It is possible to run above AI EV for 1M hands the same way it is possible to run below EV. What exactly are you trying to ask?


Are you supposing that your are guaranteed to be break even on luck after that many hands?

This is an unusual graph indeed.

Normally with a sample this large the 2 lines will be crossing many times since--despite short-run dis-convergences--they will always be tending to converge.

It's just another example of the absurd amount of variance in poker I guess.

I think there may be another small possible bias that could improve allin ev for a very good HU cash player, again I am not completely sure but...

The idea is basically that the good HU cash player is more often the winner in the final hand of any player vs player confrontation and also this final hand is biased toward the good player being the winner, (the lesser player often walks if losing a stack while the good re-buys and has a second or third shot at a winning final hand).

If a HU player is very good they will often just simply re-buy and play on when stacked possibly with enough to cover the doubled opponent ie, buy in with 200bb or more, or they will simply start off buying in with a very large stack so the opponent is often the effective stack player.

Bad players are more likely to leave the table when they are first stacked. If you stack someone in a all-in Ev context hand you are always all-in lucky in this actual hand (unless holding the absolute nuts, ie 100% equity). Even hands like AA vs Ak you still are all-in Ev 'lucky' if you win with AA and very lucky if you win with AK. ie, the winner always receives the allin Ev luck.

So if players often leave the Hero's table after getting stacked then this will usually be for a smaller pot final hand, typically a 1 BI vs 1 BI, but if the worse player wins and then keeps playing on there is now a chance this table breaks when this player ends up getting a nearly 2 BI stacking off the better player. The better player is more likely to be the one winning the final hand.

Over time and many, many games the good player (due to the skill edge) may be winning more 2x stacked final hands and so the all-in Ev is 2 times the size and positive for these final winning games. If the opponent again re-stacks Hero, the stacks are probably now too large for all-in ev hands but the good player would still play on and wear down the bad until again in range for all-in ev hands say a 300bb sized relative stack.

Would this cause a bias to show up? I am not sure and if it is I suspect it is quite small as usually people will just walk away after a certain length of game time and only some will get triggered to walk by the loss of an all-in hand.

No.

I think the precise opposite is the case. Most of the convergence (of graph lines) is at the start of a sample.

Over a small sample, it's easy to be + or - 2 buyins, so the lines will often cross a few times. Once you're 20 or 30 buyins above or below EV, it's unlikely that you'll have a "streak" of rungood/runbad that switches everything around. As has been said in the thread, your future expectation is to be EV neutral, so if you're +30 BI after 100k hands, your expectation is to be +30 BI after 200k or 1 million. It doesn't "even out" in the long run. Long-term results are on a "random walk". Half the time (?), the divergence (in absolute number of buyins) will get greater over time.

As shown by this variance simulation, the luckiest and unluckiest graph lines are getting further and further away from the mean:

Actually, no matter how far you get from the mean, you will cross again any number of times with probability 1. "Expected value" is a bit misleading here. The results average around expected value but the range of possible results widen around the expected value.

The standard deviation of results from the mean increases proportionally to the square root of the sample size. However the ratio [observed results]/[expected result] converges to 1. Even the fraction [nth unluckiest observed result]/[expected result] converges to 1.