Math and Difficulty: Percentile and Fixed Scaling: Part 2

Note: This post is about simple math and how they contribute to poor end-game design and scaling.

I mainly talk about damage in this article because damage is one of the few factors that directly contribute to gameplay. A game's difficulty rises or falls based on how dangerous his opponents are and damage plays a large role in determining player and monster longevity.

1. Small percentile bonuses are insignificant early on and become oversignificant later on.

Small percentile bonuses are a trap. They *look* significant early on. However, their relative contribution to damage versus fixed stats are usually tremendously small when given out in small amounts.

For example, a sword that deals 100 damage would only gain 1 damage from a 1% bonus. One might ask if that reward is actually a bonus. Unless monsters had exactly 101 life or 202 life or something; the actual difference to the player is almost nothing.

Therein comes another design trap: Giving out large percentage bonuses so they both look and actually are relevant.

4. Percentiles are very hard to give to players in the correct porportions.

Large percentile bonuses are dangerous because they never become irrelevant. 1% may be insignificant by itself but it the player is allowed to accumulate percentile bonuses to the tens or hundreds; their contribution out-shadows anything the player could do for themselves. Lots of small %'s compound much better than lots of fixed additions.

It's important enough to reiterate: If large enough quantities, they're always relevant. They will always carry an impact. They will never be 'out-leveled' or expire. An item that adds 20% to your final damage is always relevant. The only way such items can become irrelevant is if there is an item that adds 21% damage to the game.

Percentile bonuses never become old or outgrown unless a better percentile bonus comes forth. Then comes the design intention of giving out small bonuses to better control their growth but then that comes with it the same problem of complete irrelevancy in low amounts.

As a side note, percentile bonuses come in many forms. What makes it worse is that they usually compound together multiplicatively instead of increasing additively.

Damage % bonuses, attack speed bonuses, reduction in enemy defense, number of shots fired, critical % rate, critical damage bonuses are all different sounding but in the end they are all % damage multipliers to the players combat effectiveness.

2. Small fixed bonuses are over-significant early on and become trivial later on.
+1 damage to 1 damage is insane.

This is the inverse of the problem above. Due to the fact that early on, numbers tend to be lower, and that most people do not think fractional damage constitutes an appropriate reward fixed bonuses tend to be oversignificant in their contribution to damage or statistics.

However, the problem with fixed numbers is the fact that they are almost impossible to balance correctly for late game damage. Most commonly, they are trivialized later, because character and monster HP tends to scale enormously and outrace any linear progression that fixed number bonuses give.

However, fixed bonuses have one more fatal flaw that becomes readily apparent when percentile bonuses also apply. They have the potential to compound with percentile bonuses to create tremendously unfair situations between attack types. +100 to damage might not be so bad to someone who attacks once a second. But when you factor that the character might have +100% to damage and shoots twice as fast with multiple bullets, that +100 to damage can become compounded exponentially.

An example would be in the first X-men legends game. Equipment in that game could add fixed damage. For any melee attacker, these were relatively powerful equipment that added tremendously to their attacks. However, any ranged attacker capable of firing multiple bullets saw their benefit differ by a factor of 10 as they would gain their bonus across each bullet. When the damage output between characters start differing by 900% or more, you begin to see how fixed damage is a relatively dangerous concept.

3. The order of operations matters.
+1 damage then +100% damage is 4. +100% damage then +1 damage is 3.

This is related to the problem above with the interactions between fixed and percentile damage, however this is more of a player understanding and basic math issue.

That is, when determining the relative power of statistics, one must factor in when they are applied. If a fixed bonus (+1 damage) is applied before the a percentile bonus, that fixed bonus begins to get the benefit of scaling with the percentile bonus making it much harder to manage and balance as well.

However, there are generally complex interactions that go on. Even if fixed damage is applied after percentile damage, you must consider whether the enemy might have a vulnerability that might multiply his damage or if the player has a speed increase.

Generally, there are two major crimes here. If the math is completely intuitive, generally balancing the game becomes very hard. That is, if the bonuses stack the way the player expects it to, game balance is quite difficult. However, any tradeoff you make in the manageability of bonuses you trade player understanding and usability.

Consider, for example, World of Warcraft, which has neatly bypassed all these issues by introducing the concept, Attack power. While this approach does solve each of the issues described by essentially hiding all the math from the player, that leaves a very difficult problem for the player: How exactly do I increase my damage? Do I raise a statistic that increases my attack power? Do I get a faster sword? What is going on if I choose to optimize for critical damage?

Whereas, in a game like Diablo 2, you pretty much always know what is going to happen when you get more damage buffs: Exponentially increasing damage. However, this led to exponentially increasing monster hp; which caused it to leave many players in the dust if they had not accrued the necessary bonuses to survive.