Analysis of the Battle of Five Armies (army sizes and losses).
Dwarves: We know that Dain arrived with more than 500 Dwarves, and there are 13 Dwarves in Thorin’s company. Let’s say there are 520 Dwarves in total.
Elves: We know that there are more than 1,000 Elven spearmen, let’s say there are 1,500 Elven spearmen in total. Then I guess there are 500 Elven archers, for a total of 2,000 Elves.
Men: Laketown probably had 20,000 – 30,000 people. If we say 20,000, then 15,000 survived Smaug’s attack (though a few died afterwards). I usually put the highest amount of able-bodied armed forces of Men to 10 % of the total population, so 1,500 warriors in total, of them I will guess 1,000 went to Erebor.
Eagles: I guess 50. Not an army, though.
Skinchangers (a Man, but a special one): 1 (Beorn), and that’s for sure. Also not an army (though I guess he could be called a one man/bear army).
Istari (Maiar): 1 (Gandalf), and that’s also for sure. And not an army too (though again, I guess he could be called a one man/Maia army)
Orcs: We don’t really know, but I will guess 15,000.
Wargs: We don’t really know either, but I will guess 5,000 (NB! While it seems the entire goblin army rides on wolves, the Wargs are a separate army, and they are probably bigger than those wolves the Orcs ride on).
Power relations and cavalry multiplicator:
The “power relations” are 3 dead goblins per dead Man, and 10 dead goblins per dead Elf or Dwarf. And 1 dead Warg per dead Man and 3.(1/3) dead Wargs per dead Dwarf or Elf.
For the Eagles it is 100 dead goblins per dead Eagle, or 33.(1/3) dead Wargs per dead Eagle. However, I do not think any Eagles die.
The stronger Orcs like Bolg’s bodyguard are of the same strength as a Man, I think, and Bolg himself is probably on level with a Dwarf or Elf, but those “15,000 Orcs” are the goblin army “converted” into “common Orcs”.
The usual “cavalry multiplicator” is 10, but I don’t think wolf-cavalry should get such a high multiplicator, so they get 5.
Army sizes and losses in each “part” of the Battle of Five Armies:
First part of the battle (until some goblins attack the spurs from above):
Good guys: See above.
Losses: 100 El., 50 D., 50 M.
Bad guys: See above.
Losses: 2,370 O.
Explanation: I think the Elven archers kill 1,000 Orcs. The spearmen neutralize the cavalry multiplicator (would probably be more than neutralized if it were horses, but I don’t know how wolves react to spears, so here it is just neutralized), so further 1,000 Orcs die.
The 50 Dwarves + 50 Men correspond to 650 Orcs, so with the cavalry multiplicator it will be 130 Orcs. That will be 2,130 Orcs in total, but since the Orcs were a bit routed, I will put the number of dead Orcs to 2,370. The Wargs seem arrive later.
Second part of the battle (until Thorin’s company join the battle) (cavalry in parantheses):
Good guys: 1,900 El., 470 D., 950 M., 1 I.
Losses: 575 El., 195 D., 300 M.
Bad guys: 12,630 (7,630) O., 5,000 W.
Losses: 2,950 (1,950) O., 1,250 W.
Explanation: I think those goblins, who scaled Erebor, dismounted. I have put their numbers on 5,000. Since it says the goblins, who attacked from the Mountain, later attacked with redoubled force, I think the first wave of attack was on 2000 goblins, of those 1,300 went to attack the Elves, and 700 went to attack the Men and Dwarves.
Of those, who attacked the Elves, 700 died, and of those, who attacked the Dwarves and Men, 300 died. Furthermore 1,800 wolf-riders, who attacked the Elves, died, and 150 wolf-riders, who attacked the Men and Dwarves, died. Of the Wargs 1000 died attacking the Elves, and 250 died attacking the Dwarves and Men.
The Elves have many spearmen, so the cavalry multiplicator is again neutralized, so the goblins kill 250 Elves, and the 1000 Wargs kill 300 Elves, therefore 550 Elves die in total. But the attack from two sides might have created some confusion, so further 25 Elves die, for an actual total of 575.
The cavalry multiplicator isn’t neutralized for those wolf-riders, who attack the Men and Dwarves, so those 150 wolf-riders “correspond” to 750 infantry Orcs. That means 1,050 “infantry Orcs”. The 250 Wargs correspond to 750 Orcs. So 1,800 “infantry Orcs” in total.
The Dwarves and Men would probably suffer even more confusion, since they can’t neutralize the cavalry multiplicator, so those 300 (actual) infantry Orcs will actually correspond to 600 infantry Orcs, and the Wargs to 1,500 infantry Orcs, so 2,850 infantry Orcs in total. The 195 Dwarves correspond to 1,950 Orcs, and the 300 Men to the remaining 900 Orcs.
I usually put the multiplicator created for the infantry, because of confusion created by cavalry on the same side as the infantry to a half of the cavalry multiplicator (i.e. 2.5 here), but I have put it to 2 here, as the goblin cavalry had to ride uphill.
Third part of the battle (until the Eagles join the battle) (cavalry in parantheses):
Good guys: 1,325 El., 275 D., 650 M., 1 I.
Losses: 515 El., 75 D., 240 M.
Bad guys: 9,680 (5,680) O., 3,750 W.
Losses: 3,620 (1,600) O., 1,000 W.
Explanation: There are now 4,000 Orcs, who attack the Elves and Men from the Mountain (The Dwarves all go down to fight in the valley). Of them 2,600 attack the Elves, and 1,400 attack the Men. Of those, who attack the Elves, 1,600 die, and of those, who attack the Men, 420 die.
That means 160 Elves and 140 Men die fighting against the goblin infantry. The remaining 355 Elves and 100 Men die fighting the goblin cavalry and the Wargs. The Elven spearmen and the heightened morale from the arrival of Thorin’s company on the battlefield neutralize the cavalry multiplicator. So the 355 Elves correspond to 3,550 Orcs, the 75 Dwarves to 750 Orcs, and the 100 Men to 300 Orcs. That means 4,600 Orcs in total, but 3,000 of those Orcs are “in form of” 1,000 Wargs, so actually 1,600 Orcs of the Orc cavalry die.
Fourth part of the battle (until Beorn joins the battle) (cavalry in parantheses):
Good guys: 810 El., 200 D., 410 M., 50 Ea., 1 I.
Losses: 110 El., 50 D., 60 M.
Bad guys: 6,060 (4,080) O., 2,750 W.
Losses: 4,905 (2,925) O., 1,450 W.
Explanation: The Eagles kill 200 Orcs on each spur, and 100 Orcs in the valley. Gandalf further kills 100 Orcs on the spur, where the Elves are. The attack of the Eagles (and also of Gandalf on the southern spur) brings massive confusion and panic to the Orcs on the spurs, so 0 Elves and only 5 Men die in liberating the spurs (the Elves kill 700 Orcs, and the Men kill 780 Orcs). The Eagles also kill 100 Orcs in the valley (or the corresponding to 100 Orcs). That would correspond to 5 dead Eagles, but I don’t think any Eagles died, so it can instead be “distributed” in form of more wounds on the Eagles in general. The neutralization of the cavalry multiplicator by the spearmen, and the confusion created by the Eagles in the valley means, that the 110 Elves correspond to 5,500 Orcs, and the 55 Men correspond to 825 Orcs. So the Elves and Men kill 6,325 Orcs in the valley. Of those Orcs 3,900 are in form of 1,300 Wargs, so they actually kill 2,425 Orcs. Since the Elven spearmen probably don’t break through the encirclement around the Dwarves, the 50 Dwarves correspond to 1,000 Orcs (the Elves still halve the cavalry multiplicator, and the Eagles further makes it possible for the Dwarves to kill 5 times as much per dead Dwarf than they else would have been able to). 500 of those Orcs are actual Orcs, while the remaining 500 Orcs are in the form of 150 Wargs. So of the Orc cavalry 2,925 die and 1,450 Wargs die.
Thorin, Fili, and Kili are among those 50 Dwarves.
Fifth part of the battle (until Bolg is slain) (cavalry in parantheses):
Good guys: 700 El., 150 D., 350 M., 50 Ea., 1 I., 1 S.
Losses: 20 El., 10 D., 20 M.
Bad guys: 1,155 (1,155) O., 1,300 W.
Losses: 720 (720) O., 1,160 W.
Explanation: The Eagles, Beorn, and Gandalf together kill 600 Orcs (or what corresponds to 600 Orcs). The cavalry multiplicator is neutralized by the Elven spearmen (who now are able to neutralize it for the Dwarves too), and because of the Eagles, Beorn, and Gandalf, the Elves, Dwarves, and Men get a multiplicator of 10. So those 20 dead Elves correspond to 2,000 dead Orcs, the 10 Dwarves to 1,000 Orcs and the 20 Men to 600 Orcs. That gives a total of 4,200 Orcs. Of them 3,480 are in form of 1,160 Wargs, and thus 720 Orcs are slain.
The remaining after Bolg is slain and the Orcs and Wargs are thus completely routed:
Good guys: 680 El., 140 D., 330 M., 50 Ea., 1 I., 1 S.
Bad guys: 435 O., 140 W.
I don’t think anyone of the good guys died after that (well Thorin did technically, but I have put his death in the fourth part of the battle).
