Game design mathematics question

[That Strange Girl]

Forgive my ignorance, but I've been reading some books and articles on game design because I'm very interested in the field, particularly as relates to MMO's. One of the things I was reading talks about player power as a function of level, and how it can be represented as a continuous curve in various ways, where power is effectively the raw killing power of a player at a given level. Common types of power curve mentioned are linear, exponential, and logarithmic. In an exponential model, the higher level you are, the more power you have, theoretically to infinity. In a logarithmic curve, as your level increases, your power increase slowly decreases so you gain less and less power per new level until you'd essentially not gain any at all. And in a linear model it's just that, a straight line relationship: more level, more power.

I'm trying to think of games that use examples of these. For an exponential curve, I would think most of the common MMO's, such as EverQuest follow this curve. At low levels you're pathetically weak, but post 60 or so players seem to become ridiculously powerful. However, I can't think of any games where a linear or logarithmic model would be in use. (Would pen & paper D&D have used a linear line? I can't recall.)

As a player, of course I find the exponential model more rewarding so I can see why most of the games I can think of use this. But would a logarithmic model be preferable to a designer perhaps? I'm having trouble understanding why using logarithmic or linear curves would ever be preferable, or why one model might be better than another for PvP vs PvE and similar.

Just wondered if anybody in here was more familiar with the theory behind game design than I am, and could explain it in small words that a molecular plant pathologist could understand. I'm more familiar with fungi than the secret background machinations of game designers. :wink:

[andrew_fm]

Diablo 2 was logarithmic, sort of. A level 60 character is obscenely stronger than a level 1 character, but a level 99 character isn't SO much stronger than a level 60 character.

I would think a model where you gain power quickly and then it drops off in the higher levels would be preferable since a) it allows new players to advance quickly, which is paramount to keep them interested, and b) it allows hardcore players to keep leveling with some benefit, but they don't continue to get more and more and more incredibly powerful which would break the game for everyone else.

[shang]

Yeah, in theory, with a logarithmic power progression you wouldn't need a level cap at all. Even D2 is closer to linear progression IMHO.

[DeepT]

However, as a player it is a big dissapointment, that getting higher level means little to nothing after you are used to gaining lots of power.

If your game is grind, then why would I bother getting to level 60 if I am only a little bit more powerful then a level 50. In WoW asside from gear, the power differance is very minor between a 50 and 60 (with repsect to equal level monsters). As a player once I got 1 level 60 and seen what it is, I have no desire to get another 60.

[Michael Fitch]

Greetings:
There are probably a lot of people lurking about who would have a better answer to this than I do (Lum, I'm looking at you), but I've been coincidentally doing some thinking about this recently, so I'll go ahead and take a stab.

In general, power curves in RPG's aren't strictly exponential, linear, or logarithmic. For example, if you look at hit points in D&D, they're pretty much on a linear curve. Or, in Diablo II, you get an additional 5 stat points every level, so your overall stat points are on a linear progression. However, linear stats don't translate into a linear power curve.

With most RPG's, for example, you also get better equipment as you go up in levels. Throw in new skills, and you get a multiplier effect where the stat increases, the equipment increases, and the skill power increases combined basically produce an exponential progression.

The story doesn't end there, though. Power doesn't exist in a vacuum, so the player's power curve is relative to the opponents' power curve (usually monsters). In order to keep monsters competitive, they increase in stats, powers, equipment, etc. as well. Generally, I think you want monsters to be on an exponential curve so that the difficulty relationship between player power and monster power (of equivalent level) shifts gradually. At the same time, an exponential power curve for monsters will help to keep monsters of relatively close levels differentiated, which is another benefit. Same goes for player power, but by having them both be exponential, the differential between them ends up looking more linear.

But wait, there's more. Experience also comes into play, because the dynamics of the power curve have to do with changes over time, and since level is the key to stat and skill increases, experience is the gating factor. For this reason, you usually see experience progressions for level requirements as exponential curves. Experience granted for monsters is also exponential, but the difference between the two curves is generally larger than the difference between the player power curve and the monster power curve. This creates a relationship that often comes out as less linear and more logarithmic. The higher the player level, the more monsters of the same level he has to kill, with the difference accelerating faster the higher the level.

So, the overall player power curve usually looks exponential at the start (giving the player a strong sense of progression in the early game), more linear in the middle, to keep the tension of gameplay fairly steady, and more logarithmic at the end, as the player reaches the upper ranges of available power (to extend the gameplay experience).

Of course, you also need to keep in mind that player progression generally isn't a smooth curve. It happens in key jumps, at level-up, at finding a new, more powerful piece of equipment, at gaining a new skill. Keeping monster progression smooth can help create a solid gameplay experience because the player experiences alternating phases of weakness and strength relative to the monsters he's encountering.

So, it's not as simple as whether a power curve has one of these three mathematical dynamics. These are all tools for scaling the experience over time, but most game systems are based on the interactions between them, getting complex and "fuzzy" relationships that feel right but aren't necessarily derivable from equations, which is part of why balancing a game system like this can be incredibly complicated, often much more of an art than a science.

I could segue into a discourse on chaos theory and complex interactions from simple conditions, but I've already rambled on enough, and I'm hoping some of the RPG specialists around here will actually read this and point out where I'm wrong. :wink:

Michael.

[Alan Au]

I think it'd be interesting to have an inverted S-shape power curve, with character customization substituting for gains in raw damage output at the low and high ends of the scale. The idea is that there's a ramp up period before a character has enough basic skills, and that there's a diminishing return when approaching mastery. Customization would also give players immediately visible rewards, allowing new players to quickly establish a unique identity, and long-time players to visibly show off their status.

- Alan

[Andrew Mayer]

Let's not forget combos.

With some good combos hidden in the power tree as you rise in levels and gain new powers the curious player will be rewarded with new ways to make 2+2=OMFG. Well, at least until the mana runs out.

[andrew_fm]

Yeah, I was playing Lord of the Rings pinball today and I realized that getting from 20 million to 40 million is about as hard as getting from 40 million to 100 million. Pinball is a definite case of an exponential curve where the higher you go, the faster you keep rising.

[That Strange Girl]

Thank you for the insightful explanations, Michael in particular. I hadn't even considered the power curve of the monsters and how that could affect the perceived power curve of the players. :)

[Alan Au]

The exponential system is often used to make it so that low-level monsters no longer provide sufficient XP for the player, which is to say that killing rats will no longer (effectively) get you from level 47 to level 48. Even with scaled monster difficulty, I think the system is flawed, but that's another matter. (This has to do with exploiting the system to take advantage of monster difficulty.)

- Alan

[BaconTastesGood]

Michael did a great discussion. I wrote an article (http://bookofhook.com/Article/GameDesign/InputandOutputCurveRelati.html) about this a while ago that addresses "input vs. output curves".

[Rollory]

D&D is about as classic an example of a logarithmic curve in gaming as you can get.

Take a level 1 character, of any class. Compare that character's base combat abilities (face it, D&D is mainly about the combat) with level 2. The character has doubled in power over that interval - twice the hit dice, twice the attack bonus (if you're a fighter type), twice the saving throws in the area that your class is good at, etc. It's a huge jump (and needed; L1's die at the drop of a hat). Then look at advancement to level 3, and compare that to the level 2. Now it's just a 50% increase in power. Still pretty big. Keep going up the level scale, and each increase in power is comparatively smaller. By the time you're level 10 going to 11, your hit points are increasing by only 10%, as are your attack bonus and saving throws - a far cry from the huge jumps at low levels. There really is not that much difference in power between a level 10 character and a level 12, or a level 13 and a level 15. Sure, on average the higher-level character is more powerful, but it isn't the total massacre that you'd get putting a level 3 against a level 1.

The farther up the level chain you go, the more it levels out. It _feels_ like you're getting hugely powerful, and you are indeed far more so than you were at level 5, but each individual step means less and less.

This leaves out the effects of equipment, which can substitute and supplement for the decreased improvements provided by the game system. But that tends to be at the DM's discretion, so allows the DM to tweak the power level to their satisfaction.

[Alan Au]

Take a level 1 character, of any class. Compare that character's base combat abilities (face it, D&D is mainly about the combat) with level 2. The character has doubled in power over that interval - twice the hit dice, twice the attack bonus (if you're a fighter type), twice the saving throws in the area that your class is good at, etc. It's a huge jump (and needed; L1's die at the drop of a hat). Then look at advancement to level 3, and compare that to the level 2. Now it's just a 50% increase in power. Still pretty big. Keep going up the level scale, and each increase in power is comparatively smaller. By the time you're level 10 going to 11, your hit points are increasing by only 10%, as are your attack bonus and saving throws - a far cry from the huge jumps at low levels. There really is not that much difference in power between a level 10 character and a level 12, or a level 13 and a level 15. Sure, on average the higher-level character is more powerful, but it isn't the total massacre that you'd get putting a level 3 against a level 1.
Er, wouldn't that be a linear progression, at least as far as hitpoints are concerned? (I'm not counting the part where the hitpoint formulas change ~ level 10.)

- Alan

[infimum]

D&D is about as classic an example of a logarithmic curve in gaming as you can get.

I don't know D&D, but based on what you are describing, this is a *linear* increase in absolute power. The relative increase in power varies as 1/n, where n is the experience level. Neither curve may properly be described as logarithmic.

If the relative increase in badassery were constant at every level gain (e.g., 50% increase every time), then that would in fact be an exponential growth in badassery.

If it takes time t to increment badassery by a constant p, then a logarithmic badassery curve would require time t^n to increment badassery by np.

Edit: beaten to the punch.

[Rollory]

Hm, you might be right ... but the XP requirements for each level are not constant, they increase at an accelerating rate as you go. I think that makes it less than linear, plotted against time. (Though again, there is some DM discretion here - feed the players XP in big enough chunks and it'll take whatever shape you want)

[Shadari]

D&D is about as classic an example of a logarithmic curve in gaming as you can get.

I don't know D&D, but based on what you are describing, this is a *linear* increase in absolute power. The relative increase in power varies as 1/n, where n is the experience level. Neither curve may properly be described as logarithmic.
Formally speaking, is that really linear? The slope is not constant. What you're describing, I believe, is an arithmetic progression. Maybe it's a progressive curve then?

[infimum]

Formally speaking, is that really linear? The slope is not constant. What you're describing, I believe, is an arithmetic progression. Maybe it's a progressive curve then?
Arithmetic progressions are linear, and the slope is constant. Simple example: let the power of a character at level n be equal to n. That is about as linear as you can get (at least if you consider the independent variable to be XP level), but fits Rollory's scenario. The sequence of *relative* power gains from level n to level n+1 is not constant, but that doesn't make the power vs. XP level curve nonlinear.

You could get a logarithmic curve by taking for each n the sum of the first n terms of the relative power gain sequence, but I cannot provide a meaningful interpretation for this quantity.

(As Rollory later pointed out, a more relevant choice of independent variable might be time, and the curve of power versus time may be sublinear depending on the DM's discretion.)

Anyway, the point I really want to make is one needs to be careful to define which quantities one is talking about before talking about whether things are logarithmic or exponential or linear. Also, "logarithmic" and "exponential" have precise meanings, and just because a curve exhibits "diminishing returns" (has negative second derivative), for example, doesn't make it logarithmic.

[BaconTastesGood]

Anyway, the point I really want to make is one needs to be careful to define which quantities one is talking about before talking about whether things are logarithmic or exponential or linear. Also, "logarithmic" and "exponential" have precise meanings, and just because a curve exhibits "diminishing returns" (has negative second derivative), for example, doesn't make it logarithmic.

This is true. Unfortunately there are no concise terms for "like exponential" and "like logarithmic" -- positive/negative second derivative is the closest, but even saying "PSD" and "NSD" sounds 'bleh'.

"This is a PSD curve"
"What, you mean you made it in Photoshop?"

[MightyMooquack]

http://suad.org/~mooquack/dndprogression.png

This image maps the fighter's base attack bonus (which just happens to be the same as the class's level) to XP. The base attack bonus is as good a stat to look at as any, as they all (hit dice, saving throws, &c) progress more or less the same way.

Now, it should be noted that the rate of XP gains isn't usually constant over time; you tend to gain more XP at later levels. This means that if the above graph were base attack bonus vs. time, it would actually be smoothed out a bit more and a bit more linear. However, it still wouldn't be completely linear: the additional XP you gain at later levels usually doesn't completely offset the additional XP needed to level (in other words, it still takes longer and longer to gain additional levels).

[Misguided]

By trying to distill such a complex collection of systems down to a single stat, you can't possibly get the full picture. D&D has many components. You have to consider not only the hp, but the attack and defense bonuses (including the ability to acquire and use more effective equipment, which isn't such a simple thing to measure), changes to saving throws, etc. Plus there's feat acquisition, not to mention spells.

Now one interesting thing about D&D is that character power changes dramatically over the lifetime of a character such that a high level character could practically prance naked among a throng of things that once would have given them nightmares without getting so much as a scratch. Not all RPGs work this way (though the MMOs I have experience with do).

Back to the original question City of heroes is the game I have been playing since last year and while I'd say the curve is, in many respects, linear, there are some HUGE milestone levels where massive increases in power occur. Most notable among these is level 22. 32 and 38 are also big because those are when characters can access their ultimate powers in their primary and secondary sets.

[Flowers]

Hey strange person, be the bold one and use different curves for different level segments. 1-10 use one style for players, then 11-20, switch it up, because you can, and it might be fun. Or just use smaller 1-2 levels of exponential curves before you return to linear. Make it different for each class. Just cut up the choices and pick em out of a hat. If it doesn't work, you can just do it the old ways.

Do the same thing with monsters, but don't make it uniform across all monster types. That makes all the monsters seem too similar.