Classroom Companion: Business

18.4 
 Models for Massive Multiplayer Online Games
.
Figure 
18.6
 is a simple model for a massively multiplayer online game (MMOG) 
such as World of Warcraft. The model may also be used to analyze services in 
which the users may leave the service with a certain probability. Examples include 
social networking services and newspaper subscriptions. An individual or a player 
18.4 · Models for Massive Multiplayer Online Games

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may be in one of three possible states: potential player (B), player (P), or a player 
who has quit the game (Quitter) (Q).
There are three flows, in which it is assumed that the rate of each flow obeys the 
Bass equation:
5
New players enter the game with rate (p + qP)B.
5
Players leave the game with rate (r + sQ)P.
5
Players rejoin the game with rate (u + vP)Q.
The dotted lines in the figure show the network effect. For simplicity, we will call 
this model the BPQ model (Øverby & Audestad, 
2019
).
The coupled set of differential equations is now:
dB
dt
p qP B
 



,
dP
dt
p qP B
r
sQ P
u vP Q












,
dQ
dt
r
sQ P
u vP Q








.
Adding the three equations results in dB/dt + dP/dt + dQ/dt = 0. This leads to the 
obvious conservation law B + P + Q = N, in which N is the total population of 
potential players. As usual, the model is simplified by assuming that N is constant 
(no birth or death processes). The number of independent differential equations is 
then reduced to two since the conservation equation can be used to eliminate one 
of them. These equations can then easily be transformed into a single, rather intrac-
table, nonlinear second-order differential equation for P. There are a few cases in 
which analytic solutions can be found. However, we shall not pursue this here.
.
Figure 
18.7
is an example of a typical solution of the differential equations 
of the BPQ model. The differential equations were solved using numerical meth-
ods. The figure shows the share of the population that are potential players (B), 
Potential players
New players
Players
Quitters
Readoption
Leaving
. Fig. 18.6 Model of a massively multiplayer online game. (Authors’ own figure)