Interactive self-play tuner
Game refinement & the engagement zone
What makes a game engaging? Game-refinement theory gives a surprising answer: it’s measurable. Model any game as a contest whose outcome stays uncertain over time, and you can compute a single number — the game-refinement value, GR = √B ⁄ D, from its branching factor B and its length D. Sophisticated, enduring games — chess, Go, soccer, table tennis — all cluster in a narrow comfortable zone around 0.07–0.08. Too far below and a game feels flat; too far above and it feels chaotic.
Here an abstract game plays itself hundreds of times. From the games it generates, the demo measures the average branching and length, computes GR, and places it on the spectrum beside real games. It also reads the shape of the game-progress curve — its acceleration (the “thrill”) and its jerk (the addiction signature). Tune the game and watch its value drift toward the zone — or past it.
This interactive is a faithful re-design of the research, simplified for illustrative and teaching purposes. It conveys the idea and behaviour of the model — it is not the paper’s full method, data, or results.
Top: the game-refinement value GR = √B ⁄ D on the engagement spectrum, with real games as landmarks and the comfortable zone (~0.07–0.08) shaded. Bottom: outcome certainty over the course of a game — flat-then-late means tense, straight-up means a runaway. Tune branching and pace to slide GR into the zone; balance shapes the tension, thrill, and addictiveness.
How the model works
Each self-play game is a tug-of-war: an advantage drifts toward one side at a rate set by the pace, while balance adds see-saw wobble that keeps the contest close and can force late swings. The game ends when one side is decisively ahead. As it plays, the number of meaningful choices (the effective branching) shrinks the more decided the outcome becomes. Averaging the branching B and the length D over many games gives the refinement value GR = √B ⁄ D — so a game is engaging not because it is simply long or short, but because its complexity and its length are in the right ratio.
The “motion in mind” view treats the game’s outcome certainty as a moving object. Its speed is the pace of resolution, its acceleration is the thrill, and the rate of change of acceleration — the jerk— is the signature of an addictive experience, where late, sudden swings keep players hooked. A game can sit in the engaging zone yet still be highly addictive if its jerk runs hot. This mirrors the game-refinement and motion-in-mind work in the papers below, spanning board games, sports, arcade games, and the engagement-versus-addiction distinction.
The research behind it
This demo is a teaching stand-in for a body of work on game-refinement theory and the “motion in mind” model — measuring engagement, sophistication, and addiction across games and sports.
Landmark values (chess, Go, soccer, table tennis) are representative figures from the game-refinement literature and vary by dataset. See also the full demo gallery.