Table 2.1 the Designer Boundary Kernel

TABLES

Table 2.1 The designer boundary kernel

Functional form Kc(t) / domain in Xi / Support on t
¾ [ (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 t
3/2[(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

Boundary Biweight Kernel Method /

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

Two Modes (h = 0.222) / Five Modes (h = 0.067)
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

Date / Session / Game in run 1 / Game in run 2
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

n1 / 0.347
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

% of A choices / % of B choices / % of C choices
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 / 0.934
f / 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 / 1.000
f / 0.507
d / 0.341
N0 / 50.084
l / 3.326
Log-likelihood / -673.44
MSE / 0.157

Session 19.1

r / 1.000
f / 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 / 0.972
f / 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 / 0.441
f / 0.477
d / 0.586
N0 / 59.337
l / 3´10-4
Log-likelihood / -1,257.90
MSE / 0.122

Session 19.1

r / 0.803
f / 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

4.6.2. Estimation by SMSE

All 7 sessions

S / 5.952
e / 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 / 6.413
e / 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 / 12.318
e / 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

Game-session / % choosing A / % choosing B / % choosing C / Likelihood
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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