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n ( ( d (

; ( (�x (�x (

x (�x (�x ( (

E d ( ( (

� ( ( d (

�7e (�n�x (�1x (�n�x (

Is�s (�nM (�nd (�1� (�(;

�Y$t,$t� (�;E (�1n (�1M (�(n

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xM

}jsk��R wR$t{� $� $R t�%�� s A�R� ��R}BtR� s�s$tR� }jsk��R bYB �stIB8$9�D� bY${Y ��t��s��R

s R8sjj s8BZt� BP {YB${� �Ys� 8s�{Y�R �Y� Is�s� WB�${� sjRB �Ys�� A�R� ��R}BtR�R ��tI

�B jZ�{Y s�BZtI sR �Y$t,$t� R��}R {Yst��� GY�t �Y�R� s�� s%��s��I B%�� �Y$t,$t� R��}R

P��JZ�t{$�R s }B}Zjs�$Bt 8$-�Z�� ��RZj�R� 3Z��Y��8B��� Bt� BP �Y� JZ$�,$�R� P�s�Z��R BP

8$-�I �JZ$j$A�$Z8 $R �Ys� Bt� }jsk��:R 8$-�Z�� I�}�tIR Btjk Bt �Y� B�Y�� }jsk��:R }skB&R�

qY$R �&�{� sjRB B{{Z�R $t �Y� �Y$t,$t� R��}R 8BI�jR A�{sZR� s �TR��} �Bb }jsk��:R }skB&R

s&�{� �Bb:R A�R� ��R}BtR�R� bY${Y �Y�t s&�{� s �LdTR��} {BjZ8t }jsk��:R A�R� ��R}BtR��

7B Bt� }jsk��:R }skB&R s&�{� �Y� B�Y��:R R��s���$�R $tI$��{�jk� qY� �sAj� sjRB RYBbR �Y�

�7e w8$-�I �JZ$jA�$Z8D }��I${�$Bt� �Y� Is�s P��JZ�t{$�R� stI B%��sjj P��JZ�t{$�R P�B8

�Y� �Y$t,$t�TR��}R 8BI�j bY�t + ) 131� qY� 8BI�j h�R �Y� Is�s {jBR�� �Yst �7e PB�

�Bb }jsk��R w$� s{{BZt�R PB� ZtI��}jsk BP �Bb R��s���k 1 stI B%��}jsk BP R��s���k nD stI

$R �JZsjjk s{{Z�s�� sR �7e PB� {BjZ8t }jsk��R� VR tB��I $t �Y� ��-�� �Y� }B$t� $R tB�

�Ys� �Y� �Y$t,$t� R��}R 8BI�j BZ�}��I${�R �7eu $� {sttB�� A�{sZR� �7e YsR RZ{Y s

Y$�Y {B���js�$Btu AZ� R$8}jk �Ys� �Y� Rs8� 8BI�j bY${Y �-}js$tR A�Ys%$B� $t IB8$tst{�T

RBj%sAj�� 8s��$-� stI �t��k �s8�R sjRB ��t��s��R 8$-�Z��R BP }jsk��R }jsk$t� t�s�T}Z��

R��s���$�R bY${Y s�� {jBR� �B BZ�{B8�R $t 8$-�I �s8�R�

E�1 �s�,�� �t��k �s8�R

VtsjkR$R BP �Y� R$8}j� �t��k �s8� I�R{�$A�I $t �Y� ��-� }�B{��IR sR PBjjBbR� 7��} (:R �stT

IB8$9� RB Ww(Db1 j�%�j (:R �t���� *�ht� �Y� ��js�$%� }�B}B��$Bt BP �t��k sP��� s{{BZt�$t�

Z} �Y�BZ�Y j�%�j , sR ]w>D� *�ht� s BBj�st PZt{�$Bt 4w�D ) d $P � ��Z�� 4w�D ) (

$P � PsjR�� N�%�j d:R �t��� $& db1 c �� qY���PB��� �B�sj �t��k "sP���" j�%�j d �k}�R s��

s{{BZt��I PB� $R ]wdD ) Ww(Db1L4>]w(DbwWw(Db1D k �HWwdD qB�sj �t��k sP��� j�%�j , �k}�

$R �Y���PB�� ]w>D ) Ww(Db1 L�>

z)dWwzD4>]wz� dDbw

�>

E)dWwEDD k �H� V �$%�t { stI +

�Y�t ��t��s��R s R�JZ�t{� BP {Z8Zjs��I �t��k �s��R bY${Y sRk8}�B��R sR , ��BbR js����

*�ht� s PZt{�$Bt ]wyqq/+Dw�D sR �Y� B%��sjj �s�� BP �t��k� �$%�t + � PB� %s�$BZR {s}s{$�$�R

{�

qY� Is�s ��}B���I $t �Y� ��-� 3$�Z�� {B8� P�B8 �-}��$8�t�R Ak 7ZtIsj$ �� sj wd��xD

stI 7�sj� stI is}B}B�� wd���D� qY�$� �s8� $R tB� }��{$R�jk �Y� Rs8� sR �Y� Bt� stsjk9�I

A�{sZR� $t �Y�$� �s8� �t��st�R �s�t dL 1w�� �D wbY��� � $R �Y� tZ8A�� BP �t��st�RD stI

tBt�t��st�R �s�t d� qY�k ZR�I 1( RZA`�{�R b$�Y BII %sjZ�R BP { wd�n����d�D� qB {B8}Z��

�t��k �s��R ��}B���I $t �Y� 3$�Z�� b� s%��s��I �t��k PB� sI`s{�t� { %sjZ�R w$���� s%��s�$t�

d stI n k$�jIR s h�Z�� PB� � ) 1 �B 8s�{Y {)�d� s%��s�$t� n stI x k$�jIR s h�Z�� PB�

Page 60: ‡Ys%$B—sjps8‡qY‡B—kU qY$t,$t˙ N‡s—t$t˙ stIq‡s{Y$t˙ · ‡Ys%$B—sjps8‡qY‡B—kU qY$t,$t˙ N‡s—t$t˙ stIq‡s{Y$t˙]Bj$t3˚]s8‡—‡—d]sj$PB—t$sytR˝$˝Z˝‡BPq‡{YtBjB˙k

x�

� ) ; {B���R}BtI$t� �B {)�1� ��{�D !A%$BZRjk �Y� stsjkR$R stI Is�s s�� tB� }��P�{�jk

8s�{Y�I� AZ� b� {Bt`�{�Z�� �Ys� �Y� �Y$t,$t� R��}R 8BI�j {st R�$jj 8s�{Y Is�s {jBR�jk

stI ��}�BIZ{� �Y� �Y��� �-}��$8�t�sj ���Zjs�$�$�R I�R{�$A�I $t �Y� ��-�� bY��Y�� �Y$R $R

��Z� $R �Y� RZA`�{� BP Bt�B$t� ��R�s�{Y�

Page 61: ‡Ys%$B—sjps8‡qY‡B—kU qY$t,$t˙ N‡s—t$t˙ stIq‡s{Y$t˙ · ‡Ys%$B—sjps8‡qY‡B—kU qY$t,$t˙ N‡s—t$t˙ stIq‡s{Y$t˙]Bj$t3˚]s8‡—‡—d]sj$PB—t$sytR˝$˝Z˝‡BPq‡{YtBjB˙k

Figure 1: Poisson distributions for various τ τ

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0 1 2 3 4 5 6

thinking steps

freq

uen

cy

tau = 0.5

tau = 1

tau = 1.5

tau = 2

Page 62: ‡Ys%$B—sjps8‡qY‡B—kU qY$t,$t˙ N‡s—t$t˙ stIq‡s{Y$t˙ · ‡Ys%$B—sjps8‡qY‡B—kU qY$t,$t˙ N‡s—t$t˙ stIq‡s{Y$t˙]Bj$t3˚]s8‡—‡—d]sj$PB—t$sytR˝$˝Z˝‡BPq‡{YtBjB˙k

Figure 2: Fit of thinking-steps model to three games (R2=.84)

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.2 0.4 0.6 0.8 1

thinking steps model (common ττ)

dat

a

Stahl-Wilson data (3x3 symmetric)

Cooper-Van Huyck data (2x2asymmetric)

Costa-Gomes-Crawford-Broseta(2x2-4x2 asymmetric)

Page 63: ‡Ys%$B—sjps8‡qY‡B—kU qY$t,$t˙ N‡s—t$t˙ stIq‡s{Y$t˙ · ‡Ys%$B—sjps8‡qY‡B—kU qY$t,$t˙ N‡s—t$t˙ stIq‡s{Y$t˙]Bj$t3˚]s8‡—‡—d]sj$PB—t$sytR˝$˝Z˝‡BPq‡{YtBjB˙k

Figure 3: Nash equilibrium predictions vs data in three games

R2 = 0.49

-0.2

0

0.2

0.4

0.6

0.8

1

1.2

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Data

Eq

uili

bri

um

pre

dic

tio

n

Page 64: ‡Ys%$B—sjps8‡qY‡B—kU qY$t,$t˙ N‡s—t$t˙ stIq‡s{Y$t˙ · ‡Ys%$B—sjps8‡qY‡B—kU qY$t,$t˙ N‡s—t$t˙ stIq‡s{Y$t˙]Bj$t3˚]s8‡—‡—d]sj$PB—t$sytR˝$˝Z˝‡BPq‡{YtBjB˙k

Figure 4: How entry varies with capacity (c), data and thinking-steps model

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1 2 3 4 5 6 7 8 9

capacity (out of 10)

% e

ntr

y entry=capacity

experimental data

tau=1.5

tau=2

Page 65: ‡Ys%$B—sjps8‡qY‡B—kU qY$t,$t˙ N‡s—t$t˙ stIq‡s{Y$t˙ · ‡Ys%$B—sjps8‡qY‡B—kU qY$t,$t˙ N‡s—t$t˙ stIq‡s{Y$t˙]Bj$t3˚]s8‡—‡—d]sj$PB—t$sytR˝$˝Z˝‡BPq‡{YtBjB˙k

Figure 5: The EWA Learning Cube

Cournot Weighted Fictitious Play

Fictitious Play

Average Reinforcement

Cumulative Reinforcement

Page 66: ‡Ys%$B—sjps8‡qY‡B—kU qY$t,$t˙ N‡s—t$t˙ stIq‡s{Y$t˙ · ‡Ys%$B—sjps8‡qY‡B—kU qY$t,$t˙ N‡s—t$t˙ stIq‡s{Y$t˙]Bj$t3˚]s8‡—‡—d]sj$PB—t$sytR˝$˝Z˝‡BPq‡{YtBjB˙k

Figure 6: Predicted Frequencies for p-Beauty Contest

1

3

5

7

9

0

1~10

11~

20

21~

30

31~

40

41~

50

51~

60

61~

70

71~

80

81~

90

91~

100

0

0.1

0.2

0.3

0.4

0.5

0.6

Prob

Period

Strategy

Empirical Frequency

1

3

5

7

9

0

1~10

11~

20

21~

30

31~

40

41~

50

51~

60

61~

70

71~

80

81~

90

91~

100

0

0.1

0.2

0.3

0.4

0.5

0.6

Prob

Period

Strategy

Thinking fEWA

1

3

5

7

9

0

1~10

11~

20

21~

30

31~

40

41~

50

51~

60

61~

70

71~

80

81~

90

91~

100

0

0.1

0.2

0.3

0.4

0.5

0.6

Prob

Period

Strategy

Adaptive EWA

1

3

5

7

9

0

1~10

11~

20

21~

30

31~

40

41~

50

51~

60

61~

70

71~

80

81~

90

91~

100

0

0.1

0.2

0.3

0.4

0.5

0.6

Prob

Period

Strategy

Belief-based

1

3

5

7

9

0

1~10

11~

20

21~

30

31~

40

41~

50

51~

60

61~

70

71~

80

81~

90

91~

100

0

0.1

0.2

0.3

0.4

0.5

0.6

Prob

Period

Strategy

Choice Reinforcement with PV

1

3

5

7

9

0

1~10

11~

20

21~

30

31~

40

41~

50

51~

60

61~

70

71~

80

81~

90

91~

100

0

0.1

0.2

0.3

0.4

0.5

0.6

Prob

Period

Strategy

Quantal Response

Page 67: ‡Ys%$B—sjps8‡qY‡B—kU qY$t,$t˙ N‡s—t$t˙ stIq‡s{Y$t˙ · ‡Ys%$B—sjps8‡qY‡B—kU qY$t,$t˙ N‡s—t$t˙ stIq‡s{Y$t˙]Bj$t3˚]s8‡—‡—d]sj$PB—t$sytR˝$˝Z˝‡BPq‡{YtBjB˙k

Figure 7: Predicted Frequencies for Continental Divide

1

3

5

7

9 11

13

15

S1

S4

S7

S10

S13 0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

Prob

PeriodStrategy

Empirical Frequency

1

3

5

7

9 11

13

15

S1

S4

S7

S10

S13 0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

Prob

PeriodStrategy

Thinking fEWA

1

3

5

7

9 11

13

15S1

S4

S7

S10

S13 0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

Prob

PeriodStrategy

Belief-based

1

3

5

7

9 11

13

15

S1

S4

S7

S10

S13 0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

Prob

PeriodStrategy

Adaptive EWA

1

3

5

7

9 11

13

15

S1

S4

S7

S10

S13 0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

Prob

PeriodStrategy

Choice Reinforcement with PV

1

3

5

7

9 11

13

15

S1

S4

S7

S10

S13 0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

Prob

PeriodStrategy

Quantal Response

Page 68: ‡Ys%$B—sjps8‡qY‡B—kU qY$t,$t˙ N‡s—t$t˙ stIq‡s{Y$t˙ · ‡Ys%$B—sjps8‡qY‡B—kU qY$t,$t˙ N‡s—t$t˙ stIq‡s{Y$t˙]Bj$t3˚]s8‡—‡—d]sj$PB—t$sytR˝$˝Z˝‡BPq‡{YtBjB˙k

Figure 8: Predicted Frequency for Traveller's Dilemma (Reward = 50)

1

3

5

7

9

80

81~

90

91~

100

101~

110

111~

120

121~

130

131~

140

141~

150

151~

160

161~

170

171~

180

181~

190

191~

200

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

Prob

Period

Strategy

Empirical Frequency

1

3

5

7

9

80

81~

90

91~

100

101~

110

111~

120

121~

130

131~

140

141~

150

151~

160

161~

170

171~

180

181~

190

191~

200

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

Prob

Period

Strategy

Choice Reinforcement with PV

1

3

5

7

9

80

81~

90

91~

100

101~

110

111~

120

121~

130

131~

140

141~

150

151~

160

161~

170

171~

180

181~

190

191~

200

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

Prob

Period

Strategy

Quantal Response

1

3

5

7

9

80

81~

90

91~

100

101~

110

111~

120

121~

130

131~

140

141~

150

151~

160

161~

170

171~

180

181~

190

191~

200

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

Prob

Period

Strategy

Thinking fEWA

1

3

5

7

9

80

81~

90

91~

100

101~

110

111~

120

121~

130

131~

140

141~

150

151~

160

161~

170

171~

180

181~

190

191~

200

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

Prob

Period

Strategy

Belief-based

1

3

5

7

9

80

81~

90

91~

100

101~

110

111~

120

121~

130

131~

140

141~

150

151~

160

161~

170

171~

180

181~

190

191~

200

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

Prob

Period

Strategy

Adaptive EWA

Page 69: ‡Ys%$B—sjps8‡qY‡B—kU qY$t,$t˙ N‡s—t$t˙ stIq‡s{Y$t˙ · ‡Ys%$B—sjps8‡qY‡B—kU qY$t,$t˙ N‡s—t$t˙ stIq‡s{Y$t˙]Bj$t3˚]s8‡—‡—d]sj$PB—t$sytR˝$˝Z˝‡BPq‡{YtBjB˙k

12

34

56

781

23

45

67

89

0.0000

0.1000

0.2000

0.3000

0.4000

0.5000

0.6000

0.7000

0.8000

0.9000

1.0000

Freq

PeriodSequence

Figure 9c: Predicted Frequency for No Loan

12

34

56

781

23

45

67

89

0.0000

0.1000

0.2000

0.3000

0.4000

0.5000

0.6000

0.7000

0.8000

0.9000

1.0000

Freq

PeriodSequence

Figure 9a: Empirical Frequency for No Loan

12

34

56

781

23

45

67

89

-0.1000

0.0000

0.1000

0.2000

0.3000

0.4000

0.5000

0.6000

0.7000

0.8000

0.9000

1.0000

Freq

PeriodSequence

Figure 9b: Empirical Frequency for Default conditional on Loan (Dishonest Borrower)

12

34

56

781

23

45

67

89

-0.1000

0.0000

0.1000

0.2000

0.3000

0.4000

0.5000

0.6000

0.7000

0.8000

0.9000

1.0000

Freq

PeriodSequence

Figure 9d: Predicted Frequency for Default conditional on Loan (Dishonest Borrower)

Page 70: ‡Ys%$B—sjps8‡qY‡B—kU qY$t,$t˙ N‡s—t$t˙ stIq‡s{Y$t˙ · ‡Ys%$B—sjps8‡qY‡B—kU qY$t,$t˙ N‡s—t$t˙ stIq‡s{Y$t˙]Bj$t3˚]s8‡—‡—d]sj$PB—t$sytR˝$˝Z˝‡BPq‡{YtBjB˙k

Figure 10: Fit of data to equilibrium & thinking steps model (common ττ=1.5) in mixed-equilibrium games

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

0 0.2 0.4 0.6 0.8 1

actual frequencies

pre

dic

tio

ns

thinking steps (r=.86)

equilibrium (r=.91)