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The Metrics of Space: Molecule Design

January 15, 2013 Article Start Previous Page 4 of 5 Next

If we begin to use level geometry to reduce the amount of possible Steiner points, we are beginning to ask much more of the player. By giving them fewer options (Figure 26), we are asking the player to exercise better strategy than the other players on the map. The upside to this is that players who do well in this environment will take much more satisfaction from its successful completion, as they perceive the lack of options to be indicative of a more complex problem. To demonstrate how a reduction in Steiner points relates to increased difficulty, we need look no further than the Steiner tree problem that we see in the lower, east quadrant of the Fallout 3 overworld (see Figure 27.)

Figure 27 (Click for larger version.)

Spatial navigation problems in the early parts of Fallout 3 are negotiated via simple spanning trees where you have many possible Steiner points. This is most noticeable in the areas to the south of main Vault as this is the first (and easiest) part of the map that the player is expected to explore. As difficulty increases, though, these Steiner points are vastly reduced; this can be seen in the subway system of DC, which the player encounters later in the main quest of Fallout 3.

A Practical Implementation

So far we have examined the basic principles of graphing theory and applied this to the analysis of a number of commercial examples, but how does graphing theory stack up as a tool to concept game spaces? The following is an example of a practical implementation of the theory of molecule design created by Nassib Azar.

In this example, a molecule concept is tested, implemented, and refined in order to create a balanced, multiplayer space, which despite its simplicity, offers players with a significant amount of interesting strategic possibilities to explore.

The core idea Nassib decided to explore was a map design which had three layers of experience, represented as three concentric circles. The game space is a deathmatch style map within the default game type of Unreal Development Kit. The outer layer comprises low intensity zones designed to "feed" players into the innermost section of the game space.

For the purposes of this design, "intensity" is measured by the amount of players actively trying to kill each other within each zone. Figure 28 is one of the preproduction sketches of the map. This diagram explores how choke points, intersections, spawn points and weapon pickups could be used to increase the intensity of the play experience as the player nears the center of the map.

Figure 28

After some initial paper prototypes and feedback, the core idea of three concentric play spaces of varying intensity eventually developed into a more concrete molecule which defines the space as a whole. Figure 29 is an iteration of the early concept. In this iteration, we still have the same set of concentric circles representing intensity of play; however, edges have been added to describe how the outer sections feed into the middle.

To achieve this goal, Nassib applied the notion of Compression and Funnelling, a simple tool which looks at how forcing the player around a game space using various game elements can create heightened emotional states. In Figure 29, each edge represented additional vectors of compression on the nodes they led to. In the case of this example, the nodes represented spaces for conflict; the more the edges leading into a node the higher the compression on that node (and as a result, the higher the intensity of game experience). In this application, node size was used to represent increased compression, and subsequently, intensity of play.

Figure 29 (Click for larger version.)

Although the application of molecule design is meant to create a distinction between play experience and level geometry, Nassib chose to explore whether pure geometrical representations of space have inherent player experience value. The hexagonal attributes of Nassib's molecule prototypes were worthy of further investigation. The question was: Would the molecule translate to actual level geometry and still retain the original design intent?

The prototype molecule used to define the overall game space went through a number of iterations in the form of grey box levels developed within UDK. It was clear through prototyping that the experiment had merit; the intensity of the player's experience increases as they work their way towards the center of the map. Nodes became generic play spaces (rooms) and edges became corridors that would feed into these spaces.

Figure 30 is one of the more advanced iterations of the grey box. It shows the implementation of the original molecule into a playable space. During testing, it was found that for intensity of play to increase, the room sizes needed to increase in order to accommodate the increased play intensity.

Room sizes are designed to create the most optimal zone sizes for the desired amount of play intensity. The original molecule design translated well in this regard. Play zones became progressively larger as they player moves towards the center of the map, yet the zones are also small enough to force the players into close proximity combat, hence increasing play intensity.

Figure 30

In order to create a syphoning of players towards the center of the map, a molecule was designed to aid in the placement of various weapon pickups. There are two main molecules used to define token placement. Weapon pickups were embedded in a molecule that forced the player to move quickly towards the center of the map. Health pickups were embedded in a molecule that forced the player to explore the circular boundaries of each play zone. The differing nature of these two molecules not only adds to creating clearly defined and different movement tactics for offensive and defensive play, but also aids spreading play over the entirety of the map rather than the central most zones.

Figure 31 (Click for larger version.)

Figure 31 breaks down the graphing further. In the close up of the medium node (upper left), a differentiation is made between two different edge types leading into it. Edges 1 and 2 come from the spawn point while 3 and 4 are fed from other medium nodes. This suggests a difference in danger level and is therefore represented by expressing the edges differently. Although the initial design hypothesis suggested that there would be some type of discernable difference between edges one and two AND three and four, it took several revisions of the grey box to observe this hypothesis the real world, seen in Figure 32.

Figure 32

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James Castile
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Don't mind me. I'm still only half way through my first coffee...

A valuable perspective to have in the tool bag. Thank you. :cheers:

Simon Ludgate
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Random thought, but when I saw that map on page 4 of this article, I instantly thought of the arena layout on page 4 of Yureka chapter 204 (a Korean Manhua). I wonder how that fictional layout would survive under your analysis?

Luke McMillan
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That's not one I have ever seen before. I'll take a look and get back to you :)

Raph Koster
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Love love love. :)

As a note on further research, there is a good case to be made for the idea that a given dominion space can have a formal, quantifiable difficulty rating assigned to it based on metrics and completion rates -- in fact, multiple axes of analysis could give playstyle differentiators as well. With that, you could at a minimum get a good way to assess your levels' difficulty, and given sufficient data and some AI programming, even in theory build agents that can tell you that same info without needing to spend playtesters on it.

I am fairly sure some of those ideas, as well as some of those in the article, were presented by Andrew McLennan at GDC a few years ago.

Luke McMillan
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Thanks for the kind words Raph - it means a lot!

I have been taking an interest in some of the 3D engineering applications and how they are used to model and analyze thermal dynamics; specifically things like exhaust flow and heat distributions.

I would really like to try and see if game levels can be analyzed in the same way, basically simulating what would happen under "normal" conditions and leaving the play testers for the emergent stuff :)

I'll be sure to write a follow-up to this if when I figure out how to do this!

Raymond Ortgiesen
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Excellent article. It's refreshing to see talk about the technical aspect of level design rather than just discussing environment art and spatial relations. Both are important but I think you've helped develop an area that is woefully under studied by most level designers.

Daneel Filimonov
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This really is an awesome article! I love the fact that you can visualize the gameplay and flow of a level without wasting valuable time play-testing every single revision when all you need is just a pen and some paper (and some patience)! This has opened a new avenue for me when designing levels, thanks Luke! :)

Kenneth Blaney
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You've played it fast and loose with the math and notation of graph theory, but it is all in service of a larger point aimed at a non-mathematical audience so it is forgivable. If you are interested in fleshing this out more in a significantly more concrete and rigorous way I could recommend a few texts for you. (Teaching math and being strict about people's use of language in math proofs is my day job. Please pardon me if that came off rude in any way.)

I'm not sure if they are aware of it, but the development of Portal 2 heavily depended on the type of level design you are working with here. Specifically, they created individual levels with the idea that certain areas would be linked (either physically or just visually). The advantage they gained by designing
They then used portals to link the areas together. The final step in the level design was to stitch the various pieces together, add bridging geometry where appropriate and remove the portals.

I'm currently working on a project for which I've taken a similar level design strategy as what you mention here. That is, using directed graphs that represent the moment to moment player experience (and not just the level geometry). We too are working in UDK and I've found extensive use of the UTPortal class for this purpose. The directed graphs have been helpful to estimate player performance within a setting thus defining a good baseline for some of the goals and game balance issues well before testing has begun.

Finally, this method lends itself to a fractal-like expansion fairly readily. That is, since each of your nodes is representative of a player experience, it follows that the entire graph is representative of a larger player experience. Those larger player experiences can then, just as easily, be linked in a directed graph. If you were really clever, you could then force a certain consistency to the experience of your game by having the large experience graph be similar to the smaller experience graph which in turn is similar to an instantaneous experience graph. I recently did a study of Cyan's "Myst" along these lines which I would be happy to share with you if you are interested in doing some follow up to this.

This is way too long a response, but you hit upon a combination of subjects I'm intensely interested in. :)

Luke McMillan
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Hi Kenneth, thanks for reply.

In regards to the math - accessibility of these models was really high on my agenda. Plus, I rely on Jesse Schell's 10th rule of what game designers need to know about math and probability :)

What you mention in regards to your UDK project sounds really interesting. One of my students ( who used this model created a recursive space death match map. What I really like about it is for all intensive purposes it is a single, arena space. But as you mentioned, the portal class in UDK allows this space to be much, much more.

I'll add more of a reply when I get a chance to use a real keyboard!

Luke McMillan
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Actually, there was a section from this article which I needed to cut because I couldn't concisely explain it's application. I was discussing node distribution based on Voronoi Diagrams & Delaunay Triangulation. If you know of a good layman's explanation for Delaunay triangulation and how it can be used for better node distribution then I would love to hear your thoughts.

Kenneth Blaney
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If you want to talk about triangulation I'm not sure if there is anyway to get around talking about the circumcircle of a triangle. That said most people will follow you if you tell them that any three points not in a line forms a unique circle (the visual explanation of which would be to show that you can draw two circles with 0 points in common, 1 point in common or 2 points in common, but 3 just won't happen). You might then want to suggest that a designer will want to place nodes and edges that try to obey the Delaunay condition: when you scribe the circumcircles no nodes end up inside of a circle.

This sort of goes in with what you were saying about the edge lengths earlier in the paper which confused me earlier. That is, normally in graph theory we don't care too much about edge lengths, however in computational geometry we do. (More properly, computational geometry falls into computer science, but we are being interdisciplinary here so I'll step a little bit out of my comfort zone.) If the length of an edge represents something like the difficulty of an area in the level setting up a graph this way will help balance the experience. A graph that obeys the Delaunay condition will have a property that all of its triangles will tend towards equilateral (it maximizes the minimum angle of the triangles and the maximum value the smallest angle can have is 60 degrees, which only happens in the equilateral case) and so avoid stretched triangles. Avoiding stretched triangles minimizes long edges, which in turn minimizes inequity in paths through a level. If an area of the game is too hard and optional, players will quickly learn to avoid that area unless you make it worth the time. (For instance, speed runners of Super Metroid always avoid Spore Spawn because killing it takes a long time and it gives a useless reward. So that miniboss would be represented by a long edge and a fairly unattractive node if we were to create a graph of Super Metroid.) That said, there is nothing to prevent a good designer from breaking this rule and creating a truly grueling set of paths if the game design allows for it. (Example there could be the purely optional Red Knight in the first level of Demon's Souls.)

As I'm sure you know because you grouped them, calculating a Voronoi diagram is functionally similar to confirming that a graph obeys Delaunay. That is, connect the centers of the circumcircles you drew and then, where necessary, bisect the outer edge of the triangle (much easier to explain with a drawing). Now, the only thing that springs to mind about how a Voronoi diagram would be useful for level design would be it would start to inform you where level geometry might need to go to restrict player movement and more clearly define the zones of play from your Half-Life 2 example. You can see that Valve did this to an extent by using bridges, tunnels, curved roads and water giving you your figure 12 from figure 11. That said, this feels a little more like something you'd end up doing to fix problems with the pacing of the map after the fact.

I'd be interested to hear more about how you use Voronoi diagrams in level design. I really don't understand it at a deep enough level to do anything great with it.

Elif Bugdaycioglu
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I'm not designing a PvP game right now, but you gave me a whole new vision to use a modified molecular design in a single player adventure game! Thanks for this thorough article. :)

David OConnor
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Very interesting and worthwhile, thank you for sharing Luke! :)

Frank Washburn
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Excellent article. Do you have any caveats for designing level flow for a game that is literally in 2D space, like a metroidvania? Trying to keep in mind verticality is obviously very helpful for the in-depth examples you've listed, but mentally I'm having a difficult time trying to avoid any of these "linear, boring level traps" that you illustrated. Any notable examples of games and levels in 2D games that would you say use this kind of methodology in a 2D plane?

Raymond Ortgiesen
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It sounds like using nodes to represent major areas of the world, then distributing nodes that represent each of the types of pick ups you get and mapping where those allow you to go and where you can't go would be really helpful for establishing the flow of that type of game. In a way the items functions as gates or keys and you could represent that using the molecule.

Luke McMillan
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Thanks for the comments Frank. In one of the lectures I do about this topic, I apply the methodology to Sonic the Hedgehog II from an analytical perspective to examine the level layout and the notion of public vs. private spaces. I have the materials in lecture format so it is no problem for me to put together a quick blog post.

In regards to games which do this well in 2D space one comes immediately to mind however I haven't played it in about 10 years so I might be remembering it to be better than it was! The game is Another World. It was a really slow paced platformer with rotoscoped animations. The game makes good use of the space by not using scrolling. Instead, each section of the level is represented in a new window. The game space is classified as "Adjacent spaces displayed one room at a time" (Mark Wolf - The Medium of the Video Game).

What is cool about Another world is that each screen is not a node, but rather contains several spatial nodes. For instance, you might be in one screen with two levels, but there is no way to get to the other level in the screen you are currently in. This means that you will need to go to other screens and figure out the spatial relationships for your self. The game play is largely based around this concept, especially in the later levels.

Raymond is also on the money. According to Koster, humans are pattern recognition monsters! We get great pleasure from discovering patterns and using them to preempt risk & reward scenarios. This is why I really like the molecule approach because it immediately lends itself to this human desire.

Frank Washburn
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Thanks for the thoughtful response Luke. I haven't heard of Another World but I will surely have to look it up. If you were to put together a blog post about public vs. private spaces I would be eager to see what you came up with! Looking forward to any more of your analyses! :)

Axel Cholewa
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Excellent article on a very interesting topic!

But I have one little problem with the AOEs and their possible overlap in figs. 10+. The AOE you defined to represent "intensity of play". Meaning, the bigger the circle, the more intense the challenge. But this means that, when you combine it with a physical map like in the HF2 maps, the overlap of AOE doesn't mean anything. The radius of the circle is a measure of play intensity, while the distance between the circles is a measure of physical distance. You can just draw smaller circles without changing their meaning, because you can anyway only compare the AOEs with each other, but not with the edges.

Of course, if you use the AOE (as you probably did here) to show the actual, physical areas of activity (maybe that would be a good name for it: AOA), then your method works.

I hope I made myself clear :)

Axel Cholewa
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Oh, and since your using an abstract method, you can also use this in completely different areas. For example, a game like FTL doesn't have much level design, but it contains challenges which can be represented by graphs. Teachers could use a method like this to design exercises for their students, with the edges representing the time it takes for a student to go from one problem to next (if they have a choice).

Robert Casey
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Great article.

Lewis Wakeford
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Awesome article. I wonder, has anyone tried incorporating these theories into procedural level generation?

Nick Harris
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Exactly my thoughts on reading this article. However, I feel that this "London Underground Map" has no relevance to the automatic creation of interesting maps for a first-person shooter because it fails to represent sight-lines. It really is only a refinement of the old adventure games drawn out on grid, or hex paper for late night Dungeons & Dragons sessions. Leaning out from behind minimal cover for a rapid expert headshot just isn't simply represented here.

The article did make me think about the problem of procedurally generated combat maps though and my initial notion involves pitting AI driven teams against each other on a variety of maps generated for their atmosphere and credibility - i.e. the algorithm's sole concern is the creation of, say, multiple war-torn Beirut-style maps. Due to the inevitable asymmetry to these (as a bunch of ruined streets wouldn't happen to have convenient reflectional symmetry for the purposes of deathmatch balance), you would have to limit your battles to Attrition, Bomb, Conquest, Duel, or Espionage with one team being the attackers and the other attempting to hold onto their defensive position. The game profiler could adjust the equipment, armament, transportation and number of players independently for each team in order to rebalance the game if the defenders used their "hill" (or somesuch topographical, or topological, advantage to their strategic advantage), it could make the attackers more numerous, or allow for fewer 'tickets' on the defenders side, or none: letting them get permanently eliminated by only giving them one life, whilst the attackers drew on a substantial pool of respawn tickets, Alamo style, some of which could be taken by deceased defenders who were tired of being neutral observers.

Another, perhaps more pragmatic course is to leverage User Generated Content, something which has already proven itself with the Halo 3 Forge and over seven million maps made for Little Big Planet.

Lewis Wakeford
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Well no they don't take complex geometry into consideration, but I was thinking they would be the very starting point of generation. So you build a rough plan for the level first with molecule design which just outlines how intense certain areas will be and ensures they are enough calm areas in between. It also specifies the minimum number of exits/entrances to an area and in which general direction they are. Then a different algorithm and/or prefabs are used to generate those areas as mostly self contained challenges.

Max Quin
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Hi Luke this is a very late post but you mentioned this.

"When your imaginative space is two-dimensional, your maps will be two-dimensional also. As such, there are no interesting vertical spaces and, more importantly, the objective is not clearly visible from the beginning of the map."

I'm wondering if you could extrapolate on that. Especially concerning something like a 2D Game, and how that can be made spatially interesting given it is literally two-dimensional. Also regarding the visibility of objectives, what is the relavance and importance of this? I'd like to know more I'm intrigued.

Also I find it funny that designers arive at similair ideas, I'd always used graph theory and this idea of connections aswell but never formalized it in anyway. Also that's partially because Valve utilized the concept in many of their maps, they called it a 'Bounce' I believe in HL2 where one area connects back to another and creates interesting and flowing spatial arrangements (which makes games more fun).

I think by far the best use of connectivity is the mod Minerva: Metastasis, it's no wonder that guy was hired as a Level Designer by Valve.

Sorry for blathering on thank you very much for your insightful article and your time.