My first instinct was to make the table above, which may or may not be readable here (EDIT: mostly readable, some tab glitches).
I first calculated each sword’s damage per minute (obviously, in general you want the highest value here), and then worked it out as applied to each armour type.
Here’s where it gets… tricky, as you want the sword that maximises damage to all armour types, and the armour that minimises damage from all sword types. Do we look at average values? That could leave you open to being gamed by someone whose choices are poor except against your specific choices.
The best average choice here, by the way, is clearly Sword 1 (Blue) and Armour 3 (Yellow). But Sword 1 is not the best choice against Armours 2 and 4, and Armour 3 is not the best choice against Swords 1 and 3.
Nevertheless, the difference is small enough that I would still go with them; in the end, I think, being optimal like that is still the best strategy, as players with worse choices will be outcompeted. I am niggled by the thought that the optimal armour is not actually optimal against the optimal sword, but I don’t think you can really do anything about that
With that payoff matrix, you can quickly see that both the red and blue armors are dominated. You should never use either. If the opponent has a blue weapon (s1) you want green armor (a4) so you take the least damage. If they have a red weapon (s2), you want yellow armor (a3). If they have a yellow weapon (s3), you want green armor (a4). If they have a green weapon (s4), you want yellow armor (a4).
This shrinks the strategy space considerably. In equilibrium, everyone will have yellow or green armor. If they have yellow armor (a3), then you want the yellow weapon (s3) to damage them the most. If they have the green armor (a4) then you want the green weapon (s4).
So everyone should be randomizing between yellow and green armor, and yellow and green weapons.
My first instinct was to make the table above, which may or may not be readable here (EDIT: mostly readable, some tab glitches). I first calculated each sword’s damage per minute (obviously, in general you want the highest value here), and then worked it out as applied to each armour type.
Here’s where it gets… tricky, as you want the sword that maximises damage to all armour types, and the armour that minimises damage from all sword types. Do we look at average values? That could leave you open to being gamed by someone whose choices are poor except against your specific choices.
The best average choice here, by the way, is clearly Sword 1 (Blue) and Armour 3 (Yellow). But Sword 1 is not the best choice against Armours 2 and 4, and Armour 3 is not the best choice against Swords 1 and 3.
Nevertheless, the difference is small enough that I would still go with them; in the end, I think, being optimal like that is still the best strategy, as players with worse choices will be outcompeted. I am niggled by the thought that the optimal armour is not actually optimal against the optimal sword, but I don’t think you can really do anything about that
I got the same numbers.
With that payoff matrix, you can quickly see that both the red and blue armors are dominated. You should never use either. If the opponent has a blue weapon (s1) you want green armor (a4) so you take the least damage. If they have a red weapon (s2), you want yellow armor (a3). If they have a yellow weapon (s3), you want green armor (a4). If they have a green weapon (s4), you want yellow armor (a4).
This shrinks the strategy space considerably. In equilibrium, everyone will have yellow or green armor. If they have yellow armor (a3), then you want the yellow weapon (s3) to damage them the most. If they have the green armor (a4) then you want the green weapon (s4).
So everyone should be randomizing between yellow and green armor, and yellow and green weapons.