Complexity of Popularity and Dynamics of Within-Game Achievements in Computer Games

TL;DR

Problem: quantify how task difficulty and persistence shape achievement completion in computer games using Steam data. Approach: analyze distributions of total achievements $N$, total players $G$, and engagement metrics $F$ and $U$, fit log-normal models, and model progress decay with exponential forms $f(x)=\exp(-\alpha x)$ and $g_u(n)=G\exp(-\beta n)$. Findings: both $N$ and $G$ are well described by log-normal laws; there is a tendency toward intermediate persistence; the average fraction of players declines exponentially with progress; memoryless fragmentation explains many patterns. Significance: patterns illuminate how task fragmentation and player persistence evolve on large platforms, with potential implications for game design, player engagement, and educational applications.

Abstract

Tasks of different nature and difficulty levels are a part of people's lives. In this context, there is a scientific interest in the relationship between the difficulty of the task and the persistence need to accomplish it. Despite the generality of this problem, some tasks can be simulated in the form of games. In this way, we employ data from a large online platform, called Steam, to analyze games and the performance of their players. More specifically, we investigated persistence in completing tasks based on the proportion of players who accomplished game achievements. Overall, we present five major findings. First, the probability distribution for the number of achievements is log-normal distribution. Second, the distribution of game players also follows a log-normal. Third, most games require neither a very high degree of persistence nor a very low one. Fourth, players also prefer games that demand a certain intermediate persistence. Fifth, the proportion of players as a function of the number of achievements declines approximately exponentially. As both the log-normal and the exponential functions are memoryless, they are mathematical forms that describe random effects arising from the nature of the system. Therefore our first two findings describe random processes of fragmenting achievements and players while the last three provide a quantitative measure of the human preference in the pursuit of challenging, achievable, and justifiable tasks.

Figures (4)

• Figure 1: Probability distribution of game and player variables. Probability distributions of the number of games as function of the (A) number of achievements, (B) number of players, (C) fraction of completists, and (D) completed fraction. Panels (E) and (F) show the relationship between the fraction of completists and number of players and between the completed fraction and number of players, respectively. In panels (A), (B), (C), and (D) the markers represent the probability on the y-axis and the center of the bin on the x-axis. Dashed lines in (A) and (B) refer to fitted log-normal distributions, with parameters $\mu_A = 3.20$ [$95\%$CI: $3.18$, $3.22$] and $\sigma_A = 0.66$ [$95\%$CI: $0.64$, $0.67$], and $\mu_B = 3.96$ [$95\%$CI: $3.91$, $4.00$] and $\sigma_B = 2.18$ [$95\%$CI: $2.13$, $2.23$], respectively. In panels (E) and (F) the markers represent the average per window and the shaded area indicates the error relative to the average. In each case, the interval size was chosen to maintain $10$ markers.
• Figure 2: Exponential decay of the fraction of players as a function of the unlocked achievements. Examples of decay of the fraction of players as a function of the number of unlocked achievements for (A) completed games and (B) games where no user completed all achievements, in mono-log scale. In panel (B), we disregard achievements that no player has reached, given that we are working with the logarithmic scale. Markers refer to the data and straight lines are the corresponding exponential fits. R-squared coefficient for exponential fits according to ranking for (C) completed games and (D) games where no user completed all achievements. Colored markers correspond to R-squared values of curves shown in panels (A) and (B), and the light gray curve shows the behaviour for all games.
• Figure 3: Mean exponential decay for completed games. (A) Examples of decay of the average fraction of players as function of normalized achievements in mono-log scale. The games were grouped in windows containing varying fractions of completists, in increments of $0.1$. Within each group, a mean series was constructed with $10$ equally spaced markers. Additionally, the first point of the mean series was fixed as $(0,1)$ and the last point as $(1,\mu_{F})$, where $\mu_{F}$ is the mean of the fractions of completists in the group. At the end of this procedure, the mean series of all groups had $12$ markers. Blue markers represent the average of the group of games with fraction of completists between $0$ and $0.1$, the orange markers greater between $0.1$ and $0.2$, the green markers between $0.2$ and $0.3$ , red markers between $0.4$ and $0.5$ and purple markers between $0.7$ and $0.8$. The dashed lines represent the exponential model fitted to each different group of completists. (B) Relationship between the exponential decay rate and the fraction of completists. In this way, we calculated the mean slope $\alpha$ and the mean of the fraction of completists $F$ for each group of games. The colors correspond to the groups shown in (A), that is, the blue maker is the slope of the group of games with a fraction of completists between $0.0$ and $0.1$, the orange marker is the slope of the group of games with fraction of completists between $0.1$ and $0.2$, and so on. Gray markers represent games grouped with fractions of completists not exemplified in (A), but calculated in the same way. The gray shaded area shows a standard deviation band and the dashed line is the logarithmic function $\alpha =-\log(F)$.
• Figure 4: Relationship between the exponential decay rate multiplied by completed fraction and total players. For each game, the exponential decay rate $\alpha$ was calculated using the fraction of players on the y-axis and normalized achievement on the x-axis (all null values in the series were removed). Next, the ordered pair $(G, \alpha U)$ was taken, and the games were grouped into windows containing varying total players in increments of $10$. Within each window, the mean on the x and y axes was calculated. The gray markers represent the ordered pair of these means for each group. The gray shaded area shows standard deviation, and the dashed line represents the logarithmic function ($\alpha U = \log(G)$).