We generally use the term “Engine-building Game” to refer to games in which players develop some construct which they leverage to gain an advantage. This game structure is ubiquitous, but in this article I will focus on a handful of classic examples, such as *Through the Ages*, *Agricola*, *Terra Mystica*, *Catan*, *Dominion* and *Race for the Galaxy*. In *Dominion*, the prototypical “deck-building game,” players add more and more powerful cards to their deck, and leverage their deck to gain more useful cards as well as cards which grant victory points. In *Race for the Galaxy*, players add to a tableau of cards which they leverage to add more powerful cards to their tableau and gain victory points chips. Both of these games are characterized by player states which grow exponentially in value as the game progresses. In this article, I will formally define engine-building games as those which exhibit exponential player state growth. Thereafter I will examine the consequences of exponential growth on the properties of games and how this poses challenges which game designers have either addressed or failed to address in well-known engine-building games.

**In Part 1, I discuss the concept of growth in games in order to define engine-building games. **In Part 2, I discuss the concept of diverging player odds and the relationship between exponential growth and player odds divergence. In Part 3, I discuss other solutions to the problem of diverging player odds.

# Growth in Games

## The Value of an Asset

For the purpose of this analysis I will model all games as being “victory point games.” This analysis will apply most directly to games which are in fact victory point games such as *Ticket to Ride*, *Dominion* or *Castles of Burgundy*, but it is not too much of a stretch to model some other games as though they were determined by latent victory points which are collapsed at the end of the game into a win/loss game state.

Let us define the “value” of an asset to be the number of victory points that an asset will gain for a player by the end of the game. For some assets this “value” is clear, such as a Province card in *Dominion* (with a value of 8 points). For most assets, however, this “value” is highly contextual and difficult to calculate.

For instance, a sheep in *Caverna* has a value of at least 1 point, since this is the value of a sheep at the end of the game. However, the value is increased depending on whether it is the player’s first sheep (in which case it is worth an extra 2 points), whether eating that sheep would allow the player to feed his family and thus avoid many negative points, and whether that sheep likely has the mate and space necessary to breed, as well as countless additional considerations. Even more abstract is the value of assets which have no victory point value, because their value depends on the leverage they provide in eventually gaining assets which do have victory point value.

Many assets have value which is dependent on time. For example, treasure cards are more valuable than Victory cards early in a game of *Dominion*, whereas near the end of the game, Treasure cards become less valuable. Here I propose a potentially informative way to decompose the value of an asset which pertains to that value’s relationship with time. First let’s look at three archetypal “kinds” of assets. (Note that these types of assets don’t usually exist in games in their purest forms, but I will use them for demonstrative purposes).

**Asset Growth Archetypes**

The first kind of asset is one which is simply worth a certain number of points at the end of the game. That includes victory point tokens (as in games like *Race for the Galaxy*, in which they cannot be spent) and Noble tiles in *Splendor*. These assets should be considered to have “constant” value – their value is invariant with respect to time. Therefore I will call them **constant-type assets**.

The next type of asset is one which generates assets of constant value at a certain rate. Consider, for instance, a card which can be exhausted once per round in order to gain a victory point. Some “Familiar” cards in *Seasons*, for instance, which steal victory points from your opponents at a relatively constant rate exhibit approximately this behavior. I will call these assets **linear-type assets** because their value has a linear relationship with the amount of time remaining in the game.

We could extend this principle uncomfortably far if we desire, by considering quadratic-type assets (those which generate linear-type assets at a constant rate), cubic-type assets (those which generate quadratic-type assets at a steady rate), and in general any sort of polynomial-type asset. However, it’s very difficult to think of assets in real games which exhibit these behaviors.

Therefore, the third but most important type of asset is the asset which generates assets of its **own** growth type. This recursively valued object sounds exotic but is actually one of the most common types of asset – consider again Treasure cards in *Dominion* which can be used to purchase treasure cards, or workers in *Agricola* which can be used to gain additional workers. The mathematical representation of this form of growth is the exponential function; indeed the value of these assets has an exponential relationship with the remaining game time.

To review, there are assets which exhibit three important different “rates of growth.” Constant (or no growth), linear growth, and exponential growth. Exponential growth is “faster” than linear growth, which is of course “faster” than no growth. I put “faster” in quotation marks because I am using it in the mathematical sense, where it means that an object with a faster growth rate will *eventually* be worth more than an object with a slower growth rate, as long there is enough time left in the game. One can certainly imagine a constant asset worth 1000 points which will never be overtaken by a linearly growing asset worth 1 point per round. However, if the game were to go on long enough, that linearly growing asset would eventually worth more, just as any exponentially growing asset would eventually be worth more than the linearly growing asset.

(For fun, we could even imagine assets which grow at a faster-than exponential rate, such as hyperbolically growing assets, which have a value that diverges to a singularity in a finite amount of time. I can’t think of an example of assets with this property in victory-point games).

Although I have given examples of assets in real games which approximately exhibit the behavior of these archetypal assets, in reality very few assets are truly constant-type, linear-type or exponential-type assets. These concepts are useful however, because we can think of most assets as some combination of those three archetypal assets. Many assets have some amount of constant value, some amount of linear value, and some amount of exponential value. While almost every asset in almost every game can technically be considered an “exponential-type” asset because they can in some small way be leveraged to gain assets of their own growth rate, often the exponential value of an asset is eclipsed by its constant or linear value.

## A **Race for the Galaxy** Example

**Race for the Galaxy**

Consider the following three Race for the Galaxy cards.

**Plague World** is rich in exponential value because the green goods (or “genes”) it produces can be sold or consumed for cards (which act as money) which allow players to add more cards to their tableau, including other cards with exponential value. It may also have some linear value, because a player can use that card once per round to gain a constant amount of victory points. It has a relatively low amount of constant value because it is worth 0 victory points at the end of the game. However, it may still have some constant value because, for instance, it may cause a player to gain points from a 6-cost development which rewards players for having genes worlds.

**Galactic Salon**, on the other hand, has low exponential value because it usually won’t generate any buying power for the player. This card is rich in linear value because it likely gains the player points every round, and it has some constant value as it is worth 2 victory points at the end of the game.

Finally, **Rebel Homeworld** has little to no exponential or linear value, but is rich in constant value since it is worth many victory points at the end of the game.

# Player Growth

## Acquisitions vs. Endowments

Before I finally define an Engine-Building game, I must first I must take a detour to differentiate between two different types of assets in victory point games. I argue that assets can be classified either as “Endowments,” or “Acquisitions.” Endowments are those assets which are bestowed upon a player just for signing up to play. Usually these endowments include some starting resources and perhaps some automatic income, but usually the most important endowments bestowed upon players are their turns. In many victory point games, turns are the most valuable asset, and can be used to acquire other assets (like resources) for free. In other games like *Terra Mystica*, turns are unlimited and players are instead limited by resources. Therefore in *Terra Mystica*, the most valuable endowments are a player’s starting resources and their access to 6 income phases.

A player in a victory point game is an agent who trades her endowments and acquisitions for other acquisitions. Acquisitions include all assets which are not endowments. If, on a player’s turn in* Le Havre*, she takes 3 wood, she has traded the turn which was endowed upon her for 3 wood. Transactions may also involve trading a combination of endowments and acquisitions such as when a player decides to buy the Marketplace building for 2 wood, thus trading her endowed turn and acquired two wood for an acquired Marketplace. In *Agricola*, by the time in a round that a player is placing his third worker, he is purely trading acquisitions, as players are only endowed with two workers.

## The Value of a Player: Types of Games

We might then take a conceptual leap and consider players themselves to be assets. The value of a player is, for the purposes of this article, the sum of the value of his endowments. As a consequence of these definitions, the value of a player’s total endowments will decrease over the course of the game as they are converted into acquisitions. Therefore, just as was the case with all non-constant assets previously described, the value of a player grows with T, the amount of time left in the game. Therefore we can ask “what type of asset is the player?” It depends on how fast a player’s value grows with T. It is then informative to categorize games by the asset type *of its players*.

Consider first a game wherein players are endowed with a number of constant valued assets proportional to T. For instance, consider a rather boring game in which players may, on their turn, gain approximately 3 points. Each turn therefore is a constant-type asset, and the number of turns a player receives is proportional to the time remaining in the game. Therefore the value of the player (their final score) is about 3T, and thus has a linear relationship with T. Therefore I propose that this archetypal game should be called a **Linear Game**.

Actually, there are some games which I believe are best modeled as linear games. Take for instance *Trajan*; although technically there are assets which have exponential functionality (extra-actions tiles), the game is rich in constant valued assets; each tile and action tends to be worth about 3-5 points regardless of the time remaining in the game. A good action will always be worth around 5 points, and a poor action will be worth around 3 points.

Now consider a game wherein players endowed with about T linear valued assets. For instance, consider a game in which a player has T turns, and players can use those turns to acquire at best linear assets which produce about 3 points per turn. Therefore the game is rich in assets of value 3T, and thus the player’s value is roughly T × 3T which is 3T². This growth rate is called “quadratic” therefore this game should be called a **quadratic game**.

Quadratic games are rarer in the wild than linear games, but in this article, I discuss at least one game –* Le Havre* – which should be modeled in this way. We could consider cubic (T³) or quartic (T⁴) games, but I surmise that none exist. (We could even consider “constant-type games,” however boring those might be).

Finally, we arrive at** exponential games**, in which players are endowed with assets which grow exponentially. Workers in *Agricola*, cards and turns in *Dominion*, and money in *Power Grid* are all exponential assets because they are invested in gaining additional exponential assets. These games can alternatively be called “investment games” or “engine building games.”

Once again, most real games do not quite simply fit into the category of “linear,” “quadratic” or “exponential” games. Most games are technically exponential games in that there exist assets which can generate assets of like type, but in many games this aspect of exponential growth is dominated by more significant constant and linear valued assets. Therefore we might describe *Trajan* as a “highly linear game” because it is rich in constant valued assets despite the existence of exponential assets, and *Dominion* as a “highly exponential game,” because it is rich in exponential valued assets.

# The Appeal of Engine-Building Games

The purpose of this article is to argue that exponential games inherently pose a challenge to game designers that they must address using at least one of the solutions I will describe in Parts 2 and 3. However, before describing in detail this “problem” of engine building games, I will take a moment to outline some of the reasons that designers are willing to take on this challenge in the first place. I propose that engine building is a successful innovation in modern board games for the following reasons.

First and most obviously, engine-building games give players a **feeling of growth** throughout the course of the game which they don’t necessarily feel in a linear game. To use a silly physics metaphor: players can’t feel velocity in games, they can only feel acceleration. Players begin with few resources and limited options and throughout the game grow their acquisitions exponentially such that by the end of the game they feel powerful relative to their starting state. Because our position near the end of the game is partially the result of our decisions, we feel we have earned our newfound power. These properties seem to be psychologically desirable, but I can only speculate as to why.

Second, engine building games have **inherently dynamic choices. **I propose that one issue game designers must address is how to present players with dynamic choices. I will write more about this general problem in the future, but in an engine building game, the values of different types of assets are dependent on time. Assets which are purely investments are preferable early in the game, pure victory assets are preferable late in the game, and other assets lie somewhere on that spectrum and their value must be assessed partially as a function of time. In fact, the calculation of when linear assets become preferable to exponential assets, and then when constant assets become preferable to linear assets often constitutes the thrust of the strategic content of an engine building game.

Third, for much the same reason, engine building games have **inherently challenging decisions. **Because assets can be invested, their value is not as easily calculable. Engine building games require players to assess the value of assets at a future time, which is especially difficult in cases where the value of the asset grows unpredictably due to player interaction and dynamic game mechanisms.

Fourth, because of the exponential nature of engine-building games, they can result in **impressive and surprising outcomes. **Because small perturbations to a player’s early game performance can result in large variability in end-game outcomes, players can achieve outcomes which seem (though perhaps misleadingly) impressive or unlikely, despite the fact that they may have arisen due to relatively minor feats. If players repeatedly play a game which exhibits highly exponential player growth, eventually they will come away with a story to tell. (“Did I ever tell you I once beat Daniel at A Feast for Odin by over 100 points?”)

Finally, engine-building games fit well in the low-player-interaction regime of modern board games. As I explain in my previous post, modern board games have de-emphasized direct player interaction and have instead developed mechanisms which provide challenging decisions to players which are largely based on their own player state. Engine-building is therefore a natural fit for modern board games because complex player states can support intricate player engines which in turn provide challenging, dynamic decisions for players that don’t entirely depend on the actions of others. Consider, for instance, how many modern games have introduced personal player boards in order to track players’ complex game states (*Agricola*, *Terra Mystica*, *Castles of Burgundy*, *Scythe*, *Terraforming Mars*). Players are now able to track ten or more dimensions of personal income and investments.

So what could be the problem? In Part 2, I discuss the problem of diverging player odds and how exponential games are likely to suffer.

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