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Faculty Working Paper 91-0158
330
B385
1991:153 COPY 2
The Good, The Bad and The Ugly:
Coalition Proof Equilibrium In Infinite
Charles
Kahn
Department of Economics
University of Illinois
Games
Dilip Mookherjee
SUU is til nl Institute
Seu Delhi
Bureau of Economic and Business Research
College of
Commerce and
Business Administration
University of Illinois
at
Urbana-Champai^n
BEBR
FACULTY WORKING PAPER MO. 91-0158
College of
Commerce and
University of
Illinois at
Business Administration
Urbana -Champaign
August 1991
The Good, The Bad and The Ugly:
Coalition Proof Equilibrium
In Infinite
Charles Kahn
Department of Economics
University of
Dilip
Illinois
Mookherjee
Statistical Institute
New
Delhi
Games
Forthcoming:
Games and Economic Behavior
THE GOOD, THE BAD AND THE UGLY:
COALITION PROOF EQUILIBRIUM IN INFINITE GAMES
by
Charles M. Kahn
University of Illinois, Urbana- Champaign
and
Dilip Mookherjee
Indian Statistical Institute
Current Version: October
1990
*We thank
Carlos Asilis, Joseph Greenberg, Bentley MacLeod, Alvin Roth,
William Thomson, and an anonymous referee for useful comments, as well as
participants at the summer game theory conferences at Ohio State, 1988 and
1989.
This work was funded by NSF grant SES-8821723.
Addresses:
Kahn: Department of Economics, University of Illinois, 1206 S.
Sixth Street, Champaign, IL 61820, USA.
Indian Statistical
Mookherjee:
Institute 7 S.J.S. Sansanwal Marg, New Delhi 110016, INDIA.
THE GOOD, THE BAD, AND THE UGLY
Running Head:
Abstract
This
provides
paper
techniques
for
using
definition of Coalition Proof Nash Equilibrium.
recursive
number
of
definition of
players
characterization,
definition
infinite.
simplicity
and
a
the
new
our
are
new
characterize
to
Our new definition reduces to
Bernhein-Peleg- Whinston
finite
a
one
used
strategy
maintains
non- recursive
Examples
of
and
sets
We demonstrate the techniques by extending the
equilibria in infinite games.
the
stable
an
space.
equivalence
characterization
to
definition,
demonstrate
even
for
in
when
with
agents
JEL Classification Number: 026
Mail correspondence to:
Charles M. Kahn
Department of Economics
University of Illinois
1206 S. Sixth St.
Champaign, IL 61820
with
Unlike
between
a
finite
the
spaces
and
infinite
old
recursive
a
strategy
advantages
the
games
games
are
relative
numbers
of
THE GOOD, THE BAD AND THE UGLY:
COALITION PROOF EQUILIBRIUM IN INFINITE GAMES
I.
Introduction
Recently
notion
of
Greenberg
stable
sets
as
theoretic solution concepts.
a
used
has
[6,7,8]
framework
von
the
for
Neumann-Morgenstern
examining
and
[13]
comparing
game
The idea of a stable set applies to any abstract
system of objects with an arbitrary dominance relation defined on the set.
partitions
objects
into
two
subsets
which can be denoted
--
"good"
and
with the property of internal stability (no element of the good set
--
"bad"
the
It
is
dominated by another) and external stability (every element of the bad set
is
dominated by some element of the good set)
By varying the type of objects that constitute potential candidates for a
solution (such as strategy- tuples or coalitional arrangements), and the nature
of
the
dominance
relation among
them,
the
analysis
assumptions about the environment in which the game is played.
allows
systematic extension of known solution concepts
contexts.
For example,
alternative
can reflect
to
The approach
new and difficult
the approach is of use in defining renegotiation proof
equilibria in repeated games (Asheim [1], Asilis, Kahn and Mookherjee [2]) and
in
extending
cooperative
solutions
to
with
games
information
private
(Boyd-Prescott [4], Myerson [12], Marimon [11], and Kahn-Mookherj ee [9,10]).
A
difficulty
with
the
approach
is
that
a
guaranteed to exist for finite abstract systems.
demonstrates
that
the
Bernheim- Peleg- Whinston
Coalition
[3]
can
be
Proof
Nash
characterized
stable
only
be
For example,
Greenberg
[7]
Equilibrium
proposed
by
by
the
set
can
stable
set
of
an
abstract system of coalitional agreements with a suitable dominance relation
--
so
long as
a
stable set exists.
We provide examples with
infinite action
spaces where the stable does not exist and the equivalence breaks down.
the
might
approach
seem
usefulness
limited
of
in
economic
Thus
contexts,
where
players typically have an infinite number of actions available.
The aim of this paper is to suggest procedures for extending the stable
set approach to games with infinite action spaces and/or
employ
We
players.
procedures.
two
The
first
infinite numbers of
follows
Roth's
generalization of the von Neumann and Morgenstern solution concept,
tripartite division of the objects in the abstract system.
partition"
exists
abstract
any
for
finite
system,
or
[14]
using
This "semi-stable
infinite.
second
The
procedure involves modifying the objects examined in the abstract system,
developing
Coalition
procedures
two
Equilibrium
this
coalition-proof
partition,
equilibrium
applying
by
(CPNE)
partition of Greenberg's
From
game.
the
Proof Nash
semi-stable
We
.
abstract
generate
we
is
show
a
according to B-P-W's recursive definition.
any
for
player
of
unique
finite
player
is
definition
games
the
set
show
we
However,
the
we also provide
S-CPNE
reasonable"
fails
outcomes
of
the
self -enforcing agreements is open.
game
with
procedure
compact
handles
strategy
this
This
the
these
in
useful refinement.
despite
game.
that
With infinite action spaces,
examples where
exist,
to
a
of
of equilibria
recursive definition can generate intuitively unreasonable outcomes;
cases our non recursive concept provides
of
a
there
that
case
the
to
non- recursive
subset
a
them
system
finite
For
(S-CPNE).
this non-recursive equilibrium concept
and
by
notion of "near- rationality" of coalitional actions.
a
illustrate
We
a
recursive CPNE are unreasonable
the
existence
problem
arises
of
when
"intuitively
the
set
of
We show that the set may be open even in a
spaces
difficulty,
and
by
continuous
modeling
payoffs.
Our
second
"near- rationality "
for
coalitions.
model
We
coalitions
as
choosing not
single
action- tuples
,
but
convergent sequences of actions, with payoffs defined by the limit points of
these sequences.
interpretation is that they choose actions "sufficiently
The
thus gaining little by going to the limit.
far along" these sequences,
With this modified formulation of the set of coalitional agreements,
stable
partition
exists
derive
partition,
we
(ES-CPNE),
which
any
in
"extended"
an
provides
the
We also provide an alternative,
Since
stable
the
the set of players,
player case.
players
therefore
notion
desired
From
game.
coalition
of
refinement
of
the
this
stable
proof
equilibrium
B-P-W
definition.
recursive definition of ES-CPNE.
approach does not depend on finite recursions on
sets
the natural candidate for extension to the infinite
it is
Although the stable set need not be unique when the number of
infinite,
is
player
finite
the
propose
minimal
the
natural
the
as
corresponding
equilibrium,
stable
to
set
can be easily characterized.
extension
Roth's
coalition
"strong"
a
"supercore"
and
We
proof
characterized
as
number
of
belonging to every stable set.
A natural
economic
application
for
players is any situation with free entry.
proved particularly
useful
in
game
a
with
an
infinite
The procedures we develop here have
applications
to
market
games
players,
with
with
imperfect
information (see Kahn-Mookherj ee [10]).
II.
Simple Agreements
Let
N
be
typical player.
the
(possibly
Let A
,
i
functions of
i
infinite)
set
of
i£N
denoting
a
respectively denote the strategy space and payoff
U
i
and let A = Y
A iGN„ A
,
i
and U
:
i
A
-»
R.
Define an agreement to be
subset of
a
pair
(a,S)
where a 6 A and
is
S
non-empty
Let A denote the set of all agreements.
N.
Note that an agreement specifies actions for all players,
not just those
The reader may find it helpful to interpret an agreement as specifying
in S.
the
a
actions
parties
the
for
to
agreement,
the
given
the
actions
other
of
players
Agreement (b,S) dominates agreement (a,T) (denoted (b,S) > (a,T)
(i)
S
(ii)
a
if
C T
= b
j
for all j £ N\S
j
U (b) > U (a)
(iii)
)
i
for all
e S.
i
1
In other words,
a
dominating agreement is one
in which
a
subset of the
original parties break away and find actions which are strictly better for all
of the subset,
the rest of the players leaving their actions unchanged.
The pair (G,B)
G denotes
the
is a stable
partition of the set of agreements A
stable set of "good" agreements,
the set of "bad" agreements
--
and
B
denotes
where
--
complement,
its
if
(a,T) e B
«
There exists (b,S) 6 G
(a,T) e G
<=•
There does not exist (b,S) E G such that (b,S) > (a,T)
In other words,
such that (b,S) > (a,T)
every agreement in G is dominated only by agreements in
every agreement in B is dominated by some agreement in
It
is
whether
it
not clear
is
unique.
that
a
stable
However,
if
partition always
a
stable
B,
and
2
G.
exists,
partition
(G,B)
or
if
does
it
does,
exist,
it
generates a solution concept in the following way:
the set of solutions to the
game is the set of strategy vectors a such that (a,N)
in the set G.
is
In the case of a finite number of players and strategies,
that there always exists a unique partition;
finite -player
Example
it can be shown
this is established below.
infinite strategy games, a stable partition may not exist.
-
,
Consider
1
player game N = (i), A
a one
iii
= (0,1)
and U (a
i
If
G
non-empty,
is
contain
must
it
agreement
single
a
Let G =
agreement in G will dominate another.
((a
--
therefore must be dominated by some agreement
)
otherwise
= a
.
one
Then (a +c,{i))eB,
,(i))).
i
i
and
But in
in
G
,
cannot be empty, because then every agreement would be in B
a
contradiction.
G
and by definition
,
every agreement in B must be dominated by some agreement in G.B
Example
with
But
space.
would
1
longer
hold
if
players,
this
problem
no
more
the
player
may
had
arise
a
even
strategy spaces and continuous utility functions.
Example
Suppose N = (1,2,3), A
2
=
[0,1]
for all
i
112a
U (a
,
,
a
)
U (a ,a
12 ,a
2
= a
-
I
2
3
)
a -a
= 2a
a
-
-
la -a
I
'
3
1
3
U(a,a,a)=2a
-a
12
3
I
2
1
'
3
1
'
-
2
I
2
3
a -a
1
I
3
(
'
The best response correspondences are:
a (a
12
,
a
)
a (a ,a
2
13
a (a ,a
3
= a
2
3
12
)
= a
and
3
)
-
[0,1]
=
if a
1
1
otherwise
a
l
=
and a
2
1-a +a
12
)
.
i
compact
and
with
strategy
compact
Suppose
((a
123a
,
exists
there
a
,
)
,
(
i
)
G G
)
players other than
as
--
i
ii-i
G a (a
this
where a
),
First
{G,B}.
denotes
Nash equilibrium
strategy- tuple of
the
because such agreements cannot be dominated,
is
otherwise
--
claim
We
((b,b,b),N)
G
(
<
(a+e a+e a+e
,
dominated
(
,
both
(
,
- a
)
,
{
2
a+e
)
(a ,0 a+e
,
(
But
.
(
(a+e ,0,-y)
(
(a+e ,0,7)
Finally,
,
{
,
{
it
,
2
,
would dominate
latter
the
3
)
)
3
}
)
coalition
,
1
{
if
)
.
G
>
@
a+e
former,
the
requires
dominated
by
(
/9
(
,
,
=
(a
a+e),N)
,/3
,
G
which
B,
requires
Clearly it cannot be
by
nor
a+e
a+e a+e
)
,
)
{
,
,/9
show
,
1
,
2
,
a+e
)
,
that
)
((a,a,a),N).
So
Suppose
a+e)
is
but a+e
G
G,
<
3
(
(a+e a+e a+e
)
,
,
{
,
1
,
2}
)
by
GG
.
dominated
by
implies
that
1
contradiction.
a
{
)
dominated
ls
)
(<x,0,
(
N)
< a+e,
7
(a+e
to
a
For this it is necessary that a and
(a+e
(
and
G
such that ((a,a,a),N) G G.
agreement,
G G.
)
that
This
G G.
is
2
,
straightforward
is
((a+e, a+e,
that
((a,a,a),N)
contradicting the hypothesis that
e G,
)
)
alternatively,
Suppose,
Suppose
G.
- a
must be dominated by some pair-coalition agreement.
N)
exceed
(a,a+e ,a+e
then
singleton
it is dominated by
P
a;
G
(a,N)
to be dominated by some agreement in G.
any
)
no
follows
it
N)
,
by
(a+e a+e a+e
,
)
is
must be a
a
dominated by a singleton-agreement
can be
it
Suppose there is a unique a G [0,1]
then
1,
,
with b >
G
contradiction.
a
there
that
that
- i
involving a best- response strategy. Hence (a,N) G G implies a
If
note
playing a best response. Next, note that (a,N) G G implies
is
i
if a
partition
stable
a
)
,
cannot
N)
be
dominated by any agreement involving coalition (1,3).
It
remains
agreement
to
for
consider
N
that
possibility
the
in
is
But
G.
that
((1,1,1),N)
((1,1,1) ,N)
is
is
the
unique
dominated
by
((1,0,0),(2,3))GG.
Hence there exists no (a,N) G
in
B,
and
therefore
argument as for
(
(a+e
dominated by
,
a+e a+e
,
)
,
N)
,
G.
For any a <
some
1,
agreement
however,
then,
in
G.
((a,a,a),N) must be
Repeating
this can be ruled out.B
the
same
now
We
propose
following
the
modification
of
the
notion
of
a
partition. A semi - stable partition (G,U,B} of the set of agreements A
an ugly set U, and a bad set B
a good set G,
(a,T) e B
<=>
(a,T) G G
o
f>
-
is one
--
into
where
such that (b,S) > (a,T)
There exists (b,S) G G
(b,S)
-
stable
only if (b,S) G B
(a,T)
U = A \ (G u B)
In other words B consists of all agreements dominated by agreements
G
consists
of
agreements in
agreements
all
which
are
dominated,
not
except
in G,
and
possibly
by
3
B.
A semi-stable
partition
is
weaker
than
a
stable
partition in that
good and the bad sets do not exhaust the set of all agreements:
agreements that are neither good,
nor bad.
The
the
there may be
following lemma establishes
some properties of this ugly set.
Lemma
(a).
1
Any (a,S) in U is dominated by some (b,T)
in U.
Hence U is
either empty or infinite.
(b)
.
In finite player,
finite-strategy games, U is empty. Hence
a
semi-stable partition is a stable partition in such games.
(c)
.
(a,S) G U cannot be dominated by any (b,T) G G.
strategy sets and continuous utility functions,
Hence with compact
(a,N) G U implies a is a Nash
equilibrium.
Proof (a)
it would be
(a,S)
in G.
If
G U must be dominated by some agreement
(b,T)
G G then (a,S) would be
in B
(b,T),
instead.
If
otherwise
(a.S)
is
not dominated by another agreement in U,
then (b,T) e
But then (a,S) e G, a
B.
contradiction.
(b)
Follows from (a), since A is finite in finite games.
(c)
is obvious.
Ugly
sequences
sets
are
therefore
utility
responses
--
involve
they
functions,
they
i.e.,
another,
one
With
dominated by a good agreement.
sense
the
in
dominating
agreements
of
"open"
members
of
self -enforcing
are
of
spaces
and
coalition
the
the
in
none
but
strategy
compact
containing
of
strict
infinite
which
are
continuous
playing
best
non-cooperative
sense
Theorem
Proof:
4
1
A semi-stable partition always exists.
Define:
G
= ((a,S) e A
|
there is no (b,T)
B
= l(c,V) 6 A
|
(c,V)
For any
is
transfinite)
(finite or
given the definitions of G
B
It
is
a
= {(a,S) G A
=
a
((c,V) E A
clear that
/3
I
I
dominated by some (a,S) e G
ordinal a >
and B. for all
P
G
(c,V)
is
a
inductively
< a:
pa
n G
a
in A
=0
,
U
-
p<a
B. dominating (a,S)
p
pa
and thus B. C B
for all a.
suppose it is not true for some ordinal >
ordinal.
and G
a
dominated by some (a,S) G G
< a implies G. C G
a
define B
0,
}
P
there is no (b,T)
We first show that B
in A dominating (a,S))
Call it a and let (a,S) e
B
Q
n G
0.
.
Q
It
is
a
)
.
obvious for a =
0.
Now
Then there must be a first such
Thus
(a,S)
is
dominated by some
(c,V) E G
a
moreover (c,V) G
;
for some 8 < a
B_,
.
This means there is (b,T) G G„
8
P
which dominates (c,V); this (b,T) must be in
Thus
for some 7 < a.
B
G„ n B
B
7
*
Let
0.
6
larger
the
be
of
Then
8,-y.
n
G
*
B
7
contradicting
0,
the
6
assumption that a was the first such ordinal.
There are at most
Let
=
U
A
u
(G
-
7
7
partition.
and only
J
B
it
agreement is in
is
B
previous
distinct sets G
establish
We
).
so for some ordinal 7,
a
= G
G
7
that
{G
,U
7
7
By the above,
if
The
IAI
2
LL
£<7+l
in
semi-stable
a
is
}
7
An agreement
the sets are disjoint.
dominated onlyj by
agreements
b
j
,B
7
7+1
-
B„
..
in G
is
3
B
if
.
An
7
if and only if it is dominated by an agreement in G
theorem
semi-stable partition.
gives
next
The
inductive
an
procedure
theorem establishes
that
generating
for
in
a
game
a
with
a
finite number of players there is no other serai-stable partition.
Theorem
2
Proof
See appendix.
Theorems
For finite player games,
1
and
2
give rise to the following definition:
In a finite player game,
Proof
Equilibrium
the semi-stable partition is unique.
(S-CPNE)
if
a
strategy vector a is a Semi Stable Coalition
-
G
(a.N)
G
in
the
semi-stable
partition
of
agreements
Corollary
Proof
is
in G
1:
Every strong equilibrium is an S-CPNE.
In the proof of theorem
1,
the strategy vectors a such that (a,N)
are precisely the strong equilibria.
It
should be
kept
mind
in
from the existence of
issue
partition always exists,
exists.
good
set
((a, a, a),
In example
of
1
games
CPNE
a
partition.
All
--
definition of
of
form
the
in the set B.
is
equilibria
the
in the
(a,N)
equilibria
due
B-P-W
to
generates
it
the
in
case
player
finite
of
.
For any singleton coalition
Recursive Definition of CPNE
to be optimal for
if
(i)
defined
Having
is
i
playing a best response in
optimality
for
define optimality for a coalition
Say that a is optimal for
S
Finally,
optimal for
The
if
N
is
finite,
S
say
i
define a
,
}
a.
size
(k-1)
or
is
self -enforcing for
S,
and there does
such that (b,S) dominates (a,S).
that
a
an
is
R
(recursive) -CPNE
if
illustrates
improvement on the stable partition.
that
It
the
shows
semi-stable
that
partition
there are cases
as
1
is
is
an
in which
the S-CPNE coincides with the R-CPNE, but there is no stable partition.
corollary
a
N.
following example
S-CPNE
less,
if it is optimal for every TcS
S
if it
of
{
as follows:
of size k (>2),
S
not exist any b self -enforcing for
coalitions
all
Say that a is self -enforcing for
the
--
We now explore the connection of S-CPNE with the
CPNE
a
A semi-stable
there is no
Nash
separate
a
is
of either variety
are in the ugly set, but ((1,1,1), N)
are identical to the S-CPNE.
recursive
precisely the case:
is
when a stable partition exists,
Obviously,
of
may be that no CPNE
it
semi-stable
the
for a <
N)
this
2,
existence
the
stable or semi-stable partition.
a
but
that
is
a
strong
equilibrium
in
this
example,
it
also
shows
Since
that
would fail to hold if we used stability rather than semi- stability
the basis of the definition.
10
Example
N=
3:
(1,2),
112
U (a
this
In
)
= U (a
2
example,
involving
agreement
any other
a
=aa.
12
12
)
,
However
N.
Theorem
Proof
Let
(a,S)
€
method to establish that
singleton
k-1
,
So
S.
suppose
and let #S =
in
G
a
belongs to G n and dominates
is
C
S
for which
Therefore
it
is
we
are neither good nor bad,
intuitively plausible,
if a is a S-CPNE,
semi-stable
a
is
optimal for
it
is
true
a
is
not
for
partition.
S.
all
This
a
for
by an
and
two
the
is
is a R-CPNE.
it
We
use
inductive
an
obviously true
for any
coalitions of size not exceeding
By
inductive
the
S.
If not,
hypothesis,
But then (b,V)
B.
there exists
(a,T)
€
G.
dominates (a,S). This
G.
show there cannot be any d which
We claim that
to
optimal.
dominated by (b.V) g
that (d,S) dominates (a,S).
lead
partition;
k.
contradicts (a,S) G
Next,
stable
no
First, we establish that a is self -enforcing for
T
the unique R-CPNE.
is
the S-CPNE are a subset of the R-CPNE.
In general
In a finite player game,
3
it
1
equilibrium is
the
definitions coincide.
there
a ),(2))
argument identical to that of example
example,
strictly Pareto dominates all
(-2,-1)
((-2,-l),N)
since
agreements of the form ((1,
In this
[-1,1),
Since it is self enforcing,
also the unique S-CPNE,
is
=
A
[-2,1],
strategy vector
example,
other strategy vectors.
It
a
,
=
A
Suppose there is.
there exists
contradiction.
Since
is
self -enforcing for
Since (a,S) € G,
€ G which dominates
(e,S)
(d,S)
11
e
B,
there
exists
(d,S)
(a,S),
(f,T)
S
is
such
in B.
which would
e
G
which
dominates (d,S).
T.
If T C S,
Che
Then d cannot be optimal
self -enforcing
for
we
So
S.
therefore also (a,S). Putting
The
following
theorems
induction hypothesis implies
for
must
e
contradicting
T,
=
have
e
G
optimal for
is
hypothesis
the
that
which dominates
d
is
(d,S)
and
S-CPNE and
the
we are done.
f,
establish
(f,S)
f
circumstances
which
in
the
R-CPNE coincide:
Theorem 4
player
finite
For
games,
if
a
partition
stable
exists
the
S-CPNE and R-CPNE coincide.
Proof
R-CPNE
Given the result of Theorem
:
an
is
S-CPNE,
i.e.,
semi-stable
partition.
serai-stable
partition
agreement
in
agreements
in
agreements,
G
and by
B
are
By
is
if
a
is
theorem
stable.
Theorem
3
non optimal.
optimal
if
2,
for
agreement
Any
agreements
all
Since
then
N,
stable
a
these
in
in
two
to
show that every
(a,N)
partition
is
B
G
are
sets
e
G
in
exists,
dominated
optimal,
exhaust
the
the
the
by
so
set
an
all
of
the set G equals the set of optimal agreements.
In order
to
demonstrate the equivalence of the various versions of CPNE
we require that the stable partition exists.
is
we only have
3,
In one
important case existence
easily established:
Theorem
5:
In
any
finite
player,
finite
strategy
game,
a
stable
partition exists.
Proof
:
By Theorem
1,
a
semi-stable partition exists,
the set U is empty.
12
and by Lemma
1
(a)
Next
provide
we
with
coincide
not
do
—
N =
4:
Example -
- (0,1), A
(1,2), A
112
,
a
)
a
=aa.
12
12
= U (a
)
,
2
there is a unique Nash equilibrium (0,0).
In this game,
only self -enforcing strategy vector
therefore a R-CPNE.
We claim that
dominated by (a,N) where
exists
(c,S)
(0,N)
—
a
a a
12
£ G dominating
.
for
> 0.
So
must
S
This is dominated by
( (
1
c +e
,
that
then
be
2
optimal
is
must be
it
Since this is the
for
and
(1,2),
Suppose otherwise. Now (0,N)
£ G.
If S = N,
(a,N).
it
(1,2),
(0,N)
Clearly
J
contradiction.
((l,c),(2)) £G
= [0,1),
2
1
U (a
)
,
{
12
c =
and
2
)
)
there
would also dominate
(c,N)
(2)
and
£ B,
(a,N)
is
>
c
1,
a
So
.
2
which must therefore be
,
2
*
*
Any agreement that dominates
in B.
R-CPNE;
the latter is "unreasonable."
moreover,
,
S-CPNE
where
example
an
must be
this
((l,a ),{2))
with a
there
((l,a ),{2))
is
with a
6 G,
> c +e
> c
.
- -
so
it
it
has
dominates
2
2
2
2
((l,c ),{2)) £ G,
+e
2
*
*
Therefore,
> c
2
2
a contradiction.
2
since
The R-CPNE in this example appears unreasonable,
choosing
a
dominated
dominated strategy,
this
in
actions,
allow
is
conjectures
2
example,
close to
that
1
will
not
some
they
fashion
self -enforcing,
our
and
approach
play
a
The importance of self
1.
prevent players
from enjoying mutually beneficial
solely because there is no "best" action of this sort.
in
1
that they protect a player from being "double-crossed"
"near- rational"
would be able to describe
extend
If
should play some a
2
enforcing agreements
but
strategy.
player
=
actions a with a
1
therefore
to
behavior
allow
superior
for
to
such
agreements
13
1
0.
forms
for
the
and a
Were we
to
(1,2),
we
coalition
close to
1
as
"almost"
2
In
the
of
following section,
"almost"
we
self -enforcing
With
slightly more complicated example,
a
can show
we
R-CPNE need
that
even in a game with compact strategies and continuous utility
not be S-CPNE,
functions
Example
5:
N = (1,2,3,4)
-
A
,
[0,l]x{0,l)
i
tya)
- p p p p,[x
2
x
2 3
iv x
-
U (a) = p iP2 P 3P J2x
2
i
x
-
l
3
4
|x -x |]
-
2
1
-
I
2
3
x -x
'
I
'
(1-x +x
12
3
1
)
]
pppplx]
2
U (a) =
H
where x
2
|]
3
U(a)-pppp[2x
-x
_
r r r r
3
2
M
H
3
1
denotes the real number in [0,1] chosen by
i
component of i's decision.
example
the second
and p e(0,l)
i,
i
p=p
12
k
Call any agreement with
2:
agreement;
if all
extension of the game in
(Think, of this game as an
1c
=p
k
p=la
k
=
3
"participative"
i>
players 1-3 play the game in example
four players agree,
2;
otherwise, all players receive 0.)
It
obvious
is
p=p=
k
k
with
strategy
any
that
k
=
P
p=0is
k
self -enforcing for N. We shall establish that any such agreement is an R-CPNE
but not a S-CPNE.
To show that any such strategy a
agreement
must
self -enforcing
equilibrium,
for
Suppose
N.
and x
-
self -enforcing
those
in
example
deviate profitably,
x,
3
c
with
(1,2,3).
2:
otherwise;
=
x
2
strategy
for
-
x
1
participative
However,
participative.
be
neither
This
of
say,
J
then
the
=1.
If
<
x
1,
is
Nash
a
then any
4
components
three
by
(x+e x+e x+e
,
arguments
players
can
,
,
1)
analogous
is
to
unilaterally
and neither can any pair-subcoalition of (1,2,3)
14
b
constitute
must
it
established
first
participative
no
while x
real-valued
is
note that any dominating
an R-CPNE,
is
engineer
a
coordinated deviation that
as
for N,
for
(b,N)
dominated by
is
=
If x
(1,2,3).
1,
then
participative strategy with x
We next show that
x <
=1.
and x
1
k
- x
k
dominated by
is
where
(d,{2,3)),
- 0, which is optimal for
Now
rr
participative strategy
oj vector and x
is a
If (e, N) € B then (a,N) € G,
d
is
a
(2,3).
semi-stable partition.
in any
J G
self -enforcing
So b cannot be
.
implying that b cannot be optimal
(1,2,3)),
(c,
(b,N)
(a,N)
dominated by (e,N) where —
e
=»
self -enforcing
is
(a,N)
12
k
- x
k
is
= x
k
3
and we are done.
4
suppose that (e,N) e
Therefore,
dominates it.
it
It cannot
include player 4.
in G
must
it
,
involve
Therefore (e,N) G U.
two
is
,
12
k
-
x
k
=
x
k
=
something in G which
or paired coalitions,
involve singletons,
x
*
where
x
Further,
(1,2,3).
<
x
<
x*
1
neither can
.
to be
any
J
But
3
such
by reasoning which is identical to that of example
2.
Contradiction.
examples
above
establish
that
the
formulations are not identical in all cases.
by a stable partition are S-CPNE,
characterization
there
So it must involve the coalition
agreement must be in U
The
If so,
B.
terms
in
of
example
stable
5
recursive
Since
the
demonstrates
partitions
is
not
and
non- recursive
equilibria generated
fortiori
a
that
equivalent
to
the
the
recursive formulation.
Since our results
[1989]
it
finds
that
is
seem to be
different from those claimed in Greenberg
important to indicate the source of the discrepancy.
R-CPNE
can
be
characterized
by
using
stable
Greenberg
partitions.
problem is that he implicitly assumes that a stable partition exists.
have seen,
this cannot be demonstrated in general,
finite -action games.
15
The
As we
except for finite-player,
III.
Extended Agreements
The recursive and non- recursive definitions are equivalent for games with
finite strategy spaces and finite numbers of players.
down
in
other
cases.
example
In
would
equivalence
4
The equivalence breaks
restored
be
if
we
modified the game by adding strategies which corresponded to the limit points
of
the
strategy
resolution will
space.
work
not
example
But
5
general.
in
demonstrates
this
In
more satisfactory resolution for the case of
that
section
this
will
we
strategy
infinite
proposed
examine
a
spaces.
In this section we assume that the set of players is finite and that each
player's utility function
continuous on the compact strategy space ^ A..
is
U.
An extended agreement (a,S) consists of
of strategies a = (a
1
a
,
2
1:
The sequence converges.
2:
For all
t
€ S,
a
k
= a
t
In
actions
fixed.
other
by
words,
the
an
members
Extended
of
coalition
S
C N and a sequence
which satisfies the following conditions:
...)
,
a
k'
for all k,
k'
t
extended
agreement
coalition
agreements
include
S,
holding
simple
a
is
sequence
of
coordinated
actions
of
non-members
the
agreements
as
a
special
case:
identify a simple agreement with an extended agreement in which the sequence
of strategy vectors is constant.
Let A
denote the set of extended agreements.
An extended agreement (b,T) dominates (a,S) if
(a)
TCS
(b)
There exists k such that b
= a
j
(c)
lim U (b
k
)
> lim U (a*)
for all
j
for all ieT.
16
j
€ N\T
It is useful to note that (b,T) dominates
for any VDT
(a,V)
if
(a,V)
in A
is
initial extended agreement,
implies that it also dominates
Dominating agreements are coordinated
.
deviations from initial agreements:
(a,S)
At any step in the process that forms the
say k,
subcoalition can agree to break away and
a
follow their own coordinated deviation.
definition
The
of
partition
serai-stable
a
modification, as do the proofs of theorems
1
and
2
without
over
carries
establishing the existence
and uniqueness of the semi-stable partition.
A
vector
strategy
Equilibrium (ES-CPNE) if
a*
it
where (a,N) is in G for the
For
finite-player
serai
-
Proof
Nash
a,
stable partition of extended agreements.
there
games,
Coalition
Stable
limit of a sequence of strategy vectors
the
is
Extended
an
is
a
is
recursive
definition
in
extended
agreements analogous to the recursive definition of the previous section:
For any singleton coalition
Recursive Definition of Extended CPNE
that (a,(i)) e A
dominates
(a,
Having
(
i
if there does not exist any
is optimal
)
)
defined
optimality
Say that (a,S) e A
is
S
that
(a,S)
is
coalitions
all
for
in A
which
of size k (>2),
of
size
(k-1)
or
less,
as follows.
self -enforc ing if there does not exist an optimal
that dominates (a,S), with T C
Say
{i))
say
.
define optimality for a coalition
(b,T)
(b,
(i),
optimal
if
S.
it
is
self -enforcing
and
there
does
not
exist any self -enforcing (b,S) that dominates (a,S).
Finally,
if N
is
finite,
Recurs ive -CPNE (ER-CPNE)
a
if
it
say that
is
the
strategy vector a*
is
an Extended
limit of a sequence of strategy vectors
the
such that extended agreement (a,N) is optimal.
1
7
The next result
major result of the paper.
the
is
define Coalition Proof equilibria
all
finite
player
games
finite strategy spaces
those
--
in
terms
with
It
says
that
extended agreements,
of
infinite
strategy
spaces
if we
then for
well
as
as
the recursive and the non-recursive definitions are
--
equivalent.
Theorem
Using
6:
extended
agreements,
there
partition, which is also a stable partition.
6 G if and only if (a,S)
optimal;
is
is
a
unique
serai-stable
In this stable partition,
(a,S)
hence the limit of a is an ER-CPNE if
and only if it is an ES-CPNE.
Proof:
A
See appendix.
consequence
of
proof
the
equivalent, somewhat simpler
Corollary
:
of
theorem
this
is
that
there
is
an
recursive definition.
The following definition is equivalent to the other definitions of
Extended CPNE.
Alternative
Recursive
Definition
coalition (i), say that all (a,(i)) e A
of
Extended CPNE
For
are self -enforcing
Having defined self -enforcing for all coalitions of size
define it for a coalition
Say
that
(a,S)
G A
S
if
N
is
(k-1)
or
less,
of size k (k>2) as follows.
is
self -enforcing
finite,
say
that
there
if
enforcing (b,T) that dominates (a,S), with T C
Finally,
singleton
any
does
not
exist a self-
S.
the
strategy
vector
a*
is
an
Extended-CPNE if
it
is
the extended agreement
the limit of a sequence a of strategy vectors such that
(a,N)
self -enforcing and not dominated by any self
is
enforcing agreement.
Proof:
See appendix.
It
is
instructive
to
examples
use
2
and
equilibrium definition with the initial definitions.
extended equilibrium is the point (1,1,1).
contrast
to
5
In example 2,
Although it
the unique
(x,x,x)
and is the
There was no R-CPNE and no S-CPNE.
optimal member of this class.
5
extended
is not self enforcing,
it is the limit of a sequence of self enforcing agreements
Example
the
shows the power of our new definition.
The fact that the set
of optimal three player agreements is empty allows unreasonable R-CPNE in the
four player game.
but
does
Our definition of an S-CPNE eliminates
suggest
not
an
alternative.
example is the point (1,1,1,1):
it
is
The
extended
these equilibria,
equilibrium
for
this
"almost" self enforcing, and is maximal
among such points.
IV.
Games with a Countably Infinite Number of Players
Theorem
1,
demonstrating the existence of
a
semi-stable partition,
not depend on the number of players in the game being finite.
an infinite number of players,
the
does
But if there is
semi -stable partition need not be unique.
This fact gives rise to the following definitions:
The
strategy vector aeA
serai-stable partition of A.
It
is
is
a
a
Weakly
Stable
CPNE
if
(a,N)eG
for
some
Strongly Stable CPNE if (a,N)eG for every
semi-stable partition.
19
The following example illustrates the distinction:
Example
N =
6:
{1,2,3...};
=
A.
for
(0,1)
an infinite string of zeros and ones.
payoff of zero to all players,
a
table below.
For
corresponding payoff vector u
x
x
x
x
1
2
3
4
x
In
- (1
In this game any strategy vector gives
except
strategy vector x
a
=
=(100000...)
u
= (1
...)
u
=(101000...)
u
=(101010
u
1
1
1
game
this
1
1
1
1
...)
there are
two
(x..
complete
1
.
.
.
in
)
2
3
4
-(1,3,
the
the
table,
the
3, 3, 3, 3,
- (1,1,4,4,4,4,
- (1,1,1,5,5,5,
= (1,1,1,1,1,1,...
stable partitions;
Weakly Consistent CPNE.
are
,
in
- (2,2,2,2,2,2,
in one
all
the
even
in the other all the odd numbered
numbered strings are Weakly Consistent CPNE,
strings
,x_ ,x_
listed
,u.,u.,...) is indicated to the right:
(u.
u
1
strategy vectors
for
=
...)
1
Thus a strategy vector is
e N.
i
There
is
no
Strongly Consistent
CPNE.
00
Note that x
might
It
not an equilibrium in either partition.
is
seem
a
difficult
every semi-stable partition;
simple.
check whether
agreement
an
the procedure for doing so
*
There exists a minimal semi-stable partition {G
7
*
*
(G
in fact
to
is
in
relatively
is
This is due to the following theorem:
Theorem
is
matter
,
U
*
*
,
U
,
B
)
--
that
*
,
B
}
is
semi-stable partition
a
(G,
semi-stable
U,
B}
G
partition
c G,
20
and B
such
C B.
that
for
every
other
In the derivation in theorem
Proof:
G
,_
7*
the
= G
7*+l,
.
set G
,
= G
and let G
_
7*
B
= B
.
.
7*
any serai-stable partition,
in
the first ordinal such that
let 7* be
1,
*
•*•
Note that the set G n will belone to
will belong to
B_
semi-stable partition, and if for all
a < 8
G and B in all semi-stable partitions,
then G
G
,
a
and B
Thus
the
partition
semi-stable
minimal
generated
in
the
of
set
in
any
belong,
to
° respectively
J
P
partition
procedure
B
must as well.B
and B
P
a
the
is
precisely
theorem
1.
the
As
semi-stable
immediate
an
consequence, we have the following characterization:
Corollary
:
a
to some set G
a
is
a
strongly stable CPNE if and only if agreement (a,N) belongs
where the sets G
a
are defined as follows:
G- is the set of agreements not dominated by any agreement in A.
For a >
G
is
the set of agreements which are dominated by no agreement
in A,
except for agreements which are in turn dominated by some agreement in
V
<a
"
Note
-
that
this
B-P-W definition,
the
is
agreement.
it
is
a
recursive characterization as well,
does not require
unlike the B-P-W definition,
naturally applicable to games with infinite numbers of players.
provides a natural hierarchy among equilibria;
are
more
"salient"
than
equilibria (those in G
)
equilibria
the
recursion on the number of members of
this new characterization,
Thus
but unlike
in
G..
P
are most "salient."
21
if a < B,
When
It
also
then equilibria in G
they
exist,
the
strong
IV:
Summary and Comments
In
concept,
order
to
propose
yet
modification
another
yet
of
another
solution
the proposer ought to give three sorts of justifications.
demonstrate
that
it
is
indeed
modification,
a
the
new
First,
definition
and
to
the
original ought to be posed in such a form that the similarities are apparent.
This we have done by comparing semi-stable partitions with stable partitions,
and extended agreements with agreements.
Second,
the
proposer
ought
to
show
examples
in
that
the
concept yields more "intuitive" results than does the original.
this
as
Compare
well:
the
that
hold
We have done
for
the
the proposer ought to demonstrate
definition
original
the
of
5.
Third, and in our view most important,
theorems
solution
equilibrium extended agreement with any
initial equilibrium agreements for example
that
new
generality or more regularity with the new definition.
hold
The
with
greater
result we
have
examined is the fact that Coalition Proof Equilibrium as recursively defined
can be characterized by a stability criterion.
For
the
original definition
this is true when action sets are finite, but not when they are infinite.
For
our definition with extended agreements the characterization holds generally.
When we extend the
solution
extended agreements
in
countably infinite numbers of players,
be unique,
to
situations
with
semi-stable partitions will no longer
leading to weak and strong versions of the definition in terms of
semi- stability
.
In this case,
we would argue
greater
claim to our attention,
gives us
an explicit method
for
if
for no other
finding the
given the following interpretation:
that the strong version has the
reason than that
solutions.
theorem
This method can be
An agreement is a good one if
1)
no other
agreement blocks it (strong equilibria would fall in this category) or if
9?
1
2)
no agreement blocks it except for agreements that are blocked by agreements of
the
first kind,
or
if
3)
no
agreement blocks
it
except agreements
that are
blocked by agreements of the first or second kind, and so forth.
Future
concepts
papers
will
apply
these
two
modifications
to
other
solution
Appendix
In general dominance is not a transitive relation.
Nonetheless,
for both
agreements and extended agreements the following is valid:
Lemma
A.l:
dominates
(c,S)
If
and
(b,S)
(b,S)
dominates
then
(a,T)
(c,S)
dominates (a,T).
Proof
Parts
verified.
Part
and
(i)
In particular,
:
agreements involving
Proof of Theorem
the
definition of dominance
in
the
case
k
if b.
=•
obvious
(ii)
agreements it is nearly so:
Note
of
(iii)
a
by
k
setting
fixed
a.
for
=
S
T
i
€ S,
we
have
k
then c. =
(a,
i
i
G S.B
coalition.
B
)
and {G
U_,B „).
€ N,
o
{I}) G G
(a,
(i)) 6 G
x
and (a,
for
2
For any
1
k
a.
extended
for
transitivity on any set of
Consider any two semi-stable partitions {G.,U
Claim
agreements;
simple
of
immediately
are
2
{i}) € B
o (a,
(i))
B
2
Proof:
any
If
(b,(ij)
(a,{i})
which
G G^
dominates
dominated by (c,{i)) in G
Since
(a, (i))
is
then it cannot be dominated by any agreement.
not
Reversing the argument,
it
be
in
B
But
then
(b,(i))
must
be
which then also dominates (a,(i)), a contradiction.
dominated
(a,
must
For
by
(i)) G G
any
«
24
agreement
(a,
it
(i)) G G
must
also
be
in
G„
.
Suppose
Then there exists (b,
€ B
(i))
(a,
must also be
in
Reversing the argument, the claim
is
it.
Since
Claim
2:
(a,
and
(b,
(i))
G
it
follows
that
(a,
o
in
is
B„.
it is true that
G G
T)
(a,
(i))
(a,
established.
Suppose for all coalitions T with #T < k-1,
T) G G
which dominates
in G
(i))
L
2
o (a, T) B
G B
T)
then the same is true for coalitions of size k
Proof
:
Suppose
for
T
size
of
dominated by (b,V) which
is not in B
9
induction hypothesis implies
The
dominates
other
possibility
Now
(a,T).
that
(a,T)
e
dominated by some (c,W) G G
dominate
(a,T).
contradiction.
W
c
T
V
is
=
the above reasoning,
in
in
implies
(c,W)
(b,T)
g
B
which
W cannot equal T;
We conclude that (a,T) G G
Suppose (a,T) G B
T,
not
implies
G
(a,T)
G
G
e
,
so
Then
G
(a,T)
(b,T)
case,
B
So
implying
(b,T)
(a,T)
not
in
must
(b,T)
(c,W)
is
then the
contradicting
'
otherwise
e
B
be
would also
G
B
a
implies (a,T) G G
Then there exists (b,W) G G
(b,W) ^ G
€
If V is a proper subset of T,
.
(b,V)
that
is
G
(a,T)
k,
(a,T) G B
Reversing the argument completes the proof.
which dominates
it.
By
Proof of Theorem
6
The proof proceeds through a series of lemmas. We begin with a notational
convention, some definitions, and a preliminary observation.
Convention:
Notational
d
If
IR
let x > y mean x.
,
If A
Proof of Observation
points
in
:
an
is
a.
1
,
above observation,
Pick
= 1,
.
We will say
n.
.
set
x,
then there
bound
upper
is
upper bound for
> a.
ra.
for
point
a
z
in
A.
Define the set B
=
is
a
either there is a point
x
i
1,
.
B
.n.
in B to maximize 2.Z..
z
for
if
it
pick
is
is
or
A
IR
there
,
is
sequence
a
of
strictly
an upper bound for A.
an upper bound for A,
z.
> a.
l
l
,
a.
is
l
less than
stop.
Otherwise,
using the
which is an upper bound for A and which
is
can pick
a.
-
limit of a sequence of points
such that
is
strategy
Define a sequence recursively as follows:
Given
the
i
There exists a in A such that a > x.
increasing points in A whose limit
Proof
is an
z
For any non-empty, bounded set A in
2
which
A
> x and
z
closure of A such that
the
in
:
non-empty, compact set.
,
of
x is an upper bound for A if A does
bounded and A dominates
is
the closure of A such that
Lemma A
sequence
x.
Observation:
to be
for
> y.
A dominates x if a > x for some a in A.
not dominate
convergent
a
to represent the limit strategy vector.
vectors, we will use the notation d
For x, y in
is
From that sequence,
in A.
strictly
greater
than
b
J
a.
,
l-l
we
and the distance between
ii
a.
and
z.
2
If the process terminates the theorem is proved.
26
If the process does not
take a convergent subsequence of z.'s.
terminate,
The corresponding a.'s form
the desired sequence.
The key result is the following lemma:
it rules out "openness" of the set
of self -enforcing agreements.
Lemma A.
Suppose
3
T)
(a,
not optimal, but is self enforcing (if #T >
is
2)
then there exists (c, T) which is optimal and dominates (a, T)
Proof
The result is obvious for a singleton coalition.
:
£
-
{(d,T)
self enforcing
d.
=»
for
a.
i
For #T >
2
,
let
€ T}
In other words any self -enforcing agreement which dominates (a,T) must be
in £
Consider
.
members of
t5
.
the
set
Im £
,
the
image
Note that an extended agreement in
either
bound in Im £
there
sequence
members of
is
an
extended
of
T,
.
optimal
"Q
is
Applying lemma
agreement
agreements
with
whose utility converges
(d(l),T),
(d(2),T),
(d(3),T)
...
an
the
space
of
payoffs
for
in
optimal if and only if its image
to Im S,
dominates
strictly
to
,
for all extended agreements
A. 2
that
extended agreements will have a subsequence
such that
IK
from the limit strategy vectors
T,
is an upper
in
we can conclude
(a,T)
increasing
upper bound.
or
there
utility
This
that
is
for
sequence
a
all
of
d*(l), d*(2), d*(3),
is
a
.
.
.
convergent sequence (recall the notational convention).
strategy vector
c
Call
limit
the
.
Now choose a sequence of strategy vectors, one from each d(i), such that
this
also
sequence
converges
to
c
Call
.
Agreement (c,T) dominates (a,T) and no element of
is
agreement (f ,V)
*
dominates (c,T).
c.
If (c,T)
Then it
with V c
,
T.
is
dominated by an optimal
That means that all members of V strictly prefer
*
to c
,
k
and that for some k
k
f = c
i
1
for all
i
€ V
k'
That is to say,
.
f
k '=
d
k
'
(m)
.
That
for all
is
to
,
c
=
d
k'
(m)
.
And for all
i
k
6 V,
f
for k'
€
i
V.
And since all members of T prefer
d (m)
k
and some k'
But for some d(m)
f.
t?
sequence
self -enforcing, we are done.
Suppose (c,T) is not self enforcing.
f
"diagonalized"
this
say,
c
(f,V)
to d
(m)
,
all members of V prefer
dominates
(d (m),T),
to
f
contradicting
the
assumption that all agreements in £ were self -enforcing.
Lemma
A.A_j_
An
optimal
extended
agreement
is
not
dominated
by
any
self
enforcing extended agreement.
Proof
Suppose
otherwise
that
(a,T)
9R
is
optimal
and
dominated
by
a
self -enforcing (b,S). Since one optimal extended agreement cannot be dominated
by another,
exists
must not be optimal.
(b,S)
optimal
an
which
(c,S)
But
dominates
then by the preceding lemma there
And
(b,S).
Lemma
by
A.l,
(c,S)
dominates (a,T), a contradiction.
Lemma A
Proof
For any
5j_
stable partition,
serai
(a,T) e G implies
Identical to the proof for simple agreements in theorem
:
Lemma A
If (a,T)
in U it
is
Then by lemma
the lemma.
(a,T)
If
is
in B,
Lemma A
it
is
(b,V)
T,
is
that
G.
Since the size of V
in U.
self enforcing by the hypothesis of
is not optimal.
(a,T)
A. 4,
7j_
dominated by (b,V)
Given a semi-stable partition and
(a,T) G U implies that (a,T)
Suppose #T
:
player
e U implies
(a,T)
2),
that
By lemma A.
in G.
5
(b,V)
is
therefore (a,T) is not optimal.
optimal,
Proof
dominated by (b,V)
is
greater than the size of
is no
(>
Then (a,T) is optimal implies (a,T) €
is self -enforcing.
Proof
of the text.
2
Suppose we have a semi-stable partition with the property
6j_
for all coalitions T of size not exceeding k-1
(a,T)
is optimal.
(a,T)
i
in
single -player
T
=»
such
2
is
coalition T
agreement
(a,T)
to
dominated by
is
be
agreement, and therefore (c,(i)) e
optimal,
G,
Now suppose the result is true
If
(with #T > 2)
,
self -enforcing
Then (a,T) not self -enforcing
.
that
the case where #T = k.
a
it
an
,
means that there is
optimal
cannot
be
contradicting (a,T) G
for
any T with #T < k
(c,(i)).
dominated
For
by
a
a
any
U.
-
1,
and consider
(a,T) e U is not self -enforcing then there exists an
optimal (b,W), wich W c
Since #W < k
such that (b,W) dominates (a,T).
T,
application of the preceding lemma implies that (b,W)
is
-
1,
contradicting
in G,
the assertion that (a,T) G U.
Lemma A
8j_
(a,T)
optimal
is
and only
if
(a,T)
if
G
G
semi-stable
the
in
partition.
Combining
Proof
implies (a,T) G
The reverse
G.
semi-stable
the
preceding
two
the
partition
lemmas
we
conclude
implication is lemma
follows
from
fact
the
that
The uniqueness of
A. 5.
that
optimal
(a,T)
set
the
of
optimal
agreements is constructed without any reference to stable partitions.
Lemma A
9j_
U is empty;
Suppose
Proof:
(a,S)
preceding lemma,
i.e.
G
U.
the semi stable partition is stable.
lemma
By
(a,S) € G implies
A7
(a,S)
(a,S)
Proof of Corollary
U,
a
self -enforcing.
is not optimal.
which is optimal and which dominates (a,S).
dominates (a,S) G
is
Thus there is (c,S)
contradiction.
:
(a,T)
is
self enforcing
if and only if it is not dominated by any self enforcing (b,S) with
a
(c,S).
S
C T.
follows from the fact that optimal agreements are a subset of self
enforcing agreements.
by
the
By the preceding lemma (c,S) G G
The equivalence follows from the following fact:
"If"
By
self enforcing
To prove
(b,S)
with
"Only if" we show that if (a,T)
S
c
then
T,
This follows by applying lemma A.
30
3
it
is
is
dominated
dominated by an optimal
and lemma A.l.
References
[1]
G.
"Renegotiation- Proofness in Finite and Infinite Stage Games
Asheim,
B.
Through the Theory of Social Situations," working paper,
Economics and Business Administration, July,
[2]
C.
Asilis,
Equilibria,"
and
Kahn,
M.
C.
working paper,
D.
1988.
"A Unified Approach
Mookherjee
University of
Norwegian School of
Illinois,
-Proof
to
Urbana-Champaign
,
July,
1990.
[3]
D.
Bernheim,
Equilibrium: Part
J.
Boyd and
C.
of Economic Theory.
38
[5]
B.
Dutta,
Sengupta
Set," Journal of Economic Theory
[6]
J.
solution
Greenberg,
"Stable
concepts"
Stanford
Report 484, March,
[7]
,
1-12.
221-232.
(1986),
K.
(1987),
42,
Nash
"Financial Intermediary Coalitions," Journal
Prescott,
Ray,
D.
"Coalition- Proof
Whinston,
M.
Concepts," Journal of Economic Theory
I,
E.
[4]
and
Peleg,
B.
49
.
and R.
Vohra,
(1989),
113-134.
Standards
of
University,
Behavior:
"A
A unifying
Economics
IMSSS
Bargaining
Consistent
approach
Series,
to
Technical
1986.
"Deriving Strong and Coalition Proof Nash Equilibrium from an
Abstract System,"
Journal of Economic Theory
[8]
The
,
Theory
of
Social
.
49
(1989),
195-202.
An
Situations:
Alternative
Game-Theoretic Approach, Cambridge University Press, 1990.
[9]
C.
M.
Kahn
and
Under Moral Hazard"
[10]
and
D.
Mookherjee,
"Decentralized
Exchange
working paper, University of Illinois,
,
and
April 1990.
"A Game Theoretic Approach to Decentralized
Markets with Moral Hazard," working paper, University of Illinois,
1990.
Efficiency
September,
[11]
R.
Marimon,
"The Core of Private Information Economies," Paper presented
at the Mid-West Mathematical Economics Conference, Minneapolis, MN 1988.
[12]
R.
Myerson,
Matching
"Sustainable
Plans
with
Adverse
Selection,"
Northwestern University working paper, 1988.
[13]
J.
Behavior,
[14]
A.
von
Neumann
Princeton:
E.
Roth,
and
0.
Morgenstern,
Theory
and
of
Games
of
Cooperative
Economic
Princeton University Press, 1947.
"Subsolutions
Mathematics of Operations Research
and
.
the
Supercore
.1,(1976),
32
43-49.
Games,"
Footnotes
See also Dutta et al.
2
In other words,
[5],
dominion of G
the
is
precisely the complement of
The
G.
definition would be unaffected by writing it with one-way rather than two-way
implications
3
In a semi-stable partition it is possible
This is the case in example
therefore the bad set as well).
the
good set will not be
individual
players'
empty
strategy
for the good set to be empty
if
the
sets
are
payoff
functions
closed.
In
1
are
above.
However
continuous
definition
the
(and
and
of
a
semi-stable partition, unlike the definition of a stable partition, the double
implications are crucial.
4
This proof is a special case of the existence proof for subsolutions in Roth
(1976).
If
there
are
a
finite
number of players,
then
the
proof does not
require transfinite induction.
Again,
having an open set of strategies
extension of
this
not necessary;
more complicated
strategy spaces and continuous payoffs.
examples can be derived with compact
A natural
is
approach will
handle
games
with discontinuous
payoffs by modifying (iii) in the definition of dominance as follows:
lim inf U.(b
To
handle
k
)
> lim sup U.(a
non-compact
action
k
)
.
spaces
as
well,
it
will
be
necessary
to
substitute raonotonic sequences for convergent sequences of strategy vectors,
and to use the overtaking criterion for comparison of limit payoffs.
An alternate proof of uniqueness is provided in theorem
6.
HECKMAN
BINDERY
INC.
JUN95
„
1T
„,
1Uj
JN
"
MANCHESTER
INDIANA 46962