
Jeff Warrender(jwarrend)United States
Averill Park
New YorkCome talk design at the Jeff's World of Game Design blog! 
Because of the perceived importance of a game's balance, designers usually feel the need to try to get the numbers right in their game, and that means math. That can lead to the perception that you have to be a math whiz yourself to be able to design games, and of course nothing could be further from the truth. But the opposite extreme is to just slap some numbers down on some cards and adjust until it feels right, and not only is this inefficient, it too inefficiently gets to the point of what those numbers are for.
So, here is the simplest of simple approaches for how to get started with basic numbers for a game, for, say, the number of resources each player should receive each turn.
First, assume that the game should run about 8 turns per player.
This assumes that yours is a "do a few things on your turn" game, which many games tend to be. Players playing such a game at a leisurely pace will probably take about 2 minutes per turn. In a 4p game, eight 2minute turns is about an hour. This is a good goal to shoot for.
Now if players can only achieve 2 minute turns by playing with a hyperfocus akin to that required to landing a plane whose landing gear is broken or performing surgery in the back of a taxi cab, your turns aren't really 2 minute turns!
And, yes, yes, of course, simpleturn games like Carcassonne or Acquire or Ingenious or lots of others run more than 8 turns. But a game with resource conversion is most likely a complexturn game, and those will often have 610 turns. Tikal and El Grande each have 9, The Princes of Florence has 7, Wallenstein has 6, Sidereal and Chinatown each have 6, although those are trading games and thus not quite the same thing.
Second, quantify the amount of stuff that's supposed to happen over the course of the game by the winner. If it's a game about building castles, and you need (say) four castles to win, then players should build a castle (on average) every other turn.
Third, figure out how many resources or actions should be required to accomplish each increment of stuff, and divide things up accordingly.
So if we decide that players should need 6 resources per castle, that's 24 resources, and so players need access to about 3 resources per turn to be able to win in 8 turns.
Fourth, make simplifying approximations to make the math easy.
This is just a starting point after all, so it doesn't have to be rigorous. Round numbers up or down, use numbers that allow you to get whole numbers from division operations, learn simple relationships (e.g. 1in6 is 16%), whatever makes things simpler.
If, in our above example, we instead wanted five castles and each were to cost 6 resources, that's 30, which is close enough to 32 that we'd be on level ground saying players need 4 resources per turn. Again, it's a starting point, keep it simple.
Now, as Ch. 20 of YSTWBF states, costs are, fundamentally, a problem for players. Thus, our game design efforts should not merely consist of performing mathematical computations and then dutifully giving players the calculated number of resources to complete their tasks. Rather, it should be in finding a problem that makes the above formula difficult to achieve in some way.
A masterclass in this comes from Citadels. The math is simple. You want to build about 8 district cards, and winning scores tend to be in the 20s, so you want your average district to be about 3 gold, but you only get 2 free gold per turn (at most). Thus, your problem as a player is how to generate gold faster than everyone else.
Citadels has a brilliant answer to this. By choosing a role that matches a color of districts you've built, you can get bonus gold. But because those districts are on public display, they make your role selection somewhat predictable,thus making you a target for the targeting roles. But on the other hand, those targeting roles have no intrinsic bonuses, and so if you pick a targeting role and miss hitting your target, you're worse off.
Taken together, this simple bit of resource design creates all sorts of opportunities for doublethink, which is where the game's friction lies.
Friction doesn't come from having just the right formula, but from finding an interesting way to subvert the formula. If players need about 25 resources, maybe it's only easy to get about 10 and the rest have to be acquired with risky ventures, or maybe you have 30 but they're mostly the wrong kind and you have to engage in trades to get 25 usable resources. Your game math shouldn't just give you a recipe for what to give the players, it should provide you with guideposts for what to withhold from them.
Let me offer a practical example of this. I'm finally solo testing a game I talked about last year, "the Cause", a 2p sortofcoop, in which the players share a win condition but are not working together. How do you get them to not want to work together, if they share a goal? The only way is with incentives, otherwise players will try to cooperate, whatever the game says about that they're supposed to be rivals. And so, the game uses math against whatever instincts to play nice they might have! Players allocate resource cubes to action cards, but resources add value to those actions triangularly. Thus, if we play nice and I take two cubes and you take two cubes, we have two value3 actions between us, whereas if I grab all four cubes, I have a value10 action, so it's in my interest, if I want to take actions that will succeed, to not play nice, and not share the resources.
(I understand that the followup question will be, well, then why don't we just agree to let you have all the resources this turn and me have all the resources next turn, and "play nice" that way? The short answer is that while we share a win condition, we have asymmetric loss conditions, and so letting you take a poweredup action may advance my loss condition.)
Of course there are lots of forms that these problems, and the interaction that necessarily attends them, can take. The "correct" approach is to start with simple numbers and then tweak them so as to bring that friction more to the forefront, to make the interactions between players more vigorous and more pointed. The tendency of designers is to tweak the math with an eye toward smoothing things out, making things more "fair", making scores closer. But that's not always the right move. Sometimes "X is too powerful" doesn't indicate a change in the direction of making X weaker, it indicates making X more hotly contested. See how far you can get with letting the players themselves provide balance, which frees you up to worry more about how to use incentives and friction to guide your refined set of numbers from that simple starting point.
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