The 2+2 Forums Older Archives: Rebuttal to the magic 2/1 rule. (Long)

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.

Edited by AtticusFinch (02/10/05 12:43 PM)

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