Tuesday, July 13, 2010

FMB Artifact Thoughts

Fantasy Monster Beatdown is a game currently being designed/developed by my friend Sky. I was over at his place on the weekend to test the 4 player version of the game with Pounder and Aidan with a lot of good testing related results coming from it. This post is not about those.

As some of you may be aware, I'm a big fan of perfection. When I have a choice to make I want to figure out the optimal thing to do. Even if the choices are close, I want to crunch the numbers and decide the 'best' one. I don't need to do this at the time, of course, but I want to justify my spur of the moment choice. Or, if it's wrong, I want to know why so I can make better spur of the moment choices in the future.

One of the game mechanics of FMB has it so you get a powerful artifact every 3 turns. When you do you get to choose between 3 different artifacts drawn off of a shuffled deck. Ostensibly all these artifacts are the same power level despite doing drastically different things. Balancing the artifacts given they do very different things is not a trivial task, and a lot of artifacts have seen hefty changes over different iterations of the game. My contention has always been that a certain mechanic on artifacts is 'best' (and as such always take those ones when given the choice) but it's hard to really justify that stance. I don't play enough to have more than a gut feeling here (this effect is doubled since I like to win and therefore the few times I do play, I go for the throat with what I think is best).

On the weekend, however, I had a choice between 3 very similar artifacts. I want to compare the effects each will have on a game and then compare them to the artifact I think is the best in the deck. Some of this will be hand-wavy, so it's not meant to be a definitive proof across all situations in all games. I will spell out my assumptions in advance, however, to allow you to draw your own conclusions.

  • You know enough about FMB that I don't need to describe mechanics in huge detail. If this isn't true, you may want to click the link above and read Sky's blog. Alternatively you can ask a question in the comments.
  • This is the first artifact you're going to take. By making this assumption I can avoid dealing with artifact combos which can certainly increase individual power levels.
  • You're playing a 1v1 game, so there are 7 units per side.
  • Both teams have an even split between 1 and 2 power units. (Not realistically possible since we have an odd number of units, but I want to simplify the math a little.)
  • As such, any given attack has a 25% chance of being 1v1, 25% of being 1v2, 25% of being 2v1, 25% of being 2v2.
  • Both teams have an even split of melee/ranged and of slow/fast. They have en even split between the two types as well, so there's a 25% change of being slow/melee, 25% slow/ranged, 25% fast/melee, 25% fast/ranged.
  • The first 'hit' of a turn will negate an incoming attack. So will every other odd hit. (This is from stunning a unit, preventing it from attacking.)
  • If a slow/melee guy is killed he loses an extra 2 turns. (Sometimes he'll only lose 1 if the opponents are deep into your territory but sometimes he'll lose 3 if you control the middle node.)
  • If a slow/ranged guy is killed he loses an extra 1.4 turns. (Similar arguments as above, but he can hit people in the 7-8 range on turn 2.)
  • If a fast/melee guy is killed he loses an extra 1 turn. (He can pretty much always find a guy to hit on turn 2, but can only hit on turn 1 if you've lost a back node. In that case they're apt to try for a perma-stun instead of a kill on the fast/melee guy.)
  • If a fast/ranged guy is killed he loses an extra .5 turns. (He can attack anyone threatening your back nodes on his first turn back.)
  • Now, you could just stun an extra guy instead of killing the fast/ranged guy, but a lot of the time he's going to be in a dangerous position and you want to kill him for positional reasons and not just to prevent incoming attacks. As such, I think there's value in keeping that number at .5 instead of bumping it to 1 and assuming mass stuns.
  • The second hit (and all subsequent even hits) therefore negates (2+1.4+1+.5)/4 = 1.225 attacks.
  • The formula for attacks negated, on average, given the above is n = h + (floor(h/2)+floor(h)mod2*(h-floor(h)))*.225
  • I set up a circular reference formula in Excel and iterated it a couple hundred times. It looks like the equilibrium point for balanced armies given all of the above is ~4.5 attacks per round giving ~2.25 hits per round.
Default with no artifacts, what is the outcome of my turn going to be? (I know, I spoiled it above, but bear with me here. The method used here will be reused.) Basically what we want to do is build a 3 dimensional truth table showing all possibly outcomes along with their weights and then we can sum up the expected values.

I hope that ends up legible and sensible to everyone. The numbers down the left correspond to the cards in your deck of dice. The numbers at the very top are the attacker's strength. The numbers in the next row are the defender's strength. The third row is the relative odds of that attacker-defender matchup occuring and should sum to 1. Then the truth table has a 0 if that card would be a miss for that attacker-defender matchup and a 1 if it's a hit. The EV column on the right works out the relative worth of a given card. As you can see, the -2 and -1 always have no value in the default setup. 2 and 3 always hit. The 0 and the 1 care about the difference in strengths. The most important number is the one in the lower right hand corner, which indicates the average chance to hit long-term. As expected, it's 50% with no artifacts in play.

Artifact A - Remove all -2's from your deck.

Our average hit chance on an attack goes from 50% to 60%.

Artifact B - Remove all 3's from your opponent's deck.

Here we're looking at what the opponent's hit table becomes. He goes from hitting 50% of the time to hitting 40% of the time.

Artifact C - Once per turn, reflip one of your attacks.

Initially my idea was to build out a huge truth table trying to take into account order of attacks and when you might miss and such, but in addition to being very time consuming I'm not even sure that's a good way to try to model this effect. Thinking about the card logically, all it does is give you an extra attack every turn, with the caveat that you have to have missed at least once. So, in the final analysis, I can model this artifact by adding in an extra percentage of a swing. The question then becomes how big a percentage? What are the odds that I'm going to hit on every attack I make in a turn? Unfortunately this question is non-trivial. To work out a good estimate you'd need to build out the odds of each number of attacks on a turn and handle the different attack strengths and such. Worse, you need to account for the deck of dice mechanic. (Deck of dice approximates rolling an actual die perfectly on a single attack. Long-term it models average dice. But short-term, looking at a small number of consecutive swings it can get wonky.)

What I decided to do was just look at a few numbers and then make a reasonable guess. In particular, you'd get at least one miss 99.8% of the time with 5 1v2 attacks, 89.7% of the time with 5 2v1 attacks, 98.1% of the time with 5 2v2 attacks, and 89.1% of the time with 3 2v2 attacks. Something around 95% is about right, so that's what I used.

So, the effect of this artifact on hits is to add an extra .95 swings per turn.

Artifact D - Free strength 3 attack during spell phase

Having one of your attacks each turn be at strength 3 changes the weighted odds a little. As such, your chance to hit over all of your attacks long-term goes up to 54.5%. In addition, you get an extra attack every round.

Now, what do these things actually do from a hits standpoint? There are two ways to look at it. One is to just take the expected swings and apply the new hit chances to see the net difference. The second is to rerun the iterative relation to work out the overall impact on the game if it continued into infinity with you having the artifact as the only change. (Hitting him more means he swings at you less which means he hits you less which means you swing more...)

As well, there are two ways to look at the change. (Needed because one is a 'defensive' artifact.) You can look at net change, or as a ratio of your hits to his hits.

Simple one first:
A - +.45 hits for
B - -.45 hits against
C - +.475 hits for
D - +.75 hits for

A - +20% hits
B - +25% hits
C - +21% hits
D - +33% hits

Complicated one next:
A - +.99 hits for
B - -.91 hits against
C - +.96 hits for
D - +1.65 hits for

A - +51% hits
B - +54% hits
C - +50% hits
D - +98% hits


From a strict hits standpoint the first 3 artifacts are all very comparable. B rates to be a little better, but on the whole they're all in the same ballpark. D is way out in left field. Depending on how you look at it it could be worth twice as much as the others or merely 50% more. From a sheer hits standpoint, I take D every day over these options.

Hits aren't the only thing that matters in the game, though. It's not a game of just rushing the middle and fighting for no reason... There are positional parts to the game that take serious consideration as well. Weighting which is important between power and position is not something I'm really equipped to do and is really where the interesting strategies occur in the game. Passing on an attack to get better position on a node is strictly wrong from a hits standpoint but can be the way to win the actual game. But what we can try to do is rank the artifacts on a positional impact as well. We can't compare power to position, but we can order them in both respects.

It should be clear A and B have no positional impact. Everything they deliver they deliver through sheer power. There's no choices involved, they just happen.

C has positional impact. Generally speaking you get to choose the order of your attacks, so if you do them in descending order of importance you rate to get your free attack on a good target. Re-roll the first miss! C is approximately as powerful as A and B in terms of hits but it gives you some ability to aim the hit. If attacks ever have varying importance then C is better than either A or B.

What about D? Well, you get to point that extra swing anywhere you want. You can aim it at a unit you can't even realistically get at, allowing you to stun a unit in a very dangerous position. C gives you .95 of an extra attack at strength 1.5 probably somewhere good. D gives you a full extra attack at strength 3 anywhere you want. Power-wise this is just better. Positionally this is just better.

So, given a choice between just these 4 artifacts as my first artifact I'd choose them in order D->C->(A or B). The choice would be made between A or B based on how far through the decks of dice the players are, to see if I couldn't squeeze some edge out between them that way.

For the record, I chose C at the time. Vindicated!


Sky said...

Nice analysis. As you said though, there are lots of tricky mitigating factors, in particular that pushing attacks are usually really valuable, moreso than stunning/killing attacks. Also there is a disadvantage to having an attack be forced at the start of a turn instead of during the flow of combat since it reduces the amount of information you can have access to when choosing it. That might still mean that the Rod of Lightning is better than the other combat effects, but those considerations are quite powerful.

Ziggyny said...

Having Rod of Lightning go before any unit attacks is certainly worse than letting it go at any point in time, but I'm not convinced it's actually a very big penalty. It lowers the flexibility but doesn't really hurt the raw power which was my biggest concern.

I do think I disagree with pushes being more valuable than stuns/kills. In a specific example, when Pounder was using the pushy artifact against us last weekend it didn't seem like much of a concern. Your lightning artifact was much stronger, I think. But then, that's comparing always push to never push when having a choice between the two is where the real power comes.

At any rate, I actually think the next time I play I'm going to go for an exclusively ranged army. Pushing is useful in the little 1v1 skirmishes for outlying nodes but if any killing is going on the cavalry will come to reinforce the node soon enough. With kills being worth a gold each I think a just killing strategy could be very powerful and you want ranged for that. (Can I use 7 wizards or am I restricted to 2 by the game pieces?)