Gamification of Workplace Activities - Smart Information Systems: Computational Intelligence for Real-Life Applications - page 259

According to the empirical risk minimization principle, this approach is statisti-
cally consistent [ 33 ], meaning that the approximation

f ∗

converges to the true mini-

mum f ∗ with increasing amount of data. The learning problem consists in predicting

the missing values.
This setting reduces the gamification design problem of finding the best design
element for each user to the problem of regression learning for which a plethora of
powerful mathematical methods are available.

9.4.3 Matrix Completion

The gamification design problem as proposed in this section can be regarded as
a special case of a recommendation problem [ 16 ] for which matrix factorization
constitutes a state-of-the-art solution [ 3 , 19 , 20 ].
Matrix factorization characterizes users and game design elements by k factors
(properties) inferred from the utility-score patterns hidden in the user-design matrix
S

k

= ( s ug ) .Users u and game design elements g are associated with vectors x u ∈ R

k , respectively. The k elements in y g measure to which extent design
element g possesses these factors. Similarly, the elements in x u measure to which
extent user u prefers these factors. High correspondence between factors of user u
and factors of design element g indicate high utility. Correspondence between user
and design factors is modeled as inner product such that

and y g ∈ R

x u y g s ug ≈ (9.1)

for all known utility-scores s ug . In matrix notation, Eq. 9.1 takes the form

S ≈ X · Y ,

where X is the user matrix and Y is the game design element matrix. The rows x u of
X and the columns y g of Y describe the users u and design elements g , respectively.
Figure 9.7 illustrates how the user-design matrix S is factorized by low-rank matrices
X and Y .
Figure 9.8 shows a fictitious example of how the six users and seven game design
elements from Table 9.9 are associated to vectors from the two-dimensional vector
space

2 . The latent factors are inferred from the utility-score patterns hidden in
the user-design matrix S . In this example, the two discovered factors refer to the
preferences according to the player typology proposed by Bartle [ 2 ]. In practice,
however, there may be additional

R

( k > 2 ) or different factors, which may not be

interpretable for humans.
After all users and all game design elements have been embedded into the joint
latent factor space

k , missing values s ug of the sparse matrix S can be predicted in
a straight forward way by

R

x u y g s ug =

ˆ