pokeit methodpoker + econometrics

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.

Calculating the expected value of even the most simple bet in poker can be an elaborate process of brute mathematical force and artful inductive reasoning. As Dan Harrington outlines in the introduction to Harrington on Hold’em: Volume 1, there are four factors a player must consider when making or calling bets in poker:

1. The likelihood that their hand will improve as more cards are dealt, which is pretty much a straight mathematical exercise.

2. An estimate of the hand their opponent may hold, which is an exercise in inductive reasoning, based on hands he has held in the past, his general style of play, and the bets he has made thus far.

3.  The likelihood their opponent’s hand will improve, another mathematical exercise but complicated by the fact that their opponent’s hand is not known for sure.

4. The money odds being offered by the pot.

Let’s try to map these factors onto our expected value formula with a simple example from No-limit Hold’em:

You, the Hero, are facing a $100 bet into a $400 pot on the flop. Ignoring for now ‘implied odds’ of winning extra bets on future streets, the expected value of calling this bet is given by the formula:

<Hero, call> = p(win)*(new pot size) – (cost of bet)
<Hero, call> = p(win)*($500) – ($100) = ?

As Action Dan alluded to, breaking down the p(win) is the tricky part. All you know for sure are 2 things:

1. Your own hand

2. The board (in this case the flop)

Factors 1 & 3, the odds of your hand improving and the odds of your opponent improving are jointly determined – that is, in order to estimate your ESPN showdown win percentage you need to know your hand, the board, and your opponent’s hand. But unless there is some sort of malfunction, you won’t know your opponent’s hole cards until the showdown. No hole cards, no p(win), no p(win) no estimate of expected value.

In lieu of facts about our opponent’s hole cards we make estimates of what he is holding and then calculate the odds of winning against those hands. Taken to its logical conclusion, just knowing your own hand and the board gives you the ability to calculate the probability of your hand winning at showdown against every possible hand he could be holding. On the flop, there are 1081 possible hole card combinations our opponent could be holding (n!/r!(n-r)!). We can use a 1081-dimensional vector p = (p1, p2, …, p1081) to represent this collection of win percentage probabilities (click here for a nice primer on matrix algebra). Ignore for a second the impracticality of running 1081 calculations in a live game and assume you’ve rigged up one of the many available poker hand evaluators to do the hard work for you.

Provided we can get our evaluator to spit out thousand dimensional vectors we’re left with only one remaining item on Mr. Harrington’s list — our opponent’s hand. Of course, the problem still remains that we don’t know what our opponent’s hand is unless it is revealed at the showdown. However, if we think of it in a different way, a probabilistic way, until our opponent’s hand is revealed, it really isn’t one particular hand. Rather, it’s a probability distribution of every possible hand weighted towards the most likely hands. We can define y = (y1, y2, …, y1081) as the vector of probabilities of our opponent holding every possible hand on the flop. Here, each individual piece of vector y matches up with a piece of vector p. For example, there is a probability yj that our opponent has Aces and there is a probability pj that our hand will win against the aces at showdown. Multiplying each pj by its respective yj and summing the product gives the dot product of the vectors p and y:

y*p = SUM(yjpj) = y1p1+y2p2 +…+ y1081p1081

This linear function of y and p can be thought of as the probability of the hero’s hand winning against our opponent’s hand distribution. This is in fact the p(win) that we were looking for. Provided we know the product of the vectors y and p and the pot odds, we would be able to identify the expected value of any bet in poker.

Everything in the expected value function can be calculated in a rather straightforward mathematical way save for the hand distribution vector y. In practice, we arrive at an estimate of y by incorporating observable ‘tells’ such as betting patterns, general style of play, and the frequency of being dealt certain hands, to place an opponent on a range of likely hands. Let’s define these observable tells as the set of variables in vector x = (x1, x2, …, xn). In calculating the expected probability of an opponent having a particular hand we are implicitly constructing a linear function yj = Bxj whereby xj are the things we observe, yj is the probability of having a particular hand in our distribution, and B is our estimate of the relationship between yj and xj.

Done at the poker table it is called hand reading. Done with reams of data and a statistical package and it is called econometrics…

-chaz