Expected Value

Understanding the concept of Expected Value (EV) is very important. The EV of an action if how much you expect to make on average by doing that action if you do it many many times. For example, if you're betting on coin flips, your EV is zero, because over a long period you will always break even. All of gambling is about EV. You want to make moves that have EV > 0, this is called "Positive Expectation" or +EV for short. Moves that are -EV are "negative expectation" or losing moves. Simplistically, your entire goal in Hold'Em is to get into +EV situations. That is, win more on average than you lose. Now, any one move may lose you money, but it still may be a winning move in the long run. A lot beginners make the mistake of paying too much attention to how things turn out in individual hands. For example, let's say you get AA in the hole and raise. Against 10 players AA has something like a 30% of winning. That means 7 out of 10 times you're going to lose (if everyone goes to the river) !! So, you might play it, and lose on the river, and think "that was dumb, I wasted my money by raising". No! Raising is a +EV move, it will pay off in the long run, just because you lose money one time doesn't make it wise. The opposite happens too, players will limp in (just call) to see the flop with something terrible like 83o ("o" means "off suit", not of the same suit). Now the flop comes 883 and you just made a full house. You're going to make a lot of money on this hand, but does that mean it was wise to play 83o ? Probably not, it's a -EV move in the long run!!

The basic idea of EV goes like this. Lets say you are wagering some bet "b". You have a probability of winning "p" and if you win, you will get paid "w". The long-term EV of this wager is : EV = p * w - b This just says : you always put in your bet "b", so that's a term of "-b", and with chance "p" you win "w", so that's "p * w". So, for example, on an even coin flip p = 0.5 (50%), if b = 1, and w = 2, then EV = 0. What if you're flipping a coin and someone offers to pay you 3 if you win, and you only have to put up 1? Then your EV is 0.5 * 3 - 1 = 0.5 bets. Each time you take that wager, you win 0.5 on average, though on any one flip you will either lose 1 or win 2 (3 minus the 1 you put in).

One important thing about EV that many newbies miss is that very unlikely gambles can be very profitable. For example, if you have a 10% chance of winning, but it only costs you $1 to play and you get paid $100 if you win, then your EV is : 0.1 * 100 - 1 = $9 ; that's a very good proposition, even though you will lose money 9 out of 10 times.

To compute the EV of some complicated future possibilities, you have to sum over the branching tree (see the section on probability).  If there are various outcomes, or even various ways to get to the same outcome, you have to look at the EV of each branch and the probability of each branch, and do :

EV = Sum[i] P(i) * EV(i)

where the P's are probabilities which should add up to one.  This is a "probability weighted sum" which is a common thing in poker theory.  When you're thinking about some action in a hand, you need to consider all the ways the hand can finish, and what the result of each finish is, and what the pot is in those ways.  If you're considering your own future actions, you need to consider all your choices (f,c,r) and take the best one.  When considering opponents' future actions, you need to consider the probability of each action for them (f,c,r) and sum over all three, weighted by the probability.

Let's consider a simple example.  You're on the river, heads up.  Your opponent has bet at you and you now need to decide whether to fold, call, or raise.  If you fold, your EV is zero.  The pot is size A, and it's B more to call, so the pot is (A+B) after you call.  Let's say you can estimate your chance of winning if you call as P0.  Your EV is thus P0 * (A+B) - B.  Now, what if you raise?  We have to consider his options, he may fold or call, or re-raise.  We should estimate a probability that he will do each.  For now, let's just assume he will either fold or call, no re-raise, and that his chance of each is P1 and (1-P1).  If he calls, your chance of winning is P2.  Thus your EV in this case is : - B - R + P1 * (A+B+R) + (1-P1) * P2 * (A+B+2*R) .  You now have to choose the best of the three EV's.

Now, this may all seem like a lot of funny math, but the important thing is the concepts that this illustrates.  In this simple case we can compare calling and raising.  If you just call, you must show down the best hand to win, and you have some chance that your hand is best (this is a probability because you don't know exactly what his cards are, not because of a draw).  If you raise, he may fold some hands (some hands you beat, and some that beat you), and he will call (or raise) with others.  The key point is that in each case, the pot is a different size, and your chance of winning is different.  Generally, the bigger the pot, the lower your chance of winning.  That means that bigger pots don't necessarilly make you more money.

EV doesn't just go with the most likely winning hand.  In poker, you want to have hands that you can be confident with, and make early on (on the flop or preflop).  The best types of hands are in two categories - 1) pretty big pairs that you can be confident with the whole way and play aggressively, and 2) big drawing hands that can make the nuts and get paid by lower hands that are also big hands.  If you have the winning hand and you can't be confident with it, you won't make much money on it.  Thus, it may be winning, but it's still not a good hand.

Let's consider, for example, the pocket pairs.  You have a pocket pair, and you raise it up pre-flop.  You get two callers, and you're pretty sure they're just playing high cards (not a pair), like AK or AQ.  Now the flop comes.  What kind of flops are good for you?  Obviously a set is very good, but that won't pay very well unless they also hit, so if you have 88, something like 83A would be good, since you make a set of 8's and they pair their ace.  Most of the time you won't make a set, your hand needs to be profitable in those cases to be worth playing.  The flop can come with either 0,1,2, or 3 over cards to your pair.  The more over cards the more likely it is that your pair is beat.  With 0 over cards you can be very confident that it's good, and you can bet it and get paid.  With lots of over cards, you can't be confident, so you can't get paid, and in fact you may lay down the best hand.  You're still favored to have the best hand, but you can't know it.  For example, if you have 55 and flop comes 3TJ.  Your opponent may not have hit that flop at all, they might have KQ, AQ, AK, etc.  Or they might have hit it, with AJ,AT,QJ.  You can't know, so you can't play aggressively and make a big pot.  The higher your pocket pair, the more likely you can be confident and make a big pot, even though your chance of winning a showdown is still the same!

The clearest example of this is to consider two cases.  Your opponent has AK (but you don't know it).  In case 1, you have 55, and in case 2 you have QQ.  Your chance of winning a show-down is nearly identical in both cases, but you will make a lot more money on average with the QQ.  The reason is there are a lot more boards where you can be confident early, because you know that only an A or K on the board can help over-cards.  With pocket fives, you have to be scared of all overcards, and you may even fold if he makes a re-raise bluff!

This kind of EV logic comes from looking at all the branches and what will happen in each; consider TT, we have to consider

+ flop comes all under
I bet
he probably folds
I win a small pot
+ flop comes with one over-card
I bet
+ he may re-raise
 I call
 + now consider turn & river cases
+ he just calls
 + etc.
+ he folds
 I win a small pot
+ flop comes with two or three over-cards
I check
+ he may bet
 I fold or call
+ he just checks
 etc.

In your head you have to imagine this tree, what the pot size is in each case, and the chance of each case happening.  Obviously you aren't doing the full math all the time, but it is your guiding principle.

Another big way that EV comes into play is with position.  In two different positions, you have the exact same chance of winning a "race" where all cards are exposed to a showdown.  In late position, however, you make much more money.  The reason is that you can control the pot size better, you have more information so you know when you are winning or losing, so you can act to make a big pot when you win and a small pot when you lose.  Poker is all about manipulating your opponent and the size of the pot to favor you.