16. Balance

The system of nature, of which man is a part, tends to be self-balancing, self-adjusting, self-cleansing. Not so with technology.

—E. F. SCHUMACHER

Is chess a balanced game?

That question is impossible to answer if you do not understand what balance means in this context. The concept of balance is a highly “squishy” one—one prone to handwaving instead of definition. A person can mean many things by “balance.” One definition is that players have an even chance to win, all else being equal. However, balance can be tricky. If all players have an equal chance to win, does that speak to the game’s mechanisms or the player’s abilities? What reason does a player ever have to get better at a game if the game’s balance always gives him the same chance of winning?

Symmetry

Symmetry is a match between the options of players. In chess, both players have the same number of pieces and those pieces have identical moves. Therefore, chess is mostly symmetrical. The only asymmetry is that the white player moves first. Assuming that the white player does not get a significant advantage or penalty associated with moving first, theoretically the white player should win approximately 50 percent of the time, and the black player should win approximately 50 percent of the time. If that is true, then is chess balanced?

Try playing against a chess champion if you are a novice. According to the Elo rating system used by the United States Chess Federation, a class E player with an Elo rating of 1000 will have a probability of winning of around 1 in 318 games when playing against a class A player with a rating of 2000. If you have a 1 in 2500 chance of winning, would you say that chess is balanced?

It seems obvious to say that a balanced game is not necessarily one in which each player has an equal probability of winning. For example, Pandemic is a popular asymmetric cooperative board game where the players are trying to beat the system so that they all win together. Most of the time, the players lose, but every once in a while they win. The game does not have an exactly 50 percent win rate, yet to many, it still feels as if it is in balance because, among other reasons, it feels fair.

Symmetry is not necessarily balance. Balance is not necessarily symmetry. Why should a game be asymmetric? An asymmetric game can help players get back into flow by using the asymmetries to nudge players back into a competitive or advantageous position. In some cases, having the deck stacked against the player is part of the aesthetic, such as in games like Dark Souls and games that attempt to model real-world situations, such as Pandemic or sports games. If challenge is the targeted play aesthetic, then failure should be overweighted. In other words, players should lose more often than they win. The game should be balanced in such a way as to achieve these ends.

Following the fundamental game design directive, a single-player or co-op game where a player or players work against the system should be set up so that the players have the ability to win with effort, but not with so much effort that they give up or become frustrated. This usually means toning down the difficulty level at which a true expert (say, the designer) plays the game to adjust it for the differing skill levels of novice players.

Likewise, a game that pits players against one another should signal that players are close to one another in terms of likelihood of winning until the very end in order to keep the players in flow. Some designers attempt to model sports where the higher skilled player may show dominance early. This is fine if the goal is to be fair at the expense of player satisfaction. For instance, if I go up against a professional sumo wrestler at sumo, I expect to lose. That is fair. But a game set up this way is not fun for me to play. If your goal is player satisfaction, all players should feel “in it” until as close to the end as possible in order to provide ample opportunities for flow. This may mean introducing asymmetries.

Self-Balancing Mechanisms

Ancestral Recall is a card in the Magic: The Gathering card game. It has what, to the casual or uninitiated player, seems like a simple instruction: Draw three cards. The cost is listed in the upper right: one blue mana and the discard of the card itself (which is implied) (Figure 16.1). The card was quickly cycled out of print because it was vastly too powerful. An updated version, called Inspiration, upped its cost to one blue mana, three(!) other mana, and the discard of the card; in addition, it changed the effect to drawing only two cards. Needless to say, Ancestral Recall is dominant to Inspiration.

Figure 16.1 The original Ancestral Recall card allowed for more card drawing than Inspiration at a much cheaper mana cost.

Balance is a difficult trait to get correct. It may not even be possible to be “correct”; it may be a concept of balance maximization. Often, concerns of balance are framed as an open-ended quantitative question. How much should the sword cost? The designer could arbitrarily decide that the sword costs 100 gold. But what if that is too low? Or what if that is too high? Most of these decisions are done by gut-feel based on understanding the systems of the game and heuristics that provide clues as to what things should cost. What if balance could be determined for individual players based on their skills and values instead of being dictated by a designer?

The question of balance is another way to frame the question of price. How much something should cost is not a simple answer that you can reveal upon meditation and gut-feel. Different players have different values based on wants and needs. How much something should cost has been solved in a number of ways that you can leverage for games. Let’s take a look at two main types and some variations on one of them:

• AUCTIONS—These are the most salient example. Power Grid is a board game that uses this mechanic. In it, each player bids on power plants to fuel their energy-production empire. Some power plants are dominant over others; they are clearly out of balance with all things being equal. The #15 power plant powers three cities for 2 coal, but the #25 power plant powers five cities for 2 coal. Why would anyone ever choose to buy the #15 power plant? In Power Grid, players bid up individual power plants to the highest bidder. If #15 and #25 cost the same, the first person to have an opportunity to get #25 would clearly have an advantage. But since it goes to the highest bidder, the “balanced” price is whatever a player is willing to pay in that situation. The power plant itself does not need to be balanced; the players do the leg work.

As a designer, you can use auctions during the playtesting phase to do a kind of price discovery. Find out what prices players are willing to pay for various effects and then, when the time comes for the game to be played in a nonsupervised environment, remove the auction mechanism and hard-code the value into the game. This approach is more accurate than a blind guess, but it has the efficiency of fiat prices.

Auction mechanisms come in various types:

• The English auction, also known as the open ascending price auction, is the type used in the Power Grid example. In this auction, all bidders know the current bid and publicly bid higher and higher until only one bidder remains.

• The Dutch auction, also known as the open descending price auction, is an alternative. In it, the bidding starts at a high number and then goes down until one bidder accepts the price. In general, Dutch auctions are considerably faster than English auctions, but they also encourage less participation.

• Vickrey, also referred to as second-price auctions, are a bit more complicated. Each participant secretly bids the highest that they are willing to pay, and the winner (highest bidder) pays what the second-highest bidder bid. eBay uses a form of this type of auction. This is harder to implement in a nonautomated context.

• SUPPLY AND DEMAND—This is another method of self-balance. It can be time-consuming to set up an auction for every balanceable interaction. In Power Grid, resources become more expensive as fewer of them remain in the game. As a result of there being fewer resources, players buy the resource that provides them with the most bang-for-the-buck and balance themselves into buying efficiently.

Remember also that costs do not have to be framed in explicit terms. Money changing hands is only one type of cost. Any limited resource can be a cost. Time, soldiers, units, cards—even information—can all be costs that can be balanced by mechanism.

Progression and Numeric Relationships

The purpose of any progression mechanic is to keep the players in flow. For instance, a player might “level up” in a role-playing game so she can access bigger and scarier monsters that provide additional spectacle and challenge. I use generic role-playing game framing in my examples here, but any game’s progression is generally in response to the designer’s need to keep a player in flow given her increase in skills and her satiety with content.

Often a designer uses progression because he feels he needs to reward the players explicitly. These rewards then make the game easier for the players because they have some tie to the game’s mechanics. As the game gets easier, the players are taken out of flow, and thus the designer needs to establish some measure of progression to push the player back in the direction of challenge.

As an example, say that a sword-fighting fantasy game is fairly challenging to new players when the sword does 1 point of damage and the enemies have 5 points of life. As a reward for killing the tenth enemy, the game gives the player a new sword that does 3 points of damage. Now the enemies have to be hit only twice instead of five times to be defeated. This makes the game a lot easier. Additionally, it is likely that the player’s skills have increased. If the difficulty of the enemies does not increase in kind, the player will become bored because the fundamental game design directive was ignored.

So how much should the sword increase in power over time? Many relationships between power and time can be used, but remember that these relationships are applicable to any resource in a game: power, cost, points, and so on.

• FLAT RELATIONSHIP. This is easiest relationship to map (Figure 16.2). In a flat relationship, the output is always the same no matter what the input. This usually occurs in the case of fungible resources, where each additional resource has a set utility. Examples include that 100 coins always gain you an extra life in Super Mario Bros., and every trip around the board gains you $200 in Monopoly.

Figure 16.2 A flat relationship.

• LINEAR AND LINEAR INVERSE RELATIONSHIPS. In a linear relationship (Figure 16.3), the output scales in a linear manner based on the input. A designer uses a linear relationship if she wants the outputs to scale so that later outputs are more valuable than early ones. For instance, in the board game Power Grid, the designers wanted to create a supply-and-demand style economy and to decrease the hoarding of particular materials; they solved this problem by making it so that as players buy materials, the materials become more expensive. The first units of coal are cheap: only 1 Electro each. However, once three have been bought, the cost jumps to 2, then 3, then 4, and so on. Any player who wants to corner the market on a particular resource will have to pay dearly for it.

Figure 16.3 A linear relationship.

A linear inverse relationship follows a similar principle. However, it starts with a base value and counts down (Figure 16.4). Each output is worth less than the one before. For instance, the first player to score a monument in Roll Through the Ages gets bonus points for getting there before his opponent. Future players score less. In Le Havre, the value of boats goes down as more and more players build them.

Figure 16.4 A linear inverse relationship.

• TRIANGULAR RELATIONSHIP. Often, resources become vastly more valuable the more of them there are. Similarly, attaining a vast number of resources becomes progressively harder. For these items, a form like the triangular numbers is useful. In the triangular numbers, the gap between the numbers increases with every step. For instance, the fifth step is five higher than the fourth step. The sixth step is six higher than the fifth step (Figure 16.5).

Figure 16.5 A triangular relationship.

One example of a triangular relationship is in the popular board game Ticket to Ride. Fulfilling a route with a length of four yields 7 points. A route with a length of five yields 10. A route with a length of six yields 15 points. Since more and more resources are required to build longer and longer routes, the player is rewarded for putting in the additional work to build longer and longer routes. If a player scored only 1 point for each length of route, then it would be easier for players to accrue points by completing the shortest routes rather than triangulating (ha!) between the ease of scoring short routes and the difficulty but high reward of scoring long routes.

Another popular use of triangular-style relationships is the “level up” curves in most role-playing games. Leveling up from 1 to 2 is easy, much easier than leveling up from 29 to 30. A triangular relationship helps reflect the decreasing marginal utility of power in games, where a little bit more power should require progressively more and more resources, allowing a natural asymptote at some maximum power level.

• FIBONACCI SEQUENCE. This is a sequence of numbers similar to the triangular where the first two numbers in the series are 1 and then the next number is the result of the previous two added together: 1, 1, 2, 3, 5, 8, 13, 21, 34, 45, and so on. The Fibonacci sequence is also a good model for role-playing experience curves, especially if level 1 starts a few steps into the sequence. Realize that since the distance grows faster with every entry in the series in the Fibonacci sequence than it does with the triangular numbers, the farther you go in the series, the distance between steps will grow much more quickly.

• EXPONENTIAL RELATIONSHIP. The exponential relationship is another prototypical relationship that is generally used until the designer or team realizes how quickly it can get out of control (Figure 16.6). For example, would you rather be given $1,000,000 today or be given just a penny today and then be given double the amount every day for 30 days? For instance, a penny today, two tomorrow, and so on. At first glance, the choice seems simple: a million dollars is a lot and a penny is a little. However, because of the power of exponential doubling, the second offer is actually worth a total of $10,737,418.23.

Be very careful when using an exponential relationship because the quick escalation of values involved makes it easy to create a combo that can throw a game out of balance.

Figure 16.6 An exponential relationship.

Situations in which a player’s resources grow at a faster than linear rate are generally more appropriate for games in which the player’s opponent is a computer that does not care emotionally about balance. In the book Laws of the Game, Nobel laureate Manfred Eigen studies relationships within closed systems and determines that “exponential and hyperbolic growth result in a clear selection of one [player] unless stabilizing interactions among different [players] enforce their coexistence.”¹

¹ Eigen, M., & Winkler, R. (1981). Laws of the Game: How the Principles of Nature Govern Chance. New York: Knopf.

A Triangular Relationship Example

I have worked on a number of games that required experience tables or a list of progressively increasing quantities of challenge to receive rewards. Generally, when you need to complete such a task, you enumerate all the principles that you need to adhere to and then try your best to apply a reasonable numerical relationship that simultaneously meets all the principles.

Here’s an example: You are working on a mobile farming game. The main mechanic is to wait for a crop to grow, and then harvest it successfully without letting it wilt; if he can manage this, the player gains money and experience. The principles are as follows:

• The game will have enough content to unlock 30 player levels.

• A player with perfect coordination will be able to harvest 12 times a day. However, most will harvest on average only four times a day.

• The average player should take 28 to 30 days to reach max level.

• The perfect player should take 12 to 14 days to reach max level.

• An average player should be able to level up every three days.

Your first task is to make a spreadsheet that tracks level, experience points, and the time to reach the maximum level with an average and an expert player. For column A, use the triangular numbers to start. In the columns to the right, track the day versus the level of the average and expert players on that day. On the first pass, for simplicity, make every harvest worth 1 Experience Point (XP) (Figure 16.7). The XP will determine the player’s level.

Figure 16.7 An experience table spreadsheet structure.

Now you are able to compare what effects a simple triangular relationship has on an average player and an expert player based on the assumptions. Here you see that without any adjustments, the average player hits level 15 after a month (cell G32), and the expert player hits level 17 (cell J16). That is close on many of the requirements. Luckily, it’s easy to massage numbers in a spreadsheet. For instance, if you changed the XP per harvest from 1 to 4, you end up with a result where the average player hits level 30 on day 30. However, the expert player is at level 30 far too quickly, hitting it at day 10.

What you want is a slower advancement for the expert player. You can handle this in a number of ways, but the easiest is probably to exploit what makes the expert player different. The average player harvests four times a day and the expert player harvests more. What if the first four harvests were worth a different value than the subsequent ones?

A little spreadsheet magic allows you to change the formula so that the first four harvests point to one cell and all subsequent ones point to another. Keeping the 4 XP per harvest for the first four and making each subsequent harvest 1 XP should not change the average player at all since he harvests only four times a day, by the assumption of the average player’s behaviors listed above. Therefore, keeping the first four harvests at 4 XP keeps the average player hitting 30 at just the right time. Now you can adjust the subsequent XP to make your expert player hit 30 at the right time.

Using 2.5 XP per subsequent harvest, the table looks like Figure 16.8.

Figure 16.8 Staggered values for XP.

Decimal points are ugly, though. Although every other directive has been met, the UI designers may not wish to implement decimal points or the design may not call for something as geeky looking as high-precision values. By scaling everything by 10, which is easy in the spreadsheet, you ensure that the game balances on nice whole numbers, and it’s not nearly as obvious that the XP table is simply the triangular numbers (Figure 16.9).

Figure 16.9 Experience table spreadsheet with XP scaling. Both players hit level 30 at around the desired time.

Balance Heuristics

When working on game balance, here are a few helpful heuristics to keep in mind:

• ALWAYS CONSIDER EXTREMES. What if a player does nothing but one type of action? Is the game out of balance if people are always lucky? What if they are always unlucky; will the game still be in balance? Test all formulas with very high and very low numbers to see if they make sense.

• FIND “GOOD ENOUGH.” Very rarely is a question of game balance about finding an exact, balanced answer. Most commonly, a balance is sufficient over a range of values. The goal of determining balance is to hit upon the easiest answer that is in balance. For instance, at a damage rate of 10 damage per second (DPS), a gun is too underpowered to include, but a damage rate of 20 DPS would make the gun too overpowered. As a designer, you don’t know this innately. You have to make educated guesses supplemented by playtest feedback. Nudging a value a small amount may not result in a difference perceivable by playtesters. This is why many designers believe in a rule of doubling or halving values until they find a sufficient answer. Say the value range where most playtesters were happy was from 14 DPS to 16 DPS. If the designer started at 4 DPS and slowly nudged the gun values upward, it would take 10 revisions of 1 DPS to get to a suitable value of 14 DPS. If the designers tested 4, then 8, then 16, they reach the suitable value sooner. If they overshoot, it’s easier to determine the edges of the suitable range.

• KEEP YOUR GOALS IN MIND. Often it’s okay that something is out of balance if it doesn’t break the game’s aesthetic values. For instance, Arkham Horror is not a balanced game. Some elements are wildly more powerful than others, and some scenarios vary in difficulty and complexity. Nonetheless, many players find it enjoyable because they do not expect to win and they do expect to face something where their victory is uncertain.

You can find additional coverage of the topic of balance in Part 5, on game theory and rational decision-making.

Summary

• A symmetric game is not necessarily balanced. A balanced game is not necessarily symmetric.

• You can have players self-balance the values of game items with a few types of mechanics. Auctions are one option. Pricing by supply and demand mechanisms are others.

• Different mathematical relationships can be applied to the amount of resources gathered or lost over time. These relationships are estimates of dynamics to best preserve player flow.

• When testing for balance, always test extreme behaviors. They may not be common in players, but they can identify ways to break the game.

• Balance is rarely about solving an equation. It is more often an exercise in finding “good enough” results.