I looked at a lot more than that, for multiple table sizes and multiple sites. What you are referring to is a small test run that is published, and it was at full ring tables.

The bias is caused by the decisions of all players to decide whether to see the flop or fold, and then to see the turn or fold, and then to see the river or fold. The cards that are in the deck stub to deal to the board are biased by player decisions, because not all flops are seen, and even fewer turns are seen, and even fewer rivers are seen. A preflop all-in equity calculation is only 100% accurate in a heads-up game (but still pretty close in all games).

Actually one way you could easily create an underdog effect with HU PFAI is just to introduce some collusion between players:

I ran a 10K all-in simulation with the condition added that 3 players are sharing hole card information and never go all-in if one of their hole cards is shared with one of the other colluders. This creates a slight swing as demonstrated in the graph below:



I am still struggling to come up with a mechanism to create a swing in 'normal' play as affecting HU PFAI's so look forward to your ideas. Having built the simulator it is quite easy to introduce some rules to influence which hands end up all-in against each other. Personally, though, collusion seems much more likely than any rules based on betting behaviour.

Incidentally I did also run a 10K simulation without collusion just to check that there was no bias being caused by my software and there was none.

Just saw your latest response, Spadebidder. I don't see how what you are saying is relevant to pre-flop all-ins with only one caller. Can you possibly clarify?

Other people fold. And they don't do so always and they don't do so at random.

If you analyse a large sample you will end up with a complete mix of different playing styles in all possible positions on the table. One guy shoves then each of these other players will have some rule best known to themselves to decide whether to call or not. But if none of them have knowledge of the other hole cards then how on earth is this ever going to lead to a bias to make worse hands win more often than they should?

I have thought about it and have some theories but they have not been tested by simulation. What has been tested is actual results, across multiple unrelated sites in multiple game styles and stakes, for large numbers of hands. And it is a pretty consistent result in large samples, where in preflop all-ins underdog hands in the aggregate tend to win *slightly* more often than a preflop equity calculation would predict.

It should not be surprising that there is some kind of equity bias one way or the other in multiplayer games, as we know positively that seen flops are biased by folded hands, due to card removal effects, and we know that this bias is that low cards show up on the board more often than high cards. That isn't in dispute. Relating that directly to a specific equity skew overall for all-in hands is a pretty complex question. We can easily think of some examples where the underdog does gain equity, however. Like the Barry Greenstein example I've written about and tested. Certainly there are also cases where the underdog loses equity due to folded hands. The net result of all the various scenarios (innumerable ones) seems to be that underdog hands get a boost a little more often than favorite hands do.

I have no problem with the idea that seen flops contain more low cards than high cards, it is easy to see why that should happen.

The Barry Greenstein example says that if the board folds round to you and you have AK then your equity against a PP rises slightly. But I don't believe it is true once someone has shoved preflop because most Ace rag combinations will fold too. And most players would never try to exploit this slight bias anyway in deciding whether to call an all-in or not. Finally, it only is affecting equities close to 50% and it is in the 60/40 and above matches I see the underdog effect most pronounced.

If anyone can come up with any kind of theory then I can easily modify my code to change the algorithm which determines which 2 hands out of the 10 end up head to head. If you allow cheating its very easy but if you don't then I would be amazed if any algorithm that was realistic as to how your average player decides whether to call or shove preflop would create a bias.

It should be somewhat more pronounced. With the exact same effect size (say 0.5% more hands won), the SD offset will be greater for a 70/30 than for a 52/48, because the unit 1SD is almost 10% smaller for the 70/30. Put another way, the closer the expectation is to 50% the higher the normal variance.

No one is going to call with 27o. Ok...someone is...but not very often. Yes, people play differently...but not that differently. You can make some pretty safe assumptions about the hands people will fold. And some reasonable assumptions about the hands that will call.

For example, consider a situation where player A goes all in. Player B calls. Player C has two cards. If player C folds, this ends up as a Headsup all-in preflop hand and you analyze it. If player C calls, it doesn't end up in your little database and you ignore it. Will player C call and fold at random? Let's say player C is very tight and will only call with AA. Even still you are going to see a bias in your sample. Because an A is more likely to flop in your database where player C has folded.

Maybe I don't understand your complaint though, because this seems like pretty obvious stuff.

C folding means an Ace is more likely to come out on the board, yes. But how does that on average help the underdog? If the underdog is the guy holding Ax and he is vs a pp or Kx then he gains equity. If the underdog is 88 vs 99 then it makes virtually no difference. If he is holding KQ vs AQ then his equity goes down. On average there is no change to the underdog's equity.

The only way I can possibly see the equity changing without cheating is if the caller can gain some information by the play preceding his call as to the most likely distribution of other hole cards which will sometimes change his decision to call or fold. But more often than not the caller is the overdog anyway, which means it should cause a minor favouring of the overdog not the underdog.

You have an awful lot of bad and/or incomplete assumptions in that short post.

In Laughing Assassin's data set for PFAI he was coming up with an average equity for the caller of 52% which I found surprising. With my own data set it was also over 50% but by a smaller amount (I think about 0.5%). That is my basis for saying that the caller's equity is more likely to be higher than the shover's.

You can also not deny that the caller is acting on more information than the shover, and this can only be to his advantage.

I would therefore argue that there is more likely to be an overdog effect than an underdog effect.

Please be specific about how you think Sherman's example can lead to an average increase in the underdog's equity.

Getting it in good has absolutely zero to do with all-in results compared to equity. Zero. Statements like the above make it clear that you are thrashing around without a real grasp on the subject.

The outcome of preflop all-ins is 100% controlled by the makeup of the deck stub. The deck stub for seen flops is not totally random, which is very easily proven with simple thought experiments and confirmed with real experiments. You've accepted that this is known to be true. The question you have been pursuing in the latter part of this thread is if that known non-randomness favors the underdog in preflop all-ins more often than the favorite. Common sense says yes (slightly) from what we do know about the bias. Empirical results seem to confirm our intuition. Proving it is difficult.

I am not talking about 'getting it in good' when I spoke of acting with more information. I am talking about information regarding the folded hole cards. So it theoretically could affect what cards are in the stub. Personally I think its highly unlikely, but if it did happen it would favour the caller.

Frankly I don't think abuse is a very good substitute for a counter-argument.

Then you go on to talk about seen flops which we both know is totally irrelevant to this discussion....trying to derail the argument.

Finally you say 'Common Sense' says the underdog effect should exist without cheating or rigging. Yet you are unable to come up with a single mechanism which could account for it in preflop all-ins. Empirical results confirm the effect but not your cause for it.

The Underdog Effect exists, but unless anyone can show me how card removal can possibly affect PreFlop All-ins I am going with what to me is a far more common-sense conclusion - it is due to either collusion, another form of cheating, or rigging. I challenge you to disprove me.

Correcting obvious errors in logic is not an attempt at counter-argument. I never attempt to argue with riggies, and that has come out clearly now. You're on your own.

I've unsubscribed, so don't bother.