Lizrds! analysis (kinda long)

From: jriechel@cc.gatech.edu (James Alan Riechel)

Date: Sun, 29 Jan 1995 19:02:24 +0000

To Whom It May Concern,

I've done substantial analysis of FBI's Lizards! game,
which I plan to start in the near future.  The following analysis
is still incomplete, and will probably remain so until I play
and see a real game (so that I can estimate the frequency of
the different terrain types, etc.)

I plan to use the custom start-up sheet to select my
starting dens, lizards and terrain.  The one I intend to use
(and for which I'd appreciate comments regarding) can be
found near the bottom of this document.

[1] General considerations in choosing a custom start-up.

One may chose to do a custom start-up in the Lizards! game
for one of three reasons:  (1) one has done some analysis or
consideration and has an idea not provided by one of the
standard start-ups; (2) based upon the likelihood that
most people will choose one of the standard start-ups, one may
design a start-up position to exploit this knowledge, or (3)
one may have a cute/nifty idea that may not stand-up very
well to analysis, but may be fun to play.  In this section
I only address number (1) and (2) from above.

The general considerations when choosing an opening position
are:

(a) A larger template gives more certainty of terrain types,
but will both cost more itself and cost more for specifying
additional terrain types.

(b) The more starting dens one chooses, the less warrior
lizards one has to defeat free dens, and vica versa.

(c) Choosing a less dense (see below) arrangement for initial
dens may lead to greater productivity, but will either cost
in terms of uncertainty of terrain types or in the expense of
larger templates.

(d) Choosing a more dense (see below) arrangement for initial
dens may lead to lesser productivity, but will certainly
require a less expensive template, and may strategically lead
to a more defend-able position.

(e) Grey warrior lizards are free to immediately begin construction
of new dens but are expensive.  Choosing one den as a grey den
(of high productivity, i.e. lots of fertile) will eventually
produce large numbers of grey lizards that can construct new dens
(granting there's space available) at an arithmetically (or even
geometrically) increasing rate.

(f) Concentrating on constructing new dens in the vicinity of the
initial position may lead to a very strong, defend-able position
at the expense of space (i.e. other players will grab it).  However,
concentrating on taking-over/constructing new dens too far away from
the home den may lead to an indefensible position (i.e. your new
periphery dens are easily taken by opponents.)

(g) Not having at least 10 warrior lizards of each type (especially
Black!) leaves to chance whether you'll ever have lizards of that
type.  Choosing at least 10 lizards of each type ensures that a
den can be constructed with 50 Grey lizards for each type.

And I'm sure there are others...

[2] Choice of den arrangement, both initial and long-term

The following analysis ignores the fact that many terrain
types are unable to support the construction of dens or
cannot be converted to fertile land.  However, the results
are still interesting and important.  For each of the
following types of den arrangements, three numbers are
listed.  Each is a measure of efficiency (which is what
we want!)  The name of each type contains a number, e.g.
"Six" in ``Tight-Six''.  This number refers to the number of
fertile lands that surround each den in the arrangement.
Of course, the more fertile lands surrounding each den, the
higher population productivity they have and the higher
population they can support.

Cost		Den		Pop
		Efficiency*	Efficiency**	Efficiency***
Type		(dens/hex)	(dens/100 pops)	(pops/hex)
----		----------	---------------	----------

``Tight-Six''	0.33		1.85		18

``Loose-Six''	0.25		1.85		13.5

``Comprise-
   Five''	0.38		2.22		18

``Straight-
   Four''	0.5		2.78		18

``Straight-
   Three''	0.58		3.70		15.74

``Straight-
   Two''	0.75		5.56		13.5

``Straight-
   One''	0.86		11.1		7.7

Note:

* We don't necessarily want to minimize or maximize this number.  A smaller
number means that it'll cost less (in custom start-up points or in time) to
fill a given number of hexes with the indicated pattern.  However, since
space is not infinite in the Lizards! world, a larger number may be more
preferable, as we can pack more dens in the same number of hexes.  Another
name for this metric could simply be "Den Density."

** This number we definently want to minimize.  The fewer dens required to
create 100 population the better.  Here I assume that each den is operating
at peak performance.

*** This number we definently want to maximize, especially since space in
the Lizards! world is very, very finite.  Again, I assume that each den is
operating at peak performance.

Below I give the arrangement ``Loose-Six'', ``Tight-Six'' and ``Compromise-
Five''.  I leave the reader to discover the arrangements for the others.

``Tight-Six''

*---*       *---*
       /     \     /     \
      *  Den  *---*  Den  *
       \     /     \     /
        *---*       *---*
       /     \     /     \
  *---*       *---*       *---*
 /     \     /     \     /     \
*  Den  *---*  Den  *---*  Den  *
 \     /     \     /     \     /
  *---*       *---*       *---*
       \     /     \     /
        *---*       *---*
       /     \     /
      *  Den  *---*
       \     /
        *---*

``Loose-Six''

*       *---*       *---*       *
 \     /     \     /     \     /
  *---*  Den  *---*       *---*
 /     \     /     \     /     \
*       *---*       *---*       *
 \     /     \     /     \     /
  *---*       *---*  Den  *---*
 /     \     /     \     /     \
*       *---*       *---*       *
 \     /     \     /     \     /
  *---*  Den  *---*       *---*
 /     \     /     \     /     \
*       *---*       *---*       *
 \     /     \     /     \     /
  *---*       *---*  Den  *---*
 /     \     /     \     /     \
*       *---*       *---*       *

``Compromise Five''

*---*       *---*       *---*
 /     \     /     \     /     \
*  Den  *---*  Den  *---*       *
 \     /     \     /     \     /
  *---*       *---*       *---*
 /     \     /     \     /     \
*       *---*  Den  *---*  Den  *
 \     /     \     /     \     /
  *---*       *---*       *---*
 /     \     /     \     /     \
*  Den  *---*       *---*  Den  *
 \     /     \     /     \     /
  *---*       *---*       *---*
 /     \     /     \     /     \
*  Den  *---*  Den  *---*       *
 \     /     \     /     \     /
  *---*       *---*       *---*
 /     \     /     \     /     \
*       *---*  Den  *---*  Den  *
 \     /     \     /     \     /
  *---*       *---*       *---*

Discussion:

Well, given the different measures of efficiency, one might conclude that
``Tight-Six'' is the best.  However, the ``Tight-Six'' arrangement is
*very* sensitive to anomalies in the terrain, i.e. hexes that cannot
support dens or fertile land.  The ``Loose-Six'' would be a fine choice
if it weren't true that it's Population Efficiency is as bad as that
of ``Straight-Two''.  I summarily determine that ``Compromise-Five''
is the best choice because: (1) it's efficiency numbers are close to
optimal, and (2) it reacts will to anomalies in terrain.  On this
last point, a ``Compromise-Five'' can easily be converted locally to
a ``Straight-Four'' configuration, which itself has high efficiency.

Note, however, that in *all* cases it is better to build a new den in
newly available space that it is to convert a portion of an existing
den configuration to a more dense configuration.

[3] Finally, here's my start-up position idea

My Idea:

*---*
                   /     \
              *---*   8   *---*
             /     \     /     \
        *---*   5   *---*  1 1  *
       /     \     /     \     /
      *   2   *---*   9   *---*
       \     /     \     /
        *---*   6   *---*
       /     \     /     \
  *---*   3   *---*  1 0  *
 /     \     /     \     /
*   1   *---*   7   *---*
 \     /     \     /
  *---*   4   *---*
       \     /
        *---*

Hex	Terrain		Occupants
---	-------		---------
 1	Scrub		80 Red, 20 Green
 2	Fertile
 3	Grey Home Den	30 Den
 4	Scrub		10 Green
 5	Green Den	30 Den
 6	Fertile
 7	Temple		50 Grey, 10 Yellow, 10 Black
 8	Scrub		10 Green
 9	Red Den		30 Den
10	Fertile
11	Scrub		80 Red, 30 Green

Comments:

(a) I chose three dens so that I can initially match the maximum number
of dens available from any of the standard start-ups.

(b) I chose a grey den because once it's at maximum population production
it will produce an average of 45 lizards per turn.  It takes 5 or 6
turns to get up to this level of efficiency.  But once reached I'll be
able to construct an enormous number of dens.

(c) I chose a green den for two reasons:  (1) none of the standard start-ups
have green dens, and (2) the green lizards compliment the grey lizards
by being able to bring new dens up to full productivity.

(d) I chose 50 grey so that new dens can be constructed before the grey den
starts producing lizards at its maximum level.

(e) The 10 yellow and 10 black will settle the first dens constructed by
the 50 grey.  Having 10 of each of these ensures that I'll have *some*
yellow and black lizards.  A black den will be constructed in the hex
SE of hex 7, which is adjacent to the temple at 7.  The proximity of a
Black den and a temple to my grey home den speaks well for its
survival.

(f) 30 of the green lizards in hexes 1, 8 and 11 are for home improvement.
i.e. they will bring up the starting dens to maximum productivity.  From
there they can work on the newly constructed dens.

(g) The groups of 80 reds and 20 green will be responsible for pushing
out and capturing free dens.  The battle plan is:  have the 20 green
destroy fertiles around the free den so that the defending militia
will decrease, and then attack.  Once captured I will settle my red
lizards there and use my green lizards to improve the productivity of
the den and as scouts (so that I can see enemies nearby which could
possible take my officially undefended den.)

James
jriechel@cc.gatech.edu