(This is in response to John Wick's "Chess is not an RPG" post. Read that first lest you succumb to my ramblings mid-stream).
Part of my game design learning has been working out mental models of how to translate certain Role Playing Game concepts into computer game concepts. I think part of what John was trying to elaborate in his initial post can be explained through thinking about how RPG mechanics can and cannot be described as algorithms to a computer. I think the computer-as-game-arbiter can help codify some of these concepts.
Computers have long had a history with both RPGs and board games. Take a game like Chess for instance. Chess can be easily represented to a computer: the board is a simple 8x8 checkerboard with various pieces that have different pre-determined roles. Each of those roles can be codified so that even a simple computer like the Atari 2600 could be made to play chess. Similarly there are whole areas of board games where the mechanics can be explained to a computer. Games like Monopoly, Go, Star Realms, Magic the Gathering, etc. have computerized versions that can arbitrate the rules effectively for players, and in some cases the computer can play relatively competent versions of those games against human players. The mechanics of the games are algorithmic: a bishop can only move diagonally on its own color, a rook moves in straight lines, neither can move through another piece, the king can't move into check, etc.
D&D has similar rules. I've written a grossly over-simplified version of D&D's third edition combat. In this scenario a fighter who always wins the initiative roll uses a long sword against a goblin. The goblin retaliates with its weapon. Play continues until one party loses all hit points.
Now one might argue this doesn't take into account the myriad of ways one could perform combat in D&D 3E. Indeed I had to simplify my example a great deal in order to keep my own example simple (and for the record I was sorely tempted to make this a GURPS example since I'm more familiar with GURPS and I think GURPS keeps things much simpler than D&D 3E. But I digress. :) )
What you'll notice even with all of the options for D&D 3E is combat is still algorithmic. There may be more choices for the players, and they can narrate those choices however they wish but without the narration it's still "roll dice, add modifiers, compare against target number, roll damage, add modifiers, subtract from hit points, check if below zero".
And therein lies the rub: You can narrate yourself the most epic combats with characters swinging from chandeliers while swishing swords and lopping off monster-noggins hither and yon, but that's just window dressing for mechanics that don't care one whit if you narrate glorious tales or simply say "I swing my sword. I roll a 12. Add 3 and that's 15. What's the AC?"
Combat is actually one of D&D's strengths. Let's say you're planning on charming a noble to do some favor for your character. You add up your modifiers and 1d20 roll and compare it against a number (generally all of the NPC's modifiers and 1d20 roll) and see who is higher. The player with the higher number wins the contest. Let's say you're looking to model the character of Balric from Black Adder. He's not that smart, has relatively low charisma, but is capable from time to time of saying rather profound and insightful things. The rules say he needs to play a simple high/low game in order to have a successful speech, but if the narrative would be better served by him successfully giving that speech then the GM has to either ignore the rules (which is possible) or explain away how the speech failed. At that point it becomes tea-leaf reading: the results are what they are but it's up to the players and GM to determine what happened.
One result of D&D having rather algorithmic combat and interactions is borne out by the myriad of computer games that use a form of D&D as their engine. Computerized RPGs can rely on simple combat mechanics and menu selection to give a reasonable RPG-like experience. RPG games like Wizardry, The Dark Spire, and Eschalon don't even hide how much they've sourced from D&D mechanics. The computer can take care of the bookkeeping and dice rolls and the player is free to immerse themselves (or not) in the world. How well the players role-play with those computer games is irrelevant to the success or failure of the mechanics. Similarly the mechanics of D&D don't directly lend themselves to role-playing.
So if games like D&D can be turned into algorithms are there games that don't lend themselves to this sort of conversion? Absolutely. Games like Dungeon World, Fudge and Fate don't lend themselves to tidy conversions. Fudge and Fate in particular are very loose with their mechanics so a simple dice roll of "Great" can give narrative license for a whole slew of actions. Plus Fate has the Fate Point economy where certain actions (if thwarted) can result in the rewarding of a point, and can be spent later on to affect the outcome. Fate and Fudge also aren't tied to maps the same way that D&D is (you're in an area in the building, not on a particular square or hex). Dungeon World also introduces the concept of picking one, two, or three things that can happen to your character, of which most are rather broad strokes descriptions that don't translate well into algorithms. The interplay between player and GM is more organic in these games so current computer technology would need more advanced algorithms in order to replicate these experiences.
I won't go so far as to argue that algorithmic or rote play in D&D and similar games somehow defeats role-playing; folks have been using these systems for decades to tell some amazing stories. But I can see the point where these algorithms can impede and stifle role-play and story-telling. We've all heard the stories about failed dice rolls derailing the story, or the character that tries multiple times to overcome some obstacle while the rest of the players stall out and get bored. With the more modern designs in games like Dungeon World and Fudge / Fate there's opportunities to go beyond rote and methodical play and let the engine encourage the players to participate more in role-playing their characters instead of watching the mechanics of the game.
When your encounters and interactions with the world can be reduced to scripts and decision trees there's little left for your character to do but be driven by forces outside of their control and hope that a random-number generator rolls numbers that equal success. And if your table talk can boil down to "I rolled a 14, did I hit it?" then you have to ask yourself if the mechanics are really permitting role-play or if you're just tacking them on because it's more fun that way.
As always comments and corrections are welcome. My D&D knowledge is pretty rusty so I know I probably messed up some of the specifics, but I hope the underlying message is in there somewhere. Be gentle, and thank you for reading this far.