GT-NO: A Three-Player ‘Toy’ Poker Game

As many of you know, GTO is usually not an unbeatable strategy if there are more than two players. On the other hand, a player who uses GTO precepts in a multiplayer game is almost always going to have the best of it. But on the third hand, someone who uses exploitive strategies in that game will almost always have MORE the best of it.

While contemplating all this, I came up with a very simplified three-player “toy” poker game that would be helpful both in thinking up strategies and useful in teaching readers some basic poker math/logic that they may not be aware of. (There probably will be one or more columns about this game in the future.)

There is $100 in the pot that fell from the sky. Three players are each dealt one card with a number (including a possible fraction) from 0 to 100. The players can either bet $100 or fold. But they bet simultaneously, not in order.

If no one bets they all break even, and the sky gets its money back. If only one bets, he profits $100. If two bet, one player breaks even, one player loses $100, and one player wins $200. If all three bet, two players lose $100 each and one player wins $300.

Got it?

The GTO strategy is to bet hands very slightly above 70 or better. And there is an easy way to calculate that. But first let’s see what happens if players don’t obey GTO recommendations (which henceforth will be rounded to “bet 70 or above”).

It is true that a poor recommendation won’t cost anyone money if they all take it. But if one or two don’t, they will lose to those who call 30%, and perhaps even more money to players who exploit them.

For instance, to take an extreme example, suppose your two opponents ALWAYS folded. If you knew that, you would, of course, always call and thus have an EV of $100. If you called only with your top 30% your EV would be $30.

What if they ALWAYS called? If you stuck with betting the top 30% of your hands, there are various possible outcomes:

1. You are all in the top 30%, which happens 2.7% and you win one-third of those. EV = .9% of $300 minus 1.8% of $100 = 90 cents

2. You are in the top 30%, one of them isn’t, and one of them is. This happens 30% x 30% x 70% x 2 = 12.6%. When it does, you win $300 half of the time. That EV is therefore (6.3% times $300) minus (6.3% times $100) which is $6.30.

3. You are in the top 30% and they are both not. That happens 30% x 70% x 70% = 14.7% and you always win $300. That adds an EV of $44.10.

4. 70% of the time you don’t bet and break even.

Altogether your EV is 44.10 + 12.60 + .90 or $57.60. Way better than if they had played 70% like you, but a lot worse than if they always folded and you took ultimate advantage of that.

But what if you loosened up against these two players who always call? Say you bet your top 40% instead of 30%.

1. You will all be in the top 40% 6.4 % of the time. You will win a third of those; about 2.13%. EV is 2.13% x $300 minus 4.27% x $100 or about $2.13

2. You’re in the top 40% along with one of them. 40% x 40% x 60% x 2 = 19.2%. You win half. EV 9.6% x $300 – 9.6% x $100 or $19.20

3. Neither of them is in top 40% while you are. 40% x 60% x 60% = 14.4 % and you win $300. That’s $43.20

Total EV is 43.20 +19.20 + 2.13 = $64.53

Once again, the exploiter does better.

Next time we will look at more reasonable scenarios. ♠

David Sklansky is the author of The Theory of Poker, as well as nearly two dozen other guides on gambling, poker, and other games. The three-time WSOP bracelet winner’s latest book, Small Stakes No-Limit Hold’em: Help Them Give You Their Money, is now available on Amazon. You can contact Sklansky at [email protected].