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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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