True logarithmic augmentation

From: Benedict Adamson <yahoo_at_...>
Date: Mon, 24 Mar 2003 21:35:18 +0000

Nick Brooke wrote:

...

> I think we'll have to accept that, as is so often the case when
> the "log" word emerges, Issaries Inc simply can't perceive the
> problems we all see.

...

Innumerates of a nervous disposition might care to look away now.

Well (in TeX like form)

log_n (x_1+x_2)

= log_n (x_1 (1 + x_2/x1))
= log_n x_1 + log_n (1 + x_2/x1)
= log_n x_1 + log_n (1 + n^log_n(x_2/x1))
= log_n x_1 + log_n (1 + n^(log_n x_2 - log_n x_1))
= log_n x_1 + f( log_n x_2 - log_n x_1 )

where f(y) = log_n(1 + n^y)

Changing to logarithmic ability ratings:

A_1 = k log_n x_1,
A_2 = k log_n x_2,
A_t = k log_n (x_1+x_2),

where k and n depend on our chosen ability rating scale;

A_t = A_1 + kf(A_2 - A_1),
where A_t is the ability rating resulting from combining ability ratings   A_1 an A_2.

Now, imagine a Narrator has kf(y) tabulated for a range of ys, she can work out the ability rating resulting from combining two abilities, A_1 and A_2 (i.e. augmenting one with the other), thus:

1. Subtract one ability from the other (giving A_2 - A_1).
2. Look up that number in the table (giving kf(A_2 - A_1)).
3. Add the table value to the first ability rating (giving A_1 + kf(A_2 - A_1)).
4. The result is the total, augmented ability rating (A_t).

Note that the Narrator does not have to calculate logarithms at any point; she only has to integer arithmetic and look up a number in a table. Indeed, she can be wholly ignorant that the ability ratings have a logaithmic basis.

In practice, the table can be quite small, because we are only interested in integer kf(y) values, and we can insist that the the larger ability is used as the base ability (so A_1 >= A_2), so that table need only have k values. For Hero Wars, k seems to be about 10.

How practical is this scheme for game use? There did exist a game that did precisely this: Striker, the old Traveller minatures game. It measured armour and weapon penetrations using a logarithmic scale, so combining armour used this approach.