Balance problems: 25 points per problem (total of 50 points)

1.  Imagine that we are developing a game where players can choose to
use rockets, planes or dirigibles as their weapon.  Here is my initial
assessment of their speed, maneuverability and firepower.
 
weapon     speed    maneuverability  firepower

rocket      high     low             high
plane       medium   high            low
dirigible   low      medium          medium

I am assuming the plane is like a WWI plane with a machine gun on it.
The dirigible can carry large cannons and lots of cannonballs.  The
rocket has to be pointed at its target and it cannot change its course
much after it is launched.  It can carry a large payload, i.e., an
atomic warhead.  It can be used only once, so it might be better to
give the player a multiple rocket launcher that can be maneuvered to
point in the desired direction and fire rockets one at a time.

Further assume that we have playtested the game and found that the
plane and dirigible appear to be equally effective.

1a) Make a model (simple formula) that computes a weapon's
effectiveness.

1b) Adjust the parameters for the weapons so that they are all
equally effective.  Make as few changes as possible.

2. Suppose we are developing another game where the weapons available
are whips, boomarangs and slingshots. We'll assume that a 1d6 die is
used to determine whether target is hit and the to-hit values are
shown below.  The parameters we can change to get balance are the
frequency of use and the damage that will be done when target is hit.
Frequency of use = 1/(number of turns between uses of weapon).  So the
boomerang has a frequency of use of 1/3 because it can be used only
every third turn.  This is because it travels slower to target than
the other weapons; it returns when target is missed, but player has to
go to the boomarang and retrieve it wheh target is hit.

weapon     frequency  damage               to-hit
            of use             range:  1  2  3  4  5

whip           1        2              1  1  1
boomerang     1/3       6              1  2  3  3  4
slingshot      1        4              1  2  3  4  5

Compute average damage for each distance and weapon, and overall
damage for each weapon.  Make some changes to the frequency of use and
possibly the damage values to make these weapons more equally
effective.  Try to make very few changes.  Show that the changes you
made have made the weapons about equally effective.  If you want to
show the data graphically, you can draw the graphs by hand.