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Message: <blockquote data-quote="mmadsen" data-source="post: 3747198" data-attributes="member: 1645">Anyone designing a war game, or a roleplaying game with a lot of combat, should be familiar with <a href="http://en.wikipedia.org/wiki/Lanchester's_Square_Law" target="_blank">Lanchester's Laws</a>: In 1916, during the height of World War I, Frederick Lanchester devised a series of differential equations to demonstrate the power relationships between opposing forces. Among these are what is known as Lanchester's Linear Law (for ancient combat) and Lanchester's Square Law (for modern combat with long-range weapons such as firearms). In ancient combat, between phalanxes of men with spears, say, one man could only ever fight exactly one other man at a time. If each man kills, and is killed by, exactly one other, then the number of men remaining at the end of the battle is simply the difference between the larger army and the smaller, as you might expect (assuming identical weapons). In modern combat, however, with artillery pieces firing at each other from a distance, the guns can attack multiple targets and can receive fire from multiple directions. Lanchester determined that the power of such an army is proportional not to the number of units it has, but to the square of the number of units. This is known as Lanchester's Square Law. It relies on the fact that when either side has more units it means that they have also more surface-area that opponent can hit (thus diluting the amount of fire hitting each unit, reducing their rate of attrition) and also more firepower (thus increasing the enemy's rate of attrition). Note that Lanchester's Square Law does not apply to technological force, only numerical force; so it takes an N-squared-fold increase in quality to make up for an N-fold increase in quantity.It's that last paragraph that's the most important to our current discussion.  If we measure troop quality with a single variable -- let's say that ogres kill orcs twice as fast as orcs kill ogres -- then two ogres might seem like they'd defeat four orcs easily, but really they'd be overpowered, because multiplying the number of troops multiplies its offense and its defense.  More orcs have more attacks, and there are more orcs to kill.What confuses this is that D&D level includes multiple measures of offensive stength (to-hit and damage) and defensive strength (AC and hit points); it's not a single linear measure.  For instance, in going from first to second level, an NPC fighter might multiply his to-hit chance by 1.1, his damage by 1.0, his avoid-a-hit chance by 1.0, and his hit points by 2.0, for a total quality factor of something like 2.2.  As you can see, at lower levels, without better equipment, it's almost entirely about improved defense through extra hit points.  As characters accumulate magic weapons, armor, etc., they can improve across all four of those dimension, and a 10-percent improvement in everything isn't a 10-percent improvement in fighting ability; a 10-percent improvement across four factors is a 46-percent improvement.  Now compound that over multiple levels.</blockquote>

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