I took the time to write a fairly simple hand to show how my EV calculations match up with actual results. The hand is as follows:
JsTs BTN $200
AdAc SB $400
Qh6h BB $400
Preflop: BTN raises to 40, SB calls, BB calls.
Flop: Jh Th 2c
SB Checks, BB Checks, BTN Bet 160 (all in), SB calls, BB calls
Now this is where we can see different results. If the turn is not a heart the SB will bet all in and the BTN will fold. If the turn is a heart, the SB will check, the BB will bet all in then the SB folds.
I will now input this hand into CardRunnersEV which uses a monte carlo
method to determine EV. It ran the hand 5,000,000 times to get an average for the EV amount.
So let's see if the EV method I'm proposing would arrive at that same value $168.41.
The EV calculted would depend on the card we saw on the turn. The following list is of the equity JsTs against the remaining hand (there is no situation where JsTs is not all in against a single hand on the turn).
As: 0% (1 card)
Ks, Kd, Kc, Qs, Qd, Qc: 73.81% (6 cards)
9s, 9d, 9c, 8s, 8d, 8c, 7s, 7d, 7c, 5s, 5d, 5c, 4s, 4d, 4c, 3s, 3d, 3c: 80.952% (21 cards)
Jc, Jd, Tc, Td: 95.238% (4 cards)
2s, 2d: 9.524% (2 cards)
6s 6d 6c: 83.333% (3 cards)
Ah, Kh, 9h, 8h, 7h, 5h, 4h, 3h: 9.524% (8 cards)
2h: 14.386% (1 card)
So if the turn was a As, our EV would be -$200 because we invested $200 to have a 0% chance of winning $600.
As - EV would be -$200
Ks, Kd, Kc, Qs, Qd or Qc - EV would be $242.86
9s, 9d, 9c, 8s, 8d, 8c, 7s, 7d, 7c, 6s, 6d, 6c, 5s, 5d, 5c, 4s, 4d, 4c, 3s, 3d, 3c- EV would be $285.71
Jc, Jd, Tc, Td: - EV would be $371.428571
2s, 2d - EV would be -$142.857143
6s 6d 6c - EV would be $300.00
Ah, Kh, 9h, 8h, 7h, 5h, 4h, 3h - EV would be -$142.857143
2h - EV would be -$113.685714
In our EV calcualtions we would never see the value $168.41. But what happens when we take the weighted average of these values?
There is a small difference of $0.04 but this is due to not running enough simulations and rounding errors. I can run more simulations if you request or I could send you the CardRunnersEV file so you can see it for yourself.
The EV method I propose would work perfectly because its average matches with the actual results. There is no bias caused by calculating EV. In the current PT4 program, no EV would be calculated at all because they mistakenly believe it would be biased somehow. Holdem Manager correctly calculates EV in these situations. The EV calculation would not match the EV we knew, but its average would. It converges faster than actual results, which is what we want.