Rebuttal to the magic 2/1 rule. (Long)

[AtticusFinch]

I've been told many times that you should always call a push that gives you 2/1 odds for <= 1/3 of your stack with any two cards. I've always been dubious of this rule, and after reading the blind-stealing article, I've put together a rebuttal. In particular, note wmajik's absolutely critical point that just because a play is +$EV across an entire range of hands does not necessarily mean that it is $EV for each individual hand in that range.

Consider the following scenario:
You are five-handed with blinds at 100/200. All stacks are even except the short stack. The short stack pushes, offering exactly 2/1 pot odds. You are on the big blind, and it's folded to you. Calling costs exactly 1/3 of your remaining stack after posting the blind. I will consider two cases: first, when the short stack is the small blind, second, when he is anywhere else. I will adjust the stack sizes to make the math easier, but it doesn't matter -- only the ratios of sizes to each other and the blinds are important. In both cases, the short stack could have pushed with any two cards.

1) Short stack on small blind

Beginning stacks:
1100 (You), 400, 1100, 1100, 1100

Stacks if you fold:
900, 600, 1100, 1100, 1100
Your $EV: .1924

Stacks if you call and win:
1500, 0, 1100, 1100, 1100
Your $EV: .287

Stacks if you call and lose:
700, 800, 1100, 1100, 1100
Your $EV: .1565

To calculate the odds you need to have to win the hand for this to be a +$EV move, use the following formula:

Fold$EV < WinProb*Win$EV + (1-WinProb)*Lose$EV

Which simplifies to:

WinProb > (Fold$EV - Lose$EV)/(Win$EV - Lose$EV)

In this particular case, that gives:

WinProb > (.1924 - .1565)/(.287-.1565) = .0359/.1305 = 27.5%

Note that for a random hand, your odds are exactly 50/50 here, so this would appear to be a highly profitable call. However, consider the odds of winning against a random hand for the following hands:

72o: 34.9%
42o: 32.96%
23o: 32.2%

Still profitable, but not nearly so much as the full range. Still, in this scenario, it looks like a profitable move to call with any two cards, although calling with real trash is a very high-variance move.

2) Now consider the case where the short stack is not in the small blind.

Beginning stacks:
1700 (you), 1700, 700, 1700, 1700
Fold:
1500, 1600, 1000, 1700, 1700
$EV = .2016

Call and lose:
1000, 1600, 1500, 1700, 1700
$EV=.1449

Call and win:
2500, 1600, 0, 1700, 1700
$Ev = .2981

WinProb > (Fold$EV - Lose$EV) / (Win$EV - Lose$EV)
(.2016 - .1449)/(.2981 - .1449) = .0567/.1532 = 37%

37%! That's a much higher win percentage needed to break even. Now the same hands listed above are all big losers. Also consider:

82o: 36.75
73o: 36.51

Close, but no cigar.

After running through poor hands one at a time, I've determined that in this situation you should muck the following hands:

Offsuit:
82 down to 32
73 down to 43

Suited:
32, 42

Note that few players would push with any two, here, as their folding equity is so low. If dealt 72o, most will muck and hope to do better on the blind. Thus there are likely to be more hands that should be folded than just those listed above. This is only the beginning, not the end of the analysis. At a later date, perhaps together we can develop more sensible and flexible calling standards for this and other scenarios. However, as only one counterexample is needed to disprove a mathematical theory, consider the magic 2/1 rule debunked.

I welcome all comments and/or corrections.

[sofere]

Good Post Atticus.

I would like to add one more situation where it may be better $EV to fold than call. If you are big stack on the bubble, you would have far more fold equity with shorty in the game.

[Mr_J]

"If you are big stack on the bubble, you would have far more fold equity with shorty in the game."

Yep. I'll happily keep shortstack (barely) alive unless a hand comes up that calls for me to eliminate him. If there is no shortstack, I'm happy to create one.

[Pokerscott]

I think you are saying

"you want a better hand to call with the larger % of your stack you are committing"

In both cases you are getting 2:1 but in the second case you have to call with a bigger part of your stack.

If that is your point I completely agree [img]/images/graemlins/cool.gif[/img]

pokerscott

[microbet]

I'd guess you were reading from this thread.

There are several posts in there that come to a similar conclusion (although I think most or all were talking about pushing instead of calling) that it was a mistake to assume that because a play is +$EV for a random hand, that it was +$EV for every particular hand.

It is a really good thread. I read through it once fairly quickly and will definately reread myself.

[spentrent]

Let's pretend we can see the hole cards: opponent pushes Qx|Kx|Ax. You have one card bigger than x. 2 to 1 rule applies, yeah, because you'll win 34% of the time?

Now let's pretend a gnome stole your cards and he'll only show them to you if you call. Opponent pushes and "2 to 1 rule conditions" exist. Do you call?

[AtticusFinch]

[ QUOTE ]
I think you are saying

"you want a better hand to call with the larger % of your stack you are committing"

In both cases you are getting 2:1 but in the second case you have to call with a bigger part of your stack.

[/ QUOTE ]

That's certaintly part of it. I'm also saying that 1/3 is not the magic cutoff for when you can call with any two.

I intended to have calling cost 1/3 of your stack in both cases, but it seems I messed up the small-blind case. The corrected numbers there are:

Start:
1100, 500, 1100, 1100, 1100

Fold:
900, 700, 1100, 1100, 1100
$EV: .1886

Call and win
1600, 0, 1100, 1100, 1100
$EV: .2945

Call and lose
600, 1000, 1100, 1100, 1100
$EV: .1349

(.1886 - .1349)/(.2945-.1349) = .0537/.1596 = 33.6

Now it costs exactly 1/3 of your stack to call, the same as with the other example, yet you still need less of a margin to be +$EV. Now you're right to muck 23o and 24o, but call with any other hand.

Also, as another poster suggested, the ICM does not take into account the value of extending the bubble, so the folding range may be even wider.

[ColdestCall]

No way I'm even playin if that friggin gnome is in the room....

[AtticusFinch]

If you can't see your two cards, then yes, call. Your odds are exactly 50/50. This will be hugely profitable in the long run.

But you have more information than that, because you CAN see your cards, and can improve your expectation using it. This is much like the "Monty Hall" problem we all had so much fun with in stats.

[11t]

Are the 1100, 400, 1100, 1100 before or after posting the blinds? Either way your numbers are off. I suspect that your original mistake threw off your calculations afterwards. Re-run the numbers and repost to see if it is correct.