Note: this image does not belong to me; I found it on 4chan. It presents an interesting exercise, though, so I'm posting it here for the enjoyment of the Less Wrong community.

For the sake of this thought experiment, assume that all characters have the same amount of HP, which is sufficiently large that random effects can be treated as being equal to their expected values. There are no NPC monsters, critical hits, or other mechanics; gameplay consists of two PCs getting into a duel, and fighting until one or the other loses. The winner is fully healed afterwards.

Which sword and armor combination do you choose, and why?

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

εpoints per duel (if a win is 1 point and a loss is −1 points), Gambit says forε≤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:

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

It performs 30 iterations of ... (read more)

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... (read more)

I haven't checked your math, but you have the psychology right for MMOs. Basically, figure out what the Flavor of the Month is, and then play what counters it.

See Dave Sirlin's discussions of yomi layer 3 and rock, paper, scissors.

[...]

... (read more)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, ... (read more)

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?

My first instinct was to make the table above, which may or ... (read more)

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

You mean this table? :)

(This and the one I made below can be seen properly at http://tinyurl.com/lwgttable , along with the ATT vs DEF tables I worked out the outcomes from)

Hmm. Unless this has gone wrong, the best combo is Sword 1 and Armour 4, with Sword 1/Armour 1 being close). But if you bank on people choosing 1/4, then 1/1 will beat them.

NB: Yes, I made a lot of mistakes and edits to get here, and probably have still made some...

... (read more)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 th... (read more)

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... (read more)

Strategy:

Implementation:

http://codepad.org/KEqxSa4j

Results:

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

thosefour 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:

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... (read more)

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 t... (read more)

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:

... (read more)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.