Visualizing Expectaion Value

Expectation Value is probably one of the more poorly understood terms in poker. While many will know what it means in general, few probably could calculate it for a specific hand. I have been playing mostly MTTs lately, and I started thinking about how important the series of cards that you are dealt in an MTT is to your odds of success. Think about the raw expectation value of each hand that you are dealt. If you get dealt a bunch of +EV hands your stack should move up, while a bunch of -EV hands should send you stack south. Its a lot more complicated that that though. Different people play the same hands differently and for different expectation. Also when you say a hand is +T207 in expectation value, that does not mean that that you win T207 each time you are dealt that hand, it means that's what you will win on average when playing the hand. The average is developed from all the times you fold preflop, and all the time you win or lose a small pot, get stacked or double up. T207 may be a small percentage of your stack but it includes the times you are stacked playing the hand as well as rare instances when you triple up.


Now lets look at Tournament poker. On any hand you can be eliminated, double or triple through, or win/lose a small or big pot. So even if the expectation value is small compared to your stack, playing any hand in a MTT has the potential to knock you out. What I am trying to get to is that the variance in expectation value is HUGE for all hands. Understanding this variance level is vastly important for success in MTTs. Folding a bunch early is not a great way to build a stack, but it is a great way to reduce the variance that can knock you out in an MTT before the cash payouts begin.


So now lets try to visualize Expectation Value for a simple throwaway type hand. I will use 84o as an example here, but any hand that you would autofold from outside the blinds and play with extreme caution from the blinds will have a similar curve. I am starting with a throwaway type hand because it will be easy to deal with. Playable hands will take much more work to figure out the expectation curve. Since I am developing this as a way to look at MTT results, I am going to use percentages instead of raw numbers. That way as the blinds go up, the curves stay the same. The pots get bigger later in MTTs because the stack size you are leveraging becomes bigger. To simplify things a bit I will assume that you can lose your entire stack on any given hand (somebody at the table has you covered), and you can win up to three times your starting stack (quadruple up). While in theory you can more than quadruple up, in practice that is so rare that we can simply ignore it. So you end up with a graph that goes from -100% of your stack to +300% of your stack. This is how much you can win or lose on a given hand relative to your stack. For each percentage that you may win or lose on a hand there is a percentage likelihood that that will happen. The sum of the product of percentage likelihood times percentage won/lost will give you the overall Expectation Value for the hand. Below is a chart for 84o.

Looking at the above chart you can see that 78% of the time you will be outside of a blind, and will autofold your 84o and have EV=0. In the SB you will fold most of the time, but play on when very priced in. From the BB you will play if you can limp, or could call a min raise with enough pot odds. When you fold your SB you will lose 1% of your stack (assuming a stack size of 50x the BB). When you fold your BB you will lose 2% of your stack. When you play on you are most likely to win or lose a small pot with this hand as shown by the small band around 0%. It is also more likely to double up or get eliminated than to win or lose 30-90% of your stack. You can see the small bumps at -100%, 100% and 200% showing this. When you sum up the EV for this hand, you will find you will lose 0.15% of your stack on average when dealt it, with a small chance to win or lose a massive pot. This is the chart for a conservative player who will not get out of line with this hand. An aggressive player who may steal from the button or cut-off with 84o would have a very different chart. The results would be more spread out indicating much higher variance. Playing aggressive simply increases the variance. Playing 84o aggressively will most likely reduce the expectation value to say -0.25% as well. I am going to stop here for now, but will try to put together charts for suited connectors, small pairs, big pairs, AK and other playable MTT hands. When I get enough of these complete I will start looking at how all of this actually determines your stack size as tournaments play themselves out which will be pretty interesting.