## Let Them In Or Keep Them Out? The Horserace Paradox & Poker

### Poker Concepts, Strategy

November 15, 2007

A while back I wrote about the horserace paradox. I want to continue that discussion specifically in the context of poker.

A common decision in poker arises when you have the option of betting aggressively to drive out opponents or being passive as a means of keeping them in the hand. It should be obvious that this is in some ways related to the scenario in the horserace paradox article, where additional opponents with a limited chance to win cause a frontrunner to perform worse. I got to thinking about this issue because of a number of columns in Roy Cooke’s Real Poker II in which he advocated letting additional opponents in preflop in limit holdem when holding pocket aces because by calling they were making a mistake and therefore he must be benefiting by conservation of money. My intuition at the time was that Cooke was wrong and that those players were often not making a mistake by calling. But I wasn’t certain. Subsequently I’ve thought about it more, and I’m convinced Cooke was wrong but the situation is much more complicated than I originally thought.

The Analogy

In the classic horse race scenario, you’ve got one medium speed but steady horse, and an increasing number of horses that are occasionally fast but usually slow. An analogous situation arises in poker when you’ve got one made hand with no (or very little) chance to improve facing multiple drawing hands. When you have a large number of hands, all with the possibility of improving in various ways then the horse race paradox mathematics do not really apply because there is no “steady” horse.

The Differences

Unfortunately, the horserace paradox is not really a good model for most poker hands for several reasons.

1. The horserace paradox assumes that the amount of money wagered is the same for each horse. In poker, that’s not true thanks to implied odds. In particular, the drawing hands have more control over the final pot size (and hence amount of money that each individual “horse” wagers) than the made hands do.
2. Unlike our horse race, where the pool of money to be won in the race is zero before the wagers are place, there is always money in the pot in poker. The desire to defend this money makes it more attractive to have fewer opponents when you’re playing a made hand.
3. It’s essentially impossible to have a herd of hands chasing made hand in the way that would be required for a true horserace paradox. This is especially true after the flop when there aren’t enough available draws to the nuts to have each drawing hand on a separate draw. In that case, some of the weaker draws are usually drawing dead and have no chance to get any money out of the pot.
4. When a made hand drives some drawing hands out, it gets more money from the other drawing hands in the process.

The Raw Numbers

If you ignore these differences for a bit and just look at the horserace paradox math, you’ll find that a made hand with a 70-80% chance of winning against any single opponent continues to make more money as opponents are added up to 5 or more opponents. If the probability of the made hand winning against one opponent is P(w) you can get the best number of opponents N (ignoring the issues above) by finding the N that maximizes E in following equations:

define Z = P(w)^N

E = Z*N-(1-Z) or E= Z*N + Z – 1

However, as I stated above, the apparent optimal N is surprisingly high in most common scenarios.

Putting It All Together

As you might have guessed, the differences between a pure horserace paradox and a real poker hand almost all bias in favor of made hands getting opponents out of the pot. Specifically, differences 1, 2, and 4 above bias you towards wanting to get opponents out, while only 3 makes it attractive to keep them in. The net result is that with made hands and cards to come, in most forms of poker, it’s usually attractive to drive opponents out unless you havet a near-certain lock. There’s one other issue not discussed in the list above: if you keep your opponents in, you create the possibility that they will bluff you. If you will fold to a bluff given the current cards, that’s bad. But if you’re in a position to call them, that’s good. So the general rule of thumb is as follows:

If you have a made hand that can be drawn out on, it’s generally best to take reasonable steps get most of your opponents out of the pot unless you know they are drawing very slim.  If you are vulnerable to a bluff, it’s particularly important to get them out.