Asymmetric Information in Simple Bargaining Games: An Experimental Study

University of Tübingen Working Papers in Economics and Finance

No. 97

Asymmetric Information in Simple Bargaining Games: An Experimental Study

by

Charlotte Klempt, Kerstin Pull, Manfred Stadler

Faculty of Economics and Social Sciences www.wiwi.uni-tuebingen.de

⇤

⇤

DG U G

⇡ 8 ¹₃

20 ²₃

x 0< x < 20

G1 ⇡ x

y

G2 ⇡

DG U G

DG1 DG2 U G1 U G2

DG1 DG2 x <⇡

x y=⇡ x x ⇡

DG2 ⇡

U G1 U G2

y x

U G1

⇡ y

x y 8 U G2

⇡ x

y

U G2 x 8

U G1 U G2 DG1 DG2

e e

• DG1 U G1

x_G1(8) xG1(20)

⇡ DG2 U G2

x ⇡

⇡

G2 x > ⇡

x=⇡ U G2

• U G1

y y <8

y 8

U G2 y

x

x G2

G2 x

y x

U G2 x

xDG xU G > 0

G2 G1 xDG2 xU G2 > xDG1 xU G1 >0

G1 x

y y  4

U G1 DG1

U G1 G1

U G1

G1(20) G1(8)

G1(20) G1(8)

x 16 G1(20)

G1

G1(20) G1(8)

x 16 G1(20) G2

G1(20) G1(8)

xG1(20)/20 xG1(8)/8 > 0 U G DG

x_{U G1(20)}/20 x_{U G1(8)}/8 x_DG1(20)/20 x_DG1(8)/8 x 16

G1(20) G2 U G

DG

G2 xG2 8

xG2 8 1/3

U G2 xG2 8

DG2 U G2

(xDG1(8) xU G1(8))/8 > (xDG1(20) xU G1(20))/20 >0

G1(20) G1(8) G2

G2 G1

x 16 G1(20) G2

U G DG x 16 U G1(20)

U G1(20)

U G2 U G1

DG2

x_DG2 8 DG1

xDG2 xDG1(20) >0 xDG2 8 U G2

xDG2 xDG1(20) xU G2 xU G1(20) xDG2 8 xU G2 8

x G1

• G1

x U G1 DG1

DG1(8) U G1(8)

DG1(20) U G1(20)

• G2

x U G2 DG2 DG2

U G2

G1 G2

x x 8 G2

G1(8) x <8

• G1

G1(8)

x U G1(8) DG1(8)

U G1(20) DG1(20)

xDG2 xU G2 >

x_DG1(20) x_{U G1(20)} x 8 DG2

G2 G1(20)

⇡ = 8 ⇡ = 20

x

⇡= 8 ⇡ = 20

• G2

U G2 x <7 DG2

U G2

x 17 DG2

x

x

x DG1 U G1

x DG2 U G2

x G2 G1 G1

xDG xU G

G1 G2

xU G1(20) xDG1(20) xU G2 xDG2 xU G1(8) xDG1(8)

xU G2 xDG2

⇡ = 8 ⇡= 20 ⇡= 8 ⇡ = 20 x <7

x= 7 8x <17 x 17

x <7 x= 7 8x <17 x 17

G1

xG1(8)/8

xG1(20)/20 U G1

DG1

⇡ = 20

⇡ = 8 U G1 DG1 U G1

U G1(8) U G1(20))

DG1(8) DG1(20)

U G1 DG1

U G1

U G1

DG1 U G1

DG1

U G1 DG1 U G1

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 UG1

Demanded share

Frequency 051015

UG1(8) UG1 (20)

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

DG1

Demanded share

Frequency 051015

DG1(8) DG1(20)

U G1 DG1 xG1(8)/8 xG1(20)/20

x 16 U G1(20) U G2 x 16

x 16

U G1(20) U G2

U G1(20)

x 16

U G1(20) U G2

x 16 G1(20) G2

x 16 DG1(20) DG2

y4

y= 3 y= 4 U G1(20) U G2 DG1(20)

DG2 U G1(20) DG1(20) y= 3 y= 4

U G2 DG2

y= 3 y= 4 G1(20) U G2 DG1

G1(20) G2

U G1(20) x 16

U G2 DG1(20)

G2 U G2

DG2

x G1(20)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19

UG

Demands

Frequency 051015

UG1(20) UG2

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19

DG

Demands

Frequency 051015

DG1(20) DG2

U G2 DG2 U G1(20) DG1(20)

x 8 G2 x 8

G1(20) x 8 DG2

x 8 DG1(20)

DG1 x 8

DG2 x 8

DG2 x 8

x 8 DG1

U G2

G1

U G1 x 8

U G2 x 8

x 8

x 8 U G1

x U G1 U G2

U G1(8)

U G2(8) U G1(8) U G2(8)

U G2(8)

U G1 U G1

⇡ = 8

⇡ = 20

U G1 y 4

y 8

y  3 y = 4

5  y  7

y >17

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19

UG1

Offer Rejection frequency 0.00.40.8

UG1

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19

UG2

Offer Rejection frequency 0.00.40.8

UG2(20) UG2(8)

U G2

U G2(8) y <8 U G2(8)

U G2(20)

y = 10 U G2(8)

4 y <8 U G2(20) U G1

y <4 4  y < 8

U G1 U G2(20)

U G2(20) U G1

U G1 U G1 U G2

U G2

U G1 U G2

U G1 U G2

U G2 x= 6

y = 2 U G2(8) y = 14 U G2(20)

yU G2(20)/20 > 1/2

yU G2(20)/20 yU G2(8)/8

U G1 U G2

U G1 U G2(20)

(0,1/8] (1/8,1/4] (1/4,3/8] (3/8,1/2] (1/2,1) Relative 0ffers

Share of subjects 0.00.20.40.60.81.0

type (1) responders type (2) responders type (3) responders

U G2 yU G2(8)/8 yU G2(20)/20

yU G2(8)/8 yU G2(8)/8 yU G2(20)/20

G2

U G2 U G1

x= 6 2/8 = 0.25

x = 15

5/20 = 0.25 0.25

14/20 = 0.7 0.25

x = 7 G1(8) x = 19 G1(20) G2

x = 19

µ2 (0,1) µ

µ

µ µ

DG1(⇡) ⇡ = {8,20} x

px = e^(1/µ)x P⇡ 1

k=1e^(1/µ)k ; x= 1, ...,⇡ 1. E(x) =P⇡ 1

x=1pxx

DG2 x <8

x 8 x

p_x = 8>

<

>:

e^(1/µ)x P7

k=1e^(1/µ)k+P19

k=8e(1/µ)(2/3)k x= 1, ...,7

e(1/µ)(2/3)x

P7

k=1e^(1/µ)k+P19

k=8e(1/µ)(2/3)k x= 8, ...,19. E(x) = P19

x=1pxx

U G1(⇡) ⇡ = {8,20} y =

⇡ x

qx= e^{(1/µ)(⇡ x)}

e^{(1/µ)(⇡ x)}+ 1 ; x= 1, ...,⇡ 1, x

px= e^(1/µ)qxx P⇡ 1

k=1e^(1/µ)qkk ; x= 1, ...,⇡ 1. E(x) = P⇡ 1

x=1pxx

U G2 x

E(qx) = (1/3)q_x^{U G2(8)}+ (2/3)q_x^{U G2(20)} ,

q_x^{U G2(8)}= 8<

:

e^{(1/µ)(8 x)}

e^{(1/µ)(8 x)}+1 x= 1, ...,7

0 x= 8, ...,19

q_x^{U G2(20)} = e^{(1/µ)(20 x)}

e^{(1/µ)(20 x)}+ 1 , x= 1, ...,19. x

px = e^(1/µ)E(qx)x P19

k=1e^(1/µ)E(qk)k ; x= 1, ...,19. E(x) = P19

x=1pxx

µ 0 1 2 3 3.5 4 5 1 DG1(8)

U G1(8) DG1(20) U G1(20) DG2 U G2

x

x µ= 0

µ! 1 µ={1,2,3,3.5,4,5}

µ⇡3.5 x

µ= 0 µ! 1

U G2

µ= 3.5 µ

DG1(8) U G1(8) DG1(20) U G1(20) DG2 1 U G2

e

X Y

X Y

X Y X

Y Y

e e

e e

x X

y X Y

X (x, y

e X

(x8, y8

x₈ 1 2 3 4 5 6 7 y₈ 7 6 5 4 3 2 1

e X

(x₂₀, y₂₀ Y

x20 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 y₂₀ 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1

X X

e e

X x

e (x8, y8

e (x20, y20

x 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19

x₈ 1 2 3 4 5 6 7 ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥

y8 7 6 5 4 3 2 1 ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥

x20 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 y₂₀ 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1

X x

Y Y

(x= 5;x8 = 5;y8 = 3;x20= 5;y20 = 15) x= 5

X x=x8 =x20= 5 Y

y8 = 3 e y20 = 15

e X e

(x = 10;x8 = 0;y8 = 0;x20 = 10;y20 = 10)

e e

x20=y20 = 10

Y Y

X

Y

(x8, y8 e (x20, y20 e

Y

x₈ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ 1 2 3 4 5 6 7

y8 ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ 7 6 5 4 3 2 1

x20 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 y₂₀ 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1

X (x₈ = 3;y₈ = 5;x₂₀ = 15;y₂₀ =

5) e X e e

e Y y8 =y20 =

e X (x₈ =⇥;y₈ =⇥;x₂₀ =

5;y= 15) e

e X e Y e Y

Y Y

X e

(x₈, y₈

x₈ 1 2 3 4 5 6 7 y₈ 7 6 5 4 3 2 1

e (x₈, y₈

x20 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 y₂₀ 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1

Y Y

e e

Y

X Y

X