Date: Sat, 6 Dec 1997 13:50:55 +0000
From: Tournament Headquarters  (by way of Robert F. Heeter)
Subject: W2WT: WarBOT tip (D)

Warlords -

The WarBOT gurus and some Moderators have noticed an interesting
effect in the battle odds, and we feel this information is worth
sharing with all tournament players as tutorial information.

    ----- Panagos' Law:  "Sometimes less is really more." -----

Suppose you have a stack of three wolfriders, and you want to
take out a defense-1 city occupied by a griffin.
Your units are each strength 4, and the griffin, with the city
bonuses, is 6 + 2 + 1 = 9.  According to WarBOT, if you send those
three wolfriders against the griffin, you win 51.448%.  Cool!

Now suppose you want to defeat two griffins, and you have six
wolfriders.  The odds are the same, right?

Wrong!  Six units of strength 4 will *lose* (48.015%) to a pair of 9s!
(Try it if you don't believe us!)

This is directly contradictory to the sense of the Warlords manual,
which encourages you to stack units together so that they will fight
better.  Most of the time the manual is right, and more units fight
better.  But that's not the case when the odds are very close.

How can this be?


    ----- The Panagos Effect:  If a given battle is very evenly matched,
      and you double the number of units on each side, the odds shift
      in favor of the smaller stack with stronger units. -----

Originally noticed by Jim Panagos, this is *not* a bug in WarBOT,
or a bug in Warlords.  It's a very bizarre feature of the Warlords
combat system.  The origin of the effect cannot (yet) be explained
without several pages of mathematics, but you can verify it easily
for yourself by trying various close battles, and then scaling each
battle to larger size.  One 9 loses to three 4s, but two 9s will
beat six 4s, and three 9s will beat nine 4s even more often.
Similarly, four 5s will beat a 12, but eight 5s will lose to two 12s.
The simplest explanation is that when two stacks fight, there
are various "paths" that the battle can take through the various
possible "states" of each stack (each time a hit is scored,
a step is taken along the path, and the state of the battle changes).
WarBOT works by calculating the probability of each possible state.
When you double the size of the battle, you increase the number of
possible states, and the probability of each state can shift due to
the extra units in the battle.  Three 4s fighting a 9 is one battle,
but six 4s fighting two 9s is *not* the same as three 4s fighting one 9,
and then three more 4s fighting another 9.

    ----- Corollary:  An attacker with a large number of weak
      units can sometimes bring down a defender with a small stack of
      strong units by choosing small groups of weak units to just barely
      kill one strong unit at a time. -----

    ----- Corollary:  An attacker with a small number of strong units
      attacking a large number of weaker defenders will get the best odds
      by grouping all the strong units together for a single battle. -----

The shift in the odds due to the Panagos Effect was probably not
noticeable to SSG when they created Warlords, but in the Tournament,
where the margin of victory can be razor-thin, this shift can sometimes
provide a decisive advantage in an important battle.  We thought it
was important for everyone to be aware of this.  The Warlords manual
is generally right that larger stacks are usually better, but
when the odds are very close, it might be better to have a larger
number of weaker stacks.

Happy WarBOTting!

-- Tournament Headquarters