What is the best mathematical principle upon which to base an experience system? To explore the mathematics of XP we must first look at why such systems can be useful, and then examine the properties of the mathematical functions upon which they are most commonly built.

Experience systems have their origins in tabletop war games, but became more significant when they were used as the central progression mechanic in *Dungeons & Dragons* back in 1974. Since then, their significance to tabletop role playing games has diminished to irrelevance, but they remain of central imp

This can be seen as the central transaction between the player and the game in the context of cRPGs: “Continue to play me, and you will continue to progress.”

Compare this to the deal offered by adventure games (“Solve my puzzles to progress”) or tactical and strategic games (“Try to beat me!”) and the strength of the experience system (and its equivalents) becomes more apparent.

Central to the implementation of an experience point (or XP) system is the gearing of the progression mechanic – that is, the relationship between the XP required for on

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Progression Ratios

In examining these differences, we should consider in particular the *progression ratios* – which are determined by dividing the XP required to reach on

- the
*basic progression ratio*: [XP difference between level (n+1) and level (n) / XP difference between level (n) and level (n-1)] - the
*total progression ratio*: [XP for level (n+1) / XP for level (n)].

Note that these ratios are rarely fixed values, but usually *decrease* as the game progresses as a consequence of the mathematics of the XP systems that are used.

In general, the lower the basic progression ratio the less difference between the time the player must spend to reach on

The total progression ratio on the other hand shows how the *total XP* required for a level increases. If this value is large, it implies that the total XP the player requires for a level increases significantly as the game progresses. To a reasonable approximation, this indicates that the rewards for killing monsters (the usual basic source of XP) increases significantly between levels – with the consequence that players will progress much faster through the game if they choose to stay “ahead of the curve” (to fight monsters much tougher than themselves).

We are now ready to examine some specific mathematical choices in the context of XP.

1. Exponential Progression

In an exponential progression, the relationship of XP for level (n) and level (n+1) is a fixed number e.g. in the sequence 100, 150, 225, 337 the exponential factor is 1.5.

A quirk of these exponential systems is that (in general terms) the basic and total progression ratios are the same. For instance, if the total experience increases by a factor of 1.5 per level, the basic progression ratio will also work out at 1.5. Similarly, if the experience difference per level increases by 2.0, the total experience required will rapidly approximate to 2.0.

The original *Dungeons & Dragons* mechanics were based on a broadly exponential progression with the exponential factor changing slightly by class, but being around 2.0. This element of the rules was lost when Wizards of the Coast/TSR revamped the system in 2000.

Exponential progression might be falling out of fashion in games design, in part because of the high total progression ratio it implies. In order to give the player a smooth experience, the rewards must also rise exponentially as the player’s level increases – often meaning that the game can be ‘rushed’ by staying ahead of the curve. The extent to which on

Where exponential progression does survive in cRPGs, it tends to do so by having a very low exponential factor. For example, in *RuneScape* (Jagex, 2001) the exponential factor is a relatively low 1.1 e.g. total XP required = 83, 174, 276, 388, 512, XP differences = 83, 91, 102, 112, 124. With progression ratios around 1.1, the difficulty curve of the game is relatively gentle.

2. Polynomial Progression

In a polynomial progression, the XP required for a level is proportional to level^f, where f is the polynomial factor e.g. 2 for squared, 3 for cubed etc. It is a property of these systems that both the progression factors decline as level advances, but that total progression ratios fall more slowly than basic progression factors.

An example of such a system can be found in the Dreamcast game *Armada* (Metro 3D, 1999) which uses the formula of 8 x (current level)^3 to determine the XP requirements.

Polynomial systems generally have gradation in their early levels akin to an exponential progression, and with similar problems. The difference comes in the effect at later levels. In an exponential system, the XP values become astronomical – beyond any reasonable expectation of achievement. But in a polynomial system, the difference actually gets smaller as the level increases; at higher levels, the basic progression ratio gets closer and closer to 1.0 (the total progression ratio does the same, but the effect is slightly less pronounced).

These systems are broadly speaking an improvement over strict exponential progression, but the flattening out of the curve in the upper reaches tends to lead to a flattening out of the progression process – there becomes no appreciable difference between the actions required and time taken to advance a level on

A related problem is that the lowering total progression ratio implies there will be little difference in rewards in fighting stronger foes – in effect, the player would do better to fight weaker foes (which presumably die faster) than stronger on

3. Fibonacci Progression

In a Fibonacci progression the relationship of XP for level (n) and level (n+1) is the sum of the XP for level (n) and level (n-1) e.g. 100, 200, 300, 500, 800, 1300, 2100 etc. In such a progression system, both of the progression ratios are a fixed 1.618034, that is, *the golden ratio* (also referred to by the Greek character ‘phi’).

I know of no game which uses such a progression system, but we can speculate about such a game. Given the values of the ratios are fixed, we can expect that such a game would remain relatively constant in its rate of progress – the time taken to level up would be directly proportional to the players choice of relative difficulty. If they stay ahead of the curve, they will advance more quickly and vice versa. It would, in short, inherit all the properties of a regular exponential progression mechanism.

I am forced to conclude that Fibonacci progression, while initially seeming of interest as an alternative to exponential progression is in fact just a special case of it.

4. Near Arithmetic Progression

In cases of arithmetic progression (AP), the relationship of XP for level (n) and level (n+1) is a constant value. This is never used in practice, because it contains no variability at all, but a closely related form does occur - what we might call the *near arithmetic progression* (NAP). In this, the value is not constant, but increases very slightly between levels. For example, in *World of Warcraft* (Blizzard, 2004) the differences between level XP values are (for instance) 400, 500, 700, 700, 800, 900, 900, 1100, 1100, 1200 etc - i.e. a near arithmetic progression.

When NAP is used, the progression ratios begin relatively high but fall gradually to just above 1.0. This may result in the same problems as polynomial progression in the later stages of such a game – a problem generally avoided by setting a maximum level (60 for *World of Warcraft*).

However, arithmetic progressions are much more forgiving than polynomial progression at lower levels, since their progression ratios are always relatively low. This creates a much more gentle curve than exponential or polynomial systems, and this gentle curve is becoming popular with players because it delivers rewards relatively regularly, and while advantages can be gained by pushing “ahead of the curve”, they are generally limited. (In fact, *World of Warcraft* enforces various rules on the acquisition of XP to limit the players ability to push ahead).

Conclusion

An examination of the basic patterns in the mathematics of XP seems to show that it is not the choice of mathematical system which is the key issue, but rather the setting of the constant values that determine the steepness of the experience curve. Whether it is *RuneScape* with its exponential system with a low factor of 1.1, or *World of Warcraft* with its gentle near arithmetic progression, the focus seems to be on creating smooth curves with low progression ratios. These low progression ratios mean that there is a gentle pressure on the player to advance within the game space, and that the gaps between levels are retreating on

The fact that these progression mechanics can be fiendishly addictive to players – by virtue of creating a reward schedule which trains the player towards constant activity – is perhaps the reason we see so much of their use in modern games, and we can expect that XP systems will be with us for a long time to come.

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