@Ozymandiac:
Interesting read. I/we’ve been estimating battles based on comparing 3 factors listed in order of importance.
- number of units
- power
- ‘skew’, the concept that you can throw away low-power-units (infantry) while the defender has to sacrifice high-power-units (infantry).
This was heavily influenced by an old article on these forums by HolKann, and still the way I determine whether to attack/defend/retreat (http://www.axisandallies.org/forums/index.php?topic=19854.0). Interesting difference between your ideas is that HolKann claims the number of units is more important than the power; while your calculations show metapower is a multiplication of power and the number of units so they are symmetrically important.
Hmm, well in the case of 3x attacking inf vs 2x defending tanks, OP’s calculation would give the attacker a metapower of 9x^2 against 12x^2 for the defender, and the calculator gives 99% chance to the defender. This supports OP’s model over HolKann’s.
To further prove the model, at least in the case where every unit on each side has the same power, assume the attacker has x units of power i, and the defender has y units of power j. On the first round of battle, the attacker will kill xi/6 defending units, and the defender will kill yj/6 attacking units. If one side loses a greater proportion of its units than the other side, they will clearly lose the battle (on average), so the equilibrium, when the battle has a 50-50 chance, is when (xi/6)/y=(yj/6)/x, or when ix^2=jy^2. Note that ix^2 and jy^2 are the metapowers of each side, respectively.
I’m working on calculating the number of rounds of battle and remaining attacking units assuming the attacker has more metapower. I’ll try to post that tomorrow.