2026 Winter Omakase: Mathematics in League of Legends
Introduction
Games are, at their core, virtual worlds constructed on countless numbers and operations. Players immerse themselves in flashy graphics and satisfying hit feedback, but beneath that, intricate formulas mesh like gears to support game balance. We often hold the linear intuition that “if a stat doubles, the effect will double.” However, game designers sometimes intentionally twist that intuition or introduce complex nonlinear functions. The main course of this “2026 Winter Omakase” is the stat design of League of Legends.
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Ability Haste
In League of Legends (LoL), Ability Haste is a stat that reduces an ability’s cooldowns. With Ability Haste, players can use abilities more frequently and gain an advantage in combat. The formula for cooldown reduction as a function of Ability Haste is as follows. If the base cooldown is $t$ and Ability Haste is $\text{AH}$, the new cooldown $t'$ is calculated as follows.1
$$ t' = t \times \frac{100}{100 + \text{AH}} $$
Therefore, the cooldown reduction (CDR) $\text{CDR} (\%)$ can be computed as:
$$ \text{CDR} = \left(1 - \frac{t'}{t}\right) \times 100 = \left(1 - \frac{100}{100 + \text{AH}}\right) \times 100 = \frac{\text{AH}}{100 + \text{AH}} \times 100 $$
If you have a desired cooldown reduction, substitute that value into $\text{CDR}$ in the above expression to obtain the required Ability Haste $\text{AH}$. Solving the equation for $\text{AH}$ gives:
$$ \text{AH} = \frac{ 100 \times \text{CDR}}{100 - \text{CDR}} $$
From the shape of the graph, you can see that the required Ability Haste increases exponentially with the target reduction. For example, achieving a $50\%$ cooldown reduction requires $100$ Ability Haste, whereas achieving $90\%$ reduction requires a staggering $900$ Ability Haste. This is a system designed to prevent attaining abnormally high cooldown reductions.
To get an intuitive grasp of the Ability Haste stat itself, it is easier to think in terms of uses per unit time rather than cooldown. Since the cooldown $t$ is the time it takes to use a skill once, the uses per unit time $N$ can be expressed as its reciprocal. This is analogous to frequency and period being reciprocal quantities.
$$ N = \frac{1}{t} $$
Then the relationship between uses per unit time with Ability Haste $\text{AH}$ applied ($N'$) and $N$ is:
$$ \dfrac{N'}{N} = \dfrac{t}{t'} = \dfrac{100 + \text{AH}}{100} = 1 + \dfrac{\text{AH}}{100} $$
According to the equation, if Ability Haste is $50$, the uses per unit time increase by $50\%$ (a factor of $=1.5$), and if Ability Haste is $100$, uses per unit time increase by $100\%$ (a factor of $=2$). In other words, Ability Haste is the increase rate of uses per unit time.
Armor
In LoL, Armor reduces physical damage. The formula for physical damage reduction as a function of Armor is the same form as the Ability Haste relation above. If the physical damage is $D$ and Armor is $\text{AR}$, the actual physical damage taken $D'$ and the physical damage reduction (PDR) are computed as follows.
$$ D' = D \times \frac{100}{100 + \text{AR}}, \qquad \text{PDR} = \left(1 - \frac{D'}{D}\right) \times 100 = \frac{\text{AR}}{100 + \text{AR}} \times 100 $$
By the definition of physical damage reduction $\text{PDR}$, since $\dfrac{\text{AR}}{100 + \text{AR}} \lt 1$, no matter how high Armor is raised, you cannot offset physical damage by $100\%$.
To see the efficiency of physical damage reduction with respect to Armor intuitively, differentiate it. Computing the derivative of $\text{PDR}$, by the properties of differentiation $(f/g)' = (f'g - fg')/g^2$, yields:
$$ \begin{align*} \frac{d (\text{PDR})}{d (\text{AR})} &= \left( \frac{\text{AR}}{100 + \text{AR}} \times 100 \right)^{\prime} \\ &= 100 \left( \frac{\text{AR}}{100 + \text{AR}} \right)^{\prime} \\ &= 100 \left( \frac{1 \cdot (100 + \text{AR}) - \text{AR} \cdot 1}{(100 + \text{AR})^2} \right) \\ &= 100 \left( \frac{100}{(100 + \text{AR})^2} \right) \\ &= \frac{10000}{(100 + \text{AR})^2} \end{align*} $$
This is a rational function of the form $y = \dfrac{1}{x^2}$; because the denominator contains a squared term, the incremental increase in physical damage reduction rapidly diminishes as Armor increases. Initially, Armor converts to damage reduction at the same rate, but before reaching Armor $50$ the conversion rate of Armor to PDR falls below half. When Armor is $100$, the conversion rate to PDR drops to $25\%$.
The damage reduction formula itself is similar to Ability Haste, but here there is an additional consideration: unlike cooldowns, maximum health is also something players can increase. Including magic damage would make this more complex, but for now let us consider only physical damage and health. If current health is $\text{HP}$ and physical damage reduction is $\text{PDR}$, the maximum damage you can withstand $D_{\max}$ is calculated as follows.
$$ D_{\max}(100 - \text{PDR}) = \text{HP} \implies D_{\max} = \frac{\text{HP}}{100 - \text{PDR}} $$
Thus $D_{\max}$ can be regarded as effective health considering Armor — i.e., effective HP (EHP). To compute how much effective HP actually increases when raising Armor — intuitively, how much “health” increases — compute the rate of change of $D_{\max}$.
$$ \dfrac{D_{\max, \text{new}}}{D_{\max, \text{old}}} = \dfrac{\dfrac{\text{HP}}{100 - \text{PDR}_{\text{new}}}}{\dfrac{\text{HP}}{100 - \text{PDR}_{\text{old}}}} = \dfrac{100 - \text{PDR}_{\text{old}}}{100 - \text{PDR}_{\text{new}}} $$
An Armor value of $1$ is worth roughly $20$ gold2. Suppose your current Armor is $100$ and you invest all $1,000$ gold into Armor to raise it to $150$. The calculation below shows that this is effectively equivalent to increasing health by $25\%$.
$$ \dfrac{100 - \text{PDR}_{\text{old}}}{100 - \text{PDR}_{\text{new}}} = \dfrac{100 - \operatorname{PDR}(100)}{100 - \operatorname{PDR}(150)} = \dfrac{100 - 50}{100 - 60} = \dfrac{50}{40} = 1.25 $$
Since $\text{HP}$ has a value of $2.67$ gold per $1$ points3, investing $1,000$ gold entirely into health would increase health by about $375$ points. Therefore, if your current health satisfies the following inequality, investing $1,000$ gold into health is more efficient than investing it into Armor.
$$ \dfrac{\text{HP} + 375}{\text{HP}} \gt 1.25 \implies \text{HP} \lt 1,500 $$