Monday, June 6, 2011

40k Theory: Properly Valuing Units


I’ll be blunt: I’ve read a bunch of articles lately on both sides of the fence about sacrificial/suicide/throw-away units, and they have all sucked.  If you’ve written one of those lately, don’t get mad, I’m not trying to call you out.  The reason why those authors failed is neither because they suck at warhams (they don’t) nor that they haven’t considered the issue thoroughly enough (they have): it’s because they aren’t using game theory to properly value the units in their army as the game state changes.  

It's OK to use your brain in a 40k match
People tend to assign values to units during list creation, because it’s obvious.  The units cost points.  It doesn’t take a rocket scientist to figure out how easy it is to compare the relative value of units BEFORE the game starts.  However, once the first unit is deployed on the table and the opponent’s list is revealed, the relative value of units based on points goes completely out the window.  The game before the game (building the most efficient list possible) is a game based on managing scarce resources (points and FoC slots) to maximize your potential. 

That game ends when the real game begins.  It doesn’t matter which FoC slot a unit came from, or how many points it cost, once the game starts.  What matters is how I can make my units win the game for me.  Before I get into the implications of this to you on the tabletop, let’s go into some game theory, so you can see I’m not just talking completely out of my ass.

Units are a consumable/disposable resource.  You don’t get a bonus in the next game for saving some this game, unlike a real army.  According to accepted game theory, the value of a consumable/disposable resource is the cost to replace the resource upon consumption, not the cost you originally paid for it. 

Don’t believe me?  Let’s go to an example.  You and some buddies are going to take a single day road trip.  The night before, you gas up the car for $3.00 a gallon for 10 gallons.  Overnight, the price of gas becomes $4.00.  You go on the trip, and when you get home, the tank is on empty.  Your friends, being good guys, offer to pay for the gas since you drove.  Do you charge them the $30 you paid for the gas the night before, or do you charge them the $40 that you’re about to pay now?  It’s a complicated question.

The actual gas they used on the road trip cost $30, so in absolute fairness, they should fork over $30.  But then you have to eat $10 that you wouldn’t have spent had you not taken them on a road trip.  That isn’t a good outcome! 

To show you the correct answer, let’s change it somewhat.  You fill up for $30 the night before.  Take the road trip.  You guys pull back into town and go to fill up.  Only gas dropped to $2 a gallon.  Now you fill back up your tank for $20.  Do you suspect your buddies are going to give you $30 for gas when they just saw it only cost you $20?  I’d wager that 999/1000 times your friends would laugh in your face if you asked for $30. 

The conclusions to take from this are…
1.  Don’t volunteer to drive on road trips.
2.  A consumable resource’s actual cost in the real world is the cost you have to pay to replace it, not the cost you originally paid.

So now that we have established this, how do we apply it to 40k?

The points you paid for the unit before the game is a prediction of the relative value of the unit during the game.  The 60 points is what the MM Land Speeder cost before the game.  During the game, the Land Speeder “costs” whatever it takes to replace the Land Speeder’s functionality.  To properly cost it, you need to know what it’s functionality is.  This requires answering a few questions.

1.  What does this unit do?  It shoots an AP1 Melta weapon at a tank.
2.  How many other AP1 Melta weapons do I have?
3.  How many AP1 Melta weapons do I need to win the game in the current game state?

At the beginning of the game, when you have a lot of melta weapons on the table, each melta weapon isn’t so valuable.  At the end of the game when you only have two melta weapons left alive, they are much more valuable.  Conversely, at the beginning of the game, when your opponent has 12 tanks on the table, they are more valuable.  At the end of the game when he only has 2 left, they are less valuable.  There is one exception to this general rule: the game ends at a finite time, not when one person is tabled.  If our games continued on until one player or the other was tabled, we could probably eventually figure out mathematically what the units were actually worth with a number.

Unfortunately, victory conditions are more varied and the game length is finite.  Those create subjectivity.  You might have 6 melta units left alive and your opponent only has one tank.  Under normal conditions, each melta unit is not that valuable.  However, if you lose the game if you don’t pop that tank this turn, the ratio is skewed.  To keep this aside brief, just let it be said that victory conditions create a subjective factor and should be used to adjust your valuation.

Let’s put some numbers into it to describe what I’m saying.  In a game state where you have lots of melta, but no need for it, each melta unit would be valued at, say, 0.1.  In a game state where you had only one melta unit but your opponent had lots of tanks, your single melta unit would be valued at 1.0. 

So turn 1, after deployment, you have a bunch of melta units, and your opponent has a bunch of units that need to be melta-ed.  They are in approximate balance, so we’ll call that a 0.5 value for each melta unit. 

Fast forward to turn 3.  You’re doing well.  You’ve knocked out 3/4ths of his tanks and haven’t taken many casualties in return.  You have plenty of melta and he only has a few tanks left.  Melta might be worth 0.3 right now.  Thus, melta weapons are cheap in the current game state.  At this point, you can profitably sacrifice a couple of melta units, in order to advance your position.  Melta is cheap to replace, you can ‘consume’ it faster.

Fast forward to turn 5.  You opponent has no more tanks.  You have three melta units left.  At this game state, melta is super cheap.  You have plenty of melta left but no targets.  Melta is worth 0.15.  I wouldn’t say you should expose the units to unnecessary danger, but they are so cheap, that using one unit as a sacrificial speed bump or as bubble wrap for a more valuable unit becomes an optimal play.  The formally valuable melta unit is now so “cheap” that it can be used as bubble-wrap.

This obviously works in reverse, the first few turns could be rough, and by turn 5 you only have two melta units left and your opponent has 7 tanks alive.  Melta might be worth 0.8 right now.  That means each melta unit has to be protected and used extremely wisely, as they are very difficult to replace and very useful.

These are extremes.  Most of the time, between two skilled players with similar quality balanced lists, melta will fluctuate only a bit, from 0.4 to 0.6 through the ebb and flow of the game.  The game state is the “market” for melta.  Sometimes the game state dictates melta is very expensive, other times cheap.  You win when you trade it profitably.  Hold it when it’s valuable; “sell” it when it’s cheap.  Since most times it will be in a near balance, I’d like to wrap up the melta example by saying that it takes a lot of play experience to tell the difference between 0.4 and 0.6.  Practice until you can tell instinctively.

I'm bullish on melta by this point in the article

You can use this mental exercise for a lot of stuff beyond melta-units.  Scoring units come to mind as an obvious candidate.  It would be very valuable for a player to know when a scoring unit is worth throwing away to achieve some minor advantage, or when you have to throw away an Elite or Heavy Support choice to protect an ultra-valuable Troops choice.  Heck, there might be times when you reason that a 5 man combat squad is more valuable than a Land Raider, and it’s worth “sacrificing” a Land Raider to protect the combat squad.  At that point, a mighty Land Raider is a “throw-away unit” even though it didn’t seem so when you bought during the pregame. 

8 comments:

  1. You reference Game Theory and Psychology in back to back articles, and do so intelligently. I knew there was a reason this is rapidly becoming my favourite blog. Keep it up!

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  2. How would you handle valuing multipurpose units? Such as melta veterans (melta/scoring)

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  3. Very well written... great to see the application of market theory on 40K.

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  4. In a way, subconsiously, that's always how I see my units. For example, my 40 point Grot unit has won me more games than I care to remember. So for some games this unit is worth infinite points (when it can do some cheeky objective scoring). In some other games (mainly kill points) they just sit out of the way and do nothing, so really worth zero points. So it all depends on the missions and what the opponent brings on the table. Great article!

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  5. @MadPersian I would value units with duality highly, simply because there are fewer game states where they are useless. How likely is it that a unit with good anti-tank and good anti-infantry capability will have nothing to do? Pretty rarely. So they will rarely dip in value very low as the game goes on.

    The exception is on objective games. If they aren't a scoring unit and they aren't in position to contest an enemy objective then what good are they?

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  6. I love reading these type of articles from you. I far prefer your educational style of writing than I do the force-feeding other blogs do. As the saying goes, you're teaching someone to fish.

    You put into words very well what I've done all along. I have always said that any unit in my list is fair game when it comes to being a sacrificial unit. As you said, if the unit holds little value at that point in the game then there's no sense trying to preserve it if you can gain an advantage by sacrificing it. By sacrifice I don't mean just tossing it out there for inevitable destruction.

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  7. Nike- This article made me whip up a blogger account real quick just so I could comment. I've been reading your blog for a while but with this and the Blink articles you hit 100% of the buttons I'm looking for in a blog right now. Insightful, informative and a great way of thinking about our game.

    Please keep this up!

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  8. Thanks for the reply. Also, I agree with 200Ponies. Your Blink post (and follow up) and this post are very high quality. I also just put in my order for Blink.

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