TL;DR: Let's say that the Spearman line can defeat Eagles cost effectively. Would that make Eagles less crazy or useless? Is there a specific line where it might be a good idea? Any arguments that I haven't taken into account?
WARNING: LOOOOONG READ AHEAD
This idea can go very bad. Halbs are ubiquitous early Imp for many civs, so Aztecs stand to lose alot. I think most of us think Mayans need a few small nerfs, and they have cheap Archers that can make up for their Eagles new hypothetical weakness.
But maybe it can go very good. Most civs go some combo of Halbs, Onagers, Arbs, or Rams in early Imp. Mesos go Eagles, Arbs, Rams. If this works, then the rock papers scissors game will be brought closer: Halbs > Eagles > Arbs, Onagers > Halbs. Maybe Aztecs will have a reason to use Jags. Nobody would create Halbs vs a civ without Cavalry, and Champs are underused. If Halbs are viable vs Aztec Eagles, then Jags may have a use.
This also makes sense in terms of game lore: the manual states Spearmen are medium infantry. Champs are heavy and Eagles are light. Heavy > Medium > Light, right?
Also keep in mind that the Aztecs can swarm their Eagles very quickly, and Eagles are still Town Center and Villager annihilators with 4+4 pierce armor and very fast so even if Halbs can beat Eagles 1v1, it might not matter much.
WARNING: LOTS OF NUMBER CRUNCHING AHEAD
Halb (no Blast Furnace, yes Plate Mail) vs Aztec Eagle (yes Garland Wars, Plate Mail no Blast Furnace)
35f, 25w vs 50f, 20g
60 HP vs 60 HP
6+2 atk vs 9+6 atk
0+3 armor vs 0+3 armor
Halb kills Eagle in 10 hits (Halb swings every 3 seconds so 30 seconds)
Eagle kills Halb in 5 hits (Eagle swings every 2 sec so 10 sec)
Halb (fully upgraded) vs Aztec Eagle (fully upgraded)
Halb kills Aztec in 9 hits (27 sec)
Eagle kills Halb in 5 hits (10 sec)
Halb (no Blast Furnace, yes Plate Mail) vs Mayan Eagle (yes El Dorado, Plate Mail, no Blast Furnace)
60 HP vs 100 HP
6+2 atk vs 9+2 atk
0+3 armor vs 0+3 armor
Halb kills Eagle in 20 (60 sec)
Eagle kills Halb in 8 hits (16 sec)
Halb (fully upgraded) vs Mayan Eagle (fully upgraded)
Halb kills Eagle in 15 (45 sec)
Eagle kills Halb in 6 hits (Eagle swings every 2 sec so 12 sec)
Now let's add in the hypothetical attack bonus:
+3 attack bonus
vs Aztec (early Imp): 8 attacks (24 seconds)
vs Aztec (fully upgraded): 6 attacks (18 seconds)
vs Mayan (early Imp): 13 attacks (39 seconds)
vs May (fully upgraded): 10 attacks (30 seconds)
I think this has the most minimal, but still kind of meaningful impact.
+4 attack bonus
vs Azt (EI): 7a (21s)
vs Azt (FU): 6a (18s)
vs May (EI): 12s (36s)
vs May (FU): 10a (30s)
This is more optimal than the last because it doesn't affect the Aztec matchup very much but affects the Mayan one.
+6 attack bonus
vs Azt (EI): 6 (18)
vs Azt (FU): 5 (15)
vs May (EI): 10 (30)
vs May (FU): 8 (24)
+8 attack bonus
vs Azt (EI): 5 (15)
vs Azt (FU): 4 (12)
vs Azt (EI): 8 (24)
vs May (FU: 7 (21)
+12 attack bonus
vs Azt (EI): 4 (12)
vs Azt (FU): 4 (12)
vs May (EI): 6 (18)
vs May (FU): 6 (18)
This is the upper limit if we want to prevent Halbs from beating Aztec Eagles 1v1, and it will also affect the Mayans the most.
+13 attack bonus
vs Azt (EI): 4 (12)
vs Azt (FU): 3 (9)
vs May (EI): 6 (18)
vs May (FU): 5 (15s)
+18 attack bonus
vs Azt (EI): 3 (9)
vs Azt (FU): 3 (9)
vs May (EI): 5 (15)
vs May (FU): 4 (12)
This might be too much. A Fully Upgraded Halb now beats Aztec and Mayan Eagles 1:1. However...Elite Skirmishers also beat Arbalests 1:1, and the resources are fairly similar (25f, 35w vs 25w, 45g).
I have one more related idea:
Garland Wars should give +5.
That will allow Aztec Champs to beat Japanese Champs 1:1 in late Imp, which makes sense since Japanese get their bonus for free.
Aztec Eagles will kill Foot Archers and most Siege the same as they did. They'll kill Cavalry Archers in one less hit, which I think is a good thing. The most important thing is that when they are Fully Upgraded, they will kill Halbs in one less hit. If that were to happen, then Halbs can have a much bigger attack bonus vs Eagles and still lose 1v1, and then they'll be much more effective vs Mayan Eagles.
Also I'd personally get rid of Plumed Archers attack bonus vs Infantry. With their fast firing rate, movement speed, etc. they are already good vs Infantry so the attack bonus is overkill.
WARNING: LOOOOONG READ AHEAD
This idea can go very bad. Halbs are ubiquitous early Imp for many civs, so Aztecs stand to lose alot. I think most of us think Mayans need a few small nerfs, and they have cheap Archers that can make up for their Eagles new hypothetical weakness.
But maybe it can go very good. Most civs go some combo of Halbs, Onagers, Arbs, or Rams in early Imp. Mesos go Eagles, Arbs, Rams. If this works, then the rock papers scissors game will be brought closer: Halbs > Eagles > Arbs, Onagers > Halbs. Maybe Aztecs will have a reason to use Jags. Nobody would create Halbs vs a civ without Cavalry, and Champs are underused. If Halbs are viable vs Aztec Eagles, then Jags may have a use.
This also makes sense in terms of game lore: the manual states Spearmen are medium infantry. Champs are heavy and Eagles are light. Heavy > Medium > Light, right?
Also keep in mind that the Aztecs can swarm their Eagles very quickly, and Eagles are still Town Center and Villager annihilators with 4+4 pierce armor and very fast so even if Halbs can beat Eagles 1v1, it might not matter much.
WARNING: LOTS OF NUMBER CRUNCHING AHEAD
Halb (no Blast Furnace, yes Plate Mail) vs Aztec Eagle (yes Garland Wars, Plate Mail no Blast Furnace)
35f, 25w vs 50f, 20g
60 HP vs 60 HP
6+2 atk vs 9+6 atk
0+3 armor vs 0+3 armor
Halb kills Eagle in 10 hits (Halb swings every 3 seconds so 30 seconds)
Eagle kills Halb in 5 hits (Eagle swings every 2 sec so 10 sec)
Halb (fully upgraded) vs Aztec Eagle (fully upgraded)
Halb kills Aztec in 9 hits (27 sec)
Eagle kills Halb in 5 hits (10 sec)
Halb (no Blast Furnace, yes Plate Mail) vs Mayan Eagle (yes El Dorado, Plate Mail, no Blast Furnace)
60 HP vs 100 HP
6+2 atk vs 9+2 atk
0+3 armor vs 0+3 armor
Halb kills Eagle in 20 (60 sec)
Eagle kills Halb in 8 hits (16 sec)
Halb (fully upgraded) vs Mayan Eagle (fully upgraded)
Halb kills Eagle in 15 (45 sec)
Eagle kills Halb in 6 hits (Eagle swings every 2 sec so 12 sec)
Now let's add in the hypothetical attack bonus:
+3 attack bonus
vs Aztec (early Imp): 8 attacks (24 seconds)
vs Aztec (fully upgraded): 6 attacks (18 seconds)
vs Mayan (early Imp): 13 attacks (39 seconds)
vs May (fully upgraded): 10 attacks (30 seconds)
I think this has the most minimal, but still kind of meaningful impact.
+4 attack bonus
vs Azt (EI): 7a (21s)
vs Azt (FU): 6a (18s)
vs May (EI): 12s (36s)
vs May (FU): 10a (30s)
This is more optimal than the last because it doesn't affect the Aztec matchup very much but affects the Mayan one.
+6 attack bonus
vs Azt (EI): 6 (18)
vs Azt (FU): 5 (15)
vs May (EI): 10 (30)
vs May (FU): 8 (24)
+8 attack bonus
vs Azt (EI): 5 (15)
vs Azt (FU): 4 (12)
vs Azt (EI): 8 (24)
vs May (FU: 7 (21)
+12 attack bonus
vs Azt (EI): 4 (12)
vs Azt (FU): 4 (12)
vs May (EI): 6 (18)
vs May (FU): 6 (18)
This is the upper limit if we want to prevent Halbs from beating Aztec Eagles 1v1, and it will also affect the Mayans the most.
+13 attack bonus
vs Azt (EI): 4 (12)
vs Azt (FU): 3 (9)
vs May (EI): 6 (18)
vs May (FU): 5 (15s)
+18 attack bonus
vs Azt (EI): 3 (9)
vs Azt (FU): 3 (9)
vs May (EI): 5 (15)
vs May (FU): 4 (12)
This might be too much. A Fully Upgraded Halb now beats Aztec and Mayan Eagles 1:1. However...Elite Skirmishers also beat Arbalests 1:1, and the resources are fairly similar (25f, 35w vs 25w, 45g).
I have one more related idea:
Garland Wars should give +5.
That will allow Aztec Champs to beat Japanese Champs 1:1 in late Imp, which makes sense since Japanese get their bonus for free.
Aztec Eagles will kill Foot Archers and most Siege the same as they did. They'll kill Cavalry Archers in one less hit, which I think is a good thing. The most important thing is that when they are Fully Upgraded, they will kill Halbs in one less hit. If that were to happen, then Halbs can have a much bigger attack bonus vs Eagles and still lose 1v1, and then they'll be much more effective vs Mayan Eagles.
Also I'd personally get rid of Plumed Archers attack bonus vs Infantry. With their fast firing rate, movement speed, etc. they are already good vs Infantry so the attack bonus is overkill.