There's More to Expected Value Than Dollars and Chips

I was playing pot-limit hold'em at Hustler Casino a couple of weeks ago, and the blinds were $5-$5, which seemed like pennies compared to the $2,000-$4,000 game going behind us. As Johnny Chan, Doyle Brunson, Chip Reese, and Bobby Baldwin headlined a table of poker giants, I paused long enough to look down and see 9-2 offsuit and push it into the muck. Andy Bellin, the author of Poker Nation, was to my immediate left and limped in for $5. There were a couple of other callers, a small raise, and with Bellin in second position, three players took the flop with $100 in the pot.

The flop came 7-5-3 rainbow, and the first player bet the size of the pot. Bellin flat-called, and the player to his left raised the size of the pot. The initial bettor (who later said he had A-7) folded, and it was $400 for Bellin to call.

Holding 6-4 (suited, but with no backdoor flush draw), Bellin had about $1,500 in front of him, as did his opponent. He was all but certain this player to his left had a set, but that was just a small part of the dynamics of the situation. More importantly, the player was a business associate, and what started out as a relatively small, friendly game now put Bellin in the position of winning (or losing) a substantial pot.

As is the case in most pot-limit hold'em games – even forgetting the fact that this was his business associate – his decision wasn't straightforward. You could argue that if he was convinced his opponent had a set and was equally convinced he wouldn't lay it down even when faced with a pot-sized raise, the best play would be to call the $400, thus allowing Bellin to lay the hand down on the turn if the board paired – or get all of his money in with way the best of it if it didn't. Others would say he should raise the size of the pot.

Poker strategy aside, to Bellin's credit, he saw the big picture. He thought about other factors beyond the expected value of this particular hand that would determine his best play. So, after some thought, he said, "I know you have a set, but I flopped the nuts. Why don't I just call and we'll run it out?"

With $1,200 now in the pot, his opponent, who indeed had flopped a set of threes, replied by saying, "Sure. How many times do you want to run it?"

Once again, Bellin was faced with a decision, and his answer had nothing to do with expected value. At the expense of being vilified by Sklansky and Malmuth, let's keep the math simple and say there is $1,000 in the pot and the chances of Andy's hand holding up are 65 percent.

If you aren't familiar with the concept of multiple boards, it's not uncommon in a big-bet game, when a player is all in, for players to agree to deal out more than one flop – or in this case, more than one turn and river. Using the assumptions I listed above, if they ran it just once, Bellin's expected value would be $650. If they ran it a million times, Bellin's expected value would still be $650. And this is where the law of large numbers comes in: The larger the sample size, the closer you'll likely get to the expected value. If they ran it only once, Bellin would win $1,000 65 percent of the time and $0 35 percent of the time. His expected value would still be $650.

In his wonderful book, Telling Lies and Getting Paid, Michael Konik has a story entitled "Who Really Wants to be a Millionaire." When discussing the principles of expected value, Konik subscribes to the theory of most gamblers – take the course of action with the highest expected value. If you're on Regis' show and you're at the $500,000 level with one question standing between you and $1 million, as long as there is at least a 50 percent chance that you can guess the right answer, you should guess. If you're right, you will win $500,000 more, and if you're wrong, you will lose only $468,000 (you get to keep $32,000). So, if you're exactly 50 percent sure, any believer in expected value would say you have an easy decision to guess.

Now, let's pretend the contestant is George, a 42-year-old father of two teenage kids who are straight-A students. George is in debt a hundred dimes to a mobbed-up bookie with a penchant for breaking limbs; his mom needs another hundred dimes for life-threatening cancer surgery; and he doesn't have a college fund for his kids. Does this information change his decision?

According to Konik, it shouldn't. In his book, Konik wrote, "Either you believe in the concept of expected value or you don't; it doesn't apply only when you're comfortable." Thus, Konik would seemingly advise George to guess. What would you do?

Before you answer, let's go back to the question Bellin faced. If his goal was to win the most money possible irrespective of risk for that session, his best move would be to deal the last two cards only once. If the cards were dealt twice, the odds of him winning all of the $1,000 would decrease from 65 percent to 42.5 percent. If the cards were dealt three times, his odds would go down to 27.5 percent. Of course, if he had a short bankroll and was interested in hedging, the odds of him getting none of the money with three deals would be only 4.3 percent.

In this particular case, Bellin was looking for the outcome that would lead to the greatest likelihood of the money being divided. You could argue that if he simply wanted to see the money divided fairly (as opposed to equally), they could have made a deal in which Bellin took $650 and his opponent took $350. But, after all, this is poker, and that would detract from the spirit of the game. If he chose two flops, then 54.5 percent of the time, one player would still be getting all the money (42.25 percent for Bellin, 12.25 percent for his opponent). But when you go to three deals, that number drops to 31.75 percent, and to 19.35 percent for four deals.

Bellin chose three flops, and on all three, the board didn't pair. Thus, he ended up with the best of all words: looking like a mensch and scooping the pot. Or, you could argue that he left a lot of money on the table by making a deal in the first place and not maximizing his expected value by getting all of his money in the pot with the best of it.

As for George, before you give your answer, think about the law of large numbers. Sure, if you could get $51 for heads and only pay $49 for tails for the flip of a fair coin, you'd bet everything you had - provided that you could make the bet forever. The reality is that in poker, and even more so in life, you rarely get the chance to make a decision more than once. That's why even casinos with enormous bankrolls have betting limits.

It's all but a certainty that George will never be faced with this decision again, and $500,000 is life-changing money to George – so much so that even if he had a 99 percent chance of guessing right, it would be hard to argue with taking the money. In the world of expected value, there is only one right answer. In the game of life, it's much more complex.

If I were in George's shoes, I'd try to use one of my lifelines to buy insurance from an enterprising poker player and hedge my bet. In lieu of that, before going for the million, I'd try to visualize my legs being broken, my mom on her deathbed, and my kids commuting to vocation school – with nothing to offer but the immortal words of Michael Konik.

Greg Dinkin is the co-author of Amarillo Slim in a World Full of Fat People and The Poker MBA (www.thepokermba.com). He is also the co-founder of Venture Literary, where he works with writers to sell their work to publishers and studios (www.ventureliterary.com).