Dave Computes Simple Five-Card Draw Poker Strategies

We're considering a 5-player game of Five-Card Draw Poker with the following simple structure:

1. Each player pays a $1 ante and is dealt 5 cards.
2. A round of betting occurs, with betting in increments of $1 up to a cap of $4.
3. Active players exchange 0 - 5 cards.
4. Active players participate in a second round of betting, still with increments of $1 and a cap of $4.
5. Active players show their cards and the winner takes the pot.

It seemed to me that the simplest interesting strategy would be to ignore everything but the strength of my own hand, folding weak hands, calling middle hands, and raising strong hands. To that end, I programmed a computer to play 100 million rounds using counterfactual regret minimization. Here's the mixed strategy that emerged:

With probability 66%, always raise AA or better, folding all else.
With probability 14%, always raise QQ or better, folding all else.
With probability 10%, always raise JJ or better, folding all else.
With probability 10%, always raise any pair or better or 4-flush or open-ended straight draw, folding all else.

A player using this mixed strategy chooses one of these pure strategies at the beginning of each hand (before seeing any cards), and then uses that strategy through the end of the hand, choosing the next pure strategy at the beginning of the next hand.


After this experiment, I went on to use a much more comprehensive machine learning algorithm (explained in the next post). Here are some things I learned from it (and from other experiments), or that I suspect are true after watching a number of bots play and performing some rudimentary calculations:

1. Before the exchange, generally bet high pairs, raise 2-pairs, reraise trips, and cap with straights or better. Call for 1 bet with any pair or 4-flush or open-ended straight draw. It probably takes 2-pair or possibly a high pair to call 2 bets.

2.  Exchange 0 cards if you've got a 5-card hand (straight, flush, full house). Exchange 1 card if you've got quads, just to let your opponents think you might have 2 pairs. Exchange 2 cards if you've got trips. Exchange 1 card if you've got 2 pairs. Exchange 3 cards if you've got a pair. Exchange 1 card if you've got a 4-flush or open-ended straight draw and no pair. Exchange 4 cards if you've got an ace-high or king-high hand. Exchange all 5 cards if you've got a queen-high hand or worse. Sometimes it's advantageous to exchange fewer cards in order to appear stronger. Doing so may prevent an opponent from betting and allow you to see a showdown for free, or it may allow you to bluff an opponent out of a pot if they show weakness. For example, you might exchange 3 with a high-card hand or exchange 2 with a pair.

3.  Raise all the way with a straight or better. Otherwise, notice how many cards each player exchanges. If someone exchanged 0 (representing a straight or better) or 2 (representing trips), generally don't bet. But in case they're bluffing, be prepared to call 1 opponent with a high pair or multiple opponents with 2 pairs or better. Otherwise, if someone exchanged 1, they probably have 2 pairs that did not improve. Bet with trips and a strong 2 pair. Be prepared to call with a high pair. It's possible that an opponent who exchanged 1 is on a flush or straight draw. In that case, it's probably best not to bet. If they made their draw, betting or raising will cost you more than if you just call their bet. If they missed their draw, they probably won't call your bet anyway. Otherwise, if someone exchanged 3, you can probably bet high pairs safely. If many opponents exchanged 3, it's likely someone will make 2-pair or better, so now you want to be more careful with what you bet. If everyone exchanged 4 or more, you can probably bet high pairs.