bet, then the IP player should logically value bet only the top half of those (so, the top 22.5% of hands).

2. If OOP folds 60% instead of 55%, the IP player should now bluff with the bottom 20% of hands (they make 50%+ of better hands fold) and value bet only the top 20% of hands.

I was close. Any bluff candidate that was indifferent between x and bet, should be pure bets. Also, don’t bet as thin for value.


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I understand the first half of the sentence and I believe it is correct, but are you sure the top 22.5% against OP's top 45% is 50% equity? It just sounds weird to me and I think IP should only bet top 20% for value at NE.

Also, can you please explain little more of how you get the answer of betting bottom 20% of the hands?

I understand what I_lose said, that IP needs to deviate from the very bottom of the range first for exploit OP's overfolding, but can you talk more about how did you get the exact number?

Thank you

NVM, i got it, the top 22.5% against top 45% must have at least 50% equity. but why then answer is 20% instead of 22.5%?

I’m not sure what exact number you are looking for. But I reread your initial post and picked up something I missed. You specified IP can make a PSB. So your frequencies come from that same formula from earlier. Bet/bet+pot (which gives us the ratio, not a fraction which threw me for a loop at first)

So, both players ante $1 and IP is allowed to bet pot, or $2. So, bet/bet+pot 2/2+2 or 1/2 so for every 1 bluff we should have 2 value bets. Which scales up to your 20% and 10% values. Now, I’m not an expert on sizing and frequencies. So, I’m not sure where the exact range is derived from, only the real ion ship between value and bluff.

As for exploits based on over folding: there will be some hands that are pure bluffs, a 0 in this game, and some that are indifferent to bluffing and checking at equilibrium. If a player is over folding, calling only half as often as they should, those indifferent hands become pure bluffs. So we aren’t Devi’s ing at the bottom of our range, but at the margins.

I prefer the simpler 1/1+S (where S is the size of the bet in PSB) in this game 1/1+1 or 1/2. So OP has to defend 50% of the time to make IP indifferent to bluffing. Since they both have the same starting range, yes that 50% would include the top 20% that IP is value betting along with an extra 30% bluff catchers.


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No idea. To me it seems that the GTO answer should be around 22 - 22.5%.

Let's say that OP folds 60% of hands. This means that if IP has "the 0.2 hand" (bottom 20%), the OP will fold exactly half of better hands (he'll call 0.6-1.0 and fold 0.2-0.6). This makes the EV of the bluff equal to the EV of checking back. Which means that a bluff is the optimal play for the bottom 20% of the IP's hands.

The reason why I didn't use 1-alpha to calculate OP's defending range, is that the definition of 1-alpha should not only be simply MDF but also like you need to call this amount of hand in your "beating bluff" range, in this case, the bottom 10% of OP's hand is not in "beating bluff" range, because IP bluff with the bottom 10%, thats why I got (100%-10%) * 50%=45%, and 45% is the answer in this example.

If OP defend 50%, he is overcalling, IP can just simply not bluff at all.

Got it, so I can think that the bottom 20% of IP's range targets OP's bottom 40%, and that's 10%win and 10%lose, so EV(check)=EV(bet).

If OP calls little more than 40%, then the 10%win decrease and 10%lose increase, EV(check) greater than EV(bet), and IP check those 20%. vise versa