You know, I'm not too good with math, but I punched the calculator a few times and it tells me you might be wrong here.

0.1*0.1=0.01
0.1*0.2=0.02

0.8*0.8=0.64
0.8*0.9=0.72

So if I have a skill dictated by two variables multiplied together. Raising one of two skill, both at 10%, up to 20% gives me a 1% overall increase in the skill.
Raising one of two variables, both at 80%, up to 90% gives me an increase in the overall skill by 8%. I could be crazy here, but that sort of seems like increasing returns.

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Original post by JasRonq You know, I'm not too good with math, but I punched the calculator a few times and it tell me you might be wrong here. 0.1*0.1=0.01 0.1*0.2=0.02 0.8*0.8=0.64 0.8*0.9=0.72 So if I have a skill dictated by two variables multiplied together. Raising one of two skill, both at 10%, up to 20% gives me a 1% overall increase in the skill. Raising one of two variables, both at 80%, up to 90% gives me an increase in the overall skill by 8%. I could be crazy here, but that sort of seems like increasing returns.

It sure looks like it, doesn't it? Those two input numbers are growing, and the product is increasing as a result. So where then is this Diminishing Returns?

Looking at the last example input line, and we see that the product is very close to the first input. But on the first line, the product is not close to the same size as the first input. And this nearness of size started approaching quickly, and slows down throughout the example.

Taking the first input from 0.1 to 0.2 will make the product approach the same size much faster than going from 0.7 to 0.8.

Maybe Strength and Endurance are the componants of the player's Running Speed. Individually, they effect linear changes in Speed, EDIT but from the point-of-view of running speed, as we add one unit of speed, the changes to Strength or Endurance diminish.

[Edited by - AngleWyrm on May 4, 2008 12:37:13 AM]

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Plus, the growth in the numeric value may appear to diminish but that's not the same as saying its growth in game utility terms has diminished. There's no problem here that is inherent to the numbers or in multiplying them together, but in how you choose to use them to dictate gameplay.

Kylotan is correct - there is no necessary law here, nor do I believe this "Emergent Bug" crops up with any significant degree of regularity.

If my damage function is strength * weaponpower, doubling my strength always doubles my damage output, whether my strength is 0.5 or 5 million. Adding 2 strength always increases damage by twice my weaponpower, no matter what my strength was before. No matter how many dimensions you multiply together, increasing one of them always results in a constant increase, not a diminishing increase.

I think where you may have been confused is in your initial example where you give 'critical hit %' as one of your dimensions. In many interpretations, a critical hit requires a hit, so it already gets multiplied against 'chance to hit'. It's a complex factor in itself, so you can't just add it in like it was an ordinary dimension.

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Maybe Strength and Endurance are the componants of the player's Running Speed. Individually, they effect linear changes in Speed, but to change them both imposes not additional cost for the player, but multiplicative cost. The player must pay in two dimensions, instead of one.

You just said they individually effect linear changes in Speed. Therefore the cost to increase speed is linear. If the player changes them both, they get a "multiplicative" increase in speed, for a "multiplicative" cost. I think you just have to let go of this one..

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Original post by Argus2 If my damage function is strength * weaponpower, doubling my strength always doubles my damage output, whether my strength is 0.5 or 5 million.

Doubling your strength doubles your damage output, but doubling your damage output does not double your strength or your weaponpower. Your strength and your weaponpower grow as the square root of your damage output. You have implicitly said so by multiplying them together.

The volume of a box doubles if I double it's length. But if I double it's volume, the length, width and height do not double. They grow as the cube root of the volume.

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Doubling your strength doubles your damage output, but doubling your damage output does not double your strength or your weaponpower.

That depends entirely on how you distribute that output. If strength stayed constant, weaponpower does actually double, and vice versa. If you distribute equally between them, then of course they don't increase to the same extent. But there are no diminishing returns here. How does this affect a player?

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Original post by AngleWyrm It sure looks like it, doesn't it? Those two input numbers are growing, and the product is increasing as a result. So where then is this Diminishing Returns? Looking at the last example input line, and we see that the product is very close to the first input. But on the first line, the product is not close to the same size as the first input. And this nearness of size started approaching quickly, and slows down throughout the example. Taking the first input from 0.1 to 0.2 will make the product approach the same size much faster than going from 0.7 to 0.8. Maybe Strength and Endurance are the components of the player's Running Speed. Individually, they effect linear changes in Speed, but to change them both imposes not additional cost for the player, but multiplicative cost. The player must pay in two dimensions, instead of one.

I have no idea what you are trying to say here.

Lets just assume that we increase both variables equally, this gives you an x^2 (that is, X squared) equation as you go from 0.1*0.1 to 0.2*0.2. The result is this:



please point to where the diminishing returns are. Id like to know at what point the effort to gain a point in each variable (two points) gives a diminished return on that effort. From what I can see, the more effort I put in, the higher rate of return I end up with.

Even if I ignored one variable and maxed the other, you have a linear graph, x instead of x squared, thats still not diminishing, just staying stable.

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Original post by Argus2 Quote: Doubling your strength doubles your damage output, but doubling your damage output does not double your strength or your weaponpower. That depends entirely on how you distribute that output. If strength stayed constant, weaponpower does actually double, and vice versa. If you distribute equally between them, then of course they don't increase to the same extent. But there are no diminishing returns here. How does this affect a player?

What you have just described is not a direct scaling up of damage output. It is equal in quantity, but unequal in shape. The shape has been changed on purpose, by modifying the relative sizes of the two sides that make up the whole rectangle. This 'on purpose' is the part that the player should be doing; the player can pick and choose where in the 2D space of strength vs weaponpower they wish to go, in order to achieve a given damage output.

And for the designer, making that space interesting makes the game interesting. Perhaps there are weaponpower upgrades that are available to a weapon, giving a spread. But the next weapon requires different upgrades, and so a question of invest now, or hold off until the next weapon comes up. Where in the strength-weaponpower space is it worth going? A voyage of discovery for the player, in a landscape engineered by the designer.

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Original post by JasRonq Lets just assume that we increase both variables equally, this gives you an x^2 (that is, X squared) equation as you go from 0.1*0.1 to 0.2*0.2. The result is this please point to where the diminishing returns are.

If we increase the variables at a steady rate (as is done in the above illustration), the product increases as in your picture.
BTW, Nice graph - how did you do that?
If on the other hand we increase the product at a steady rate (as is done in the below illustration), the change in the variables shrinks.



And judging by the level of "That can't be right!" in the responses, (as opposed to just argumentativeness), it appears that this observation is quite amazing.

Another example: Suppose we have a factory that produces units. The player can boost factory production by adding Personnel. What if we want to put a diminishing return on this influence of personnel on factory output?

We re-tool Personnel to be multidimensional; instead of just personnel, we have personnel and equipment.

EDIT:Then the formula is changed to Input = Sqrt(Personnel * Equipment), to be distributed amongst Personnel & Equipment as the player chooses:



[Edited by - AngleWyrm on May 4, 2008 12:59:00 AM]

That last graph is a good demonstration of the lack of diminishing returns there. Take a look along the diagonal. Do you see negative curvature? And as one adds a third variable, the curvature becomes positive. The moral of the story is that if you have stats which are multiplied together it's stupid to only increase one of them.... but that's hardly revolutionary.

[Edited by - Sneftel on May 1, 2008 9:39:43 PM]

I think you have it backwards. If I don't have to double my strength to get double the out put then that is not a diminishing return. A diminishing return would be I double my strength once and get double the output then double it again and only get a 75% increase. See a diminishing return requires an output to become smaller for equal increases of input. What you are talking about is going from output to input and is called a multiplicative increase, that is a small increase in base numbers produces a large increase in results.

The center diagonal is growing linearly, but the legs are growing with the tell-tale sideways parabola.

The player is NOT investing in the legs. The player is investing in the TOTAL, and getting a choice of how to distribute a change between the legs.

If the player goes for the central diagonal, then they choose to get a 50/50 split on the legs. OR they could go for 100% on one leg, but not be getting the center diagonal as a result.

[Edited by - AngleWyrm on May 4, 2008 12:52:51 AM]