A Statistical Analysis of Player Improvement and Single-Player High Scores

Player learning in single-player games creates a tension for high-score systems: even when players steadily improve, the probability of setting a new personal or all-time high drops rapidly as play continues. This study examines that tension through both mathematical modeling and empirical analysis, applying linear and power-law learning frameworks alongside an analytical expression for the chance of beating a previous score. Its central argument is that improvement in skill is often too slow to offset the declining odds of exceeding past results, so high-score rewards become progressively harder to earn over time.

The analysis rests on simulated data and game analytics from two mobile titles, Canabalt and Drop7 Hardcore. Canabalt exhibits roughly linear skill improvement, while Drop7 Hardcore follows a power-law learning curve. Under an independent and identically distributed (i.i.d.) baseline with no learning, the probability of beating the previous score is 1/2 and the probability of a new all-time high on game n is 1/n, with the expected number of high scores following the harmonic series, approximately ln(n) plus the Euler-Mascheroni constant. Real data closely track these predictions, with only small deviations (fitted correction factors of roughly 1.01 to 1.11) attributable to learning, whose influence diminishes as games accumulate.

Several practical conclusions follow. Periodic resets, such as rolling windows and daily or weekly leaderboards, can keep high-score rewards attainable. The probability of achieving a leaderboard placement of m out of n also holds approximately under near-i.i.d. conditions. Score streaks of length k occur with probability 1/(k+1)! regardless of the underlying score distribution, and learning effects are most visible in small leaderboards. The study further proposes a decaying "score to beat" design mechanism and reports that the probability of beating the mean is constant for exponential (about 0.368) and Rayleigh (about 0.456) score distributions, with a summary table consolidating the analytical and empirical findings.