My friend Rob did a blog post the other day that was ostensibly about an over-the-top wild woman at the table. But along the way, he said this:

I was paying more attention to Natalee’s comments and her outrageous behavior than the poker, but I was following enough about the poker to quickly figure out how to play in this situation. With Natalee being so loose-aggressive and so many players going on tilt because of her presence—playing even crazier and more aggressive than they ordinarily would—I knew that the only way to play was to be extremely tight. I’m a tight player anyway, but now I became uber-tight. I wasn’t going to play any speculative hands, any borderline hands. No, I was going to wait for a true premium hand to play. That’s the only way to play at a table full of maniacs and that’s what this table was.This is a common reaction, but it's wrong in terms of maximizing profit. I submitted a comment to that effect. I first appealed to authority. In one of the very first blog posts I ever wrote (November 6, 2006), I said this:

Q-10 offsuit is a hand I would ordinarily either toss or try to play very cheaply, with a high willingness to throw it away unless I get a lot of help from the community cards, or a good bluffing situation arises. And against a loose-aggressive player, my usual reaction has been to re-tighten my already fairly tight game. Recently, however, I've read two things that are making me re-think that strategy. I include them here for the possible benefit of readers, though I didn't want to interrupt the story above.

First, I finished Barry Greenstein's superb book, "Ace on the River." On p. 204, he has a table of "opponent's tactics," the "typical incorrect adjustment" that we mediocre players make, and the "Better adjustment" that he recommends. The first entry in the table is "Extremely loose play." The incorrect adjustment is listed as "Wait for a good hand." The better reaction is given as "Loosen your standards and reraise frequently." Seems like cogent advice, though it takes more nerve than I've usually been willing to bring to bear. I'm trying to change that, a little at a time.

Second, just a few days after reading that, I read a column in Card Player magazine by Matt Lessinger, who I think is one of their best writers. He wrote,

Overaggressive Ozzie sits down in the same game, and in six of his first 10 hands, he raises to $20 preflop. Five times, he wins the blinds uncontested. One time, someone calls him and the hand goes to the river, at which point Ozzie produces Q-8 and wins with a pair of eights. I don't have to be Phil Ivey to put two and two together, and conclude that he probably did not have premium hands when he made his other raises. Therefore, I'm going to wait for something good and try to pick him off.

You can read the rest of my comment in the comments section of Rob's blog post, linked above.But am I going to wait for pocket aces or kings? I could, but why would I want to play so timidly? In a cash game, I want to take advantage of all favorable opportunities that come along. Here, I have an opponent who is making oversized preflop raises with substandard values. If I pick up a hand such as 9-9, or even A-J, and think I can get heads up with Ozzie, I'm probably going to play it, because it figures to be the best hand against his typical raising values. And I'm certainly not going to be deterred by the fact that he is raising to $20 rather than a more normal raise to $6 or $8.

If you do, you might go on to read that some others piped up doubting and/or flatly disagreeing with me, including Rob himself. So late last night I whipped out an even longer reply (again, see the comments thread). I did a quick semi-quantitative illustration of the effects of tightening up versus loosening up one's starting hand standards when playing against a table of opponents who play basically any two cards and who are willing to commit lots of money without much of a hand. My conclusion:

Here's the Reader's Digest version: You want to expand your starting hand range enough that your profits go up by winning more pots, but not so much that your average starting hand range is the same as theirs. As long as you're reasonably judicious about it, the increased frequency of wins will more than compensate for the smaller profit per hand. Against a bunch of people who are playing any two cards, you can expand your range quite a bit and still on average have the strongest starting hand, which translates into the highest probability of winning the pot, assuming that you don't play worse than they do post-flop.

Again I stress that this is not the kind of question on which reasonable people can have opposite but equally valid opinions, like whether strawberries are better than raspberries. The math (the real math, not just the back-of-the-envelope version I'm doing here) clearly favors widening your starting hand range. It does necessarily bring higher variance, but if the question is just long-term average profits, there's no serious question about the direction of the correct adjustment. Unlike most poker questions, here there is a factually, objectively right answer: Loosening up is more profitable, tightening up is less profitable. Those who disagree are simply wrong.When I woke up this morning, I was thinking about this some more, and wondered if I could make the math a level or two more rigorous than I did in that comment, without spending all day on it. (I have an apartment full of stuff to pack up, you know, and I'm planning to leave in less than a week.) I thought of a way to use PokerStove to do it.

With PokerStove you can enter your hand range as either specific cards, or use the slider to change the percentage of starting hands you'll play. It assumes that if you're playing, say, 20% of hands, it's the

*best*20%, which is a well-defined set.

Here are the assumptions I used: I'm playing at a 10-handed table. The hold'em is played by everybody looking at their two down cards, then deciding to play or not. To play, you put in your whole stack of exactly $100, after which the dealer puts out the five board cards and pushes the pot to the winner. After each hand, everybody either removes their profit from the table or reloads, so that everybody has exactly $100 to play each hand. (Obviously this is unrealistic, but it simplifies and clarifies the math.) There is no rake (yay!) and we can get out 40 hands per hour. My nine opponents are all willing to play any two cards. I, however, am being selective. I want to maximize my profit per hour. Note that pots are $900 when I don't play, $1000 when I do play.

Here's a spreadsheet showing the results of a variety of strategies:

(Right-click on the image to see it full size.)

Column D is simply the percentage of hands played times 40 hands per hour. Column E shows how much I'm putting into the middle of the table per hour, calculated as the value from Column D times $100. Pot size is always $1000, because the hands I don't play cost me nothing and profit me nothing. Column H is the product of the pot size, my equity, and the number of hands per hour I'm playing. Finally, Column I show my profit, which is Column H minus my investment from Column E.

In the top row, you see what happens if I play nothing but A-A. My equity (i.e., the percentage of the average pot that I can expect to win) is very high (31.1%), but my profit sucks because I'm playing so few hands.

At the opposite extreme, consider the bottom row. There I'm using the same strategy as my goofy opponents, playing every single hand I'm dealt. Unsurprisingly, my profit is zero. The ten of us are just exchanging chips across the table, with nobody having any advantage. (Good thing there's no rake, or we'd all lose everything to the house over time.)

Clearly the sweet spot for maximizing profit is somewhere in between those two extremes.

For the few lines below the A-A line, I added specific starting hands, down to any two Broadway cards. The trend is perfectly clear: I lose equity in each pot I'm contesting because I'm adding sequentially weaker starting hands to my repertoire. However, this is more than compensated for by the greater number of pots I'm playing, so my profit per hour rises dramatically.

For those not familiar with PokerStove, here's how the hand-selection tool looks like, in this case set for any two Broadway cards:

And here is what the equity calculation page looks like:

After that, I just used the PokerStove slider to select higher fractions of all possible starting hands, in 5% increments, then calculated the outcomes.

At first, going looser continues to increase hourly profit. After a while, though, as you can see, the profit trend reverses. The marginal loss in equity as I add weaker and weaker hands to my starting lineup is not compensated for by the fact that I'm making myself eligible to win more pots.

The point of maximal profit isn't entirely clear because the trend isn't perfectly linear. (It might be if I were to let PokerStove run longer for each calculation and therefore got more precise equity values. But it takes a long time to settle on these values when the software has to account for nine opponents' completely random hands.) Nevertheless, you can see that it's going to be somewhere in the range of playing the best 35-45% of starting hands I'm dealt. If you want to see what that range includes, fire up PokerStove, click on the "Player 1" box, and play with the slider. It will highlight what specific cards you're selecting. (You can find a nice tutorial on how to use PokerStove here.)

Translating the math from this over-simplified version of hold'em to the real thing requires assuming that I don't make more mistakes than my opponents do. I have to be able to read them at least as well as they read me, fold second-best hands at least as well as they do, size my bets to maximize profits when I have the best hand at least as well as they do, etc. But if that isn't a valid assumption, then I simply shouldn't be playing at this table. I shouldn't play anywhere that I don't have an edge in skill over most of my opponents.

The upshot is, as I said, that the math unequivocally supports being looser than one's usual default strategy. It would be foolhardy to try to play 35-45% of starting hands against a typical table. You'll get eaten alive. But that is demonstrably the correct strategy adjustment to make in the extreme scenario I've constructed. No real-world table will be quite that wild, but the point still stands that loosening up at least moderately is the way to go. Tightening up means that you leave lots and lots of money on the table that should have been yours.