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 importance to computer RPGs where they
provide a centralised progression mechanic effectively brokering the time the
player spends with the game into rewards. By relating a character’s power level
to experience (and hence indirectly to time spent playing) cRPGs allow any and
all players to progress in return for their time commitment.
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 one level and the next. This gearing can be arranged according to a number of different mathematical models, with different implications.
It is important to note that we will be
examining solely the XP values for levels in this discussion (because this data
is readily available) and not the XP rewards the player receives for, say,
killing monsters. This data is harder to come by, but we can estimate the
effect of this element, as we shall see shortly.
In examining these differences, we should
consider in particular the progression ratios – which are determined by
dividing the XP required to reach one level by the XP required to reach the
previous level. There are two such ratios of note:
- 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 one level and the next, that is, the less steep the progression curve of the game. It should be remembered that the basic progression ratio need not be constant, and in fact generally recedes as the game progresses. This doesn’t mean that progress gets faster as the game proceeds, however, owing to the cumulative effect of multipliers over a number of levels.
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 one can remain ahead of the curve determines how rushed the game can be –
in the worst case, the player can dip into a future area, kill a monster for
phenomenal XP and level up through a large part of the game in seconds. This
problem can be ameliorated by various means, however.
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 once the level becomes sufficiently high.
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 ones, as the XP difference between the two will generally not be significant next to the number of foes that must be defeated, and so racking up easy kills becomes a faster means of progression than fighting foes of equivalent level.
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
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).
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 only slightly – the next level is always in sight, encouraging the player to play just that little bit more.
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.
Feel free to share your perspective on
the experience systems of cRPGS you have played. Can you think of specific
games in which progression was too slow or too fast? Have you played games
where the struggle to advance eventually became insurmountable? I welcome your views! Those with a more mathematical bent are encouraged to comment on the mathematics, especially if I have made any errors!