MrEff

20

I think using a 16x16 strategy matrix just obscures the issue. Your choice of sword is never affected by the opponent's choice of sword; it seems that way because switching swords could turn a loss into a win, but that is a simple optimization, not a game theoric calculation. Your opponent changing his sword will never alter your best choice of sword (taking armor as fixed). What we really have is two 4x4 games that happen to be identical, requiring a single solution.

So I drew the 4x4 sword vs armor type matrix and solved for dominated strategies. Going through the table and marking best responses, I reduced it like this:

Red is never the best sword for any armor. Although it isn't dominated by any other single strategy, the mixed strategy "blue or yellow or green sword" does dominate it. While it seems that a strategy could be "good enough" in any situation while never being the best, in fact such a strategy is never optimal. You can Google this reasoning method as "never a best response".

Similarly, neither the red armor nor the blue armor is ever a best response to any type of sword. Eliminate both choices.

Redraw the matrix with the remaining choices, discarding the red sword and the red and blue armors.

With the blue armor nor longer a viable choice, the blue sword is never a best response to the remaining choices (yellow or green armor). Eliminate.

We now have a 2x2 matrix of (green, yellow) and (green, yellow). If you assume that the margin of victory is unimportant, the equilibrium strategy is to mix green sword and yellow sword 50/50 and to mix green armor and yellow armor 50/50.

However, in any actual MMO, the margin of victory is important. You require less healing, or you can do more damage to a second opponent before you die, or you are less likely to suffer an unexpected defeat due to randomness. Using the actual damage/second numbers, you will choose the green sword only 35.7 percent of the time, and the green armor 49 percent of the time. The final answer will be between these two situations and requires more information to calculate.

Choosing blue/green because it has the most wins vs. the field is not a great solution method, because "the field" will simultaneously be choosing optimal strategies as well. You will certainly not face an equal distribution of opposing strategies. It is possible to calculate what you will face if you assume the opponents are all close to rational.

Because each weapon and each armor must be the best response to something to ever be worth choosing, the number of viable armors and the number of viable swords must be equal.

I didn't add in a stylin' bonus, but you have a 50% chance to be stylin' already so it shouldn't make a big difference.

I am not quite certain how Steve Rayhawk got a different answer out of Gambit. I assume it's because he used wins/losses instead of dps numbers, which turned some strongly dominated strategies into weakly dominated strategies and kept them alive.

A very interesting article, both informative and clearly explained.