To explain a little better, I own a pinball machine (Medieval Madness, Williams 1997), so I really do play an insane amount of pinball. And I've gotten fairly high scores; but not every game is a new high; not every game is exciting. So i wanted to come up with a meta-game to play over the course of many pinball games. originally, it started as just "how long can i keep getting a better score than i did the previous game?" -- but monotonic increase does not encourage playing to the best of ones ability (because a really good game can break a streak early), and i didn't like that.

So I came up with a few other ideas, and i wanted to test them out -- but they'd take hundreds of games to see if they were fair and interesting. Which is, finally, where perl comes in.

first, i needed to model a single game of pinball. now, my top score is 178 million, and some games come in as low as (approximately) 178 thousand, but it's not a linear scale; the average is not 89 million, or even 17.8 million, but about 10 million. and, of course, i'v emade 178 million my top score in under 2000 games; given 10000 or 100k games, there's no hard upper limit. so just using rand() was right out of the question. on the other hand, i didn't want to map out every switch on the table and figure out probabilities for everything. so a model is in order. it turns out that the score increases faster thee longer you play, with a few large jumps which also get bigger as you go. i poked and tweaked, and here is what i ended up with:

sub score {
  # score is reported in millions. 

  my $t = .1;    # there is an actual minimum score.  
  my $b = 0;     # this counts the balls. 

  my $round = 0;
  while (++$round) {
  # it's a 3 ball table, so ...
    last if $b > 3;
  # get some points, 
    $t += int(rand $round*10)/10; 
  # risk draining
    (++$b && redo) if (int rand int sqrt (9*$round));
  # possibly get an extra ball
    $b-- unless int rand(5);
  # get the big points if we made it this far
    $t += 2 * $round;

  
  # the rest of this isn't interesting
  # let me know if something too unlikely happens...
    print ("Divine Intervention: $t points in $round rounds.\n") 
      and last if not $round % 20;

  #  ... and stop it if it gets out of hand
    print ("Apocolypse!") and last if $round > 400;

    }

  # point out statistically unlikely games;
  # otoh, they should range up to 200 ~ .01% of the time,
  # and some real games do fail to get off the ground.
  warn "Great game: $t in $round!\n" if ($round > 12 or $t > 150);
  warn "Horrid try: $t in $round!\n" if ($round <  1 or $t < .2);

  return $t;
  }
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a simple "run this a million times and tell me the average" eventually revealed that it was doing pretty much what i wanted. so i wrote a few meta-games, and ended up with this as my basic game:

###
#
# this sub was originally separated out so i could try averages vs. 
# weighting the averages toward the most recent game.
# i chose to weight them because it won't allow any one game to be too
+ hard.

sub game {
  my $historic = shift;

  my $game = score;

  return (($historic+$game)/2);
}

###  the meta game itself.
#
# basically, each round you are trying to keep your (weighted) average
+ score above 
# the target (set at the start of the metagame, and incremented with e
+ach game).
# $spares is the number of games you can "do over";
# pretend that didn't happen and play that game again.

sub meta {
  my ($historic, $target, $spares) = @_;
  my $games = 0;
  my $archive = 0;

  do {
    {
      $archive = $historic;
      ($historic) = game($historic);
      if ($historic < $target and $spares) {
        --$spares;
        $historic = $archive;
        redo;
        }
      }
    } while (++$games and $historic > $target and ++$target);

  return ($archive, --$target, $games);
}
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i usually play meta(score+score, 10, 0);

Clarification: this isn't a whole program, this is just a couple subs. if you want to see what it's doing, you need a wrapper of some sort. this one will play the meta-game as i currently play it on my real machine, and tell you some stuff:

sub show_meta {
    my ($first, $target, $spares) = @_;
    my ($lastgood, $made, $games) = meta (@_);

    print <<OUTPUT;
First two games averaged to $first.
Played for $games (using $spares spares)
(Making target $made with $lastgood and then having a bad game)
OUTPUT
}

show_meta (((score + score)/2), 10, 0);
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