How their bad calls can cost you chips - Implicit Collusion

I ran across this while doing some research for a coaching session I’m giving to a new student later this week. Its called Morton’s Theorem and relates to how expectations of the player with the best hand change when other players either fold correctly or chase incorrectly. Interestingly enough, depending on the size of the pot, having players chase incorrectly can still result in negative expectations.

This theorem seems particularly relevant to the games being played here (loose). I think it also helps explain why people perceive that chasing pays off here more than the basic math suggests it should. If the theory is correct, it is not that draws hit more here than they should or because of any flaw in the RNG. Instead, it shows that the player with the best hand may still lose expected value even with being called incorrectly by 2 or more others at certain pot sizes.

Its interesting to me and I’d love to hear what other players think about both the validity of the theorem and how it may apply to the games being played here. Here is the 1st paragraph about the theorem and a link to the full article:


My impression of this theory is that it is simply describing the proportional amount of the best hand’s equity that is being transferred to the 3rd player by the incorrect call of the 2nd player. However, this doesn’t explain why there are differences in expectations across differing pot sizes. I honestly don’t know if this theory is valid at all and that is why I am presenting it for discussion.

If the theory is valid, does it help explain the perception that chasing pays off? It wouldn’t mean that its paying off for the chaser but instead that it is negatively impacting the player with the best hand and distributing some of that equity across the other players. Most importantly, if the theory is valid, what would be the best counter-strategy to employ?

Warlock, excellent post. Great food for thought. I’d say the best strategy (which I’m sure you’re already employing) would be to open fairly large and be willing to 3- and 4-bet preflop to knock weaker hands that are more likely to have draws on later streets out of the pot. That way you’re more likely to be playing heads-up after the flop.

If you still end up heading multiple ways to the flop, probably the best advice is to bet your strong hands (overpairs, sets) heavier and deny equity, then throw on the brakes for any turns/rivers that complete straight and/or flush draws.

That would be the conventional way of dealing with loose play but this theory suggests that increased pot sizes create the issue to begin with. If that is the case then inflating the pot is not going to be effective as a counter-strategy. In fact, this theory would seem to suggest decreasing open raise sizes from EP would help. Keeping pot sizes in that paradox range or smaller would appear to be optimal but I’m just not sure if that is true or if it is, how best to get there.

Would love to hear from any players/staff who have considered this before and from anyone with any thoughts at all about the subject. I think this could be a great open conversation and could possibly shed some light on a lot of the issues players seem to be having here.

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I believe this is the original thread in which the concept was proposed and later named. I suggest reading it through to see how it was developed and brainstormed. It looks like it was started in April 1997 and was meant to refute part of Sklansky’s Theorem and some of Mike Caro’s work:!topic/

For me it looks like a simple poker wisdom is put in so many words: Bet low if you have good cards to encourage as my betters as possible. If you have mediocre cards bet high to jam as many players as possible.

It works well in tournaments where the stack is limited and the blinds get high. But not in a ring with more than six players where the stacks are unlimited. Mediocre hands (high pair, good kicker or two low pairs) covers only 30% probability, you will find some callers and they are more likely to win. It breaks even with high pairs which covers 50% probability. In other words, there will be chasers and they will be rightly chasing, if your probability is less than 50%.

Hi I was worried when I saw the title, but I don’t think this is encouraging players to collude

This is my take out too.

At the risk of stating the obvious, the question is: Is it best to take on one player with a 60% chance to win or 2 players with a 50% chance?

An edge case may be in tournaments where our tournament life has value, but conversely setting multiple players all-in may be significant too

The expected reward is higher and the risk is usually the same, (unless contestant #2 has more chips), the long term business proposition says we usually want more players along for the ride.

The second or subsequent player could be making positive expectation calls from their perspective and this could be due to our bet sizing.

Poker is Deep.


I have thought about this issue a lot (without knowing what it was called) because it really is the central dilemma when playing on Replay, especially in ring games where stacks are deeper and there are potential implied odds. There was a post here once where somebody called it “fish schooling”. This term may apply more to preflop (if everybody calls then everybody gets a great price) than Morton’s theorem, but the dilemma for someone looking to bet for value is similar.

I really disagree with the idea that making the pot bigger when you have a made hand and playing cautiously with more marginal hands is the right approach. In fact, being willing to make the pot huge gives your opponents great implied odds to chase their draws. Many players at the highest stakes on Replay have built their bankrolls by betting huge for value and getting called by opponents getting bad prices, but this approach is extremely exploitable without balance.

I noticed that the article discusses an example from limit hold’em. Perhaps the ability to make bigger bets in no-limit would reverse the effect because the price given to continue drawing would be worse than is possible in limit. I still think that massively inflating the pot with a vulnerable made hand is not the optimal strategy.

A similar logic applies preflop. If you raise AA from UTG, you would rather have 1-2 callers than 7-8, even though you would be creating a bigger pot with the highest amount of equity. I need more time to wrap my head around this idea because it is not about how multi-way pots give opponents the correct price to draw, but rather that the bettor wants some opponents to fold correctly in multi-way pots, at least in some situations. From a practical standpoint, I am not sure how relevant the theorem itself is because you cannot know whether or not both opponents are drawing to separate draws as opposed to having top pair, or both opponents may be drawing to the same flush (decreasing their individual and overall equity).

Edit: the article explicitly mentions “schooling”. Also, it is interesting that the incorrect chaser’s action most directly benefits the correct chaser, rather than the original bettor. This also makes sense according to the idea of schooling, which, again, explains why there is so much chasing on Replay.

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Not at all Rob - the condition was called implicit collusion because the effect was the same as explicit collusion but the players weren’t actually working together consciously or formally. I did include it in the title because I knew the phrase would get eyeballs :slight_smile:

As to your thoughts - it looks like this was the debate he was having with Mike Caro although Caro had some really weird thoughts on it (what else is new?). The debate between raising and not raising from EP seemed to be a main point of contention, with Caro suggesting not raising, in lower stakes limit games at least. I don’t see much on the topic directly relating to NL though he addressed Sklansky in the NL context so I’m confused on this point. Poker is deep and many great minds have given it serious thought over the years. I’m just trying to figure out what those great minds meant.

It does but it differentiates between schooling (those calling correctly) and implicit collusion (calling incorrectly) “Schooling occurs when many opponents correctly call against a player with the best hand, whereas implicit collusion occurs when an opponent incorrectly calls against a player with the best hand.”

The example is also from Limit but the application I believe was supposed to apply to NL as well. I think the variables become too great to run a proof in NL so he used Limit instead. This is just my best reading of it though so I could be wrong.

Cool stuff indeed and when I read it I immediately thought of the conversations going on here. Many of us have thought about this yet this was the 1st time I ever heard of the name or the theory. It seems to apply - now if I could just find a similar article on how to address it :slight_smile: I’ve looked and haven’t found a definitive counter-strategy yet.

The biggest head-scratcher for me so far is the thought that opponents folding correctly can actually be beneficial to us. Total 180-degree shift from Sklansky.

Not sure if this is directly related, but I was looking at Equilab to look at shifts in equity in multi-way drawing situations and here is a spot I found interesting.
Player A holds AdAh
Player B holds 9h8h

The flop comes Th7h2c. In this scenario player A has 47.4% equity and player B has 52.6%. Pretty standard flip. But if you add a player C who holds KsTs, this new player has 10.4% equity, Player A’s equity goes down to 35.7% and player B’s equity actually goes up to 53.9%.

This seems like a prime example of Morton’s theorem in which player A may want player C to fold correctly (at least in some spots depending on the bet size) because player C’s equity share disproportionally goes toward player B. This seems to be due to players who are drawing having clean outs (e.g., any heart, J, or 6) while players with made hands are unlikely to improve (e.g.,to a boat by the river), so their equity is divided among additional players as long as those players have live outs. It also shows the beauty of combo draws, especially in multi-way pots.

This example makes me think that Morton’s theorem is true (the equity share of player C is not divided equally among players A and B, altering the expected value of bets). Is it fair to say that Sklansky’s theorem only holds for heads up pots? But from a practical standpoint, player A still wants to bet when they are likely to have the best hand, and they are able to re-evaluate if a heart or other scary card hits the board.


Thanks Joe. This looks like a very good example, so long as we define “best hand” as best made hand. Otherwise we would have to start looking at it from the perspective of the player holding 9h8h after the flop. I am starting to get a clearer picture of how this plays out in limit games but still not sure how to reconcile it with NL. The best I’ve got so far is that at a certain point SPR’s become so low that there is no folding certain made hands anymore. I don’t see how varying pot sizes come into play otherwise, though I may just be being thick here.

Unfortunately there doesn’t seem to be a lot of material on this theorem. Andy Morton died at an early age (motorcycle accident I think) and it doesn’t look like his work was picked up by anyone afterwards. Caro somehow managed to come up with a scenario where Morton wasn’t actually disagreeing with him and left it at that. I’m not quite as convinced as he seemed to be and Morton wasn’t around to continue the discussion.

This seems to be fertile ground to explore, especially in the context of the play on this site at most levels. I am enjoying thinking about it and hope we get more input from others who may have thoughts on it as well. I’m going to continue to work on it and once I have a decent grasp of the dynamics, start working on scenarios with more than 2 callers. I need to figure out how to work in pot sizes first for NL applications. Assuming we get that far, then the fun will begin in figuring out a solution.

Added: Morton’s premise was that Sklansky’s theorem only applied to HU and to some multiway pots but not all. I’m amazed that I never heard of exceptions to Sklansky until now. You’d think that if a concept as well known as the Fundamental Theory of Poker had exclusions, almost everyone would have heard about them. Morton’s theory seems to have been mostly forgotten or ignored though.

I could be completely misunderstanding this, but it seems like in NL the theorem is true (as in the example in the wikipedia article, someone could use the same big bet size in NL), but in NL the bet can be in any proportion to the pot, so there is an upper limit to the “paradoxical region” meaning that at a certain bet size the player with the made hand (and the lead in overall equity) will want the additional player to call.

Take this example using real cards (hands omitted for simplicity). Player A has 60.91% equity heads up vs player B who has 39.09%. The pot is 1000 and a bet of 500 has 718 EV for player A. Adding player C with 13.07% equity lowers player A’s equity to 46.73% (still leading) and player B’s is now 40.2%. The EV of a half pot bet from player A is now only 668, a decrease of 50. Morton’s theorem holds because player C is getting the wrong price to call but player A still does not want a call.

But if player A bets 2000 into the same 1000 pot, their EV heads up vs player B is 1046, but including player C raises their EV to 1271. So maybe I was wrong before and big bet sizes actually are the way to deal with this dilemma. The variability of equity share between players alters the EV of a bet multi-way, but having the lead in equity means a bet that gets additional callers will always be optimal if it is large enough. Does this make sense?

Just running in and out and caught this. I believe in the original article there was an upper limit to the paradoxical region where player A wants player C to fold correctly so this fits. I think you are zoning in on it but I’m not sure we are at optimal quite yet. At some point the bet sizes become so large that we likely aren’t getting any calls at all, whether we want them or not. Therefore, if we make outsized bets we would lose EV by losing all incorrect calls. I’m still not sure how the pot sizes play into this in the first place. More to think about. How do we locate the sweet spot where we shed the 1 caller but keep the other? More bothersome, how does this play out with more than 2 callers? I’m starting to wonder if there is actually a solution to this at all or whether it simply is an exception to Sklansky and has no optimal solution?

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An excellent article. I am glad you posted it.
Isolating a problematic situation using the psychology of poker and finding the best solutions using math. It reminds me of something I saw a long time ago about playing sixth street in seven card stud.

Thanks for the post.

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The fundamental theorem is about playing as if you could see your opponent’s cards, but in practical terms it seems more relevant to making deceptive plays that disguise your hand (which you know) than to trying to play as if you know your opponent’s hand (which you cannot see).

It seems impossible to come up with an optimal strategy when you never know your opponent’s hand (only range assumptions) and you cannot know what bet size they may be willing to call with any particular hand type.

You could be thinking at meta level 4 while your opponent is on level 2, and you could make the optimal play according to the theorem for the wrong reasons. I think the issue that morton’s theory raises about how equity share of a bet is distrubuted is the most interesting part, but in terms of actually making a bet with a made hand, I think the range-based decision making isn’t altered by the validity of either theorem.

Both these theories were put forth long before hand v range and then range v range were even being discussed. They may have limited applications when competing with players using more GTO-based strategies. Still, there are hoards of players who have not made that shift (and never will), especially at lower levels. I’d think the older strategies and theories are probably as applicable to these games today as they were 20+ years ago.

I’m sorry if my thought processes were a bit foggy earlier. Feels like I’m swimming through Jell-O in my head today. So, in the 1st example we had a situation where it would appear that player A would always want player C to fold, regardless of bet size because of how C’s hand redistributes A’s equity. In the second example, player A still maintains the equity lead, even with player C in the pot. His margin decreases but I’m not seeing how his EV as a percentage of the incremental bet should change based on the size. I’m spit balling here but something about the math seems off to me.

I wish there were more discussions about Morton’s Theorem from the time so that we could see specifically where he thought this condition manifested most prominantly. All we have is the one example from limit and not even a robust discussion about that. It does seem that he found an exception to Sklansky and maybe that was all he was trying to do? When I’m up to it, I’ll try to play around and see if I can find any groups of hands that are more susceptible to this condition than others. You may have picked up on that right away in selecting strong made hands that have very little room to improve post flop vs multiple unique drawing hands. Will we see the same equity redistribution for 2nd pair hands looking to make 2-pair or for pocket pairs looking to pick up sets or is this mostly applicable to conventional drawing hands such as straights and flushes?

Very glad you are enjoying it @SSeville. It was so nice to find something like this - seemingly long forgotten but fairly powerful and relevant (IMO). It may turn out that this was only meant to describe something that had not been previously quantified and that is as far as its usefulness goes. Even if that is the case and there is no optimal counter-strategy, just airing it out and giving it thought would be worthwhile.

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A post was merged into an existing topic: Some reflections on a parallel case

Well, it seems that this topic has been fairly well-received so far. Lots of views, some great comments and I’ve even received some PM’s about it. I am very happy that this bit of poker-theory from the past has generated so much interest. I hope the thread continues to attract attention and that the discussion moves forwards.

To that end, I would like to make sure that everyone knows that their thoughts are welcomed here. No need to PM me, just post directly and lets all enjoy it. I understand that some people may be reticent to post for whatever reason but I’d like anyone and everyone with something to say to participate. If people think this theory is valid, let us know. If others think this is total fallacy, post that. Don’t think you have to be a theory-wonk to get in on the discussion, please. The more the merrier and hopefully we can all enjoy the discussion that follows. All I ask is that posters stay on topic and even if they think a comment is wrong, refrain from making the counter-point personal.

Some of you know that I am experiencing a bit of a medical thing. This leaves me at times unable to fully engage in what I think is an interesting subject here. I would appreciate any and all help in keeping the discussion going and to developing a clearer grasp of what Morton put forward. Does it help explain certain behaviors or perceptions? Does it apply as much to NL as it seems to for limit games? If it has validity, are there counter-strategies to maximize our expectations? Lots of questions so far and not many answers so there are many ways to contribute if you’d like.