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Table 9.9 Sparse user-design matrix of utility-scores consisting of six users and seven game design
elements
g1
g2
g3
g4
g5
g6
g7
Ann
-
2
-
-
1
-
4
Bob
-
-
-
-
4
1
-
Col
-
3
4
2
-
-
-
Don
-
-
-
4
3
-
2
Elk
5
5
-
-
-
0
-
Flo
2
-
4
4
-
5
-
Higher scores indicate higher utility and vice versa
) = ˆ
s 2
( ˆ
s
,
s
s
.
Our goal is to find a function that minimizes the expected loss
E
[
f
]=
(
f
(
u
,
g
),
s ug )
dP
(
u
,
g
,
s ug )
where P
(
u
,
g
,
s
)
denotes the joint probability distribution on
U × G × R
.
Suppose that we know a function (ground truth) f
that minimizes the expected
loss E
. Then we are in a similar situation as in the above scenario, where each user
has explored all game design elements. The complete user-design matrix S
[
f
]
= (
s ug )
has elements of the form
s ug =
f (
u
,
g
).
We can assign each user u a game design element g u according to the following rule
g u =
argmax
g
f (
u
,
g
).
In practice, we neither know f
nor the joint probability distribution P
(
u
,
g
,
s ug )
.
Therefore, we cannot find a minimum f
of E
[
f
]
directly. Instead, we try to approx-
f
imate f
by a function
that minimizes the empirical loss
n
1
n
E
[
f
]=
1 (
f
(
u
,
g
),
s ug ).
i
=
on the basis of a sample of observed data
(
u 1 ,
g 1 ,
s 1 ),...,(
u n ,
g n ,
s n ).
The sparse user-design matrix shown in Table 9.9 is an example of a sample of
observed data.
 
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