Gambit said the only equilibrium was mixed, with 1⁄5 each of (blue sword, blue armor), (blue sword, green armor), (yellow sword, yellow armor), (green sword, yellow armor), and (green sword, green armor).
With a stylin’ bonus of ε points per duel (if a win is 1 point and a loss is −1 points), Gambit says for ε≤1/4 the equilibrium is: (blue sword, blue armor): 1/5−(4/5)ε (blue sword, green armor): 1/5−(3/5)ε (yellow sword, yellow armor): 1/5+(4/5)ε (green sword, yellow armor): 1/5+(3/5)ε (green sword, green armor): 1⁄5
>>2vc// >Note: this image does not belong to me; I found it on 4chan. >>2vc//2sbt >Yellow yellow, because it looks the most awesome and seems like a generally decent combo, >>2vc//2scc >Gambit says >(yellow sword, yellow armor): 1/5+(4/5)ε
I do not see the logic behind this. Why would you ever choose to wear blue armor? No matter what weapon the opponent has, the best armor is either green or yellow. The blue weapon is only optimal against blue armor, but nobody should be wearing blue armor.
EDIT: The deleted comment this responds to claimed that you should never use the blue armor, because against any weapon, either the green or the yellow armor is better.
It can make sense to choose the armor that is not optimal in any case if it is good enough in more cases. There are no bonus points for winning by a larger margin.
This post is incorrect. You do need to consider the individual matchups, not the average damage taken and given. My suggestion loses 2-3.
I understand the same-color bonus, but it seems to break the results when you set the stylin bonus to 0. For each (blue, yellow, green) sword, I get the damage against (blue,red,yellow,green) armor to be:
Blue Sword: (6240, 6160, 6400, 6080)
Yellow Sword: (6150, 5975, 6500, 5700)
Green Sword: (5940, 6210, 5400, 6840)
For a sword distribution of (.4,.2,.4) and an armor distribution of (.4, 0, .4, .2) , I get expected values for swords of (6272, 6200, 5904) and for armors of (6102,6143,6020,6308).
That’s not an equilibrium, is it? Against that population, why would I not pick the Blue Sword with the Yellow Armor, for an expected payoff of (6272,6020)?
Gambit said the only equilibrium was mixed, with 1⁄5 each of (blue sword, blue armor), (blue sword, green armor), (yellow sword, yellow armor), (green sword, yellow armor), and (green sword, green armor).
FWIW, my calculations confirm this—you beat me to posting. One nitpick—this is not the only equilibrium, you can transfer weight from (blue, green) to (red, green) up to 10%.
Gambit said the only equilibrium was mixed, with 1⁄5 each of (blue sword, blue armor), (blue sword, green armor), (yellow sword, yellow armor), (green sword, yellow armor), and (green sword, green armor).
With a stylin’ bonus of ε points per duel (if a win is 1 point and a loss is −1 points), Gambit says for ε≤1/4 the equilibrium is:
(blue sword, blue armor): 1/5−(4/5)ε
(blue sword, green armor): 1/5−(3/5)ε
(yellow sword, yellow armor): 1/5+(4/5)ε
(green sword, yellow armor): 1/5+(3/5)ε
(green sword, green armor): 1⁄5
>>2vc//
>Note: this image does not belong to me; I found it on 4chan.
>>2vc//2sbt
>Yellow yellow, because it looks the most awesome and seems like a generally decent combo,
>>2vc//2scc
>Gambit says
>(yellow sword, yellow armor): 1/5+(4/5)ε
File : Gambit_sez_small.jpg
Uh-oh, LessWrong is turning into 4chan! :)
I do not see the logic behind this. Why would you ever choose to wear blue armor? No matter what weapon the opponent has, the best armor is either green or yellow. The blue weapon is only optimal against blue armor, but nobody should be wearing blue armor.
EDIT: The deleted comment this responds to claimed that you should never use the blue armor, because against any weapon, either the green or the yellow armor is better.
It can make sense to choose the armor that is not optimal in any case if it is good enough in more cases. There are no bonus points for winning by a larger margin.
This post is incorrect. You do need to consider the individual matchups, not the average damage taken and given. My suggestion loses 2-3.
I understand the same-color bonus, but it seems to break the results when you set the stylin bonus to 0. For each (blue, yellow, green) sword, I get the damage against (blue,red,yellow,green) armor to be:
Blue Sword: (6240, 6160, 6400, 6080)
Yellow Sword: (6150, 5975, 6500, 5700)
Green Sword: (5940, 6210, 5400, 6840)
For a sword distribution of (.4,.2,.4) and an armor distribution of (.4, 0, .4, .2) , I get expected values for swords of (6272, 6200, 5904) and for armors of (6102,6143,6020,6308).
That’s not an equilibrium, is it? Against that population, why would I not pick the Blue Sword with the Yellow Armor, for an expected payoff of (6272,6020)?
FWIW, my calculations confirm this—you beat me to posting. One nitpick—this is not the only equilibrium, you can transfer weight from (blue, green) to (red, green) up to 10%.