Mathematics & Game Design, Part One

I’ve written more than once that I don’t like game designs that require me to do math. More specifically I’m talking about games like SantiagoPower Grid, and several others which have a strong mathematical basis and a strong ability to analyze that math during game play.

You see, I think games should be entertaining: it’s why I play them — to enjoy myself and to have fun. And, I don’t have a lot of fun when I sitting around adding, multiplying, and dividing (except, perhaps, in the case of a game of Primordial Soup).

Worse, I increasingly think that games which have a strong mathematical component have a core flaw in them related to victory. This flaw comes about because there are generally three types of players who might be playing these games:

  1. Some players will be totally oblivious to the underlying mathematical basis of the game, and will play by gut because it’s their only way to do things.
  2. Some players will understand the mathematical basis of the game, but will choose to largely or entirely ignore it because it detracts from their fun to carefully figure things out.
  3. And finally, some players will embrace the math, carefully calculating and recalculating every move against the mathematical basis which is laid bare.

Now the flaw arises from the fact that player type #3 will generally win these games if the underlying mathematics are actually sound. Worse, he’ll probably do so by minutely analyzing all of the options, each turn, to the point where his turns might take two or three times as long as his opponents’. I don’t find it particularly endearing for a game’s main path to victory to be raw time put into the game, yet for many mathematically based games, that’s exactly what happens.

Now don’t get me wrong, I don’t think that the flaw is having mathematics at the basis of a game. On the contrary, that’s often good game design as mathematician Reiner Knizia has proven through many a game. Instead the problem is placing the mathematics so close to the surface, and making them so static — so unchangeable — that you can calculate them without any fudge factor, revealing exactly the valuation of any move.

Auction games are in particular danger of hitting this pitfall, since you’re expending limited resources in order to gain victory in some form. Things get even more dangerous when you’re actually making your purchases with the commodity that will ultimately be used for victory points, as this allows for a simple apples-to-apples comparison if the math lies too near the surface.

However, there are ways to combat all of these problems. One of the easiest answers is to introduce chaotic player interactions, so that valuations depend upon the actions of other players. Another is to give players more explicit ways of changing valuations. A third is to introduce a few levels of abstraction between purchase and victory.

A few different games reveal how these methods can work, and when they didn’t …

Santiago: In this game a set of plantation tiles is revealed, and then players bid for selection order among that lot with money (victory points).

The problem for me arises in the very mathematical results of the tile placement. Whenever he makes a placement a player scores an amount equal to the number of his control markers times the number of connected plantation markers of the same type.

Thus it’s pretty easy to make a calculation that goes like this: “There’s just one good tile for me, and that’s the double-marker banana. If I purchase it the overall banana field goes from size 6 to 7, and my number of control markers goes from 2 to 4. Thus the tile represents a 4×7-2×6=16 point gain for me. The next best tile (after carefully running through calculations for all of them) just nets me 6 points, so going first represents a 10 point gain. Thus, if I bid 10 I breakeven. The bid’s gotten to me, and it looks like I need to bid 4 to take it. I guess that’s worthwhile, since it’s the same gain as my second choice.”

Now granted, there’s chaos here, because other players going after you could mess up your calculations, and a tile can become worthless if water doesn’t flow to it (though as a first player, you’ll also often have the ability to place where there’s already water). Nonetheless, the math is so close to the surface, that it seems pretty clear that a careful calculator will do better than a gut-feeler.

Don’t get me wrong, I think Santiago is a fine game, with quite a few interesting gameplay elements. However, I can only play Santiago when I’m pretty tired, because when my brain is fuzzy I don’t automatically start doing all the calculations. When I’m tired enough, Santiago is fun.

Boomtown: In this game, a set of cards are revealed, and then players bid for first-place selection among them with money (victory points). Last-place selectors also get rewarded with some bonus cash.

On the base level, this game’s a lot like Santiago. You’re bidding with victory points (money) for victory points that are pretty directly related. However after that it goes further afield in some good ways.

First, Boomtown doesn’t require the same type of calculation as Santiago. You’re bidding on mines, and the value of each mine is printed on the mine. However, each mine also has a secondary value: it can produce money (victory points) during the game if it “produces”, and that’s determined by a 2-12 roll each turn.

Now, I suppose you could try and calculate a mine’s valuation if you wanted:

Value of mine = value + (value * probability * number of expected turns remaining)

However, probability of this sort just isn’t as natural of a calculation as the simple multiplication of Santiago. I don’t have to stop myself from doing it, and I suspect it doesn’t even cross the minds of most folks. (It really didn’t cross my mind until I started in on this article.)

Better, Boomtown further obfuscates value by giving additional points for majority control of the five different colors of mines. This results in both short-term payments from other players and a payout from the bank at the end. Thus, certain colors of mines are much more valuable to certain players, but the exact amount of that value isn’t clear.

This sort of inclarity (or abstraction, to paint it in a more positive light) can really benefit a game because it takes away the advantage of the math-hounds. Sometimes Boomtown is still a little too mathy for me, but I’m more likely to play it and enjoy it when I’m not tired.

Ra: In this games, sets of tiles are revealed, and then players bid for entire lots with special “sun” tiles.

First, it’s clear that Ra does have a mathematical basis. You just have to look at the list of scores for different items, which vary widely — and were probably very carefully considered — to see that. However, Ra does a lot to abstract those tile valuations by introducing multiple levels of uncertainty.

First, unlike both Santiago and Boomtown you’re not bidding with victory points. Instead, your sun tiles affect your buying power in future rounds, but in a way that’s not entirely direct. At the end of the game sun tiles are turned in for victory points, but in pure competition to other players — a competition where you won’t know your standing until the last moment unless you really carefully monitor all the plays.

Second, there are a few tiles that have direct and immediate point values (gold and a first civilization), but other tiles have entirely speculative value (additional civilizations, monuments). You’re counting on being able to make additional purchases later in the game in order to give them value — and again while this valuation could be determined to some extent by a probabilistic calculation, it’s not ever done.

Third, some victory points (for pharaohs and final money) come about through player competition, and as with the colored-mine competition in Boomtown, this introduces a lot of uncertainty into the calculation.

Fourth, I think Ra makes a very good game design deicision by giving people the ability to buy an entire set of tiles rather than singletons. With a singleton purchase the human instict is much more to try and make a valuation, but with a lot everything starts to get lost in the static, and you just start looking at the high points and low points.

And then you make a decision by gut.

Some people complain about the randomness of Ra, and this goes to the exact strengths that I see. There is uncertainty, granted — chaos and speculation — but it’s that same thing that keeps the game from becoming a number-crunching exercise that you could set up on a spreadsheet, where he who takes the longest turn wins.

Reviews: Boomtown (B+), Ra (A), Santiago (B)

Next Time: Though I think math should largely be hidden from players, designers absolutely need to think about it. In my next column on this topic I’m going to talk about a few game designs that failed the sound-underlying-mathematics test, and I’m even going to show the math why.


Author’s Note: Given my dislike for raw math in games, it’s perhaps not a surprise that Boomtown and Santiago have both gone out of my collection while Ra remains (and gets a few plays a year). The next math article is a couple of articles down the line. —SA, 8/30/12

3 thoughts on “Mathematics & Game Design, Part One

  1. I know I’m late to this article, but the topic is timeless, so here goes:

    Your premise seems to be that static values lead to an undesirable level of analysis paralysis, because they reward raw computation time rather than other non-AP-inducing aspects of gameplay. (I left out the part about math being too close to the surface, because that actually prevents AP by decreasing the amount of time it takes for each player to do the necessary calculations. Pushing the math further from the surface doesn’t change the fact that the math can be done, it just makes it more difficult, and thus more time-consuming).

    The real problem with games that feature static values is not that the valuations take too long, but that they become the entire point of the game. Without another interactive element with which to muddy the valuation, the game becomes a simple, uninteresting math contest. Thus, the only way to make the game interesting is to muddy the valuation in another manner: obfuscation. This is a cheap design trick which leads to one of two players winning: the player with the most accurate mathematical intuition or the player who takes the most time to do the math. Neither of these outcomes is particularly rewarding.

    That said, I don’t think Santiago is a good choice with which to illustrate your point. Santiago’s values are far from static, as they largely depend upon the effects of the other players’ actions. I’ll go through your hypothetical example and point out other gameplay features that must be considered, none of which are trivially calculable:

    “There’s just one good tile for me, and that’s the double-marker banana. If I purchase it the overall banana field goes from size 6 to 7, and my number of control markers goes from 2 to 4. Thus the tile represents a 4×7-2×6=16 point gain for me.”

    This is the first, and most fundamental, flaw in reasoning. The points are not scored at the time of placement, but at the end of the game. So, future placements, most of which will be made by other players, greatly affect the final value of the plantation, and thus render it incalculable.

    “The next best tile (after carefully running through calculations for all of them) just nets me 6 points, so going first represents a 10 point gain. Thus, if I bid 10 I breakeven. The bid’s gotten to me, and it looks like I need to bid 4 to take it. I guess that’s worthwhile, since it’s the same gain as my second choice.”

    You also have to take into account the final value of the tiles that you would be leaving for the other players. Relative score is what matters, not absolute score. So, the worth of going first is contingent upon not just your net gain relative to going second (or third, or fourth…), but also the other players’ net losses.

    “Now granted, there’s chaos here, because other players going after you could mess up your calculations, and a tile can become worthless if water doesn’t flow to it (though as a first player, you’ll also often have the ability to place where there’s already water).”

    This is too important of a point to dismiss. The entire second phase of each turn is the placement of new water links, with cash incentives thrown in by the other players. Without knowledge of what is going to happen in this phase, there is no way to know if the supposedly calculable value of your plantation will come to fruition.

    “Nonetheless, the math is so close to the surface, that it seems pretty clear that a careful calculator will do better than a gut-feeler.”

    The careful calculator may do better at calculating the present value of a particular plantation tile, but a gut-feeler more skilled in manipulating the other players’ incentives will ultimately do better at extracting value from his or her purchases.

    It’s hard for me to comment on Boomtown without recent experience playing the game (it’s been at least a few years), but I’ll try:

    “Now, I suppose you could try and calculate a mine’s valuation if you wanted:

    Value of mine = value + (value * probability * number of expected turns remaining)

    However, probability of this sort just isn’t as natural of a calculation as the simple multiplication of Santiago. I don’t have to stop myself from doing it, and I suspect it doesn’t even cross the minds of most folks.”

    This is precisely what makes it a lesser design choice than that of Santiago. If no one does the math, it rewards the player with a better “gut feel” for probability distribution, rather than someone with skill at a more interesting part of the gameplay. On the other hand, if someone does the math, it both bogs the game down (because the math is more obfuscated than that of Santiago) and rewards the player who did the math, rather than someone with skill at a more interesting part of the gameplay.

    “Better, Boomtown further obfuscates value by giving additional points for majority control of the five different colors of mines. This results in both short-term payments from other players and a payout from the bank at the end. Thus, certain colors of mines are much more valuable to certain players, but the exact amount of that value isn’t clear.”

    This is the much better game design choice. Games with a conflict between short-term value and long-term value reward accurate projections of future game states, which is infinitely more interesting than a simple math contest.

    Your analysis of Ra is spot-on. The math is trivially easy, but ultimately secondary to other aspects of valuation (relative worth to all players, long-term versus short-term payout, accurate forecast of round length, etc.) This, and not obfuscation of the math behind the valuation, is what makes Ra interesting.

    I guess what I’m trying to say is that good game design doesn’t hide the math. Rather, it provides a context in which the math becomes meaningful beyond raw computation.

    • Though I don’t agree with all of your points, I appreciate the well-considered discussion! (Our core disagree, I think, is whether being able to play a game by gut is better or worse.)

  2. Agreed. I have a feeling that this core disagreement would put us on opposing sides of the hidden-but-trackable information debate as well. The reasoning is similar, with raw calculation being replaced by raw memorization.

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