Tuesday, 16 July 2013

The Systematic Overvaluing of Critical Hits

People love critical hits, usually because a huge number pops up on the screen.  I think because people love critical hits so much, game designers have been fooled into dramatically overcosting abilities and equipment that add to them.  Much in the way that people would rather have a "no taxes" sale than a 20% off sale, people seem to prefer 10% more crit chance to 10% more damage - even when their critical hit multiplier is less than two.

What got me thinking about this is the Knave/Assassin class in Rogue Legacy.  The Assassin deals only 75% as much damage as the Paladin, but gets 15% more crit chance and 125% more crit damage.  That sounds like a lot, but since your base crit chance is zero, the average damage the Assassin does is actually only 95% of what the paladin does.

If you buy up the skills that give crit chance and damage and wear items that give you crit chance and damage then you might get the Assasin to average about 20% more damage than the paladin.  Of course this is with more variability, which is bad for players.

Let's look at two characters.  One does four damage per swing, the other does three but has a one-in-three chance to critical hit for double damage.  These two average the same damage.

If you are attacking an enemy with 12 health, which one would you rather be?  One of them kills the enemy in 3 hits every time.  The other one kills the enemy in two hits one in nine times, in three hits four out of nine times and in four hits every other time.  That's an average of three and a third hits.  Doing the same damage is less useful when the damage is more random.

This difference starts to disappear when we give the monster a lot more health.  If the enemy has 120 health instead of 12 then the average hits for the critical damage character is only 30 and three sixteenths, barely more than the 30 hits for the higher damage character.  All this confirms, though, is that randomness does not favour the player, it's just that more hits make the random criticals average out.

When criticals can help is when they can eliminate difficult enemies in a small number of hits.  But for this to be helpful you have to know you are only going to do it a small number of times.  If there is a boss that gives you are a lot of trouble and you can choose between taking 30 hits or kill it or having a one in thirty chance to kill it outright, the latter might be preferable if you know you will never live long enough to get the thirtieth hit in.  This is of no help, however, if you are traipsing through a dungeon killing enemy after enemy.  In most games you have to kill many enemies for every time you die in order to stay afloat.  To be successful at that you need to perform consistently, not have bursts where you do well and bursts where you do poorly.

Now the assassin would be fine were it not for the fact that they pay dearly for their high critical rate.  They have only 75% of the health and 65% of the mana of a paladin.  Less health and mana is a very large price to pay for something that may not be an advantage at all.

Of course none of this is even addressing the fact that on the skill tree you are choosing between 2 more damage or 2% more chance to critically hit.  When your base damage starts at 25, this isn't really a choice at all.  You probably have to invest about 50 points into the extra damage skill before the critical skills become worthwhile.

This same motif of criticals being overcosted appears in many games.  In Diablo 3 an item may have up to 200 of your primary stat or up to 10% critical chance as a modifier.  If your gear is already fantastically powerful, it is possible that the critical chance is the better pick between those two, but it would be very rare.

World of Warcraft had an even more extreme design, where in the Burning Crusade critical hit rating could be worth as little as a ninth of a spell power for some classes.  In Wrath of the Lich King there was rarely any meaningful choice in gear, but when it came to trinkets where you did have choice critical hit rating might be worth a fifth of an intelligence, strength, or dexterity at most.  That often meant using a much lower level trinket simply because the primary stat was available on that trinket, and it meant virtually never matching gem colours.

Generally, our desire to see big numbers pop up on the screen is fooling us into taking items of and skills of less value.  The problem with critical hits is that the chance they happen has to stop at 100% while player damage in many games scales to tens, hundreds or thousands of times the starting amount.  Under this model there is no way that a chance to do larger damage can compete with scaling the base amount until the critical multiplier becomes absurd.  There is certainly a problem with having a 10% chance to crit for 10,000 times your normal damage.  That is a lot more like having a 90% chance to miss.

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