Swords and Armor: A Game Theory Thought Experiment

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

My general heuristic for these sorts of games is to play the option that beats the option that beats the option that looks best to me at first blush. In this case that means I play green sword, yellow armor. It's a reasonably fast heuristic that does reasonably well.

Deleted earlier comment due to a bug in the code.

Here's the result of a naive brute force program that assumes a random distribution of opponents (i.e. any combo is equally likely), sorted by number of wins:

185: red/blue
269: red/red
397: yellow/blue
407: yellow/red
438: red/yellow
464: red/green
471: yellow/green
483: yellow/yellow
512: blue/yellow
528: green/green
539: green/red
561: green/blue
567: green/yellow
578: blue/red
635: blue/green
646: blue/blue

The program is here: http://pastie.org/1217024 (pipe through sort -n)

It performs 30 iterations of all 16 vs 16 matchups. Note that the player that attacks first has an advantage, so doing all 16 vs 16 balances that out (everyone is player 1 as often as he is player 2).

I signed up today to comment in this thread, so don't mock me too heavily. :)

Edit: Bumped iterations to 30 and hit points to 80,000 to try to smooth out randomness in the results.

3: blue/red
5: blue/green
7: blue/blue
7: green/green
7: green/red
9: green/blue
11: green/yellow

7: blue/red
7: red/green
9: green/green
9: green/red
11: blue/blue
11: blue/green
11: blue/yellow
11: green/blue
11: yellow/yellow
13: green/yellow

2: red/blue
8: red/red
13: yellow/blue
13: yellow/red
15: red/yellow
15: yellow/green
17: blue/yellow
17: red/green
17: yellow/yellow
19: blue/red
19: green/blue
19: green/green
19: green/red
19: green/yellow
21: blue/blue
23: blue/green

63: red/blue
247: red/red
391: yellow/blue
397: yellow/red
450: red/yellow
463: yellow/green
507: yellow/yellow
510: blue/yellow
518: red/green
566: green/red
567: blue/red
568: green/yellow
569: green/blue
570: green/green
633: blue/blue
661: blue/green

After running through a 4x4 matrix, I figured that blue/blue was the best overall. Nothing can outdamage you unless it's by x/green armor, which then gets owned by everyone with a green/x so everyone else starts playing green/green and ties, until eventually some smart guy swaps to green/blue. Thus, you laugh for a while as everyone else follows and lose to your original blue/blue, starting the cycle again.

Matrix + human psychology for laughs [and wins?]

Personally, I'm caught between how awesome the green sword looks and the style of going blue/blue [plus blue/blue deals with everyone else's odd choice of yellow/yellow]. And I'm sure others probably know how to work the dominance form better e.g. Gambit or rosyatrandom.

This is a zero-sum game with the same 16 strategies for each player. If player 1 does damage A1 (for attack), S1 times per (say) hour, and player 2 has armour that cancels D2 points from a successful attack and gives E2 (evade) percent change of dodging, the rate of damage by player 1 on player 2 is S1*(1-E2/100)*(A1-D2). Subtract from this the same expression with 1 and 2 swapped and that is the differential rate. Given the conditions of the problem, the only thing that matters about this rate is its sign. So there's a 16*16 payoff matrix with entries 1, 0, and -1.

I've computed it, and it turns out that no strategy dominates any other: every row contains at least one -1 in a column containing at least one 1. However, I forget how to compute a minimax mixed strategy for such games. I think it's a linear programming problem, and for 16 variables it can't be done manually unless the data have some regularity to be exploited. Eyeballing the matrix, I don't see any.

16 possible combination's of Weapon/Armor available, 256 possible combination's of Weapons/Armor between two players.

Only factor in deciding a winner between players is the player with the higher DPS, the margin between the two players is irrelevant.

So you work out the winner in each of the 256 match-ups (16 mirror matches will be stalemates). Armed with the knowledge of what-beats-what, it's just a matter of knowing the distribution what combo's other players went and selecting the best armor/sword combo for that population.

Do these steps seem right?

I'd actually kind of like to work this out, but there has to be a better way to work out the 256 combo's than doing it all by hand. Any suggestions?

    D    S    D*S        a1        a2        a3        a4    
s1    100    80    8000    6336    6256    6400    6080    6268
s2    80    100    8000    6120    6120    6000    6080    6080
s3    150    50    7500    6210    6035    6500    5700    6111.25
s4    50    180    9000    6156    6426    5400    6840    6205.5
                    6205.5    6209.25    6075    6175    


                Da    -12        -8        -20        0    
                Do    0.1        0.15    0        0.24    

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

The red sword is strictly worse than the blue sword. Beyond that I'm too lazy to figure it out.

VS        a1         a1    a1    a1    a2    a2    a2    a2    a3    a3    a3    a3    a4    a4    a4    a4
        s1        s2    s3    s4    s1    s2    s3    s4    s1    s2    s3    s4    s1    s2    s3    s4
a1    s1    0.5        0    0    0    1    0    0    0    0    0    0    0    1    1    1    1
a1    s2    1        0.5    1    1    1    0.5    1    1    1    1    1    1    1    1    1    1
a1    s3    1        0    0.5    0    1    1    1    1    0    0    0    0    1    1    1    1
a1    s4    1        0    1    0.5    0    0    0    0    1    1    1    1    0    0    0    0
a2    s1    0        0    0    1    0.5    0    0    1    0    0    0    1    1    1    0    1
a2    s2    1        0.5    0    1    1    0.5    0    1    1    1    1    1    1    1    0    1
a2    s3    1        0    0    1    1    1    0.5    1    0    0    0    0    1    1    1    1
a2    s4    1        0    0    1    0    0    0    0.5    1    1    1    1    0    0    0    0
a3    s1    1        0    1    0    1    0    1    0    0.5    0    1    0    1    0    1    0
a3    s2    1        0    1    0    1    0    1    0    1    0.5    1    0    1    0    1    0
a3    s3    1        0    1    0    1    0    1    0    0    0    0.5    0    1    1    1    0
a3    s4    1        0    1    0    0    0    1    0    1    1    1    0.5    0    0    0    0
a4    s1    0        0    0    1    0    0    0    1    0    0    0    1    0.5    0.5    0    1
a4    s2    0        0    0    1    0    0    0    1    1    1    0    1    0.5    0.5    0    1
a4    s3    0        0    0    1    1    1    0    1    0    0    0    1    1    1    0.5    1
a4    s4    0        0    0    1    0    0    0    1    1    1    1    1    0    0    0    0.5
_    _    10.5    1    6.5    9.5    9.5    4    6.5    9.5    8.5    7.5    8.5    9.5    11    9    7.5    9.5

This has more to do with human psychology than strict mathematical game theory:

As an obsessive gamer and game designer: when fighting a random opponent, unless there is a ladder system and you end up in the top 2% or so of the population, the optimal strategy is to counter whatever is the optimal strategy vs. a null, average, or un-equipped person. That is to say, the vast majority of players who do not make a nearly-random selection will calculate the ideal strategy against a percieved "average" or "typical" set of values for damage/speed/armor/dodge, and then stop exactly there. So to win, you need to go exactly one step beyond that and then stop exactly there.

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:

1. 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".

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

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

4. 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.

5. 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.

I'd play green/yellow; it seems like it would do well among the top choices, plus I like weapons that attack fast.

I began working out the 16x16 matrix but then I realized I could just read the thread and somebody would have already worked it out. Machrider's simulations seem like the most convincing since they're the only ones that account for multiple rounds of selection.

Since you're fully healed at the end of the fight, the actual amount of damage you do or take is less relevant; whether you did more than the opponent is more relevant. To that end, I worked out the survival time in each armor against each sword and ranked them from worst case scenario to best case scenario. Now each armor has four different lifespans (worst to best) - and, for each sword, I worked out how many of the four armors could be killed in each lifespan. (4 armors x 4 enemy swords = 16 lifespans, 16 lifespans x 4 your swords = 64 different amounts of damage). So in the end, for each of the 16 combinations of equipment, the table gave me 4 numbers of wins (worst case, second worst, second best, best case). Note that the mirror match comes somewhere in between worst case and best case since it's a draw.

By these calculations, blue sword / green armor is the best (0, 3.5, 3.5, 4), closely followed by blue sword / blue armor (1.5, 3, 3, 3). There's a four-way tie for third, with blue sword / red armor (0, 2.5, 3, 4), green sword / blue armor (2, 2, 2.5, 3), green sword yellow armor (1, 2, 3, 3.5) and green sword / green armor (0.5, 3, 3, 3). Depending on how strong trends are, any of these could be the temporary best.

Incidentally, the worst option is red sword / blue armor - it has the dubious distinction of being unable to win any matchup, and tie only two.

Strategy:

1. Minimize worst-case damage taken.
2. Maximize worst-case damage inflicted.

Implementation:

http://codepad.org/KEqxSa4j

Results:

• Selected Armor: blue with max damage 6336.0
• Selected Sword: blue with min damage 6080.0

What I find interesting is that even though this post is being downvoted, there are an awful lot of people still taking the time to think about it :)

Yellow yellow, because it looks the most awesome and seems like a generally decent combo, and also because it might be a slightly suboptimal but odd combo that people wouldn't prepare for, and also because I'm too lazy to do math.

I'm not familiar with PvP MMOs or how this kind of gameplay works. What do damage/speed/dodge signify? Is it turn-based or continuous?

Out of those four colors? Blue/blue, of course, because I'm RAVENCLAW!

I've given the problem a quick pass and I can conclude this much:

1) Using the red weapon is weakly dominated, so definitely don't do that.

2) There's no Nash equilibrium in pure strategies, so if there's an equilibrium, it's in mixed strategies. If I have time over the weekend, I might re-learn how to do Lagrange Multiplier calculus and then I'll have a go at calculating a mixed strategy equilibrium.

Some pagelong computer program I wrote says that these are the dominant strategies:

(sword damage, sword speed) (armor damage, armor dodge)
(100,  80) (-12,  10)
(100,  80) (-20,   0)
(100,  80) (  0,  24)
( 80, 100) (  0,  24)
(150,  50) (-20,   0)
( 50, 180) (-20,   0)
( 50, 180) (  0,  24)

I suppose I'd pick one at random because I'm too lazy to figure out what the right mixed equilibrium is.

So, along with a lot of other people I did a win/loss matrix, and I find a different answer from anyone else's. By my calculation, Yellow Sword / Green Armour strictly dominates.

Under the given hypothesis of very large HP, the winner of a duel is simply the one that does most expected damage per minute; the formula is (damage-reduction)(speed(1-dodge)).

I think rosyatrandom's table must be wrong. He says that B/B beats B/G, which is false; consequently there must be a mistake in his calculation. Consider: The blue sword does 100x80x0.76=6080 damage to the green armour, and 88x80x0.9=6336 damage to the blue armour; consequently blue sword and green armour beats blue sword and blue armour.

Edit: Never mind, I made a mistake in my script. Back to the drawing board until I find the bug.

Edit2: I'd put in the wrong speed values for the yellow and green swords. Now I don't find a dominant strategy.

A lot of different answers have been posted. Are we being marked on this?

It always depends on the choice of the opponent. If the damage caused by the opponents weapon is less than or equal to 80 ({50,80}) choose the yellow armour, for the other available choices ({100,150}) choose the green armour to minimize damage per attack.

Regarding the choice of weapon, always choose green to inflict maximum damage in a given amount of time.

So the best strategy is to go with a yellow armour and green weapon if one has no prior knowledge.

So, uh, after this post, knowing that some people are going to see the mixed strategy equilibrium, some are going to calculate the best combo given all combos are represented equally, some are going to pick based on style, etc etc... the best strategy would then be assign a prior probability to the chances of each analysis being made by your opponent, weight each analysis's "best picks" according to the prior, and build a representation of the field you are likely to face - then pick the best combo against that specific field?

If that's correct then the critical question is how accurate your priors are. So my answer looks something like "Pick the member of the equilibrium that performs best against other members that might be chosen for other reasons, cause they will be over-represented compared to other equilibrium members." That looks like Rayhawk's ε points for style calculation in gambit, except you consider ε to be a constant for "irrationally disposed towards x".

Of course, over time you'd gather statistical data and you could use it to refine your priors on analysis/irrational dispositions, but with that data you don't need any of this, you can just do simple statistics.

I once played a game where a similar kind of setup was arranged with armour - however, the armour selection was not necessarily fixed over each battle. Players could effectively change armour at specific points throughout the combat.

If you want to try an even more interesting arrangement, imagine players get to switch one item of their setup every, say, 10 or 15 rounds.

This problem seems to get a bit more complicated in the game theoretical sense of trying to guess what the average opponent will be equipping. A naive assessment of the expected damage inflicted and expected damage taken led me to think green/green is the best combo.

Then I wrote this stupid brute force program that fights every combo against each other 10 times and tallies up wins: http://pastie.org/1216986

Running it through | sort -n, I get:

10: yellow/yellow
30: blue/yellow
50: red/yellow
72: yellow/blue
105: blue/blue
105: yellow/red
133: red/blue
136: green/yellow
177: blue/red
183: red/red
210: green/blue
242: yellow/green
247: green/red
271: blue/green
279: red/green
310: green/green

So against a truly randomized field of players, green/green is indeed the combo that will most likely win. However, if the opponents typically favor only a few of the above combos, the result might be different.

Edit: Funny observation about yellow/yellow having 10 wins (after 10 trials of every matchup): yellow/yellow only wins when it is against yellow/yellow. Someone has to win that fight, so 10 is the minimum wins any combo can have. Against all other combos, yellow/yellow always loses.

The letter of the question doesn't say you and your opponent have common knowledge of each other's rationality, and it's not clear that that's the spirit of the question either. I'd probably just play whatever does best against a random choice.

Why do all these answers assume the opponent is rational?

If you assume the payoffs are damage and not just victory, you can treat this as a 4x4 zero sum game since weapon choice has no effect on armor effectiveness and vice versa. Then you can calculate the damage per minute for every weapon vs. every armor, and dominance reasoning yields: blue sword, green armor.

But that assumption is probably not accurate and I'm too lazy to compute the 16x16 win/loss payoff matrix, so what Steve Rayhawk said.

Everyone will choose the armor which minimizes the maximum damage you can take from an opponent, and choose the weapon which maximizes the minimum damage you can inflict on an opponent. Green/green.

I'm assuming the equation is (speed(1-dodge))(damage-dmg_reduction) expected HP loss per minute.

Green sword and yellow armor.

There is one mixed Nash equilibrium with lopsided majority blue armor, minority green, and nearly even majority green sword, minority yellow. I expect people to be biased more towards handling the majority of the cases.

EDIT: this is using the damage per time as the utility. This decouples the choice of armor and sword so that it's really one 4x4 game where you play both sides. Time for survival would be another reasonable utility, which can be had just by inverting the damage numbers.

Green & Green/Yellow

Green sword has the best damage*speed. For armor, assume the average hit does 100 damage. Then p% dodge is as good as -p damage, and green wins there, too.

But! If everyone else reasons that the green sword is best, too, then the average hit would do 50 damage, and the best armor would be yellow.

Everyone will choose the armor which minimizes the maximum damage you can take from an opponent, and choose the weapon which maximizes the minimum damage you can inflict on an opponent. Green/green.

I'm assuming the equation is (speed x (1-dodge)) x (damage-dmg_reduction) expected HP loss per minute.

Oddly enough, the armor doesn't specify whose damage is reduced, and by the usual notational convention, wearing red blue or yellow armor would reduce your own damage dealt, not damage taken. This is counterintuitive, but it makes green/green completely dominant.