作者：Shannon Appelcline

我曾多次表示我不喜欢涉及数学元素的游戏设计。具体来说，我谈论的是《Santiago》、《Power Grid》及其他基于数学运算及需要在体验过程中进行数学分析的游戏作品。

我觉得游戏应具有娱乐性，这是我玩游戏的主要原因：进行享受，收获乐趣。进行加减乘除的运算无法让我收获众多乐趣（游戏邦注：除非是在《Primordial Soup》之类的游戏中）。

更糟的是，我日益发现，融入强烈数学元素的游戏在获胜方面存在核心缺陷。出现这一缺陷是因为体验这类游戏的玩家通常属于如下3种类型：

1. 有些玩家会完全没注意到游戏的潜在数学基础，他们会凭直觉进行操作，因为这是他们的唯一操作方式。

2. 有些玩家清楚游戏基于数学元素，但选择忽略它们，因为刻意进行计算会减少他们的游戏乐趣。

3. 最后，有些玩家会接受数学内容，基于游戏的数学基础，仔细计算每个步骤。

缺陷源自于：如果潜在数学机制颇为稳固，那么第3类玩家通常会胜出。更糟的是，这类玩家多半会仔细分析各个选项、各回合，以至于他们的游戏时间要比对手多出2-3倍。我觉得如果游戏的主要获胜渠道纯粹取决于所投入的时间，那将很难吸引玩家的眼球，而这就是多数融入数学元素的游戏作品所采用的模式。

不要误解我的意思，我并不是说缺陷在于游戏基于数学元素。相反，这是个杰出的设计模式，桌游设计大师Reiner Knizia推出的众多杰出作品就是最佳证明。这里的问题在于，将数学元素置于表面层次，将其设置成静态形式，这样玩家无需考虑附加因素就能够完成运算，获悉各操作步骤的价值。

动作游戏尤其容易陷入这一误区，因为玩家通过消耗有限资源，以获得某种形式的胜利。更糟糕的情况是，购买虚拟商品能够提高胜利点数，如果数学元素过于表面化，那么玩家就能够进行简单的同类比较。

但我们完全能够克服所有这些问题。一个最简单方式就是引入混乱的玩家互动，这样计算就取决于其他玩家的具体行动。另一方法是让玩家能够更明确地调整计算方式。第三就是设置多层次的消费和胜利抽象关系。

若干不同作品清楚说明如何巧妙运用这些方法，什么时候它们缺乏可行性。

Santiago from polarplaygames.com

《Santiago》：游戏融入系列种植地块，玩家通过投标争取地块的选择顺序（游戏邦注：通过资金，也就是胜利点数）。

在我看来，问题存在于地块的放置位置完全基于数学运算。当玩家确定地块位置时，他获得的积分=控制标记数量X同类种植标记数量。

因此我们很容易就会进行这样的运算：“这里有块好地，是双倍标记的香蕉。如果我将它买下，那么整块香蕉地的尺寸就是6到7，我的控制标记就是2到4。因此地块代表的积分是4×7-2×6=16。下块耕地只能够给我带来6个积分，所以选择前者能够带来10个积分。因此，如果我竞价10点，那么我就能够实现盈亏平衡。我突然想到，自己应该竞价4点，将其买下。我觉得这将非常值得，因为选择后者，这还是同样的游戏。”

就算如此，这里依然存在混乱元素，因为同你进行角逐的玩家会打乱你的计算工作，如果没有水资源，那么地块也就变得毫无价值（虽然首个玩家通常能够将地块放置在有水资源的地方）。尽管如此，这里的数学元素依然非常肤浅，因此显而易见，精心计算的玩家要比凭直觉操作的玩家表现更突出。

不要误解我的意思，我觉得《Santiago》是款非常不错的作品，包含众多有趣的玩法元素。但我只会在非常疲惫的状态下玩这款游戏，因为当我的大脑一团混乱时，我就不会无意识地进行所有运算。当我非常疲惫时，《Santiago》就是款非常有趣的作品。

Boomtown from droidmill.com

《新兴都市》：游戏融入系列卡片，玩家通过竞价争取第一选择权（通过资金，也就是胜利积点）。最终选择者还会获得若干现金奖励。

从基础层面来看，这款游戏和《Santiago》有些相似。你通过胜利积点（资金）竞价胜利积点（它们存在直接关联性）。但之后的游戏内容则就截然不同。

首先，《新兴都市》无需进行《Santiago》类型的运算。你就矿山进行投标，矿山的价值直接印在上面。但每座矿山还存在次级价值：如果“投入生产”，它将能够在游戏中带来收益，而这由每个回合中的2-12次骰子投掷决定。

现在你可以尝试计算矿山的价值：

矿山价值=价值+（价值X概率X预期余留回合数量）

但这里的概率运算并不像《Santiago》中的简单乘法那般自然而然。我无需阻止自己陷入这样的运算，我猜多数玩家根本就没想到这点。

值得庆幸的是，在《新兴都市》中，控制5种不同颜色的矿山能够得到额外积点，游戏凭此进一步模糊其中的价值元素。这带来玩家的短期性消费，最终是银行付款。特定颜色的矿山对某些玩家来说更有价值，但其具体的价值数量并不明确。

这种模糊性有利于游戏，因为它能够排除擅长数学运算的优势。有时我觉得《新兴都市》依然存在太多数学元素，但我更愿意并不疲惫的状态下体验这款游戏。

Ra from book-of-ra-game.net

《Ra》：游戏融入系列板块，然后玩家投标包含特殊“太阳”板块的整片地块。

首先，显而易见的是《Ra》基于数学元素。你需要查看不同道具的积分列表，它们之间的差异很大，多半是经过精心设置。但《Ra》通过引入不同层次的不确定性将这些地块的价值抽象化。

首先，和《Santiago》及《新兴都市》不同，你不是通过胜利积点进行投标。相反，你的太阳地块会影响你在未来回合的购买力（游戏邦注：但不是以直接方式）。在游戏结尾，太阳地块会被用于换取胜利积点，但纯粹是为了同其他玩家进行竞争，在这个竞争中，你到最后一刻才会清楚自己的地位，除非你仔细观察所有体验回合。

其次，有少数地块存在直接的即时积分价值，而其地块则只有投机价值。你寄希望于能够在随后的游戏中购买额外商品，进而赋予它们价值，同样，虽然这一价值在某种程度上取决于概率运算，但事实情况其实并非如此。

第三，有些胜利积点来自于玩家的比赛活动，就和《新兴都市》中的彩色矿山竞赛一样，这给计算带来众多不确定性。

第四，《Ra》的巧妙之处在于，游戏允许玩家购买整片地块，而非只能购买单个地块。在购买单个地块的过程中，玩家倾向进行计算，但面对众多地块，玩家就会迷失自我，纯粹锁定于高积分和低积分。

那么你就会依靠直觉做出决策。

有些玩家会抱怨《Ra》的随机性，我觉得这其实正是游戏的优点所在。这里存在不确定性（混乱和投机因素），但这让游戏不会成为纯粹基于数据运算的活动，在纯粹的数据运算活动中，玩家可以依靠电子表格，投入最多时间的玩家通常都能够胜出。（本文为游戏邦/gamerboom.com编译，拒绝任何不保留版权的转载，如需转载请联系：游戏邦）

Mathematics & Game Design, Part One

by Shannon Appelcline

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 Santiago, Power 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 opponent’s. I don’t find it particularly endearing for a game’s main path to victory 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 game 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 solely 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 move 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.（Source：boredgamegeeks）