A song of ice and fire vs The lord of the rings (read)

This is not a thread of which is the better literary piece of work, to each his own opinion. Both are great books (series of books) full with great characters and enjoyable moments, to say the least. Having said that, this thread is more about who would win in a battle between the armies of westeros and the middle earth, lead by some some specific characters, if they decide to wage war. To decide the outcome we are going to use the book version, unless stated otherwise.

armies:

• -Lord of the rings:

Gondor and Rohan. Both kingdoms are at their prime. Both armies are being led by king king Aragorn and king Théoden. Gimli and and Legolas are fighting with the realm of men.

• -infantry: 10,000
• - cavarly: 10,000
• -archers: 5000

• - A song of ice and fire:

the westerlands and the north. both armies are being led by Robb Stark and Tywin Lannister. the hound and Jaime lannister are commanding a small pack of knights.

• -infantry: 10,000
• -knights: 10,000
• - archers: 5000

• Rules:
• Battle takes place in a neutral ground with no natural advantage.
• The middle earth army would not have the help from the elfs and dwarfs
• no magic users are allowed
• Lets assume that the Lannister and Starks have no emnisty, on the contrary, they have a powerful alliance in search to expand Westeros.
• Aragorn and Theoden are looking to repel the invasion of the foreigners and they would fight until the last man.

Round 2: Siege warfare

1. The middle earth army tries to lay siege and take Storm's end. Same army, same general but with the siege engines the Orcs used to lay siege to Minas Tirith.
2. The westerosi army tries to lay siege and take Minas tirith. Same army, but this time they have the siege engines Renly had in his camp at Bitterbridge.

Round 3:

1. Sandor vs Gimli
2. Aragorn vs Jaime Lannister

• Rules:

1. Standard equipment for both characters in both.
2. Aragorn have the sword from the return of the king
3. we can use both movie and television version for the lord of the rings and game of thrones, respectively

(If someone can help me with some extra information about the lord of the rings characters and the westerosi characters, i would appreciate it.)