Understanding repeat playing behavior in casual games using a Bayesian data augmentation approach

Abstract

With an estimated market size of nearly $18 billion in 2016, casual games (games played over social networks or mobile devices) have become increasingly popular. Because most casual games are free to install, understanding repeat playing behavior is important for game developers as it directly drives advertising revenue. Game developers are keenly interested in benchmarking their game versus the market average, and understanding how genre and various game mechanics drive repeat playing behavior. Such cross-sectional analysis, however, is difficult to conduct because individual-level data on competitors’ games are not publicly available, and that the casual gaming industry is highly fragmented with each firm making only a handful of games.

I develop a Bayesian approach, based on a parsimonious Hidden Markov Model at the individual level in conjunction with data augmentation, to study repeat playing behavior using only publicly available data. After applying the proposed approach to a sample of 379 casual games, I find that the average daily attrition rate across game is around 36.5%, with an average “play” rate of 47.9%, resulting in an average ARPU (average revenue per user) across games of around 20.5 cents. Certain genres are linked to higher attrition rates and play rates. In addition, giving out a “daily bonus” or limiting the amount of time that gamers can play each day are associated with a 17.7% and 16.4% higher ARPU, respectively.

Appendix

1.1 I. MCMC computational procedure

I describe the MCMC procedure used to calibrate the model. In each iteration, I draw from the full conditional distributions of model parameters in the following order:

• $\left({S}_{(i)},{Y}_{(i)}\right),{\pi}_{i t},{\theta}_{i j},{\phi}_{i j},\left({a}_i^{\left(\pi \right)},{b}_i^{\left(\pi \right)}\right),\left({a}_i^{\left(\theta \right)},{b}_i^{\left(\theta \right)}\right),\left({a}_i^{\left(\phi \right)},{b}_i^{\left(\phi \right)}\right)$. An independent Metropolis-Hasting algorithm is used to sample each row of (S _(i), Y _(i)); given (S _(i), Y _(i)) , π _it  , θ _ij  , ϕ _ij , are drawn using a Gibbs sampler, and the hyperparameters $\left({a}_i^{\left(\pi \right)},{b}_i^{\left(\pi \right)}\right),\left({a}_i^{\left(\theta \right)},{b}_i^{\left(\theta \right)}\right)$, and $\left({a}_i^{\left(\phi \right)},{b}_i^{\left(\phi \right)}\right)$ are drawn using a random-walk Metropolis-Hastings algorithm (Gelman et al. 2013). Each step is outlined as follows.

• 1) Drawing each row of (S _(i), Y _(i)):

• For each representative gamer j (i.e., the j-th row in (S _(i), Y _(i))), I simulate a “proposed play history” (including both the time series of her play history and state transitions) using the HMM specified in Eq. [1]-[5] with the current draw of θ _ij and ϕ _ij . Then, I compute the likelihood of the proposed play history given Eq. [8] and [9], and accept or reject the new draw based on the Metropolis-Hastings acceptance probability (Gelman et al. 2013).

• 2) Drawing π _it :

• Denote the number of gamers (in the representative sample of size R) who are in the “Unaware” state at the start of the t-period by ${n}_{it}^{(U)}$, and denote the number of transitions from the “Unaware” state to the “Active” state during the t-th period by ${n}_{it}^{\left( U\to A\right)}$. Then, π _it can be sampled from a $Beta\left({a}_i^{\left(\pi \right)}+{n}_{i t}^{\left( U\to A\right)},{b}_i^{\left(\pi \right)}+{n}_{i t}^{(U)}-{n}_{i t}^{\left( U\to A\right)}\right)$ distribution.

• 3) Drawing θ _ij :

• For gamer j, denote her total number of “Active”➔“Active” transitions by ${n}_{ij}^{\left( A\to A\right)}$, and denote her total number of “Active”➔“Dead” transitions by ${n}_{ij}^{\left( A\to D\right)}$ (by definition, ${n}_{ij}^{\left( A\to D\right)}$ takes either the value of 0 or 1 since “Dead” is an absorbing state). Then, θ _ij can be sampled from a $Beta\left({a}_i^{\left(\theta \right)}+{n}_{i j}^{\left( A\to D\right)},{b}_i^{\left(\theta \right)}+{n}_{i j}^{\left( A\to A\right)}\right)$ distribution.

• 4) Drawing ϕ _ij :

• Denote the number of days that gamer j stays in the “Active” state (excluding the first day when she first becomes “Active”) by ${n}_{ij}^{(A)}$, and denote the number of days that gamer j plays the game (excluding the first day) by ${n}_{ij}^{(P)}$. Then, ϕ _ij can be sampled from a $Beta\left({a}_i^{\left(\phi \right)}+{n}_{i j}^{(P)},{b}_i^{\left(\phi \right)}+{n}_{i j}^{(A)}-{n}_{i j}^{(P)}\right)$ distribution.

• 5) Drawing $\left({a}_i^{\left(\pi \right)},{b}_i^{\left(\pi \right)}\right),\left({a}_i^{\left(\theta \right)},{b}_i^{\left(\theta \right)}\right)$, and $\left({a}_i^{\left(\phi \right)},{b}_i^{\left(\phi \right)}\right)$:

• Because standard conjugate computations are not available to sample the hyperparameters $\left({a}_i^{\left(\pi \right)},{b}_i^{\left(\pi \right)}\right),\left({a}_i^{\left(\theta \right)},{b}_i^{\left(\theta \right)}\right)$, and $\left({a}_i^{\left(\phi \right)},{b}_i^{\left(\phi \right)}\right)$, I use a random walk Metropolis-Hastings algorithm to sample from their posterior distributions. To sample $\left({a}_i^{\left(\pi \right)},{b}_i^{\left(\pi \right)}\right)$, I use a bivariate, independent Gaussian random walk proposal distribution with the mean centered on the value of the previous draw; negative draws are “reflected” off the origin. The variance of the proposal distribution is adjusted to achieve an acceptance rate close to 50% (Gelman et al. 2013). Similarly, the hyperparameters $\left({a}_i^{\left(\theta \right)},{b}_i^{\left(\theta \right)}\right)$ and $\left({a}_i^{\left(\phi \right)},{b}_i^{\left(\phi \right)}\right)$ can be sampled using an analogous procedure.