Monday, March 26, 2012

Philosophizing About Probability

Over at Digital Orc he posted a video arguing about the distinction between "experimental" probability versus "theoretical" probability.  I disagree with his definitions.  Probability is always theoretical.  What he calls an "experimental" probability is better be described as an estimate of the probability subject to an experimental uncertainty.  From the rules of error propagation, it is actually possible to derive a formula for the uncertainty associated with one's estimation of a probability. 

Exactly how this distinction would be valuable to gamers is unclear to me, except maybe for assessing the fairness of one's dice.  Even then, though, to really do it properly you'd need to invoke Bayes theorem and calculate the probability that the dice are fair given that you rolled some number of successes.

What's valuable to gamers isn't a largely philosophical discussion of the nature of probability.  What's more useful is concrete tactical advice based on mathematical analysis.  That's more difficult, or at least more tedious.  Let's face it, multiplying all the little probabilities together to make the probability tree for a combat round is mind numbingly dull.  I still haven't filled out that darned transition matrix.  Maybe I should just give up and write the Monte Carlo.  Or maybe I should write a program for populating the matrix.  Huuuum...

Sunday, March 11, 2012

Dungeons and Geometry

I was recently looking through an archive of early Dungeon magazines, and found a fascinating article in issue No. 6, July/August 1987, which featured a great example of a non-Euclidean dungeon called The Forbidden Mountain by Larry L. Church. If you all can find a copy of that issue, I highly recommend it. The article begins with a discussion of spherical and hyperbolic geometries, and their implication for how a dungeon mapped on an ordinary piece of graph paper would in fact play (see figures 1, 2, and 3).  Next he imagines briefly the possibilities of dungeons in arbitrarily curved space (figure 4), and produces a diagram of a dungeon that is effectively on a Moebius strip (figure 5)!  Gary Gygax was known to have had a great deal of respect for gamers' minds, believing them to have above average intelligence and creativity. It's striking to me that so early in the magazine's history they would put such a sophisticated thought in their magazine.  It speaks to the great deal of respect the magazine had for the intelligence of their readers.  I'm suspect that respect is gone today.

I also can't help but wonder if their inclusion of a non-Euclidean dungeon in the first year of publication suggests that at least among homebrew dungeon designers, such machinations were more common than one might guess from early modules.   

Both H.P. Lovecraft's The Dreams in the Witch House and The Call of Cthulu used non-Euclidean geometry as a device to make their horrors seem alien, with access to unfathomable knowledge which humanity is only able to touch upon. It was a common theme in all his stories.  Since even in his time, the ideas of alternative formulations of geometry were well developed, it's implicit in his stories that humanity had already gone "too far," in its search for knowledge.  The Theory of Relativity was something we were not "meant" to know.  The ability to manipulate space itself would almost certainly be a hallmark of powerful magic or divine intervention. I was also thinking about when I was eleven and learning to play D&D while also teaching myself to program in Ye Old BASIC with Lyne Numbers on my ATARI XE computer.  I remember carefully keying in a program called "Hunt the Wumpus" and taking pride in how I'd made the Wumpus' lair into a Moebius strip.  Dr Who's TARDIS would be impossible without non-Euclidean geometry.

As a dungeon master's tool, geometrical tricks are probably about the dirtiest, least expected trick you can pull.  Particularly with all the advanced graphics tools we have now, there's no reason for us to feel constrained to two or three dimensions.  None the less, it seems that we've all succumbed to the tyranny of our graphics software, which invariably seems to produce a flatter, less, mindbending world.  Non-Euclidean dungeons could be examples of weird "funhouse" sorts of magical effects, or else inspire horror and awe in the tradition of H.P. Lovecraft, were a dungeon becomes a weird arrangement of higher dimensional objects, time and space lose their conventional meaning, and serves to make humans feel weak and pitiful in the face of superior beings from the outer reaches.

Maybe it's just me, but growing up I felt like my friends and I all shared an uncomfortable fascination with the alternative formulations of space and time.  In highschool I read Albert Einstein's The Theory of Relativity, and Rucker's Geometry, Relativity and the Fourth Dimension. In college one of our textbooks was Spacetime Physics by Taylor and Wheeler.  Contemplating non-Euclidean geometries was an important part of my intellectual growth as a young person.  Dungeons and Dragons was one place where people could perform gedankenexperiments  and develop a visceral feel for them.  I suspect the experience was more widespread than is commonly acknowledged.  Maybe it's also just me, but it seemed like non-Euclidean geometry was everywhere then.  Maybe I've just lost touch with geeky kids.  So... my next dungeon will be non-Euclidean.

Sunday, March 4, 2012

More On Backstabbing

A recent discussion of ways to play out the backstabbing mechanic has led me to review the rules, because they greatly influence the probabilities of success.  The Labyrinth Lord ruleset, as written, is extremely uncharitable to thieves in this one respect.  "[He] must catch an opponent unaware of his presense, using move silently and hide in shadows." The requirement to do both makes a successful backstab exceptionally improbable for low level thieves.  After a complaint from a player I've been trying to get interested in old school play, I've been exploring ways to adjust so that thieves could be a little more exciting to play.  Some of those ideas were discussed here.  I also decided to look it up in my old rule books.

According the Rules Cyclopedia:

If a thief can sneak up on a victim, completely unnoticed, the thief may backstab - if he is using a one-handed melee weapon, he may strike at particularly vulnerable points of his target's body. (Though the ability is called "backstabbing," the weapon doesn't have to be a stabbing weapon.  A thief can use this ability with a club, for example.)
If the intended victim sees, hears, or is warned of the thief's approach, the thief's attack is not a backstab; it is an ordinary attack, doing the damage appropriate for the weapon used.

When no battle is in progress, a backstab attack may require a Move Silently ability check.  The DM will make all the necessary decisions on that matter.

Furthermore, in my old, red box, Mentzer, D&D Player's Manual it describes the backstabbing ability this way:
If a thief can sneak up on a victim, completely unnoticed, the thief may Backstab.  If the intended victim sees, hears, or is otherwise warned of the thief's approach, a Backstab may not be taken but the thief may still attack normally.
EXAMPLE: An Apprentice is carrying a sword, and sees an ogre approaching the party.  The player says "I'll Hide in Shadows."  The DM rolls 19 on d%, so the ogre does not see the thief (but the DM does not announce the fact). During the battle, the ogre gets turned around, with its back towards the thief.  The player says, "I'll try to move in for a Backstab!" The DM decides that the ogre doesn't notice the thief's approach (no roll is made; it depends on the situation, and the DM's judgement) and says "The ogre doesn't notice you; roll for a Backstab." The thief player then makes a Hit Roll...
So originally, the issue wasn't necessarily the game mechanics, but rather the DM's discretion. Depending on the situation, a DM might require a Move Silently check, a Hide in Shadows check, both, or none.  While I definitely am a big fan of the Labyrinth Lord rule set, I'm coming to appreciate it's differences from D&D.  In this one little respect, I think Labyrinth Lord desperately needs to be tweaked.  One of the brilliant features of old school gaming is how much is left to the dungeon master.  The way it's worded in Labyrinth Lord specifically, though, is excessively binding and needs to be house ruled. Otherwise, no rational player would backstab until achieving a fairly high level. Not only that, house ruling it would be more in keeping with the original game rules.