Jam-or-fold: (near)optimal for stacks up to 10 or 11BB?

[JocK]

In "The mathematics of Poker" Chen and Ankenman state:"

"Jam-or-fold is at least near-optimal for stack sizes up to about 10 or 11 blinds"

However, based on some calculculations (see: http://base.google.com/base/a/112163...41895488656518 ) I have reasons to believe that for stacks deeper than about 8 BB jam-or-fold is no longer (near-)optimal.

I am keen to get your (and in particular Jerrod's) comments on this. Did I make some computational error? Or is my assumption that (when not limited to jam-or-fold) playing from the small blind in headsup gives an edge (for any effective stacksize) flawed?

Johannes

[jukofyork]

[ QUOTE ]
In "The mathematics of Poker" Chen and Ankenman state:"

"Jam-or-fold is at least near-optimal for stack sizes up to about 10 or 11 blinds"

However, based on some calculculations (see: http://base.google.com/base/a/112163...41895488656518 ) I have reasons to believe that for stacks deeper than about 8 BB jam-or-fold is no longer (near-)optimal.

I am keen to get your (and in particular Jerrod's) comments on this. Did I make some computational error? Or is my assumption that (when not limited to jam-or-fold) playing from the small blind in headsup gives an edge (for any effective stacksize) flawed?

[/ QUOTE ]
Not sure if it's related, but for Lee Jones' SAGE system 8BBs is also the point where the system becomes -EV. I'm not sure if this is related to other factors though, such as his point scoring system or possibly because he may have excluded some of probabilistic pushes/calls from the mixed NE strategy to simplify his system.

Juk [img]/images/graemlins/smile.gif[/img]

[Jerrod Ankenman]

[ QUOTE ]
In "The mathematics of Poker" Chen and Ankenman state:"

"Jam-or-fold is at least near-optimal for stack sizes up to about 10 or 11 blinds"

However, based on some calculculations (see: http://base.google.com/base/a/112163...41895488656518 ) I have reasons to believe that for stacks deeper than about 8 BB jam-or-fold is no longer (near-)optimal.

I am keen to get your (and in particular Jerrod's) comments on this. Did I make some computational error? Or is my assumption that (when not limited to jam-or-fold) playing from the small blind in headsup gives an edge (for any effective stacksize) flawed?

Johannes

[/ QUOTE ]

From the linked page:

[ QUOTE ]

Each time you jam with a non-AA hand, you would win 1 BB due to Chris folding with an approximate probability of 1322/1326, whilst with a probability 4/1326 Chris would call your all-in,

[/ QUOTE ]

Evidently these should be 6/1225 (when you don't have an ace) and 3/1225 (when you do). It seems from your results that you might have done this correctly in computing the value and this is just an oversight. I'll assume this.

You claim:

[ QUOTE ]
Based on the considerations mentioned in the section 'positional advantage', it is difficult to believe that playing from the button would carry no advantage (or even represent a disadvantage) in case the player at the button is not limited to jam-or-fold actions, and post-flop action can be exected to be present.

[/ QUOTE ]

At a stack size of about eight, I'm not sure that the button having additional actions is really very helpful to him at all. He can't really gain much from raising a smaller amount, based on the argument we present in the book -- succinctly, that if the BB just plays a naive strategy of jamming whenever he would have called a jam, the SB can't gain by raising a smaller amount, and the BB has the additional option of calling a raise preflop. It's fairly unclear from an analytical point of view what happens if the SB plays a limping strategy; my intuition is that there's isn't much to gain here either, as the BB doesn't call jams particularly loosely (only about 50% IIRC), so giving him free flops with half his hands seems wrong. I'm not certain about this, so I would be interested to hear well-reasoned arguments that there is a lot of value in limping in this game.

So, while it's true that the SB must have higher equity in the full game than he does in the jam-or-fold game, I suspect that at stacks of 8-10, it's not very much higher, hence justifying the "near-optimal" designation we gave to jam or fold. Notably, I think it's highly likely that for some stack size above 5 that the SB on the button has negative equity from playing this game.

[JocK]

thanks for the reactions.

[ QUOTE ]
[ QUOTE ]
[..] is my assumption that (when not limited to jam-or-fold) playing from the small blind in headsup gives an edge (for any effective stacksize) flawed?


[/ QUOTE ]

[..] I think it's highly likely that for some stack size above 5 that the SB on the button has negative equity from playing this game.

[/ QUOTE ]
My intuition must be failing me. I find it difficult to believe that for the full game the equity of the SB on the button can drop to negative values. A hand-waving argument for this is that when the SB would follow a (non-optimal) strategy that for each hand starts with limping, the result is that the BB and SB have contributed symmetrically to the pot, and that the BB is first to make a move. Effectively, the SB would have position on each betting round (including the pre-flop betting). Could this ever turn into a -EV proposition for the SB?

If indeed the EV of the SB can not drop below zero, the full game would have an EV roughly as depicted (red curve) in attached picture (yellow curve is the computed EV of jam-or-fold game - if picture doesn't show: click on below link).



http://picasaweb.google.nl/jkoelm/NLHE


Apparently my argument is wrong, and the full game can lead to negative EVs for the SB on the button?


PS. Indeed, my calculations do include card-elimination effects. It was just wrongly represented in the text. Have corrected that, thanks for pointing out.

[Jerrod Ankenman]

[ QUOTE ]
My intuition must be failing me. I find it difficult to believe that for the full game the equity of the SB on the button can drop to negative values. A hand-waving argument for this is that when the SB would follow a (non-optimal) strategy that for each hand starts with limping, the result is that the BB and SB have contributed symmetrically to the pot, and that the BB is first to make a move. Effectively, the SB would have position on each betting round (including the pre-flop betting). Could this ever turn into a -EV proposition for the SB?

[/ QUOTE ]

Well, this game isn't symmetrical, right? If the big blind checks after the small blind limps, the small blind doesn't get an action. If he did, I would cheerfully concede that the game must be positive for the SB. Instead, however, the BB has the option of raising or checking and if he checks, there's a flop immediately. This is a pretty big advantage.

Let's talk about the small blind's equity in the jof game at a stack size of eight. We know that his equity in the game is around zero, because we've calculated it. At 8, he jams with like around 65% of hands (i dunno what the exact number is, it's not that important).

We know that:

EV(game) = EV(jamming hands) + (-.5)(% fold) = 0

right? He loses .5 units on the hands when he folds (35%), or .175 units. So the EV of jamming with all those jamming hands must be about .175.

Now suppose that the BB decides to play this strategy after the button limps. He's going to pretend that he's the small blind in a jam-or-fold game, and jam all the 65% of hands that the small blind would jam. But now instead of folding the other 35%, he'll just check and take a flop. So to the .175 he gains from jamming his good hands, he adds some fraction of the pot. As long as his total equity from the asymmetrical game where he has his bottom 35% of hands and the SB has his whole set of hands is at least .5 units (out of the two unit pot), he has positive equity playing this game.

A quick pokerstove of some 35% crosssection that I just picked out of the air shows him having about 40% allin equity, which is a good 15% higher than he needs. Maybe he loses that much postflop, maybe he doesn't. But on the other hand, he can additionally jam with fewer hands preflop, especially the marginal ones, in order to strengthen his postflop distribution, etc.

Anyhow, I think the point is that with stacks of eight units, position on the later streets is worth hardly anything at all. Hence the dude on the button can have negative equity.

[JocK]

OK, I think I have to refine my argument.

(Yes, stubborn enough to continue to be pretty much convinced that being on the button heads-up constitutes an advantage for any stack size... [img]/images/graemlins/wink.gif[/img])

The EV of the SB in a heads-up no-limit game is a function of the stacksize S, and can be written: EV = f(S/BB)*BB.

Now, suppose the SB follows a (definitely non-optimal) strategy of always making a minraise. The BB now faces the situation the SB was facing: he needs to double his stake in the pot in order to stay in the game. The only difference is that the blinds are now twice as high. So, the EV for the BB in response to SB's inferior startegy is: EV = 2*BB*f(S/(2*BB)). In this zero-sum game, the corresponding EV for the SB is then: EV = -2*BB*f(S/(2*BB)).

As the 'always minraise startegy' is sub-optimal, it follows that:

-2*BB*f(S/(2*BB)) < BB*f(S/BB)

or:

-2f(x) < f(2x)

If for a certain SB strategy, a violation of this inequality results, this particular strategy is proven to be suboptimal.

So, what about SB's jam-or-fold strategy? For this strategy, the function f(x) can be derived (see yellow curve in the plot posted in an earlier message in this thread), and it can easily be seen that for x in excess of 6, the above inequality gets violated.

Does this prove that for S > 6BB jam-or-fold is suboptimal?

Not really. Strictly seaking, the above only proves that jam-or-fold is suboptimal for S > 12BB. However, one may strengthen this statement a bit: if jam-or-fold is near-optimal for S around S = 6BB, then for S > 12BB jam-or-fold on the SB is strongly suboptimal, as it fares badly against the very poor strategy of always min-raising on the SB.

If jam-or-fold is strongly suboptimal for S > 12BB, it is unlikely near-optimal for S up to 10 or 11 BB.

Or not...? [img]/images/graemlins/confused.gif[/img]

[Jerrod Ankenman]

[ QUOTE ]
OK, I think I have to refine my argument.

(Yes, stubborn enough to continue to be pretty much convinced that being on the button heads-up constitutes an advantage for any stack size... [img]/images/graemlins/wink.gif[/img])

The EV of the SB in a heads-up no-limit game is a function of the stacksize S, and can be written: EV = f(S/BB)*BB.

Now, suppose the SB follows a (definitely non-optimal) strategy of always making a minraise. The BB now faces the situation the SB was facing: he needs to double his stake in the pot in order to stay in the game. The only difference is that the blinds are now twice as high.

[/ QUOTE ]

This is not true. There is a big, big difference between "if i call, the other guy can raise" and "if i call, we see a flop."

[ QUOTE ]
So, the EV for the BB in response to SB's inferior startegy is: EV = 2*BB*f(S/(2*BB)). In this zero-sum game, the corresponding EV for the SB is then: EV = -2*BB*f(S/(2*BB)).

[/ QUOTE ]

So this BB equity is too low, because it assumes that if the dude calls, the other guy can raise. But that's not true.

[ QUOTE ]
<snip> However, one may strengthen this statement a bit: if jam-or-fold is near-optimal for S around S = 6BB, then for S > 12BB jam-or-fold on the SB is strongly suboptimal, as it fares badly against the very poor strategy of always min-raising on the SB.

[/ QUOTE ]

I'm not sure this statement is true either. The optimal strategy doesn't have to fare well against bad strategies necessarily. It sometimes will. But that's not an inherent property. Consider the optimal strategy in Roshambo, which doesn't fare well against the lame strategy of scissors 100%.

[JocK]

Thanks Jerrod, you're right. In the above description I erroneously extended the functional expression for the EV in the jam-or-fold game to the full game.

However, when limited to situations in which the strategic choices are limited to jam-or-fold, the same argument can still be used, and leads to the conclusion (I'll try to be more careful in formulating this statement...!):

1) in a situation where the opponent could raise a call, a jam-or-fold strategy is suboptimal for a stack larger than 12 times the opponent's 2/3rd stake in the pot,

and/or:

2) in a situation where a call would end the pre-flop betting, a jam-or-fold strategy is suboptimal for a stack larger than 6 times the opponent's 2/3rd stake in the pot.

[Do you agree with this? I don't think your last remark (and the comparison with Roshambo) is relevant: I am not comparing two different strategies against a presumed suboptimal strategy of the opponent. Rather, I am comparing two strategies for the same player (the SB): 'jam-or-fold' versus 'always-minraise'.]

From our discussion so far, I presume you would conclude that 2) is likely correct, but that 1) does not hold true...?

I don't think I can argue against that conclusion.

So, if theory isn't going to settle the question up to which stack size jam-or-fold from the SB is near-optimal, one could look for empirical data. Indeed, it would be interesting to have statistics on heads-up EV for the SB as function of stack size. The big question is whether the EV observed in practice indeed drops negative for stacks deeper than 8 BB.

Anyone who has such statistics available?

[Jerrod Ankenman]

[ QUOTE ]
Thanks Jerrod, you're right. In the above description I erroneously extended the functional expression for the EV in the jam-or-fold game to the full game.

However, when limited to situations in which the strategic choices are limited to jam-or-fold, the same argument can still be used, and leads to the conclusion (I'll try to be more careful in formulating this statement...!):

1) in a situation where the opponent could raise a call, a jam-or-fold strategy is suboptimal for a stack larger than 12 times the opponent's 2/3rd stake in the pot,

and/or:

2) in a situation where a call would end the pre-flop betting, a jam-or-fold strategy is suboptimal for a stack larger than 6 times the opponent's 2/3rd stake in the pot.

[Do you agree with this? I don't think your last remark (and the comparison with Roshambo) is relevant: I am not comparing two different strategies against a presumed suboptimal strategy of the opponent. Rather, I am comparing two strategies for the same player (the SB): 'jam-or-fold' versus 'always-minraise'.]

From our discussion so far, I presume you would conclude that 2) is likely correct, but that 1) does not hold true...?

I don't think I can argue against that conclusion.

So, if theory isn't going to settle the question up to which stack size jam-or-fold from the SB is near-optimal, one could look for empirical data. Indeed, it would be interesting to have statistics on heads-up EV for the SB as function of stack size. The big question is whether the EV observed in practice indeed drops negative for stacks deeper than 8 BB.

Anyone who has such statistics available?

[/ QUOTE ]

Well let's be clear; I don't know what *OPTIMAL* is, and so it's entirely possible that at like 10 or 11, optimal is not jam-or-fold. However, my claim is that the difference in nemesis equity between the best jam-or-fold strategy and the optimal strategy (which might include limping or raising to a smaller amount and postflop play) is small enough that we don't care that much, particularly because the optimal strategy is basically intractable while the jam-or-fold problem is tractable.

My Roshambo remark was just to clarify about that you can't make comparative goodness comparisons between strategies with an eye toward claiming they are or are not close to optimal. It wasn't a very good example. It seems that the rest of my post was better. [img]/images/graemlins/smile.gif[/img]

Anyway, the question of up to what stack size jof is near-optimal basically depends on what you want to call near-optimal. I mean, it seems in your original post you used "same side of zero" as a qualitative measure of near-optimal, which is probably okay for the actual case. But I mean, if the first stack where you do something besides JoF is 8.5, then at 9, JoF is still going to be "near-optimal" by my definition. The structure of the problem just isn't such that there will be big cliffs of equity where a strategy with limping is a big quantitative leap forward. This happens in other games, but I'm pretty sure it doesn't happen here - JoF is very smooth and the second guy still has the option of putting his money in preflop if he wants.

[SilentNoise]

omg