Table 2.1 the Designer Boundary Kernel
TABLES
Table 2.1 The designer boundary kernel
Functional form Kc(t) / domain in Xi / Support on t¾ [ (c1+1) – 5/4(1+2c1)(t-c1)2][t-(c1+2)]2 / [b1 , b1 + h) / [c1, c1 + 2)
¾ [(-c2+1) – 5/4(1-2c2)(t-c2)2][t-(c2 -2)]2 / (b2 – h , b2] / (c2 – 2, c2 ]
(1-t2)2 / [b1 + h , b2 – h] / [-1, 1]
------
Table 2.2 The first derivative of the designer boundary kernel
Functional form K’(t) / Domain in Xi / Support on t3/2[(c1+1) – 5/4(1+2c1)(t-c1)2][t-(c1+2)] – 15/8(1+2c1)(t-c1)][t-(c1+2)]2 / [b1 , b1 + h) / [c1, c1 + 2)
3/2[(1-c1) – 5/4(1-2c1)(t-c1)2][t-(c1-2)] – 15/8(1-2c1)(t-c1)][t-(c1-2)]2 / (b2 – h , b2] / (c2 – 2, c2 ]
-15/4(1-t2)t / [b1 + h , b2 – h] / [-1, 1]
Table 2.3. Bootstrap Results
2.3.1. First data set
Boundary Biweight Kernel Method /Gaussian Reflection Method
One Mode (h=0.771) : / One Mode (h=0.224)1 0 / 1 0
P-value = 0.26 / P-value=0.392
Two Modes (h=0.683) : / Two Modes (h= 0.223)
1 0 / 1 0
0 1 / 0 1
P-value =0.71 / P-value=0.056
Three Modes (h=0.462) : / Three Modes (h=0.114)
1 0 / 1 0
0 1 / 0 1
0 0 / 0.34 0.34
P-value = 0.02 / P-value = 0.336
Four Modes (h=0.418) : / Four Modes (h=0.109)
1 0 / 1 0
0 1 / 0 1
0 0 / 0.34 0.34
0.52 0.48 / 0.68 0.20
P-value = 0.04 / P-value = 0.108
Five Modes (h=0.401) : / Five Modes (h=0.102)
1 0 / 1 0
0 1 / 0 1
0 0 / 0.34 0.34
0.52 0.49 / 0.65 0.18
0 0.33 / 0.28 0.71
P-value = 0.01 / P-value = 0.022
Six Modes (h= 0.273) : / Six Modes ( h=0.092)
1 0 / 1 0
0 1 / 0 1
0.36 0.35 / 0.34 0.34
0.48 0 / 0.64 0.15
0 0 / 0.26 0.73
0 0.44 / 0 0.26
P-value = 0.215 / P-value = 0.042
2.3.2. Second Data Set
Gaussian Reflection Method
One Modes (h=0.758) / One Mode (h=0.2649)1 0 / 1 0
P-value = 0.076 / P-value = 0.0350
Two Modes (h=0.7208) / Two Modes (h=0.2126)
1 0 / 1 0
0 0 / 0 1
P-value = 0.052 / P-value = 0.0100
Three Modes (h=0.4506) / Three Modes (h=0.17301)
1 0 / 1 0
0 0 / 0 1
0 1 / 0 0
P-value = 0.02 / P-value = 0.0000
Four Modes (h=0.3424) / Four Modes (h=0.11)
1 0 / 1 0
0 0 / 0 1
0 1 / 0 0
0.43 0.28 / 0.45 0.29
P-value = 0.0475 / P-value = 0.1100
Five Modes (h=0.3244) / Five Modes (h= 0.073)
1 0 / 1 0
0 0 / 0 1
0 1 / 0 0
0.43 0.29 / 0.43 0.29
0 0.50 / 0 0.50
P-value = 0.015 / P-value = 0.375
Six Modes (h=0.2703) / Six Modes (h= 0.065)
1 0 / 1 0
0 0 / 0 1
0 1 / 0 0
0.43 0.33 / 0.43 0.29
0.49 0 / 0 0.50
0 0.52 / 0.34 0.65
P-value = 0.075 / P-value = 0.315
2.3.3. Subset of second data set with the Gaussian Reflection Method
1 0 / 1 0
0.44 0.44 / 0 1
P-value = 0.04 / 0 0
0.33 0.34
Three Modes (h = 0.154) / 0.50 0
1 0 / P-value = 0.37
0 1
0.35 0.35 / Six Modes (h = 0.06)
P-value = 0.015 / 1 0
0 1
Four Modes (h = 0.09) / 0 0
1 0 / 0.33 0.34
0 1 / 0.50 0
0 0 / 0 0.5
0.33 0.34 / P-value = 0.335
P-value = 0.13
2.3.4. Subset of second data set with NEÇMM = Æ: Boundary Biweight Kernel Method
Six Modes (h = 0.208)1 0
0 1
0 0
0.34 0.34
0 0.48
0.53 0
P-value = 0.086
Table 2.4 Test Results for Local Mode Test on a subset of the data
Mode Excess Mass P-value
0, 0 0.015 0
0, 1 0.010 0.150
1/3, 1/3 0.497 0
1, 0 0.010 0.065
0, 0.5 0.000 0.345
0.5, 0 0.003 0.080
Table 4.1 The experimental sessions
2-17-1998 / 1 / 16 / 9
2-19-1998 / 2 / 19 / 12
4-7-1998 / 3 / 14 / 1
4-9-1998 / 4 / 13 / WG
6-18-1998 / 5 / 16 / 13
7-16-1998 / 6 / 16B / 13B
7-30-1998 / 7 / 16B / 13C
1-28-1999 / 8 / 16 / 19
3-2-1999 / 9 / 16B / 19
Table 4.2 Parameter Estimates for 20 one-shot games, using the HS population heterogeneity model
n2 / 1.031
nopt / 1.031
nNE / 1.031
g / ------
w1 / 0.131
w2 / 0.050
wNE / 0.045
m / ------
a0 / 0.076
a1 / 0.435
a2 / 0.066
aopt / 0.093
aNE / 0.066
aH / 0.265
Log-likelihood / -1177.76
Key: In each column, the unique Nash evidence enters the model through both the archetypal Nash rule (the row with nNE) and the Hybrid rule (the row with wNE). The UNE column is the base model in which each Nash equilibrium is equally likely. The other columns represent the Nash equilibrium selection principles of payoff dominance (PD), risk dominance (RD), and security (SEC). Since numbers are represented to the third decimal point, 0.000 can be thought of as some number smaller than 0.001.
nk is the scaling parameter for evidence vector k in archetypal rule k; wk is the weight parameter for evidence vector k in the Hybrid rule; g is the strength parameter for the specific equilibrium selection principle (of the corresponding column) for the archetypal Nash rule; m is the strength parameter for the specific equilibrium selection principle (of the corresponding column) for the Hybrid rule; and at is the proportion of population using rule t, where t Î{0,1,2} denotes the level-t rule, t = NE denotes the archetypal Nash rule, t = opt denotes the archetypal optimistic rule, and t = H denotes the hybrid rule.
Table 4.3 Initial Point Predictions for 20 games using HS98
1 / 0.790 / 0.073 / 0.137
2 / 0.026 / 0.063 / 0.912
3 / 0.467 / 0.506 / 0.027
4 / 0.137 / 0.804 / 0.058
5 / 0.911 / 0.026 / 0.063
6 / 0.197 / 0.026 / 0.777
7 / 0.813 / 0.038 / 0.149
8 / 0.075 / 0.899 / 0.026
9 / 0.829 / 0.115 / 0.056
10 / 0.064 / 0.063 / 0.873
11 / 0.660 / 0.314 / 0.0258
12 / 0.711 / 0.055 / 0.234
13 / 0.668 / 0.105 / 0.228
14 / 0.026 / 0.911 / 0.063
15 / 0.225 / 0.609 / 0.166
16 / 0.787 / 0.153 / 0.060
17 / 0.026 / 0.063 / 0.912
18 / 0.502 / 0.197 / 0.301
19 / 0.091 / 0.773 / 0.137
20 / 0.062 / 0.063 / 0.876
Table 4.4 Parameter Estimates for SPA (and replicator dynamics with entropy)
Table 4.5 Parameter Estimates on EWA.
4.5.1 Estimation by Maximum Likelihood
All 7 sessions
r / 0.934f / 0.634
d / 0.533
N0 / 17.324
l / 0.684
Log-likelihood / -1220.90
MSE / 0.184
Sessions 16.1 , 16.2, 16.3, 16.4
r / 1.000f / 0.507
d / 0.341
N0 / 50.084
l / 3.326
Log-likelihood / -673.44
MSE / 0.157
Session 19.1
r / 1.000f / 0.889
d / 0.405
N0 / 60.890
l / 0.741
Log-likelihood / -168.88
MSE / 0.019
Table 4.5 Parameter Estimates on EWA.
4.5.2 Estimation by SMSE
All 7 sessions
r / 0.972f / 0.653
d / 0.540
N0 / 49.728
l / 0.503
Log-likelihood / -1,612.83
MSE / 0.113
Sessions 16.1 , 16.2, 16.3, 16.4
r / 0.441f / 0.477
d / 0.586
N0 / 59.337
l / 3´10-4
Log-likelihood / -1,257.90
MSE / 0.122
Session 19.1
r / 0.803f / 0.808
d / 0.795
N0 / 40.381
l / 0.117
Log-likelihood / -214.593
MSE / 0.009
Table 4.6 Parameter Estimates on RE Reinforcement
Table 4.6 Parameter Estimates on RE Reinforcement
4.6.2. Estimation by SMSE
All 7 sessions
S / 5.952e / 0.292
f / 0.634
r1 / 0.043
w- / 0.000
w+ / 0.000
Log-likelihood / -1,457.44
MSE / 0.113
Sessions 16.1 , 16.2, 16.3, 16.4
S / 6.413e / 0.668
f / 0.452
r1 / 0.105
w- / 1.000
w+ / 0.000
Log-likelihood / -1,258.96
MSE / 0.126
Session 19.1
S / 12.318e / 0.008
f / 0.014
r1 / 0.000
w- / 0.645
w+ / 0.000
Log-likelihood / -350.069
MSE / 0.009
Table 4.7 Likelihood of observed choices in the final period of repeated game 16 under the replicator dynamics model with parameters estimated by log-likelihood maximization, using kernel density estimates
16-1 / 0.083 / 0.917 / 0.000 / 0.6163
16-2 / 0.000 / 1.000 / 0.000 / 1.734
16-3 / 0.800 / 0.120 / 0.080 / 9.643
16-4 / 0.040 / 0.960 / 0.000 / 1.448
Combined game 16 / ---- / ---- / ---- / 14.918
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