3xBB + 1xBB per limper

[_TKO_]

The 3/1 rule seems to be the generally accepted preflop raising strategy in NL games, and I've been wondering why that is.

I worked around some numbers in Excel, and I noted that the 3/1 rule has the following properties:
<font class="small">Code:</font><hr /><pre>3BB +1BB
Limpers Pot Raise Pot+Raise CC odds x Pot % of Stack
0 3 6 9 1.50:1 2 3.00%
1 5 8 13 1.63:1 1 1/2 4.00%
2 7 10 17 1.70:1 1 1/2 5.00%
3 9 12 21 1.75:1 1 1/4 6.00%
4 11 14 25 1.79:1 1 1/4 7.00%
5 13 16 29 1.81:1 1 1/4 8.00%
6 15 18 33 1.83:1 1 1/4 9.00%
7 17 20 37 1.85:1 1 1/4 10.00% </pre><hr />

So, you are always betting between 1P and 2P, thus offering odds of less than 2:1 to anyone who wants to coldcall your raise. The interesting part is that after 2 limpers, any coldcaller slips into the range of the 5/10 rule. A solid player can still afford to call your raise (with a full 100BB stack) with speculative hands. It appears that the rule is centered around isolating two or more limpers. But, where else does this rule come from?

In pot-limit holdem, the maximum bet after any number of limpers is very similar to the 3/1 rule. Actually, it is more of a 2.5/1 rule:
<font class="small">Code:</font><hr /><pre>Pot betting (aka 2.5xBB + 1xBB)
Limpers Pot Raise Pot+Raise CC odds x Pot % of Stack
0 3 5 8 1.60 1 3/4 2.50%
1 5 7 12 1.71 1 1/2 3.50%
2 7 9 16 1.78 1 1/4 4.50%
3 9 11 20 1.82 1 1/4 5.50%
4 11 13 24 1.85 1 1/4 6.50%
5 13 15 28 1.87 1 1/4 7.50%
6 15 17 32 1.88 1 1/4 8.50%
7 17 19 36 1.89 1 9.50%</pre><hr />

But why did the 3/1 come from this? 2.5/1 could also move down to being a 2/1 rule:
<font class="small">Code:</font><hr /><pre>2BB +1BB
Limpers Pot Raise Pot+Raise CC odds x Pot % of Stack
0 3 4 7 1.75 1 1/4 2.00%
1 5 6 11 1.83 1 1/4 3.00%
2 7 8 15 1.88 1 1/4 4.00%
3 9 10 19 1.90 1 5.00%
4 11 12 23 1.92 1 6.00%
5 13 14 27 1.93 1 7.00%
6 15 16 31 1.94 1 8.00%
7 17 18 35 1.94 1 9.00%</pre><hr />

Of course, the 2/1 rule only effectively isolates against 3 limpers, and 3 limpers makes for at least a 4-way pot, which is usually the borderline between shorthanded and multiway. Ok, so if we really want to isolate limpers, why not adopt a 4/1 rule?
<font class="small">Code:</font><hr /><pre>4BB +1BB
Limpers Pot Raise Pot+Raise CC odds x Pot % of Stack
0 3 8 11 1.38 2 3/4 4.00%
1 5 10 15 1.50 2 5.00%
2 7 12 19 1.58 1 3/4 6.00%
3 9 14 23 1.64 1 1/2 7.00%
4 11 16 27 1.69 1 1/2 8.00%
5 13 18 31 1.72 1 1/2 9.00%
6 15 20 35 1.75 1 1/4 10.00%
7 17 22 39 1.77 1 1/4 11.00%</pre><hr />

Following the 4/1 rule, we would be mostly stabbing the blinds, but now we are risking a larger portion of our own stack to do so. Furthermore, we limit the amount of value we gain from our raises, since anyone calling behind us would want to do so with a stacking-type hand.

But why always +1BB per limper? Why not less? Surely, we'll get more value this way. If we were only to add half a BB per limper, we would often be offering better than 1:1 on a call, which is usually an incentive to play almost anything, especially in a larger pot. In order to avoid offering better than even-money with a +0.5BB/limper strategy, our base bet would need to be at least 5BB:
<font class="small">Code:</font><hr /><pre>5BB +0.5BB
Limpers Pot Raise Pot+Raise CC odds x Pot % of Stack
0 3 10 13 1.30 3 1/4 5.00%
1 5 11 16 1.45 2 1/4 5.50%
2 7 12 19 1.58 1 3/4 6.00%
3 9 13 22 1.69 1 1/2 6.50%
4 11 14 25 1.79 1 1/4 7.00%
5 13 15 28 1.87 1 1/4 7.50%
6 15 16 31 1.94 1 8.00%
7 17 17 34 2.00 1 8.50%</pre><hr />

With a large base bet and small incremental bet, you will be putting a large amount of money into small shorthanded pots, and a small amount of money into large multiway pots. Contrast this to having a small base bet with a large incremental bet:
<font class="small">Code:</font><hr /><pre>2BB +2BB
Limpers Pot Raise Pot+Raise CC odds x Pot % of Stack
0 3 4 7 1.75 1 1/4 2.00%
1 5 8 13 1.63 1 1/2 4.00%
2 7 12 19 1.58 1 3/4 6.00%
3 9 16 25 1.56 1 3/4 8.00%
4 11 20 31 1.55 1 3/4 10.00%
5 13 24 37 1.54 1 3/4 12.00%
6 15 28 43 1.54 1 3/4 14.00%
7 17 32 49 1.53 2 16.00%</pre><hr />

Following this strategy, you will tend to shut down pots as the number of limpers increases. Now, you are putting a large amount of money into more multiway pots.

So, back to my original question: why the 3/1 rule?

[Moose747]

I prefer 3x +0.5x per limper. But then I also generally try to raise as a function of pot size and not big blind size, so it ends up being 2xpot + 1/3xpot per limper.

This is a situation where I don't really trust common knowledge. I'm waiting to see some really good analysis/arguments.

[_TKO_]

[ QUOTE ]
I prefer 3x +0.5x per limper. But then I also generally try to raise as a function of pot size and not big blind size, so it ends up being 2xpot + 1/3xpot per limper.

[/ QUOTE ]

3/0.5:
<font class="small">Code:</font><hr /><pre>3BB +0.5BB
Limpers Pot Raise Pot+Raise CC odds x Pot % of stack
0 3 6 9 1.50:1 2 3.0%
1 5 7 12 1.71:1 1 1/2 3.5%
2 7 8 15 1.88:1 1 4.0%
3 9 9 18 2.00:1 1 4.5%
4 11 10 21 2.10:1 1 5.0%
5 13 11 24 2.18:1 3/4 5.5%
6 15 12 27 2.25:1 3/4 6.0%
7 17 13 30 2.31:1 3/4 6.5%</pre><hr />

Note that many people should be coldcalling your raises with this strategy, especially with many limpers.

The comment on raising according to the size of the pot is interesting. One of the interesting things I noted about my results was that the pot odds you end up offering are different compared to how limpers have entered the pot in front of you. I would think that as the number of limpers increases, so does the importance of protecting your equity in the pot.

[Moose747]

TKO, I'm not sure I follow entirely what you're doing in your charts (for one thing the tabs are off on my screen) but I think you're on the right track here.

If I understand correctly, you're looking at the pot odds callers are getting given the number of limpers and size of the raise. One concern I have is that this number is different for players who haven't acted yet, for the SB and for the BB/limpers.

[Moose747]

Assuming that we want to offer callers the same pot odds regardless of how many limpers have entered the pot, I find that the following formulas achieve this for each of the three groups I mentioned in the last post.

3xBB/2xBB per limper preserves pot odds for players yet to act

3/1.25 preserves for the SB

3/0.8 preserves for the BB/limpers

[_TKO_]

[ QUOTE ]
TKO, I'm not sure I follow entirely what you're doing in your charts (for one thing the tabs are off on my screen) but I think you're on the right track here.

[/ QUOTE ]

I've posted the charts as Code, so the tabs shouldn't be off. It's no matter, though.

Anyway, the charts are based on a 1/2 blind structure. Using the rule stated, I've calculated what the size of your raise would be given the number of players that have limped in front of you. I then added the immediate odds for a palyer to coldcall your raise (assuming no one else has coldcalled in front of her), the ratio of your total bet to the size of the pot, and the percentage of a full stack that your total bet represents.

There are many other considerations to be taken, but the charts I compiled were a start on comparing the 3/1 rule to possible alternatives.

[_TKO_]

[ QUOTE ]
Assuming that we want to offer callers the same pot odds regardless of how many limpers have entered the pot, I find that the following formulas achieve this for each of the three groups I mentioned in the last post.

3xBB/2xBB per limper preserves pot odds for players yet to act

3/1.25 preserves for the SB

3/0.8 preserves for the BB/limpers

[/ QUOTE ]

Note that this last line is very close to the 3/1 rule. I did consider the odds for the BB to call, but decided not to include the numbers in my charts since a) play from the blinds is different from when a player has no money invested; and b) you have position on the players in the blinds, so you are slightly less concerned about protecting your hand against them. I also thought about how this strategy changes when you yourself are in the blinds, but decided not to include it because play from the blinds has its own considerations.

"we want to offer callers the same pot odds regardless of how many limpers have entered the pot"

This particular statement intrigues me. Is this our ultimate goal? If so, then what actual pot odds do we want to offer players preflop?

[RagnarPirate]

It seems to make no sense to me, but recently more good players are making the minimum raise preflop in both unopened and hands with a limper or even 2. It doesn't seem to make sense to me, but I'm definitely seeing it. Is this to reopen betting purely for information from early limpers? I've seen it from a couple of online pro mtt players, so I assume there is a reason that I'm missing.

[Moose747]

So here's a question: how does your raising range change once players start limping in front of you? I know mine changes enormously, particularly when I'm in late position. Hands I would normally raise (pocket pairs, suited big cards, occasionally suited connectors) become calling hands and other hands I might open (A9, KQ) become folds.

The only hands I feel compelled to raise in late position after one or more limps are high pocket pairs and unsuited high cards. Given that my raising range is so much tighter, it makes sense that I would want to give opponents better pot odds to call, since they are further behind my range.

[RagnarPirate]

The reason that you have a tighter range is that early position standards should be higher than late position standards. They are claiming that they have a card in that narrower range. So, if you're going to play you want to have a card in the range that you would play from THEIR position or better. I think that I am going to play tighter AND more aggressively to the limper. It is either a fold or a bigger raise to the limper.