Efficiency

The Old School RuneScape Wiki also has an article on: osrsw:Efficiency

For other uses, see Efficiency (disambiguation).

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Efficiency is a measure of the cost per experience of a training method, factoring in opportunity cost (the amount of profit that was sacrificed by pursuing the training method instead of spending the same amount of time earning money). Efficiency is often used in the context of evaluating, based on how valuable you perceive your time to be, two different methods to train a skill for which their experience per hour and profit per hour rates are known or calculated. If one method gives more experience per hour than the other and also costs less or gives more profit, then there is no question which method is more efficient. However, if one method gives faster experience than the other but costs more or gives less profit, then the most efficient method depends on how much the player perceives their value of time to be. It is possible to calculate which method is more efficient for a player with a known value of time, or at what value of time the two methods are equally efficient.

Efficiency cannot take into account enjoyment of the game and therefore a method considered "most efficient" may not necessarily be the preferred method.

Comparing two methods

Finding the better method for a known value of time

One way to compare two methods for training a skill is by using the equation:

$E={\frac {V-P}{X}}$

where:

• E is the efficiency ratio, in coins per XP. A lower value indicates a more efficient method. (A negative value indicates an incorrect value of V, as explained in the following.)
• V is your value of time per hour. This comprises not only the profit per hour if you spent the time moneymaking, but may also include the value of experience.

  For example, if Graahk nature runes at 91+ Runecrafting is 900,000 profit per hour, someone with 91+ Runecrafting should value their time at least 900,000 per hour – but may wish to add on some additional amount to take into consideration runecrafting experience gained. (Remark: At current prices, and with a few more requirements the profit is closer to 3,700,000 per hour, but please bear with the example numbers.)

• The quantity E, with units of coins per xp, denotes how many coins one experience point of that skill is worth, assuming that coins and time are freely converted in both ways. For example, if E equals 1, then one is indifferent between 1 coin and 1 xp in that skill.
  • If E is negative, then actually, one has not correctly assigned the value of V. For example, suppose that one set V = 700,000 and wanted to compute the efficiency of Graahk nature crafting. One would compute E = (700000-900000)/(42000) or about -4.8. This is nonsensical because one (normal person) would always prefer a gain of 1 Runecrafting experience to losing 4.8 coins. A value of V = 1,100,000 (or any amount over 900,000) is plausible; it yields a rating of about 4.8.
• P is the positive profit per hour of the method being considered. If it costs money, P is negative.
• X is the experience per hour of the method being considered.

This equation primarily models methods that give experience in a single skill. Methods that give experience in multiple skills can only be used if only one type of experience is being considered - however, the experience in the other skill (ie the one not being modelled) would give the method an edge in value, compared to methods that only give experience in one skill. Thus the equation would not be as effective. The equation also cannot be used to compare training methods from different skills; it can only be used to compare methods to train the same skill.

Values used in the equation should either be used in full, or all multiplied or divided by the same constant. In practice the easiest way to use the equation is to express experience, profit, and value of time in terms of thousands (k) of coins or experience per hour. Essentially the k can be ignored then. Note that it is also possible to convert the hourly rates to some other unit (eg day) but this may result in a loss of precision.

Time efficiency example

For example, suppose that you are considering burning maple logs to train firemaking, and can burn 1200 per hour at 135 firemaking experience each. (This example completely ignores bonfires.) Maples cost 229 each. Based on these numbers, burning maples would give 162,000 firemaking experience and cost 274,800 per hour. To calculate the efficiency of this method you also need a value of time per hour. Suppose that time per hour is valued at 400,000. The efficiency rating of the method would be:

$E={\frac {400,000+229*1,200}{162,000}}=4.165$

Now suppose that you are also considering burning yew logs. Let's assume you can also burn 1200 per hour, and that yew logs give 202.5 firemaking experience each and cost 381 each. Then, burning yews would give 243,000 firemaking experience and would cost 457,200 cash per hour. For burning yews based on these numbers and also based on a 400,000 per hour value of time, the rating is:

$E={\frac {400,000-(-381*1,200)}{243,000}}=3.528$

Based on these calculations (and current prices), burning yews would be slightly more efficient than burning maples, in other words costing approximately 15.3 percent less gold per experience point. (If this quantity is negative, then at current prices, maples are actually more efficient). As one may expect, as time is valued higher, the more useful fast methods become.

Finding a value of time at which two methods are equally efficient

Now suppose that you're not sure what to value your time at, as many people don't have any specific idea. It is possible to compare two methods and get a value of time per hour at which one method becomes better than the other. This could help you choose which method to use.

The value of time at which two methods for training the same skill become equally efficient is given by the equation

$V={\frac {P_{1}X_{2}-P_{2}X_{1}}{X_{2}-X_{1}}}$

where:

• V is the value of time per hour at which the two methods are equally efficient.
• P₁ is the profit per hour of method 1.
• P₂ is the profit per hour of method 2.
• X₁ is the experience per hour of method 1.
• X₂ is the experience per hour of method 2.

Typically the faster and more expensive method is used for method 2.

Going back to the maple and yew logs example, taking maples as method 1 and yews as method 2, the two methods would be equally efficient at a value of time of 90,000 per hour. Therefore, based on these calculations, burning maple logs would be more efficient for a player who values time below this amount per hour, and burning yew logs would be more efficient for a player who values time above this amount per hour. If the result is 0 or lower, the faster method is always more efficient.

 template = :Efficiency/Calc
 form = efficiencyCalcForm
 result = efficiencyCalcResult
 param = experience1|Experience per Hour (Method 1)||int|
 param = drop1|Choose||select|Profit per Hour,Coins per XP
 param = profit1|Profit (Method 1)||number|-100000000-100000000
 param = experience2|Experience per Hour (Method 2)||int|
 param = drop2|Choose||select|Profit per Hour,Coins per XP
 param = profit2|Profit (Method 2)||number|-100000000-100000000

Please wait for the form to load. If it does not load, try Refreshing the page Checking that you have JavaScript enabled in your browser Checking your custom JavaScript does not interfere with the calculator script in any way

• Note: When inputting values into the calculator, be sure to input a negative number for methods that result in a loss.

Finding a value of time at which two methods are equally efficient during Double XP Weekends

Similar to above, we can find the value of time at which one method becomes better than the other during a Double XP Weekend.

The value of time at which two methods for training the same skill become equally efficient during Double XP Weekends is given by the equation

$V={\frac {X_{1}{(P_{1}-P_{2})}}{2(X_{2}-X_{1})}}$

Using the maple and yew logs example, they would be equally efficient at a value of time of 182,400 per hour on Double XP. Therefore, maple logs would be more efficient to burn over Double XP for a player who values their time under this amount, and yew logs would be more efficient for a player who values their time above this amount.

Why the equations work

Consider the first equation

$E={\frac {V-P}{X}}$

The units in the numerator are coins per hour. The units in the denominator are experience per hour. The units for E are therefore coins per experience. E is a measure of how much one experience costs, but also valuing time. Suppose that you can gain 250k crafting experience per hour at a loss of 1M coins per hour, and value time at 1M coins per hour. The number that most people would consider the "cost per experience" would be 1000k/250k = 4 coins per experience. However, this value does not take into account the time spent training. Instead of training, you could be making money! Essentially, by training, you are losing the opportunity cost of your value of time. In this example, to gain that 250k crafting experience, you actually would be losing 1M worth of time in addition to the 1M of coins lost. Therefore, you actually are paying 8 coins per crafting experience if both time and money are considered. This is equal to the efficiency rating for the method.

This idea is where the equation for the efficiency rating comes from. V / X is equal to the cost per experience in time. P / X is equal to the profit per experience in money, or -P /X is equal to the cost per experience in money. Therefore, the total cost per one experience, considering both money and the opportunity cost of time, is equal to

$E={\frac {V}{X}}-{\frac {P}{X}}={\frac {V-P}{X}}$

Now consider the second equation

$V={\frac {P_{1}X_{2}-P_{2}X_{1}}{X_{2}-X_{1}}}$

This is simply a solution for value of time when setting the efficiency ratings for two methods equal to each other:

$E_{1}=E_{2}={\frac {V-P_{1}}{X_{1}}}={\frac {V-P_{2}}{X_{2}}}$

Cross-multiplying gives $X_{2}(V-P_{1})=X_{1}(V-P_{2})$

Expanding gives $X_{2}V-P_{1}X_{2}=X_{1}V-P_{2}X_{1}$

Getting V on one side gives $X_{2}V-X_{1}V=P_{1}X_{2}-P_{2}X_{1}$

Factoring V out gives $V(X_{2}-X_{1})=P_{1}X_{2}-P_{2}X_{1}$

Dividing gives $V={\frac {P_{1}X_{2}-P_{2}X_{1}}{X_{2}-X_{1}}}$

The equation for the Double XP value is derived in a different way separate from E. Instead we start by asking how much experience we would miss out on (ΔX) and how much money we would save (ΔP) in an hour by doing method 1 instead of method 2 on Double XP. Then we find the number of hours (H) of method 1 outside of Double XP required to make up for ΔX. Then ΔP can be divided by H to give the profit per hour (V) at which they are equally efficient during Double XP.

To get this we start with $\Delta {X=X_{2}-X_{1}}$and $\Delta {P=P_{1}-P_{2}}$

Experience will be doubled , so we multiply ΔX by 2 giving $\Delta {X_{D}=2(X_{2}-X_{1})}$

Now we divide ΔX_D by X₁ giving $H={\frac {2(X_{2}-X_{1})}{X_{1}}}$

Dividing ΔP by H gives $V={\frac {P_{1}-P_{2}}{({\frac {2(X_{2}-X_{1})}{X_{1}}})}}$

Multiplying by ${\frac {X_{1}}{X_{1}}}$ gives $V={\frac {X_{1}{(P_{1}-P_{2})}}{2(X_{2}-X_{1})}}$