Two Kinds of Difficulty

There are two reasons accomplishing something might be difficult.

The first is intrinsic difficulty. Imagine you wanted to hold your breath for ten minutes straight. This is a challenging task because of the limitations of the human body. It’s exceptionally hard to do without intense training.

The second is relative difficulty. Now imagine you wanted to have the world record for holding your breath. Here, what is important isn’t so much the absolute difficulty of holding your breath. Instead, it’s the level of competition. If there’s a lot of serious competitors, this will be very hard. The world record for breath-holding is twenty-two minutes, so in this case it certainly qualifies as very difficult.

I think it’s important to separate these two types of difficulty, particularly with respect to learning accomplishments, because they have completely different features and expectations.

What Makes Learning Hard?

I think an illustrative example of the difference between intrinsic and relative difficulty is quantum physics and chess.

Quantum physics is a hard subject. The math required to understand requires a steady ascent of increasing abstraction starting from counting to arithmetic to algebra to calculus to differential equations and waves. It’s a trek that usually takes years to complete, and many people fall off along the way.

Chess, in comparison is dead simple. I wager that I could teach almost anyone how to play chess in less than half an hour, with no prerequisites. Perhaps slightly more time would be necessary if the person had never played a board game before, but not much.

Quantum physics, in my opinion, is intrinsically difficult. It’s difficulty is owing to the fact that, in order to understand it at all, you need to have decades of specialized training layering abstraction upon abstraction.

Playing chess, to the standards we typically expect, however, still isn’t easy. To play chess competitively, you need to invest a lot more time with it than to understand quantum physics. Grandmasters can log tens of thousands of hours of brutally intense coaching. The average person could spend hundreds of hours and still not be able to win a local tournament.

The reason for this discrepancy, however, is obvious: competition. Chess is a popular game and therefore the relative difficulty is quite high, even if the intrinsic difficulty is somewhat less so.

What’s Harder: Chess or Go?

Go is a Chinese game played on a grid, similar to chess. The difference is that, in Go, the board starts empty and one places identical pieces one at a time along the intersections of the grid. If your piece finds itself surrounded by enemy pieces, it gets removed. This leads to its more descriptive Chinese name, 围棋 (wéiqí), which loosely translates to “encircling game”.

Recently, I had an email conversation with a reader about learning to play competitive-level Go. Like chess, this involves memorizing thousands of position variants. Some of which may be dozens of moves long and must be executed perfectly or the position is lost.

My correspondent remarked that many competitive Go players essentially train on the game their entire lives, starting in special schools from childhood, easily putting in 30,000+ hours into the game. He suggested that this could be due to the fiendish complexity of the game.

This leads to an interesting question. Which is harder: chess or Go? And why?

Now that we’ve discussed relative vs intrinsic difficulty, this question has new meaning.

In terms of intrinsic difficulty, it’s likely that Go is indeed more difficult than chess. For one, chess has a smaller board and fewer positions. Therefore, at any given point in the game, there are a much smaller set of possible moves in chess than there are for Go.

This is one of the reasons why computers have had a much harder time learning Go. Computers can outperform humans in chess mostly through brute-force. A computer can quickly dive down several moves deep, examining most fruitful possibilities. In Go, the movespace expands so much faster that computers only were able to beat humans by employing sophisticated pattern-recognition algorithms that are likely similar to the ones used by human brains.

However, what about the comment about “30,000+ hours” to become world-class at Go? Here, of course, we’re not dealing with intrinsic difficulty at all, but relative difficulty. If Go were an obscure game that almost nobody in the world played, the rules could be exactly the same but it might only take a few hundred hours to become the world champion.

This difficulty—how long does it take to become a grandmaster—almost entirely depends on the relative competitiveness of chess and Go. My feeling is that this gives chess a slight upper hand, as there are probably more serious chess players than Go players worldwide.

Unless the game itself has a lower-bound of complexity that prevents further mastery (say Tic-Tac-Toe, which can be played perfectly after minutes of instruction), relative difficulty will be the most important factor when sizing up your ability.

How Does Relative Difficulty and Intrinsic Difficulty Affect Your Plans?

We’re very used to thinking about relative difficulty for tasks. So much so, that we often confuse it for intrinsic difficulty. In school, if everyone was getting 100% on every test, the teacher would make it more difficult until some people weren’t getting perfect scores.

Because we’re conditioned to focus on relative difficulty, people quickly deem themselves incapable or incompetent at learning certain things, simply because they fall lower on that end of the spectrum. If you were in the bottom 20% for mathematics, you might tell yourself you can’t learn math. What you really mean, however, is that you’re not as good at math as others, and therefore the relative difficulty is too high.

This actually has some interesting force behind it. There’s some clever argument showing that the same student who was bright in his or her high school may perform differently depending on the exclusiveness of the university he or she enters. Basically, if you’re smart, but everyone around you is even smarter, that may convince you that you’re not actually all that smart and you may downgrade your ambitions either through choosing less competitive majors or not graduating at all.

I’m not sure that this necessarily suggests that the average person shouldn’t try to get into the most exclusive school, but it does at least highlight the possibility that relative assessments of ability often get confused for intrinsic difficulty.

Relative difficulty is important. It certainly serves a purpose of sorting people into different buckets of talent and effort invested. I don’t mean to deny that. However, perceiving the intrinsic difficulty (or lack thereof) is often underrated simply because it is so absent in the formal schooling.

In many cases, what you want is an absolute level of ability, not merely a relative one. If I can fix my plumbing, ask for directions in French, do my taxes, cook a meal, write a useful program, perform first aid, drive a car, start a campfire, give a presentation, make friends, manage my time or any number of other things, my absolute performance matters a lot more than where I fit on a bell curve.

Next time you find yourself saying a certain subject is too difficult, ask yourself whether it’s intrinsically difficult or only by comparison to other people? The answer may change how you approach it.

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  • rif a. saurous

    I think you have some inconsistency in your definitions. You first suggested the intrinsic difficulty of chess was low because “you could teach almost anyone to play it in 30 minutes.” By the definition of “how long does it take to learn to make legal moves,” chess is *more* complex than Go: Go has only one type of piece and only one type of move (place a stone on a board), with a couple of edge cases, while chess has six different types of pieces that move differently, and at least as many edge cases. Go is certainly more difficult than chess for computers to play well because it has a higher branching factor and evaluating the quality of a position is more difficult, but that’s not the definition you used earlier in the article.

    I agree with the general point that the distinction between relative and intrinsic difficulty is important and people would do well to understand it, but I think using inherently competitive games to illustrate both simultaneously is problematic.

  • Scott Young

    “Amount of time to teach to perform the skill at all” is one aspect of intrinsic difficulty–it isn’t the definition of it.

    Chess, by that standard, might be somewhat harder than Go, but not by much. They’re both easily teachable within a short time frame.

    Higher numbers of possible moves would be another valid way of assessing intrinsic difficulty. As would, albeit imperfectly, the ability for a computer to play the game. Computer difficulty of a game is somewhat more challenging to apply as evidence for intrinsic difficulty if computers happen to be very good at something people are bad at (say if the game involved doing lots of square roots in your head).

    If I were to summarize my opinion on the matter: Chess is easier to learn at the start, but Go becomes more difficult more quickly (due to the wider variety of possible moves). Chess is, however, likely more relatively difficult due to the somewhat higher competition.

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