[MUD-Dev] [DGN] The psychology of random numbers

[J C Lawrence]

On Tue, 20 Jan 2004 20:44:07 -0500
Dave Bacher <DaveB at battlebazaar.com> wrote:
> On Fri, 16 Jan 2004 08:06:13 -0700, Ted L. Chen wrote:
>> On Jan 8 2004, Sean Kelly wrote:

> 10 consecutive rolls -- 1 in 10240000000000

Before you take this too far, it isn't just 10 consecutive, it is the
probability of any 10 value sequence, be it all 1s, or 1/2/3/4/5/6.../20
or 2/4/6/8/10...20/1/3/5/7...29.  All number sequences are of equal
probability once they are defined, and it doesn't matter what the
sequence is.

> If I attack 6 on 3 once and lose, I'm likely to complain a little, but
> write it off as chance.  If I fail twice, I'm likely to complain a
> little and write it off as chance.  If I fail 20 turns in a row, then
> I probably have a valid complaint -- either your dice are rigged or
> you are impossibly lucky.

The problem here is that bell curves don't just vanish at the edges.
With straight dice sure it won't happen often, but that's the point.  It
will happen, just not often.

  ObTrueStory: In Risk I've marched a three-strong army all the way
  across Russia, across Africa and up through Brazil into North America
  fighting all the way without losing a man until I hit Texas.  Then I
  died.  I doubt I'll ever do that again, but it did happen.  I even
  swapped and mixed dice with the opponent several times just in case
  they were off.  I still lived.  At one point my measly little one man
  defender successfully stood off and decimated an army of over 20.
  I'll be rather surprised if I ever see either happen again, the odds
  are against it, but it isn't impossible.

Probability means just that, its a question of probabilities.  It isn't
very likely that all the air molecules in the room will simultaneously
exit the building via random brownian motion, or that the hostess'
undergarments will suddenly rematerialise three feet over to the left
via quantum tunnelling, but it is possible, just very VERY improbable.

  Of course a cup of hot tea helps.  I'm still waiting for the
  undergarments of my hostess to leap any observable distance in any
  direction at a party I'm attending, but I'm sure that if I can just
  get to enough parties it will eventually.  Perhaps I'm not attending
  the right parties.

Where this gets messy is human perception.  The apparency (I haven't
studied this) is that people tend to think that any behaviour more than
a few percentiles out from the centre of a standard distribution is "too
improbable" and that any sort of visible pattern in a "random" sequence
that persists for more than a handful of values is "obviously not
random", belieing the fact that such will necessarily occur in any
random system.  This view persists even when you demonstrate the fallacy
for them.

  Flip a coin a hundred times.  What was the longest streak of heads you
  got?  What is the probability of such a streak of heads?  Pretty small
  huh?  Yet it happened...  Flip it another 100 times.  Are the numbers
  of heads and tails in each hundred even close to each other, or even
  close to 50?  The odds are quite good they won't be, and yet the coin
  is (relatively) true.  

It all comes down to the fact that the value of the next coin flip isn't
affected by the value of the prior flip, or the one before that.  Yeah,
I know you know that, but most people look at a sequence of 30 heads and
say, "It can't be heads next time, that's just TOO improbable!"  And yet
it has the same improbability as flipping another head ... or tails.

> If every time I play with you the fights go like this, I'm going to
> believe strongly the dice are rigged.  If every time I play, you are
> impossibly lucky, the simplest explanation is that you are cheating.
> And so, I will call you on it.

Precisely.

> The C PRNG is a LCG.

AIR there is no formally standardised implementation of the PRNG for
libC.  Compilers have shipped with Mersenne based PRNGs.  Modulo/LCG
implementations are simply the most common (they're cheap).

> If you show me the results of the rolls, then you have to be confident
> there isn't actually a problem, because if I perceive a problem, the
> first thing I'm going to do is run the logs through an analyzer and
> see what numbers come out.  In the most probably case, I'll find out
> there really is a skew, and I'll proceed to post it on a web site and
> it will become a much bigger issue than if it had been owned up to in
> the first place, especially on a large game.

Again the perception point enters: The sample sizes in such empirical
reports are (statistically) horribly small.  Any result generated from
them can't be trusted.  This isn't however the problem.  The problem is
that the small sample size is ignored, most people don't understand
probability and thus there is a PR problem.

The PR problem is the killer, not the distribution accuracy, and that's
where the engineering problem definition comes in.  In general we're not
engineering for statistically accurate probability models, we're
engineering for something that is perceived well, and in general that
means a system which appears unpredictable to the untrained eye for
almost all small sample sizes, and which, if anything, "fails" to the
player advantage more than it should.  "Appearing random" and "being
truly random" are different problems with different solutions.

I haven't known a player yet who would complain about being lucky.

--
J C Lawrence
---------(*)                Satan, oscillate my metallic sonatas.
claw at kanga.nu               He lived as a devil, eh?
http://www.kanga.nu/~claw/  Evil is a name of a foeman, as I live.

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