Game Theory Resolution

[jay_shark]

For the second solution , it should read

raise with [0.01,0.1975] . Hope that's clear now .

[mykey1961]

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As you can see , there are infinitely many solutions, all of which cannot do better than EV = 0.421875

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And that is what makes this [0,1] problem different than the 100 #'s problem, less solutions there, and this one doesn't require the SB to raise with the lowest possible value to be optimal.

[jay_shark]

Interesting enough , this strategy shows that the sb should raise with about 56.25% of all numbers .So if we were to rank our cards in nl, then raising with the top 56.25% of all hands is a pretty darn good strategy .

We should not rank our cards in terms of equity ,hot and cold , but in terms of your average EV gained by those cards .

[TNixon]

In this game, you are going to be raising with a certain percentage of numbers that are lower than what the BB will call with.

You're raising everything higher than .625, and .1875 lower than .625, so 1/3 of your hands are lower than .625.

But it *does not matter* where that 1/3 comes from. It could be 0-.1875, it could be .4375-.625, it could be 0-.09375 and .531-.625. It simply does not matter, because anytime you're raising with a number lower than .625, you only win if your opponent folds. The solutions are all identical in this game.

And since you're still playing the same number of hands, (and since he can safely assumes you're not being stupid, playing 0-.5625 and folding .5625+) if the big blind calls with anything lower than .625, then he is no longer playing optimally.

But as I mentioned originally, at that point, this game becomes more about third-level thinking than it is about the numbers. If you can get the BB to play sub-optimally, then you can exploit his adjustments. But playing 0-.1875 instead of .4375-.625 should not cause him to adjust, unless believes you're playing more than 43% of the time. The problem here is that you have to believe he's capable of adjusting to your play, and willing to, because if he simply calls .625 no matter what you do, he will end up ahead in the long run, since the SB has a very clear disadvantage in this game.

If we equate these numbers to percentage hand ranges, this is all very different from poker, where there are certain "bluffing hands" that are going to be better than others, because they have a better chance to win against the top 62.5%, for example, and do better when you do get called. In poker, 56 suited has a better chance to beat AA than just about every other 2 cards. But in this game, 44 loses to 100 every single time.

[mykey1961]

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But as I mentioned originally, at that point, this game becomes more about third-level thinking than it is about the numbers. If you can get the BB to play sub-optimally, then you can exploit his adjustments. But playing 0-.1875 instead of .4375-.625 should not cause him to adjust, unless believes you're playing more than 43% of the time. The problem here is that you have to believe he's capable of adjusting to your play, and willing to, because if he simply calls .625 no matter what you do, he will end up ahead in the long run, since the SB has a very clear disadvantage in this game.


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You wouldn't play this game unless you expected to be in the BB as often as you would be in the SB.

So if both of you are playing optimal (and there isn't a rake) the long run wouldn't be so bad.

EV_SB/2 + EV_BB/2 = 0

If in the SB you bet with 7/16 -> 1, and the BB plays non-optimal, you will do better than if you bet with 0 -> 3/16, 5/8 -> 1 against the same player.

[TNixon]

The point wasn't that you wouldn't want to play if you had to be the SB all the time, but that because the BB has an inherent advantage, it is profitable to call .625+ from the big blind, no matter what the player in the SB is doing, so there's no need to modify the strategy at all.

Which means that there's some chance that any attempt to get a player to modify their strategy in the BB is going to do absolutely nothing except lose you more money.

Think of it like a game of roshambo (which can involve many levels of thinking), except that in this case, calling rock is always slightly +EV, no matter *what* your opponent is doing.

[mykey1961]

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The point wasn't that you wouldn't want to play if you had to be the SB all the time, but that because the BB has an inherent advantage, it is profitable to call .625+ from the big blind, no matter what the player in the SB is doing, so there's no need to modify the strategy at all.

Which means that there's some chance that any attempt to get a player to modify their strategy in the BB is going to do absolutely nothing except lose you more money.

Think of it like a game of roshambo (which can involve many levels of thinking), except that in this case, calling rock is always slightly +EV, no matter *what* your opponent is doing.

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There isn't a need for the SB to change from his optimal strategy either, since whatever he loses as SB he will gain as BB.

[vmacosta]

jayshark and others,
is it tru that the way to apply these results to holdem is that the sb should push with any hand that has 43% equity vs. a random hand? In that case it looks like SB has to push with something like T4o and better. Then I guess the BB has to call with everything except perhaps the worst (32o - 82o).

Or is it not so straightforward...

[jay_shark]

Well it's not quite the same situation . In this game , when your number is higher than you win 100% of the time . In hold em , when your card is better(hot and cold) then you don't necessarily win 100% of the time .

This is essentially how the SAGE system was derived but you would need to compare hand ranges for various possible subsets of hands amongst the two players .

[TNixon]

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There isn't a need for the SB to change from his optimal strategy either, since whatever he loses as SB he will gain as BB.

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Yeah. There was some thought buried somewhere about how it *might* be worthwhile to try to get the BB to play sub-optimally and then exploit that, but the more I think about it, the less sense it makes. There's no reason for him to play sub-optimally, and the only thing you can do to try to get him to play a wider range is by playing a wider range yourself. But while you're doing that, even if he doesn't adjust, he's not as well off as he *could* be, but he's still better off than he was before you opened up, and it's going to be fairly obvious when you re-adjust to try to exploit.

So yeah, I guess there's really no third-level thinking involved here at all.

What a silly, silly game. I'm sure glad we have poker instead.

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