Unit size = unbalanced??

[econ21]

It doesn't assume that. The larger unit will always have more men engaged than the smaller unit. The smaller unit will loose men at a faster rate than the larger unit. The 60 man unit will not loose 2/3 of it's strength in beating the 40 man unit.

That's all true, but it's just quantifying the advantage of the larger unit that is tricky. I think all of what you have said above could be true and the 40 man unit be say, 70% as strong as the 60 man unit rather than less than 2/3 as strong as you stated. I don't see how it overturns my point that the larger unit typically cannot bring all its numbers to bear meaning that the combat strength of the smaller unit may be more rather than its numerical size.

To take the issue further, I think we need a mathematical measure of a units' combat strength. But I confess I don't know how to construct that. I can see how you could quantify a unit's offensive strength (e.g. expected casualities inflicted) and even its defensive strength (expected casualities suffered versus a benchmark attacker). But how do you put them together to measure overall combat strength? If it was just casualities inflicted minus casualities suffered, it would not be very useful for solving this argument as it might average around zero, so it would be hard to say if a unit is more or less than 2/3 as strong as another.

Strikes at a flank or rear of a man get a substantial combat bonus which further increases the advantage of the larger unit.

That's a good point - I am not sure if outweighs my point about inability to bring numbers to bear, but it certainly needs to be weighed against it.

[Puzz3D]

That's all true, but it's just quantifying the advantage of the larger unit that is tricky. I think all of what you have said above could be true and the 40 man unit be say, 70% as strong as the 60 man unit rather than less than 2/3 as strong as you stated. I don't see how it overturns my point that the larger unit typically cannot bring all its numbers to bear meaning that the combat strength of the smaller unit may be more rather than its numerical size.

Assuming equal combat stats, if you put the two units on hold formation, the combat will always be one-on-one. In this case, the 40 man unit will be exactly 2/3 as strong as the 60 man unit. On average, the 40 man unit would kill 1 man for every 1 that it looses. If the 40 man unit fights until it looses all 40 men, the larger unit will also loose 40 men and have 20 men remaining. This is the definition of 2/3 as strong, i.e. the stronger unit loosing 2/3 of it's strength.

If you put the two units on engage-at-will formation, the 60 man unit will not loose 2/3 of its strength in beating the 40 man unit. Men in the larger unit will wrap around the smaller unit and strike at the flanks of enemy men. Even if it's only 2 men extra of 10 men engaged head-to-head, those 2 represent much more than a 20% advantage. I don't remember my test results, but as I recall the 60 man unit can win with as many as 40 men remaining.

To take the issue further, I think we need a mathematical measure of a units' combat strength. But I confess I don't know how to construct that. I can see how you could quantify a unit's offensive strength (e.g. expected casualities inflicted) and even its defensive strength (expected casualities suffered versus a benchmark attacker). But how do you put them together to measure overall combat strength? If it was just casualities inflicted minus casualities suffered, it would not be very useful for solving this argument as it might average around zero, so it would be hard to say if a unit is more or less than 2/3 as strong as another.

I don't think you can develop an algorithm to measure this because you don't know the correlation coefficient that binds the men together. Each man in a unit does not move independently (Poisson distribution) as you've pointed out, so not all the men are randomly engaged. Even so, the smaller unit has no advantages as far as I can see, so the rate of losses in the smaller unit will always be greater than in the larger unit, and it will always be less effective than the ratio of the starting strength of the two units.

You could develop a table of losses vs size difference with empirical tests. Based on my testing in the past the effect is very strong, but i can't remember the results. One thing I do remember is that with two 60 man units a win is almost guaranteed if one unit can gain a 15 man (25%) advantage in men with say a very good charge. The other thing I clearly remember is that, with 60 man units fighting head-to-head and with equal combat stats and charge bonus, a 1 point (20%) combat advantage produces a win 6 times out of 10, and a 2 point (44%) advantage produces a win 9 times out of 10.