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