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?

Solvability

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.

Boredom

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!

 

If you enjoyed this article, consider contributing to my Patreon campaign.

  • MichaelSinsbeck

    Hey Keith, nice article. Thank you for pointing out the difference between solvability (as in computer science) and the type of solvability game designers are interested in (maybe we should call it “practical solvability” or “player solvability”).

    I disagree with your conclusion “We don’t want as much depth as possible”. Your reasoning says that if a game is too deep then “the amount that [players] can learn about the system feels futile compared to what they can
    sense is there”. I think this is not the correct reason. What makes the players lose interest is not the large depth (or complexity or un-solvability), but the lack of accessibility. In the example of the game of Go, you are correct that people are frequently overwhelmed by the game. I believe, however, that is not because of the large depth of the game, but because player have to put in quiet some effort before they can play only slightly better than random. The learning curve for absolute beginners is very steep (so steep in fact that a beginner cannot even understand explanations by more experienced players).

    So, is a game imaginable that has a very large depth, but that people do not loose interest in? Going back to your iceberg analogy: Image that the part under water were divided into bite-sized chunks of insights (or understanding) and that learning one of these chunks improves the players solution strength in a notable way. I claim that such a game would be interesting for players for a long time. Players would not get overwhelmed by the games overall complexity, if the way to the top is divided into many small steps.

    So instead of asking “How much is there to learn in the game overall?” one should rather ask “Can a player at any skill level learn something meaningful (=something that will improve her game) in a reasonable time?” or in other words “Is the learning curve reasonably flat at any skill level?”. This, by the way, is the definition of “depth” by Lantz et al (Depth in Strategic Games, http://julian.togelius.com/Lantz2017Depth.pdf ), it is different from the definition you give in this article.

    The chunks I mentioned, of course, are not constant in size. Typically, when I pick up a new game, I learn new things quickly and with time my improvements become smaller. The moment when players get bored of a game is probably the first time they face a chunk that is too large.

  • Léo Panda

    I was about to say the same thing. There’s a difference between complexity and depths. That’s the point of Blizzard / Riot’s games (Easy to play, hard to master) and one of the main differences between UX and game design.

  • Max Hospadaruk

    I don’t want to dig in too much to the post above (which I think may have a bit of merit… not sure how much though) but I’d like to push back a bit on something you mentioned:

    “That’s the point of Blizzard / Riot’s games (Easy to play, hard to master)”

    the problem with this comparison (and with relating it to the article above) is that with the exception of hearthstone anll blizz/riot games are heavily contaminated with “execution.” Which is to say, it is very hard to access or even analyze the *strategy* components of these games in any robust way simply because most matches can be won solely with superior “twitch” (muscle memory, hand-eye-coordination, reflex response time… etc).

    As an example: lets say in a moba that it is generally agreed that character A is a counter to character B. However, if the player using character B is simply far more physically skilled at the execution of the game, they can often win even against character A a large majority of the time. Neither player gets *any* useful learning or feedback from these games (strategically); even though the player who chose hero A made a sound strategic choice, the *feedback* they get is “you lost.”

    I don’t mean this as a criticism of these games (execution is often super fun!), simply to point out that execution barriers are *not* equal to strategic depth; they’re simply a way to delay access to whatever strategic depth a game offers. A better phrase for blizz and riot games might be “easy to play, very hard to play competently, then somewhat easier to master”

  • Léo Panda

    Hello Max,

    I think that we need to separate the theory from it’s execution.

    Depths is in the theory part. How long will it take to know everything there is to know about this game and identify the best way to play the game?

    Complexity is in the execution part. How long it will take to be able to play that game correctly. Being able to act and move the way you intend to.

    That’s why I say that those are 2 different things.

    If core gameplay is too complex and just being able to move, shoot or navigate in the menus is a pain, that’s incredibly frustrating for the players.

    On the other hand, depths is what makes the game interesting, you “just” have to balance it right.

    That’s why I like Blizzard & Riot’s philosophy. It’s quite easy to navigate, move, use the controls and so on. You go right into the the core of the gameplay without losing time on things that mean nothing to the players or the designers. Is creating an account supposed to be a mini puzzle game ? No. So make it quick and simple, dammit.

    On the other hands, learning how to adjust your aim to the moving targets, always get the last hit on the minions, handle the minion waves and so on is incredibly complicated. But those are some core aspects of the gameplay and it’s depths!

    To come back on your example if champion A counters champion B, generally in lane phase, it means that if both players play their best and with equal resources, champion A should take the advantage over champion B in terms of resources.

    In practice, indeed you also have to take into account things such as skill, knowledge of the champions, knowledge of the match-ups, hand-eye coordination… But knowledge of the theory will still make people go toward specific champions, team compositions, strategies… and thus make everyone feel the fact that the meta is solved.