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The Math of Khan

Jeff Warrender
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Averill Park
New York
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"You Said This Would Be Fun", a book about game design, available at Amazon and DriveThruRPG
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Disclaimer: This post contains math! Continue reading at your own peril!


Tau's comments in the previous post are important and I think worthy of a quick post in their own right. In fact Tau should be the one to write it. Maybe more to say on that at a future time. But for now you're stuck with me so here goes.

I use Sidereal Confluence as an example quite a lot here and in my design book, both because it has a lot to teach us as a design and because the process that led to it also has much to teach. Some of the things I learned while watching Tau's process for designing and balancing the game figure in to Ch. 20 of my book, about costs and balance. Let's illustrate some of these principles with a worked example from the game I mentioned in the last post, The Cause, a 2p not-really-co-op.

From gallery of jwarrend


indigo Setting the stage: Action cards.

On your turn, you play an action card to the table and allocate 1-4 cubes to the boxes in the top row of the card. Then, you exhaust a worker to roll a d12. If the result is covered by a cube, then it's a success. (e.g. you allocated 3 cubes, and rolled a 5, that's covered by the cube in the "4-6" box).

On a success, you move the leftmost cube in the row down to one of the rewards (rewards in the "1 cube" row require 1 'success cube', rewards in the "2 cube" row require 2). You keep rolling till you stop, run out of workers to exhaust, or go bust (a roll of 11 or 12).

coffee The basic equation

A worker costs 2 cubes to acquire, and each worker gives you one die roll per round. A worker's value is N die rolls, where N is the number of remaining rounds after the one in which the worker is acquired.

The opportunity cost of acquiring a worker is that in the current round, a particular action will cover 2 boxes not 4.

sugar Expectation value

A brief diversion. When we're dealing with a random element, the math becomes more complicated, because the randomizer can take on different values. We can use the "expectation value", akin to the weighted average, as a stand-in.

Say I have a d6 that has faces {1,1,2,2,3,4}. The expectation value is (2*1 + 2*2 + 1*3 + 1*4)/6 = 13/6 or about 2. That's the same as the median, in this case, but if my die was {1,1,2,2,3,20}, that's 29/6 or about 5, even though no individual roll will produce 5. But I can use 5 in the equations for the game's math.

coffee

Ok, back to The Cause. When we go from two cubes to four, the expectation value goes from 3/12 to 10/12, so whatever we get from having spent those four cubes should be three times as good (approximately) as two would be.

That means, then, that the value of N additional rolls should be about as good as a factor of 3 boost in the expectation value of a single roll.

corn Growth curve

Let's say we desire that a full investment of 4 cubes should beget 5 cubes over 4 rolls. Forget for now how we achieve this. The bottom line is that our growth curve is that 4 cubes become 5, or that we get 1.25 cubes for 1 cube and 1 roll.

This in turn means that N rolls are worth {4, 3, 2.5, 2, 1.5} cubes if N is {6,5,4,3,2}.

But of course, we also need to have cubes to invest to fuel that growth curve. If you spend 2 cubes to get an extra worker early on, you're getting more rolls, and thus potentially more cubes, over the course of the game, but you're also taking cubes out of your engine, and thus the growth you can generate is reduced.

tobacco Reward baseline

Now we can use all of this to analyze the rewards on the card above and see if they're commensurate with the formula. On this card we have three rewards, two that require one success each, and one that requires two successes.

Without going into the gory details[*], to get {1,2,3,4} successes I'd expect to need {1.2, 2.5, 4.2, 7.2} rolls, statistically.

If we said that 4 cubes should beget 5 cubes of value in 4 rolls, then it should be that 3 successes (which should take 4 rolls to achieve) should pay out about 5 cubes.

Sidebar: From a player standpoint, setting the "pivot point" at 3 rewards introduces a nice tension. If I've achieved 2 rewards, do I stop so as to avoid the risk of going bust (and burning rolls) or do I try to squeeze the most from my investment and go for that 3rd reward? And if I get 3, do I stop there or go way off into inefficiency land and try to get a 4th?)

Now the card shows three rewards:
- 2 grey cubes
- Get a worker out of jail (worth about 2 cubes, but it's a little better than that)
- Move "the courts" tile up one, which has no intrinsic value but is important for game-winningness.

The implication of this math, then, is that moving a tile up one is worth about 3 cubes. That's very useful, I can calibrate other cards with this knowledge.

[*] Gory details available on request, but please don't request them, unless you really want to.

coffee Full circle moment

And just as a consistency check:

If I had invested 3 cubes, to get {1,2,3} successes I'd expect to need {2, 4.4, 8.4} rolls.

If had I invested 2 cubes, to get {1, 2} successes I'd expect to need {4, 10} rolls.

So by investing 4 cubes instead of 2, with about 4 rolls I expect 3 successes instead of 1, a result that is 3 times as good, exactly what we had set as our target.


bluetaj:

Now to make an interesting game, not all rewards on a given card will be of equal value, of course. Maybe a card gives three cubes, or a reduction of one space on the "violence" track, and those three cubes sure seem better but if I've got to reduce that violence track I've got to do it, even if it's technically inefficient.

So the math helps us to look at what the average behavior ought to be. Of course, it's more complicated than all this because The Cause isn't a resource converter or an engine builder, the point is more to show how to use analysis like Tau's to set parameters.

bluetaj:

Let it be observed that this was by no means an easy set of numbers to run, and I'm not 100% sure I'm thinking about this correctly anyway! So, games with challenging math aren't necessarily a good place for a new designer to start, unless you're good at math, in which case, go for it!

But I do think it nicely illustrates Tau's point, that an internally-consistent set of numbers can help tremendously in creating a set of cards that is nominally balanced. Then, playtesting data can point to changes in the formula, which allows you to adjust all the numbers and maintain internal consistency, as opposed to just buffing some numbers and nerfing others and seeing what happens.

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