Poker is a game. Poker is a game of negotiation. Poker is a game of negotiation where money is the score of winning and losing.

For a while now, I have been interested in a model of No Limit Texas Hold'em that views concealed hole cards as "stock options" in a potential showdown. Revealed hole cards are actual stock holdings that have actualized equity in a pot that will be awarded by the dealer to one or more winners.

Now, applying this model to Hold'em should be relatively straightforward. There is a solid foundation of mathematics that can compute the most profitable course of action for almost any "stock option" in a given preflop or postflop scenario of Hold'em. The solvers are pushing the boundary of human understanding of "GTO" poker. As the computing power of the average home computer continues to expand exponentially the capability of "bots" is quickly ushering in an era of "solved" No Limit Texas Hold'em.

So, is poker dying? Online poker, at high stakes, will have to come up with a security solution against bots. However, as long as the participants are all human, as in live poker, there will always be poker to be played.

Why is this? No matter how many hands of "GTO" poker are played, no matter how user friendly and instructional the simulators become, in the end the players are human. Humans make decisions which are not optimal. Humans "misbehave".

According to 2017 Nobel Prize winner for economics Richard Thaler, actual people are not perfect economists (econs). People make incorrect economic decisions all the time. The nature of these mistakes is the foundation for Thaler's work in economics. Thaler is the father of Behavioral Economics. As a social science, economics is supposed to lead to a better understanding of society and thus lead to ways to improve society.

What is the point of understanding with fine detail the decisions people "should" be making if those people continue to make wrong decisions? If the world was populated with "econs" then simple dissemination of the latest understanding of economic theory should lead to increased optimization and a society that is closer to equilibrium. Thus the rub. Humans. Thaler's groundbreaking work is about NUDGES. Society can "nudge" humans into more optimal decisions by something as simple as placing the most optimal option at the top of a list of options that a free society allows its members to choose from. Or, society can make an obviously beneficial option (such as a company matched 401k retirement plan) a default, where the free members of society must actually visit the HR department to opt out, if they so choose.

You may have noticed a couple of words from the previous paragraph that are common among poker theorists. Optimization and Equilibrium. This is not just a coincidence of similar ways of thinking. The presence of "Optimal" decisions and the balance of "equilibrium" in poker theory comes directly from economics, and we even refer to equilibrium as "NashE" in honor of another Nobel Prize winner in economics, John Nash (The father of game theory).

So what does all this have to do with current poker theory? How does any of this put more money into your bankroll? Firstly, there is such a thing as perfect GTO poker. It is there, wether or not WE know it, believe in it, or act upon it.
Secondly, human beings will NEVER play perfect GTO poker.

These two statements are the core of what I am proposing. By studying the mistakes humans make and why they make them, poker players can be "nudged" into making even more sub-optimal decisions that they would make in a vacuum. Instead of making the correct decision the first of only two possible decisions (polar), a poker player can leave the optimal decision buried in a heap of decisions that will overwhelm a lesser player and lead to increased sub-optimization.

Instead of endeavoring to become a human "GTO Bot" totally indifferent to our opponent, I propose that we can better understand our human opponents and the mistakes they make. We can "nudge" them into further mistakes.

Make no mistake, I am not trying to redefine what is "optimal" in poker. Nash Equilibrium is a thing, it exists. Decisions that meander away from NashE are by definition sub-optimal.

However, poker is a game of negotiation among human beings. There is no extra money awarded to the most optimal player. Only the player who wins the most money, most often with the unanimous consent of all his human opponents, is the winner.

Loss Aversion: Mathematically equivalent gambles are treated differently by humans.

Economically, a 100 percent chance to lose 25 dollars is equal to a 50 percent chance to lose 50 dollars. Either holding is identical. This should be easy for cash game poker players. Losses and winning come and go, back and forth. However, theory and research has shown that humans don't think this way.

Human beings (by a wide margin) prefer to have the 50/50 chance of twice as much loss, so long as there is the equal chance to avoid the loss entirely. This is called Loss Aversion, and is central to all the further concepts in Behavioral Economics. What I find particularly HUMAN about loss aversion, is that the very fear of loss is what motivates humans to further expose themselves to greater and greater potential losses, all in the hopes of avoiding loss entirely. So human.

Personally, if I have to take a 25 dollar hit, I would hope that I would just accept a lower variance 25 bucks deduction rather than add unnecessary variance with a 50/50 chance to lose twice as much. But that is quite probably bulls___t. I am human, and I really really REALLY hate to book a session as a loss, even though sessions are irrelevant and all things being equal, walking away and getting some sleep is usually +EV versus chasing losses while tired.

It goes further. Theory and research show that humans hate losses TWICE as much as they like winnings.

In a game of ever shrinking edges, Human beings are experiencing losses twice as negatively as they experience winnings. How 'bout another round of coin-flips for stacks eh?

A LAG (loose aggressive) poker player who can conquer this irrational human preoccupation with avoiding losses, embrace variance, while maintaining the correct odds to gamble with, can emotionally destroy a nit (a very loss averse poker player). Add in carefully crafted wordplay to emphasize that a player is "stuck" and that player will be highly motivated to recover his losses. That is when mistakes happen.

The topic of irrational, exploitable population tendencies is a good one to explore.

But based on some things you're saying, I feel the need to add a caveat: market participants are a mixture. There are indeed the masses of irrational actors who are the farthest thing from Econs, but that doesn't mean there are no Econ-like actors. If there weren't, then sportsbetting markets (for instance) would be far less efficient. The reason sports lines are mostly accurate is the presence of a few deep-pocketed Econs whose wagers have a much larger effect on the line than many other participants combined. I'm pretty sure financial markets tend to work similarly.

Stating the obvious, the Poker economy is also a mixture. There are the masses of live-tards whose "thought" processes expose the listener to the risk of losing IQ points, but there are also the Doug Polks.

One doesn't need to be a GTO wizard to behave rationally at the poker table. A player who lets Math rule at all times is acting rationally, even if playing exploitatively. To your example of risk aversion, if you understand growth and the Kelly Criterion, when presented with two equal-EV options you'll choose the lower-variance one without batting an eye. As for being stuck, not everyone tilts.

That said, I look forward to your next installments. Certainly these behaviors you describe apply to a large chunk of the poker-playing population.

Risk Aversion: Buying and Selling Poker Insurance

As can be watched in classic episodes of televised high stakes poker, the practice of poker players exchanging "insurance" is a thing. When a player is all-in and no more betting can occur, the players involved in the hand can buy and sell "insurance" (so long as allowed by the house) from any player at the table.

Basically, the holder of the majority equity of the pot can make a side bet with anyone willing to gamble, that will cover losses if that player winds up not winning the hand.

Odds are negotiable, and favor the seller of the insurance. This is all simplified to say that the holder of X percent EV is willing to accept <X percent actual money from the hand, to avoid the possibility of losing the hand and losing all equity.

This is closely related to the human irrationality of Loss Aversion (remember that most applications of Behavioral Economics in poker will be somehow related to Loss Aversion) but deals with the other side of things. The irrational fear of losing something that you hold is called Risk Aversion.

Even though the hand is not over and there are still cards to be dealt, poker players understand expected value and know what a given holding is worth versus the random cards yet to be dealt. Poker players can feel the pain of losing money that is not technically part of their endowment. The money is actually in escrow with the dealer and does not belong to either player. Weird huh?

Now any self respecting econ would never buy poker insurance, and always sell it to anyone willing to overpay for it. I suppose two econs could agree to split the pot exactly by equity, using an appropriately sided die to award the remainder chip. That would be a particular phenomena among gamblers called variance aversion, I suppose. Many players (although some top pros decline) like to have the dealer split the hand into two or more runouts and "run it twice" to soften the variance, a much more simple method.
Yet, as seen on televised poker, some top pros like to buy insurance and some top pros decline to sell it. This is not optimal behavior by either side. Remember, these are professional poker players, practitioners of supposedly smart decisions involving money.

If you have made it this far down in this post, you will be rewarded. Lets take this one more step, and show how precisely we can use Behavioral Economics to separate misbehavers from their money.

It is not just that you should always sell the insurance and never buy it. The buyer of the insurance is not just giving away money. He is getting something out of the purchase in exchange for a loss of equity. This is called Transactional Utility, and is exclusive to Behavioral Economics. The risk avoider is paying Sklansky Bucks* to purchase peace of mind, while the seller of the insurance is receiving Sklansky Bucks and giving away peace of mind. In the mind games of poker this is real. Of course, WE (enlightened perfect econ poker players) should sell it and pocket those SB's. But how do regular human beings value peace of mind? What value is there in the poker insurance? How do you make sure to gouge the buyer of the insurance? When should they pay the MOST?

How to correctly price poker insurance....

Based on Behavioral Economics, there are several combinations of players who might exchange insurance.

Lets call them player A and player B. Lets say that both players are "up" and the outcome of the hand will not change that for either player. Player A is ahead in the odds of this pot but is willing to pay player B for insurance. This should be the lowest price for the insurance. Personally I would want a ten percent equity boost in exchange for some irrational peace of mind (But will accept any amount).

Next scenario, player A is "down" for the session and player B is "up". The size of the pot will erase player A's losses and bring him back to "even" but not change the standing of player B. This should be more expensive. Player A not only wants peace of mind, but also wants to erase his losses and be able to book a "win" in his mental accounting. A two-for-one misbehaving bargain!

Now the next two scenarios involve the seller of the insurance (player B) being "down" for the session. If you feel this should be irrelevant, as a perfect econ player, then consider it as simply bargaining advice to use against the buyer, since the buyer will obviously understand these considerations. However, I remind you that we are all human and humans make transactional utility exchanges all the time. It is in our nature, it is how we are wired.

Player A is up for the session and player B is down. Player A wants to buy insurance from player B. The size of the pot will erase player B's losses for the session. This should be more expensive than the first scenario but less than the second scenario above. The reason is that player B is giving up the chance to get back to even. That is big indeed! However, when it comes to exchanges of peace of mind, it is a buyers market. Player B should point out how important the pot is and ask for the extra equity.

Our last scenario is where both players are down for the session. This should be the most expensive. The size of the pot will erase the losses of either player. Player A wants to buy insurance from player B. Player B should want a huge bump in equity for such a transaction. By reminding player A of these tragic circumstances, player B should be entitled to twice as much as in the very first scenario above, when both players were up.
BONUS: Any of the above scenarios happen near the end of a preset time when the game will end. Player B should want more equity in any of the above scenarios.










* We call EV dollars "Sklansky Bucks" to honor and tease the founder of modern poker theory, David Sklansky.

Interesting posts. I am just getting into GTO. You mentioned Utility. Is it possible, as in other econometric / finance theories, to use different utility functions throughout the GTO calculations? EG replace expected value with expected utility - perhaps differently per player - to come up with a different set of rules? Or does that not factor into GTO.

Hi windsurfer1,

Thanks for the question. GTO is a very complex and advanced method of making poker decisions. Learning GTO and practicing it at the tables will make you a better player, more balanced, with a goal of totally non-exploitable play. But, by definition, there is no transactional utility. No player is any different than any other. GTO theory is firmly based in fundamental economic theory of perfect decisions. Well, maybe more correct is to say that GTO poker theory is within the particular subset of economic theory that deals with perfectly negotiated compromises between optimal agents with competing self interests.

That is why I labeled this thread "Beyond Nash" because perfect non-exploitability is not the way humans play poker, especially live poker.

So, I encourage you to study GTO, it will expand your understanding of the modern game, especially the way it is played online. It has its own math which should not be altered. All I am proposing is an additional way to analyze a hand of poker.

Traditional Modern: I got it in good, or made the right fold. I maximized my equity and the result was +EV in the long run.

Post-Modern GTO: I played it right, my ranges are correct and my bet sizing and bet frequency was spot on, my opponent can not counter this.

Behavioral Model: My opponent turned over (or folded) cards that are not supposed to be part of a rational range, he misplayed the hand. I could have won more money or lost less money if I factored that possibility into this particular players decisions.

-Rob

Hi heehaww,

Everything you added was spot on, thanks for adding it to the thread. Your observation is one of the strongest counters against behavioral economics: Basically that participants don't have to be experts, so long as they lean towards the correct decisions, as if they are experts.

This is extra complicated by poker players who are supposedly always learning and always becoming more expert at making poker decisions.

Without some form of experimental data or data mining of past poker decisions, the Behavioral Economic model of poker may never be able to totally push aside the "as if" objections.

-Rob

Very interesting, thanks for the response.

I just (re) read the Wikipedia article on Nash Equilibrium. I think the broader point that you are making is that when you know another player is behaving non-GTO-optimal, you can improve your results. For example, in the prisoner's dilemma game, if you knew your opponent would always cooperate (sub-optimally), you would also cooperate to improve your result.

Seems to me that thinking about players' biases is a pretty important part of the game. An analogy is the financial markets where despite legions of economists insisting that markets are efficient and optimal, things like the internet bubble still happen, or the financial crisis, and if you don't adjust your strategy you can not only make less money but get your ass handed to you. In general I am a big fan of behavioral economics in that context, not sure if what you are talking about would classified under the same umbrella?

I notice that the Nash equilibrium examples do indeed use Utility; the units of the maximization don't seem to need to be anything particular. Can I ask, in the current applications of GTO to poker, what are the "units" that are being measured? Just profits? If so you might use a different unit, perhaps one scaled by volatility, for example.

For example when you say:

This is true in terms of EV but it would not be true to someone who is risk-aware. Specifically the former has a variance (risk) of zero whereas the latter has a non-zero positive risk. For example after 4 trials the maximum loss of the former is 100 but it is 200 for the latter. I know you were making a point about prospect theory but I am just cherry picking the example (sorry).

How is GTO actually being applied to the poker problem - it is mainly computer simulations? I imagine there is no formulaic reduction of the problem?

I enjoyed reading your posts thanks much for the info!!
Rob (also)

Von Neumann and Morgenstern prove in their seminal work that requiring each player to maximize his/her expected value of the payoffs in order to realize a GTO (Nash equilibrium) is not as restrictive as first appears.

Actual dollar payoffs can be replaced by utilities of those payoffs. Under reasonable conditions, rational players then attempt to maximize his/her expected utility. Under this framework, all of the standard game theory machinery carries through (existence of equilibria, etc.) using utilities rather than dollar payoffs. Of course, if different people use different utility functions, then some of the traditional results (such as for zero-sum games) may need to be amended.

The conditions under which this holds are quite reasonable. An entire literature has sprung up discussing and debating these conditions. For our purposes in the poker community, we invariably ignore such subtleties and use the standard EV machinery.

Wow! For a new member of the forum, you are contributing some really thoughtful ideas. Welcome!

The applications to Wall Street are huge, especially from a governmental regulatory perspective. Richard Thaler points out that the tech bubble was an example of a failure of the fundamental model, but much less destructive than the housing bubble, since the former was comprised largely by overly valued stocks, and the money lost was easier for society to absorb. Where the housing bubble actually put a lot human misbehavers out of a place to live, and simultaneously reduced the wealth of practically everyone. The stakes are much higher when humans borrow against the inflated value of their house, versus "letting it ride" when it comes to overly valued tech stocks.

Nash of course uses Utility, I should have made clear that I was only ruling out any form of "Transactional Utility" which is when humans like to "get a good deal" and make irrational choices accordingly.

As for the deep math of GTO poker theory, I am not an expert. However this particular sub-forum has the some best poker mathmatizers I have ever read. They sometimes will repeat themselves so feel free to start a GTO thread, but I recommend that you first search and read the threads that already exist.

The math involves balanced betting from ranges that contain mixtures of value and bluff, such that your opponent is "indifferent" and can not exploit you, even if they called or folded 100 percent of hands (or a balanced subset of bluff catchers). The ratio is derived from the size of the bet relative to the size of the pot. And this is just a description of last-to-act bluff catching on the river! There is a lot more.

The units are BB, the Big Blind. The scenarios usually start out by describing stack depth as 100 BB deep, etc. The level of variance might be described as SPR, whcih is stack-to-pot ratio. SPR defines the largest amount the pot could grow to given the stack sizes involved of the remaining players, and is used to keep a running number on the best or worst odds that a shove will offer to a potential caller. There is so much more to it than this, and I probably messed up some of this. I have a tendency to move on from ideas that I am less passionate about and get really addicted to ideas that are new (at least new to me).

I really do think this science of human error in decision making is tailor-made for poker!

-Rob

Gotcha, thanks, I expected as much. It'd be really interesting to see how everything comes out if you used a risk-adjusted EV. I'll first try to read up on the basics I guess!

There are many writings by Mike Caro which hit on the point of Loss Adversion. Some of his advice on this is very useful. He wrote a lot about human behavior, and much of it is quite good. Also, GTO is the fondation, not necessarily and endpoint for how to play poker at the local establishment. Ideally you'd start with a GTO foundation, and then add-in the behavioral adjustments, and then re-calculate the game (which would be impossible because behavioral adjustments would add more dimensions than is practical to compute). I still recommend reading Mike Caro's theories on poker psychology.

It's also good to know some economics stuff such as Keynes' Beauty Contest, and incorporate it into your poker knowledge.

The Framework to Applying Behavioral Economics to Poker.

When applying principles of Behavioral Economics to poker, some beneficial ground rules can be helpful.

Among the social sciences, economics has been the most reluctant to incorporate the advances made by other branches of social science, such as psychology and sociology. Economists have been obsessed with providing the "correct" decisions. Whether or not the correct decision is implemented by humans was almost left out, or simply an afterthought.

The major problem with this is that economics makes predictions. Big predictions.

In Keynes' Beauty Contest, for example, the sophisticated solution does not require the solver to know which specific people he is competing against. The winner is the one who can separate out his own bias, and select based upon a purely objective accepted definition of beauty, while expecting every other judge to do likewise, and to also know that every other judge is doing the same. Crunch the numbers and one or more equilibria are established.

But, what if one judge is particularly social, and knows several other judges preferences? That information alone does not matter, so long as they also eliminate their bias which is important to achieving equilibrium. But now, what if that one judge is highly social and also a trained psychologist. He can categorize the level of naiveté of his opponents AT SOLVING PROBLEMS and simultaneously know their preferences. He can make predictions about how each person will decide and add them up. THIS judge will be better at winning the contest yet will vary from equilibrium according to what judges are in the room.

All people, including poker players, vary in degrees of decision making from "sophisticated" to "naive". A sophisticated participant is highly econ-like, but still needs a tiny nudge to be perfect. A naive participant is almost totally ignorant of economic principles in decision making.

Poker has always been about people. Poker theorists have made great lasting contributions to poker strategy that emphasize this to varying degrees.

But if we want to use specific tools and make specific predictions about poker using Behavioral Economics, then we have to know the difference between past poker theories and categorize them as "fundamental" economic theory or "behavioral" economic theory.
We have to look backward with a special lens that can differentiate the two. Further complication is that most theories have been a mixture of the two.

How many times does a poker author give advice for a particular situation of poker, but then add the caveat "unless your opponent is a nit" or a maniac or a novice or a drunk, etc. The author is giving you sophisticated advice but reminding you that your opponents are sometimes naive. The overall premise is that your level of sophistication is your "edge" and your future decisions will be +EV versus playing poker without the knowledge imparted to you by the author.

If we consider sophisticated poker strategy to be predicting our opponents decisions, then the author is teaching you about the decisions your opponent makes, by making predictions about your opponents' predictions.

This can be leveled infinitely. There is no end to predictions about predictions.

Well, eventually a gambler wants some percentages to go along with those predictions. Eventually serious math gets involved, by seriously talented theorists.

You can take any logical and true poker statement and generate a mathematical function which describes it. Then, you can ascribe this behavior to all poker players, and describe how this true logic poker statement will be used by every player against each other. The function repeats as every player attempts to "level" each other. When the mathmatizers solve this for infinite decisions it generates one or more equilibria, which we tend to call Nash Equilibrium.

That is good and well. That describes perfect play which perfectly incorporates the true poker statement in a worst case scenario where every poker player on the planet is in the hall of fame. Whenever we come up against an anomalous player who does not belong in the hall of fame, we profit. I have no problem with this, as has been pointed out this does not have to be the end of your poker decisions.

Here is where we get the lens to differentiate poker theories, past and present.

If the theory can be number crunched in a function and solved for infinity, where balance is achieved, that is fundamental economics theory.

If the theory requires extra information, gathered from specific events among agents, which can not be generalized across all agents, then that is not fundamental economics. That can not be placed in a function and solved for infinity because that theory is experiential. It is happening AS YOU PLAY and the predictions it makes vary. It is living and fluctuating. That is how I will categorize a theory as behavioral economic instead of fundamental.

Its easier to identify the fundamental first, and leave the rest as behavioral or for later debate as to just what degree of behavioral.

Any model of poker theory that can, in any way, be useful against any given player such that you do not need any information about that player is fundamental. So you see how the vast majority of poker theory is in fact fundamental.

Isn't this just another way to describe Exploit Poker Theory? Maybe, but that depends on the particular exploit theory. It is not enough to say that "you should never bluff a call station". If an exploit theory leaves it at that, then that is incomplete and is in fact an actual exploit instead of a robust theory that can be proven. There are times when it is +EV to bluff a notoriously sticky opponent, just fewer than against a more balanced opponent.

In hold'em, you should open raise a much wider range on the button than you open raise under the gun. This is purely mathematical, and can be solved by various criteria such that every player knows that every player knows how to open raise a hand of hold'em. This is fundamental, we do not NEED to know who the opponents are. We don't need to know anything other than what chair we are sitting in, and maybe the size of our stack relative to potential callers (various criteria).

But we know intuitively that there is more to this story. We want to know WHO is in the big blind, or on the button still yet to act behind us. Any theory that deals with "who" is behavioral.

It can be debated as to whether or not we should care who the opponents are. Most poker players want some info on the tendencies of their opponents. This comes naturally since we are social creatures. The question becomes what info do we want, and how much stock do we put into it when we already have fundamental theory to guide us. Are we even capable of keeping all of this straight such that it puts extra dollars in our bankroll? Don't we already know this intuitively?

Yes, it is possible.

No, we do not know this intuitively.

The more I read about GTO (which admittedly I've just started doing) the more it reminds me of the debate between efficient vs. inefficient capital markets in finance / economics. In my humble opinion a key element missing from efficient market theories is the TIME that it takes for a given "inefficiency" to be arbitraged out of a market and the fact that by definition that arbitrage requires market participants (who are either HUMAN or humans programming computers) to be trading it in expectation of profits arising from inefficiencies.

I think the poker version of it is: GTO assumes that people are playing optimally without mistake, presumably (?) because they've been playing long enough / so many hands that their mistakes / tendencies have been "picked up on" and exploited by other players over time. That's fine, and it seems to me it's not an unreasonable to expect a very seasoned, good player to play a game which is somewhat close to this. These players have been in every situation so many times and have had their weaknesses picked up upon and taken advantage of. But is this really true of all players? What about those at lower stakes? What about the tourist who stumbles in to the poker room drunk from the neighboring nightclub?

It seems to me that the key thing not considered in GTO is that "optimal" in the GT sense ignores the fact that real life players will only get to optimal when they play sub-optimal over an extended period of time against players who are trying to EXPLOIT those mistakes. The presence of an equilibrium depends on a suite of market participants attempting to profit from deviations from that equilibrium.

Here's a hypothetical. Say the poker universe consists of only GTO players. Is it possible for a non-GTO player to enter the game and extract profits? It seems to me it would be. Because you would know exactly (or at least on average) how every player would behave in every situation. Then you could simply take advantage of that. In the short run before your non-optimal tendency is picked up upon you could do things like push all in disproportionately and force more folds than warranted. (This seems like what Tom Dwan did in his early years). Eventually this would stop working, but has GTO distinguished the long-term equilibrium from the short term?

There are several things wrong in the above post.

Since you seem to be new, curious, smart, willing to learn, a good poster, etc., you should consider it a pleasant challenge to read more on GTO in poker and discern for yourself where you are wrong.

Ha, ok, fair enough. My knowledge of markets and finance is good so it must be my analogy to GTO which is imperfect.

Always willing to learn. Any hint? Which sentence(s) are the offending ones?

Imho, your four paragraphs started out very well, but each paragraph subsequently varied more from established GTO theory more than the previous paragraph.

Let me switch around the premise in your last paragraph, and see if you can tell the difference.

Behaviorists like to pick on GTO since they know the world is made up of bad actors that play poker. However, there is another equally plausible extreme scenario which can cast a more reasonable light on what GTO can accomplish in poker.

What if, a player is the ONLY person in the world who can play perfect GTO? Every other poker player on the planet is biased in some way. Some are tighter, some are looser, some more aggressive, some more cautious, etc.

Now, as in your temporal decision making exploit, our perfect GTO player only gets to play ONE HAND against every other player in the world, one at a time, until she has played every player in the world.

Now. Which player on the planet is THE MOST PROFITABLE versus the tendencies of other poker players?

Thanks for the continued replies. In general, as suggested, I need to take a step back because I have waded into an area with a lot of preexisting research, terminology, and debate and need to better understand the context in which a lot of these statements are being made. So I will review what is out there. When I can find the time...

However. With regards to your example. Someone told me that against people with biases, it is actually NOT most profitable to be GTO. It is most profitable to be exploitative. Meaning someone who could perfectly sniff out the biases and be exploitative would play everyone in the world and win more. Is that not correct?

Also with regards from varying from established GTO theory that was kind of my point, I am interested in things like, how does optimal play work in a world full of non-optimal actors, how many iterations does it take to get to GTO. I understand on paper the Nash equilibrium for a lot of simple games can be figured out recursively in a small number of iterations but in the real world does that happen with poker, and if not, does that need to be taken into account.

Anyway, thanks for the replies and I will try to read up.

It's correct but incomplete. Such a perfectly exploitive strategy would be quite exploitable.

Looking for some math to use in assessing the skill of an imperfect poker player, which is every human poker player.

Looks like Bayes’ Theorem is the shizzle of bias math. So many different ways to look at it though. This is fun, I will muddle around in it and see what might be concise and understandable to poker players.

Also, I should have put an extra word in my paragraph regarding Keynes’ Beauty Contest.

Our hero judge needs two bits of info on the fellow judges to exploit. Hero judge needs the preferences of other respondents and simultaneously the sophistication of other respondents at solving problems.

But, all problems are not equal. Some sets of choices are harder to distinguish between. Some sets of pictures will contain highly similar faces. Hero judge must also be able to recognize this and predict how well known respondents will exercise their sophistication at THESE particular faces.

We see this in hold’em poker.

Let us suppose a particular flop perfectly favors neither players range, in the betting order. It is perfectly balanced such that the first player to act should be equally likely to “have a piece” of this flop as compared to the second player to act. There are lots of considerations, but what I am getting at is that any action by either player reveals the least possible information.

Let us call this a “wide decision gate” since poker players can usually apply the term wide as meaning many options.

As a poker player steps through decision gates of varying complexity they get “narrower” as the complexity increases.

All flops are not equal. Some favor certain ranges much more than others.

So, when a player with unfavorable range steps through a wide decision gate, that does gives off usable information but not the most information.

However, when a player with unfavorable range steps through a narrower decision gate, this gives away more information.

There are way fewer combinations of holdings for a player who is range disadvantaged to use to step through the narrow gate.

So, to win the most money from a player, we have to know their preferred style of poker, what level of sophistication they exercise, AND what complexity of decision we just asked them to make in our poker negotiations.

pffft!

Yes, but to exploit it you would need to collect a series of observations to back out the pattern, which would take time, possibly a long time in human hands and which even in computer time would not be instantaneous. Also, the presumption was that there exist non-optimal actors, and probably they have a lesser ability to exploit others (or to avoid being exploited).

The Concept of Fairness in Poker

What is fair vs what is unfair.
Ask an econ “what is fair” and then ask a human the same question. The answers from econs will vary depending on the data you provide. The answers from humans will vary depending on the data you provide, but also with the WAY you provide it. This is because fundamental economics theory has no use for fairness. Meanwhile, fairness is important to humans and it depends upon the way a transaction is performed, not just the actual transaction itself.

Experimental economics has tested this extensively. Data shows that humans vary in their level of sophistication, of course. However, humans tend to value fairness very highly, and make irrational decisions because of it.

Some major corporate mistakes can happen when a company run by econs forgets that the vast majority of customers are naive humans.

But what about poker? Can an econ poker player win less money because they overlook the irrational perception of fairness among human opponents?

Yes, of course.

Using the concept of fairness to predict “tilt”.

Let us say player 1 is a perfectly balanced GTO-Econ player. Every negotiation of poker from player 1 is fundamental within that hand of poker, and indifferent to any other previous hand of poker, and indifferent to any particular poker player.

Player 2 is oddly human, such that player 2 is equally balanced between perfectly sophisticated and perfectly naive. Further, player 2 is skilled in poker as well, but not as skilled as player 1. This weirdness is arbitrary, but serves to show that player 2 is a learner, and is more sophisticated than an average human.

The players negotiate a series of pots between them, which we can say varies by N where N=number of pots negotiated.

Now, lets say that because of pure luck, player 1 wins every pot, sometimes with showdown and sometimes without. Player 2 is sophisticated enough to know that unlucky streaks happen, and that player 1 is also bluffing some hands and value betting some hands.

However, player 2 does make small mistakes sometimes. Player 2 accepts this and is aware of this. Losing a hand of poker by making a mistake or just being unlucky is an acceptable form of playing poker, and player 2 views such a transaction as “fair”.

At what number N will player 2 begin to react irrationally based on the human concept of fairness? When will it become unfair? What is player 2’s tolerance for all this losing?

Now you see why we have gone out of our way to describe player 2 as a pretty sound poker player, not a novice and not a “fish” in poker jargon.

What about N=10. Player 1 has won ten pots in a row and player 2 has endured this tragedy and reloaded several buy-ins without making a single irrational mistake. Quite a feat of mental self discipline!

However, every human has a limit. This will vary among humans but each human can only take so much N pots before irrationality takes over and they go on “tilt”.

The truth is that nothing has happened differently in any particular N pot. The next pot is no more or less likely to be a win for player 1, EVEN THOUGH player 2 has learned by negotiation that player 1 is likely to be a better player, which is acceptable to player 2 as player 2 is a learner and wants to get better at poker. That much is “fair”.

So, for ever player there is a number N at which they are on the brink of disaster and a number N+1 when they begin to make irrational decisions based on fairness. During pot N+1 player 2 is “on tilt”.

Lets call the last rational pot Nt for tolerance such that the N was the limit of tolerance by player 2. Now, the next pot will be Nt+1.

Thus far we have simply applied the basics of Behavioral Economics to poker to describe tilt, and show that the likelihood of tilt increases as N increases and N approaches Nt.

But, let us dive further.

Lets first set up some a scenario in pot Nt. What was the last straw that broke player 2?

The signals will vary from player to player. Some players stand up and mumble and walk around the table after almost any lost hand of poker. Is such a player more or less likely to have a lower N number? Well, it depends. Such a player may be performing a ritual that re-centers their mind and processes the emotion productively. Or, such a player may be already on the verge of tilt and just reached Nt.

But that is almost purely psychological and could get quite tricky.

What about the raw data of the hand. What hands are more likely to be Nt and predictors of an imminent Nt+1 meltdown?

Lets try to imagine the most extreme Nt hand.

The size of the pot is large, such that it accounts for a large portion of player 2’s losses up to this point, but not all the losses. Player 2 is deeply stuck and no single pot will recover the losses.
Further, player 2 has “got it in good” and is all in with a better hand. However, player 1 wins the hand. Player 2 can no longer tolerate such tragedy. Player 2 breaks, quite dramatically and displays new signs of discomfort not displayed in any previous hand. Simply put, this is just not fair. Often player 2 will say or exclaim these exact words, having never said such a thing during any previous N hand.

Now, assume player 2 does not quit immediately but continues and rebuys.

Now we are in pot Nt+1. Remember that no egregious errors have been performed by player 2 prior to this particular pot.

This is where GTO-Econ player has the opportunity to make more money, or lose less money, by taking notice of all this human drama.

Study shows that player 2 has switched gears from being a conditional cooperator to a vengeful punisher. The reason player 2 has not quit is to seek revenge and punish player 1, along with any other innocent players that get in the way.

Even within behavioral economics, irrationalities vary. Along the way, player 2 has skillfully avoided other more immediate fallacies of loss aversion. But now, player 2 wants REVENGE.

What will pot Nt+1 look like?

It will be large, and will go to showdown. Player 2 is done with folding. In fact, if player 2 does fold the hand for any large amount, it is unlikely to be Nt + 1 and player 2 is not in fact on tilt.

Such a player is highly unbluffable. Any bluffs by by player 1 should be big draws with significant equity. There is virtually no +EV folding equity to be found here. However, player 1 should be willing to bet larger with more of her value range. Player 2 is much more likely to “pay it off” with a weaker hand. Player 2 will deliver the punishment, even if it means putting a large amount of money on the table with range disadvantage, meaning player 2 has fewer combinations of winning cards than player 1.

Humans will give up money to punish those who they perceive as unfair, even when the unfairness is not proven to be any single offender. People will punish groups of others so long as the offender is guaranteed to be among this group. This is proven repeatedly by experimental economists.

Now. Lets say that player 2 does in fact win a big hand in Nt+1 or very soon thereafter. They are still down for the session, and still unlikely to recover all the loss this session. What will be that players fairness bias in hand Nt+2 after winning Nt+1?

The research is surprising.

They almost totally return to the previously level of neutrality. They are cured of fairness bias, but still human and vulnerable to OTHER biases. It will be hard for player 2 to make perfect judgements going further, but player 1 should absolutely not expect player 2 to be virtually unbluffable or willing to risk large sums at disadvantage. Player 2 got to punish someone and feels that justice has been restored.

I tend to think GTO is more important if you want to play mid to high stakes poker, and not lose to the pros, and psychology is more important if you want to exploit the fish. That said, of the two, the math and game theory component is the one you can't do without. You can play poker with no psychology. You cannot play poker with no math.

This is a good distinction to make. Improving fundamental decisions should always be the priority, since those improvements will make money regardless of the opponent. Any behavioral theory of poker must admit to this logic, and not purport otherwise. Thanks for reminding us, since my long walls of text might be misinterpreted to imply otherwise, even though I make several statements to remind that behavioral theory is an add-on to fundamental theory. I think Matt Janda puts it perfectly when he says:

“The further your line is from GTO, the stronger your read needs to be to justify it.”

Guys, hate to sound obtuse, but you can play poker without math, and letting math rule at the Poker tables is, in a way, very much an irrational thing to do. The mind already has its own calculative process, a workable framework for making decisions which can be converted into a mathematical equation, but, Poker is a mind game. It is only rational to use the strategic method already built into your mind. To convert your deliberations into math can be useful to help you achieve clarity but it is somewhat ironic to say its the rational way to think through a poker play.

This natural strategic method is the one that we use to exploit. It was actually only recently I cracked this bit - In our mind the way that we concoct our plays is by weighing everything up. We weigh up our risk against our rewards. We weigh up our different options. We seem to be able to just weigh up everything. It's weird, because to describe the method used by the mind we need to explain it as a mathematical sum, but in our mind we don't use math. We just, kind of, see the "weights" and see how likely a certain future might appear.

And guys, Learning GTO and practicing it at the table might make you a 'better' player but it might not, interfering with our natural thought process might cause problems. And I would bet my left leg that any person studying exploitative poker would make far more money far quicker than somebody learning GTO. (Provided good coaches etc etc). Not only is exploiting easier to learn and more profitable than GTO. Learning to exploit will gradually teach you GTO, while learning GTO will make it more difficult for you to exploit... There is a clear winner here. In Poker and in society both. We use this same "exploitative" thought process to strategize through all our deliberations. In Poker we can use it to get the most from our opponent, but in the real world we use it to get the most from others which is not a bad thing! A society where everyone used GTO would be a society where everyone acted the one specific way to minimise the risk to themselves from alllll the different types of people in the world. Screw that! The "exploitative" thought process is all about empathy, its about acting differently based on different people, a society where everyone "exploited" would be a society where everyone gradually learnt to respect each others differences. (Don't let the end product confuse you here. If you we're respectful of others eventually we would all be acting the same way. As with Poker, it is not the end product that is important, it the way we get there. And the exploitative study route gives us more profit)