Investigation of Battle Odds

[Atilius]

I spent some time investigating how unit experience influences battle odds.

I have two armies, each consisting of 20 units of hastati. At first I vary the experience of the attacker ( note exp 1 = 1 bronze chevron, exp 4 = 1 silver
chevron and so on):

Attacker Exp Defender Exp Battle Odds
0...................0 ............. 1:1
1...................0 ............. 6:5
2...................0 ............. 6:5
3...................0 ............. 6:5
4...................0 ............. 7:5
5...................0 ............. 7:5
6...................0 ............. 7:5
7...................0 ............. 8:5
8...................0 ............. 8:5
9...................0 ............. 9:5

Reverse the experience values just to make sure odds reverse also:

0...................9 ............. 5:9

It looks like we are getting some round off, but taking this into account it appears that a unit becomes ~10% more effective for each additional level of experience.

If this is the case then if one army has experience 8 and the other 4, we'd expect the odds to be 1.8:1.4 (9:7). If I try this I find the odds are actually 9:8. Rounding? If I try exp 9 and exp 4 I'd expect 1.9:1.4 (9.5:7), I actually get 9:7.

Anyway, this seems to indicate that the ~10% incease per experience level (perhaps slightly less) is approximately correct.

I did one other check on this: How many exp 9 units does it take to make an even match for 20 exp 0 units? If the above is correct, the answer should be 20/1.9 which is slightly closer to 11 than to 10. If I set up a battle with 11 exp 9 units against 20 exp 0 units, I get battle odds of 1:1.

So a good rule of thumb is that each level of experience increases the unit's effectiveness by 10% of the strength of a 0 experience unit.

[Atilius]

I was looking for something useful to do this evening, but decided instead to have a look at the effect of armor on battle odds.

Setup: RTW 1.3, two armies of 20 hastati each.

Possible armor values (AV) are:
0 (no shield),
1 (bronze shield),
2 (silver shield),
3 (gold shield)

Results:

Attacker AV Defender AV Battle Odds
0 ............... 0 .................... 1:1
1 ............... 0 .................... 6:5
2 ............... 0 .................... 7:5
3 ............... 0 .................... 8:5

I also reversed the AVs (attacker AV 0, defender AV 1,2,3 ) and got reversed odds as you'd expect.

These numbers indicate that each armor upgrade increases the effectiveness of the unit by about 20% of the effectiveness of an AV 0 unit - at least to the battle odds calculator.

I ran another test to check this conclusion. If the above statement is true then how many AV 3 units are required to balance 20 AV 0 units? The answer should be 20 * 1.0 / (1.0 + 3 * 0.2 ) = 20 /1.6 = 12.5.

# AV 3 units # AV 0 units Battle Odds
11 ............... 20 .................... 4:5
12 ............... 20 .................... 1:1
13 ............... 20 .................... 1:1
14 ............... 20 .................... 6:5

SO: each armor upgrade increases the effectiveness of the unit by about 20% of the effectiveness of an AV 0 unit.

[magnum]

Any chance that you could run the same test(s) with different attack/defensive values than the Hastiti does? Wondering if its 20% or if the calculation is actualling feeding in the numbers, in which case a lower defense the armor upgrade would be more than 20%, a higher defense the armor upgrade would be less than 20%.

[Atilius]

Magnum,

You are correct. The unit's effectiveness appears to be proportional to the armor modifier plus unit's raw armor value.

Here are the results for Hastati previously posted.

Attacker AV Defender AV Battle Odds
0 .................. 0 ....................... 1:1
1 .................. 0 ....................... 6:5
2 .................. 0 ....................... 7:5
3 .................. 0 ....................... 8:5

As I concluded before, these results are consistent with adding 20% effectiveness per armor upgrade:

1.2:1 = 6:5
1.4:1 = 7:5
1.6:1 = 8:5

However, the base armor value for hastati is 5, so these results are also consistent with adding 1 to the base armor value for each armor upgrade:

(5+1):5 = 6:5
(5+2):5 = 7:5
(5+3):5 = 8:5

So I experimented with armored hoplites, which have a much higher armor value (11):

Attacker AB Defender AB Battle Odds
0 .................. 0 ....................... 1:1
1 .................. 0 ....................... 1:1
2 .................. 0 ....................... 1:1
3 .................. 0 ....................... 1:1

where AB is the Armor Bonus due to upgrades

These results come nowhere near my earlier 20% per upgrade rule.

With +1 per level we would have expected:
(11+1):11 = 1.09:1
(11+2):11 = 1.18:1
(11+3):11 = 1.27:1

The raw numbers look strange, but I could tell that the units with the armor bonus were getting more effective by the kill ratios I got when I autoresolved the battles. It's simply that the base armor value is high enough that the +1 bonus for each upgrade doesn't have much of an effect (as you speculated).
Throw in the rounding that's necessary to get a ratio of integers and these results are reasonable.

Next I tried the finding the number of AB 3 units required to match 10 AB 0 units: this should be 10 * 11/(11+3) or roughly 7.8.

# AB 3 # AB 0 Battle Odds
6 .................. 10 ....................... 4:5
7 .................. 10 ....................... 1:1
8 .................. 10 ....................... 1:1
9 .................. 10 ....................... 7:6

Which is just about right. My 20% rule would have predicted 6.25 which is clearly wrong.

I also used Principes (base armor value 7) as an intermediate case.

Attacker AB Defender AB Battle Odds Expect(+1) Expect(+20%)
0 .................. 0 ....................... 1:1 ......... 1:1 .......... 1:1
1 .................. 0 ....................... 7:6 ......... 6.9:6 ......... 7.2:6
2 .................. 0 ....................... 4:3 ......... 3.9:3 ......... 4.2:3
3 .................. 0 ....................... 3:2 ......... 2.9:2 ......... 3.2:2

Both rules do a good job here, but the +1 rule is a bit better.

And finally, how many +3 Principes are needed to match 20 plain ones?
The +1 rule says 20 * 7/(7+3) = 14, the +20% rule predicts 20/1.6 = 12.5:

# AB 3 # AB 0 Battle Odds
12 .................. 20 ....................... 5:6
13 .................. 20 ....................... 1:1
14 .................. 20 ....................... 1:1
15 .................. 20 ....................... 7:6

The +1 rule wins again.

A note of caution:
I also tried this with mercenary bastarnae (armor value 2) and it didn't work at all. The upgrades hardly increased their battle odds at all. These guys are running around with metal helmets and almost nothing else, but are upgradable by a blacksmith. It may be that the battle odds calculator knows somehow that, while their heads are a bit safer, the rest of them isn't any better off.

But it seems that a good rule of thumb for the effectiveness of a unit with upgraded armor is proportional to the modified armor value (base value + bonus) divided by the base armor value.

This rule of thumb probably only applies to fairly well-armored units.

This was an excellent suggestion Magnum. I'll look into the weapon bonus issue with this same general approach in mind, and revisit the experience bonus in light of this.