I have been working on a diceless combat mechanic for Pathfinder (it could be applied to any system in which combat involves a roll to hit and a roll for damage, but Pathfinder will be used for examples) which applies basic concepts of probability theory.
Since every dice roll has an expected value (E[1d6]=3.5, E[1d8]=4.5, etc), every weapon has an expected value for damage. By determining the probability of hitting, we can determine the damage of a given attack by multiplying the probability of a hit by a weapon's expected value.
This leads to a diceless attack mechanic. When an attack is declared, rather than rolling for attack and damage, the attacker would automatically deal damage to the defender based on his expected damage and probability of hitting.
For example, a character wielding a weapon that does 1d8 damage (or 4.5 damage) attacks a goblin. He needs to roll an 11 or higher on a d20 to hit the goblin. Thus when attacking, the character would automatically deal 0.5*4.5=2.25 damage to the goblin (rounding up or down depending on preferences).
I know this would remove critical hits from the game. Perhaps characters could hold critical tokens based on their level, traits, etc which they could spend to attain a critical hit (which could mean double damage, the maximum dice value of damage, weapon's expected damage not multiplied by probability, or something along these lines).
I'm also not sure how this could be applied to saving throws vs death, sleep, etc where there is no expected value of damage, but simply hit or miss.
What other side effects, advantages, or downsides would this averaging method have on gameplay? It's likely some game out there has done similar in the past. Does any one have any experiences playing in a system like this and can report on its effect in play?
The suggested mechanic uses similar maths that optimisers use to build combat-focussed characters. Also White Dwarf magazine presented an optional Monster Mark system of experience built around expected damage to a standardised fighter (I think this may even of made it into a TSR product, but I may be mis-remembering).
In the unmodified game, there are lots of gambling decisions to make, where the results are not guaranteed either way (although usually in favour of the PCs). In essence this is a large part of the combat game, and attempts to mirror risks and unknown factors in fictional combats. It would change almost beyond recognition with dice rolls removed.
There would also be common situations where either the DM or the players could guarantee specific results. For example, if two goblins attack a player character it would reduce the character to 0 hit points. As DM, knowing this, would you play that action or not? Seems you are damned either way - playing it is a decision to take the character out of combat, not playing it is not really trying when the combat tactics to choose are now practically the only part of combat remaining.
Pathfinder battleboard tactics and choices of what actions to take, which spell to cast etc, are not rich enough by themselves to make for a compelling combat game. Without uncertainty due to dice rolls, the game would at best be an exploration to find the end result - not clear until resolved due to free choices of action and position. In my opinion it may well become a joyless, numerical grind with number crunching replacing intuition and certainty replacing risk.
There are diceless RPG systems (Amber is one I have heard of but not played) - the difference is that have been designed as a game from the ground up without needing dice. Removing dice rolls from a system initially built around them is unlikely to lead to something as playable.