1
p
.8
.75
pm
pe
.6
.4
.2
0
.2
.4
.6
.8
Figure 1. FPM: Equlibrium bid function(s)
d i(p) 1
Ti
p
1
2
0.958
1
(0.204)
(0)
0.812
0.952
.05
(0.307)
(0.01)
0.819
0.902
.1
(0.221)
(0.01)
0.781
0.85
.15
(0.217)
(0)
0.776
0.777
.2
(0.122)
(0.112)
0.704
0.729
.25
(0.17)
(0.102)
0.668
0.7
.3
(0.111)
(0)
0.584
0.65
.35
(0.193)
(0)
0.575
0.6
.4
(0.102)
(0)
0.543
0.542
.45
(0.022)
(0.032)
0.51
0.5
.5
(0.129)
(0)
0.45
0.45
.55
(0.108)
(0.001)
0.449
0.4
.6
(0.162)
(0)
0.354
0.349
.65
(0.075)
(0.005)
0.36
0.3
.7
(0.168)
(0)
0.249
0.25
.75
(0.119)
(0)
0.196
0.2
.8
(0.02)
(0)
0.148
0.15
.85
(0.057)
(0)
0.097
0.096
.9
(0.016)
(0.018)
0.114
0.049
.95
(0.211)
(0.007)
0.054
0
1
(0.207)
(0)
Figure 2. ICM: Average aggregate bids
0
1
3
4
1
1
(0)
(0)
0.95
0.95
(0)
(0)
0.9
0.9
(0)
(0)
0.854
0.85
(0.02)
(0)
0.8
0.775
(0)
(0.122)
0.75
0.75
(0)
(0)
0.7
0.7
(0)
(0)
0.65
0.65
(0)
(0)
0.6
0.6
(0)
(0)
0.55
0.55
(0)
(0)
0.5
0.5
(0)
(0)
0.45
0.45
(0)
(0)
0.4
0.4
(0)
(0)
0.35
0.354
(0)
(0.02)
0.3
0.3
(0)
(0)
0.252
0.248
(0.01)
(0.01)
0.2
0.2
(0)
(0)
0.15
0.15
(0)
(0)
0.1
0.098
(0)
(0.01)
0.05
0.05
(0)
(0)
0
0
(0)
(0)
(with standard deviation)
Ti
p
0
.05
.1
.15
.2
.25
.3
.35
.4
.45
.5
.55
.6
.65
.7
.75
.8
.85
.9
.95
1
1
0.941
(0.143)
0.858
(0.224)
0.871
(0.203)
0.821
(0.202)
0.885
(0.169)
0.739
(0.24)
0.804
(0.196)
0.747
(0.195)
0.69
(0.19)
0.68
(0.203)
0.632
(0.206)
0.515
(0.177)
0.476
(0.131)
0.451
(0.176)
0.381
(0.128)
0.316
(0.112)
0.22
(0.086)
0.222
(0.195)
0.108
(0.045)
0.107
(0.158))
0.051
(0.128)
2
3
0.948
(0.14)
0.947
(0.154)
0.943
(0.143)
0.95
(0.108)
0.906
(0.169)
0.894
(0.184)
0.855
(0.205)
0.802
(0.201)
0.776
(0.2)
0.669
(0.19)
0.652
(0.211)
0.638
(0.193)
0.496
(0.143)
0.458
(0.134)
0.353
(0.105)
0.283
(0.122)
0.21
(0.038)
0.156
(0.03)
0.098
(0.018)
0.059
(0.026)
0.066
(0.036)
0.944
(0.2)
0.96
(0.128)
0.945
(0.155)
0.944
(0.163)
0.918
(0.181)
0.888
(0.185)
0.863
(0.198)
0.864
(0.186)
0.832
(0.208)
0.751
(0.211)
0.692
(0.21)
0.6523
(0.211)
0.493
(0.141)
0.482
(0.168)
0.382
(0.134)
0.267
(0.07)
0.206
(0.028)
0.173
(0.133)
0.097
(0.016)
0.052
(0.01)
0.0005
(0.003)
4
0.977
(0.114)
0.972
(0.105)
0.976
(0.08)
0.959
(0.117)
0.96
(0.132)
0.944
(0.123)
0.911
(0.187)
0.904
(0.166)
0.84
(0.188)
0.821
(0.209)
0.741
(0.209)
0.634
(0.209)
0.561
(0.175)
0.513
(0.185)
0.39
(0.126)
0.286
(0.086)
0.208
(0.032)
0.163
(0.04)
0.109
(0.074)
0.051
(0.007)
0
(0)
3
Figure 3. FPM: Average aggregate bids (with standard deviation)
1
p
.8
.6
.4
d *i
T1
.2
T2
T3
T4
0
.2
.4
.6
.8
Figure 4. FPM: Evolution of aggregate bids
d i(p) 1
π0
.7
π*0
.6
T1
.5
T2
.4
T3
.3
T4
.2
.1
p
0
.2
.4
.6
.8
1
Figure 5. FPM: Evolution of the seller's profits
Ti
α
I
T1 − T4
II
T1
III
T2
IV
T3
V
T4
.968
(.006)
.921
(.016)
.995
(.003)
−1
(.0004)
.997
(.002)
β
−.966
(.006)
−.871
(.022)
−.995
(.004)
1
(.0007)
−.996
(.004)
ρ
R2
.037
0.9240
-
.108
.735
-
.044
.9873
-
0
.9997
-
0
.9919
γ
.004
(.001)
Figure 6. ICM: Panel Data Estimation
5
p
1
0.8
0.6
d *i
0.4
T1
T2
0.2
T3
T4
0.2
0.4
0.6
Figure 8. FPM: estimated bid functions
0.8
d i(p) 1
FPM
Tk
ICM
p ≥ 0.75
p ∈ [pe , 0.75)
p < pe
1
31.3
61
148.2 46
37.5
2
818.7
81
182 79
70.6
3
201200
358
146.5 98
76
Figure 9. Maximum-Likelihood estimations of µ
7
4
1325.4
401
126 150
129
17.5
f(di )
T1
p=.2
T2
17.5
p=.65
f(di )
17.5
f(di )
15
15
12.5
12.5
10
10
10
7.5
7.5
7.5
5
5
5
2.5
2.5
2.5
15
p=.8
T3
12.5
T4
0.2
0.4
0.6
0.8
di
1
0.2
0.4
0.6
0.8
Figure 10. ICM: estimated QRE distributions
di 1
0.2
0.4
0.6
0.8
di 1
p
1
0.8
0.6
0.4
d *i
T1
T2
0.2
T3
T4
0.2
0.4
0.6
0.8
Figure 11. ICM: estimated QRE average bids
d i(p) 1
1
1
T1
T2
µ=818.7
µ=31.3
0.8
0.8
0.6
0.6
0.4
0.4
d *i
0.2
Actual
0.2
QRE
0.2
0.4
0.6
0.8
1
1
0.2
0.4
0.6
0.8
1
T3
T4
µ=1325.4
µ=201200
0.8
0.8
0.6
0.6
0.4
0.4
0.2
0.2
0.2
0.4
0.6
0.8
1
1
0.2
0.4
0.6
0.8
Figure 12. FPM: evolution of estimated QRE average bid functions
1
p=.2
15
5
f(di )
T1
p=.65
4
12.5
10
7.5
3
T2
12
T3
10
T4
8
2
6
5
2.5
0.7
0.75
0.8
0.85
0.9
0.95
p=.8
14
4
1
2
0.2
0.4
0.6
0.8
Figure 13. FPM: QRE distributions
di 1
0.2
0.4
0.6
0.8
1
1
p
0.8
0.6
d *i
0.4
T1
T2
0.2
T3
T4
0.2
0.4
0.6
0.8
Figure 14. FPM: estimated QRE average bids
d i(p) 1
1
1
T1
T2
µ=79
µ=46
0.8
0.8
0.6
0.6
0.4
0.4
d *i
0.2
Actual
0.2
QRE
0.2
0.4
0.6
0.8
1
1
0.2
0.4
0.6
0.8
1
T3
T4
µ=150
µ=98
0.8
0.8
0.6
0.6
0.4
0.4
0.2
0.2
0.2
0.4
0.6
0.8
1
1
0.2
0.4
0.6
0.8
Figure 15. FPM: evolution of estimated QRE average bid functions
1