Thursday, April 23, 2009

Everyone is Wrong

At my weekly poker lunch meeting yesterday, an interesting topic came up and has caused a lot of heated discussion and confusion. It all started when SolarTed, one of the regulars at the group, brought in a new book by Alan Schoonmaker, titled "Poker Players Are Different". In the opening chapter, the author asks the following question: if you push all-in PF in a full-ring cash game with pocket aces, how many callers do you want to get?"

We went around the room and asked everyone what they thought, and the majority of players said that getting one caller was ideal. This certainly agrees with the conventional wisdom espoused in most poker books that states you want to get heads up with AA, because this maximizes your chance of winning.

The author, however, pointed out that maximizing your chance of winning isn't the goal. The goal is to maximize your expected value. These two things are not the same. In other words, the author said you want all nine opponents to call, as this maximizes EV.

This topic intrigued me, so I figured I should run the actual EV calculation to see who's right. (my original verbal answer at the time of the meeting, by the way, was I thought one wanted to see 2-3 players, as I figured this would maximize EV).

I ran pokerstove to calculate how AA does against 1, 2, 3, etc. opponents who hold random hands (I also ran it against opponents who held better than average hands, but the results didn't change significantly enough to matter).

As we all know, expected value equals the percentage of the time we win, multiplied by the amount we win, minus the percentage of the time we lose, multiplied by the amount we lose. Writ another way:

EV=(%winrate)($won) - ($loserate)($lost).

The classic coinflip problem best illustrates this: if we flip a coin, its chance of coming up heads is 50%, and tails 50%. If I bet a buck on the outcome, the EV for the coin toss is:

EV=(50%)($1) - (50%)(1$) = zero. If it's a trick coin, that comes up heads 75% of the time, and I bet on heads, my EV is:

EV=(75%)($1)-(25%)($1)=$.50 positive expected value.

Now, per PokerStove, AA holds up against a variable # of players as follows:

#opp win%
1 85%
2 73%
3 64%
4 56%
5 49%
6 44%
7 39%
8 34%
9 30%

Against one opponent with $100 stack sizes, the expected value with AA is therefore:


Similarly, against two opponents:

EV=(73%)($200)-(27%)($100)=$119, and so on...

I put all this into a spreadsheet and created the chart shown above. The funny thing is that it turns out everyone was wrong, including Schoonmaker. Note how the curve maxes out around 7 players and then slowly starts falling back down. Said another way: to maximize EV when pushing with AA, you want 7 opponents, or in other words, the more the merrier... up to a point. If you're in a 9-handed game, if everyone calls, it's a good thing, but you're losing a tiny bit of EV vs. when all but one call. In even larger games, the effect would be larger.

In any case, one opponent is the worst possible answer, and Alan Schoonmaker's 9 opponents, while pretty damn good, is not ideal. The best result when shoving preflop with AA is to get 7 callers, thereby maximizing your EV. And maximizing EV, of course, is the ultimate goal of any poker decision you make at the table.

All-in for now....

1 comment:

  1. Mark, at 1st I thought I disagreed with you, but then I did a simple Excel spreadsheet with the PokerStove stats you provided and indeed your largest net profit is with 7 other callers .....$21,200 vs $20,000 with 9 callers.

    YOU ARE THINKING LIKE A WINNER I'd say. :-) Finding where your best edge is. You might want to drop Schoonmaker a line and let him know. :-) He has two email addresses in the book ( and If you don't want to I'd consider doing it with my Excel stats and give you credit, of course, for pointing it out to us.

    The important thing to remember what Schoomaker is trying to do with his book is to help YOU compare YOURSELF with what he has concluded is what make "big poker winners" tick. There aren't right/wrong answers to his questions for you. You don't answer his questions with what you think is the most correct answer. The question is what would YOU do, how do YOU think, then when you later look at the answers he provides you can see how you fit in terms of what he sees as characteristics of "big time poker" winners and what it takes for them to be that way.

    The question we had last Wednesday is one of six from his 1st chapter. We are to reply to all six before we look at any of the answers in the Appendix. After we have done that and looked at the answers we are then ask to write "What Have You Learned About Yourself." That's what I'm trying to do with this book and where/how I feel it will be most useful to me.

    The rest of the book has a couple of questions at the end of each section and we are to note our responses in Chapter 24. When you finish the book you'll look at all your responses and see how you compare.