Saturday, November 28, 2009

Ballhype, Golden Picks and EV, Part 2: Explaining Expected Value to the Uninitiated through a poker example

A few days ago, I recalled my poker research and the common theme of expected value (EV). Similar to the microeconomic concept of marginal utility, EV focuses around taking the expected gain of a positive outcome and multiplying by its probability of occuring... then subtracting the expected loss of a negative outcome multiplied by its probability of happening... to get a net expected value. If the final EV is negative, taking that route will fail in the long run, while the play will succeed in the long run if the net EV is positive.

To better illustrate the EV concept, here's a simplified example: Let's say you're playing $4/$8 limit Texas Hold'Em poker, and you have two pair on the turn, having paired your Ace and your Ten, with one card yet to come. There's $46 in the pot, one other player in the pot and he has made a single big bet of $8. You can close the action with a call, close the hand with a fold or keep the turn going with a raise to $16.

Let's say there's three of one suit on the board, neither of which match your cards, and whether or not you are a master reader, you know the guy making this bet well enough that you're fairly sure he has a flush (let's say the three cards are far enough apart that a straight flush is impossible), so to win this hand the river card has to improve your two pair to a full house (the only hand that will beat the flush). Therefore let's say the only options you will consider here are calling the $8 bet or folding. Is calling the $8 bet profitable?

With four cards on the board, and two in your hand, there are 46 other possible cards to come on the river. We need to find the probability that our needed full house card will come on the river. Let's never mind the cards other players have folded, cards that were burned between each street and cards in your opponent's hand, as the forthcoming odds will compensate for the chance that your needed cards are among the dead cards.

There are four cards that will score the full house: The two remaining Aces (there are four total in the deck, one is in your hand and one is already on the board), and the two remaining tens (ditto). Since four of the 46 possible remaining cards will win the hand on the river, our odds of winning the hand on the river are 4 out of 46 (8.7%).

Let's keep the whole implied odds concept simple and say that we get to act last and that, if we call the $8 and our river card hits, our opponent will just go ahead and make another $8 bet on the river, which we'll call. Let's also assume that, with the pot so big, the casino dealer has already pulled the maximum rake and jackpot drop, so no additional money will be taken from the pot.

With $46 in the pot plus another $8 from the opponent's bet, there's $54 total. Knowing this player will bet another $8 on the river if we hit, that's a total of $62 we will win if our hand hits. That's our expected positive outcome if we call: We will get $62.

If we call and the hand doesn't hit, we lose $8. We ignore all other money we've put into this pot: That's a sunk cost which you're not getting back whether you fold this hand or call and lose. Thus if we fold, we have a 100% chance of netting 0 dollars on that decision.

The expected value of calling is determined by the chance of hitting the full house and winning $62 minus the 91.3% chance of missing and our $8 call going to waste:

(0.087 * $62.00) + (0.913 * -$8.00) = -$1.91

If you hypothetically got into this exact same decision a million times, and you made the exact same decision to call every single time... over the long run you would average a loss of $1.91 for every time you called the bet. Thus the decision to call is not a profitable one: the expected value of calling is negative.

The decision to fold, even though its expected value is $0.00, is more profitable over the decision to call by $1.91. Yes, it's guaranteed you win nothing, but is a more relatively lucrative decision that the negative EV decision of calling. The times you hit and win money will not offset all the times you call and lose money.

Many experienced poker players make decisions involving expected value all the time, and (provided they have requisite skill and experience) over the long run win money because they don't invest in bets, calls and raises unless doing so has a positive expectation. As they get into these situations time and again, the positive EV decisions mean that they lose, but what they win when they invest offsets those losses and nets them a profit over the long run.

The reason I wasted your time with this long poker example is because expected value is a concept you can apply to everything in life.

... just as I decided to apply to Ballhype's Golden Picks contest. More to come in Part 3

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