Solvability In Games

What does it mean to say that one game is “more solvable” than another? Is there a relationship between solvability (of any sort) and the point at which players get bored of games?


I should start out by making it clear that in game design, we are not usually concerned with true or mathematical solvability. We are not really concerned with the same kind of solvability that AI researchers are concerned with while trying to solve larger and larger Go boards.

This article talks about “solved games”, and classifies solved games primarily in two ways:

Solved: Provide an algorithm that can produce perfect moves from any position, even if mistakes have already been made on one or both sides.

Weakly Solved: Provide an algorithm that secures a win for one player, or a draw for either, against any possible moves by the opponent, from the beginning of the game. That is, produce at least one complete ideal game (all moves start to end) with proof that each move is optimal for the player making it. It does not necessarily mean a computer program using the solution will play optimally against an imperfect opponent.

This is what mathematicians and AI researchers are concerned with, and this is what they’re talking about when they talk about solved games. It also mentions that the 5×5 go board is weakly solved for all opening moves as of 2002, and the 7×7 board has been weakly solved as of 2015. (Interestingly, last time I checked this information out was in 2014, and at that time the latest board weakly solved was 7×6.)

Game Design

This kind of solvability is not what game designers are, or at least should be thinking about in their process of game design. At best, this kind of solution is only adjacent to what we should be thinking about.

That’s because your game getting solved (or even “weakly solved”) is not something that happens very often. Players will stop playing your game long before they come anywhere near actually solving your game.

A more useful metric for game designers is depth, which can be described as the loose quantifying of the number of viable strategies and tactics in your game. While there aren’t a lot of games that are easily solved or weakly solved out there, there are a lot of games for which a small handful of strategies/tactics are viable. In fact, I’d say that that’s the norm.

Where people get tripped up is that there is a relationship between solvability and depth. This leads people to thinking that they’re sort of the same thing. In the most technical sense, they are, but in a practical sense, they are not.

Depth vs. Solvability

It’s not hard to make a game that is extremely unsolvable. Take just about any simple abstract game and multiply the board size by 10, or 100, and you’re there. You can even take something as simple and mega-solvable as Tic Tac Toe and multiply its board size (perhaps adding an extra rule or two) and you quickly get something mega-unsolvable.

But that’s clearly not what we want. We want enough un-solvability (could also be called “complexity” perhaps) to facilitate enough depth so that a game is replayable, surprising and interesting in the way that we consider good strategy games to be such things.

But we also don’t want 200×200 Tic Tac Toe, because (among other reasons) it’s simply too much complexity. In a sense, it’s too un-solvable.

The way games work is that players make inputs and get a final win/loss binary bit of feedback, which informs the value of the sequences of moves they made that match, and through this process, they gain some heuristic understanding of the system. But players can really only gain this heuristic understanding if the system is not too complex.


If a player has learned the rules of a game and has been playing it, and then quits, this is not going to be because they solved the game. The most likely reason is that the player has gotten far enough through the solution process that they have a sense of what it would take to complete the solution process, and they lose interest. They feel as though the system will not surprise them from here on out, and in most cases, they’re probably right to feel that way.

Imagine the total solvability of a game to be an iceberg floating in water. The part of the iceberg that is visible (above the water) is the part that the player has already learned—they have solved this part of the iceberg, if you will.

As players play, they are also getting a rough sense of how big this iceberg is. If they get the sense that the iceberg is insanely massive (as I did with, say, Go), they will lose interest because the amount that they can learn about the system (in a single match, in 10 matches, or even in a year) feels futile compared to what they can sense is there.

Above: a game that has too much complexity (un-solvability)

On the other hand, there are times where, even though you haven’t got a game even 1/3 solved, you can sense that the project of solving this thing wouldn’t be all that hard. (I got that sense from the board game Hive, as an example.)

A game with not enough complexity

We don’t want systems that are too unsolvable, and we don’t want systems that are too solvable. Said in this way, it sounds kind of obvious. But I think we need to be explicit about this and change how we think about—or at least, how I think about—depth.

(Another component to depth and complexity is the degree to which any two given strategies are similar. We can factor this into quantifying overall complexity: if two strategies are “50% similar”, they would perhaps count as 1.5 strategies, for example. Possibly another article could cover the degree to which difference-between-strategies is a component of understanding depth in games, but for the purposes of this article, I think it is sufficient to talk about blanket complexity, because I think with enough complexity comes enough distinct strategies.)

Depth as Balanced Complexity

For any given system, there is some middle point of solvability where you have an ideal amount of depth—enough depth to keep a game playable and interesting for as long as possible (which hopefully, could be years), but not so much that it feels unlearnable.

Above: balanced level of complexity (don’t take the actual volume ratios here too literally!)

To summarize, here are the main takeaways.

  1. In game design we often tend to want other disciplines to come rescue us, whether it be psychology, game theory, or in this case, computational theory. While all of these have something to teach us, none of these will offer us real, applicable answers to the questions of game design theory.
  2. Since “depth” really just means “complexity”, we don’t actually want as much depth as possible.
  3. Players do not get bored when they solve a game, they get bored when they either feel like there is too much or too little to learn here.

Thanks for reading!


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