Experimental Business Research II springer 2005 phần 8 potx

D
YNAMI
C
S
C
TABILITY
OF
N
F
ASH
N
N
-E
FFICIENT
E
E
P
T
U
BLI
C
G
C
OODS
M
S
ECHANISMS
MM
18
7
Th

e
id
ea o
f
us
i
ng supermo
d
u
l
ar
i
ty as a ro
b
ust sta
bili
ty cr
i
ter
i
on
f
or Nas
h
-
effi
c
i
ent mec
h

an
i
sms
i
s not on
l
y
b
ase
d
on
i
ts goo
d
t
h
eoret
i
ca
l
propert
i
es,
b
ut a
l
so on
s
tron
g

experimental evidence. In fact it is inspired b
y
the experimental results of
Chen and Plott (199
6
) and Chen and Tang (1998), where they varied a punishment
p
arameter
i
n t
h
e Groves-Le
d
yar
d
mec
h
an
i
sm
i
n a set o
f
exper
i
ments an
d
o
b
ta

i
ne
d
t
otall
y
different d
y
namic stabilit
y
results
.
I
n t
hi
s paper, we rev
i
ew t
h
e ma
i
n exper
i
menta
l

fi
n
di
ngs on t

h
e
d
ynam
i
c sta
bili
ty
o
f
Nas
h
-e
ffi
c
i
ent pu
bli
c goo
d
s mec
h
an
i
sms, exam
i
ne t
h
e supermo
d

u
l
ar
i
ty o
f
ex
i
st
i
ng
Nash-efficient public
g
oods mechanisms, and use the results to sort a class of experi-
m
enta
l

fi
n
di
ngs.
S
ect
i
on 2
i
ntro
d
uces t

h
e env
i
ronment. Sect
i
on 3 rev
i
ews t
h
e exper
i
menta
l
resu
l
ts.
S
ection 4 discusses supermodular
g
ames. Section
5
investi
g
ates whether the existin
g
m
echanisms are supermodular games. Section
6
concludes the paper.
2

. A PUBLI
C

GOO
D
S
ENVIR
O
NMENT
W
e
fi
rst
i
ntro
d
uce notat
i
on an
d
t
h
e econom
i
c env
i
ronment. Most o
f
t
h

e exper
i
menta
l
i
m
p
lementations of incentive-com
p
atible mechanisms use a sim
p
le environment.
U
sua
ll
y t
h
ere
i
s one pr
i
vate goo
d

x
,
one pu
bli
c goo
d

y
,
an
d
n
≥
3 p
l
ayers,
i
n
d
exe
d

b
y
s
u
b
scr
i
p
t
i
.
Pro
d
uct
i

on tec
h
no
l
o
gy

f
or t
h
e pu
bli
c
g
oo
d
ex
hibi
ts constant returns to
s
cale, i.e., the
p
roduction function
f
(
·) is given by
y
=
f
(

f
f
x
(
(
)
=
x
/
b
f
o
r
so
m
e
b
> 0
.
P
re
f
erences are
l
arge
l
y restr
i
cte
d

to t
h
e c
l
ass o
f
quas
ili
near pre
f
erences, except Harsta
d
an
d
Marrese (1982) an
d
Fa
lki
n
g
er et a
l
. (2000). Let
E
represent the set of transitive,
E
c
omplete and convex individual preference orderings,
ՠ
i

,
and initial endowments,
ω
x
i
. We
f
orma
ll
y
d
e
fi
ne E
Q
as
f
o
ll
o
w
s
.
D
EFINITI
O
N 1.
E
Q
=

{(
ՠ
i
,
ω
x
i
)
ʦ
E
:
ՠ
i
is representable b
y
a
C
2
utilit
y
functio
n
o
f
t
h
e
f
orm

v
i
(
y
)
+
x
i
suc
h
t
h
a
t
Dv
i
(
y
)
> 0 an
d
D
2
v
i
(
y
)
<
0

f
or a
ll
y
> 0
,
an
d

ω
x
i
> 0},
x
wh
er
e
D
k
is the
k
k
th
or
d
er
d
er
iv

at
iv
e
.
Falkin
g
er et al. (2000) use a quadratic environment in their experimental stud
y
of
th
e Fa
lki
nger mec
h
an
i
sm. We
d
e
fi
ne t
hi
s env
i
ronment as
E
QD
.
D
EFINITI

O
N 2
.
E
QD
=

{
(
ՠ
i
,
ω
x
i
)
ʦ
E
:
ՠ
i
is representable b
y
a
C
2
utilit
y
function
o

f
t
h
e
f
orm
A
i
x
i
−
1
2
B
i
x
i
2
+
y

wh
ere
A
i
,
B
i
>
0

an
d

ω
x
i
> 0}.
x
A
n economic mec
h
ani
sm

i
s
d
e
fi
ne
d
as a non-cooperat
i
ve game
f
orm p
l
aye
d

by
the a
g
ents. The
g
ame is described in its normal form. In all mechanisms considered
i
n t
hi
s paper, t
h
e
i
mp
l
ementat
i
on concept use
d

i
s Nas
h
equ
ilib
r
i
um. In t
h
e Nas

hi
mp
l
e
-
m
entat
i
on
f
ramewor
k
t
h
e agents are assume
d
to
h
ave comp
l
ete
i
n
f
ormat
i
on a
b
out
the environment while the desi

g
ner does not know an
y
thin
g
about the environment.
3
. EXPERIMENTAL RE
SU
LT
S
S
even exper
i
ments
h
ave
b
een con
d
ucte
d
w
i
t
h
mec
h
an
i

sms
h
av
i
ng Pareto-opt
i
ma
l
Nas
h
equ
ilib
r
i
a
i
n pu
bli
c goo
d
s env
i
ronments (see C
h
en (
f
ort
h
com
i

ng)
f
or a survey).
188 Ex
p
erimental Business Research Vol. I
I
Somet
i
mes t
h
e
d
ata converge
d
qu
i
c
kl
y to t
h
e Nas
h
equ
ilib
r
i
a; ot
h
er t

i
mes
i
t
did
not.
Sm
i
t
h
(1979) stu
di
es a s
i
mp
lifi
e
d
vers
i
on o
f
t
h
e Groves-Le
d
yar
d
mec
h

an
i
sm w
hi
c
h
balanced the bud
g
et onl
y
in equilibrium. In the five-sub
j
ect treatment (R1) one out
o
f
t
h
ree sess
i
ons converge
d
to t
h
e stage game Nas
h
equ
ilib
r
i
um. In t

h
e e
i
g
h
t-su
bj
ect
treatment (R2) ne
i
t
h
er sess
i
on converge
d
to t
h
e Nas
h
equ
ilib
r
i
um pre
di
ct
i
on. Harsta
d

a
nd Marrese (1981) found that onl
y
three out of twelve sessions attained approxim-
a
te
l
y Nas
h
equ
ilib
r
i
um outcomes un
d
er t
h
e s
i
mp
lifi
e
d
vers
i
on o
f
t
h
e Groves-Le

d
yar
d
m
ec
h
an
i
sm. Harsta
d
an
d
Marrese (1982) stu
di
e
d
t
h
e comp
l
ete vers
i
on o
f
t
h
e Groves
-
Led
y

ard mechanism in Cobb-Dou
g
las economies. In the three-sub
j
ect treatment one
out o
f

fi
ve sess
i
ons converge
d
to t
h
e Nas
h
equ
ilib
r
i
um. In t
h
e
f
our-su
bj
ect treatment
one out o
f

f
our sess
i
ons converge
d
to one o
f
t
h
e Nas
h
equ
ilib
r
i
a. Mor
i
(1989)
c
ompares the performance of a Lindahl process with the Groves-Led
y
ard mechan-
i
sm. He ran
fi
ve sess
i
ons
f

or eac
h
mec
h
an
i
sm, w
i
t
h

fi
ve su
bj
ects
i
n eac
h
sess
i
on. T
h
e
a
ggregate
l
eve
l
s o
f

pu
bli
c goo
d
s prov
id
e
d

i
n eac
h
o
f
t
h
e Groves-Le
d
yar
d
sess
i
ons
were much closer to the Pareto optimal level than those provided usin
g
a Lindahl
process. At t
h
e
i

n
di
v
id
ua
l

l
eve
l
, eac
h
o
f
t
h
e
fi
ve sess
i
ons stoppe
d
w
i
t
hi
n ten roun
d
s
w

h
en every su
bj
ect repeate
d
t
h
e same messages. However, s
i
nce
i
n
di
v
id
ua
l
mes-
sa
g
es must be in multiples of .2
5
while the equilibrium messa
g
es were not on the
g
r
id
, convergence to Nas
h

equ
ilib
r
i
um messages was approx
i
mate. None o
f
t
h
e
ab
ove exper
i
ments stu
di
e
d
t
h
e e
ff
ects o
f
t
h
e pun
i
s
h

ment parameter, w
hi
c
h

d
eter-
m
ines the ma
g
nitude of punishment if a pla
y
er’s contribution deviates from the
m
ean o
f
ot
h
er p
l
ayers’ contr
ib
ut
i
ons, on t
h
e per
f
ormance o
f

t
h
e mec
h
an
i
sm.
C
hen and Plott (199
6
) first assessed the performance of the Groves-Ledyard
m
echanism under different punishment parameters. Each
g
roup consisted of five
p
l
ayers w
i
t
h

diff
erent pre
f
erences. T
h
ey
f
oun

d
t
h
at
b
y vary
i
ng t
h
e pun
i
s
h
ment para-
m
eter t
h
e
d
ynam
i
cs an
d
sta
bili
ty c
h
ange
d

d
ramat
i
ca
ll
y. T
hi
s
fi
n
di
ng was rep
li
cate
d
b
y
Chen and Tan
g
(1998) with twent
y
-one independent sessions and a lon
g
er time
ser
i
es (100 roun
d
s)
i

n an exper
i
ment
d
es
i
gne
d
to stu
d
y t
h
e
l
earn
i
ng
d
ynam
i
cs. C
h
en
a
n
d
Tang (1998) a
l
so stu
di

e
d
t
h
e Wa
lk
er mec
h
an
i
sm (Wa
lk
er, 1981)
i
n t
h
e same
e
conomic environment
.
Fi
gure 1 presents t
h
e t
i
me ser
i
es
d
ata

f
rom C
h
en an
d
Tang (1998)
f
or two out o
f
fi
ve types o
f
p
l
ayers. T
h
e
d
ata
f
or t
h
e rema
i
n
i
ng t
h
ree types o
f

p
l
ayers
di
sp
l
ay very
similar patterns. Each t
y
pe differ in their mar
g
inal utilit
y
for the public
g
ood. Each
g
rap
h
presents t
h
e mean (t
h
e
bl
ac
k

d
ots), stan

d
ar
d

d
ev
i
at
i
on (t
h
e error
b
ars) an
d
stage game equ
ilib
r
i
a (t
h
e
d
as
h
e
d

li
nes)

f
or eac
h
o
f
t
h
e two
diff
erent types average
d
over seven independent sessions for each mechanism. The two
g
raphs in the first
c
o
l
umn
di
sp
l
ay t
h
e mean contr
ib
ut
i
on (an
d
stan

d
ar
d

d
ev
i
at
i
on)
f
or types 1 an
d
2
p
l
ayers un
d
er t
h
e Wa
lk
er mec
h
an
i
sm (
h
erea
f

ter Wa
lk
er). T
h
e secon
d
co
l
umn
di
s-
pla
y
s the avera
g
e contributions for t
y
pes 1 and 2 for the Groves-Led
y
ard mechan-
i
sm un
d
er a
l
ow pun
i
s
h
ment parameter (

h
erea
f
ter GL1). T
h
e t
hi
r
d
co
l
umn
di
sp
l
ays
t
h
e same
i
n
f
ormat
i
on
f
or t
h
e Groves-Le
d

yar
d
mec
h
an
i
sm un
d
er a
hi
g
h
pun
i
s
h
-
m
ent parameter (hereafter GL100). From these
g
raphs, it is apparent that all seven
sess
i
ons o
f
t
h
e Groves-Le
d
yar

d
mec
h
an
i
sm un
d
er a
hi
g
h
pun
i
s
h
ment parameter
c
onverge
d
3
very qu
i
c
kl
y to
i
ts stage game Nas
h
equ
ilib

r
i
um an
d
rema
i
ne
d
sta
bl
e
,
D
YNAMI
C
S
C
TABILITY
OF
N
F
ASH
N
N
-E
FFICIENT
E
E
P
T

U
BLI
C
G
C
OODS
M
S
ECHANISMS
MM
189
Mean contribution of type 1 players
30
20
10
0
–10
–20
100
0 20 40 60 80 100
Walker Mechanism
Nash Equilibrium
Round
Mean contribution of type 1 players
30
20
10
0
–10
–20

0 20 40 60 80 100
Groves–Ledyard Mechanism
(low punishment parameter)
Nash Equilibrium
Round
Mean contribution of type 1 players
30
20
10
0
–10
–20
0 20 40 60 80 100
Groves–Ledyard Mechanism
(high punishment parameter)
Nash Equilibrium
Round
Mean contribution of type 2 players
30
20
10
0
–10
–20
0 20 40 60 80 100
Nash Equilibrium
Round
Mean contribution of type 2 players
30
20

10
0
–10
–20
0 20 40 60 80 100
Nash Equilibrium
Round
Mean contribution of type 2 players
30
20
10
0
–10
–20
0 20 40 60 80 100
Nash Equilibrium
Round
Figure 1. Mean Contribution and Standard Deviation in Chen and Tang (1998).
190 Ex
p
erimental Business Research Vol. I
I
w
hil
e t
h
e same mec
h
an
i

sm
did
not converge un
d
er a
l
ow pun
i
s
h
ment parameter; t
h
e
Wa
lk
er mec
h
an
i
sm
did
not converge to
i
ts stage game Nas
h
equ
ilib
r
i
um e

i
t
h
er.
Because of its
g
ood d
y
namic properties, GL100 had far better performance than
GL1 an
d
Wa
lk
er, eva
l
uate
d

i
n terms o
f
system e
ffi
c
i
ency, c
l
ose to Pareto opt
i
ma

l
l
eve
l
o
f
pu
bli
c goo
d
s prov
i
s
i
on,
l
ess v
i
o
l
at
i
ons o
f

i
n
di
v
id

ua
l
rat
i
ona
li
ty constra
i
nts
a
nd conver
g
ence to its sta
g
e
g
ame equilibrium. All these results are statisticall
y
hi
g
hl
y s
i
gn
ifi
cant (C
h
en an
d
Tang, 1998).

T
h
ese resu
l
ts
ill
ustrate t
h
e
i
mportance to
d
es
i
gn mec
h
an
i
sms w
hi
c
h
not on
l
y
h
ave
g
ood static properties, but also
g

ood d
y
namic stabilit
y
properties like GL100.
O
n
l
y w
h
en t
h
e
d
ynam
i
cs
l
ea
d
to t
h
e convergence to t
h
e stat
i
c equ
ilib
r
i

um, can a
ll
t
h
e n
i
ce stat
i
c propert
i
es
b
e rea
li
ze
d
.
F
alkin
g
er et al. (2000) stud
y
the Falkin
g
er mechanism in a quasilinear as well as
a
qua
d
rat
i

c env
i
ronment. In t
h
e quas
ili
near env
i
ronment, t
h
e mean contr
ib
ut
i
ons
m
ove
d
towar
d
s t
h
e Nas
h
equ
ilib
r
i
um
l

eve
l

b
ut
did
not qu
i
te reac
h
t
h
e equ
ilib
r
i
um.
I
n the
q
uadratic environment the mean contribution level hovered around the Nash
e
qu
ilib
r
i
um, even t
h
oug
h

none o
f
t
h
e 23 sess
i
ons
h
a
d
a mean contr
ib
ut
i
on
l
eve
l
e
xact
l
y equa
l
to t
h
e Nas
h
equ
ilib
r

i
um
l
eve
l

i
n t
h
e
l
ast
fi
ve roun
d
s. T
h
ere
f
ore, Nas
h
e
quilibrium was a
g
ood description of the avera
g
e contribution pattern, althou
g
h
i

n
di
v
id
ua
l
p
l
ayers
did
not necessar
il
y p
l
ay t
h
e equ
ilib
r
i
um.
I
n Section 5 we will provide a theoretical explanation for the above experimental
results in li
g
ht of supermodular
g
ames.
4. SUPERMODULARITY AND STABILITY
We first define supermodular

g
ames and review their stabilit
y
properties. Then we
di
scuss a
l
ternat
i
ve sta
bili
ty cr
i
ter
i
a an
d
t
h
e
i
r re
l
at
i
ons
hi
p w
i
t

h
supermo
d
u
l
ar
i
ty.
Supermo
d
u
l
ar
g
ames are
g
ames
i
n w
hi
c
h
eac
h
p
l
a
y
er’s mar
gi

na
l
ut
ili
t
y
o
f

i
n-
c
reasin
g
her strate
gy
rises with increases in her rival’s strate
g
ies, so that (rou
g
hl
y
)
t
h
e p
l
ayer’s strateg
i
es are “strateg

i
c comp
l
ements.” Supermo
d
u
l
ar games nee
d
an
or
d
er structure on strate
gy
spaces, a wea
k
cont
i
nu
i
t
y
requ
i
rement on pa
y
o
ff
s, an
d

c
omplementarit
y
between components of a pla
y
er’s own strate
g
ies, in addition to
t
h
e a
b
ove-ment
i
one
d
strateg
i
c comp
l
ementar
i
ty
b
etween p
l
ayers’ strateg
i
es. Suppose
e

ac
h
p
l
a
y
er
i
’
s strate
gy
set
S
i
is a subset of a finite-dimensional Euclidean s
p
ace
R
k
i
.
T
h
en
S
≡
×
n
i
=

1
S
i
i
s

a

subset

of

R
k
, where
k
=
∑
n
i
=
1
k
i
.
DEFINITION 3
.
A
s
upermo

d
u
l
ar gam
e
i
s suc
h
t
h
at,
f
or eac
h
p
l
a
y
e
r

i
,
S
i

i
s a non-
e
mpt

y
sublattice o
f

R
k
i
,
u
i
is u
pp
er semi-continuous in
s
i
f
o
r fix
ed
s
−
i
a
n
d

co
n
t
in

uous
in
s
−
i
f
o
r fix
ed
s
i
,
u
i
has increasing differences in (
s
i
,

s
−
i
)
, and
u
i
is su
p
ermodular in
s

i
.
I
ncreas
i
n
g

diff
erences sa
y
s t
h
at an
i
ncrease
i
n t
h
e strate
gy
o
f
p
l
a
y
er
i
’s r

iv
a
l
s
raises her mar
g
inal utilit
y
of pla
y
in
g
a hi
g
h strate
gy
. The supermodularit
y
assump-
t
i
on ensures comp
l
ementar
i
ty among components o
f
a p
l
ayer’s own strateg

i
es. Note
t
h
at
i
t
i
s automat
i
ca
lly
sat
i
s
fi
e
d
w
h
en
S
i
i
s one-
di
mens
i
ona
l

. As t
h
e
f
o
ll
ow
i
ng t
h
eorem
D
YNAMI
C
S
C
TABILITY
OF
N
F
ASH
N
N
-E
FFICIENT
E
E
P
T
U

BLI
C
G
C
OODS
M
S
ECHANISMS
MM
191
i
n
di
cates supermo
d
u
l
ar
i
ty an
d

i
ncreas
i
ng
diff
erences are eas
il
y c

h
aracter
i
ze
d

f
or
s
moot
h

f
unct
i
ons
i
n
R
n
.
T
HE
O
REM 1.
(
To
pk
is, 197
8

)
Le
t
u
i

b
e tw
i
ce cont
i
nuous
l
y
diff
erent
i
a
bl
e on S
i
.
Th
en
u
i

h
as
i

ncreas
i
ng
diff
erences
i
n
(
s
i
,
s
j
s
)
if
an
d
on
l
y
if

∂
2
u
i
/
∂
s

ih
∂
s
jl
s
≥

0

f
or a
ll
i
≠
j
a
n
d

a
ll 1
≤
h
≤
k
i

a
n
d

a
ll 1
≤
l
≤
k
j
k
; and
u
i
is su
p
ermodular in
s
i
if and onl
y
i
f
∂
2
u
i
/
∂
s
ih
∂

s
il
≥

0

f
or a
ll
i
an
d
a
ll
1
≤
h
<
l
≤
k
i
.
S
upermodular
g
ames are of interest particularl
y
because of their ver
y

robus
t
s
ta
bili
ty propert
i
es. M
il
grom an
d
Ro
b
erts (1990) prove
d
t
h
at
i
n t
h
ese games t
h
e set
o
f

l
earn
i

ng a
l
gor
i
t
h
ms cons
i
stent w
i
t
h
a
d
apt
i
ve
l
earn
i
ng converge to t
h
e set
b
oun
d
e
d
b
y

the lar
g
est and the smallest Nash equilibrium strate
gy
profiles. Intuitivel
y
, a
s
equence
i
s cons
i
stent w
i
t
h
a
d
apt
i
ve
l
earn
i
ng
if
p
l
ayers “eventua
ll

y a
b
an
d
on strateg
i
e
s
th
at per
f
orm cons
i
stent
l
y
b
a
dl
y
i
n t
h
e sense t
h
at t
h
ere ex
i
sts some ot

h
er strategy t
h
at
p
erforms strictl
y
and uniforml
y
better a
g
ainst ever
y
combination of what the com-
p
et
i
tors
h
ave p
l
aye
d

i
n t
h
e not too
di
stant past.” (M

il
grom an
d
Ro
b
erts, 1990) T
hi
s
i
nc
l
u
d
es a w
id
e c
l
ass o
f

i
nterest
i
ng
l
earn
i
ng
d
ynam

i
cs, suc
h
as Bayes
i
an
l
earn
i
ng,
fi
ctitious pla
y
, adaptive learnin
g
, Cournot best-repl
y
and man
y
others.
Si
nce exper
i
menta
l
ev
id
ence suggests t
h
at

i
n
di
v
id
ua
l
p
l
ayers ten
d
to a
d
opt
diff
er-
e
nt learning rules (El-Gamal and Grether, 1995), instead of using a specific learn-
i
n
g
al
g
orithm to stud
y
stabilit
y
, one can use supermodularit
y
as a robust stabilit

y
c
riterion for
g
ames with a unique Nash equilibrium. For supermodular
g
ames with
a un
i
que Nas
h
equ
ilib
r
i
um, we expect any a
d
apt
i
ve
l
earn
i
ng a
l
gor
i
t
h
m to converge

t
o the unique Nash equilibrium, in particular, Cournot best-repl
y
, fictitious pla
y
and adaptive learnin
g
. Compared with stabilit
y
anal
y
sis usin
g
Cournot best-repl
y
d
ynam
i
cs, supermo
d
u
l
ar
i
ty
i
s muc
h
more ro
b

ust an
d

i
nc
l
us
i
ve
i
n t
h
e sense t
h
at
i
t
i
mplies stabilit
y
under Cournot best-repl
y
and man
y
other learnin
g
d
y
namics men-
t

ioned above.
5
.
SU
PERM
O
D
U
LARITY
O
F EXI
S
TIN
G
NA
S
H-EFFI
C
IENT
PU
BLI
C

GOO
D
S
ME
C
HANI
S

M
S
In t
hi
s sect
i
on we
i
nvest
i
gate t
h
e supermo
d
u
l
ar
i
ty o
f

fi
ve we
ll
-
k
nown Nas
h
-e
ffi

c
i
ent
p
u
bli
c goo
d
s mec
h
an
i
sms. We use supermo
d
u
l
ar
i
ty to ana
l
yze t
h
e exper
i
menta
l
r
esults on Nash-efficient public
g
oods mechanisms.

Th
e Groves-Le
d
yar
d
mec
h
an
i
sm (1977)
i
s t
h
e
fi
rst mec
h
an
i
sm
i
n a genera
l
equ
i
-
lib
r
i
um sett

i
ng w
h
ose Nas
h
equ
ilib
r
i
um
i
s Pareto opt
i
ma
l
. T
h
e mec
h
an
i
sm a
ll
ocates
p
rivate
g
oods throu
g
h the competitive markets and public

g
oods throu
g
h a
g
overn-
m
ent a
ll
ocat
i
on-taxat
i
on sc
h
eme t
h
at
d
epen
d
s on
i
n
f
ormat
i
on commun
i
cate

d
to t
h
e
government
b
y consumers regar
di
ng t
h
e
i
r pre
f
erences. G
i
ven t
h
e government sc
h
eme
,
c
onsumers find it in their best interest to reveal their true
p
references for
p
ublic
goo
d

s. T
h
e mec
h
an
i
sm
b
a
l
ances t
h
e
b
u
d
get
b
ot
h
on an
d
o
ff
t
h
e equ
ilib
r
i

um pat
h
,
b
ut
i
t
d
oes not
i
mp
l
ement L
i
n
d
a
hl
a
ll
ocat
i
ons. Later on, more game
f
orms
h
ave
b
een
192 Ex

p
erimental Business Research Vol. I
I
di
scovere
d
w
hi
c
h

i
mp
l
ement L
i
n
d
a
hl
a
ll
ocat
i
ons
i
n Nas
h
equ
ilib

r
i
um. T
h
ese
i
nc
l
u
d
e
H
urwicz (1979), Walker (1981), Tian (1989), Kim (1993) and Peleg (199
6
).
DEFINITI
O
N 4
.
F
or t
h
e Groves-Le
d
yar
d
mec
h
an
i

sm, t
h
e strategy space o
f
p
l
aye
r
i
i
s
S
i
⊂
R
1
w
i
t
h

g
ener
i
c e
l
ement
m
i
ʦ

S
i
.
T
h
e outcome
f
unct
i
on o
f
t
h
e pu
bli
c
g
oo
d
a
nd the net cost share of the private
g
ood for pla
y
er
i
a
r
e
Ym m

k
k
()
m
m
∑
T
Y
n
b
n
n
i
T
T
GL
ii i
()
m
()
m
b
( ) – .
m
ii i
⋅+
b
()
−
⎡

⎣
⎢
⎡
⎡
⎣
⎣
⎤
⎦
⎦
⎦
γ
µ
i
)–)–
ii
i
2
1
22
wh
ere
γ
> 0,
γ
n
≥
3,
µ
−
i

=
∑
j
∑
≠
i
m
j
m
/(
n
−
1)
i
s t
h
e mean o
f
ot
h
ers’ messages, an
d
σ
2
−
i
=
∑
h
≠

i
(
m
h
−
µ
−
i
)
2
/(
n
−
2)
i
s t
h
e square
d
stan
d
ar
d
error o
f
t
h
e mean o
f
ot

h
ers’
m
essa
g
es
.
I
n t
h
e Groves-Le
dy
ar
d
mec
h
an
i
sm eac
h
a
g
ent reports
m
i
, t
h
e
i
ncrement

(
or
d
ecre
-
m
ent) of the public
g
ood pla
y
er
i
would like to add to (or subtract from) the amounts
propose
d

b
y ot
h
ers. T
h
e p
l
anner sums up t
h
e
i
n
di
v

id
ua
l
contr
ib
ut
i
ons to get t
h
e tota
l
a
mount o
f
pu
bli
c
g
oo
d
,
Y
, and taxes each individual based on her own message, and
Y
Y
the mean and sample variance of ever
y
one else’s messa
g
es. Thus each individual’s

tax s
h
are
i
s compose
d
o
f
t
h
ree parts: t
h
e per cap
i
ta cost o
f
pro
d
uct
i
on,
Y
·
Y
b
/
n
/
,
p

l
us
a
pos
i
t
i
ve mu
l
t
i
p
l
e,
γ
/2, of the difference between her own message and the mean of
γ
o
thers’ messa
g
es, (
n
−
1)/
n
/
/
×
(
m

i
−
µ
−
i
)
2
,
and the sam
p
le variance of others’ mes-
sages,
σ
2
−
i
. W
hil
e t
h
e
fi
rst two parts guarantee t
h
at Nas
h
equ
ilib
r
i

a o
f
t
h
e mec
h
an
i
sm
a
re Pareto opt
i
ma
l
, t
h
e
l
ast part
i
nsures t
h
at
b
u
dg
et
i
s
b

a
l
ance
d

b
ot
h
on an
d
o
ff
the e
q
uilibrium
p
ath. Note that the free
p
arameter,
γ
, determines the magnitude of
γ
pun
i
s
h
ment w
h
en an
i

n
di
v
id
ua
l

d
ev
i
ates
f
rom t
h
e mean o
f
ot
h
ers’ messages. It
d
oes
n
ot a
ff
ect an
y
o
f
t
h

e stat
i
c t
h
eoret
i
ca
l
propert
i
es o
f
t
h
e mec
h
an
i
sm.
C
hen and Plott (1996) and Chen and Tan
g
(1998) found that the punishment
parameter,
γ
, had a significant effect in inducing convergence and dynamic stability.
γ
For a
l
ar

g
e enou
gh
γ
, the system converged to its stage game Nash equilibrium very
γ
quickl
y
and remained stable; while under a small
γ
, the system did not converge
γ
to
i
ts stage game Nas
h
equ
ilib
r
i
um. In t
h
e
f
o
ll
ow
i
ng propos
i

t
i
on, we prov
id
e a
n
ecessar
y
an
d
su
ffi
c
i
ent con
di
t
i
on
f
or t
h
e mec
h
an
i
sm to
b
e a supermo
d

u
l
ar
g
ame
g
iven quasilinear preferences, and thus to conver
g
e to its Nash equilibrium under a
w
id
e c
l
ass o
f

l
earn
i
ng
d
ynam
i
cs.
P
R
O
P
OS
ITI

O
N 1. The Groves-Led
y
ard mechanism is a supermodular
g
ame for
a
n
y
e
ʦ
E
Q

if
an
d
on
l
y
if

γ
ʦ
[
ʦ
−
m
i
n

i
ʦ
N
∂
∂
2
2
v
y
i
{}
n
,
+
∞]
.
P
roof:
Si
nce
u
i

i
s
C
2

on

S
i
,
b
y T
h
eorem 1,
u
i

h
as
i
ncreas
i
ng
diff
erences
i
n (
m
i
,
m
−
i
)
if
an
d

on
l
y
if
D
YNAMI
C
S
C
TABILITY
OF
N
F
ASH
N
N
-E
FFICIENT
E
E
P
T
U
BLI
C
G
C
OODS
M
S

ECHANISMS
MM
193
∂
∂∂
∂
∂
∀
22
∂
2
u
m
∂
v
y
i
ij
∂
m
∂
i
/ , ,
∀
0

∂
∂
2
v

ni
∀
0
i
/
∂
2
n
i
γ
which holds if and only i
f

γ
ʦ
[
−
min
i
ʦ
N
∂
∂
2
2
v
y
i
{}
n

,
+
∞
]
.
Q
.E.D
.
Th
ere
f
ore, w
h
en t
h
e pun
i
s
h
ment parameter
i
s a
b
ove t
h
e t
h
res
h
o

ld
, a
l
arge c
l
ass
of interestin
g
learnin
g
d
y
namics conver
g
e, which is consistent with the experimental
r
esu
l
ts. Intu
i
t
i
ve
l
y, w
h
en t
h
e pun
i

s
h
ment parameter
i
s su
ffi
c
i
ent
l
y
hi
g
h
, t
h
e
i
ncent
i
ve
f
or eac
h
agent to matc
h
t
h
e mean o
f

ot
h
er agents’ messages
i
s a
l
so
hi
g
h
. T
h
ere
f
ore,
when other a
g
ents increase their contributions, a
g
en
t

i
also wants to increase her
c
ontr
ib
ut
i
on to avo

id
t
h
e pena
l
ty. T
h
us t
h
e messages
b
ecome strateg
i
c comp
l
ements
an
d
t
h
e game
i
s trans
f
orme
d

i
nto a supermo
d

u
l
ar game. Muenc
h
an
d
Wa
lk
er (1983)
found a conver
g
ence condition for the Groves-Led
y
ard mechanism usin
g
Courno
t
b
est-rep
l
y
d
ynam
i
cs an
d
parameter
i
ze
d

qua
d
rat
i
c pre
f
erences. T
hi
s propos
i
t
i
on gen-
e
ra
li
zes t
h
e
i
r resu
l
t to genera
l
quas
ili
near pre
f
erences an
d

a muc
h
w
id
er c
l
ass o
f
learnin
g
d
y
namics.
Falkinger (199
6
) introduces a class of simple mechanisms. In this incentive
c
ompat
ibl
e mec
h
an
i
sm
f
or pu
bli
c goo
d
s, Nas

h
equ
ilib
r
i
um
i
s Pareto opt
i
ma
l
w
h
en
a parameter is chosen appropriatel
y
, i.e., when
β
=
1
−
1
/
n
.
However, it does not
i
mp
l
ement L

i
n
d
a
hl
a
ll
ocat
i
ons an
d
t
h
e ex
i
stence o
f
equ
ilib
r
i
um can
b
e
d
e
li
cate
i
n

s
ome en
vi
ronments
.
DEFINITI
O
N 5
.
F
or the Falkinger (199
6
) mechanism, the strategy space of player
i
i
s
S
i
⊂
R
1
w
i
t
h
gener
i
c e
l
emen

t

m
i
ʦ
S
i
.
T
h
e outcome
f
unct
i
on o
f
t
h
e pu
bli
c goo
d
and the net cost share of the private
g
ood for pla
y
er
i
are
Y

k
k
() ,
mm
m
k
∑
Tm bm
m
m
n
i
T
T
F
i
i
j
m
ji

m
()
mb
m
m
i
,
bm
i

−
−
⎛
⎝
⎜
⎛
⎛
⎝
⎝
⎞
⎠
⎟
⎞
⎞
⎠
⎠
⎡
⎣
⎢
⎡
⎡
⎢
⎣
⎣
⎤
⎦
⎥
⎤⎤
⎥
⎦⎦

⎥⎥
∑
β
1
wher
e
β
> 0.
β
Thi
s tax-su
b
s
idy
sc
h
eme wor
k
s as
f
o
ll
ows:
if
an
i
n
di
v
id

ua
l
’s contr
ib
ut
i
on
i
s
above the avera
g
e contribution of the others, she
g
ets a subsid
y
of
β
for a mar
g
inal
i
ncrease
i
n
h
er contr
ib
ut
i
on. I

f

h
er contr
ib
ut
i
on
i
s
b
e
l
ow t
h
e average contr
ib
ut
i
on o
f
ot
h
ers, s
h
e
h
as to pay a tax w
h
ere

b
y a marg
i
na
l

i
ncrease
i
n
h
er contr
ib
ut
i
on re
d
uces
her tax pa
y
ment b
y
β
. If
β
β
is chosen appropriately, Nash equilibrium of this mech-
β
an
i

sm
i
s Pareto e
ffi
c
i
ent. Furt
h
ermore,
i
t
f
u
ll
y
b
a
l
ances t
h
e
b
u
d
get
b
ot
h
on an
d

o
ff
th
e equ
ilib
r
i
um pat
h
.
194 Ex
p
erimental Business Research Vol. I
I
P
ROPO
S
ITION 2.
Th
e Fa
lki
nger mec
h
an
i
sm
i
s a supermo
d
u

l
ar game
f
or any
e
ʦ
E
Q
D
if
an
d
on
ly

if
β
≥

1.
P
roof:
Si
nce
u
i

i
s
C

2
on

S
i
,

b
y T
h
eorem 1,
u
i

h
as
i
ncreas
i
ng
diff
erences
i
n (
m
i
,

m
−

i
)
i
f and onl
y
if
∂
∂∂
∀
22
1
u
m
∂
Bb
n
i
ij
∂
m
∂
i
BB
) , ,
∀
10
i
=
−
1

ββ
(
w
hi
c
h

h
o
ld
s
if
an
d
on
l
y
if

β
≥
1.
Q
.E.D.
S
i
nce Pareto e
ffi
c
i

enc
y
requ
i
res t
h
at
β
=
1

−
1/
n
,
i
n a
l
ar
g
e econom
y
, t
hi
s w
ill
produce a
g
ame which is close to bein
g

a supermodular
g
ame. It is interestin
g
to note that in the quadratic environment of Falkinger et al. (2000), the game is
v
er
y
c
l
ose to
b
e
i
n
g
a supermo
d
u
l
ar
g
ame:
i
n t
h
e exper
i
ment
β

was set to 2/3. The
β
results show the mean contribution level hovered around the Nash e
q
uilibrium, even
though none of the 23 sessions had a mean contribution level exactly equal to the
Nas
h
equ
ilib
r
i
um
l
eve
l

i
n t
h
e
l
ast
fi
ve roun
d
s. T
h
e
i

r resu
l
ts su
gg
est t
h
at t
h
e con-
v
er
g
ence in supermodular
g
ames mi
g
ht be a function of the de
g
ree of strate
g
ic
c
omplementarity. That is, in games with a unique Nash equilibrium which can
i
n
d
uce supermo
d
u
l

ar
g
ames, suc
h
as t
h
e Groves-Le
dy
ar
d
mec
h
an
i
sm
f
or an
y

e
ʦ
E
Q
a
nd the Falkin
g
er mechanism for an
y

e

ʦ
E
Q
D
, as the de
g
ree of strate
g
ic comple
-
m
entarity increases, we might observe more rapid convergence to its stage game
Nas
h
equ
ilib
r
i
um
.
Three specific
g
ame forms implementin
g
Lindahl allocations in Nash equilib-
rium have been introduced, Hurwicz
(
1979
)
, Walker

(
1981
)
, and Kim
(
1993
)
. Since
T
ian (1989) and Pele
g
(199
6
) do not have specific mechanisms, we will onl
y
inves
-
ti
g
ate the supermodularit
y
of these three mechanisms. All three improve on the
Groves-Ledyard mechanism in the sense that they all satisfy the individual ration-
ali
t
y
constra
i
nt
i

n equ
ilib
r
i
um. W
hil
e Hurw
i
cz (1979) an
d
Wa
lk
er (1981) can
b
e
shown to be unstable for an
y
decentralized ad
j
ustment process in certain quadratic
e
nvironments (Kim, 1986), the Kim mechanism is stable under a gradient adjust-
m
ent process
gi
ven quas
ili
near ut
ili
t

y

f
unct
i
ons, w
hi
c
h

i
s a cont
i
nuous t
i
me vers
i
on
o
f the Cournot-Nash tâtonnement ad
j
ustment process. Whether the Kim mechanism
i
s stable under other decentralized learning processes is still an open question. Kim
(198
6
) has shown that for an
y

g

ame form implementin
g
Lindahl allocations there
d
oes not exist a decentralized ad
j
ustment process which ensures local stabilit
y
of
Nash e
q
uilibria in certain classes of environments.
P
R
O
P
OS
ITI
O
N 3
.
N
one of the Hurwicz (1979), Walker (1981) and Kim (1993)
m
echanisms is a supermodular
g
ame for an
y

e

ʦ
E
Q
.
P
roof: See Appen
di
x.
z
D
YNAMI
C
S
C
TABILITY
OF
N
F
ASH
N
N
-E
FFICIENT
E
E
P
T
U
BLI
C

G
C
OODS
M
S
ECHANISMS
MM
1
95
Th
e
f
o
ll
ow
i
ng o
b
servat
i
on organ
i
zes a
ll
exper
i
menta
l
resu
l

ts on Nas
h
-e
ffi
c
i
ent
p
u
bli
c goo
d
s mec
h
an
i
sms w
i
t
h
ava
il
a
bl
e parameters
b
y
l
oo
ki

ng at w
h
et
h
er t
h
ey are
s
upermodular
g
ames. The desi
g
n parameters used in Smith’s (1979) R1 treatment
an
d
Harsta
d
an
d
Marrese
(
1981
)
are not ava
il
a
bl
e.
O
B

S
ERVATION 1
.
(
1) None o
f
t
h
e
f
o
ll
owing experiments is a supermo
d
u
l
ar
g
ame: t
h
e Groves-Le
d
yar
d
mec
h
anism stu
d
ie
d

in Smit
h
’s (1979) R2 treatment,
Harstad and Marrese
(
1982), Mori
(
1989), Chen and Plott
(
199
6
)’s low
γ
treatment,
γ
a
n
d
C
h
en an
d
Tan
g
(1998)’s
l
ow
γ
treatment, the Walker mechanism in Chen and
γ

T
ang (1998), an
d
t
h
e Fa
lk
inger mec
h
anism in Fa
lk
inger et a
l
. (2000).
(
2) The Groves-Ledyard mechanism under the high
γ
in Chen and Plott (1996)
γ
a
n
d
C
h
en an
d
Tan
g
(1998) are
b

ot
h
supermo
d
u
l
ar
g
ames.
Th
ere
f
ore, none o
f
t
h
e ex
i
st
i
ng exper
i
ments w
hi
c
h

did
not converge
i

s a
s
upermo
d
u
l
ar game, w
hil
e t
h
ose w
hi
c
h

did
converge we
ll
are
b
ot
h
supermo
d
u
l
ar
games
.
Note t

h
at
d
es
i
gn
i
ng a mec
h
an
i
sm as a supermo
d
u
l
ar game m
i
g
h
t requ
i
re some
i
n
f
ormat
i
on on t
h
e part o

f
t
h
e p
l
anner. For examp
l
e, un
d
er t
h
e Groves-Le
d
yar
d
m
echanism, when choosin
g
parameters to induce supermodularit
y
, the planner needs
t
o
k
now t
h
e sma
ll
est secon
d

part
i
a
l

d
er
i
vat
i
ve o
f
t
h
e p
l
ayers’ ut
ili
ty
f
or pu
bli
c goo
d
s
i
n the societ
y
, i.e., mi
n

i
ʦ
N
∂
∂
2
2
v
y
i
,
for all possible levels of the public
g
ood,
y
,
which i
s
s
tate-
d
epen
d
ent
i
n
f
ormat
i
on. In Nas

h

i
mp
l
ementat
i
on t
h
eory we usua
ll
y assume t
h
at
the planner does not have an
y
information about the pla
y
ers’ preferences. In that
c
ase, even thou
g
h there exist a set of stable mechanisms amon
g
a famil
y
of mechan-
i
sms, t
h

e p
l
anner
d
oes not
h
ave t
h
e
i
n
f
ormat
i
on to c
h
oose t
h
e r
i
g
h
t one. T
h
ere
f
ore,
i
n order to choose
p

arameters to im
p
lement the stable set of mechanisms, the
p
lan-
n
er needs to have some information about the distribution of
p
references and an
e
st
i
mate a
b
out t
h
e poss
ibl
e range o
f
pu
bli
c goo
d
s
l
eve
l
. One poss
ibl

e way o
f
o
b
ta
i
n-
i
n
g
the information is throu
g
h samplin
g
(Gar
y
-Bobo and Jaaidane, 2000). If the
r
equisite information is not available, then an alternative mi
g
ht be to use “approx-
i
mate
l
y” supermo
d
u
l
ar mec
h

an
i
sms, suc
h
as t
h
e Fa
lki
nger mec
h
an
i
sm. In
l
arge econo
-
m
ies when the
p
lanner selects
β
=
1
−
1/
n
t
o induce efficienc
y
, the mechanism is

approximatel
y
supermodular.
6
.
CO
N
C
L
U
DIN
G
REMARK
S
S
o far Nash implementation theor
y
has mainl
y
focused on establishin
g
static pro-
p
ert
i
es o
f
t
h
e equ

ilib
r
i
a. However, exper
i
menta
l
ev
id
ence suggests t
h
at t
h
e
f
un
d
a-
m
enta
l
quest
i
on concern
i
ng any actua
l

i
mp

l
ementat
i
on o
f
a spec
ifi
c mec
h
an
i
sm
i
s
whether decentralized d
y
namic learnin
g
processes will actuall
y
conver
g
e to one of
th
e equ
ilib
r
i
a prom
i

se
d

b
y t
h
eory. Base
d
on
i
ts attract
i
ve t
h
eoret
i
ca
l
propert
i
es an
d
th
e support
i
ng ev
id
ence
f
or t

h
ese propert
i
es
i
n t
h
e exper
i
menta
l

li
terature, we
f
ocus
196 Ex
p
erimental Business Research Vol. I
I
on supermo
d
u
l
ar
i
ty as a ro
b
ust sta
bili

ty cr
i
ter
i
on
f
or Nas
h
-e
ffi
c
i
ent pu
bli
c goo
d
s
m
ec
h
an
i
sms w
i
t
h
a un
i
que Nas
h

equ
ilib
r
i
um
.
This paper demonstrates that
g
iven a quasilinear utilit
y
function the Groves-
Le
d
yar
d
mec
h
an
i
sm
i
s a supermo
d
u
l
ar game
if
an
d
on

l
y
if
t
h
e pun
i
s
h
ment parameter
i
s a
b
ove a certa
i
n t
h
res
h
o
ld
w
hil
e none o
f
t
h
e Hurw
i
cz

,
Wa
lk
er an
d
K
i
m mec
h
an-
i
sms is a supermodular
g
ame. The Falkin
g
er mechanism can be converted into a
supermo
d
u
l
ar game
i
n a qua
d
rat
i
c env
i
ronment
if

t
h
e su
b
s
id
y coe
ffi
c
i
ent
i
s at
l
east
one. T
h
ese resu
l
ts genera
li
ze a prev
i
ous convergence resu
l
t on t
h
e Groves-Le
d
yar

d
m
echanism b
y
Muench and Walker (1983). The
y
are consistent with the experi-
m
enta
l

fi
n
di
ngs o
f

i
n Sm
i
t
h
(1979), Harsta
d
an
d
Marrese (1982), Mor
i
(1989), C
h

en
a
nd Plott (199
6
), Chen and Tang (1998), and Falkinger et al. (2000).
Two aspects of the conver
g
ence and stabilit
y
anal
y
sis in this paper warrant
a
ttent
i
on. F
i
rst, supermo
d
u
l
ar
i
ty
i
s su
ffi
c
i
ent

b
ut not necessary
f
or convergence to
h
o
ld
. It
i
s poss
ibl
e t
h
at a mec
h
an
i
sm cou
ld

f
a
il
supermo
d
u
l
ar
i
ty

b
ut st
ill

b
e
h
aves
well on a class of ad
j
ustment d
y
namics, such as the Kim mechanism. Secondl
y
, The
sta
bili
ty ana
l
ys
i
s
i
n t
hi
s paper,
lik
e ot
h
er t

h
eoret
i
ca
l
stu
di
es o
f
t
h
e
d
ynam
i
c sta
bili
ty
o
f
Nas
h
mec
h
an
i
sms,
h
ave
b

een most
l
y restr
i
cte
d
to quas
ili
near ut
ili
ty
f
unct
i
ons. It
i
s desirable to extend the anal
y
sis to other more
g
eneral environments. The maximal
d
oma
i
n o
f
sta
bl
e env
i

ronments rema
i
ns an open quest
i
on
.
R
esults in this paper su
gg
est a new research a
g
enda that s
y
stematicall
y
investi
g
ates
t
h
e ro
l
e o
f
supermo
d
u
l
ar
i

ty
i
n
l
earn
i
ng an
d
convergence to Nas
h
equ
ilib
r
i
um. Two
stu
di
es p
i
oneer t
hi
s new researc
h
agen
d
a. Ar
if
ov
i
c an

d
Le
d
yar
d
(2003) stu
d
y t
h
e
Groves-Led
y
ard mechanism in the same environment as Chen and Tan
g
(1998), but
u
se a muc
h

l
arger num
b
er o
f
pun
i
s
h
ment parameters. C
h

en an
d
Gazza
l
e(
f
ort
h
com
i
ng)
stu
d
y
l
earn
i
ng an
d
convergence
i
n Var
i
an’s (1994) compensat
i
on mec
h
an
i
sm

b
y
s
y
stematicall
y
var
y
in
g
a free parameter below, close to, at and be
y
ond the threshold
o
f
supermo
d
u
l
ar
i
ty to assess
i
ts e
ff
ects on convergence. F
i
n
di
ngs

f
rom
b
ot
h
stu
di
es
a
re cons
i
stent. F
i
rst, supermo
d
u
l
ar an
d
“near-supermo
d
u
l
ar” games converge s
i
g-
n
ificantl
y
better than those far below the threshold. Second, from a little below the

t
h
res
h
o
ld
to t
h
e t
h
res
h
o
ld
, t
h
e
i
mprovement
i
s stat
i
st
i
ca
ll
y
i
ns
i

gn
ifi
cant. T
hi
r
d
, w
i
t
hin
t
h
e c
l
ass o
f
supermo
d
u
l
ar games,
i
ncreas
i
ng t
h
e parameter
f
ar
b

eyon
d
t
h
e t
h
res
h
o
ld
d
oes not si
g
nificantl
y
improve conver
g
ence. The robustness of these findin
g
s should
b
e
f
urt
h
er
i
nvest
i
gate

d

i
n
f
uture exper
i
ments
i
n ot
h
er games,
f
or examp
l
e, t
h
e Fa
lki
nger
m
ec
h
an
i
sm, as we
ll
as games outs
id
e t

h
e pu
bli
c goo
d
s
d
oma
i
n
.
A
C
KN
O
WLED
G
MEN
T
I
thank John Led
y
ard, David Roth and Tatsu
y
oshi Sai
j
o for discussions that lead to
t
hi
s pro

j
ect; K
l
aus A
bbi
n
k
, Bet
h
A
ll
en, Rac
h
e
l
Croson, Roger Gor
d
on, E
li
sa
b
et
h
H
o
ff
man, Matt
h
ew Jac
k

son, Wo
lf
gang Lorenzon, Laura Razzo
li
n
i
, Sara So
l
n
i
c
k
,
T
a
y
fun Sönmez, William Thomson, Lise Vesterlund, Xavier Vives, seminar particip
-
a
nts
i
n Bonn, Ham
b
urg, M
i
c
hi
gan, M
i
nnesota, P

i
tts
b
urg
h
, Pur
d
ue, an
d
part
i
c
i
pants
o
f
t
h
e 1997 Nort
h
Amer
i
ca Econometr
i
c Soc
i
ety Summer Meet
i
ngs (Pasa
d

ena, CA),
the 1997 Economic Science Association meetin
g
s (Tucson, AZ), the 1998 Midwest
D
YNAMI
C
S
C
TABILITY
OF
N
F
ASH
N
N
-E
FFICIENT
E
E
P
T
U
BLI
C
G
C
OODS
M
S

ECHANISMS
MM
19
7
Economic Theor
y
meetin
g
s (Ann Arbor, MI) and the 1999 NBER Decentralization
Conference (New York, NY) for their comments and su
gg
estions. The hospitalit
y
of
t
h
e W
i
rtsc
h
a
f
tspo
li
t
i
sc
h
e A
b

te
il
ung at t
h
e Un
i
vers
i
ty o
f
Bonn, t
h
e researc
h
support
p
rovided b
y
Deutsche Forschun
g
s
g
emeinschaft throu
g
h SFB303 at the Universit
y
of Bonn and NSF
g
rant SBR-980
55

86 are
g
ratefull
y
acknowled
g
ed. An
y
remainin
g
e
rrors are my own
.
N
O
TE
S
1
A
L
i
n
d
a
hl
equ
ilib
r
i
um

f
or t
h
e pu
bli
c goo
d
s economy
i
s c
h
aracter
i
ze
d

b
y a set o
f
persona
li
ze
d
pr
i
ces an
d
an allocation such that utilit
y
and profit maximization and feasibilit

y
conditions are satisfied. As each
c
onsumer’s consumption of the public
g
ood is a distinct commodit
y
with its own market, externalities
are eliminated. Thus, a Lindahl equilibrium is Pareto efficient. See, e.g., Milleron (1972).
2
N
ote t
h
at t
h
e a
d
apt
i
ve
l
earn
i
ng
d
e
fi
ne
d

b
y M
il
grom an
d
Ro
b
erts (1990)
d
oes not
i
nc
l
u
d
e t
h
e s
i
mp
l
e
r
einforcement learning model of Roth and Erev (1995). It includes a subset of the EWA learning
m
odels (Camerer and Ho, 1999) for certain
p
arameter combinations.
3
“Theoreticall

y
, conver
g
ence implies that no deviation will ever be observed once the s
y
stem equili-
b
rates. In an exper
i
menta
l
sett
i
ng w
i
t
h

l
ong
i
terat
i
ons, even a
f
ter t
h
e system equ
ilib
rates, su

bj
ects
s
omet
i
mes exper
i
ment
by
occas
i
ona
l

d
ev
i
at
i
on. T
h
ere
f
ore,
i
t
i
s necessar
y
to

h
ave some
b
e
h
av
i
ora
l
d
e
fi
n
i
t
i
on o
f
conver
g
ence: a s
y
stem conver
g
es to an equ
ilib
r
i
um at roun
d

t
,
if

x
i
(
s
)
=
x
e
i
,
∀
i
an
d

∀
s
≥
t
,
e
xce
p
t for a maximum of
n

rounds of deviation for
s
>

t
,
where
n
is small. For our ex
p
eriments of 100
rounds, we le
t

n
≤
5
, i.e., there could be a total of u
p
to
5
rounds of ex
p
erimentation or mistakes after
t
h
e system converge
d
.” (C
h

en an
d
Tang, 1998).
R
EFEREN
C
E
S
Arifovic, J. and Led
y
ard, J. (2003). “Computer Testbeds and Mechanism Desi
g
n: Application to the Class
o
f Groves-Led
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ard Mechanisms for Provision of Public Goods.” Manuscript, Caltech.
Boylan, R. and El-Gamal, M. (1993). “Fictitious Play: A Statistical Study of Multiple Economic Experi-
ments.
”
G
ames Econ. Be
h
avio
r
5, 205–222.
r
Ca
b
ra

l
es, A. (1999). “A
d
apt
i
ve D
y
nam
i
cs an
d
t
h
e Imp
l
ementat
i
on Pro
bl
em w
i
t
h
Comp
l
ete In
f
ormat
i
on.”

J
ournal o
f
Economic Theory 86, 1
5
9–184
.
Camerer, C. an
d
Ho, T.
(
1999
)
. “Experience
d
-Wei
gh
te
d
Attraction Learnin
g
in Norma
l
Form Games
,
”
Econometr
i
c
a

, Vol.
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.
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-
mental Research,” in
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he Handbook o
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h
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d

Gazza
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ort
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n
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h
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i
n
g

i
n Games Generate Conver
g
ence to Nas
h
E
qu
ilib
r
i
a? T
h
e Ro

l
e o
f
Supermo
d
u
l
ar
i
t
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i
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l
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bli
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h
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h
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i
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f
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h
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f
erences
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oo
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ll
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i
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h
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ilib
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oth, A. and Erev, I. (1995). “Learning in Extensive Form Games: Experimental Data and Simpl
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mith, V. (1979). “Incentive Com
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i
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.
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q

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i
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ilib
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i
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i

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n
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ll
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i
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n
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r
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–71.
D
YNAMI
C
S
C
TABILITY
OF
N
F
ASH
N
N
-E
FFICIENT
E
E
P
T
U
BLI
C
G
C
OODS
M
S
ECHANISMS
MM
199

APPE
N
DI
X
B
e
f
ore prov
i
ng Propos
i
t
i
on 3, we
fi
rst
d
e
fi
ne t
h
e t
h
ree mec
h
an
i
sms. A
ll
t

h
ree mec
h
-
a
nisms require that the number of pla
y
ers be at least three, i.e.,
n
≥
3
.
DEFINITI
O
N 6.
F
or the Hurwicz (1979) mechanism, the strategy space o
f
playe
r
i is
S
i
⊂
R
2
w
ith
g
eneric elemen

t

(
p
i
,

y
i
)
ʦ
S
i
.
The outcome
f
unction o
f
the publi
c
g
oo
d
an
d
t
h
e net cost s
h
are of t

h
e private goo
d
for p
l
ayer i are
Yy
y
n
k
k
()
y
,
∑
T
i
H
(
p
,
y
)
=
R
i
⋅
Y
(
Y

Y
y
)
+
p
i
⋅
(
y
i
−
y
i
+
1
)
−
p
i
+
1
(
y
i
+
1
−
y
i
+

2
)
2
,
wh
ere
R
i
=
1
n
+
p
i
+
1
−
p
i
+
2
.
DEFINITI
O
N 7. For t
h
e Wa
lk
er (1981) mec
h

anism, t
h
e strategy space o
f
p
l
ayer
i is
S
i
⊂
R
1
wit
h
generic e
l
ement m
i
ʦ
S
i
. T
h
e outcome function of t
h
e pu
bl
ic goo
d

a
nd the net cost share o
f
the private good
f
or player
i
a
r
e
Y
k
k
() ,
mmm
k
∑
T
n
i
W
ii
().
Ym

mm
()
m
m
i

m

⎛
⎝
⎞
⎠
i
1
1
m
m
+
i
i
m
DEFINITI
O
N 8.
F
or the Kim (1993) mechanism, the strategy space o
f
player i is
S
i
⊂
R
2
wit
h

generic e
l
ement
(
m
i
,

z
i
)
ʦ
S
i
.
T
h
e outcome function of t
h
e pu
bl
ic goo
d
a
nd the net cost share o
f
the private good
f
or player i
a

r
e
Y
k
k
() ,
mm
m
k
∑
T
i
K
ii
k
k
,
zm
i
z
k
( , ) ( , ) ( )
mz Pmz Ymmz Pmz Ym
i
Pmz Ym
P
mz Ym
z
z
Pmz Ymmz Ym

Pmz Ym
Pmz Ym
P
mz Ym
⎛
⎝
⎝
⎝
⎞
⎠
⎠
⎠
∑
1
2
2
wh
ere P
i
(
m
,

z
)
=
b
n
n
z

j
ji
ji
j
ji
.
z
j

m
j
−

+
m
j

m
j
n
i
j
n
j
i
∑∑
1
P
roof of Pro
p

osition 3:
(1) To s
h
ow t
h
at t
h
e Hurw
i
cz mec
h
an
i
sm
i
s not a supermo
d
u
l
ar game
f
or any
e
ʦ
E
Q
, it suffices to show that the pa
y
off function,
u

i
, does not have increasin
g
d
ifference in
(
y
i
,

y
−
i
)
.
200 Ex
p
erimental Business Research Vol. I
I
Sinc
e
u
i
(
p
,
y
)
=
v

i
(
y
)
+
ω
i
−
T
i
H
, we hav
e
∂
∂∂
∂
∂
2
2
2
2
1
u
yy
∂
n
v
y
i
ij

y
∂
i
,
∂
2
2
=
f
or a
ll

j
≠
i
+
1.
By
Definition 1,
∂
∂
2
2
v
y
i
<
0
, so
∂

∂∂
2
u
yy
∂
i
ij
y
∂
<
0, for all
i

a
n
d
f
o
r
a
ll
j
≠
i
+
1
.
B
y T
h

eorem 1,
u
i
d
oes not
h
ave
i
ncreas
i
ng
diff
erence
i
n
(
y
i
,
y
−
i
).
(2) To s
h
ow t
h
at t
h
e Wa

lk
er mec
h
an
i
sm
i
s not a supermo
d
u
l
ar
g
ame
f
or an
y
e
ʦ
E
Q
,
it suffices to show that the pa
y
off function,
u
i
,
does not have increasin
g

d
ifference in
(
m
i
,
m
−
i
):
∂
∂∂
∂
∂
∂
∂
∂
∂
2
2
2
2
2
2
2
11
11
11
u
m

∂
v
y
j
v
y
j
v
y
ji i
1
i
ij
∂
m
∂
i
i
i
;
1
,
ji
, ;
11
ji
, , .
1
ji i
=

+
1

1
1
i
i
1

ii
1
⎧
⎨
⎪
⎧⎧
⎪
⎪⎪
⎪
⎪⎪
⎪
⎨⎨
⎪⎪
⎩
⎪
⎨⎨
⎪
⎪⎪
⎪
⎪⎪
⎪

⎩⎩
⎪
⎪
if
if
if
B
y De
fi
n
i
t
i
on 1,
∂
∂
2
2
v
y
i
<
0
,
s
o
∂
∂∂
2
u

m
∂
i
ij
∂
m
∂
< 0
f
or a
ll

j
i
+ 1.
By
Theorem 1,
u
i
does not have increasin
g
difference in (
m
i
,
m
−
i
)
.

(3) To show that the Kim mechanism is not a supermodular
g
ame for an
y

e
ʦ
E
Q
,
it suffices to show that the pa
y
off function,
u
i
,
does not have increasin
g
difference
i
n
(
m
i
,
z
−
i
):
∂

∂∂
2
1
0
u
mz n
∂
i
ij
z
z
∂
.
0
=− <

By
Theorem 1,
u
i
,
does not have increasin
g
difference in (
m
i
,
z
−
i

).
Q
.E.D
.
E
NTRY
EE
T
Y
IMES
TT
IN
Q
N
U
E
U
E
S
WITH
E
H
NDOGENOUS
EE
A
S
RRI
V
AL
S

201
201
C
h
apter 1
1
ENTRY TIMES IN QUEUES WITH ENDOGENOUS
ARRIVALS: DYNAMICS OF PLAY ON THE
I
NDIVIDUAL AND A
GG
RE
G
ATE LEVEL
S
J
. Neil Bearde
n
University of Arizon
a
A
mnon Rapopor
t
University o
f
Arizon
a
a
n
d

H
ong Kong University of Science an
d
Tec
h
no
l
ogy
Darr
y
l A. Seale
University o
f
Neva
d
a Las Vegas
Abs
tr
act
Thi
s c
h
apter cons
id
ers arr
i
va
l
t
i

me an
d
stay
i
ng out
d
ec
i
s
i
ons
i
n severa
l
var
i
ants o
f
a
queuein
g

g
ame characterized b
y
endo
g
enousl
y
determined arrival times, simult-

a
neous p
l
ay,
fi
n
i
te popu
l
at
i
ons o
f
symmetr
i
c p
l
ayers,
di
screte strategy spaces, an
d
fi
xe
d
start
i
ng an
d
c
l

os
i
ng t
i
mes o
f
t
h
e serv
i
ce
f
ac
ili
ty. Exper
i
menta
l
resu
l
ts s
h
ow
1) consistent patterns of behavior on the a
gg
re
g
ate level in all the conditions that
a
re accounte

d

f
or qu
i
te we
ll

b
y t
h
e symmetr
i
c m
i
xe
d
-strategy equ
ilib
r
i
um o
f
t
h
e
stage game, 2) cons
id
era
bl

e
i
n
di
v
id
ua
l

diff
erences
i
n arr
i
va
l
t
i
me
di
str
ib
ut
i
ons t
h
at
d
ef
y

classification, and 3) learnin
g
trends across iterations of the sta
g
e queuein
g
g
ame
i
n some o
f
t
h
e exper
i
menta
l
con
di
t
i
ons. We propose an
d
su
b
sequent
l
y test a
s
i

mp
l
e re
i
n
f
orcement-
b
ase
d

l
earn
i
ng mo
d
e
l
t
h
at, w
i
t
h
a
f
ew except
i
ons, accounts
f

or
these ma
j
or findin
g
s
.
K
e
y
words: Queuein
g
, Endo
g
enous Arrivals, Equilibrium Anal
y
sis, Experimentation
J
EL Classification: C72, C9
2
1
. INTR
O
D
UC
TI
O
N
I
n two recent exper

i
ments, Rapoport, Ste
i
n, Parco, an
d
Sea
l
e (RSPS,
i
n press) an
d
Seale, Parco, Stein, and Ra
p
o
p
ort (SPSR, 2003) have studied arrival time and
©
200
5
Sprin
g
er
.
P
r
inted

i
n the
N

etherlands.
A. Rapoport and
R
.

d
Zwick (
e
(
(
ds.
)
,

E
x
p
erimental Business Researc
h
,
Vol. II
,
2
0
1–221
.
202 Ex
p
erimental Business Research Vol. I
I

s
tay
i
ng out
d
ec
i
s
i
ons
i
n a c
l
ass o
f
queue
i
ng pro
bl
ems w
i
t
h
en
d
ogenous
l
y
d
eterm

i
ne
d
arr
i
va
l
t
i
mes, a
fi
n
i
te an
d
common
l
y
k
nown ca
lli
ng popu
l
at
i
on o
f
p
l
ayers (

n
=
2
0

i
n
b
oth experiments), discrete strate
gy
spaces, and fixed startin
g
and closin
g
time of the
s
erv
i
ce
f
ac
ili
ty. Focus
i
ng on trans
i
ent
b
e
h

av
i
or, t
h
ese pro
bl
ems
diff
er
f
rom t
h
e ones
t
yp
i
ca
ll
y stu
di
e
d

i
n queue
i
ng t
h
eory t
h

at assume cont
i
nuous strategy spaces, stea
d
y-
s
tate behavior, and exo
g
enousl
y
determined arrival times (but see Hassin & Haviv,
2
003; Lar
i
v
i
ere & M
i
eg
h
em, 2003). T
h
e o
bj
ect
i
ve o
f
eac
h

p
l
ayer
i
n t
h
e queue
i
ng
p
ro
bl
ems stu
di
e
d

b
y RSPS an
d
SPSR
i
s to max
i
m
i
ze
h
er expecte
d

payo
ff

b
y com-
p
letin
g
service while minimizin
g
her waitin
g
time in the queue. Formulatin
g
these
q
ueue
i
ng pro
bl
ems as comp
l
ete
i
n
f
ormat
i
on, non-cooperat
i

ve games
i
n strateg
i
c
f
orm, RSPS an
d
su
b
sequent
l
y SPSR constructe
d
a Mar
k
ov c
h
a
i
n a
l
gor
i
t
h
m to com-
p
ute s
y

mmetric mixed-strate
gy
equilibrium solutions to the queuein
g

g
ames. Imple-
m
ent
i
ng a repeate
d
game
d
es
i
gn, t
h
ey t
h
en assesse
d
t
h
e
d
escr
i
pt
i

ve power o
f
t
h
ese
s
o
l
ut
i
ons
i
n severa
l
var
i
ants o
f
t
h
e game. T
h
ese var
i
ants
diff
er
f
rom one anot
h

er on
t
hree dimensions: 1) whether arrivals before the startin
g
time of the service facilit
y
are a
ll
owe
d
; 2
)
w
h
et
h
er a
ll
t
h
e
n
p
l
ayers can comp
l
ete t
h
e
i

r serv
i
ce w
i
t
h
no wa
i
t
i
ng
i
n
li
ne; an
d
3) w
h
et
h
er at t
h
e en
d
o
f
eac
h
stage game (tr
i

a
l
) p
l
ayers on
l
y rece
i
v
e
p
rivate information about their own outcome or
p
ublic information about the decisions
an
d
payo
ff
s o
f
a
ll
t
h
e
n
p
l
ayers
.

U
s
i
ng severa
l
stat
i
st
i
cs to compare o
b
serve
d
to equ
ilib
r
i
um (pre
di
cte
d
)
b
e
h
av
i
or
(
e.

g
., mean pa
y
offs, distribution of arrival times, distribution of interarrival times,
di
str
ib
ut
i
on o
f
wa
i
t
i
ng t
i
mes
i
n t
h
e queue), RSPS an
d
SPSR reporte
d
t
h
ree ma
j
or

fi
n
di
ngs. F
i
rst, w
i
t
h
one except
i
on t
h
at we
di
scuss
l
ater, t
h
ey reporte
d
cons
i
stent
p
atterns of behavior on the a
gg
re
g
ate level that can be accounted for remarkabl

y
w
e
ll

b
y t
h
e symmetr
i
c m
i
xe
d
-strategy equ
ilib
r
i
um. Secon
d
, t
h
ey o
b
serve
d
cons
id
er-
a

bl
e
i
n
di
v
id
ua
l

diff
erences
i
n arr
i
va
l
t
i
me an
d
stay
i
ng out
d
ec
i
s
i
ons t

h
at
d
e
fi
e
d
c
lassification. Most sub
j
ects often switched their decisions from trial to trial but
d
e
fi
n
i
te
l
y not
i
n a manner cons
i
stent w
i
t
h
equ
ilib
r
i

um p
l
ay. T
hi
r
d
, t
h
ey reporte
d
l
earn
i
ng tren
d
s across tr
i
a
l
s t
h
at strong
l
y
d
epen
d
e
d
on t

h
e
di
mens
i
ons ment
i
one
d
above. In particular, when the parameter values of the
g
ame were so selected that
a
ll
t
h
e
n
p
l
ayers cou
ld
,
i
n pr
i
nc
i
p
l

e, comp
l
ete serv
i
ce w
i
t
h
out wa
i
t
i
ng
i
n
li
ne (an
d
,
c
onsequent
l
y, max
i
m
i
ze t
h
e
i

r
i
n
di
v
id
ua
l
payo
ff
s), t
h
ere was on
l
y very wea
k
ev
id
-
e
nce for learnin
g
across trials re
g
ardless of the nature of the outcome feedback
(
pr
i
vate vs. group) t
h

at was prov
id
e
d
at t
h
e en
d
o
f
eac
h
tr
i
a
l
. W
h
en t
h
e parameter
va
l
ues were se
l
ecte
d
so t
h
at

i
n equ
ilib
r
i
um a su
b
stant
i
a
l

f
ract
i
on o
f
t
h
e p
l
ayers
s
hould sta
y
out on each trial, learnin
g
depended on the nature of the outcome
f
ee

db
ac
k
. I
f
eac
h
p
l
ayer was
i
n
f
orme
d
at t
h
e en
d
o
f
eac
h
tr
i
a
l
o
f
t

h
e
d
ec
i
s
i
ons
(
stay
i
ng out or arr
i
va
l
t
i
me) an
d
payo
ff
s o
f
a
ll
t
h
e
n
p

l
ayers, t
h
en SPSR reporte
d
s
tron
g
evidence of learnin
g
in the direction of equilibrium pla
y
with most pla
y
ers
fi
rst rece
i
v
i
ng negat
i
ve payo
ff
s
b
ecause o
f
congest
i

on (not enoug
h
p
l
ayers stay
i
ng
out) an
d
t
h
en gra
d
ua
ll
y approac
hi
ng equ
ilib
r
i
um p
l
ay
b
y
i
ncreas
i
ng t

h
e
f
requency o
f
s
ta
y
in
g
out decisions. If each pla
y
er was onl
y
informed of his own decision and
p
ayo
ff
,
l
earn
i
ng
did
not ta
k
e p
l
ace an
d

most o
f
t
h
e p
l
ayers en
d
e
d
up
d
eep
i
n t
h
e
negat
i
ve payo
ff

d
oma
i
n.
E
NTRY
EE
T

Y
IMES
TT
IN
Q
N
U
E
U
E
S
WITH
E
H
NDOGENOUS
EE
A
S
RRI
V
AL
S
203
T
h
e ma
j
or purpose o
f
t

h
e present paper
i
s to exp
l
a
i
n an
d
reconc
il
e t
h
ese t
h
ree
m
a
j
or
fi
n
di
ngs. Focus
i
ng on t
h
e
d
ynam

i
cs o
f
p
l
ay, we present an
d
t
h
en test a s
i
mp
l
e
m
odel in an attempt to explain 1) how the a
gg
re
g
ation of individual arrival time
di
str
ib
ut
i
ons t
h
at
diff
er cons

id
era
bl
y
f
rom one anot
h
er resu
l
ts
i
n rep
li
ca
bl
e patterns
t
h
at are accounte
d

f
or
b
y t
h
e symmetr
i
c m
i

xe
d
-strategy equ
ilib
r
i
um, an
d
2)
h
ow
outcome information (private vs. public) affects learnin
g
when the service facilit
y
c
annot accommo
d
ate a
ll
t
h
e mem
b
ers o
f
t
h
e ca
lli

ng popu
l
at
i
on
b
etween
i
ts start
i
ng
a
n
d
c
l
os
i
ng t
i
mes. A
l
t
h
oug
h
t
h
e ana
l

yses t
h
at we present
b
e
l
ow most
l
y
f
ocus on t
h
e
i
ndividual and a
gg
re
g
ate distributions of arrival time (that also include the decision
to stay out o
f
t
h
e queue), we a
l
so comment on t
h
e
di
str

ib
ut
i
on o
f

f
requency o
f
sw
i
tc
hi
ng t
h
e
d
ec
i
s
i
on
f
rom one tr
i
a
l
to anot
h
er an

d
t
h
e
di
str
ib
ut
i
on o
f
t
h
e magn
i
tu
d
e
of such switches.
T
h
e rest o
f
t
h
e c
h
apter
i
s organ

i
ze
d
as
f
o
ll
ows. Sect
i
on 2 states t
h
e assumpt
i
ons
o
f
t
h
e queue
i
ng game an
d

ill
ustrates
i
t w
i
t
h

an examp
l
e. Sect
i
on 3
d
escr
ib
es t
h
e
m
ixed-strate
gy
equilibrium distributions of arrival time for the three variants of th
e
queue
i
ng game stu
di
e
d

b
y RSPS an
d
SPSR, an
d
t
h

en compares t
h
em to o
b
serve
d
a
ggregate
di
str
ib
ut
i
ons. Se
l
ecte
d

i
n
di
v
id
ua
l

di
str
ib
ut

i
ons o
f
arr
i
va
l
t
i
me are a
l
so
presented both to illustrate the differences amon
g
members of the same popula-
t
i
on an
d
t
h
e
f
a
il
ure o
f
t
h
e m

i
xe
d
-strategy equ
ilib
r
i
um to account
f
or
i
n
di
v
id
ua
l
b
e
h
av
i
or. Sect
i
on 4
d
escr
ib
es a s
i

mp
l
e re
i
n
f
orcement-type
l
earn
i
ng mo
d
e
l
an
d
t
h
e
e
stimation of its
p
arameters. Section
5
contains a discussion of the model’s success
or failure in accounting for the three major findings mentioned above. Section
6
c
onc
l

u
d
es.
2
. THE
Q
UEUEIN
G

G
AME WITH END
OG
EN
O
U
S
ARRIVAL TIM
E
T
he queuein
g

g
ame is characterized b
y
a 6-tuple (
n
,
d
,

c
,
r
,

g
,
T
), where
n
i
s

t
h
e
n
um
b
er o
f
p
l
ayers an
d

d
is the (fixed) time required to serve a single player (same
d
f

or a
ll
n
p
l
ayers). T
h
ere are t
h
ree payo
ff
parameters, name
l
y,
c
,
r
,
an
d
g
:
c
i
s t
h
e per
u
nit waitin
g

cost,
r
is the payoff for completing service, and
r
g
is the pa
y
off for
stay
i
ng out o
f
t
h
e queue.
T
+
1
i
s t
h
e num
b
er o
f
entry per
i
o
d
s (pure strateg

i
es).
For examp
l
e,
if
t
h
e serv
i
ce
f
ac
ili
ty
i
s open
f
or exact
l
y two
h
ours an
d
entry t
i
me
i
s measured in minutes, then there are
T

+
1
=
121 possible entr
y
times
.
T
h
e
f
o
ll
ow
i
ng assumpt
i
ons c
h
aracter
i
ze t
h
e game. T
h
e serv
i
ce
f
ac

ili
ty opens at
T
o
TT
an
d
c
l
oses a
t

T
e
T
T
. Arr
i
va
l
s are ma
d
e
i
n
di
screte t
i
me un
i

ts (s
i
n
gl
e m
i
nutes
i
s RSPS
a
nd
5
minute intervals in SPSR). Decisions are made simultaneousl
y
and anon
y
-
m
ous
l
y. T
h
us, at t
h
e
b
eg
i
nn
i

ng o
f
eac
h
tr
i
a
l
, eac
h
p
l
ayer must
d
ec
id
e w
h
et
h
er to
e
nter t
h
e queue. I
f
s
h
e
d

ec
id
es to
d
o so, s
h
e must spec
ify

h
er t
i
me o
f
arr
i
va
l
(e.
g
.,
8
:01, 8:02, . . . in RSPS). If
m
pla
y
ers happen to arrive at the same time, 2
≤
m
≤

n
,
then their order of arrival is determined randomly with equal probability 1/
m
/
/
. Ba
lk-
i
n
g
(not enter
i
n
g
t
h
e queue upon arr
i
va
l
) an
d
rene
gi
n
g
(
d
epart

i
n
g
t
h
e queue a
f
te
r
a
rrival and before service commences) are
p
rohibited. One im
p
lication of the latter
ru
l
e
i
s t
h
at p
l
ayers cannot
l
eave t
h
e queue even
if
t

h
ey
k
now w
i
t
h
certa
i
nty t
h
at
serv
i
ce w
ill
not
b
e prov
id
e
d
. Ear
ly
arr
i
va
l
s
b

e
f
ore t
i
me
T
o
TT
may
(S
P
S
R
)
or may no
t
204 Ex
p
erimental Business Research Vol. I
I
(RSPS)
b
e a
ll
owe
d
. Serv
i
ce t
i

me
f
or eac
h
p
l
ayer,
d
,

i
s
fi
xe
d
, an
d
t
h
e queue
di
sc
i
p
li
n
e
i
s FIFO. T
h

ere
i
s a s
i
ng
l
e server, a s
i
ng
l
e serv
i
ce stage, an
d
no
li
m
i
t on t
h
e queue
len
g
th. Because the decisions are made simultaneousl
y
, pla
y
ers cannot observe the
s
tate o

f
t
h
e queue
b
e
f
ore ma
ki
ng t
h
e
i
r
d
ec
i
s
i
ons. F
i
na
ll
y, t
h
e payo
ff

f
unct

i
on – t
h
e
s
ame
f
or a
ll
n
p
l
ayers –
i
s g
i
ven
by
H
g
rc
i
HH
c
c
−
c
c
=
c

⎧
⎨
⎪
⎧
⎧
⎨
⎨
⎩
⎪
⎨
⎨
⎩
⎩
if player stays out
i
if player waits times units and fails to complete service
wiw
i
ii
wiw
i
if player waits times units and complets service
wiw
i
ii
iw
i
w
wh
er

e
w
i
i
s t
h
e t
i
me spent
i
n t
h
e queue unt
il
serv
i
ce commences. No wa
i
t
i
n
g
cost
i
s
c
harged for the time
(
d
) spent being served. RSPS and SPSR make the natural

assumpt
i
ons:
r
>
g
,
r
>

c
,
an
d

c
>

0
. T
h
e
v
a
l
ues o
f

T
o

T
T
,
T
e
T
T
an
d

d
,
as we
ll
as t
h
e va
l
ues
o
f
t
h
e wa
i
t
i
n
g
t

i
mes
w
i
, are measured in commensurate units. The three pa
y
off
p
arame
t
ers
c
,
r
,
and
g
,
the
p
o
p
ulation size
n
,
and the opening and closing times
T
o
TT
an

d
T
e
T
T
are assume
d
to
b
e common
l
y
k
nown. For a
di
scuss
i
on o
f
t
h
e assumpt
i
ons an
d
th
e
i
r
j

ust
ifi
cat
i
on see RSPS an
d
SPSR.
E
xampl
e
Table 1 provides an example that illustrates the queuein
g

g
ame and the
c
omputat
i
on o
f
t
h
e
i
n
di
v
id
ua
l

payo
ff
s. (See t
h
e su
bj
ect
i
nstruct
i
ons
i
n t
h
e Appen
di
x
o
f
RSPS
f
or a s
i
m
il
ar examp
l
e.) T
h
e parameter va

l
ues
f
or t
hi
s examp
l
e are
n
=

20,
d
=
4
5
,
T
o
T
T
=
8:00,
T
e
T
T
=
18:00,
c

=
1,
r
=
100, and
g
=
1
5
. The same
p
arameter value
s
are use
d

i
n two o
f
t
h
e
f
our con
di
t
i
ons
i
n SPSR (see

b
e
l
ow). P
l
ayers are restr
i
cte
d
to
arrive at 5-minute time intervals, and early arrivals (before
T
o
TT
)
are a
ll
owe
d
. Pa
y
o
ff
s
are in
p
ennies
.
C
o

l
umns 1 an
d
2 o
f
Ta
bl
e 1 present t
h
e p
l
ayer num
b
er (an
i
nteger
f
rom 1 to 20)
and the pla
y
ers’ decisions. In this example, 1
6
of the 20 pla
y
ers opted to enter (at
p
ossibl
y
different times), and 4 pla
y

ers (6, 16, 11 and 4, who are listed at the
bottom) decided to stay out. Players 13, 3, and 18 arrived at 7:05, 7:25, and 7:30
,
b
e
f
ore t
h
e open
i
n
g
t
i
me
T
o
T
T
, and had to wait in line 55, 80, and 120 minutes, respect-
i
vel
y
. All three completed service. Pla
y
ers 1
5
and 1 arrived at exactl
y
8:00, and the

t
wo-player tie was resolved in favor of player 15 (who still had to wait 45 ×
3
=
1
35
m
inutes until players 13, 3, and 18 completed service). Player 14 arrived at 14:55 an
d
was served immediatel
y
with no dela
y
. Althou
g
h pla
y
ers 2, 17, and 7 arrived more
than 45 minutes before closing time, none of them received service. Of the twenty
p
layers in this example, eight lost money. Total system idle time was 25 minutes,
from 14:4
5
to 14:
55
and from 17:4
5
to 18:00. Columns 3, 4, and
5

p
resent the
b
eg
i
nn
i
ng o
f
t
h
e serv
i
ce t
i
me, t
h
e wa
i
t
i
ng t
i
me (
i
n m
i
nutes), an
d
t

h
e wa
i
t
i
ng costs.
Th
e rewar
d

(
t
h
at cou
ld
assume one o
f
t
h
e t
h
ree va
l
ues
r
,

0,
o
r

g
)
i
s presente
d

i
n
c
olumn 6, and the pa
y
off is shown in the ri
g
ht-hand column.
E
NTRY
EE
T
Y
IMES
TT
IN
Q
N
U
E
U
E
S

WITH
E
H
NDOGENOUS
EE
A
S
RRI
V
AL
S
20
5
T
able 1. Example o
f
a Queueing Game when Early Arrivals are Possible (d
=
4
5
)
P
layer Decision Service Waiting Waiting Reward Payo
ff
S
tarts at Time
C
os
t
1

3 Arrive: 7:05 8:00 55
$
0.55
$
1.00
$
0.45
3
Arri
v
e: 7:2
58
:4
5800
.
80
1.
00 0
.2
0
1
8 Arr
i
ve: 7:30
9
:30 120 1.20 1.00
−
0
.
20

15
Arrive: 8:00 10:1
5
13
5
1.3
5
1.00
−
0.3
5
1 Arr
i
ve: 8:00 11:00 180 1.80 1.0
0
−
0
.
80
5
Arri
v
e:
8
:4
5
11:4
5
1
80

1.
80
1.
00
−
0
.
80
20
Arri
v
e: 1
0
:
00
12:
30
1
50
1.
50
1.
00
−
0
.
50
10
Arri
v

e: 12:1
0
1
3
:1
5650
.
65
1.
00 0
.
35
8
Arri
v
e: 1
3
:4
5
14:
00
1
50
.1
5
1.
00 0
.
85
1

4 Arri
v
e: 14:
55
14:
55 0 0
1.
00
1.
00
9 Arrive: 1
5
:00 1:40 40 0.40 1.00 0.6
0
1
2 Arrive: 1
5
:00 16:2
5
8
5
0.8
5
1.00 0.1
5
1
9 Arrive: 1
5
:30 17:10 100 1.00 1.00 0
2 Arrive: 16:2

5
NA 100 1.00 0
−
1
.
00
1
7 Arrive: 17:00 NA 8
5
0.8
5
0
−
0.8
5
7 Arri
v
e: 17:
00
NA
60 0
.
60 0
−
0
.
60
6 Sta
y
out None 0 0 0.15 0.15

1
6 Sta
y
out None 0 0 0.15 0.15
1
1 Sta
y
out None 0 0 0.15 0.15
4 Sta
y
out None 0 0 0.15 0.15
206 Ex
p
erimental Business Research Vol. I
I
3
. PREDI
C
TED AND
O
B
S
ERVED RE
SU
LT
S
3
.1. Ex
p
erimental Condition

s
R
SPS and SPSR to
g
ether conducted three different experimental conditions that
diff
er
f
rom one anot
h
er
i
n one or more parameters or assumpt
i
ons. T
h
ese con
di
t
i
ons
are
d
escr
ib
e
d

b
e

l
o
w
. In a
ll
t
h
ree con
di
t
i
ons
n
=
2
0
an
d
t
h
e num
b
er o
f

i
terat
i
ons o
f

t
he sta
g
e
g
ame is 7
5
. All the experiments are computer-controlled.
Condition 1
(
RSPS)
.
T
o
TT
=
8
:
00
,
T
e
TT
=
18
:
00
,
d
=

30
,
c
=

1
,
r
=

6
0, and
g
=

0
. T
i
me
i
s
m
easured in single minute intervals, and early arrivals are prohibited. This para-
m
eterization gives rise to
6
01 possible entry times, namely 8:00, 8:01, . . . , 18:00,
an
d
anot

h
er
d
ec
i
s
i
on o
f
sta
yi
n
g
out. In
f
ormat
i
on
i
s pr
i
vate. In part
i
cu
l
ar, at t
he
e
nd of each trial each player is reminnded of her decision (arrival time or staying
out); num

b
er o
f
p
l
ayers t
i
e
d
at
h
er arr
i
va
l
t
i
me,
if
any; an
d
t
h
e outcome o
f
t
h
e
t
i

e-
b
rea
ki
n
g
ru
l
e;
h
er queue wa
i
t
i
n
g
t
i
me
(
w
i
); her pa
y
off for the
t
r
ial

(

H
i
H
H
); and her
c
umulative payoff from the beginning of the session. We refer to this information
c
on
di
t
i
on as
P
rivate Outcome Informatio
n
.
Con
d
ition 2 (SPSR
)
.
T
o
TT
=
8
:
00,

T
e
TT
=

18
:
00,

d
=

30,

c
=
1,

r
=
100,

g
=
1
5
. Time i
s
m
easured in 5-minute intervals, and earl

y
arrivals are allowed. To limit the strate
gy
s
pace, pla
y
ers are not allowed to enter the queue before 6:00. In fact, this requiremen
t
i
mposes no practical limitation. This parameterization gives rise to 145 possible
e
ntr
y
times, namel
y
, 6:00, 6:05,
.

,
18:00 an
d
an a
ddi
t
i
ona
l
pure strate
gy
o

f
sta
y-
i
n
g
out.
C
on
di
t
i
on 2 was
f
urt
h
er
di
v
id
e
d

i
nto two su
b
-con
di
t
i

ons, name
l
y Con
di
t
i
on 2P
an
d
Con
di
t
i
on 2G,
i
n terms o
f
t
h
e
i
n
f
ormat
i
on prov
id
e
d
to t

h
e p
l
a
y
er at t
h
e en
d
of each trial. Condition 2P included
P
rivate Outcome In
f
ormation. Condition 2G
i
nc
l
u
d
e
d
G
roup Outcome Information w
hi
c
h
cons
i
ste
d,

i
n a
ddi
t
i
on to t
h
e
P
rivat
e
O
utcome Information, o
f
comp
l
ete
i
n
f
ormat
i
on a
b
out t
h
e 1) arr
i
va

l
t
i
mes an
d
sta
yi
n
g
out decisions, 2) service startin
g
time, and 3) individual pa
y
offs for all the
n
pla
y
ers
i
n t
h
e sess
i
on. T
hi
s was accomp
li
s
h
e

d

b
y present
i
ng a computer “Resu
l
ts” screen at
th
e en
d
o
f
eac
h
tr
i
a
l
t
h
at cons
i
ste
d
o
f
a 2
0

×
3 matr
i
x w
i
t
h
rows correspon
di
n
g
to t
h
e
t
went
y
pla
y
ers arran
g
ed in terms of the time of their arrival (sta
y
in
g
out decisions
were p
l
ace
d

at t
h
e
b
ottom rows), an
d
t
h
ree co
l
umns correspon
di
ng to t
h
e p
l
ayer’s
d
ec
i
s
i
on (arr
i
va
l
t
i
me or sta
yi

n
g
out), start
i
n
g
t
i
me o
f
serv
i
ce, an
d

i
n
di
v
id
ua
l
pa
y
o
ff
for the trial (see A
pp
endix in SPSR for details)
.

Condition 3
(
SPSR)
.
Condition 3 used the same
p
arameter values as Condition 2
w
i
t
h
t
h
e on
l
y except
i
on t
h
a
t

d
=
45 minutes. It
,
too
,
was further divided into tw
o

s
u
b
-con
di
t
i
ons, Con
di
t
i
on 3P an
d
Con
di
t
i
on 3G t
h
at
i
ncorporate
d
t
h
e Pr
i
vate an
d
G

roup Outcome Information, respectivel
y
. Note that if
d
=
30 (Conditions 1 and 2),
a
ll
t
h
e
n
p
l
ayers can comp
l
ete serv
i
ce w
i
t
h
out wa
i
t
i
ng
if
t
h

ey arr
i
ve at 30 m
i
nute
i
nterva
l
s start
i
n
g
exact
ly
at 8:00. In contrast, on
ly
13 o
f
t
h
e 20 p
l
a
y
ers can comp
l
ete
E
NTRY
EE

T
Y
IMES
TT
IN
Q
N
U
E
U
E
S
WITH
E
H
NDOGENOUS
EE
A
S
RRI
V
AL
S
20
7
service in Condition 3 without waiting in the queue, if they arrive at 45 minute
i
ntervals starting at exactly 8:00 (8:00, 8:45, . . . , 17:00), whereas the remaining 7
pla
y

ers have to sta
y
out. As we show below, this difference in service time an
d
w
h
et
h
er or not ear
l
y arr
i
va
l
s are a
ll
owe
d
strong
l
y a
ff
ect t
h
e m
i
xe
d
-strategy equ
ilib

r
ia
f
or t
h
ese t
h
ree exper
i
menta
l
con
di
t
i
ons
.
3.2. Method
S
ub
j
ects. Condition 1 included four
g
roups of
n
=
20 members each, whereas Con-
di
t
i

ons 2P, 2G, 3P, an
d
3G eac
h

i
nc
l
u
d
e
d
two groups o
f

n
=
20 p
l
ayers
f
or a tota
l
of
12 groups (240 su
bj
ects) across con
di
t
i

ons. W
i
t
h
t
h
e except
i
on o
f
Group 4
i
n
C
ondition 1, all the sub
j
ects were Universit
y
of Arizona students, mostl
y
under-
g
ra
d
uates, w
h
o vo
l
unteere
d

to part
i
c
i
pate
i
n a
d
ec
i
s
i
on ma
ki
ng exper
i
ment
f
or pay-
off
cont
i
ngent on per
f
ormance. Ma
l
es an
d

f

ema
l
es part
i
c
i
pate
d

i
n a
l
most equa
l
proportions. Group 4 in Condition 1 included twent
y
“sophisticated” sub
j
ects who
part
i
c
i
pate
d

i
n a summer wor
k
s

h
op on exper
i
menta
l
econom
i
cs t
h
at was con
d
ucte
d
a
t t
h
e Un
i
vers
i
ty o
f
Ar
i
zona. Mem
b
ers o
f
t
hi

s group were gra
d
uate stu
d
ents an
d
p
ost-doctoral fellows of economics with a keen interest in ex
p
erimental economics
a
n
d
so
lid

b
ac
k
groun
d

i
n game t
h
eory. In
di
v
id
ua

l
payo
ff
range
d
cons
id
era
bl
y,
d
epen
d
-
i
ng on the experimental condition, from
$
15.00 to
$
53.24. The conversion rate of
the fictitious currenc
y
(called “francs”) used in the experiment was doubled for the
“
sop
hi
st
i
cate
d

” su
bj
ects
i
n Group 4 o
f
Con
di
t
i
on 1
.
P
rocedur
e
. Details of the ex
p
erimental
p
rocedure a
pp
ear in RSPS and SPSR and
w
ill
not
b
e repeate
d

h

ere. Bas
i
ca
ll
y,
i
n a
ll
t
h
ree con
di
t
i
ons t
h
e queue
i
ng game was
presente
d
as an em
i
ss
i
ons contro
l
scenar
i
o w

i
t
h
a
fi
xe
d
an
d
common
l
y
k
nown num
b
e
r
o
f car owners, a station whose openin
g
and closin
g
times are fixed and commonl
y
k
nown,
fi
xe
d
serv

i
ce t
i
me per customer, an
d
a common payo
ff
structure (see a
b
ove)
.
At t
h
e
b
eg
i
nn
i
ng o
f
t
h
e sess
i
on, eac
h
su
bj
ect was prov

id
e
d
w
i
t
h
an en
d
owment o
f
1,000 francs. Francs earned durin
g
each trial were added or subtracted from this
e
n
d
owment. At t
h
e en
d
o
f
t
h
e sess
i
on, t
h
e cumu

l
at
i
ve payo
ff

i
n
f
rancs was converte
d
to US
d
o
ll
ars
(
100
f
rancs
=
US
$
1.00). Subjects who ended the session losing their
e
ntire endowment were onl
y
paid their show-up fee. Sub
j
ects were paid individuall

y
a
n
d

di
sm
i
sse
d.
Eq
uilibrium Analysi
s
.
Each of the queuein
g

g
ames in Conditions 1 and 2 has
n
!
e
qu
ilib
r
i
a
i
n pure strateg
i

es, w
h
ere p
l
ayers arr
i
ve at 30 m
i
nute
i
nterva
l
s start
i
ng
a
t 8:00. Un
d
er pure strategy equ
ilib
r
i
um p
l
ay, eac
h
p
l
ayer
h

as zero wa
i
t
i
ng t
i
me
with an associated pa
y
off of
r
.
Technicall
y
, these are coordination
g
ames wit
h
n
!
pure-strategy equ
ilib
r
i
a t
h
at are not Pareto-ran
k
a
bl

e an
d

d
o not
d
epen
d
on t
h
e
re
w
ar
d
to cost rat
i
o r
/c
. W
i
t
h
out pre-p
l
ay commun
i
cat
i
on, coor

di
nat
i
on on any one
pure strate
gy
equilibrium is practicall
y
impossible due the lar
g
e size of the
g
roup
e
ven un
d
er mu
l
t
i
p
l
e
i
terat
i
ons o
f
t
h

e stage game. T
h
e queue
i
ng game
i
n Con
di
t
i
on 3
h
as mu
l
t
i
p
l
e asymmetr
i
c pure-strategy equ
ilib
r
i
a w
h
ere 13 p
l
ayers enter t
h

e queue
208 Ex
p
erimental Business Research Vol. I
I
(with at least 4
5
minute intervals between consecutive arrivals and, consequentl
y
, no
waitin
g
time) and 7 pla
y
ers sta
y
in
g
out. A
g
ain, it is hi
g
hl
y
unlikel
y
that the twent
y
pl
ayers cou

ld
coor
di
nate on any part
i
cu
l
ar equ
ilib
r
i
um, even
i
n Con
di
t
i
on 3G, w
i
t
h
-
out pre-pla
y
communication
.
Each of the three queuein
g

g

ames in Conditions 1, 2, and 3 has a s
y
mmetric
mi
xe
d
-strategy equ
ilib
r
i
um so
l
ut
i
on. T
h
e Appen
di
x o
f
RSPS conta
i
ns a
d
eta
il
e
d
descri
p

tion of the com
p
utational method used to construct these solutions. Essen-
t
iall
y
, it consists of specif
y
in
g
the state space, the transitional probabilities of the
s
toc
h
ast
i
c process t
h
at governs t
h
e queue progress
i
on, an
d
t
h
e
i
terat
i

ve proce
d
ure
t
o compute the arrival times and sta
y
in
g
out decisions under mixed-strate
gy
pla
y
.
Fi
g
s. 1, 2, and 3 displa
y
the equilibrium solutions for the three
g
ames in Conditions
1, 2, an
d
3, respect
i
ve
l
y
.
S
everal features of the e

q
uilibrium solutions warrant discussion. In the solution
for Condition 1 (Fi
g
. 1), pla
y
ers
j
oin the queue at the earliest possible time of 8:00
(
t
= 0) with probability 0.211 and stay out with probability 0.060. They should never
j
oin the queue between 8:01 and 9:03, and then
j
oin the queue with positive prob-
abilities until 17:30 (
t
=
5
70). Because g
=
0 in Condition 1, the expected pa
y
off
un
d
er t
hi
s equ

ilib
r
i
um
i
s c
l
ear
l
y zero. In t
h
e equ
ilib
r
i
um so
l
ut
i
on
f
or Con
di
t
i
on 2
(Fi
g
. 2), pla
y

ers should alwa
y
s enter the queue startin
g
at 6:3
5
and endin
g
at 17:30.
T
he expected pa
y
off under equilibrium pla
y
is 18.3
5

>
g
=
1
5
. In contrast, th
e
e
qu
ilib
r
i
um so

l
ut
i
on
f
or Con
di
t
i
on 3
di
sp
l
ays a very
diff
erent pattern. T
h
e pro
b
a
bil
-
i
t
y
of sta
y
in
g
out is 0.409

6
, impl
y
in
g
that, on avera
g
e, 8.2 pla
y
ers out of 20 should
s
ta
y
out on each trial. The expected value is clearl
y
g
=
1
5.
0
.
0000
0
.
00
1
0
0
.
00

2
0
0
.
0030
0
.
0040
060
12
0
1
80
24
0 300 360
42
0
4
80 5
4
0
A
rr
i
va
l
T
i
me (8:00 to 17:30)
Probabilit

y
0
.21
1
0
.
060
F
igure 1. Symmetric mixed-strategy e
q
uilibrium of arrival times for Condition 1
.
E
NTRY
EE
T
Y
IMES
TT
IN
Q
N
U
E
U
E
S
WITH
E
H

NDOGENOUS
EE
A
S
RRI
V
AL
S
209
0.00
0
0.00
2
0
.
004
0.00
6
0
.
008
0
.
010
0
.
012
0
.
014

0
10 20 30 40 50 60 70 80
9
0 100 110 120 130
Arrival Time
(
6:35 to 17:30
)
Pro
b
a
bili
t
y
0
.
000
0
.
002
0
.
004
0
.00
6
0
.
008
0

.
010
0
.
012
0
.
014
0
10 20 30 40
5
06070809010011012
0
A
rrival Time
(
6:35 to 17:30
)
Probabilit
y
T
he probability of staying out is 0.409
6
A
ll
t
h
ree
fi
gures ex

hibi
t t
h
e per
i
o
di
c
i
ty o
f
t
h
e so
l
ut
i
on t
h
at
h
as a
l
so
b
een reporte
d
i
n solvin
g

for the equilibria of
g
ames with smaller strate
gy
spaces and smaller
n
umber of players. For example, in Fig. 2 players should arrive at the queue at 6:40,
6
:50, . . . , 8:00 with probability 0.0058 and at the intermediate times 6:45, 6:55, . . . ,
7:
55
with probabilit
y
0.0118, which is twice as lar
g
e. In Fi
g
. 3, pla
y
ers should enter
F
igure 2. Symmetric mixed-strategy e
q
uilibrium of arrival times for Condition 2
.
F
igure 3. Symmetric mixe
d
-strategy equi
l

i
b
rium of arriva
l
times for Con
d
ition 3
.