Test your intuition (three handed SNG play)

[Bozeman]

Suppose you have made it to threehanded in a NL SNG (prizes 50%,30%,20%). Suppose the stacks are 2K, 3K, and 5K (before putting in blinds) and the blinds are 250/500. You pay the bb. After the button folds, the small blind goes allin.

1) Which stack sizes for you (bb) and him (sb) should you be tightest with? Loosest?

2) How tight should you be?

3) Should you be tighter when you out chip him or when he out chips you?

4) Obviously you need to be tighter when neither you nor the SB is the 2K stack than when one of you is since 3K chips are at stake instead of 2K (you are getting 7:5 odds instead of 5:3). Do relative tightnesses (compared to proper ring game play, for example) rank differently than absolute tightnesses?

5) Suppose you are twice as good as the other two players, how do things change (vs. all three equal)? (I have a mathematical definition of "twice as good", but you are welcome to use any definition you like, as long as you state it.)

Craig

[M.B.E.]

Good problem.

<font color="purple">1) Which stack sizes for you (bb) and him (sb) should you be tightest with? Loosest?</font>

You would expect your opponent to be tightest when he has the 3K stack -- he won't want to risk busting out 3rd when he might get 2nd just by waiting. Thus when your opponent moves in when he had the 3K stack, he's probably got a good hand so you should be tight in your calling requirements (whether your stack is 2K or 5K).

Be loosest when your opponent has 2K and you have 5K. Now your opponent is forced to play ultra-aggressively for fear of being blinded off, so he doesn't need much of a hand. Plus you're getting 5:3 pot odds and you won't bust out of the tourney if you lose.

<font color="purple">2) How tight should you be?</font>

Intuitively I'd say that if your opponent has the 2K stack and yours is 5K, you can call with 2/3 of your hands (playing loosest). At your tightest, you'd probably call with 1/5 of your hands.

<font color="purple">3) Should you be tighter when you out chip him or when he out chips you?</font>

When he outchips you.

I won't attempt question four since I don't quite understand it.

<font color="purple">5) Suppose you are twice as good as the other two players, how do things change (vs. all three equal)? (I have a mathematical definition of "twice as good", but you are welcome to use any definition you like, as long as you state it.)</font>

Conventional wisdom is you call the all-in less (i.e. play tighter) if you are better. However, with blinds this high relative to stack size that should not be a big factor. In fact it would only be a factor at all if there's the possibility of getting action past the flop on future hands, which seems unlikely. So I'll say that it makes virtually no difference that you are twice as good as the other players.

[M.B.E.]

Thought about this some more. Given that the prize structure is 50%-30%-20%, I now realize that when you're down to the last three, it's usually worth taking a considerable risk of busting out if that will improve your chance of winning. Notionally since you're guaranteed third place money, you can put the 20% in your pocket; now you can see that first place pays three times as much as second. So you would rarely use a strategy designed to "squeak into second place". You want to shoot for first, which means playing pretty loose.

Algebraically, you want to call whenever

p &gt; (3c + d) / (3a + b)

where:
p is the probability of winning the hand if you call
a is, if you call, your probability of finishing first if you win the hand minus your probability of finishing first if you lose it
b is, if you call, your probability of finishing second if you win the hand minus your probability of finishing second if you lose it
c is your probability of finishing first if you fold, minus your probability of finishing first if you call and lose the hand
d is your probability of finishing second if you fold, minus your probability of finishing second if you call and lose the hand

Notice that a and c are both always positive, while b and d could potentially be negative. (b and d might be negative if you have a big chip lead.)

If your opponent has you covered, then a, b, c, and d can be described more simply: in that case, a and b are your probabilities of finishing first and second, respectively, if you call and win the hand; while c and d are your probabilities of finishing first and second, respectively, if you fold. Then you can see that 3c+d is your equity in the tourney if you fold while 3a+b is your equity in the tourney if you call and win (ignoring 3rd place money which is already in your pocket).

[M.B.E.]

To make things simpler, I ignored b and d in the formula, so that the formula reduces to p&gt;c/a. I'm not sure how much of a difference that will make but it shouldn't be too much, and b and d as I defined them take too long to calculate.

Anyway, the result I got was that c/a = .417 if you have 5K and the SB has 3K, or you have 3K and the BB has 5K. In all other cases, c/a = .375. That's not a big difference. However, it doesn't quite answer the question, because p depends not only on the two cards you're holding but also on the raising standards of your opponent.

[Bozeman]

Thanks MBE for posting your good answers.

First, a few general comments:

Whether or not you call depends on four things: the range of hands your opponent might have (quite variable), the stacks, your edge, and the hand you hold. (Is there any important factor I am missing?)

While there are many ranges of hands your opponent can have, the rest of the problem can be broken down into one simple measure: what win percentage do you need against his range of hands to have +EV. It is this percentage that I will refer to as tightness for this analysis.

Since your opp cannot fold (being allin), you need to play tighter than him, except to the extent that dead money increases your pot odds.

For a ring game (that is chips=money), the analysis is simple: if one of you is the 2K stack, you are getting 25:15=5:3 odds, so you should call if you have better than 37.5% chance of winning. If 3K chips are at stake, you need 35:25=7:5 odds, or ~41.7% chance of winning.

For a tournament, real money EV ($EV or EV) is not equal to chip EV (CEV). Because there is a value to survival, you always need a better hand than CEV would indicate.

The difference for the threehanded SNG is fairly small because you are effectively competing for first (+3 units) and second (+1 unit). This quite top heavy prize structure corresponds to small changes. One would get a larger difference between $EV and CEV (and therefore between proper tourney play and proper ring play) 4 handed (on the bubble in general), with a flatter structure at the top (like in most multitable tourneys), or especially near the bubble in a supersatellite.

1) Which stack sizes for you (bb) and him (sb) should you be tightest with? Loosest?

If we measure tightness in win prob.you need, from loosest to tightest they are:
<font class="small">Code:</font><hr /><pre> You(BB) Him(SB) Win Prob. (%)
3 2 38.3
2 3 39.2
5 2 41.5
2 5 42.9
5 3 47.4
3 5 52.1</pre><hr />

2) How tight should you be?

Between ~1% and ~10% tighter than CEV dictates.

3) Should you be tighter when you out chip him or when he out chips you?

When he outchips you, though this difference is small compared to the changes you should make based on who is sitting out. The person who has folded essentially always benefits when you call.

4) Obviously you need to be tighter when neither you nor the SB is the 2K stack than when one of you is since 3K chips are at stake instead of 2K (you are getting 7:5 odds instead of 5:3). Do relative tightnesses (compared to proper ring game play, for example) rank differently than absolute tightnesses?
If we look at D=winpercentagefor+$EV-winpercentagefor+CEV, we see:
<font class="small">Code:</font><hr /><pre> You(BB) Him(SB) D (%)
3 2 0.8
2 3 1.7
5 2 4.0
2 5 5.4
5 3 5.7
3 5 10.4</pre><hr />
The order remains the same.

5) Suppose you are twice as good as the other two players, how do things change (vs. all three equal)? (I have a mathematical definition of "twice as good", but you are welcome to use any definition you like, as long as you state it.)

I used twice as good because I think it is an approximate upper boundary for how much of an advantage you can have when the blinds become large. I defined twice as good as someone who does as well as someone would in an equal game with a stack twice as large as this good player's current stack.
For this situation, this means that a 5K stack for the good player is worth 42.3% of the prize money instead of the 38.6% that it is worth for an average player, a 3K stack is 37.8 vs. 32.8%, and a 2K stack is 33.5% vs. 28.6%.
This does have a significant effect on the absolute tightness (this player obviously needs to play tighter), and the realtive tightness of the various stack distributions:
<font class="small">Code:</font><hr /><pre> Good(BB) Him(SB) Win Prob. (%)
3 2 51.9
2 3 51.7
5 2 48.0
2 5 53.7
5 3 56.6
3 5 64.7</pre><hr />
Now you need to be very tight with a small or medium stack. You should be 6.5-13.6% tighter than the average player, and the biggest changes appear for the 32,23, and 35 cases.


Anything else to note or ask?

Craig

PS Do my posts stink? They get very few replies. I admittedly post mostly mathematical (boring?) stuff, since this is the stuff that can (usually) be clearly stated instead of "it depends".

[J.R.]

Your posts are excellent. I always read them, but the questions posed are unfortunately often out of my range. That just means you're really smart. Thanks for the insights.

[Che]

[ QUOTE ]
Do my posts stink? They get very few replies. I admittedly post mostly mathematical (boring?) stuff

[/ QUOTE ]

I love your posts, but I'm very mathematically oriented (and I still get a headache sometimes trying to duplicate your calculations).

I'd reply to more of them, but I'm very new to poker (been playing less than 4 months) so I often struggle when trying to make a coherent application of the math to the game. Otherwise, I'd probably respond to all of your posts since I really appreciate the info. Given a few more months of experience, I hope to be able to make a meaningful contribution on the math side. In the meantime, keep posting your boring stuff - I love it. /images/graemlins/wink.gif

[CrisBrown]

Hi Bozeman,

I enjoy your posts a lot, actually, even if I don't often reply to them. We have very different perspectives on the game and I enjoy that, because your perspective helps me to see things in a way I wouldn't otherwise have considered.

Cris

[DougBrennan]

I know that my only problem (and it's my problem, not yours) with your posts is trying to figure out how to apply the math concepts you have illuminated.
My poker brain still works along the lines of "Well, yeah, but does playing 11.2% tighter mean I should call with my AKo or not?" /images/graemlins/laugh.gif

I suspect the numbers tell you more of a story than they do some of the rest of us. I know when I look at baseball statistics I get a lot of info out of them that my friends don't e.g. "Yeah he hit a lot of homers, but he never walks, strikes out a lot, plays at Wrigley Field, he's not really very good." And my friends say "But he hit 34 homeruns, how can you say he isn't very good?"

So please, keep posting, we'll figure it out eventually.

[LetsRock]

depends....

[Bozeman]

"I suspect the numbers tell you more of a story than they do some of the rest of us."

I think this is very true. In addition, there is the annoying factor that I like to figure out what I can figure out (the lure of the puzzle), and am not always able to apply it to play at the table.

As for this question, a lot depends on what your opponent will go allin with. If he would go allin with any two cards, then 37.5% is almost all hands in holdem (probably all hands in Omaha), folding only those hands whose high card is below 9 and whose low card is below 4 except for 35s. For 38.3%, cut out 64,54,52s,62s,and 72s. The marginal hand for 52.1% is J7s. For the extreme case, 64.7%, you need to only play pairs through 77, Apaint suited, and AK. This shows why going allin every hand is an effective strategy for the underdog (skillwise) as long as the blinds are reasonably large. It also shows that it is very hard to be twice as good as the opposition (with large blinds) unless they are folding too many hands.

Using allenciox's table of matchups, if your opponent would go allin with the top 40% (all pairs, A, K, Q9-QJ, Q5s-Q8s, J8s-JTs, JTo, T9s) at 52.1% you need A4s (KJo, 44) to call. At 38.3%, the cutoff is about T6s. For 42.9%, the cutoff is around QTo (K5s). As you can see, a 5% difference in win percentage can encompass a lot of hands if you are in the mediocre hand part of the list. However, this means that if you make mistakes with mediocre hands, they are likely to be small ones.

Craig