final table balancing act

[Che]

Third hand at the final table of a Stars 10+1 rebuy.
Blinds maxed out at 40000/20000/2000a.
Avg. stack ~T540,000.

I'm in eighth with T165000, but I have very good reads on the chip leader and #3 so I think I can contend if I can get above T400,000 or so.

It's folded around to the CO (the short stack) who minraises to T80000, leaving himself with T18000. Button and SB both have T600,000+, but they both fold. Rats.

I have T7s in the BB and T120,000 after posting. I consider the following tourney concept (as explained by Greg in Guy McSucker's "Question for Fossilman" thread):

[ QUOTE ]
Contrarily, if your stack is getting so small that you are running out of play, i.e., you're approaching the point where it's all-in or fold as the only options, the less you should pass on small edges.

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A desperate short stack can push in with a wide range of hands when folded to in the CO so I estimate that I'm somewhere between a 4:1 dog (AA-JJ) and a 3-2 dog (unsuited overcards) with most of the distribution on the 3-2 end. There are a few hands that are outside this range (TT, 66-22), but most playable CO hands are in the stated range. The minraise rather than the push was a red flag that probably should have shifted me toward the top of the range, but he might do this to look stronger and I didn't want to overthink it. (BTW- no read on this player so I'm treating him as a random internet player)

I'm getting 3:1 implied odds (assuming all the money goes in, which it will since I'm betting the flop no matter what) so I'm +EV against most of his possible hands as I see it, but probably only slightly +EV against the entire range. As a short stack, I should call based on the concept quoted above.

However, there's also this to consider (from same post):

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The later it is in a tourney, the bigger edge you can give up on a single hand. This is because of the more general concept of the chips you win are worth less than the chips you lose once you're in or near the money.


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If I fold, I can continue to fold for 18 more hands before blinding out. One, two or maybe more opponents might bust in the meantime which would increase my payout by several hundred dollars. So applying this concept says fold.

The third factor (table image) isn't a factor since short stacks are pushed around regardless of image so how do I break the (apparent) tie?

If I play and lose, I'm in 9th which is the same place I'll be if I fold. But if I play and win, I'll be guaranteed 8th and I'll only be one double away from contention. I decided to play.

Results:
Flop: K [img]/images/graemlins/spade.gif[/img] 9 [img]/images/graemlins/spade.gif[/img] J [img]/images/graemlins/heart.gif[/img]
I have clubs, but it doesn't really matter. I was planning to bet the flop regardless in hopes that the short stack is a total moron. Unfortunately, even a moron wouldn't fold A [img]/images/graemlins/club.gif[/img] K [img]/images/graemlins/club.gif[/img] in this situation so I lose to his TPTK when my double gutshot draw misses the turn and river.

I went in a few hands later with KQs (getting 3:1 from the pot) and busted against AA. Out in ninth.

Final thoughts: In retrospect, I probably should have folded even though that would have been a FTOP error (3:1 for the call, only a 2:1 dog). I think concept 2 (extra value of final table chips) should have won out over concept 1 (short stacks play all +EV situations).

What do you think?

Thanks,
Che

[Che]

[ QUOTE ]
The minraise rather than the push was a red flag that probably should have shifted me toward the top of the range, but he might do this to look stronger and I didn't want to overthink it.

[/ QUOTE ]

Was this the real mistake? Given another day to think about it, it may have been.

[M.B.E.]

Che, let's assume the prize payout structure for the top 9 is 25/15/9.5/7/5.5/4.5/3.5/2.6/1.7. Now what's your equity if you (a) fold preflop, (b) call/raise preflop and win, (c) call/raise preflop and lose.

Okay, (c) is easy: you'll be in 9th place with hardly any chips, so call your equity 2% of the prize pool.

Let's try (b). If you play the hand and win, you'll have 301,000 chips, right? Well, it would help to know the total amount of chips in play. Roughly I'd estimate your equity at about 7% of the prize pool.

Now (a). If you fold, you'll have 123,000 chips, which will be last place. That should be good for at least 7th-place money if you play conservatively, maybe a bit higher on average if you don't, so let's call your equity in this case 4% of the prize pool (halfway between sixth and seventh).

You figured you were between a 4:1 underdog and 3:2 underdog. Let's say your chance of winning the hand if you play is 35%. So calculate:

(0.35 x 7%) + (0.65 x 2%) = 3.75%

That's a bit less than your 4% equity from folding. So I think the answer is to fold. But it's close. If you thought that if you won the hand your tournament equity would be 10% rather than 7%, you should call.

[M.B.E.]

By the way, the calculation I just did is the theoretically correct method, but would be very difficult to apply in real time when making a decision. Anyone have an idea for a shortcut?

[thekid]

Che,
All calculations aside, I think the idea is to give yourself the best chance to win the tournament. You have terrible cards. If you call and lose you are probably going to finish in 9th, and your opponent's hand doesn't have to be a monster for you to be behind. If you fold you might catch a monster or go on a rush in subseguent hands, so why rush here.

[Che]

[ QUOTE ]
let's assume the prize payout structure for the top 9 is 25/15/9.5/7/5.5/4.5/3.5/2.6/1.7.

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That's correct. Sorry it wasn't in my original post.

Thank you very much for the response, MBE. I'm a little confused about how the equity for scenarios a and b are determined, though. Or maybe I'm just surprised by the results.

BTW: ~T5,400,000 in play


Scenario A- Fold: T123,000. 4% equity. Reasoning: We assume an average finish between 6th and 7th. Sounds reasonable.

Scenario B- Play and win: T301,000. 7% equity. Reasoning: It wasn't explicit in your post, but I infer from scenario A's explanation that you estimate this chip count to be worth 4th place money on average.

I know extra chips are worth less than the ones you already have, but does an almost 2.5x increase really translate to just a 1.75x increase in equity? I guess I have been seriously overestimating the value of doubling up when you're already at the final table.

Maybe I just haven't spent enough time considering the implications of the top-heavy tournament payout structure, but if there's another factor I'm missing, please let me know if you have time.

Thanks,
Che

[M.B.E.]

[ QUOTE ]
Scenario B- Play and win: T301,000. 7% equity. Reasoning: It wasn't explicit in your post, but I infer from scenario A's explanation that you estimate this chip count to be worth 4th place money on average.

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Not exactly. "4th place money on average" would actually be greater; e.g. if you finished first half the time and seventh half the time, your "average" would be fourth, but your average prize would be (25%+3.5%)/2=14.5%, which is just short of second.

Basically I came up with the 7% figure by saying, okay there will be eight players left; if you add up the prizes for the top eight it comes to 72.6%, then divide by 8 to get 9.1%, but 301,000 is less than average so discount that down to 7%. Actually maybe 6% would be a better figure, considering that 301,000 would be less than half of average.

[ QUOTE ]
I know extra chips are worth less than the ones you already have, but does an almost 2.5x increase really translate to just a 1.75x increase in equity? I guess I have been seriously overestimating the value of doubling up when you're already at the final table.

[/ QUOTE ]
It's quirky because of the payout structure. Let's say that everything in your example was the same except that the chip leader had a huge portion of the chips, and the players in second through seventh each had just a little more chips than you in eighth. Well if that were the case, then doubling up would be a huge advantage to you, since it would give you a great shot at the 15% second place prize.

But given the scenario you were actually in, even if you doubled up you'd be far out of second -- you'd have to double up a second time even to get close.

[Che]

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Quote:
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I know extra chips are worth less than the ones you already have, but does an almost 2.5x increase really translate to just a 1.75x increase in equity? I guess I have been seriously overestimating the value of doubling up when you're already at the final table.


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It's quirky because of the payout structure.

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It is quirky, but maybe not as much as we thought.

We’ve been making decisions based on the total equity of a given chip count. From a wannabe economist’s viewpoint, most of the total equity is relevant to the decision-making process – but not all.

Using the payout structure at hand (25/15/9.5/7/5.5/4.5/3.5/2.6/1.7), 74.3% of the prize pool is still in play. But not really. 15.3% is actually not in play since each of the 9 players already have 1.7% in their pocket. Only 59% is still available to be won.

We wouldn’t consider the amount of the buyin when making a decision at the final table because the buyin is sunk cost. Similarly, we shouldn’t consider the 1.7% we have already won in our decision – it’s “sunk gain” so to speak.

Fortunately, reducing each total equity value by a constant leaves us with an identical result when deciding whether to play or fold.

Total Equity Calculation:

Fold: 4% Equity
Play: (0.35 x 7%) + (0.65 x 2%) = 3.75%
Differential of .25% in favor of folding.

“Marginal Equity” Calculation (Marginal Equity = Total Equity – “Sunk Gain”):

Fold: 4%-1.7%=2.3% Marginal Equity
Play: [0.35 x (7% - 1.7%)] + [0.65 x (2% - 1.7%)] = 2.05% Marginal Equity
Again, differential of .25% in favor of folding.

If I’m not mistaken, the diffential for a total equity calculation and the differential for a marginal equity calculation will always be identical since reducing everything by a constant has no impact on the relative results.

So why even bring it up?

Because using this approach (marginal equity, as I’m calling it) does change the result when you compare increasing your chip count by a multiple of X to the corresponding increase in equity.

For example, in terms of total equity, we calculated that increasing my chip stack from ~T120,000 to ~T300,000 (a multiple of 2.5) only increased my total equity from 4% to 7% (a multiple of 1.75). This was terrifying in my mind. It made me think that I would need a HUGE chip EV advantage to play any hand as a small stack due to the quickly diminishing returns of each additional chip won.

However, in terms of marginal equity, our equity still increases at a slower rate than our chip count, but the difference is much less drastic. The total equity increase from 4% to 7% is actually an increase from 2.3% to 5.3% in marginal equity (a multiple of 2.3).

Intuitively, this makes sense to me while the 1.75 multiple did not. Additional chips are still worth less than the chips already possessed, but the difference in value is not that large – even at the final table.

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Anyone have an idea for a shortcut?


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My initial conclusion (I reserve the right to change my mind again tomorrow [img]/images/graemlins/smirk.gif[/img]) is that short stacks at the final table should probably play any +EV situation just as they would a long way from the money. Perhaps this is the shortcut – just play a short stack the way you would normally play it. Again, this makes sense (at least to me) intuitively because a very short 9th place stack at the final table is still a long way from the real money – places 1, 2 and 3.

BTW- I’m wondering how this applies to the chip leader(s), especially those with substantial leads over the rest of the field. Guess that’ll have to wait for another day – my brain is tired.

As always, please feel free to punch holes in anything I post. All criticism is welcome – that’s how I learn so don’t worry about hurting my feelings if I’ve written something totally ridiculous (here or in any other post).

Finally, my apologies to any real economists who read this. (Curmudgeon – are you out there?) I hope I haven’t butchered these concepts too badly. I’m not an economist and never will be, but I have had just enough MBA classes to be dangerous. [img]/images/graemlins/wink.gif[/img]

Sincerely,
Che