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14
60
12
50
10
40
8
30
6
20
4
10
2
0
0
0
10
20
30
40
50
60
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0
2
4
6
8
10
scenarios
scenarios
Bidding Problem:
Scheduling Problem:
P =1 ,E =32
P =1 ,E = 100
Fig. 1. Time as a function of the number of scenarios
4500
4500
4000
4000
3500
3500
3000
3000
2500
2500
2000
2000
1500
1500
1
2
4
8
16
32
64
1
2
4
8
16
32
scenarios
policies
P =1, E =32
S =1, E =32
Bidding Problem
6.5 x 10 5
6.5 x 10 5
6 x 10 5
6
6
5.5
5.5
5.5
5
5
5
4.5
4.5
4.5
4
4
4
0
0
2
2
4
4
6
6
8
8
10
10
0
20
40
60
80
scenarios
scenarios
policies
P =1, E = 100
S =1, E = 100
Scheduling Problem
Fig. 2. Reward and revenue as a function of the number of scenarios and policies
tion, to find the best quality solution. Indeed, the next experiments provides evidence to
support these hypotheses.
4.2
Optimizing E , S ,and P
We tested only 243 of the 392 possible settings of the bidding parameters, and 282 of
the 12000 possible settings of the scheduling parameters because of artificially-imposed
time constraints. In the bidding problem, we ran all combinations of the parameter
settings for which we predicted the running time would be less than 3 minutes (in TAC
Travel games, hotel bidding proceeds in 1 minute rounds); in the scheduling problem,
 
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