Progression and Power Curves

Summary

Progression curves (also called power curves) are the mathematical relationships that govern how player power, item strength, and system resources scale over the course of a game. They are the primary tool for intransitive balance — ensuring that more powerful items genuinely cost proportionately more, and that the gap between early and late progression feels meaningful without becoming overwhelming.

Sellers (Advanced Game Design, Ch. 10) provides the most systematic treatment of progression curve types and economic balance in the game design literature.

(Sellers, Advanced Game Design, see source-advanced-game-design)

Resource types

Before defining curves, Sellers distinguishes three types of resources in game systems:

Type	Definition	Examples
Core resources	The primary resources the game is built around; always present and central to decisions	Gold in an RPG, health in a shooter, action points in a strategy game
Subsidiary resources	Support the core; enhance or enable core resource acquisition or use	Mana (enables gold-earning spells), food supply (enables army size)
Special-case resources	Rare, conditional, or situational resources that only matter in specific contexts	Quest items, unique crafting materials, limited-time seasonal currencies

Designing around clear resource type hierarchy prevents the common failure of overloading players with many resources of equal apparent importance. A game with twelve equally-weighted currencies is harder to reason about than one with two core resources and a handful of well-defined subsidiaries.

Curve types

Linear progression

Pattern: Power increases by a fixed amount per level or cost increment.

power(n) = base + n × increment

Properties: Simple and predictable. Players can reason easily about the value of advancement. The gap between level 1 and level 100 is exactly proportional to the gap between level 1 and level 2.

Risks: Becomes boring quickly — each increment feels identical. No sense of acceleration or milestone. Most games use linear progressions only for secondary attributes (e.g. minor stat bonuses) rather than for the primary power curve.

Polynomial (power) progression

Pattern: Power increases at an accelerating rate proportional to a power of the level.

power(n) = base × n^k   (where k > 1, e.g. k = 2 for quadratic)

Properties: Early levels feel small; later levels feel significant. Creates clear feelings of acceleration and milestone achievement. Most RPG XP-to-level curves are polynomial.

Risks: If the curve is too steep, early levels become trivially fast and late levels become oppressively slow (or vice versa). Requires careful calibration between the early and late game.

Exponential progression

Pattern: Power multiplies by a fixed factor per level or cost unit.

power(n) = base × factor^n   (e.g. factor = 1.5 means each level is 50% stronger)

Properties: Growth accelerates rapidly. Numbers become very large very quickly — this is the structure behind gacha games, idle games, and games with explicit “prestige” resets. The absolute gap between adjacent high levels is enormous.

Risks: Can create “number inflation” where the values lose meaning (numbers in the billions). Also produces runaway economic snowball effects if the exponential resource also generates more of the same resource. The Habitat crystal ball/doll arbitrage (see below) demonstrates how unintended exponential cross-rates create economic collapse.

Logistic (sigmoid) progression

Pattern: Follows an S-curve — slow start, rapid middle, plateau.

power(n) = max / (1 + e^(-k(n - midpoint)))

Properties: The most psychologically satisfying curve for long-form progression. Early progress is slow (the learner’s plateau), then accelerates sharply as the player becomes competent (the growth phase), then tapers off towards a cap (the mastery plateau). This matches the natural learning curve and Koster’s fun-as-learning model.

Best use: Overall game progression arc; player skill development; any system where you want a natural ceiling with a defined approach rate. Many mobile games use logistic curves for their core progression to prevent early churners and to extend mid-game engagement.

Piecewise-linear progression

Pattern: Multiple linear segments joined at breakpoints. Each segment has its own slope.

power(n) = { slope_1 × n  if n < breakpoint_1
           { slope_2 × n  if n < breakpoint_2
           { ...

Properties: Allows deliberate design of “acceleration zones” and “plateau zones” without a continuous formula. The breakpoints represent the designer’s explicit decisions about when progression should feel fast (rewards for getting past a difficult section) and when it should feel slow (milestones requiring commitment).

Best use: Games with distinct acts or phases where different progression pacing is desirable; games where designers want explicit control over the player’s long-term engagement curve without fitting a continuous function.

Near-arithmetic progression (NAP)

Pattern: Approximately linear, but with slight acceleration — each step costs slightly more than the previous one.

Properties: Feels intuitively fair and predictable. Players can estimate the cost of advancement without a calculator. Used in many tabletop RPGs (where arithmetic must be done mentally).

Best use: Games with player-facing cost tables; any system where cognitive accessibility of the progression curve matters.

Attribute weight coefficients

When parts have multiple attributes contributing to their overall power (attack, defence, speed, range), the designer must weight these attributes to calculate a consistent cost-to-power ratio across all parts.

Method:

1. Define a reference unit: a baseline part with all attributes at their default values and a defined base cost.
2. Assign a weight coefficient to each attribute representing how much that attribute contributes to overall power relative to the others (e.g. attack = 2.0, defence = 1.5, speed = 1.0).
3. The cost of a custom part = Σ (attribute_value × weight_coefficient) across all attributes.

Example (Sellers’ nautical game):

• A frigate with high attack and moderate speed: (attack × 2.0) + (speed × 1.0) = total cost.
• A galleon with high defence and cargo: (defence × 1.5) + (cargo × 0.8) = total cost.

If both ships cost the same in-game gold, their attribute-weight totals should be equal. If the frigate’s calculation produces 120 and the galleon’s produces 100, the frigate is undercosted — an intransitive balance failure.

This technique is the mathematical core of intransitive balance (see game-balance). It enables rapid detection of dominant builds before playtesting.

Hedonic fatigue

Hedonic fatigue (also called hedonic adaptation) is the psychological mechanism by which the same reward becomes less satisfying with repetition. The first gold coin is exciting; the thousandth gold coin is routine.

Design implications:

• Escalate reward intensity: rewards must grow in magnitude over time to maintain the same subjective impact. This is the primary driver of escalating progression curves rather than flat ones.
• Vary reward types: mixing reward types (currency, gear, cosmetics, story beats, unlocks) slows hedonic adaptation because the player’s expectation doesn’t calibrate to one type.
• Use surprise: unexpected rewards produce disproportionate dopamine response relative to expected rewards of the same value (see neurochemical-engagement). Variable reward schedules (see reward-systems) exploit this mechanism.
• Punctuate progression: see below.

Punctuated vs. continuous progression

Continuous progression — rewards arrive smoothly and frequently, often in small increments. Produces steady engagement but can feel undifferentiated; players may not notice individual rewards.

Punctuated progression — rewards arrive in distinct, separated bursts. The gaps between rewards create anticipation; the reward moments feel significant. The approach to a level-up becomes its own source of engagement.

Most successful long-form games use punctuated progression at a macro scale (discrete level-ups, unlocks, story chapters) layered over continuous progression at a micro scale (XP accumulating towards the next level, small currency drops during play).

The interval between punctuation points must be calibrated to the target session length: punctuation that requires 45 minutes to reach will not sustain players with 10-minute sessions.

Economic balance

Beyond individual progression curves, games with trade or exchange systems face systemic economic balance challenges. Sellers identifies three failure modes:

Inflation

Resources enter the economy faster than they are consumed. As resource supply grows, each unit of resource buys less. In a game with a fixed price table, inflation means players accumulate resources faster than they can meaningfully spend them — the economy loses its challenge.

Causes: Undertuned sources (too many resource drops); overtuned sources (quests that pay too much relative to prices); player strategies that exploit efficient farming loops.

Countermeasures: Resource sinks (recurring costs players must pay); time-limited resources; dynamic pricing (prices that rise with inflation); supply caps.

Stagnation

Resources enter the economy too slowly. Players cannot afford meaningful purchases; progress feels blocked. Stagnation is the more common failure mode for beginning designers, who underestimate how much currency the player needs to feel engaged.

Countermeasures: Increase source rates; add free daily resources; provide cheap “gateway” items that give the player a foothold.

Arbitrage

Arbitrage occurs when a player can exploit the difference between buy prices and sell prices — or between prices in different markets — to generate unlimited profit at zero risk.

The Habitat case study (Sellers, Ch. 10): In Habitat (Lucasfilm Games, 1986), an early online world, dolls cost 75 tokens in one vendor and crystal balls cost 75 tokens in another. A player discovered that dolls could be sold to the crystal ball vendor (and vice versa) — and that the sell price was 100 tokens. By buying dolls, selling them as crystal balls, and repeating, players could generate infinite wealth, destroying the game’s economy overnight.

This is the canonical arbitrage case study in game design. The lesson: price tables must be internally consistent. Every buy price and sell price must be checked against every other buy price and sell price for circular arbitrage opportunities.

Albion Online market channel (Sellers, Ch. 10): Albion Online uses explicit tier-to-power relationships:

power = 1.2^tier
rarity = 3^tier

The 1.2× base for power produces moderate growth (at tier 8, power is about 4.3× tier 1). The 3× base for rarity produces steep growth (at tier 8, a tier-8 item is 6,561× rarer than tier 1). The combination creates a market where top-tier items are both meaningfully stronger and genuinely scarce — avoiding both inflation (items are rare) and stagnation (power gain justifies investment).

Import and export price boundaries: Sellers also describes a price boundary model for multi-market economies. Items have an import price (the maximum a market will pay for an item brought in from outside) and an export price (the minimum acceptable for selling an item to be taken elsewhere). Arbitrage becomes possible when the export price of one market exceeds the import price of another — this gap is what traders exploit in well-designed market systems, turning arbitrage from an exploit into an intended mechanic.

Analytical balance

Sellers extends the concept of game balance into the domain of acquisition analytics: the metrics that determine whether a game’s progression design is commercially viable.

The key analytical categories:

Metric	Definition
FTUE	First-Time User Experience — the first 0–5 minutes. The most critical design window: most players who churn, churn here
Acquisition funnel	The conversion path from ad impression → store page → install → first session → return session
D0 retention	% of players who return after their first session (within 24 hours)
D1 retention	% returning at Day 1
D7 / D30 retention	% returning at Day 7 / Day 30. D30 of 5% is considered acceptable for mobile; 10%+ is strong
DAU / MAU stickiness	DAU ÷ MAU: the proportion of monthly active users who play daily. Higher = more habitual engagement
Conversion rate	% of players who make at least one real-money purchase
ARPU	Average Revenue Per User
LTV	Lifetime Value: total revenue expected from a player over their entire engagement lifetime
eCPU	Effective Cost Per User: total user acquisition cost divided by total users acquired

The iron equation

The fundamental viability test for any free-to-play or service game:

LTV > eCPU + Ops

Where:

• LTV = the revenue you expect to generate from a player over their lifetime
• eCPU = the cost to acquire that player (advertising, influencer spend, store featuring)
• Ops = the ongoing operational cost per player (server costs, support, content updates)

If LTV ≤ eCPU + Ops, the game loses money per player acquired — no volume of growth will fix this. The iron equation must be solved before scaling acquisition spend.

Progression design’s role in the iron equation: progression pacing directly controls LTV. A game that churns all players by Day 7 has LTV approaching zero. A game that sustains players through Day 30 has an order-of-magnitude better LTV. This is why retention (D1/D7/D30) is the most important analytical metric for live games — it is the leading indicator of LTV.

(Sellers, Advanced Game Design, Ch. 10; see also game-analytics for the full metrics framework)

In practice

A practical progression design workflow:

1. Choose curve types per resource: use sigmoid for primary player progression, polynomial for item power, linear for secondary attributes.
2. Define a reference unit and calculate attribute weight coefficients for all items.
3. Build a spreadsheet: one row per item/level, one column per attribute, plus a derived “total cost” column.
4. Check for arbitrage: for every exchange rate in the game, verify that no circular path from resource A back to resource A generates a profit.
5. Plot the curve: visualise the power-to-level relationship. Does the slope feel right? Are there unintended plateaus or acceleration spikes?
6. Playtest retention: after basic balancing, measure how far players actually progress. D1 and D7 drop-off points reveal where progression pacing fails in practice.

Open questions

• Sigmoid curves are psychologically satisfying but mathematically complex to tune. Is there a practical heuristic for setting the midpoint and steepness parameters for a given game length?
• The iron equation assumes LTV can be projected accurately from early retention data. How reliable is this projection in the first few weeks of launch, before long-term retention is measurable?
• Punctuated progression is well established for single-player games. For live-service games with perpetual progression, how do designers prevent hedonic fatigue from making all content eventually feel routine?

Related

• game-balance — Progression curves are the central tool for intransitive balance
• internal-economy — Sources, drains, and converters are the economic layer that progression curves govern
• game-parts-and-attributes — First/second/third-order attributes are what progression curves scale
• reward-systems — Reward schedules (variable interval, fixed ratio) interact with hedonic fatigue
• neurochemical-engagement — Hedonic fatigue maps to dopaminergic habituation
• game-analytics — Analytical balance metrics: retention, LTV, ARPU, iron equation
• fun-as-learning — Koster’s learning curve maps to the sigmoid progression arc
• dark-patterns — Exploitation of hedonic fatigue can produce manipulative progression systems
• source-advanced-game-design