This week, I’m going to talk about the different kinds of numbers you see in games and how to classify them. This is going to be important later, because you can’t really know how to balance a game or how to choose the right numbers unless you first know what kinds of numbers you’re dealing with. Sometimes, a balance change is as simple as replacing one kind of number with another, so understanding what kinds of numbers there are and getting an intuition for how they work is something we need to cover before anything else.
In particular, we’re going to be examining relationships between numbers. Numbers in games don’t exist in a vacuum. They only have meaning in relation to each other. For example, suppose I tell you that the main character in a game does 5 damage when he attacks. That tells you nothing unless you know how much damage enemies can take before they keel over dead. Now you have two numbers, Damage and Hit Points, and each one only has meaning in relation to the other.
Or, suppose I tell you that a sword costs 250 Gold. That has no meaning, until I tell you that the player routinely finds bags with thousands of Gold lying around the country side, and then you know the sword is cheap. Or, I tell you that the player only gets 1 Gold at most from winning each combat, and then it’s really expensive. Even within a game, the relative value of something can change; maybe 250 Gold is a lot at the start of the game but it’s pocket change at the end. In World of Warcraft, 1 Gold used to be a tidy sum, but today it takes tens or hundreds to buy the really epic loot.
With all that said, what kinds of ways can numbers be related to each other?
Identity and Linear Relationships
Probably the simplest type of relationship, which math geeks would call an identity relationship, is where two values change in exactly the same way. Add +1 to one value, it’s equivalent to adding +1 to the other. For game balance purposes, you can treat the two values as identical.
也许最简单的关系类型，也就是数学狂人所说的身份关系，就是两个价值观以完全相同的方式改变。把 + 1加到一个值上，就等于把 + 1加到另一个值上。出于游戏平衡的目的，您可以将这两个值视为相同的。
You would think that in such a case, you might just make a single value, but there are some cases where it makes sense to have two different values that just happen to have a one-to-one conversion. As an example, Ultima III: Exodus has Food, something that each character needed to not starve to death in a dungeon. You never got food as an item drop, and could only buy it from food vendors in towns. Food decreases over time, and has no other value (and cannot be sold or exchanged for anything else); its only purpose is to act as a continual slow drain on your resources. Each character also has Gold, something that they find while adventuring. Unlike food, Gold doesn’t degrade over time, and it is versatile (you can use it to bribe guards, buy hints, purchase weapons or armor… or purchase Food). While these are clearly two separate values that serve very different purposes within the game, each unit of Food costs 1 Gold (10 Food costs 10 Gold, 1000 Food costs 1000 Gold, and so on). Food and Gold have an identity relationship… although it is one-way in this case, since you can convert Gold to Food but not vice versa.
您可能认为在这种情况下，您可能只创建一个值，但在某些情况下，两个不同的值恰好有一个一对一的转换是有意义的。比如说，创世纪 III: 出埃及记有食物，每个角色都需要食物才不会饿死在地牢里。你从来没有得到食物作为一个项目下降，并只能买到它从食品摊贩在城镇。食物随着时间的推移而减少，没有其他价值(不能出售或交换其他任何东西) ; 它唯一的目的就是不断缓慢地消耗你的资源。每个角色都有黄金，这是他们在冒险时发现的东西。与食物不同，黄金不会随着时间的推移而降解，而且它是多用途的(你可以用它来贿赂守卫，购买暗示，购买武器或装甲... 或购买食物)。虽然这是两个明显不同的价值观，服务于非常不同的目的在游戏中，每个单位的食物成本1黄金(10食物成本10黄金，1000食物成本1000黄金，等等)。食物和黄金有一种身份关系... 虽然在这种情况下是单向的，因为你可以将黄金转换为食物，但反之则不然。
A more general case of an identity relationship is the linear relationship, where the conversion rate between two values is a constant. If a healing spell always costs 5 MP and heals exactly 50 HP, then there is a 1-to-10 linear relationship between MP and HP. If you can spend 100 Gold to gain +1 Dexterity, there’s a 100-to-1 linear relationship between Gold and Dexterity. And so on.
恒等关系的一个更一般的情况是线性关系，其中两个值之间的转换率是一个常数。如果一个治疗法术总是消耗5 MP 并且治疗正好50 HP，那么 MP 和 HP 之间有1-10的线性关系。如果你能花费100金币来获得 + 1灵活度，那么黄金和灵活度之间有100:1的线性关系。诸如此类。
Note that we are so far ignoring cases where a relationship is partly random (maybe that healing spell heals somewhere between 25 and 75 HP, randomly chosen each time). Randomness is something we’ll get into in a few weeks, so we’re conveniently leaving that out of the picture for now.
Exponential and Triangular Relationships
Sometimes, a linear relationship doesn’t work for your game. You may have a relationship where there are either increasing or diminishing returns.
For example, suppose a player can pay resources to gain additional actions in a turn-based strategy game. One extra action might be a small boost, but three or four extra actions might be like taking a whole extra turn — it might feel a lot more than 3 or 4 times as powerful as a single action. This would be increasing returns: each extra action is more valuable than the last. You would therefore want the cost of each extra action to increase, as you buy more of them.
Or, maybe you have a game where players have incentive to spend all of their in-game money every turn to keep pace with their opponents, and hoarding cash has a real opportunity cost (that is, they miss out on opportunities they would have had if they’d spent it instead). In this case, buying a lot of something all at once is actually not as good as buying one at a time, so it makes sense to give players a discount for “buying in bulk” as it were. Here we have a decreasing return, where each extra item purchased is not as useful as the last.
In such cases, you need a numeric relationship that increases or decreases its rate of exchange as you exchange more or less at a time. The simplest way to do this is an exponential relationship: when you add to one value, multiply the other one. An example is doubling: for each +1 you give to one value, double the other one. This gives you a relationship where buying 1, 2, 3, 4 or 5 of something costs 1, 2, 4, 8 or 16, respectively. As you can see, the numbers get really big, really fast when you do this.
在这种情况下，您需要一个数值关系，该关系随着您一次交换的多少而增加或减少其交换率。实现这一点的最简单方法是指数关系: 当你将一个值相加时，将另一个值相乘。一个例子是双倍: 对于每个 + 1你给一个值，双倍于另一个。这会给你一种关系，买1,2,3,4或5件东西分别要花费1,2,4,8或16。正如你所看到的，当你这样做的时候，数字会变得非常大，非常快。
Because the numbers get prohibitively large very quickly, you have to be careful when using exponential relationships. For example, nearly every card in any Collectible Card Game that I’ve played that has the word “double” on it somewhere (as in, one card doubles some value on another card) ends up being too powerful. I know offhand of one exception, and that was an all-or-nothing gamble where it doubled your attack strength but then made you lose at the end of the turn if you hadn’t won already! The lesson here is to be very, very careful when using exponentials.
因为这些数字很快就会变得非常大，所以在使用指数关系时你必须非常小心。例如，我玩过的任何一张交换卡片游戏中，几乎每张牌上都有“ double”这个单词(比如，一张牌在另一张牌上增加了一倍) ，结果都太强大了。我立刻知道一个例外，那是一个孤注一掷的赌博，它使你的攻击力翻倍，但是在转弯结束时，如果你还没有赢的话，你就输了！这里的教训是在使用指数时要非常非常小心。
What if you want something that increases, but not as fast as an exponential? A common pattern in game design is the triangular relationship. If you’re unfamiliar with the term, you have probably at least seen this series:
1, 3, 6, 10, 15, 21, 28, …
That is the classic triangular pattern (so called because several ways to visualize it involve triangles). In our earlier example, maybe the first extra action costs 1 resource; the next costs 2 (for a running total of 3), the next costs 3 (for a total of 6), and so on.
这就是经典的三角形模式(之所以这么叫是因为几种可视化的方法都涉及到三角形)。在我们前面的例子中，第一个额外的动作可能花费1个资源; 第二个动作花费2个资源(总共3个) ，第二个动作花费3个资源(总共6个) ，以此类推。
An interesting thing to notice about triangular numbers is when you look at the difference between each successive pair of numbers. The difference between the first two numbers (1 and 3) is 2. The difference between the next two numbers (3 and 6) is 3. The next difference (between 6 and 10) is 4. So the successive differences are linear: they follow the pattern 1, 2, 3, 4…
Triangular numbers usually make a pretty good first guess for increasing costs. What if you want a decreasing cost, where something starts out expensive and gets cheaper? In that case, figure out how much the first one should cost, then make each one after that cost 1 less. For example, suppose you decide the first Widget should cost 7 Gold. Then try making the second cost 6 Gold (for a total of 13), the third costs 5 Gold (total of 18), and so on.
对于增加成本，三角数通常是很好的第一猜测。如果你想要降低成本，让某些东西开始变得昂贵，变得更便宜，那该怎么办？在这种情况下，计算出第一个应该花多少钱，然后使每一个成本减少1。例如，假设你决定第一个 Widget 应该花费7黄金。然后尝试使第二个成本6黄金(总共13) ，第三个成本5黄金(总共18) ，等等。
Note that in this case, you will eventually reach a point where each successive item costs zero (or even negative), which gets kind of ridiculous. This is actually a pretty common thing in game balance, that if you have a math formula the game balance will break at the mathematical extremes. The design solution is to set hard limits on the formula, so that you don’t ever reach those extremes. In our Widget example above, maybe the players are simply prevented from buying more than 3 or 4 Widgets at a time.
请注意，在这种情况下，您最终将达到一个点，其中每个连续的项目成本为零(甚至负) ，这变得有点荒谬。这实际上是游戏平衡中很常见的一件事，如果你有一个数学公式，游戏平衡会在数学的极端打破。设计的解决方案是对公式设置严格的限制，这样你就永远不会达到那些极限。在我们上面的 Widget 例子中，可能只是简单地阻止玩家同时购买3或4个以上的 Widgets。
Other Numeric Relationships
While linear and triangular relationships are among the most common in games, they are not the only ones available. In fact, there are an infinite number of potential numeric relationships. If none of the typical relationships work for your game, come up with your own custom relationship!
Maybe you have certain cost peaks, where certain thresholds cost more than others because those have in-game significance. For example, if everything in your game has 5 hit points, there is actually a huge difference between doing 4 or 5 damage, so that 5th point of damage will probably cost a lot more than you would otherwise expect. You might have oscillations, where several specific quantities are particularly cheap (or expensive). You can create any ratio between two values that you want… but do so with some understanding of what effect it will have on play!
Relationships Within Systems
Individual values in a game usually exist within larger systems. By analyzing all of the different numbers and relationships between them in a game’s systems, we can gain a lot of insight into how the game is balanced.
Let us take a simple example: the first Dragon Warrior game for the NES. In the game’s combat system, you have four main stats: Hit Points (HP), Magic Points (MP), Attack and Defense. This is a game of attrition; you are exploring game areas, and every few steps you get attacked by an enemy. You lose if your HP is ever reduced to zero.
让我们举个简单的例子: NES 的第一款神龙武士游戏。在游戏的战斗系统中，你有四个主要属性: 生命值(HP) ，魔法值(MP) ，攻击和防御。这是一个消耗战游戏，你正在探索游戏区域，每走几步你就会遭到敌人的攻击。如果你的惠普减少到零，你就输了。
How are all of these numbers related? Random encounters are related to HP: each encounter reduces HP (you can also say it the other way: by walking around getting into fights, you can essentially convert HP into encounters). This is an inverse relationship, as more encounters means less HP.
这些数字之间有什么联系？随机遭遇与 HP 有关: 每次遭遇都会减少 HP (你也可以用另一种方式说: 四处走动进入战斗，你基本上可以将 HP 转化为遭遇)。这是一个相反的关系，因为更多的遭遇战意味着更少的 HP。
There’s a direct relationship between HP and Defense: the more defense you have, the less damage you take, which means your HP lasts longer. Effectively, increasing your Defense is equivalent to giving yourself a pile of extra HP.
惠普和国防之间有一个直接的关系: 你拥有的防御越多，你受到的伤害就越少，这意味着你的惠普持续时间更长。实际上，增加你的防御能力相当于给自己一堆额外的 HP。
Ironically, we see the same relationship between HP and Attack. The higher your attack stat, the faster you can defeat an enemy. If you defeat an enemy faster, that means it has less opportunity to damage you, so you take less damage. Thus, you can survive more fights with higher Attack.
MP is an interesting case, because you can use it for a lot of things. There are healing spells that directly convert MP into HP. There are attack spells that do damage (hopefully more than you’d do with a standard attack); like a higher Attack stat, these finish combats earlier, which means they preserve your HP. There are buff/debuff spells that likewise reduce the damage you take in a combat. There are teleport spells that take you across long distances, so that you don’t have to get in fights along the way, so these again act to preserve your HP. So even though MP is versatile, virtually all of the uses for it involve converting it (directly or indirectly) into HP.
MP 是一个有趣的例子，因为你可以用它做很多事情。有些治疗法术可以直接将 MP 转换成 HP。有些攻击法术可以造成伤害(希望比你用标准攻击造成的伤害更多) ，比如更高的攻击属性，这些结束战斗更早，这意味着它们可以保留你的血量。有些增益/减益法术同样可以减少你在战斗中的伤害。有一些传送法术可以让你穿越很远的距离，这样你就不用在战斗中一路走下去，所以这些法术可以保护你的 HP。因此，即使 MP 是通用的，几乎所有的用途，它涉及转换成惠普(直接或间接)。
If you draw this all out on paper, you’ll see that everything — Attack, Defense, MP, Monster Encounters — is linked directly to HP. As the loss condition for the game, the designers put the HP stat in the middle of everything! This is a common technique, making a single resource central to all of the others, and it is best to make this central resource either the win or loss condition for the game.
如果你把这些都写在纸上，你会发现所有的东西——攻击、防御、 MP、怪物遭遇战——都与惠普直接相关。作为游戏的损失条件，设计师把 HP 统计在中间的一切！这是一种常见的技术，使一种资源成为所有其他资源的中心，最好使这种中心资源成为游戏的输赢条件。
Now, there’s one additional wrinkle here: the combat system interacts with two other systems in the game through the monster encounters. After you defeat a monster, you get two things: Gold and Experience (XP). These interact with the economic and leveling systems in the game, respectively.
现在，这里还有一个额外的问题: 在怪物遭遇战中，战斗系统与游戏中的另外两个系统相互作用。在你击败了一个怪物之后，你会得到两样东西: 金牌和经验(XP)。它们分别与游戏中的经济系统和水平系统相互作用。
Let’s examine the leveling system first. Collect enough XP and you’ll level up, which increases all of your stats (HP, MP, Attack and Defense). As you can see, this creates a feedback loop: defeating enemies causes you to gain a level, which increases your stats, which lets you defeat more enemies. And in fact, this would be a positive feedback loop that would cause the player to gain high levels of power very fast, if there weren’t some kind of counteracting force in the game. That counteraction comes in the form of an increasing XP-to-Level relationship, so it takes progressively more and more XP to gain a level. Another counteracting force is that of player time; while the player could maximize their level by just staying in the early areas of the game beating on the weakest enemies, the gain is so slow that they are incentivized to take some risks so they can level a little faster.
让我们先检查一下水准系统。收集足够的经验值，你就可以升级，增加你所有的属性(HP，MP，攻击和防御)。正如你所看到的，这创造了一个反馈循环: 击败敌人会让你获得一个等级，这会增加你的属性，这会让你击败更多的敌人。事实上，如果游戏中没有某种反作用力，这将是一个正反馈循环，使玩家很快获得高等级的能量。这种反作用以增加 XP 到级别关系的形式出现，因此需要越来越多的 XP 才能获得级别。另一个反作用力是玩家时间; 玩家可以通过在游戏开始的时候击败最弱的敌人来最大化他们的等级，但是这个过程是如此的缓慢，以至于他们被激励去冒一些风险，这样他们就可以更快的升级。
Examining the economic system, Gold is used for a few things. Its primary use is to buy equipment which permanently increases the player’s Attack or Defense, thus effectively converting Gold into extra permanent HP. Gold can also be used to buy consumable items, most of which mimic the effects of certain spells, thus you can (on a limited basis, since you only have a few inventory slots) convert Gold to temporary MP. Here we see another feedback loop: defeating monsters earns Gold, which the player uses to increase their stats, which lets them defeat even more monsters. In this case, what prevents this from being a positive feedback loop is that it’s limited by progression: you have a limited selection of equipment to buy, and the more expensive stuff requires that you travel to areas that you are just not strong enough to reach at the start of the game. And of course, once you buy the most expensive equipment in the game, extra Gold doesn’t do you much good.
考察经济体系，黄金有几种用途。它的主要用途是购买设备，永久增加球员的攻击或防御，从而有效地转换黄金成额外的永久 HP。黄金也可以用来购买消耗品，其中大部分模仿某些咒语的效果，因此你可以(在有限的基础上，因为你只有几个库存插槽)将黄金转换为临时 MP。这里我们看到另一个反馈循环: 击败怪物可以获得金牌，玩家可以用金牌来增加他们的属性，这样他们就可以击败更多的怪物。在这种情况下，阻止这成为一个积极的反馈循环的原因是它受到进步的限制: 你只有有限的设备可以选择购买，更昂贵的东西需要你前往你在游戏开始时还不够强壮的地方。当然，一旦你购买了游戏中最昂贵的设备，额外的黄金不会给你带来多少好处。
Another loop that is linked to the economic system, is that of progression itself. Many areas in the game are behind locked doors, and in order to open them you need to use your Gold to purchase magic keys. You defeat monsters, get Gold, use it to purchase Keys, and use those keys to open new areas which have stronger monsters (which then let you get even more Gold/XP). Of course, this loop is itself limited by the player’s stats; unlocking a new area with monsters that are too strong to handle does not help the player much.
How would a designer balance things within all these systems? By relating everything back to the central value of HP, and then comparing.
For example, say you have a healing spell and a damage spell, and you want to know which is better. Calculate the amount of HP that the player would no longer lose as a result of using the damage spell and ending the combat earlier, and compare that to the amount of HP actually restored by the healing spell. Or, say you want to know which is better, a particular sword or a particular piece of armor. Again, figure out how much extra HP each would save you.
Now, this does not mean that everything in the game must be exactly equal to be balanced. For example, you may want spells that are learned later in the game to be more cost-effective, so that the player has reason to use them. You may also want the more expensive equipment to be less cost-effective, in order to make the player really work for it. However, at any given time in the game, you probably want the choices made available at that time to be at least somewhat balanced with each other. For example, if the player reaches a new town with several new pieces of equipment, you would expect those to be roughly equivalent in terms of their HP-to-cost ratios.
现在，这并不意味着游戏中的所有东西都必须完全相等才能平衡。例如，你可能希望在游戏后期学习的法术更具成本效益，这样玩家就有理由使用它们。你也可能希望更昂贵的设备成本效益更低，以使玩家真正为它工作。但是，在游戏中的任何时候，您都可能希望当时提供的选择至少在一定程度上彼此平衡。例如，如果玩家到达一个新的城镇与几个新的设备，你会期望这些大致相当于他们的 hp 与成本的比例。
You might wonder, if this kind of analysis works for a stat-driven game like an RPG, is it useful for any other kind of game? The answer is yes. Let’s examine an action title, the original Super Mario Bros. (made popular from the arcade and NES versions).
What kinds of resources do we have in Mario? There are lives, coins, and time (from a countdown timer). There’s actually a numeric score. And then there are objects within the game — coin blocks, enemies, and so on — which can sometimes work for or against you depending on the situation. Let us proceed to analyze the relationships.
Coins: 硬币: there is a 100-to-1 relationship between Coins and Lives, since collecting 100 coins awards an extra life. There is a 1-to-200 relationship between Coins and Score, since collecting a coin gives 200 points. There is a relationship between Coin Blocks and Coins, in that each block gives you some number of coins. 硬币与生命之间存在100比1的关系，因为收集100枚硬币意味着额外的生命。硬币和得分之间有1:200的关系，因为收集一枚硬币可以得到200分。硬币积木和硬币之间有一定的关系，每个积木都会给你一定数量的硬币
Time: 时间: 下午四时三十分: there is a 100-to-1 relationship between Time and Score, since you get a time bonus at the end of each level. There is also an inverse relationship between Time and Lives, since running out of time costs you a life. 时间和分数之间有100比1的关系，因为你在每一级结束时都会得到一个时间奖励。时间和生命之间也存在着逆向关系，因为耗尽时间会让你付出生命的代价
Enemies: 敌人: there is a relationship between Enemies and Score, since killing enemies gives you from 100 to 1000 points (Depending on the enemy). There is an inverse relationship between Enemies and Lives, since sometimes an enemy will cost you a life. (In a few select levels there is potentially a 敌人和得分之间是有关系的，因为杀死敌人可以给你100到1000分(取决于敌人)。敌人和生命之间有一个相反的关系，因为有时候敌人会让你付出生命的代价。(在一些选定的级别中，可能存在一个positive 肯定的 relationship between Enemies and Lives, as stomping enough enemies in a combo will give extra lives, but that is a special case.) 敌人和生命之间的关系，因为在一个连击中踩死足够多的敌人会带来额外的生命，但这是一个特殊情况。)
Lives: 生活: there is this strange relationship between Lives and everything else, because losing a life resets the Coins, Time and Enemies on a level. Note that since Coins give you extra Lives, and losing a Life resets Coins, any level with more than 100 Coins would provide a positive feedback loop where you could die intentionally, get more than 100 Coins, and repeat to gain infinite lives. The original 生命和其他一切事物之间存在着这种奇怪的关系，因为失去生命在某种程度上重置了硬币、时间和敌人。请注意，因为硬币给你额外的生命，而失去一个生命重置硬币，任何水平超过100个硬币将提供一个积极的反馈循环，你可以故意死亡，得到超过100个硬币，并重复获得无限的生命。原版Super Mario Bros. 超级马里奥兄弟did not have any levels like this, but 没有任何这样的水平，但Super Mario 3 超级马里奥3 did. 曾经
Relationship between Lives and Score: 生命与得分的关系: There is no 没有direct 直接的 link between Lives and Score. However, losing a Life resets a bunch of things that give scoring opportunities, so indirectly you can convert a Life to Score. Interestingly, this does not happen the other way around; unlike other arcade games of the time, you cannot earn extra Lives by getting a sufficiently high Score. 生活和得分之间的联系。然而，失去一个生命会重置一系列可以给你得分机会的东西，所以你可以间接地将一个生命转化为得分。有趣的是，这并不会反过来发生; 不像当时的其他街机游戏，你不能通过获得足够高的分数来赚取额外的生命
Looking at these relationships, we see that Score is actually the central resource in Super Mario Bros. since everything is tied to Score. This makes sense in the context of early arcade games, since the win condition is not “beat the game,” but rather, “get the highest score.”
看看这些关系，我们发现 Score 实际上是超级马里奥兄弟的核心资源，因为一切都与 Score 相关。这在早期的街机游戏中是有意义的，因为获胜的条件不是“打败游戏” ，而是“得到最高分”
How would you balance these resources with one another. There are a few ways. You can figure out how many enemies you kill and their relative risks (that is, which enemies are harder to kill and which are more likely to kill you). Compare that with how many coins you find in a typical level, and how much time you typically complete the level with. Then, you can either change the amount of score granted to the player from each of these things (making a global change throughout the game), or you can vary the number of coins and enemies, the amount of time, or the length of a level (making a local change within individual levels). Any of these techniques could be used to adjust a player’s expected total score, and also how much each of these activities (coin collecting, enemy stomping, time completion) contributes to the final score.
When you’re designing a game, note that you can change your resources around, and even eliminate a resource or change the central resource to something else. The Mario series survived this quite well; the games that followed the original eliminated Score entirely, and everything was later related to Lives.
当你设计一个游戏的时候，注意你可以改变你周围的资源，甚至消除一个资源或者把中心资源改成其他的东西。马里奥系列在这个游戏中生存得很好，之后的游戏完全取消了 scores，后来所有的游戏都和 Lives 有关。
Interactions Between Relationships
When you form chains or loops of resources and relationships between them, the relationships stack with each other. They can either combine to become more intense, or they can cancel each other out (completely or partially).
We just saw one example of this in the Mario games, with Lives and Coins. If you have a level that contains 200 Coins, then the 100 Coins to 1 Life relationship combines with 1 Life to 200 Coins in that level, to create a doubling effect where you convert 1 Life to 2 Lives in a single iteration.
Here’s another example, from the PS2 game Baldur’s Gate: Dark Alliance. In this action-RPG, you get XP from defeating enemies, which in turn causes you to level up. The XP-to-Level relationship is triangular: going from Level 1 to Level 2 requires 1000 XP, Level 2 to Level 3 costs 2000 XP, rising to Level 4 costs 3000 XP, and so on.
这是另一个例子，来自 PS2游戏《博德之门: 黑暗联盟》。在这个动作-rpg 中，你可以从击败敌人中获得经验，这反过来又会使你升级。XP 到级别的关系是三角形的: 从级别1到级别2需要1000 XP，级别2到级别3需要2000 XP，级别4需要3000 XP，以此类推。
Each time you level up, you get a number of upgrade points to spend on special abilities. These also follow a triangular progression: at Level 2 you get 1 upgrade point; at Level 3 you get 2 points; the next level gives you 3 points, then the next gives you 4 points, and so on.
每升一级，你都会得到一些升级点数，用于特殊能力的升级。这些也遵循一个三角级数: 在第二级你得到1个升级点; 在第三级你得到2个点; 下一级给你3个点，然后下一级给你4个点，以此类推。
However, these relationships chain together, since XP gives you Levels and Levels give you Upgrade Points. Since XP is the actual resource the player is earning, it is the XP-to-Points ratio we care about, and the two triangular relationships actually cancel with each other to form a linear relationship of 1000 XP to 1 Upgrade Point. While the awarding of these upgrade points is staggered based on levels, on average you are earning them at a constant XP rate.
然而，这些关系链在一起，因为 XP 给你级别和级别给你升级点。因为 XP 是玩家实际赚取的资源，它是我们所关心的 XP-to-points 比率，而这两个三角关系实际上相互抵消，形成1000 XP 比1升级点的线性关系。虽然这些升级点的奖励是基于级别交错进行的，但平均而言，你是以一个恒定的 XP 速率获得它们的。
How does Time fit into this (as in, the amount of time the player spends on the game)? If the player were fighting the same enemies over and over for the same XP rewards, there would be a triangular increase in the amount of time it takes to earn a level (and a constant amount of time to earn each Upgrade Point, on average). However, as with most RPGs, there is a system of increasing XP rewards as the player fights stronger monsters. This increasing XP curve doesn’t increase as fast as the triangular progression of level-ups, which means that it doesn’t completely cancel out the triangular effect, but it does partly reduce it — in other words, you level up slightly faster in the early game and slower in the late game, but the play time between level gains doesn’t increase as fast as a triangular relationship.
时间和这个有什么关系(比如，玩家花在游戏上的时间) ？如果玩家为了同样的经验奖励和同样的敌人一次又一次地战斗，那么获得一级所需的时间将会有一个三角增长(平均而言，获得每个升级点所需的时间是固定的)。然而，正如大多数 rpg 一样，当玩家与更强大的怪物战斗时，会有一个增加 XP 奖励的系统。XP 增长曲线的增长速度不如级别升高的三角形增长速度快，这意味着它不能完全抵消三角形效应，但它确实在一定程度上降低了这种效应ーー换句话说，你在游戏早期升得稍快，在游戏后期升得较慢，但级别升高之间的游戏时间增长速度不如三角形关系快。
Note, however, the way this interacts with Upgrade Points. Since the XP-to-Point ratio is linear, and the player gets an increasing amount of XP per unit time, they are actually getting an increasing rate of Upgrade Point gain!
但是，请注意这与升级点交互的方式。因为 XP 对点的比率是线性的，而且玩家每单位时间获得的 XP 数量越来越多，他们实际上获得的升级点增加的速度越来越快！
This kind of system has some interesting effects. By changing the rate of XP gain (that is, exactly how fast the XP rewards increase for defeating enemies) you can change both the rate of leveling up and the rate of Upgrade Point gains. If the XP rewards increase faster than the triangular rate of the levels themselves, the player will actually level up faster as the game progresses. If the XP rewards increase more slowly than the rate of level ups, the player will level faster in the early game and slower in the late game (which is usually what you want, as it gives the player frequent rewards early on and starts spacing them out once they’ve committed to continued play). If the XP rewards increase at exactly the same rate, the player will level up at a more or less constant rate.
这种系统有一些有趣的效果。通过改变 XP 增益率(即击败敌人后 XP 增益的增长速度) ，您可以同时改变升级速度和升级点增益率。如果 XP 的奖励增长速度快于关卡本身的三角速度，玩家实际上会随着游戏的进展更快地升级。如果 XP 奖励的增长速度慢于升级的速度，玩家在早期游戏中升级的速度会更快，而在后期游戏中升级的速度会更慢(这通常是你想要的，因为它会给玩家早期的频繁奖励，一旦他们决定继续玩下去，就会开始间隔时间)。如果 XP 奖励以完全相同的速度增加，玩家将以一个或多或少恒定的速度升级。
Suppose you decide to have the player gain levels faster in the early game and slower in the late game, but you never want them to go longer than an hour between levels. How would you balance the XP system? Simple: figure out what level they will be at in the late game, scale the XP gains to take about an hour per level up at that point, and then work your way backwards from there.
假设你决定让玩家在游戏早期快速升级，而在游戏后期慢速升级，但是你不希望他们在两级之间的升级时间超过一小时。您将如何平衡 XP 系统？很简单: 弄清楚他们在游戏后期的级别是什么，在那个级别上升一个小时，然后从那个级别开始向后推进。
Note another useful property this leveling system has: it provides a negative feedback loop that keeps the player in a narrow range of levels during each point in the game. Consider two situations:
Over-leveling: 水准过高: The player has done a lot of level-grinding and is now too powerful for the enemies in their current region. For one thing, they’ll be able to defeat the nearby enemies faster, so they don’t have to stick around too long. For another, the XP gains aren’t that good if their level is already high; they are unlikely to gain much in the way of additional levels by defeating weaker enemies. The maximum level a player can reach is effectively limited by the XP-reward curve. 玩家已经做了很多的水平磨削，现在是太强大的敌人在他们目前的地区。一方面，他们能够更快地击败附近的敌人，这样他们就不必逗留太久。另一方面，如果他们的等级已经很高了，那么经验值的提升就不那么好了; 他们不太可能通过击败较弱的敌人获得额外的等级。玩家可以达到的最高等级是有效地受到 xp- 奖赏曲线的限制
Under-leveling: 水平不足: Suppose instead the opposite case, where the player has progressed quickly through the game and is now at a lower level than the enemies in the current region. In this case, the XP gains will be relatively high (compared to the player’s level), and the player will only need to defeat a few enemies to level up quickly. 假设相反的情况，玩家在整个游戏中进步很快，现在比当前区域的敌人的等级要低。在这种情况下，XP 收益将相对较高(相对于玩家的水平) ，玩家将只需要击败一些敌人快速升级
In either case, the game’s system pushes the player’s level towards a narrow range in the middle of the extremes. It is much easier to balance a combat system to provide an appropriate level of challenge, when you know what level the player will be at during every step of the way!
How Relationships Interact
How do you know how two numeric relationships will stack together? Here’s a quick-reference guide:
Two linear relationships that combine: multiply them together. If you can turn 1 of Resource A into 2 Resource B, and 1 Resource B into 5 Resource C, then there is a 1-to-10 conversion between A and C (2×5). 两个线性关系的结合: 乘以他们在一起。如果你可以把资源 a 的1变成2个资源 b，把1个资源 b 变成5个资源 c，那么在 a 和 c 之间就有一个1-10的转换(2 × 5)
Linear relationship combines with an increasing (triangular or exponential) relationship: the increasing relationship just gets multiplied by a bigger number, but the nature of the curve stays the same. 线性关系与递增(三角形或指数形式)关系相结合: 递增关系只是乘以更大的数，但曲线的性质保持不变
Linear relationship counteracts an increasing relationship: if the linear conversion is large, it may dominate early on, but eventually the increasing relationship will outpace it. Exactly where the two curves meet and the game shifts from one to the other depends on the exact numbers, and tweaking these can provide an interesting strategic shift for the players. 线性关系抵消了日益增长的关系: 如果线性转换很大，它可能在早期占主导地位，但最终日益增长的关系将超过它。两条曲线在哪里交汇，游戏从一条曲线移动到另一条曲线的确切位置取决于具体的数字，调整这些曲线可以为玩家提供一个有趣的策略转变
Two increasing relationships combine: you end up with an increasing relationship that’s even faster than either of the two individually. 两种日益增长的关系结合在一起: 你最终会得到一种增长的关系，这种增长甚至比两种关系中的任何一种都要快
Two increasing relationships counteract one another: depends on the exact relationships. In general, an exponential relationship will dominate a triangular one (how fast this happens depends on the exact numbers used). Two identical relationships (such as two pure triangulars) will cancel out to form a linear or identity relationship. 两种不断增加的关系相互抵消: 取决于确切的关系。一般来说，指数关系将支配三角关系(这种关系发生的速度取决于使用的精确数字)。两个相同的关系(例如两个纯粹的三角关系)会相互抵消，形成一个线性或同一关系
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If You’re Working On a Game Now…
Are you designing your own game right now? Try this: make a list of every resource or number in your game on a piece of paper. Put a box around each, and spread the boxes out. Then, draw arrows between each set of boxes that has a direct relationship in your game, and label the arrow with the kind of relationship (linear, triangular, exponential, etc.).
Use this diagram to identify a few areas of interest in the balance of your game:
Do you see any loops where a resource can be converted to something else, then maybe something else, and then back to the original? If you get back more of the original than you started with by doing this, you may have just identified a positive feedback loop in your game. 您是否看到任何循环，在这些循环中，资源可以被转换为其他内容，然后可能是其他内容，然后返回到原始内容？如果你通过这样做得到了比你开始时更多的原创，你可能已经在你的游戏中发现了一个积极的反馈循环
Do you see a central resource that everything else seems tied to? If so, is that central resource either the win or loss condition, or does it seem kind of arbitrary? If not, does it make sense to 你是否看到了一个其他所有事物都似乎与之相关联的核心资源？如果是这样的话，这个核心资源是输是赢的条件，还是看起来有点武断？如果没有，是否有意义create 创造 a new central resource, perhaps by adding new relationships between resources? 一个新的中央资源，或许通过增加资源之间的新关系？
You can then use this diagram to predict changes to gameplay. If you change the nature of a relationship, you might be able to make a pretty good guess at what other relationships will also change as a result, and what effect that might have on the game’s systems overall.
If your game is a single-player game with some kind of progression system, “Time” (as in, the amount of time the player spends actually playing the game) should be one of your resources, and you can use your diagram to see if the rewards and power gains the player gets from playing are expected to increase, decrease, or remain constant over time.
Here’s your game balance challenge for this week. First, choose any single-player game that you’ve played and are familiar with, that has progression mechanics. Examples of games with progression are action-adventure games (Zelda), action-RPGs (Diablo), RPGs (Final Fantasy), or MMORPGs (World of Warcraft). I’ll recommend that you choose something relatively simple, such as an NES-era game or earlier. You’re going to analyze the numbers in this game, and as you’ve seen from the earlier examples here, even simple games can have pretty involved systems.
这是你本周的游戏平衡挑战。首先，选择任何你玩过并且熟悉的单人游戏，这个游戏有进阶机制。动作冒险游戏(塞尔达) ，动作 rpg (暗黑) ，rpg (最终幻想) ，或者 mmorpg (魔兽世界)。我建议您选择一些相对简单的游戏，比如 nes- 时代的游戏或更早的游戏。你们将要分析这个游戏中的数字，正如你们已经从前面的例子中看到的，即使是简单的游戏也有相当复杂的系统。
In these games, there is some kind of progression where the player gains new abilities and/or improves their stats over time. As the player progresses, enemies get stronger; again this could just mean they have higher stats, or they might also gain new abilities that require better strategy and tactics to defeat.
Start by asking yourself this question: overall, what was the difficulty curve of the game like? Did it start off easy and get slowly, progressively harder? Or, did you notice one or more of these undesirable patterns:
A series of levels that seemed to go by very slowly, because the player was underpowered at the time and did not gain enough power fast enough to compensate, so you had to grind for a long time in one spot. 一系列的关卡看起来进行得非常缓慢，因为玩家当时动力不足，而且没有足够快的速度获得足够的力量来补偿，所以你不得不在一个地方磨了很长时间
A sudden spike in difficulty with one dungeon that had much more challenging enemies than those that came immediately before or after. 在一个地下城中突然增加难度，这个地下城的敌人比前面或后面的敌人更具挑战性
A dungeon that was much easier than was probably intended, allowing you to blast through it quickly since you were much more powerful than the inhabitants by the time you actually reached it. 这是一个比预想的要容易得多的地牢，允许你快速的爆破通过它，因为当你真正到达它的时候，你比那些居民更有力量
The hardest point in the game was not at the end, but somewhere in the middle. Perhaps you got a certain weapon, ally, or special ability that was 比赛中最难的部分不是在最后，而是在中间的某个地方。也许你拥有某种武器，盟友，或者某种特殊的能力really 真的 powerful, and made you effectively unbeatable from that point on until the end of the game. 强大，并使你有效地不可战胜，从那一点，直到游戏结束
So far, all you’re doing is using your memory and intuition, and it probably takes you all of a few seconds to remember the standout moments of epic win and horrible grind in your chosen game. It’s useful to build intuition, but it is even better to make your intuition stronger by backing it up with math. So, once you’ve written down your intuitive guesses at the points where the game becomes unbalanced, let’s start analyzing.
First, seek a strategy guide or FAQ that gives all of the numbers for the game. A web search may turn up surprisingly detailed walkthroughs that show you every number and every resource in the game, and exactly how they are all related.
Next, make a list on paper of all of the resources in the game. Using the FAQ as your guide, also show all relationships between the resources (draw arrows between them, and label the arrows with the relationship type). From this diagram, you may be able to identify exactly what happened.
接下来，在纸上列出游戏中的所有资源。使用 FAQ 作为指南，还显示资源之间的所有关系(在它们之间画箭头，并用关系类型标记箭头)。从这个图表中，您可能能够确切地识别发生了什么。
For example, maybe you seemed to level up a lot in one particular dungeon, gaining a lot of power in a short time. In such a case, you might start by looking at the leveling system: perhaps there is a certain range of levels where the XP requirements to gain a level are much lower than the rest of the progression curve. You might also look at the combat reward system: maybe you just gain a lot more XP than expected from the enemies in that dungeon.
例如，也许你在一个特定的地牢里升了很多等级，在短时间内获得了很多能量。在这种情况下，您可以从查看级别系统开始: 也许在某个级别范围内，XP 获得级别的要求比其他级别要低得多。你也可以看看战斗奖励系统: 也许你只是从那个地牢里的敌人那里获得了比预期更多的经验。
As another example, maybe the game felt too easy after you found a really powerful weapon. In this case you’d look at the combat system: look at how much damage you do versus how much enemies can take, as separate curves throughout the game, and identify the sudden spike in power when you get that weapon. You may be able to graphically see the relationship of your power level versus that of the enemies over time.
Lastly, if you do identify unbalanced areas of the game from this perspective, you should be able to use your numbers and curves to immediately suggest a change. Not only will you know exactly which resource needs to be changed, but also by how much.
This exercise will probably take you a few hours, as researching a game and analyzing the numbers is not a trivial task (even for a simple game). However, after doing this, you will be much more comfortable with identifying resources and relationships in games, and also being able to use your understanding of a game’s systems to improve the balance of those systems.