Friday, January 29, 2010

Linearity vs. Spoilability: Part 1

In the last post, I talked a bit about why linearity and replayability are not well-correlated. Since I wrote that, I've been rethinking the whole notion of linearity in games, and concluded that it's not very useful. Much more useful in my opinion is the notion of 'spoilability', which I came up with last week.

My original intention was just to summarize my whole argument in one post, since it's fairly easy to explain. But I got a little sidetracked after getting part of it done a few days ago. So it's become a series. Here's part 1:

1. Linearity isn't a very useful concept.

This shouldn't come as a surprise, since I think the use of the word 'linear' in game reviews and criticism has dropped quite a bit over the past few years. Despite this, I think there's still a perception that nonlinearity is important, because it's the key characteristic that differentiates games as a form of storytelling from movies or books. And in an obvious sense, that's true. After all, those forms generally have a beginning, an end, and one conventional means by which to pass from the former to the latter. But what does it mean to call a game linear?

Usually, we use the term when we find a game too restrictive in one way or another. Prince of Persia: The Sands of Time is a good example. The main character has a wide range of fun acrobatic moves, yet in each room there's usually only one path that will not lead him to his death. This makes PoP more of a puzzle game than some players would have liked. So when we call the game 'too linear', we might be referring to this fact. Or we might be referring to the game's overarching narrative. After all, the game never gives us any choice in releasing the Sands of Time and causing all hell to break loose; I believe that takes place in a cutscene.

So we might differentiate between story-linearity and game-linearity. Still, there is no easy way to quantify these concepts, even in the most clear-cut examples. For instance, Knights of the Old Republic constantly offers you dialogue options that directly affect various character stats. Yet like many RPGs, there are only two viable paths for your character: the transcendent, inscrutable light Jedi or the greedy, asshole dark Jedi. As soon as you decide who you want to be, it's painfully obvious (except in interesting ambiguous cases which are far too rare) which lines to select in every single dialogue tree. This makes the game all too linear and even boring, which is the main reason I've never finished it.

Lots of games offer choices, but some choices aren't really choices at all. Some choices are explicit, and some, like Silent Hill 2's, aren't. There are games like Morrowind, where the main narrative can be avoided, but never really diverges once you decide to continue with it. What about free-form games like Civilization? Some would say they are completely non-linear given the range of possibilities. However, in narrative terms, Civ is really quite simple. You start with a society, it grows, and then either it dies or everyone else's does.

Again, I think a lot of game writers have figured out that 'linear' is an outdated term. But here's my idea, which DOES get at the real issue that people usually refer to when they use that word:

NEXT TIME: Spoilability!

Wednesday, January 27, 2010

A Straight Line of a Different Kind

Contra III is a 30 minute game that has lasted me over a month. The game has 6 levels, and none of them will take a skilled gamer more than a few minutes. But that's an easy enough thing to say. Yesterday I was overjoyed when I made it to the fifth level for the first time.

So why have I stuck with it for so long? It's true that I am a more patient gamer than ever before. I avoid using the word 'hardcore', since I respect gamers of all types, but it's true that patience and diligence count for something with this type of game. Ten years ago, I dedicated myself to adventure games because I was sick of game overs, but now I can't get enough of them.

The real reason is that Contra III is very fair. Like the Castlevania series, it's very well-crafted. I die often, but each time I know where I went wrong and how to do better next time. Unlike the Castlevania series, it's also a great adrenaline rush no matter how many times I play it.


Like most classic platformers and run'n'guns, Contra III is as linear as a game could possibly be, for good reason. Each difficult boss such as the one above (which was the hardest for me so far) has one crucial weakness: it's forced to act the same way every single time. If it didn't, I'd have given up. Linearity keeps hard games like this replayable.

Linearity and Replayability

At the time, the most common reason given for the decline of adventure games in the late 90's is their linearity. Adventure games had strong, coherent narratives that play out like movies, and this kept the games from being replayable. Players wanted more agency in their games; the future of video games was more choice, more non-linearity.

But if nonlinearity equals replayability, then why are completely linear games like Contra so endlessly replayable? One could argue that I'm not really replaying Contra III, since I haven't beaten it yet. But I have beaten Castlevania, and I keep coming back to it. It provides challenging, hypnotizing gameplay with no real time commitment. That seems to me to be the definition of replayable.


As far as adventure games go, I've replayed the Gabriel Knight series more than any other games I can think of. I've played the third game at least five times. (In fact, they're more fun when you don't get stuck on every puzzle.) It's like rewatching a favorite movie.

Some games are not very replayable to me, even when there are choices involved. Silent Hill 2 is probably my favorite console game, and there was a time when I wanted to see all six endings, which requires you to play the game six times. To this day, I've only seen two of them, for a number of reasons. Firstly, the game gets less scary when you replay it; less to do with linearity than with the horror genre itself. Secondly, the game is unique in that the endings reflect how you've chosen to play the game (do you use first aid often, how close do you stick to Maria?) and so it's tricky to force the game to a specific ending.




The third reason is more relevant to this discussion, since I generally feel this way about games with choice. To me, my first ending will always be my ending. Even though I did not consciously choose it, it is how I would've wanted the game to end (it is arguably the 'best' ending for the character). Two of my friends got a different ending (a more tragic one) that they are both happy with. When I force the game to end a different way, it feels like I'm making that choice...well, not really a choice. So I choose not to replay it, and savor my version of the story.

So in certain ways, linear games are more replayable and nonlinear games are less so. There's more to be said here, so...next time.

Screenshots from MobyGames, as if it wasn't obvious.

Tuesday, January 19, 2010

Remember When We Used to Get Nostalgic About Games?

So I couldn't finish Rocket Knight Adventures. Not because it was too difficult. Nor did I lose interest; it's still one of my favorite Genesis games. No, I couldn't finish it because my Genesis started freezing up, at random points late in the game. I'd blame the age of the system, but I remember this kind of thing happening on a sports game I used to have, College Football 1995 in, well, 1995.

I bought RKA used, a couple weeks ago, at one of the nicer retro game shops in town. I'd been looking for it specifically since I heard it praised on 1up's Retro Game Blog a while back. It was 5 bucks. Most of the games I pick up these days are under ten.


This is the first thing I saw after the Konami logo. The Rocket Knight (this game never explicitly uses his name Sparkster, at least as far as I got into it) pulls out his sword from behind him with a 'ching'. As soon as I heard it, something struck me about this game that I couldn't put my finger on. I watched the short intro, got to the menu, and started up a new game.

As soon as the Stage 1-1 song started playing (you can listen to a lot of the music here, which I recommend, because it's one of my favorite things about this game), it hit me. I'd definitely played this game before. Everything about it was familiar.

In the Genesis's heyday, I only owned three or four games. The vast majority of the games I played I rented at a mom'n'pop video store in my hometown called American Video (which closed up shop shortly after Hollywood Video came to town). American Video had pretty spotty selection, but chances are this is where I came across the game about a dog in a suit of armor with a rocket on his back. Well, at least he has a dog's nose, but the prehensile tail makes it more likely he's a monkey. Whatever.

Discovering that you've played a game over ten years ago is a bizarre experience. (In fact, the same thing happened to me when I played Dynamite Headdy recently, which is somewhat remarkable since that game sold poorly here in the States.) Video games, particularly those with dated sound cababilities, have a knack for burying themselves deep in your subconscious. I'd be interested if anyone else has had this experience. But ultimately, it's not something I seek out.

You might think that as a retro gamer, I enjoy playing games from my own past. For the most part, this is not true. This is because the majority of games I played 'back in the day' were pretty awful. The number of awful-to-mediocre movie or TV-licensed 16-bit platformers is truly staggering. I do actually remember the first three games (in order) I owned for my Genesis:

  1. Lion King. Actually not that bad - it was a pack-in with my Genesis. A little short.
  2. Home Alone 2. Laying clever traps for Joe Pesci and Daniel Stern sounds like fun at first, but after Joe trips over the water fountain spill for the sixteenth time, it gets wearisome.
  3. Batman Forever. By far the worst of the bunch. Simple goons take forever to kill, and your subweapons are harder to activate than MK fatalities. Plus you have to hear Jim Carrey say "Riddle me this, Riddle me that" at least 30000 times. At least there are no bat-nipples.
All things considered, it's a wonder I keep my Genesis at all, instead of booting it out the window for giving me such awful memories.

As you can probably tell, nostalgia ranks pretty low on my list of reasons for being a retro gamer. So what are those reasons?

  1. Price - No, I haven't played Suikoden 2. Most old games really are very cheap. And they usually work. And NES save batteries last a lot longer than you think.
  2. Knowledge about what's good - I've always thought the best time to get a console is when the next one is coming out. I may have to rethink that, since the current generation is sticking around for a while.
  3. Good games stay good. Now surely some games don't age well, while others are ahead of their time, and these are important and interesting issues. There will always be trends, and they tell us something. But games wouldn't be worth much as a medium if we let it stand that all our best games of today would become unplayable in ten years.
I have no special gripe with today's games. I do wish there were fewer space marines, but all generations have their quirks. I'm still very interested in where games are going, but I guess that's tempered since I'm generally a little late to the action.

Oh yeah, you should play Fallout 3. Good game.

Next time: ANYTHING but another Sega Genesis post, I promise.

Screenshot from mobygames.com.

Friday, January 15, 2010

Why Good Mathematicians Make Mistakes (Part 2):

In my last post, I described why I think Legendre was so determined to prove the Parallel Postulate from the other axioms. Legendre knew where the faults in his proofs were, but he believed they could be overcome if we allow some basic assumptions to be made about straight lines. Indeed, he wrote explicitly that the problems we have with this postulate are due to the "imperfection of the common language and to the difficulty of giving a good definition of a straight line."

What Legendre didn't know is that the difficulty is not particular to the "common language," but that there simply is no way to define a straight line (or for that matter, a line itself or a point) without making reference to another model, that is itself undefined. Generations of mathematicians before Legendre had tried to give a simple, self-contained definition, but the clearest of all is, in my opinion, the very first one we have, which predates Euclid.

In the Parmenides, Plato gives what is believed to be the common definition of contemporary geometers: a straight line is one in which the middle covers the ends. Of course, we cannot be sure about these things, but this is the only way anyone's been able to make sense of this:

If you, yes you, put your eye in the middle of a straight line, then you wouldn't be able to see anything of the line but the point where you put your eye.

Modern mathematicians should be up in arms at this point; giving a definition that refers to a previous model is one thing, but a definition that refers to reality, to a human being, is unconscionable. In fact, Euclid's revolutionary act was to remove the human from the divine realm of mathematics. So here's Euclid's stumper of a definition:

A straight line is one which lies evenly on the points on it.

If you're like me, then you should be wondering, what is this shit? I thought there must be a translation issue here. So I looked at the source (Did I mention I took two years of ancient Greek in college? Liberal arts schools are great.) and here's what I found. Sorry, fellow Greek scholars, but you're just going to have to accept a poor transliteration until I can figure out how to use Greek characters in Blogspot:

"ex isou tois eph' eautes semeiois keitai" (lies evenly on the points on itself)

The ambiguity is in the 'ex isou', which literally means something like 'in an equal way'. The phrase can either be taken with 'keitai' (lies evenly on the points on itself) or 'tois semeiois' (sits on its evenly spread out points). What's the difference? Here I turned to Thomas Heath's 1908 edition of Euclid, what may still be the standard source on interpreting Euclid in Greek, if only because both reading Euclid and learning ancient Greek have been in serious decline since education stopped being so aristocratic.

Anyways, he says that the first translation might mean that a straight line looks the same no matter which point you look at; you'd be hard pressed to tell the difference between picking one point or another, since the line 'around it' will look the same. The second translation depends on what you mean by 'evenly spread out points'. He might mean that a curve amounts to a kind of 'bunching' of points. But nobody really knows. Proclus, the earliest commentator on Euclid, gives the least convincing interpretation of all; that a straight line measures out the smallest distance between two points. (Here I guess 'ex isou' is taken to mean 'of equal measure? I don't know.) It's pretty obvious why Euclid is the only one who gives this definition, and Greeks afterwards go back to Plato's definition.

Ironically, Platonism may have been the reason that Euclid changed to this vague and pretty meaningless definition. If Euclid was a Platonist (he is believed to have studied at the Academy), then it makes sense that he would have attempted to remove all traces of the material world from his geometry. I've been spent more than a bit of time studying Plato, so I find this pretty fascinating. I've always held the suspicion that Plato had mathematics in mind when he invented 'forms'. If so, then it's pretty interesting that Euclidean geometry suffers from some of the same issues with Platonism as more general philosophy does:

Does the form of a straight line exist in the world of the forms? If so, then it would be pure in itself; it wouldn't make reference to anything else. But then it would be impossible to describe it.

There are bigger Platonic problems with lines than the fact that lines may not exist in nature (or that no one can draw them perfectly straight). Euclid's failure and the failure of so many mathematicians exemplify the fact that even the idea of a line isn't sensible unless we give it some kind of context, a place to live like a real two-dimensional vector space.

I've always thought of myself as a mathematical Platonist, but after thinking about these issues, I'm not so sure.

Next time: Rocket Knight Adventures!!

Wednesday, January 13, 2010

Why Good Mathematicans Make Mistakes

So my girlfriend and I have started doing math again in our spare time (she's a biologist who's a good sport). The story begins when I discovered the awesome text Mathematical Masterpieces, which sent me on a search for it's prequel Mathematical Expeditions, which is geared towards lower-division undergrads so that I could read through it with Clare (any opportunity to do math with a partner is appreciated, since it's a lonely game in general). Both books talk about individual math topics through a math-historical perspective; each chapter has you reading original sources in chronological order in order to learn about, say, Fermat's Last Theorem or set theory. It's interesting stuff, and it doesn't shy away from real math like a lot of math history does. There desperately needs to be more historical context in the standard math education.

Anyways, the first chapter which Clare and I started reading talks about the development of hyperbolic geometry after generations of mathematicians failed to prove the independence of Euclid's "Parallel Postulate" from his other axioms. At the very beginning of Book I of Euclid's Elements, Euclid first defines a point, a line, and all the standard objects that Euclidean geometry is made of. I'll say more about this later. Then he gives five axioms for geometry. The first three basically say that all objects that should exist do: there's a line between every two points, a circle with any radius and center, and any finite line can be extended infinitely. Pretty non-controversial. The fourth one says that all right angles are equal, which sounds weird, but really just gives him the right to compare angles at all. The fifth one is a doozy (here I'm kind of paraphrasing):

If a line crosses two other lines, in such a way that the interior angles on the same side add to less than two right angles, than the two lines will converge on that side.

This is Euclid's Parallel Postulate, often just called Euclid's Postulate. (Here's the way I think of it. Take two lines, and two points, one on each of the lines. If at those points, the lines look like they're going to converge, than they will.) And that's it. Euclid then lists five "common notions", which are also axioms, but really for math in general. Stuff like equality is transitive, and the whole is greater than the part.

Anyways, from the moment that Euclid wrote it, this postulate was pretty controversial. Our earliest commentator mentions it, and tries to do away with it. On the one hand, it looks ugly. Turns out there are a lot of nicer statements that are equivalent. I was always taught it as this:

Given a line and a point off the line, there's only one line parallel to this line through the point.

Euclid almost certainly knew this equivalence, but phrased it in a way that he could work with more easily. It's his book.

Here's the more important issue: why is it even necessary? We all know what a straight line is, so assuming we have straight lines, we know that if they're coming towards each other, they're going to meet. We shouldn't need to take this an axiom, because it is inherent to the very notion of a straight line. This was the thought of pretty much every mathematician until the nineteenth century, and trying to prove it from the other postulates was a big deal, one of the great unsolved problems. Even Euclid lays out the Elements in such a way as to get as far as he can without using it, proving 26 theorems before he makes use of it.

From a modern perspective, the question is whether the fifth axiom is independent of the others. This was not the perspective of mathematicians prior to the discovery of hyperbolic geometry. The historical perspective was, this axiom must depend on the others, and we have to prove it. In Mathematical Expeditions, you read a source from the last major mathematician to believe in this dependence, Adrien-Marie Legendre. Legendre gave a number of proofs, and believed that each one was correct even when the flaws were pointed out to him. With each proof, he tried to make it simpler, make it the follow the same clear form of logic that Euclid himself used. Legendre is a smart guy; he solved some problems in number theory that Euler couldn't figure out, and he gave birth to the theory of elliptic functions, which makes him partly responsible for the solution to Fermat's Last Theorem. So why couldn't he or other mathematicians at the time acknowledge that they might be wrong about this?

The book answers this with a number of theories, and they're all pretty insightful. There's a cute historical argument that basically says that in an unstable world, mathematics, especially Euclid, was a torch-bearer for eternal, implacable stability, and the possibility of alternate geometries did not sit well. But I want to add a couple of my own theories, which I'm sure have been brought up many times before:

In modern math logic, one of the most fundamental ideas is the distinction between a syntactic model and a system of semantic logical statements that might be applied to that model. In our heads, we have a perfect visual model for Euclidean geometry. So did Euclid, and that's what he set out to work with. But Euclid wasn't working syntactically, he's working with a semantic system, and thus he needed to start somewhere. I get the sense after reading Legendre that even as late as 1792, this distinction was not well-known. I think he had serious problems with the notion that one would start with this awkward postulate and then prove how a straight line behaves, rather than the other way around. Reading one of his "proofs", you can see that he knows exactly where the flaw is, even adding a footnote with an explanation of why one should take his necessary assumption (which, too, was equivalent to the parallel postulate) as given. It's an assumption that one is perfectly willing to accept if one has the visual model in mind, but it's laughably easy to spot if one is working semantically. Smart mathematicians of then and now did not lay out axioms and figure out things to prove from them; they used intuition and insight, reverting to semantics when they need to convince others.

There's another more philosophical issue that is, I think, more to the point, and perhaps the real reason why Legendre was so adamant. That is that a 'straight line' is a perfectly intuitive notion that has no fundamental definition. I can see I'm going a little bit overboard for today, so I'll talk about this tomorrow.

Tuesday, January 12, 2010

A Sega Genesis Discussion (with 100% more 'tude)

One embarrassing fact about me: I own seven game consoles. Not counting handhelds. (Interesting fact #2: None of those are current generation. Yup, I'm poor.) So you can say that I'm a guy that likes console games.

It was not always this way. This is because the very first console I received, one very special Christmas in 1994 (I think), was the Sega Genesis. Looking back on it, I think I can say that if my parents had decided to get me an SNES, I would probably know a lot less than I do about PC adventure games.

It's not that the Genesis is a bad system; in fact, it was Sega's most successful console by a long shot. It broke the NES's long-standing dominance over the hearts of gamers and forced Nintendo to design a follow-up. For one or two years there, it was king. Unfortunately for the Genesis, Nintendo did pull through with a console that was better than it in every way; the SNES had better graphics, the most popular franchises, and more third-party support than you can shake a stick at. To be a Sega child was to be a spiteful one, and I didn't stick with the Genesis for very long.

Both today and then, the problem with the Genesis is that it's hard to know what games for it are any good; there are precious few franchises you can count on. When I was a kid, I stuck with crappy movie-licensed games because I didn't know any better. Now that I have the internet to inform me, what are the best Genesis games I've played? (Note I haven't gotten to any of the good Sega RPGs yet)

  1. The Sonic games. Recently, I replayed 1,2,3, and Knuckles, beating them all for the first time. Yup, I was bad at platformers. Sonic 3 is actually my favorite, although they're all great.
  2. The Treasure classics: Gunstar Heroes, Dynamite Headdy, Alien Soldier (not released in US). Gunstar Heroes is perhaps my absolute favorite. I could not put down the controller when I first played this.
  3. Rocket Knight Adventures (hope to talk about this game later, once I beat it)
And that's pretty much it; still one of my least favorite consoles. But I have to say that the games on this list pretty much all demonstrate the Genesis' trademark, which Sega endlessly promoted with its blue furry mascot: character. These platformers/run'n'guns are so good because each level is filled to the brim with colorful backgrounds, multi-sprite multi-phase bosses, and a fun well-utilized gameplay mechanic. And they all have pretty good soundtracks. I love my NES, but the Genesis went a long way to make games exciting, and that's still true today.

You know you're a technophobe...

...when your mom invites you to join Facebook. True story. I am horrible with social networking sites. Probably because I'm horrible at social networking. So this blog is a concession, in a sense.

But I do have things to say! Mainly about retro games, math, and whatever bit of consumable media I've just enjoyed. And delicious meals.

They say blogs are going out of style, so...hello world. This is for those who enjoy the above things, and those friends I've lost touch with over the years.