I believe that there is a design oversight that is very rarely addressed or remarked upon.
This is due to the fact that many problems look like they have only one facet to be modified and so they look like impossible balancing acts. Today, let's talk about the player.
There is a fundamental tension between these two statements:
Players want to feel powerful vs. Players want to have fun / Players want to be challenged
1. Players want to feel powerful...
Games are escapism. Players don't want to feel mundane. They want to feel like they've done things right. They want to feel like their character or avatar is great. They want to feel good about themselves, in that they've done well.
In an RPG, this would mean that players don't want to feel weak. They don't want to have to constantly juggle to do impressive feats.
In an FPS, this means that the player doesn't want to feel like he's holding a pea shooter. He needs to have some visible effect on the environment around him.
In short, in a game, the player needs to feel good and have a character that can affect the game environment very strongly. If the player spends 36 hours to whack one thing until it dies, the player instead feels weak, pathetic as if he's not doing something "right" or that he is missing the point.
...and yet.. when players are TOO powerful...
Players, however, get very bored when all they have to do is plow through the same things over and over again. The most engaging game is one where you are constantly challenged a half-step beyond your skill.
If every single enemy dies in easily in a single blow with no chance of success, that may be fun for a little while as the player would have an enormous impact on the world... but if that's the entirety of the game then the game isn't going to be very fun or challenging later.
If every single monster dies in a single bullet or the player is allowed to stock 9 billion health and be utterly immortal, then all that is left for the player's enjoyment is that he "feels" immortal but in reality, there is nothing pushing him to play better or get further "into" the game.
In short, if the player is super powered what gameplay turns into is a simple mindless button pressing exercise that doesn't need the player to actively engage in. A simple robot hitting 'A' for the next 12 hours would play just as well.
...but this seems impossible to get juuuust right...
You can tweak player power all along the spectrum but too little health and the player is frustrated because he dies at everything and too much health and the player feels immortal again.
Some sections might be harder than others.
Or how about, you can set the difficulty of the entire game so that the player has less health on harder difficulties and more health on easier ones...
...but that leaves the player with little choice over the matter since he knows nothing about your game and if he can't change the difficulty level in between sections of the games he is now "locked in" by his decision at the beginning of the game.
...so what's the answer?
The core oversight here in design is: They are too focused on the player and how much he effects the world and vice versa.
When you realize that the player is an active part of the entire game world then the solution becomes easier.
If the player has a large impact on the game (which he should), then the game world has to be interesting and varied enough to handle that impact.
For example, let us take a first person shooter.
If the player can kill every enemy with a single bullet that may still be interesting. Imagine the movie 'Aliens'. If the aliens are in open terrain, they are reasonably easy to deal with. They are soft, fragile creatures. However, the trick is the aliens are intelligent and swift little buggers. The player has a huge impact on the world, he can easily end the life of an alien with a swift burst of fire but the aliens themselves aren't stupid either. They'll hide in walls, leap in weird patterns or swarm in unconventional ways.
Once you realize that making enemy's more difficult can be done without decreasing the power of the player, then you can see the solution...
I'll give another example, this time in Crono Trigger. Crono Trigger was an excellent example of good game design in this regard. If there was an enemy that seemed to take a million years for you to defeat, it was a sure sign that you were missing something tactically. The acid slimes that were immune to physical and took 12 hits could be defeated by Crono's slash technique. Every creature had counterattacks that made you stop attacking at specific instances. You could avoid monsters if you were fast enough to move around them.
The players are enormously powerful in Crono Trigger but the game itself was not a boring grind because the encounters the player faced could be dealt and often had to be dealt with in a variety of changing ways; From proper target selection, to holding back on attacks, to attacking in a correct sequence, none of the encounters artificially increased the difficulty by making monsters simply have more defense or do more damage and thus take away from player power.
To rephrase a line from Dungeons and Dragons:
Powerful heroes need Interesting encounters.
Monday, August 31, 2009
Thursday, August 27, 2009
Steal My Idea 1
Note: Going to try something out here where I do a bit of speculative game design called "Steal my Idea."
Steal my Idea
I intend for this to be a semi-regular addition to the blog where I outline a specific mechanic or game feature that could be implemented to serve a specific purpose.
Note that if you *actually* steal my idea, comment or shoot me an e-mail at kuoyen on gmail.com.
Pair Bonding / Team Training
An quest line to reward or facilitate player interactions with each other
Intended game: Casual MMOs that encourage player community.
Form: Daily Quest Chain
Original Inspiration: Ether Saga Online
Main Idea: The more quests the pair complete together, the greater the reward for each portion of the quest. Either player may decide to break their chain at any time by turning in their quest reward. In this case, both players are rewarded for how many pair quests they have completed and can restart again at Quest 1 after a day.
Quest Start: Give players tokens that they can get daily. The token given can be based on a variety of factors depending on the desired effect. Class can be an option or self-described gender or even random.
Restrictions: Some minimum level after tutorial levels.
Quest 1: Collect a token of a different type from someone. Give your token to that same person.
Restrictions: Similar level ranges are encouraged.
Alternative: Wildly differing level ranges can also be encouraged to promote old players helping new players.
After this quest, the two players are given a quest specific item that marks the two of them. Thereafter, the two, if partied, gain the option to participate in a number of bonus quests that require the two to interact
Possible Pair Quests: Parallel dungeons
A dungeon instance where the two players are isolated in two different paths. Each must manipulate switches and levers to get the other player across to the final area.
Pair Quest: Follow the leader.
A complex dungeon filled with paths and numerous doors and things to interact with. In alternating sections one player is designated as leader and is able to 'see' the correct path using a buff. The other player must follow the first player through the dungeon without being able to see which paths are safe or not. Things like ropes, platforms that you have to jump or ladders.
Pair Quest: Arena - Monster Mash
A simple kill quest where the amount of monsters are too much for one player to handle to encourage teamwork.
Pair Quest: Arena - Single Boss
Possibly at the end of the arena, where instead of dealing with a group of monsters; the player learns to face an individual threat in a pair. The boss selection should probably be based on the player's classes to strain their individual abilities.
Possible Rewards:
- An appropriate amount of in-game currency for that level.
- An appropriate amount of experience for that level.
- Pair Points that denote that you are an effective teammate!
Spinoff Idea: You can scale this up to a group of 4 or 5 but at that point groups are rather hard to form and quickly untenable for players to take full advantage of.
Steal my Idea
I intend for this to be a semi-regular addition to the blog where I outline a specific mechanic or game feature that could be implemented to serve a specific purpose.
Note that if you *actually* steal my idea, comment or shoot me an e-mail at kuoyen on gmail.com.
Pair Bonding / Team Training
An quest line to reward or facilitate player interactions with each other
Intended game: Casual MMOs that encourage player community.
Form: Daily Quest Chain
Original Inspiration: Ether Saga Online
Main Idea: The more quests the pair complete together, the greater the reward for each portion of the quest. Either player may decide to break their chain at any time by turning in their quest reward. In this case, both players are rewarded for how many pair quests they have completed and can restart again at Quest 1 after a day.
Quest Start: Give players tokens that they can get daily. The token given can be based on a variety of factors depending on the desired effect. Class can be an option or self-described gender or even random.
Restrictions: Some minimum level after tutorial levels.
Quest 1: Collect a token of a different type from someone. Give your token to that same person.
Restrictions: Similar level ranges are encouraged.
Alternative: Wildly differing level ranges can also be encouraged to promote old players helping new players.
After this quest, the two players are given a quest specific item that marks the two of them. Thereafter, the two, if partied, gain the option to participate in a number of bonus quests that require the two to interact
Possible Pair Quests: Parallel dungeons
A dungeon instance where the two players are isolated in two different paths. Each must manipulate switches and levers to get the other player across to the final area.
Pair Quest: Follow the leader.
A complex dungeon filled with paths and numerous doors and things to interact with. In alternating sections one player is designated as leader and is able to 'see' the correct path using a buff. The other player must follow the first player through the dungeon without being able to see which paths are safe or not. Things like ropes, platforms that you have to jump or ladders.
Pair Quest: Arena - Monster Mash
A simple kill quest where the amount of monsters are too much for one player to handle to encourage teamwork.
Pair Quest: Arena - Single Boss
Possibly at the end of the arena, where instead of dealing with a group of monsters; the player learns to face an individual threat in a pair. The boss selection should probably be based on the player's classes to strain their individual abilities.
Possible Rewards:
- An appropriate amount of in-game currency for that level.
- An appropriate amount of experience for that level.
- Pair Points that denote that you are an effective teammate!
Spinoff Idea: You can scale this up to a group of 4 or 5 but at that point groups are rather hard to form and quickly untenable for players to take full advantage of.
Wednesday, August 26, 2009
Math and Difficulty: Percentile and Fixed Scaling: Part 4
Note: This is part 4 of a section on Math and Scaling of item modifiers.
Let me describe... possible solutions to the points I've talked about in this series and some discussion about the results.
0. The "Safe" law about percent modifiers, if I didn't stress it enough..
Reduction modifiers should stack multiplicatively.
Enhancement modifiers should stack additively.
The reasoning being is that this is the safest way to ensure that things don't have exponential growth.
0b. Consider taking only the highest bonus or prevent things from stacking.
If you are *super* paranoid about things scaling out of control, consider using only the highest reduction or enhancement of each category. Typically however, this will simply lead players to cherry pick a whole slew of other statistics and become well rounded.
This allows you to create a whole spectrum of gear without worrying about potentially dangerous interactions allowing for better ease and flexibility of reward.
For example, if they can only have one enhancement to critical strike then they will then choose to upgrade damage, accuracy, speed, etc. Since focusing on one statistic is useless they will choose to get bonuses in a variety of statistics.
(This may be a good thing.)
Note/Beware: Make sure players understand only the highest bonus applies.
Note/Beware: Lots of nominally unrelated things can actually be related! Damage %, Attack Speed %, Number of hits, Accuracy % all increase DPS! If you ensure that only the highest bonus takes effect, be sure that you have as many offensive modifiers as defensive modifiers.
Otherwise, if there are more offensive modifiers, players that like to play defense won't be able to "catch up" or vice versa, if there are more defensive modifiers, players might not be able to do any damage.
1. Cap reduction modifiers aggressively.
Capping reduction modifiers (damage reduction, slows target, etc.) aggressively (to 50% or less) hinders their exponential growth regardless of what kind of math model you use to accumulate reduction modifiers.
That is, if you cap all reduction modifiers to a maximum of below or equal 50% then you keep the marginal benefit of each additional percent of reduction roughly equal. This is due to the fact that in reductions each percentage has increasing returns that is grows roughly exponentially.
For example, let's consider a speed reduction spell. If this spell reduces their attack speed by 10% players will survive 11% longer since his damage output will drop to 90% of what it was, a relative difference of 11%.
1/0.9 -> 1.11
1/0.8 -> 1.25
1/0.7 -> 1.42
1/0.6 -> 1.66
1/0.5 -> 2
You can see the exponential creep start here where the last 10% is 3 times as effective as the first 10% but after the 0.5 mark, it starts to climb to ridiculous levels, such as 1% being 10 to 20 times more effective at the 0.66 mark than the first 1%.
1b. Consider compensating players for capped stats - Optional! Interesting!
If players are theoretically able to reach a percentage higher than your cap then compensate them in other ways in order. If players are not compensated for capped stats then you start to hit another odd phenomenon where players stack equipment to cap as many different statistics as possible.
For example, in WoW players reach a critical strike % cap at roughly 23% chance to critical. This causes players to go for hit % chance in order to increase their DPS, causing both statistics to be maxed by both players and making equipment diversity kind of useless. The problem is that players will want to optimize themselves and since the multiple capped statistics can are obtainable and maximizable then they will seek to maximize all of them and so everyone's bonus statistics looks the same.
If additional critical strike % had been converted to critical strike bonus damage %, this would be an interesting compensation that would allow players to have builds that are more critical focused with some loss in consistency by sacrifice hit rating.
Some Examples:
Hit % Chance -> Critical Strike %
If they are able to hit a target 100% of the time then it makes sense that they could perform more deadly critical blow.
Debuff Target % -> Increased Hit % for Debuff
For example, if their ice bolt slows for the maximum 50% slow then compensate them by making the 50% slow more reliable by reducing the enemy's chance to resist the spell.
Resist Effect % -> Flat % Reduction if they fail to resist.
For example, if a character has a 50% resist against stun effects, if they go over the cap perhaps consider compensating them by making the stun they *fail* to resist last shorter.
2. Have scaling flat bonus effects based on level.
This is *similar* to having percentile effects but still allows equipment to grow obsolete when the player reaches a new tier of equipment.
A weapon could have +1 / damage per level of the player. This can, in theory, scale with the player like % effects... However, you are guaranteed that this bonus scales linearly. If your health/defense factors scale exponentially then this equipment is guaranteed to be relevant for a while and still be outgrown and irrelevant after a number of levels without any other fact.
For example, let's say that at each level, our hero (and the monsters) gain 10% a level and our hero, receives, as a prize a sword that does 10 damage and +1 additional damage per level!
Level 1: 100 hp / 11 damage sword = 10 hits to die.
Level 2: 121 hp / 12 damage sword = 11 hits to die.
Level 3: 133 hp / 13 damage sword = 11 hits to die.
Level 4: 146 hp / 14 damage sword = 11 hits to die.
Level 5: 161 hp / 15 damage sword = 11 hits to die.
Level 6: 177 hp / 16 damage sword = 12 hits to die.
Level 7: 194 hp / 17 damage sword = 12 hits to die.
Level 8: 214 hp / 18 damage sword = 12 hits to die.
Level 9: 236 hp / 19 damage sword = 13 hits to die.
The sword is slowly becoming irrelevant. In a few more levels, the sword will have to be replaced because it kills monsters far too slowly. He will soon need to switch equipment thus allowing us to simultaneously give a useful bonus and a bonus that will eventually become irrelevant.
This is better than both flat bonus and % bonus to damage in terms of how it scales and how easy it is to gift. However, this works off the following assumptions:
1. Your bonus scales linearly. (In the above example, it was x * 1)
2. The thing it scales against rises exponentially (In the above example, it was a x^10% factor).
However, beware. If your bonus is too large then you run the risk of players not bothering to replace their weapons for a long time because they don't need to. It scales "well enough" for them to skip an upgrade or two. This is good in a sense, in that players will never hit an equipment dead-end. This is bad in a sense, in that it encourages players not to upgrade.
Let me describe... possible solutions to the points I've talked about in this series and some discussion about the results.
0. The "Safe" law about percent modifiers, if I didn't stress it enough..
Reduction modifiers should stack multiplicatively.
Enhancement modifiers should stack additively.
The reasoning being is that this is the safest way to ensure that things don't have exponential growth.
0b. Consider taking only the highest bonus or prevent things from stacking.
If you are *super* paranoid about things scaling out of control, consider using only the highest reduction or enhancement of each category. Typically however, this will simply lead players to cherry pick a whole slew of other statistics and become well rounded.
This allows you to create a whole spectrum of gear without worrying about potentially dangerous interactions allowing for better ease and flexibility of reward.
For example, if they can only have one enhancement to critical strike then they will then choose to upgrade damage, accuracy, speed, etc. Since focusing on one statistic is useless they will choose to get bonuses in a variety of statistics.
(This may be a good thing.)
Note/Beware: Make sure players understand only the highest bonus applies.
Note/Beware: Lots of nominally unrelated things can actually be related! Damage %, Attack Speed %, Number of hits, Accuracy % all increase DPS! If you ensure that only the highest bonus takes effect, be sure that you have as many offensive modifiers as defensive modifiers.
Otherwise, if there are more offensive modifiers, players that like to play defense won't be able to "catch up" or vice versa, if there are more defensive modifiers, players might not be able to do any damage.
1. Cap reduction modifiers aggressively.
Capping reduction modifiers (damage reduction, slows target, etc.) aggressively (to 50% or less) hinders their exponential growth regardless of what kind of math model you use to accumulate reduction modifiers.
That is, if you cap all reduction modifiers to a maximum of below or equal 50% then you keep the marginal benefit of each additional percent of reduction roughly equal. This is due to the fact that in reductions each percentage has increasing returns that is grows roughly exponentially.
For example, let's consider a speed reduction spell. If this spell reduces their attack speed by 10% players will survive 11% longer since his damage output will drop to 90% of what it was, a relative difference of 11%.
1/0.9 -> 1.11
1/0.8 -> 1.25
1/0.7 -> 1.42
1/0.6 -> 1.66
1/0.5 -> 2
You can see the exponential creep start here where the last 10% is 3 times as effective as the first 10% but after the 0.5 mark, it starts to climb to ridiculous levels, such as 1% being 10 to 20 times more effective at the 0.66 mark than the first 1%.
1b. Consider compensating players for capped stats - Optional! Interesting!
If players are theoretically able to reach a percentage higher than your cap then compensate them in other ways in order. If players are not compensated for capped stats then you start to hit another odd phenomenon where players stack equipment to cap as many different statistics as possible.
For example, in WoW players reach a critical strike % cap at roughly 23% chance to critical. This causes players to go for hit % chance in order to increase their DPS, causing both statistics to be maxed by both players and making equipment diversity kind of useless. The problem is that players will want to optimize themselves and since the multiple capped statistics can are obtainable and maximizable then they will seek to maximize all of them and so everyone's bonus statistics looks the same.
If additional critical strike % had been converted to critical strike bonus damage %, this would be an interesting compensation that would allow players to have builds that are more critical focused with some loss in consistency by sacrifice hit rating.
Some Examples:
Hit % Chance -> Critical Strike %
If they are able to hit a target 100% of the time then it makes sense that they could perform more deadly critical blow.
Debuff Target % -> Increased Hit % for Debuff
For example, if their ice bolt slows for the maximum 50% slow then compensate them by making the 50% slow more reliable by reducing the enemy's chance to resist the spell.
Resist Effect % -> Flat % Reduction if they fail to resist.
For example, if a character has a 50% resist against stun effects, if they go over the cap perhaps consider compensating them by making the stun they *fail* to resist last shorter.
2. Have scaling flat bonus effects based on level.
This is *similar* to having percentile effects but still allows equipment to grow obsolete when the player reaches a new tier of equipment.
A weapon could have +1 / damage per level of the player. This can, in theory, scale with the player like % effects... However, you are guaranteed that this bonus scales linearly. If your health/defense factors scale exponentially then this equipment is guaranteed to be relevant for a while and still be outgrown and irrelevant after a number of levels without any other fact.
For example, let's say that at each level, our hero (and the monsters) gain 10% a level and our hero, receives, as a prize a sword that does 10 damage and +1 additional damage per level!
Level 1: 100 hp / 11 damage sword = 10 hits to die.
Level 2: 121 hp / 12 damage sword = 11 hits to die.
Level 3: 133 hp / 13 damage sword = 11 hits to die.
Level 4: 146 hp / 14 damage sword = 11 hits to die.
Level 5: 161 hp / 15 damage sword = 11 hits to die.
Level 6: 177 hp / 16 damage sword = 12 hits to die.
Level 7: 194 hp / 17 damage sword = 12 hits to die.
Level 8: 214 hp / 18 damage sword = 12 hits to die.
Level 9: 236 hp / 19 damage sword = 13 hits to die.
The sword is slowly becoming irrelevant. In a few more levels, the sword will have to be replaced because it kills monsters far too slowly. He will soon need to switch equipment thus allowing us to simultaneously give a useful bonus and a bonus that will eventually become irrelevant.
This is better than both flat bonus and % bonus to damage in terms of how it scales and how easy it is to gift. However, this works off the following assumptions:
1. Your bonus scales linearly. (In the above example, it was x * 1)
2. The thing it scales against rises exponentially (In the above example, it was a x^10% factor).
However, beware. If your bonus is too large then you run the risk of players not bothering to replace their weapons for a long time because they don't need to. It scales "well enough" for them to skip an upgrade or two. This is good in a sense, in that players will never hit an equipment dead-end. This is bad in a sense, in that it encourages players not to upgrade.
Monday, August 24, 2009
Math and Difficulty: Percentile and Fixed Scaling: Part 3
Note: This is the third in a series about math and game balance, specifically regarding to item balance.
5. Percentile bonuses that can accrue are dangerous. Percentile bonuses out of 100 that can accrue are doubly dangerous.
+400% to a statistics at late game dwarfs everything else. A small insignificant +1 damage suddenly becomes worth 5 times the initial worth. A +1% critical bonus at the beginning of the game might be worthwhile until you realize that you can't give away more than 1% or 2% because by end game, players will reach 100%.
The crux of this is when you realize that some bonuses have a maximum relevant amount. For example, once you have reached +100% damage reduction, you can't give any more because it becomes meaningless. Some attributes simply should not be raised that high. So how do you pace yourself? Do you start off giving the player miniscule bonuses to damage reduction that accumulate? Do you arbitrarily cap players to a maximum amount of any one statistic and let overflow do nothing?
How would you then give them a "better" version of say "critical strike" gear without actually increasing the amount of critical strike they get? Or do you keep making gear with the same percentages? What if you miscalculate and accidentally give them 95% critical strike? What if you need to add new content but they've already reached absurd levels of bonuses?
World of Warcraft takes the approach of factors that scale down dependant on your level. At level 1, one critical factor might give you 1% critical strike but at level 50, one critical factor gives you 0.05% critical strike. This makes it easy to create new equipment that's better than the old (Here! This one has +86 critical factor while the other one only has 43!) but this introduces another problem. Leveling up decreases the power of your equipment. Leveling up in their system simply makes all your existing gear worse and it introduces some confusion in the inner workings of the game (Just how much critical factor do I need to reach that critical % cap?)
6. Not all percentiles are equal: Reduction and additions are and must work differently.
If I have 90% damage reduction and I get a mere 5% more reduction additively; I don't take 5% less damage. I take 50% less damage because now I'm at 95% reduction. Alternatively if I have 100% bonus to damage and I get 50% more bonus to damage additively, I don't do 50% more damage, I only do a mere 20% more damage because now I'm at 150% bonus damage.
This phenomenon often pops its nasty head up when designers either scale poorly or don't reliaze the exponential growth of penalties or the possible exponential growth of bonuses.
For example, let us take a item that reduces incoming damage by 50%. This is a very nice item as it effectively doubles our lifespan. However, what do you expect should happen when I equip another item that reduces incoming damage by 25%. This new item is clearly inferior to our old one.
Well, if the math works additively... 50% + 25% -> 75%. This means that our inferior item has raised our damage reduction to 75%! This means our lifespan is now four times our original by adding the inferior item.
What gives?
What if we then equipped another item that was a mere 15% damage reduction? This is even more inferior to our other item. However, if the math works additively.. 75% + 15% -> 90% and we now have ten times our original value.
Some poor designers then take this as a sign that this statistic is too powerful in large quantities and arbitrarily cap the statistic. Or else they give out the bonuses in small measure and irk out 5 and 10% damage reductions randomly.
However, what they fail to realize is this...
A reduction scales exponentially fast after 50%.
To double your lifespan with damage reduction, you need 50%
To triple your lifespan with damage reduction, you need 66%.
To quadruple your lifespawn with damage reduction, you need 75%.
The last few % are the ones that matter if you do additive reduction.
However, let's take positive multipliers. These are the inverse. Positive multipliers suffer the inverse problem.
Positive multipliers grow exponentially if they accumulate multiplicatively.
For example, if I have a sword that doubles my damage and a necklace that doubles my damage...
If the math works like this: 2 * 2 = 4!
Then I suddenly do four times the damage! If I get another item that doubles my damage..
2 * 2 * 2 = 8! I suddenly do eight times my damage!
This quickly leads to untenable growth when you realize just how many factors can accumulate multiplicatively...
Therefore, it is best that reductions and multipliers use different math: Reductions best accumulate multiplicatively while multipliers best accumulate additively to prevent dangerous exponential growth in effect.
5. Percentile bonuses that can accrue are dangerous. Percentile bonuses out of 100 that can accrue are doubly dangerous.
+400% to a statistics at late game dwarfs everything else. A small insignificant +1 damage suddenly becomes worth 5 times the initial worth. A +1% critical bonus at the beginning of the game might be worthwhile until you realize that you can't give away more than 1% or 2% because by end game, players will reach 100%.
The crux of this is when you realize that some bonuses have a maximum relevant amount. For example, once you have reached +100% damage reduction, you can't give any more because it becomes meaningless. Some attributes simply should not be raised that high. So how do you pace yourself? Do you start off giving the player miniscule bonuses to damage reduction that accumulate? Do you arbitrarily cap players to a maximum amount of any one statistic and let overflow do nothing?
How would you then give them a "better" version of say "critical strike" gear without actually increasing the amount of critical strike they get? Or do you keep making gear with the same percentages? What if you miscalculate and accidentally give them 95% critical strike? What if you need to add new content but they've already reached absurd levels of bonuses?
World of Warcraft takes the approach of factors that scale down dependant on your level. At level 1, one critical factor might give you 1% critical strike but at level 50, one critical factor gives you 0.05% critical strike. This makes it easy to create new equipment that's better than the old (Here! This one has +86 critical factor while the other one only has 43!) but this introduces another problem. Leveling up decreases the power of your equipment. Leveling up in their system simply makes all your existing gear worse and it introduces some confusion in the inner workings of the game (Just how much critical factor do I need to reach that critical % cap?)
6. Not all percentiles are equal: Reduction and additions are and must work differently.
If I have 90% damage reduction and I get a mere 5% more reduction additively; I don't take 5% less damage. I take 50% less damage because now I'm at 95% reduction. Alternatively if I have 100% bonus to damage and I get 50% more bonus to damage additively, I don't do 50% more damage, I only do a mere 20% more damage because now I'm at 150% bonus damage.
This phenomenon often pops its nasty head up when designers either scale poorly or don't reliaze the exponential growth of penalties or the possible exponential growth of bonuses.
For example, let us take a item that reduces incoming damage by 50%. This is a very nice item as it effectively doubles our lifespan. However, what do you expect should happen when I equip another item that reduces incoming damage by 25%. This new item is clearly inferior to our old one.
Well, if the math works additively... 50% + 25% -> 75%. This means that our inferior item has raised our damage reduction to 75%! This means our lifespan is now four times our original by adding the inferior item.
What gives?
What if we then equipped another item that was a mere 15% damage reduction? This is even more inferior to our other item. However, if the math works additively.. 75% + 15% -> 90% and we now have ten times our original value.
Some poor designers then take this as a sign that this statistic is too powerful in large quantities and arbitrarily cap the statistic. Or else they give out the bonuses in small measure and irk out 5 and 10% damage reductions randomly.
However, what they fail to realize is this...
A reduction scales exponentially fast after 50%.
To double your lifespan with damage reduction, you need 50%
To triple your lifespan with damage reduction, you need 66%.
To quadruple your lifespawn with damage reduction, you need 75%.
The last few % are the ones that matter if you do additive reduction.
However, let's take positive multipliers. These are the inverse. Positive multipliers suffer the inverse problem.
Positive multipliers grow exponentially if they accumulate multiplicatively.
For example, if I have a sword that doubles my damage and a necklace that doubles my damage...
If the math works like this: 2 * 2 = 4!
Then I suddenly do four times the damage! If I get another item that doubles my damage..
2 * 2 * 2 = 8! I suddenly do eight times my damage!
This quickly leads to untenable growth when you realize just how many factors can accumulate multiplicatively...
Therefore, it is best that reductions and multipliers use different math: Reductions best accumulate multiplicatively while multipliers best accumulate additively to prevent dangerous exponential growth in effect.
Wednesday, August 19, 2009
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.
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.
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