07 October 2009

Games and Reality are Probably Different, Part 1

I've been thinking about how games represent reality. Realistic games are used to depict historical scenarios and as training for real strategy and tactics. They do it well enough that gamers get into discussions (even arguments) about how realistic a game is, and how the rules might be made better. The same applies to how games represent fantasy and fiction; A game about battling against dragons or Giant Battling Robots can represent that particular sort of fantasy very well, and players get into the same sort of discussion about how the game could be made more realistic (though in this case maybe I ought to say "more fantastic"). There is one aspect of games (boardgames, miniatures games, and RPG’s in particular) that I think is not realistic, and in all the various discussions of games I’ve been a part of no one has ever raised this complaint: Probability distributions generated by dice do not accurately represent the difficulty of real world tasks. (see footnote 1)

When you roll dice in a game to determine success or failure of an action, there is some predetermined probability of success. This probability is modified by various conditions; it might be range, terrain, type of weapon or armor, or any number of things. Each one of these things will add (or perhaps subtract) from the difficulty of the task. Add enough of these modifiers and the task becomes impossible (or impossible to fail). This is what I will refer to as “additive” probability, because difficulty modifiers add (or subtract) from the probability of success.

Now let’s consider a real world task; my example task will be shooting a weapon to hit a target of a fixed size. Suppose you are shooting a weapon (gun, bow and arrow, laser, PGMP-15, etc) at a target that has an area of 1 square meter. Hitting within that area is a success, and a miss is a failure. Also suppose the target is located at a distance D such that your probability of hitting within the 1 meter area is 50%. I’m assuming there is a bit of inherent randomness to the aiming process here that can be represented as a probability.

Now consider a second target of the same size but twice as far away. When we aim our weapon at this target, it’s apparent size is going to be 0.25 square meters, because it will appear half as tall and half as wide, and present ¼ the visible area of the closer target. At one quarter of the apparent size, it should be 4 times harder to hit the target (2,3). If we double the range again, we will get another proportional reduction in apparent target size, and a proportional increase in difficulty. The effect of increasing range has a proportional (or multiplying) effect on difficulty. We might find plenty of other example where adding difficulty has this proportional effect on the probability of success.

This is the difference I wanted to point out, that the real world often, and maybe always, has proportional probability instead of additive probability. Additive and proportional models of probability are two different ways of representing a probability that depends on other factors. (4)  So, how different are they, really? Does it make any difference? If it does make a difference, why don't we hear more about this? If it doesn't, why not?

A demonstration of these differences would be helpful, and I've made up some charts comparing different probability distributions. This is getting long, so I will save those for part 2, where I will also try to answer some of my own questions.

(1) I’m intentionally being a bit contrary in this post to make a point. Please feel free to disagree with me.
(2) The exact form of the relationship depends on certain assumptions that I am not stating, but proportionality still holds.
(3) When I write “4 times harder” this means I am representing probability on a scale where this make sense (Odds). If you have a 50% chance of success, making this 4 times easier is nonsense (200% success?). If we represent 50% chance as 1:1 odds (read as “one-to-one odds”) of  success, it now makes sense to talk about 4:1 odds. For probability p, the corresponding odds = p/(1-p).
(4)  I use these proportional representations regularly in my work, and it is a standard statistical method (logistic regression, proportional odds and proportional hazards models).
(5) Bonus points if you figured out that the title of this post is a play-on-words. :-)
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