Just finished recording a demo of the Pokeit pre-flop equity model.
The first part gives an overview of the Pokeit equity model and a brief discussion of how it can be used to estimate of your opponent’s hand range and your win probability against that range:
This question, posed to a class full of undergraduates, wasn’t interpreted quite the way Mustafa had intended. At the time, I was taking Stat 10 to satisfy a pre-requisite for my hastily put together plan to switch majors and apply to UNC’s business school. See, it turned out that physics just wasn’t my calling. Perhaps it was Dr. Yu Wu’s prickly demeanor in Physics 26, or the fact that Dr Hernandez in Physics 27 was an asshole. Whatever it was, my grades in physics had not been compelling. By the time the second Physics 27 midterm rolled around, knew I had to stop griding glass into my eyes. I spoke with my academic advisor, and by the end of the day, I had signed up for STAT 10 and ECON 10 in the Spring. Next semester, Mustafa Tural introduced me to statistics.
We are all familiar with idea that the distribution of hands shown at showdown is different from the distribution of all hands dealt. The specter of this bias has confounded the poker botting community and lead many to eschew estimating hand ranges all together. While my scouring of the pokerai.org archives was not totally exhaustive, I don’t believe that anyone else has tried to quantify the bias in showed hands in a systematic way. In this analysis we will use a two econometric models, one created from a dataset revealing every hand, and the other from a dataset limited to hands showed at showdown, to predict a player’s hand range distribution in several different game states. By comparing the showdown equity of the match-up between an arbitrary hand and the ‘all hands’ and ’showed’ hand range distributions, we can estimate the bias in terms of its affect on showdown equity. In a meta-analysis of +45,000 game states, equity estimates derived from the dataset limited to hands showed at showdown were -1.34% ± 2.1% lower on average than those derived from the full dataset – indicating a slight upward bias in the strength of observed hands.
There’s this great story one of my economics professors use to tell in order to illustrate that “correlation does not imply causation”. It’s most certainly false but it goes like this. The setting is 16th century Russia, during the latter half of Ivan the Terrible’s reign. In general, this was not a great time to be living in Russia. A combination of drought, famine, Polish-Lithuanian raids, Tatar invasions, and the sea-trading blockade carried out by the Swedes, Poles and the Hanseatic League had devastated the country. On top of that, a particularly nasty epidemic of the plague was killing between 600 and 1000 people every day. It was not known at the time how the plague spread, so efforts to fight it were subject to wild theories. Ivan, mentally unstable and physically disabled, suspected treachery. To prove it, he had his advisors gather statistics on the number of doctors and the number of dead throughout his Kingdom. Once it was discovered that regions with more doctors also had more deaths, Ivan rounded up all of the doctors and had them executed for treason.
At 9:46 p.m., blogging on his site FiveThirtyEight.com, Nate Silver called the presidential election for Barack Obama. The television networks followed suit about an hour and 15 minutes later after most polls in Western states closed.
Of course, Mr. Silver had a head start: he had forecast that Senator Obama would beat Senator John McCain back in March.
From the New York Times – November 10, 2008
Nate Silver, the prodigy behind the PECOTA system for predicting the performance of baseball players, and former economic consultant for KPMG, had developed a new statistical framework for analyzing elections. Silver had already proven its scary accuracy during the Democratic primaries in May. While every other commentator was celebrating Hillary Clinton’s resurgent momentum, Silver was skeptical of the new polls showing she would win by five in Indiana and had closed the gap to 8 in North Carolina. The fresh polls didn’t make sense when compared against the relatively stable demographic data. Blogging under than handle Poblano, he broke down the numbers in a different way – Clinton by just two in Indiana, and a seventeen point whuppin in the Tar Heel State. On May 6th, the night of the Democratic primaries, Clinton won Indiana by one and lost North Carolina by fifteen.
The Pokeit pre-flop prototype is available for download. You input the game state, your hole cards, and the action during the hand, and the Excel based tool gives you a 2-way equity estimate of your hand against each opponent’s hand range.
Hand range estimates are derived using a multinomial logit regression model. The general idea is that the model takes the input variables of game state and opponent action and then tells you the conditional probability that your opponent holds each of the 169 possible hands pre-flop. Each component of your opponent’s hand range distribution has an associated showdown equity value against your hand. To derive the estimated equity of your hand against the distribution, we multiply the probability of your opponent holding each hand, by the showdown equity, and sum the whole thing together.
The input variables of the model include:
♦ Player action on each pre-flop ‘round’ of betting (call/check or raise)
♦ Action behind the player (call, raise, 3-bet, etc.)
♦ A variable that combines position with # of players at the table
♦ Amount bet on each pre-flop ‘round’
♦ An interaction between player action and action behind
♦ An interaction between player action and position/# players
♦ An interaction between player action and amount bet
The data used to produce these estimates comes from a 181,007 hands worth of 6-max NL-Hold’em on PokerStars with limits ranging from $0.50-$1.00 NL to $3.00-$6.00 NL. The analysis is completely derived from the revealed hands of my friend Joe, who was kind enough to ship me his PokerTracker database. As you’ll notice when you test it out, Joe is pretty TAG. Some of his stats are vpip = 0.18, pf_raise = 0.12, wwsf = 0.43, and total_af = 2.77. Because the statistical model is based off of just one player, the prototype is somewhat impractical for in-game use. It should however give you an idea of what we have in store.
Just a few notes about the functionality of the Excel workbook:
The basic problem in poker is that unlike craps, roulette, blackjack, or any of the various table games you might find yourself blowing cash on at Casino Royal, the expected value of any bet in poker is uncertain. Calculating expected value in roulette is pretty straightforward in comparison. Take for example a $100 bet on black. The equation for the expected value, or average profit per bet would be:
<X, black> = p(black)*(payout if black) – $100
That is, the probability that black comes up, multiplied by payout if black comes up, minus your original bet. On a standard American table, the payout for black is 1:1 and the probability of hitting black is a little less than even:
<X, black> = (16/38)*$200 – $100
<X, black> = $94.74 – $100 = -$5.26
Almost every bet in roulette will give you the same expected value: - 5.26% with the only difference being the variance between individual outcomes. The same concept is true for other table games. The expected value of betting the Pass line in Craps is -1.41% per bet and likewise, the expected value of betting ‘Hard Eight’ is -9.09% per bet. This is of course how the casinos make their money. Poker is different from the table games primarily because the expected value of bets change and estimates of expected value are always uncertain.
