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A few months ago I tweeted an offhand thought: 4d chess is boring. But I didn’t really think about why I thought that until someone pointed out in a recent discussion somewhere (can’t find it now) that an important insight from theoretical computer science suggests that 4d chess ought to be

simpler than 2d. I went “Doh!” because that insight (which I’ll explain in a minute) was one of my favorite ideas (and go-to hacks) from grad school, and I had failed to connect the dots. Then I did a double take: wait, did that really take care of the idea?

Well… yes and no.

Yes, things get simpler in a certain sense when you add dimensions, but no that’s not the whole story. To complete the story you need to add another idea: when you add more dimensions, playing to continue the game (infinite game thinking) gets easier than playing to win (finite game thinking). So the headline idea is that players with a certain kind of simpler strategy, and an objective of continuing the game rather than winning, have an advantage. To prevail in higher-dimensional games, you not only have to keep things simple, you have to switch from playing to win to playing to not lose, ie just continuing the game. For extra credit, you should try to make the opponent go for the explicit win.

Let’s unpack that and learn how to actually play figurative 4d chess.

Deterministic versus randomized strategies to approximately calculate area

Popular Meme versus Expert Theory

The idea of 4d chess turned into a popular meme with Trump’s victory in 2016 (helped along by narratives like the breathless Master Persuader one peddled by Scott Adams).

In the popular and intuitive perception, the idea of higher-dimensional games goes along with higher-dimensional strategies. This is 4d chess in the sense portrayed by Bradley Cooper in the movie Limitless, where he takes an intelligence-enhancing drug and is able to think on more levels, more moves out, in pursuit of more complex intentions, and powered by deeper insights. This understanding of 4d chess — and what constitutes effective play within it — is why the ridiculous QAnon conspiracy theory has unironic adherents. QAnon is the theory that a Master Persuader is executing a Master Plan too subtle for lower-dimensional mortals to comprehend. There are bigger problems with this idea of 4d strategy besides the obvious one that Trump doesn’t look like he’s thinking on more levels or more moves out (in fact, it is fairly obvious he’s thinking on fewer levels and fewer moves out; if that’s acting, he deserves an Oscar).

The bigger problem is that this understanding of higher-dimensional games is at odds with our best theoretical understanding of higher-dimensional thinking, which suggests that higher dimensional games require lower-dimensional strategies. In other words, thinking on more levels, more moves out, isn’t even theoretically the right thing to do. You have to go in the other direction: fewer levels, fewer steps ahead.

Often, when a popular understanding conflicts with an expert understanding, (and it’s not obviously due to pure ignorance or a superstitious belief system), there is some sort of shallow semantic confusion. For instance, we commonly talk of “steep learning curves,” which technically correspond to easier learning (since you learn more of the subject matter in a shorter time). But here the conflict is shallow: we’re really just using the metaphor of a steep hill being harder to climb, not talking about actual learning curves. Nobody actually believes that tougher things can be learned more easily. You can fix the confusion very easily by avoiding talk of steepness and speaking instead of long versus short learning curves, where popular and expert understandings of the words coincide.

But the 4d chess meme is different, and here the reality matches our counter-intuitive theoretical understanding of higher-dimensional games better. The popular and intuitive perception is an actual, consequential, and deep misreading of things going on. Why?

Deterministic versus Randomized Strategies

Here’s the key insight from computer science: _under fairly general conditions, more random strategies are cheaper to extend to higher dimensions than comparable deterministic ones. _

A simple illustration (see picture) can be found in the problem of approximately computing the area of an irregular shape. One way is to superimpose a square grid and count the fraction of grid points that fall inside. Another way is to draw a square around it, generate random points inside the square, and count the fraction of points that fall inside (your basic Monte Carlo method).

Turns out, as you increase the number of dimensions, the first (structured and deterministic) approach rapidly becomes computationally intractable. The second approach however, easily scales to high dimensions. This important insight (which won a major prize in 1991) is at the heart of many modern algorithms of practical importance, including the latest and greatest “deep” learning algorithms in AI, and many classic adaptive control algorithms.

I don’t know of any work systematically applying this insight in the context of higher-dimensional chess in particular (since there are many ways to construct the rules of such a game), but it’s easy to see why and how the game might get simplified and why random moves might get less costly.

In regular 2d chess for instance, achieving checkmate generally requires attacking the opponent’s king in 2 ways simultaneously. The king can’t move. But in 3d, the king might be able to escape vertically, so you’d generally need three simultaneous attacks to achieve checkmate. Defending against checkmates gets easier, achieving checkmates gets harder.

This is not an arbitrary, isolated effect. A closed planar curve partitions a plane into an inside and outside, but in 3d a bug trapped inside a circle can simply hop out. You need a sphere to trap it. In 3d, knots are possible. In 4d, knots are not possible. In 2d, laying out a network of roads (or a circuit board) without unwanted intersections is hard. In 3d, you just build flyovers and tunnels.

Life is just easier in higher dimensions. Unless you want to “win”.

In general, these sorts of qualitative effects in higher dimensional games suggest it is harder to achieve deterministically defined win states like checkmate. On the flip side, the downside risk to more thoughtless action is mitigated, because there are fewer ways to get fatally trapped. Or to put it another way, the number of ways to escape from traps increases faster than the number of ways to get trapped. Antifragilistas among you might have spotted an obvious implication: Extreme downsides get mitigated and you can gain more easily from uncertainty.

In 2d, moving your king carelessly might expose it to checkmate. In 3d, the king can move more carelessly with less risk. The extra dimension weakens the attacker more, and empowers the defender more.

The short of it is this: in higher dimensions, it gets rapidly harder to win, but rapidly easier to not lose. So adding dimensions turns symmetric games into asymmetric games: people who want to win get weaker. People who just want to keep playing get stronger. Ineffective players try to “win” in some finite sense. Effective ones try to goad opponents into going for the clear win, while making sure they simply stay in the game and don’t lose.

A very nice illustration of this is in an episode of Star Trek: TNG where Data faces off against Sirna Kolrami, the “galaxy’s best strategist” in the game of Strategema. Data initially loses, but then decides to play for the draw rather than win, leading to his opponent quitting out of frustration, since as an Android, Data could effectively keep going without getting bored or frustrated, drawing on an effectively infinite reservoir of what for him (but not for Kolrami) was a free resource: time and attention.

This should remind you of guerrilla warfare.

Asymmetric Game Regimes

Let’s talk about war for a bit, though not all higher-dimensional games map to war.

Over the past century, war has gotten more asymmetric as it’s gotten higher dimensional. World War I, with weak air power and primitive tanks was all about the highly symmetric condition of attrition warfare. World War 2, where air power reigned supreme and advanced tanks combined with motorized infantry to emable the Blitzkrieg model, was less symmetric. Vietnam, where complex political dimensions and the global Cold War backdrop entered the equation, was strongly asymmetric. And by the time we get to Iraq and Afghanistan, there is no meaningful definition of winning, but there is a definition of losing: getting into nation-building and failing. And today we are in the age of cyberwarfare, which is so highly asymmetric, a small team of state-sponsored hackers could use the NotPetya ransomware to bloodlessly bring huge multi-national corporations to their knees and cause billions of dollars of damage.

During this evolution, war went from stylized conflict with rules derived from honor codes, to a pattern of “total war” with no rules, to a blurring of lines between war and peace, to software eating…

Venkatesh Rao

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