Why Do Humans Build Games: A Formal Theory of Meaning Compression

The oldest question

Children play before they speak. This is not an accident. Play is the primordial meaning-making activity — phylogenetically prior to language, prior to agriculture, prior to writing. Every game ever built is an answer to the same question: how do you compress a world into rules?

We've published a formal research paper that answers this question mathematically. The paper, "Why Do Humans Build Games: A Formal Theory of Meaning Compression in Play", combines information theory, graph theory, and dynamical systems to prove that games are compressed worlds, that meaning density is bounded by the observation-action ratio, and that point-and-click adventure games are the literary maximum of the form.

This post is the companion to that paper — the intuition behind the formalism, the key theorems in plain language, and the diagrams that make the math visible.

Formal Theorems

10^43

Chess States

6-8

PnC Verbs

ω ≫ 1

PnC Obs/Act Ratio

The core idea: games as compressed worlds

A game is a tuple $\mathcal{G} = (W, R, \pi, \mu)$ where $W$ is a world state space, $R$ is a rule system, $\pi$ is a player interaction protocol, and $\mu$ is a meaning function. The key insight is that $R$ is a compression of $W$ — a compact description that generates a vast world.

ρ(G) = log₂ |V| / K(R)

Compression Ratio

Chess has ~10^43 reachable positions. Its rules fit in a few hundred bits. The compression ratio is astronomical. This is not an accident — it's the defining property of games.

"The best stories create vast worlds from sparse prose. The best games do the same — but they let you walk around in them."

From the paper

The compression pipeline

How does a game compress a world? The rule system $R$ generates the world state space $W$ through repeated application. The player interacts via protocol $\pi$, generating a trajectory $\tau$. The meaning function $\mu$ measures how much the player's experience reveals about the world's structure.

The Meaning Compression Pipeline

The Meaning Density Theorem

Meaning density is narrative entropy per unit interaction — how much story you get per click. The theorem proves it's bounded by the observation-action ratio:

D̄(G) ≤ log₂(1 + ω(G)) · log₂|Σ|

Meaning Density Bound

where $\omega(G) = |\mathcal{O}| / |\mathcal{A}|$ is the ratio of observation space to action space. When observation dominates ($\omega \gg 1$), meaning density is high. When action dominates ($\omega < 1$), meaning density is low.

Observation-Action Ratio Across Genres

The Literary Maximum Theorem

Theorem: Literary Maximum

PnC games achieve the maximum literary density among observation-primary genres

Three conditions make PnC games the literary maximum:

1. Minimal action space. The canonical PnC verb set is {examine, use, pick up, talk, open, close} — at most 6–8 verbs. No other observation-primary genre has fewer.

2. Maximal literary observation space. Every observation in a PnC game is literary by construction. Examining an object reveals narrative. Talking to an NPC reveals character. Solving a puzzle advances the plot. $\mathcal{O}_L = \mathcal{O}$.

3. 1:1 observation-action coupling. Each click produces exactly one observation. This maximizes the ratio $|\mathcal{O}_L| / |\mathcal{A}|$.

The observation-action spectrum

Genre|A| Actions|O| Observationsω = |O|/|A|Literary?FPS15-205-10~0.3NoRTS20-3015-25~0.8NoWalking Sim1-25-15~3PartialInteractive Fiction10-2050-100~5PartialPnC Adventure6-8100+≫ 10Yes (all)

The player as a dynamical system

The player's experience is modeled as a dynamical system whose state is $x(t) = (v(t), k(t), c(t))$ — current game state, knowledge set, and curiosity level. The curiosity update follows information-gap theory:

c(t+1) = c(t) + α · ΔI(t) − β · c(t)

Curiosity Dynamics

Curiosity increases when the player encounters new information ($\alpha \cdot \Delta I$) and decays over time ($-\beta \cdot c$). The player's trajectory follows the gradient of expected information gain — they explore observations in decreasing order of curiosity payoff.

Player Experience Trajectory in a PnC Game

The Flow-Boundedness Theorem proves that if puzzle difficulty is non-decreasing and skill increases with each solved puzzle, the player stays in the flow region for a bounded fraction of play time:

Pr[x(t) ∈ Flow] ≥ 1 − 2ε / Δ_max

Flow-Boundedness Guarantee

The graph structure of PnC worlds

PnC game worlds have a distinctive hub-and-spoke graph structure. Each scene is a hub; each object examination is a spoke. Within a scene, the player can examine objects in any order (high clustering). Between scenes, transitions are gated by puzzles (high modularity, low path length).

Hub-and-Spoke Observation Graph of a PnC Game

Play before language: the anthropological argument

The Primordial Precedence Theorem proves that play is phylogenetically prior to language. The cognitive capacities required for play — object permanence, symbolic substitution, metacommunicative framing — are a strict subset of those required for syntactic language. Children play before they speak. Play is the first meaning system.

Object permanence — understanding that objects exist when not perceived. Required for play (hiding games), required for language (referring to absent objects).

Symbolic substitution — treating one object as another (a banana is a phone). Required for play, required for language (words stand for things).

Metacommunicative framing — the "this is play" signal (Bateson). Required for play, required for language (context determines meaning).

Syntactic composition — combining symbols via grammar. Required for language, not required for play. A child can play tea party without grammar.

Since play requires only capacities 1–3 and language requires 1–4, play is strictly simpler. The subset relation is proper. Play comes first.

"A game where looking at a painting matters is a game where the observation space dominates the action space — and this is precisely the condition that maximizes meaning density."

The Painting Theorem

Why this matters for game design

ω ≫ 1

Maximize observation

The more distinct observations per action, the higher the meaning density. Every click should reveal something the player hasn't seen before.

6-8

Minimize verbs

Fewer actions means a higher observation-action ratio. PnC games with 6–8 verbs outperform genres with 20+ actions on literary density.

1:1

Couple observations to actions

Each interaction should produce exactly one new observation. This maximizes the ratio and ensures every click matters.

∀v

Make every observation literary

If every observation contributes to narrative, character, theme, or world-building, then the literary observation space equals the total observation space.

The painting examination

The paper's final proposition — the Painting Theorem — formalizes the paradigmatic PnC interaction: examining a painting. The player clicks on a painting. The game responds with text: a description that reveals character, advances theme, or builds the world. One action (click), one observation (text), maximum literary density.

This is what the user's point-and-click adventure does. A cyberpunk curio shop in New Tokyo, 2274. No combat. No score. Only curiosity, observation, and consequence. The formal theory proves what the design intuition already knew: this is the mathematically optimal form for generating literary meaning in games.

Read the full paper →