Poker and EV, Part II
In my last post, I wrote about Expected Value and how it pertained to making decisions in poker. After writing that summary in response to a question on one of Microsoft’s email distribution lists, a friend of mine from Tellme sent a tremendously insightful follow-up geared towards more advanced players who might use EV to specifically make tournament decisions.
Here is the response from Michael Bodell:
The other main thing that comes up with tournament versus cash games that is generally over estimated in multitable tournaments (especially early) and underestimated in single table tournaments (or especially underestimated in large satellite events [where everyone who gets the prize will get the same prize – I.e., a flat payout to those “in the money”]) is the difference between cEV and $EV.
cEV is the expected value in chips.
$EV is the expected value in $.
In a cash game cEV == $EV. Assuming you are playing within your bankroll and seriously (so risk aversion and utility can be ignored), then you would make the play that was the most +cEV. [1] So if you take Jason’s example below you’d (assuming you are confident in your read) always make the call.
In a tournament cEV != $EV (unless it is winner take all). In an online single table sit-n-go the payout is often 50% for 1st, 30% for 2nd, 20% for 3rd. Now imagine we are in the situation Jason describes but imagine there are only 4 people left. Imagine that 2 of them are down to $1 in chips, you have $100 left to call or fold, and your opponent has $300 (in addition to what he just bet and what is in the pot). Now if you call and win the situation is 300 to 300 to 1 to 1. If you call and lose the situation is 600 to 1 to 1 and you are out. If you fold the situation is 500 to 100 to 1 to 1. Assume that this was a $100 tournament that 10 people started in. According to cEV you should still make the call (the expected number of chips you’ll have is more if you call then if you fold). According to $EV you should fold (the expected number of dollars you have is more if you fold than if you call). There is a technique using ICM (independent chip model) that calculates your $EV given certain chip configurations [it is modeled as calculating the probability each person wins followed by the probability each person comes 2nd given the other person wins, etc. and is thought to be fairly accurate if people’s skill levels are similar]. 300:300:1:1 is worth $399.17 for you and you’ll get this 40% of the time when you call. 600:1:1 is worth $0 to you and you’ll get this 60% of the time when you call. Therefore $EV(call) = $399.17 * .4 + $0 * .6 = $159.908. 500:100:1:1 is worth $331.28. Therefore the $EV of folding is $331.28. $331.28 > $159.908. So even though the cEV says to call it (it is +cEV), it is still more than twice as good to fold (-$EV).
Early on in a tournament cEV is very close to $EV and you can ignore this, but when you are near the bubble or when you are at a final table with a big ramp up in prize money for each place then cEV and $EV can really diverge. The classic “Is it ever right to fold AA pre-flop?” situation comes up when you are in the bubble of a large satellite. Say 1000 people entered and the top 50 all get a seat at the WSOP ME. There are currently 51 players left and 4 people go all in and you, on the BB, look down at AA. All of the all ins have you covered. Here is an extreme case where it would be massively +cEV for you to call but massively -$EV to do so.
[1] – One thing to note is meta-game. Most of the time people analyze EV (chip or $) a hand at a time. But really one move may be slightly EV for that hand but significantly affect your future EV (negatively affect your table image or adjust the strategy of your opponent to a more correct one). But this is a second order effect, and for most people most of the time doesn’t apply.
I especially like Michael’s inclusion of the well-known situation where it’s appropriate to lay down AA pre-flop. While most people will never experience this situation, I know someone who was presented with this exact scenario (as the small stack, he picked up AA against his remaining two all-in opponents, both with big stacks; the top two finishers received equal prizes) just a couple weeks ago during an online satellite for the WSOP Main Event. Unfortunately, he wasn’t familiar with the strategy discussed above, and without enough consideration, he pushed all-in as well. He lost the hand, and the entry into the WSOP; had he laid down pre-flop, he almost certainly would have won a seat.