On Things Big and Small

A pleasant Thursday morning before the Boston Monsoon, I was in J’s car going to Foxwoods. With us was T, an aspiring player earnest about improvement who has lamented about his recent rut. This was our first trip together, and he gave me a couple of hands to dissect on the ride. I happily obliged.

His first couple of hands were fairly standard, so a “dude you’re destroying him here, just bet” or “well bottom-two may not be good here since he’s so tight” settled those. The next hand got interesting: after he c-bet a dry Axx flop with mid-pair meh-kicker, the turn paired the A. I asked T to give his analysis, and he gave me several reasons to bet, along the lines of “I think I’m ahead” and “I bet because I didn’t want to look weak since I’ve been checking.”

To me, this was completely fine – these thoughts describe exactly how I would first approach the situation, if not how I might just make the decision. However, for this particular hand several factors bugged me (for example, I knew that his opponent is solid and balances his ranges well), so I decided to ask what I thought to be the natural next question: what is his opponent’s range? What is the range T is representing? What is the expected value in each part of the range given his river plan?

T was confused for a moment, and gave me a few more sentences like “well, I think he’s strong?” or “well, he probably doesn’t have an A.” I was in turn confused myself because he wasn’t answering my questions, but I quickly realized that I was speaking a different language. I understood at that point what his plateau was, why I would make a horrible mathematician, and why Martin Luther King Jr’s battle was so difficult.

I’ll explain. Cards first.

Poker, Chess, and Hard Work in Play

By design, poker is a game about making the best decision in each position, which we define as all the information available to a player for his move. The alluring complexity of the game is that calculating the opponent’s hand distribution in a position is difficult (in fact mathematically impossible, given the element of human psychology). Regardless, we can set an educated guess on the opponent’s possible hands (and more importantly, how he’ll play each hand given the situation) based on our experience of the common archetypes of poker players (“maniacs,” “tight-passive,” etc.). Then,¬† we can split the opponent’s hand into a few “buckets” (monsters, strong hands, air, combo draws, etc.) based on how likely they’ll react to our plays. The key idea is after making these assumptions, the problem actually becomes completely mathematical – we just sum our expected gain/loss against each “bucket” to calculate the value of each play, a mental calculation that is much easier to do than most people think.

In practice, even this is too costly in mental cycles to do for every decision during every hand. This is why we have heuristics, delicious nuggets of information that allow us to short-cut thinking in ubiquitous situations. Wisdom such as “do not go broke with top pair,” “don’t draw except to the nuts,” or as I told T earlier, “keep betting if you think you are ahead” allow us to make decisions almost instantly when a pattern matches one of our heuristics.

However, poker is ultimately a game about winning positions, not following heuristics. Heuristics are *man-made* principles used to make thinking faster, not more correct. You don’t get karma points when you follow a heuristic, even though knowing lots of them stack the odds in your favor. In difficult situations, heuristics often directly contradict each other, which can only mean that some of the heuristics were just wrong in these situations (there are times when the opponent’s betting pattern tells you that not only should you go broke with top pair, you may even want to call with ace high!). These are the situations where we must sit down and calculate, an action that, while itself an estimate, is the best we can reasonably do.

This was the real story of T’s hand. T was very comfortable using these nuggets of poker wisdom – precisely the one- or two-sentence answers I gave him on the earlier hands! – but none of these were useful enough for this particular hand, *which was what made the hand more interesting in the first place.* When I get into these situations, I try to “fall back” onto the next level of calculation. T was not used to this, and so he was stuck juggling with heuristics in a “lost” state that many players are familiar with (I’m certainly not excluding myself, who not only had been in T’s rut for at least 2 years but still gets “lost” in more complicated hands fairly regularly).

In this sense, poker is really no different from other games. Beginners in chess quickly learn to “control the center,” “create strong linked pawns,” and “develop minor pieces before major pieces.” Go has an even more elaborate set of heuristics encoded into “shapes,” with proverbs that go as far as to say “if you just play good shapes and avoid bad shapes, you will win.” Again, often these notions simply break down in specialized cases where we must just calculate – look at a list of moves, the opponent’s counter-moves, your own counter-counter-moves, etc. – basically, sitting down to do the hard work.

The Game of Life and Why We’re All a Little Bit Racist

When chewing on T’s problem, my big realization (for myself, not for the world – I’m probably reinventing a whole warehouse of wheels by now) was that the danger of only having heuristics not only extends to thinking in games, but to thinking in general – about economics, about other people, about everything.

What are heuristics when we abstract them from the games we’ve been talking about?¬† They are sweeping statements of the form “X is Y” or “X causes Y,” shortcuts in the bigger game of analyzing life’s information and decisions. When these heuristics work (and they frequently do, which is why they exist), we grok big world situations very quickly, which leaves a great aftertaste of wisdom for our egos.

But look at controversial issues that involve a lot of agents: financial crises, terrorism, health care. Much of the controversy comes down to people not really knowing things at a deep level, rather just heuristics of buzz-words and labelling of groups. Everybody knows we’re in a “financial crisis,” and I’ve talked to many people who says “it is because of the people on Wall Street,” but would be hard pressed to tell you how exactly the crisis happened (or even what subprime mortgages *are*), because the only information they had was the heuristic¬† “Wall Street people are greedy.” As true as that assessment may be, it trivializes a very important issue (I myself do not claim to know it satisfactorily, even after working for a year at a hedge fund specializing in subprime mortgages). The danger of lazy thinking contaminates more than just cocktail party arguments. For example, keen politicians can convert heuristics like the above into fuel for their own agendas from anti-Wall Street legislation to the Patriot Act. People with a bit of medical knowledge also get a “med school syndrome” where they diagnose themselves with all kinds of diseases because of a few heuristics – in this case, some knowledge of symptoms.

It actually gets a lot worse when we involve other people. While not everyone may care about philosophical debate, everyone lives life with a social identity defined by groups, and often these groups really don’t like each other that much. Animosity brews when everyone brandishes their heuristics about others as truth instead of estimations. Many of my fellow atheists are quick to denounce stereotypes of “amoral” (which has a grain of truth as an estimation) as an oversimplification, while somehow not giving equal treatment to our stereotype of the religious as “illogical.”When we replace tongues with more sophisticated weapons, we get fantastic celebrations of human nature: Crusades, extermination camps, and The War on Terrorism.

As scary as this all sounds, are we really going to stop and throw away all stereotypes we have of others? Of course not – each of us is so complex that we spend whole lifetimes trying to “find ourselves.” We don’t have nearly as much time to look at all the nuances of even our closest family and friends – why would we even think of giving credit to our “enemies?” We should treat them as statistics since it requires much less memory in our brain.

When we fight racism, we’re fighting laziness. This is why we’re going to lose.

But in games we don’t have to lose: remember, we fought laziness by calculating through all variants from a position. Can we do something similar here? Well, what we really want to do is a perfect calculation, which can only mean somehow simulating the entire world and every individual in some giantic computer and see what would happen when we make each decision – this is of course computationally impossible. The next best thing, which we can do, is to split the bigger population into smaller groups, fine enough so that one can reasonbly think of them as units (and pray that some variation of the Central Limit Theorem holds). Yes, this is still inefficient and always makes for ugly math, but it is by definition a more refined picture. And of course, on things like job hiring or college entrances, there’s no reason why we should be making these groups at all. We have so much information in each application and the interviews about that particular applicant that using heuristics like race is more stupid than amoral.

… but They’re Really not that Different

The alert reader will quickly point out that this grouping *is* heuristic, and he is completely right. There is no real definition to “smallest groups possible,” so if I chose the groups to be fairly large, say “Blacks,” we easily rediscover… racism! Not an achievement to be proud about, given all the preaching I just did. To be honest, I have already tried a sleight of hand at the beginning of this post – remember how we had to do estimates in poker for our opponent’s archetypes? That was a heuristic too: “archetype” and “stereotype” are not really different here. I haven’t even mentioned the very act of “bucketing” the opponent’s cards, which is completely analogous to the “grouping” of people I just discussed.

My real point is *not* that “heuristics = bad” and “calculation = good” (these are two horrible heuristics!), rather that while heuristics and calculations are on the same spectrum of decision-making tools, we usually *err on the side of less calculation.* The more we lean towards the “big picture” (heuristic) side, the faster we can make decisions. The more we lean towards the “small picture” (calculation) side, the more accurate our decision becomes. The art is then finding the right level of abstraction for each problem, because life is not a math Olympiad and we can rarely find perfect calculations that are elegantly short (let’s call this “The Zhang Uncertainty Principle”). However, the real bottleneck is usually our ability to do calculations since heuristics are, by nature, easier to digest. Thus, when we get stuck learning something, like T did with poker, our next step to mastery usually involves being able to do the next level of calculation.

I actually love heuristics – in every field, the heuristics are the “big lessons,” which inherently lend themselves to be more transferable to other fields and thus generate creative ideas. As a generalist, I feel I have a much harder time-allocation problem than specialists: since it is basically impossible to master any sufficiently large field, the generalist needs a good idea of when to stop learning something. I personally look for points in a skill where heuristics tend to disappear; the lessons from the skill after the point will generally be mostly useful *only for the skill itself* which loses some of its charm for me. Specialists tend to dismiss this as “lazy,” but how many of them usually go beyond the basics in areas like socialization, cooking, or fitness, each a respectable skill with lots of ideas and complexity? I think either approach is okay: we don’t have time to learn everything in the world, so we naturally choose to learn some things for their own sake and some things for raw functionality. Both generalists and specialists have spikes and flat points, just in different numbers.

This melding of heuristics and calculation is not really a surprise, because heuristics are just estimates of calculations. Actually, once we spend enough time with a game, a more beautiful phenomenon occurs. Someone with a degree of mastery in any of the games I’ve mentioned can attest that eventually players develop meta-heuristics that basically put checks and balances on heuristics – masters gain an instinct about exactly when their heuristics fail to hold (something like changing the heuristic “if X then Y” to “if X then Y, but only if Z”). Now they freely rely on heuristics to make quick decisions, because this mental layer won’t even consider heuristics that might not hold! Both Go and chess masters have known to claim that often they “only see one move – the right one,” because many of the moves were crossed off by the tremendously developed “society” of heuristics in their mind. My favorite relevant quote is attributed (dubiously) to Bruce Lee:

“Before I studied the art, a punch was just like a punch, a kick just like a kick. After I studied the art, a punch is no longer a punch, a kick no longer a kick. Now that I understand the art, a punch is just like a punch, a kick just like a kick.”

I feel the right path here is (again, a heuristic) balance, and since we usually don’t calculate enough, to balance generally means to think “smaller” (as Josh Waitzkin of The Art of Learning says, “make small circles.”). It would be horrible to only learn the heuristics in everything – one will not be able to enjoy the higher fruits of learning skills deeply. But if we refuse to learn heuristics, not only would we be burned out after overclocking our brains, we would fall into the trap of pedantry. I feel the right thing to do is to use heuristics to make decisions, but have a feel for when those shortcuts stop holding (typically, when things look complicated). Then, we have to stare at the board and play the position. Luckily, most chess games of life are not blitz.

-Y

P.S. Yes, I know a couple of you among the five or so readers I actually have for this blog are really just here for a poker hand, so here’s an interesting one at 5/10: I have AQh in the BB, cutoff (reads: tight, medium aggression) raises a limper to 40, I make it 120, cutoff calls and everyone else folds (we both have around 2k). Pot roughly 270. Flop AcJc7d , I cbet to 175, he calls, pot ~ 420. Turn a Ts, I check (I debated a long time between checking/betting here), he bets around 250, I call, pot ~900, we both about about 1.5k behind. River a 6s. I think and check. As played, river is definitely a check/call. The main question is the turn – is betting better?

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6 Responses to “On Things Big and Small”

  1. sawyertabony Says:

    Your hand is certainly strong enough to check call the turn, as you demonstrated, so let’s look at your opponents buckets to see if the raise is better:

    hands that have you in bad shape – {KQ, 98, sets, two pairs, AK} AK, JJ, 77, KcQc, AJ seem to be the most likely hands from this bucket given his action… would prefer to check/call against these hands, but a bet maybe doesn’t end up losing you much more money.

    hands that are drawing – {two clubs, flukey straight draws} only a few hands in this bucket match his action, maybe KcTc, but this sort of hand he probably wouldn’t call your pf raise with too often.

    hands that you are beating – {weaker aces, lower pairs} The board/action doesn’t allow for too many of these, since KK is probably betting preflop harder and 99, 88 can often give up on the flop. A9, A8 really have no business calling your preflop raise from a tight player. hands like KJ, QJ might be here, and occasional floats with complete air.

    So given my eyeballing these thoughts, I think I like the check/call more. I didn’t answer the question of whether you can actually save money when you are against monsters by betting and then folding to raises/ big bets on the river. If you had AK instead of AQ, the presence of AQ in his bucket of weaker aces goes a long way towards value betting.

  2. Jeffrey Says:

    With racism, there are a lot more issues going on than simply the heuristic/calculation spectrum. The problem is that the effectiveness of a heuristic as a heuristic is poorly correlated with the effectiveness of a heuristic as progress toward calculation.

    Racism gives an extremely effective first approximation, or at least more accurate than anything else I know of that can be done in half a second. The uncomfortable reality is that test scores *are* very different between races. So there is some sort of a rational basis for racism. (And I’m so glad I’m not famous and concerned about quote-mining.)

    The problem is that once you know even the smallest amount about someone, such as their socioeconomic background, learning their race no longer provides additional information and only serves to distract you from the more useful information. Most of the time, we do have more information, so trying to not allow someone’s race to influence our opinion is progress toward a more informed opinion.

    By contrast, heuristics about people in bins like “college graduate”, “raised by educated parents”, or “tight-passive” are progress toward real knowledge, and still should be kept in mind during the calculation stage.

    Perhaps the worst problem with racism is what it does with incentives. If people are to be judged based on their GPA, this encourages studying. If people are to be judged by their race, this encourages bitterness at the world. So *even if* knowing someones race gave as much legitimate information as knowing their GPA, it would still be damaging because it encourages dysfunction.

  3. Yan Says:

    @ Jeffrey: I actually basically agree with you. I think this was what I tried to say at the end of the 2nd part (admissions, etc.) but realized the bulk of my argument was in one of the many paragraphs I deleted. This is what I meant by “laziness” – we usually have a ton more information. If we were in some bizarro-world where we only had one stereotype and no other information and we weren’t even allowed to look for more, then there’s a very good mathematical argument for using the stereotype to its full potential.

  4. Jordan Magnuson Says:

    Great piece of writing Yan. I don’t really have much to add, but I really enjoyed following your thought-process here, and you’ve once again piqued my interest in poker. It makes me happy to get a glimpse at the inner workings of another generalist :)

  5. T Says:

    Yan,

    At the risk of embarrassing my future self at several levels: The key to the hand I think is basically realizing your opponent should never be floating in that spot. By calling flop and betting that turn the best he can represent is AK/AT (maybe AQ if he gives you credit for merging flop) such a hand can be easily blown off given the dozens of combos of 89/KQ/TT that are in your range but not his. Think of it this way; he’s sitting with KQ or 79 on the flop… what is his plan on a blank turn? Board pair? Third broccoli? There is just nothing that he can rep on any of those cards, except the outlandishly floated 2- or 3-outer on a boardpair and 4-outer on a T. Certainly nothing he can rep on a broccoli. Almost every broccoli draw that calls a 3bet preflop is a combo draw on that flop and is raising, because they have no showdown value and they need to get rid of air, want to put pressure on TPGK, and might I add again, must avoid capping themselves by merely calling. (Recalling that a capped range exposes itself to being multibarrelled or, worse, c/r bluffed if it fails to pot control – heat which can be combated by calling down with SDV, not a float or missed draw!)

    Thus if we check turn it’s to c/f. A flop calling range from an assumedly competent player is AT, AQ, AK, JT, and then stuff that is still TP or MP (A9s-, J9s+, QQ); turning any of the 1pair stuff into a bluff is just far too thin – too much of your range is air and not enough of it that’s value is going to fold. On the other hand, assuming it’s +EV to bet with a range of 2pair+ hands and the appropriate fraction of bluffs, it should also be +EV to add in AQ/AK plus a few more bluffs (50-60% more, to be precise).

    So let’s examine betting. You certainly have to DB bluff sometimes; but are you going to be doing so often that your range which includes most 89/QK/AA/JJ/TT/AJ/A7/some AT/JT is still balanced after adding in 1 pair? Well, turns out you only need to increase your bluff range by 50-60%. And with AQ (66% equity against the above flop calling range) we’re always showing a profit. By betting 450 into 600 to set up a conveniently sized river shove, and assuming they call with the top 4/7 of their range, we’re pretty much a dead heat when called, netting us ~$257 for our fold equity. Now compare this to the alternative of checking, hoping they’ll check back (which will happen 2/3rd of the time) and then betting X into 600 on blank rivers, which comprise very close to half the deck. They’re calling 600/(600 + X) of the time, which nets us 2/3 * (600X/(600 + X) + 600). Even when X is half pot for $300 this comes out to $533, half the time, thus even when they always fold on scares we make more than our other plan. When X is larger we make much more, in conclusion c/f > bet > c/c

    I’m not into big things these days. Especially the non poker things in this entry are much too big for me to discuss at all meaningfully. I’ll try to call you in a few hours.

  6. T Says:

    Sorry, I don’t know why I mentioned that a capped range exposes itself to c/r bluffing. It may in general, but here that would obviously rep nothing (not to mention sizing would be extremely awkward). If we had to factor in the option of turning AQ into a c/r bluff my head would explode and I may never have finished my post or ended up calling you.

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