Surviving a Combinatorial Explosion

Surviving a Combinatorial Explosion

Sid Meier once defined a game as "A series of interesting choices." This is a good way to look at games, and to look at game-like situations, because you can expand it: anything that stays interesting for a long time must have some way to create new interesting choices faster than those choices can be exhausted.

Video games sometimes take the "interesting choices" idea to its logical conclusion: create a game where every action has long-term consequences. In many video games, a character can commit heinous acts in one place, lay low for a while, and return to normal existence. It requires some suspension of disbelief to think that a spree killer could reenter polite society with no consequence other than a few minutes of evading the authorities. So some games try to give characters a persistent, realistic reputation, and to ensure that every choice has some kind of meaning.

This creates problems. Suppose the game has ten significant choices, i.e. ten ways you can permanently alter the outcome. This requires 2^10, or 1,024, discrete endings:

So the usual trick is to have a series of choices that are carefully crafted to give the appearance of significance, without actually changing much:

This approach sometimes frustrates players, who quickly develop awareness for what's a sidequest and what is the main plot. In fact, this structure is annoying enough that some designers prefer the more honest version:

And then there's the completely nonlinear approach, where the plot is either nonexistent or an emergent property of player interactions. Some multiplayer games exist to provide a framework for in-game stories, like the infamous Eve Online. It's nominally a game about spaceships, but actually a much less lucrative clone of investment banking—players use spreadsheets to execute zero-sum political schemes at their opponents' expense. Single-player games like Dwarf Fortress and Caves of Qud pride themselves on extreme levels of potential user interaction—essentially everything players do matters, and every aspect of the game's reality can be modified by a sufficiently skilled and patient player. They run into a different combinatorial drawback: one measure of how diverse a game's end states are is how limited its graphics are. If you're going to allow in-game characters to have complicated interactions, you're either going to have to imagine and draw everything, or draw almost nothing and leave the details up to the reader's imagination. (At least until DALL-E is widely available.)

Multiplayer games provide another source of combinatorial differences. For some games, the game mechanics just exist to provide a context for player-versus-player competition. In a variety of games, there are two main levels of metagame: being good at the basic mechanics, and being good at modeling what an opponent will do and countering it.

An open-ended game can support a variety of different endings, but only because the story, and the meaning, is supplied by the player. The game is providing a framework for an external story, instead of telling the story itself.

Game designers have to think about this kind of problem a lot, because they're supplying so much of the interaction, but any time someone is looking at a screen and using some kind of interface to change what that screen is displaying, they're navigating one of these structures. Which means the concept is very generalizable.

For example, some career paths seem to follow the pseudolinear model: there are a series of what look like highly important decisions starting at age 20 that all lead to business school, or to working at one of a handful of large tech companies, or leaving media to do PR instead. Some of these career paths are suspiciously linear; law firms that hire associates aren't quite committing to paying them the 9% annualized growth rate that those jobs' comp usually entails, but they're at least promising that that's the standard—which is only attainable if there's some way to jettison people who won't be worth $400k to the firm at some point before that's what they cost.

Other decisions follow the combinatorial explosion path. The investment research process can go something like this:

1. Archegos is liquidating a bunch of stocks, and I wonder if they had any good ones.
2. Turns out they were wildly bullish on streaming, and cable companies' streaming efforts are a surprisingly interesting area. Lots of new products, some of which will cannibalize the core business, but they're also streaming services with a built-in brand and content library.
3. If so many media companies are betting the business on streaming, and making their entire library available at low introductory prices to get users, what does that do to the pricing power of other streaming services?
4. I wonder if other streaming companies will see slower growth. (Netflix attributes this slower growth to fewer new releases, but that's partly a semantic distinction—if other companies are making their entire libraries streamable, those are newish releases that suck some oxygen out of the room.)

The interconnections between companies, their suppliers, and investor sentiment means that investing is a naturally free-associating kind of field, where you often follow threads in surprising directions.1 And, in a meta enough example, Netflix itself, as well as other streaming providers, offer the browser a decision tree that's pretty close to combinatorial, since they have so many criteria to search and browse by. And since Netflix adapts its recommendations to what you express interest in, it fits the original definition of a game: every decision you make is an interesting one, since its consequence is to make the app better informed about what to show you next.

What the linear and pseudo-linear paths have in common is that they're designed top-down as part of some standard path. They make sense working backwards from the endpoint. Whereas the combinatorial explosion is exhausting to design from scratch, but works very nicely when it's imposed on an even more disorderly system. 2^N can be a dauntingly big number, but it's smaller than infinity, and more tractable. The artificial phenomena that allow combinatorial explosions quickly run out of interesting outcomes, and have to prune the decision tree until they're back to some form of pseudolinearity. And once people realize this—once they know that they made an interesting-looking series of decisions that didn't get them far off the one acceptable track—they're disappointed. Meanwhile, in the messy real world, breaking processes down into branching steps is one of the only ways to contain complexity. In very nonlinear fields, the somewhat artificial distinctions between schools of thought ("lean startup" vs "fat startup," "growth investor" vs "value investor," etc.) are just ways to start narrowing tasks down until they're granular enough to actually get done.

The problem game designers are trying to solve for is replay value: can the game stay interesting after players have figured out the basics, and are the basics simple enough that players figure them out before giving up in frustration? That general idea can be applied to projects, careers, and life decisions—there's an optimal difficulty curve where the challenge escalates along with skill level, and it's important for decisions to have a meaningful payoff. A zero-decision narrative is just a novel with a different format, but a combinatorial explosion is too complicated for any predetermined narrative, so the end goal has to emerge one step at a time.

Elsewhere

Sanctions, Cautiously

The Economist has a good piece on the US's increasingly aggressive use of sanctions ($), which they note have accelerated under the Obama and Trump administrations, and seem on course to stay high under Biden. The combination of complex global supply chains and dollar ubiquity has given the US veto power over much of the world’s economic activity, especially through secondary sanctions that punish people and companies for doing business with sanctioned parties. But every time sanctions are used, it provides a stronger incentive to create alternative systems that are sanctions-resistant. These can involve entirely new financial networks, or cleverer ways to transact within existing ones. Or it can involve closer ties between different countries whose commonality is that they can't use dollars and don't like it.

Adtech Always Adds Layers

One of the wonderful things about the adtech business, at least in its free-for-all data-sharing heyday, was that it could be sliced into so many layers, each of which could add value (or at least charge something). The natural tendency is for companies at any layer in the stack to specialize, because repeating whatever they're best at is a more scalable and predictable growth strategy. But scalability often correlates with ease of copying, so the result is that each layer has multiple companies trying to grow into the same limited market. An interesting new instance of this is WhizzCo, which is creating a programmatic marketplace for content recommendation widgets. These ad units—once memorably described as chum—are modern ancestors of the ancient and lucrative "advertorial" format, a way for media outlets to sell off a little of their credibility for a lot of ad revenue. It's a tricky model to get right, because the revenue is easier to measure than the credibility hit. On the other hand, while these ads are not especially attractive, they are especially lucrative, and on the margin they make some media outlets viable that would otherwise fail.

Unfundable Mines

At one end of the cleantech supply chain, there are gleaming vehicles and glowing press releases. At the other end, there are holes in the ground to extract the essential raw materials for batteries and electronics. While the visible front-end of the business is very fundable right now, the mines that are limiting factors are harder to finance ($, WSJ). Some of this is location-based; US and Canadian mining companies have cost disadvantages, both because of where the biggest deposits are and because of more stringent safety and environmental regulations. And some of it is because it takes a while for trendiness to percolate down the supply chain.

Less Marginal Business

The WSJ highlights how SPACs are helping companies with below-investment grade debt raise funds ($, WSJ). This makes a lot of economic sense: part of the appeal of going public through a SPAC is that there's a legal safe harbor for making forward-looking projections instead of just showing backwards-looking financials. Any company whose future prospects are significantly brighter than its recent past will pay a lot for debt capital—creditors are punished in the worst-case scenario, but don't participate in the upside. So, in one sense, this actually represents improving market efficiency: SPACs are a source of equity funding for companies that probably shouldn't have been borrowing in the first place.