MTT Luck Part 2 - Positive Hands

In Part 1 I described what an Orthodox hand was and how these were essentially flips regardless of the actual cards dealt, if you can get around to this way of thinking for Orthodox hands it may well help you deal with tilt issues when you get that set of 6s cracked by AA as in my previous example. It also highlights how in MTT endgames nearly every hand can become a flip as the play becomes standardised due to shallow stack sizes. So what about hands that we wouldn't play the same as our opponents?

Let us consider another simple hand, it is folded around to Player A, who is in the SB with 10bb, player B in the BB also has 10bb. Player A looks down to see QJs and decides to shove, player B looks at their cards and sees pocket Aces so snap calls the shove, the board runs out A27 T K giving player A the pot with a runner-runner gutshot, how sick was that hand? Note that player B would not have shoved the QJs if they had been in the SB, so this is not an Orthodox hand as discussed previously so we can not consider it a 'flip'. As such, on the face of it, it looks like player A got terribly lucky and player B terribly unlucky but luck in this hand is not a zero sum equation; that is that Player A is not as lucky as player B is unlucky. Why is this the case? Well let us analyse the hand again but this time we will do some basic calculations of the expected value of each players move. If you don’t want to get bogged down with the maths, skip this bit and go to the final figures I give in the table.

For simplicity let us assume at the start of the hand each player has exactly 10,000 chips, the SB is 500, the BB is 1,000 and there are 10 antes of 100 each in the pot, so the pot before any action is (100 x 10) + 500 + 1000 = 2,500 chips.

It is folded around to player A, who after posting the SB and ante has 9,400 chips left, they expect the BB to only call a shove with the top 20% of hands, so if player A shoves, player B folds 80% of the time. Of the 20% of the time player B calls, then player A has 44% equity against the top 20% of hands (check PokerStove to work this out if you wish).

We now have 3 possible outcomes: 

1.             80% of the time player A wins the pot of 2,500 chips so their stack becomes 9,400 + 2,500 = 11,900, the EV here is 0.8 x 11,900 = 9,520     
2.       When player A calls, player B still wins 44% of the time, this happens 44% x 20% = 8.8% of the time, the EV for player A here is 1,830. Calculated as 20,800 (the total size of the pot when both players call) x 0.088
3.       The rest of the time 11.2% player A loses and has zero stack

We now add these 3 figures together 9,520 + 1,830 + 0 = 11,350, this is the prior expected stack of player A after he shoves and this is a gain of 1,950, this is an increase of over 20%! (Prior Expected Stack is the stack that player A can expect to have on average when they shove prior to player B acting).

What happens to Player’s A’s expected stack size when B calls with AA, well AA v QJs is 80% v 20% so the posterior expected stack (This is the Expected stack of player A after B calls and shows AA) of A is 20,800 * 0.20 = 4,160. 

When player B sees AA, they don't really care what player A has, but let's assume that they expect play A to shove the top 50% of hands, player B has 85% equity against this range , we can calculate player B’s Prior Expected Stack (this is prior to Player A shoving  but given that Player B knows his own cards) and this is 14,540.

Similarly once player A shoves, we can work out that Player B’s Posterior Expected Stack is 16,640. I have summarised the scenarios and outcomes in the table below:

As we can see, both players have a positive expectation at the time they make their action (their Prior Expected Stacks are above their current stacks). I will label these types of hands as Positive Hands, in which both players make plays that have a positive expectation at the time given the info they have. Of course this is subjective depending on factors such as playing styles and there may be multiple streets of action some of which maybe slightly negative or have implied odds calculations etc but it highlights a good principal.

How can both players have positive expectations in the same hand? Well there are dead chips in the pot, but also player A makes a move against an unknown random hand, which means his shove is profitable long term. The fact here is that each player does not know his opponent’s hole cards so they both have different expectation levels to those that exist once the action plays out and the cards are shown. Note that the sum of the 2 Prior Expected Stacks do not equal the sum of the 2 stacks, but the sum of the Posterior Expected Stacks do, this is because the Prior stats take account of folding and ranges of random hands, the Posterior Stats have complete information. Once the hands are known one player can only gain at another’s expense as is seen in the table above.

So that’s a lot of number crunching, how does this fit in with luck in MTT’s?  Well we can see that when player A shoves he makes a good move that is profitable long term, but all of a sudden he expects to lose lost 7,190 (11,350 – 4,160) because player B has AA, this is out of A’s control so A can rightly say he has been very unlucky.

Also, Player B obviously makes a +EV move since he has the nuts at the time, but he has been lucky to get dealt AA at the start of the hand in a spot where player A can make a +EV move, he has a big +EV situation at the start of the hand but this increases further when player A shoves, even though A does not make a mistake by shoving. Since A has not made a mistake, we cannot state that he has been outplayed, as such player B is very lucky to have found himself on the right side of a positive hand. This is an often overlooked side of luck, it is very easy to get AA in the BB and the SB finds 84o and folds, the maths shows that AA wants the shove long term, but this is not under his control so he is lucky when he gets the shove.

Naturally when player B wins with the AA here he will want to congratulate himself on doubling up, but his double up is entirely down to luck, therefore when he loses with AA in this spot he should take it in good grace, this hand is in the hands of luck much like the orthodox hands but only these spots generally favour one player over the other since both players wouldn’t play it the same way. Player B is losing out in the long run when they have QJs in the SB but fold here, but this does not mean that they should begrudge player A winning occasionally when they shove QJs, remember QJs should beat AA about 20% of the time. In the long run here if Player A shoves QJs from the SB but Player B would fold, player A will crush player B assuming a random hand in the BB. Note this means there is a skill element in the prior element of the hand for Player A, but not for Player B, as such Player A has more right to be aggrieved at being unlucky that Player B, even though player B actually has the best hand when the stacks go in!

In summary, Positive Hands are hands where both players make a move that has a positive expectation prior to them knowing their opponent’s hole cards, however if the roles were reversed each player would act quite differently so these cannot be considered Orthodox hands, if both players were to act the same way then the Orthodox hand logic from part 1 should be applied. In Positive hand situations it is not necessarily the player with the best hand when the stacks go in who should feel the most upset when they lose the hand, but often it is the player who has the worst hand but runs into a better hand. They gain the most against a random hand but also lose the most when they are unlucky to run into the top range of hands. The player with the best hand also gains when the other player makes their own positive move which in itself is down to luck, so this player is lucky in two ways, firstly that they have such a good hand and secondly that their opponent has a hand good enough that they can make a positive move, therefore when you lose with the best hand in these spots you have already used some good luck up at the start of the hand. I have intentionally given an overly simplified example to illustrate my general point but there are lot of permutations and other factors involved of course. Try to bear this example in mind next time someone shoves a positive hand into your AA which gets cracked.