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The unrelenting variance of tournament poker

Jul 19, 2019

After making my own trip to play in a few events at this year’s World Series of Poker, I tuned in diligently every day to watch the 2019 $10,000 No-Limit Hold’em Main Event. On day six(!) of the ten(!!) day grind, arguably the best remaining player found himself making a call for all his chips against one of the chip leaders. Winning this hand would put Sam Greenwood in a dominating position with a very real chance to take down the whole thing.

When the cards were flipped up, Greenwood held Aces and a more than 4:1 edge on his opponent with only the river card to dodge. But as happens dubiously too often, the dealer completed his opponent’s straight draw, and Greenwood’s Main Event run came to a gut wrenching end. Six 10-hour days and thousands of successfully navigated poker hands made all kablooey by one odds defying card.

Sam Greenwood reacting to being put all in for his Main Event tournament life. Even holding Aces here does not make you feel good about calling.

On the bright side for Greenwood, his exit came well into the money with a $211,945 payday. Greenwood also likely survived more than his fair share of flips during his deep run. But 85% of the 8,569 Main Event tournament entrants hit the felt with nothing but a (bad beat) story and a $10,000 deduction to their bank account after failing to string together enough successful flips to make the money.

To make the money in a poker tournament is pretty tough. Unlike cash games where you can sustainably win sessions 50%+ of the time, you might only cash in 15% of your tournaments[1]. And in any given tournament, you will likely have your tournament life in jeopardy numerous times before the money bubble pops. Between cashing only 15% of your events and each tournament requiring multiple flips to go your way, variance hits tournament players particularly hard. Reflecting on my own tournament experiences, I am all too familiar.

I can relate Mike. Without variance, poker would not be nearly as popular as more recreational player would find little reason to participate. But boy, does it sting.

Personal experience paints a very biased version of variance. I wanted to see what variance looks like a bit more objectively. As an experiment, I picked out four common all-in hand equity situations and simulated outcomes for each one million times. For each simulation, if your equity survives a selected number of flips, you made the money in the tournament! If your opponent comes out ahead at any point, a tournament losing streak has begun.

With such a large number of simulations, the hand equities quickly converge to expectation. For example, pitting A♦A♥ (81.55%) versus Q♦Q♠ (18.45%) three times consecutively has the Aces surviving with all the chips 54% of the time[2]. However, what we are really interested in here is how often we consecutively defy the 54%. In other words, what does variance and running bad look like over a large tournament sample size.

I put all the results in tables below with some commentary. To understand the tables, here’s our first table with term definitions.

Term Definition
Your Equity The percent of the time you would expect to win a single flip against your opponent's hand when all-in.
Opponent Equity The percent of time your opponent would expect to win a single flip against your hand when you are all-in.
Flips The number of all-ins you must survive with the corresponding equities before you are in the money.
Variance A common tournament losing streak. Playing the corresponding hand equities and number of flips, every 10 tournaments you should expect to start a tournament losing streak at least this long.
Run Bad A very rare tournament losing streak. Playing the corresponding hand equities and number of flips, somebody out there has sadly experienced a tournament losing streak at least this long.
Shaded The WSOP Main Event pays out 15% of all entrants. All players of equal skill, you would expect to cash in approximately 1 in 7 tournaments. A red shaded table cell indicates you are running worse than expected[3].

The first spot we are going to look at gives you the same 4:1 equity as Sam Greenwood. This means you have something like Aces versus an opponent’s Queens preflop (2 outs), a gut-shot straight draw on the flop (4 outs), or the open-ended straight draw on the turn (8 outs) which took Greenwood down.

Your Equity Opponent Equity Flips ITM % Variance Run Bad
81.55% (e.g. A♦A♥) 18.45% (e.g. Q♦Q♠) 1 81.47% 1 7
81.55% (e.g. A♦A♥) 18.45% (e.g. Q♦Q♠) 2 66.49% 2 11
81.55% (e.g. A♦A♥) 18.45% (e.g. Q♦Q♠) 3 54.20% 3 15
81.55% (e.g. A♦A♥) 18.45% (e.g. Q♦Q♠) 5 36.03% 6 26
81.55% (e.g. A♦A♥) 18.45% (e.g. Q♦Q♠) 8 19.58% 13 50

The good news here is a 4:1 edge with all your chips on the line does incredibly well over the long run no matter the number of times you need to flip to make the money. Even flipping a very dramatic 8 times every tournament has only slightly concerning variance of 13 tournament losing streaks[4]. But even with the confidence of a 4:1 advantage, we get a preview of things to come as some poor soul out there has immediately lost at least 7 consecutive tournaments while going all-in with their dominating Aces. I have found myself pointing to some existential poker conspiracy against me for much less.

Our next spot gives you around 2:1 equity. This is a hand like your Kings versus an opponent’s Ace King suited preflop (3 outs), or an open ended straight draw on the flop (8 outs), or a very live combo flush and open ended straight draw on the turn (17 outs).

Your Equity Opponent Equity Flips ITM % Variance Run Bad
65.89% (e.g. K♥K♠) 34.11% (e.g. A♦K♦) 1 65.83% 2 11
65.89% (e.g. K♥K♠) 34.11% (e.g. A♦K♦) 2 54.34% 5 21
65.89% (e.g. K♥K♠) 34.11% (e.g. A♦K♦) 3 28.62% 8 34
65.89% (e.g. K♥K♠) 34.11% (e.g. A♦K♦) 5 12.44% 22 79
65.89% (e.g. K♥K♠) 34.11% (e.g. A♦K♦) 8 3.53% 83 291

Again, our 2:1 edge holds up reasonably well for the ITM % across most flip count situations. However, it’s not uncommon to race all-in with something like Kings more than three times before the money. With the pickup in variance at 3 or more flips, this starts challenging the mental fortitude of a player as they both welcome seeing their 2:1 edge yet must endure the common losing streaks.

Next up is the classic coin flip situation. This is a hand like your Nines going against AK suited preflop (6 outs) or a flush draw with an over card on the flop (12 outs).

Your Equity Opponent Equity Flips ITM % Variance Run Bad
52.76% (e.g. 9♥9♠) 47.24% (e.g. A♦K♦) 1 52.77% 3 16
52.76% (e.g. 9♥9♠) 47.24% (e.g. A♦K♦) 2 27.91% 9 37
52.76% (e.g. 9♥9♠) 47.24% (e.g. A♦K♦) 3 14.71% 18 68
52.76% (e.g. 9♥9♠) 47.24% (e.g. A♦K♦) 5 4.10% 71 217
52.76% (e.g. 9♥9♠) 47.24% (e.g. A♦K♦) 8 0.60% 495 1,392

Flipping more than once per tournament in this spot, while often correct, really invites some variance pain. I also think this table compared to the others shows how just getting it in ahead is not good enough over the long run. It really matters how far ahead if you want to play tournament poker and keep your sanity.

Finally, we are going to look at what happens when you get it in very behind. This is a very appealing hand like your Ace King suited running into Kings preflop (6 outs), or hoping to complete your flush draw after the flop (9 outs), or your certainly-I-won’t-miss-everything open ended straight flush draw on the turn (17 outs).

Your Equity Opponent Equity Flips ITM % Variance Run Bad
34.11% (e.g. A♦K♦) 65.89% (e.g. K♥K♠) 1 34.09% 7 29
34.11% (e.g. A♦K♦) 65.89% (e.g. K♥K♠) 2 11.59% 24 81
34.11% (e.g. A♦K♦) 65.89% (e.g. K♥K♠) 3 3.98% 73 265
34.11% (e.g. A♦K♦) 65.89% (e.g. K♥K♠) 5 0.05% 658 1,746
34.11% (e.g. A♦K♦) 65.89% (e.g. K♥K♠) 8 0.02% 14,424 19,300

Turns out, you don’t want to get it in very behind.

Looking over all the tables as a whole, here are a couple of thoughts:

First, if you’re going to get your chips in the middle multiple times before the money of a tournament, picking even slightly better spots significantly reduces variance. For example a 2:1 spot has less than half the variance of a 1:1 spot when flipped 3 times. That’s the difference between regularly enduring eight tournament losing streaks and eighteen tournament losing streaks according to the tables.

Second, the best thing a player can do to limit variance before making the money is to limit the number of times they put all their chips at risk. This is even more important than picking good spots. You would rather only need to put all your chips in behind once at 1:2 than three times at 2:1. This adds credence to many good tournament poker players advocating to try and double up as quickly as possible rather than waiting for the most perfect spot, as playing from ahead in tournaments often outweighs any single flip’s equity deficit[5].

Variance is inevitable even with the most favorable odds. Somebody out there is about to start the most most unbelievable string of tournament coolers. Let’s just hope it’s not you or me[6].


[1] This obviously highly depends on the payout percentage of the tournaments you play and your approach to tournament poker. Nearly any cash percentage can be profitable if you occasionally run deep.[/f]

[2] The Aces have a 88.18% chance of winning a single flip against the Queens. Winning 3 consecutive flips means winning .8118 * .8118 * .8118 which comes out to .5423 or 54%.[/f]

[3] Yes, chasing an above average ITM % is often a way to end up playing unprofitable tournament poker. However, it’s still a helpful metric for understanding variance.[/f]

[4] I suspect if you find yourself flipping for all your chips eight times regularly before the money in well structured tournaments…you might have a leak in your game.[/f]

[5] The advantages of taking a chance at doubling up early even at the cost of getting it in bad include:

* You can now use your skill and chip advantage to bully the rest of the table.

* You can pick your spots far more carefully as no other player can put you at risk and you have less urgency to put your chips in the middle.

* To play tournament poker profitably, making the money cutoff is less important than making a deep run. In order to make a deep run, you typically need a mound of chips rather than merely surviving. So let’s give ourselves the most opportunities to play with a lot of chips instead of wasting time dragging out less profitable stack sizes.[/f]

[6] Actually, better you than me.[/f]

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