Ok guys, let's take a look at the stats (vanilla)
Veteran War Wagon
Attack: 42 ranged, ROF 3, 3x bonus vs cav
Resist: 20%, Melee
Attack: 30, ROF 1.5
Resist: 20%, ranged
Let's have them FU.
Both units get 3 cards:
-Cavalry Combat 10%/10%
-Cavalry Hitpoints 10%
-Cavalry Attack 10%
Cuirassier gets another 10% / 10% bonus for royal guard upgrade.
Now, a non carded, Imperial War Wagon has:
-500 + 150 + 250 HP = 900 HP
-42 + 12.6 + 21 attack = 74 attack (rounded up)
A non carded, Imperial Lamedarme has:
-500+200+250 HP = 950 HP
-30+12+15 attack = 57 attack, ROF 1.5
So, while an Imperial War Wagon has a theoretical damage output of 74 every three seconds, a gendarme has 114. WW have 20% Melee resistance, taking in only 91.2 damage.
Unfortunately, for the french horsie, the war wagon gets a 3x bonus against him, effectively tripling the output to 222. Lamedarmes have 20% Ranged Resistance, so they take only 178 Damage per shot.
So, War wagons inflict almost twice the damage on gendarmes than vice versa, they cost exactly the same and thus should clearly win.
But then there is lame splash damage. Other ranged cav can hit and run, war wagons theoretically could, but they're just too clumsy. Then again, I don't guess they're too much affected by splash damage, because they already use up much space.
I am not quite sure there, the splash damage makes the calculating a little difficult, but still, War Wagons should cleary win.
And btw: for a dragoon type of unit, war wagons have ridiculously high siege.
EDIT: Both units have the same upgrades.
+30% extra HP, 40 on gay plains. (Cav HP, Combat, Arsenal, comanche TP)
+20% extra Attack
that evens out too.
EDIT 2: How much extra damage does the arsenal tech for ranged horsies give? 10%?
EDIT 3: Depending on how many cards the frenchy and the german use to upgrade their horsies, it might take 7 war wagons or 6 war wagons to one-shot-kill one lamedarme. u certainly want to avoid having to use 7 since you can already get only 33 of them. (or worse 20, if you max out in pop)
EDIT 4: it seems unfair that a frenchy has 99 villie eco with 80 villagers, but germans can have 89 villagers for 99 villie eco. a smart player would use 60 settlers+20 settler wagons though, for a 10% bonus in economy (my opinion)