Thursday, June 30, 2005

Another Quick Calculation

Matt, prompt and thoughtful commenter he is, prompted the following analysis.

Suppose you have a heads-up freezeout (actually, many heads-up freezeouts, operating in parallel universes, so that I can just say there are infinitely many people and eschew combinatorics in favor of nice friendly continuous algebraic manipulation.) In the first round the fraction of skilled players is s (and, so, 1-s unskilled players.) Skilled players are even matches for each other (as are unskilled players for each other,) but skilled players beat unskilled players with probability w.

Let s(n) be the fraction of skilled players in round n. (So, s(1) = s.)

Now, s(n+1) = s(n)^2 + 2*w*s*(1-s) [See asterisk at bottom if this isn't clear.]
Simplifying, s(n+1) = (1-2w)*s(n)^2 + 2w*s(n)

A little manipulation shows that s(n+1) > s(n) if and only if w > .5, which is exactly what we'd want, so I have some confidence the above formula is correct.

Matt suggested (something like) finding the smallest n such that s(n) > .8 if s(1) = .5 and w = 2/3.

Turns out, unless my rounding errors took me too far astray, s(5) is a little bigger than .8, so that would suggest an answer of 32 players, in Matt's original question.

I'm a little rusty with my recursive equations, even though that one's very simple. Anyone who want to analyze how fast s grows? I've already guessed that the decay of donkeys' equity is roughly exponential; these sorts of questions are the crux of what I'm trying to get to the bottom of with these questions.



*In round n, s(n)^2 gives the frequency of skilled-skilled matchups, each of which will send a skilled player on to the next round; 2*s(n)*(1 - s(n)) is the frequency of skilled-unskilled matchups, which send skilled players to the next round with probability w.

A Quick Dead Money Calculation

So I was just arguing about "Thoughts on Dead Money" with Tom, of Delino fame, and he said many silly things, but also some reasonable things, and I ended up devising the following model to convince him.

Suppose you're playing a heads-up tournament, winner-take-all, with 2^N players. Everyone has equal skill except one guy, who wins with probability p. Since it's winner-take-all, his equity is the probability he wins times the prize, which is p^N * 2^N, which is (2p)^N.

Now, if the player is an underdog, p < .5, and 2p is less than 1, so equity E = (2p)^N falls off very fast. And if the player is a total donkey, p is something like .37 or so, so E falls off even faster.

There are obvious objections, namely that the WSOP championship isn't a heads-up event, that it isn't winner-take-all, and that skill edges aren't that well-behaved, especially at full tables.

Even so I think the model is illustrative. The inherent dilation in the progress of a larger tournament, especially when combined with the slow structure of the championship event, will handicap the truly horrible players and reduce their equity to almost nothing. Remember that exponential growth (and decay) is comparatively very fast; if the qualitative result here is correct, and who knows if it is, the dead money might be even deader than I claimed earlier.

If you think I'm overstating the badness of the bad players, I think you're wrong. Until recently I doubted it, but almost any WSOP coverage by a player narrating his progress through a tournament has included plenty of stories that would make any EV-minded person wince. Bill Chen and Matt [something] are two sharp guys playing almost every event; they write up lots of hands and it's instructive.

Friday, June 24, 2005

An Offer

Any sports gamblers out there -- I'll be glad to place bets for you while I'm in Vegas. (Unless this is illegal, in which case I'll, ahem, be taking suggestions.)

Thoughts On Dead Money (or, There's Hope For Me Yet)

The following story comes from Phil Gordon's (very entertaining) Podcasts:

Preliminary NLHE tournament (I think the $1.5K shootout.) Tight player with plenty of chips raises the T2,000 big blind to T6,000. Phil calls in the small blind with 44. Big blind moves in for over T120,000. Big stack calls quickly, Phil folds. Big stack has AA, big blind has 77.

---

Now, I usually find better things to do with the time I devote to poker than make fun of bad players and bad plays. But this leads into a conjecture I have, namely that although there's enough variance in tournament poker to give almost anyone a shot to win a tournament, even a pretty big one, that really really big tournaments are almost impossible for a bad player to win.

Both parts of the above sentence are trivial, but if I make them more precise I might get to an interesting conclusion. A decent player plays a bad player heads-up. If the bad player has a strategy even approaching reasonable, he can't be worse than about, oh, 37% to win. Now put a bad player against two reasonable players. Maybe then he has a 22% chance to win. Note that in the first case the bad player would have 74% of the equity a player with no skill advantage or disadvantage has, and in the second case he would have 66% or so. (I'd imagine the bad player is in a worse spot here.)

Remember I'm just making these numbers up, but think about what happens to that percentage as the number of players grows. The prevailing belief these days seems to hold that the percent of his buyin equity a bad player has in EV: (a) has some sort of lower limit that is not terribly low, and/or (b) in a large field, is not far below 100%.

Obviously people don't actually think that, because most people don't think that quantitatively (or carefully.) But what if large fields don't "protect" bad players? What if 300-500ish fields do, but tournaments in the thousands don't? My conjecture is that as field sizes grow very large, that equity percentage actually approaches zero. In the infinite case, with certain conditions, it'd be trivial: the applicable part is that the blind structures and field sizes in the thousands would exhibit this effect. That is, the dead money is deader than ever in the big fields. Note that Raymer and Moneymaker are both good players. In fact, I don't think there's been an actual donkey at the final table either of the last two years. With the number of horrible players in the field that's at least a shred of evidence.

Also note exactly what I'm saying: it's not that a bad player has a harder time winning a larger tournament (everyone does,) but that he loses equity as the field size grows.

Anyway, the odds against me making money, especially a big score, are still long. But I see no (a priori) reason to reject the idea that bad players have it worse than ever before.

--Nate

Wednesday, June 22, 2005

What to do with 54?

Again, the scenario: heads-up, both players have a little over 12 BBs before the hand, you get a free play in the BB with 54s, and it comes 664 rainbow.

I think the min-checkraisers (or smaller checkraisers) have it right. I checkraise and slowplay less than most players, but I think checkraising is better than betting out, because bet-raise doesn't give you the information you need; check-bet-minraise-reraise does. (The only issue is that in theory you'll need to also be min-checkraising some stronger hands in this spot, but that's doable enough.)

By the way: Ilan, you're a hell of a guy, but minraising QJo on the button 3-handed would have been horrible.

Thanks for the comments; I've set it so that anyone (blogger or not) can comment, so please do. I'm definitely going to have all my WSOP and pre-WSOP exploits up here, and if strategy discussion breaks out, all the better.

--Nate

Starting on Saturday

So today I found out I'm starting on Saturday, the third and final of the three days into which "Day 1" is split. Which means that if I survive to the second day I won't have had a day or two of rest. Other people starting on Saturday perceive this as vastly unfair. My feelings on the subject are: whatever, it was randomly generated, I got the bad day, and furthermore endurance and discipline are two of my strengths. I get in town Wednesday late afternoon; my flight out is scheduled for Monday (day 3) afternoon, but I'd love to have to change it. Good thing I don't believe in cursing myself, or "showing that I don't have faith in myself," because otherwise I'd have had the uncomfortable choice of either displaying a lack of faith in myself or being a big favorite to have to buy an expensive plane ticket last-minute.

So I've got two and a half days to hang out with any friends and family that make the trip, make my tour of Vegas' poker rooms, and see how the tournament's playing. There are worse things.

How I Won The Seat

So here are some critical hands from my Step 6. Ten players, first place gets the prize package, second and third get to try again, fourth through ninth have to drop down to lower levels, tenth gets a certainly infuriating $30.

Second hand of the satellite; everyone starts with 1000 chips. UTG+1 raises the T15 big blind to T90. Folded to me on the button with AKo. Geez, already an important decision to make. I don't like a large reraise, I don't like a small reraise, and I hate a fold, especially because those raises are often scared pairs like JJ or TT. I call. The flop comes AA8 with two hearts. UTG+1 bets 150 into the 205 pot. I can't put him on hearts, and if I can just get him to make one more bet I might be able to take his whole stack. I don't mind giving a free card to someone with two or fewer outs, and he might be able to put me on a hand that doesn't include an ace. I call. The turn's the deuce of hearts. He bets 325. I think and push. He pauses for a long time -- thirty seconds or so -- and calls with QQ (no heart.) The river's a blank and I'm up to 2005 chips.

Still in level 1, I get what is probably a bad call out of my system. The cutoff limps, I complete the small blind with JTo, and the big blind checks. Flop is 875 rainbow. Check-check-check. Turn's a deuce; check-check-check. River's a ten. Check, BB bets 45, fold, I call and he shows a flopped straight.

In level 2 I take down blinds in level 2 with 99, 66, and A9o, and don't do much of anything until level 4, when I open-push from the SB; BB posted 100, calls his other 540 with A6o, and hits an ace. I'm down to 1280.

The next hand it's folded to me on the button with K9s; I triple the blind. The big blind calls. Flop comes AJ7 with two of my suit, and the BB pushes for 515 into a 650 pot. Easy call; he has JT and I miss. Down to 465.

The blinds are about to go up and I need to find hands to push. I find one a couple hands later with K7o and take the blinds. The next hand I pick up QJs and take the blinds again; I'm up to 765. A fold, and then I push UTG with AQs. MP2 overpushes in a flash and I don't like the looks of it, but he TT, a worse hand than I feared, and I hit to get up to 1680.

Then I get QQ in the big blind, push against a late-position raiser, and see him fold. Up to 2195. A round later I open-push in the SB with TT and take it down uncontested. Gee, this game is easy when you keep getting premium hands.

Then I don't do much until it's 3-handed and I have QJo on the button. Blinds are 150-300 and I have 3295, a little too much to comfortably open-push, so I open for 900. SB pushes for 2280 total and I call. (I could have folded if the BB had pushed.) He shows KJ and the flop comes rags. Crap. Turn's the lovely Qh. I am so good at this game. River's a blank. We're heads-up and I have 60% of the chips.

A few hands in, I get 54s in the (400) big blind. Button completes and I check. Flop is 664. I check, button bets 450, I make it 2300, button pushes for 5k total, I call. He has T4o and I river an ace to chop the pot. I'd love it if a better player than I am would tell me how to play that pot. I might have gotten it right but I'm far from sure.

We push and fold for a while, and then this hand comes up: the button completes again and I check with 94s. Check-check. Turn's a 9. I'm pretty sure he's going to bet, and I check and call 600 (into a 1000 pot.) River's a beautiful 4. I check, he bets 1200, I push for another 2800ish (we started the hand almost exactly even,) he folds. I'm really not a habitual checkraiser heads-up (far from it, in fact,) but it seemed like a good idea at the time.

A few hands later I have what is almost a 4-1 chip lead and find KJo in the big blind. He open-pushes (for four big blinds) from the small blind and I have an automatic call. He shows ATo; I flop a king but he rivers a one-card flush.

But shortly thereafter I'm in the big blind again, and he open-pushes again (this time for 7 big blinds) and I have another automatic call with A9o and he shows KTo and the board ends up with five cards eight or smaller and I win. Off to Vegas, and all it took was, as a friendly fellow observing the table said, "[a] horseshoe up [my] ass."