How Mathematical Modeling Balances Game Numbers: From Heroes to Economy

1. Modeling Numeric Balance

Numeric balance is a fundamental requirement in game design, especially for competitive titles. For example, in League of Legends each champion has attributes such as attack, defense, and skill effects that must be balanced to ensure fair matches.

Assume a hero’s overall combat power can be expressed as:

Attack power (A)

Defense power (D)

Skill effect (S)

Each attribute has a weight (w) reflecting its contribution to the comprehensive combat power.

Historical match data can be used to correlate win rates with these attributes, allowing designers to adjust the weights so that, under reasonable play, all champions have roughly equal win probabilities. Sensitivity analysis then shows how changes to a single attribute affect battle outcomes, preventing any attribute from becoming overly dominant.

2. Designing Growth Curves

Player progression often relies on well‑designed numeric growth curves. Common models include linear, exponential, and logarithmic growth.

Linear Growth

Suitable for slow‑paced, long‑term accumulation games. In Minecraft survival mode, resource gathering and building progress roughly linearly over time.

If a player gains resources at a rate of r per minute, the total resources after t minutes are:

R(t) = r·t + R₀, where R₀ is the initial amount.

Exponential Growth

Used when a game wants to emphasize increasing difficulty. In Diablo III , the experience required for each level grows exponentially.

E₀ – initial experience

k – growth rate

The exponential curve forces players to invest more time or become more efficient to level up in later stages.

Logarithmic Growth

Appropriate for systems where efficiency diminishes as progress increases. In the Civilization series, research time for new technologies decreases as the number of already‑researched technologies grows.

If T(n) denotes the time to research the n‑th technology, a typical model is T(n) = a / (b + n), where a and b are constants.

Complex games often blend these models. For instance, World of Warcraft uses linear growth for early levels, exponential for mid‑game, and logarithmic for high‑level progression to keep players engaged.

3. Probability Mechanisms

Probability is essential for loot drops, gacha, and other chance‑based systems. Using Genshin Impact ’s banner as an example, we illustrate how to design a fair probability system.

Designing Probability Distributions

Assume a banner contains N different items, and item i has drop probability p_i, with Σp_i = 1. Common practice is a tiered probability: common items have high p, rare items have low p, and a “pity” system guarantees a rare item after a certain number of attempts.

Modeling the Pity System

To prevent player churn, Genshin Impact guarantees a 5‑star item if none has been obtained in 90 draws; the 91st draw is guaranteed to be 5‑star.

This can be expressed with conditional probability: the probability of obtaining a 5‑star on draw k given no 5‑star in the previous k‑1 draws is 0 for k ≤ 90 and 1 for k = 91.

4. Practical Case: Modeling an In‑Game Economy

An economy system governs resource production, consumption, and circulation, directly affecting game balance. In Animal Crossing: New Horizons , players gather resources to craft items and trade them, forming an economic loop.

Assume three resources A, B, and C with conversion rates α and β. If the player’s consumption rate is c and production rate is p, the dynamic balance can be modeled by a differential equation such as dR/dt = p – c, where R represents the resource vector.

Solving this equation reveals long‑term trends, enabling designers to adjust production and consumption rates for a stable economy.

Numeric system design exemplifies the blend of art and science in game development. Mathematical modeling provides designers with powerful tools to create balanced characters, growth curves, probability mechanisms, and economies, ultimately delivering a richer player experience.

“The most exciting part of a game is restarting it; the most thrilling part of life is getting back up each time you fall.”