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IMPLEMENTATION AND STRONG NASH EQUILIBRIUM
Eric Maskin
Number 216
January 1978
massachusetts
institute of
technology
50 memorial drive
Cambridge, mass. 021 39
IMPLEMENTATION AND STRONG NASH EQUILIBRIUM
Eric Maskin
Number 216
I
January 1978
am grateful for the financial support of the National Science Foundation.
A social choice correspondence (SCC) is a mapping which associates each possible profile of individuals' preferences with a set
of feasible alternatives (the set of f -optima)
SCC,
f,
is to construct a game
.
To implement
*n
form g such that, for all preference profiles
the equilibrium set of g (with respect to some solution concept)
coincides with the f-optimal set.
In a recent study [1], I examined the general question of
implementing social choice correspondences when Nash equilibrium is
the solution concept.
Nash equilibrium, of course, is a strictly
noncooperative notion, and so it is natural to consider the extent
to which the results carry over when coalitions can form.
The
cooperative counterpart of Nash is the strong equilibrium due to
Aumann.
Whereas Nash defines equilibrium in terms of deviations
only by single individuals, Aumann 's equilibrium incorporates
deviations by every conceivable coalition. This paper considers
implementation for strong equilibrium.
The results of my previous paper were positive.
isfies a monotonicity property
If an SCC sat-
and a much weaker requirement called
no veto power, it can be implemented by Nash equilibrium.
The results
for strong equilibrium, on the other hand, are on the whole
negative.
I
show (theorem 2) that SCC's satisfying no veto power
cannot in general be implemented for strong equilibrium when the
number of alternatives is at least three.
There are, of course, some circumstances in which one is willing
to forego no veto power, weak though it is.
If one can identify some
alternative as the status quo, for example, then the property individual
rationality may have some appeal.
Individual rationality implies
R
,
nonetheless that non-status-quo alternatives can be vetoed.
(theorem 3) that one in fact can implement the SCC
I
show
which selects all
f
Pareto optima that no one finds less desirable than the status quo.
Unfortunately, as theorem
demonstrates,
3
rational SCC which is implementable.
I
f
n is the only individually
begin with a section on
terminology.
1.
Notation and Definitions
Let A be a set of social alternatives containing, to avoid
trivialities, at least two elements.
For convenience,
I
shall
assume A to be finite throughout, but all results may be extended to
LetQ. be the class of all orderings of the elements
the infinite case.
of A.
where
An n-person social choice correspondence (SCC) on
C
,... J&
(X..
f
:
where V
(R.
0^
,
.
X
.
.
G\
.
,R
.
)
*
* 01
e
,(£
)
IT(R
A
f(R_,...,R
,
)
is nonempty.
shall be concerned primarily with the case wherevX
In this studv, I
=
\)s
for all i;
priori restriction can be placed on
n
For any profile (R.R ) E
JI
the set f(R_,...,R
,
J.
n
-1=1
i.e., with the case where no
preferences.
, . . .
is a correspondence
,
.
(GL
a_
,
.
.
&
)
-J
is called the set of f -optima .
u
f (&!_,...
)
shall assume throughout that
= A,
n
CR,,...,R
1
n
I
n
)
E
Jl%
j-1
j
Otherwise we may delete those elements from A which can never be f-optima.
.
Three properties which shall concern me and which may he desirahle
in SCC's are monotonicity, no veto power, and individual rationality.
Monotonicity:
L
f
In
a e f(R, ,...,R
A
e
then a
if 3
e
f (R,
)
f
, .
e
.
.
n
a e f(R',
satisfies no veto power iff
V(R l5 ...,R
i
,R
n
The SCC
such that
n}
{1
V i ^
1
,
.
.
.
.
n
,R'
n
)
e
...R').
e
)
n&.
j
for all b,
aR b
j
(R'
1
),
in
and [Vi Vh aR.b =>aR'b] imply
)
Individual Rationality
status quo.
V(R_,...,R
1
i
j
1
is monotonic iff
j
No Veto Power:
V a
± A
11$
:
f
Let some alternative a
:
II
:
be identified as the
Q
t A
satisfies individual rationality
IKS
V a
iff
V(R ,...,R
n
1
The alternative a
to
R
(R,
)
e
)
£
e
fC^,...^)
aR^
iff there does not exist b e A such that bP
g
:
S,
x
.
.
1
is agent i's strategy space,
S
i
i.
A is (weakly) Pareto optimal in A with respect
An n-person game form for the set A is a mapping
where
for all
.
x
S
-*•
n
A
a—
for all i.
II
$.
3
Nash Equilibrium
g
:
s
:
E
II
* A with respect
ns,
is a Nash equilibrium for the game form
S
in
to the preference profile
(R,
J
V
V Sl
i
i
±
Strong Equilibrium
form
V C
c.
g
IIS.
:
{l,..
.
,
->
A
e
n} V s
n
e
C
(R_,...,R
n
1
g
S
)
* A
IIS
:
ai
,
V(R, ,...,R
1
for strong equilibrium iff
In
)
where
SE
iff
)
ill
C such that
g(s)R g(s
n
)
e
]gl
Hi
NE
(R.,,...,R
n
)
s
,
3/
)
JIR
:
-»
A
= f(R, ,...,R ),
n
1
is the set of Nash equilibria for the game form g
,
1
f(R, ,...,R
...,R
is said to implement the SCC f
with respect to the preferences (R
f
e
,
3
jeC
iff
for Nash equilibrium
*
NE
is a strong equilibrium for the game
IIS
with respect to the profile (R
The game form g
where
s_ )
i
,
i
s
:
iff
)
2/
g(s)R g(s
S
e
,...,R
...,R ).
n
V(R, ,...,R )
1
n
gin
(R,,...^
)
e
Analogously, g implements
IIS
.
SE
(R,
g
j
is the set of strong
the game form g with respect to the preferences (R
,
,
.
. .
1
,R
n
)
=
equilibria for
...,R
).
I should note that if g implements f for strong equilibrium, it
does not necessarily implement it for Nash equilibrium.
The reason for
this apparent anomaly is that g may possess Nash equilibria which are
not strong and which, furthermore, do not lead to outcomes in f(R.,...,R ).
-L
n
For example, consider the following two person game form, where Player
chooses rows as strategies, and Player 2, columnH.
s
2
s
2
a
a
a
b
1
.
This game form implements for strong equilibrium the SCC f*
f*
:
where
(R
x(^ t {a,b}
^-
aj and
(b.
(b,
a.)
'*
where
(b, bj = f*(b, aj=f*(a, b
f
It does not implement
(s
,s
lb
)
{a}
,
and
b
a, al = {b}.
f
for Nash equilibrium, however, because
f
constitutes a non
f
-optimal Nash equilibrium with respect
bl
to [a, aj
|to
2.
Veto Power
A principal theorem in Maskin
[1]
Is the assertion that an SCC
satisfying no veto power can be implemented for Nash equilibrium
iff it is monotonic.
In this section I show that the picture is
quite different for strong equilibrium.
Monotonicity remains a
necessary condition (theorem 1), but if the number of alternatives exceeds
two, no veto power and implementabillty become mutually incompatible,
at least when preferences are unrestricted.
n
Theorem 1;
If f
nGL * A can be implemented
:
for strong equilibrium,
then f is monotonic.
Proof:
If f is not monotonic then there exist a e A and (R, ,
1
ln..j
(R..
R
In
n
,
,
)
£
II(£.
such that a e f(R,
j=l
n
H S'
j=l 3
'
yet a
i f(R-
R
).
Now if
g
, . . .
:
-*
,R
and [Vi
)
A implements
Vb
.
.
.
n
II
aR.,b
f for
,R ),
*>aR.bl,
strong
n
equilibrium, there exists
s
e
IT
S
such that
s
is a strong equilibrium
In
for (R, ,...,R
and g(s) = a.
)
But observe that s is also a strong
equilibrium with respect to (R',...,R'), a contradiction of the defin
1
nition of implementation.
Q.E.D.
Theorem
2
If
:
|A|
an d if
3,
_>
satisfies no veto power,
^a ^ ^
f
:<
it cannot be implemented for strong equilibrium.
Proof
ments
(R,
1
Write A = {a(l)
:
f
, .
.
.
,a(m)
e
&"
A
Suppose that g
.
Assume first that m
for strong equilibrium.
R )
n
}
:
1 S
>_
n.
+ A
imple-
Choose
so that
a(l)P a(2)
...
P a(n)P a(n+l)
a(n)P a(l)
...
2
...
1
1
1
P a(n-l)P a(n+l)
2
P a(m)
...
2
2
a(n-l)P a(n) ... P a(n-2)P a(n+l)
3
P a(m)
...
3
3
P a(m)
3
a(2)P a(3) ... P a(l)P a(n+l)
n
n
n
...
P a(m)
n
Now suppose that s* is a strong equilibrium with respect to
(R_,...,R
1
n
for
If g(s*) = a(p) where p
).
<
any
i e
i =
n-p+2
i = 1
{1,
.
.
.
e
{l,...,m) then,
,n} if p
>_
n+1
if 2
<_
p
if p = 1
£
n
.
4/
g(s*, s
f a(p-l)
)
for all
s
.
e
s* is an equilibrium and a(p-l)P a(p)
Now consider R
for any
R
~*X
,
such that
e 6s>.
X
S
II
for all
V a ^ a(p-l)
.
a(p-l)
,
A
f(R
£
XI
,
R
^ i.
j
1
because player i can block
.),
But this is a contradiction of no veto power.
a(p-l) by playing s*.
t
Suppose next that n
Observe that
aP.a(p-l).
,
A.
e Q^
because
,
> m.
For i=l,...,m choose (R
n
"i
,
...,R
such
e (^
)
that
a(l)P^a(2)
...
a(m-l)pja(m)
a(2)P*a(3)
...
aCnOpJaCl)
...
a(m)P„a(m-l)
i
a(m)P a(l)
m
„i
and
= t,1
R
R.
i
)
= a(q) * a(i), then
Vi
because a(q-l)P.a(q)
3
ji
1
V
a
.
e^"
^"A
-q
q can block a(q-l) by playing s
(1)
|c|
V R
)
= a(i).
= m-1,
c
q
1
(S,
f(R
ft
,
R
q
+ a(q-l)
such that
,
a
-q
Now,
because player
)
This is a violation of no veto power.
.
3 R_ c
Vj
e
e C
<3^"
a(i-l)P .a(i)
m+1
.
This means that
such that a(i-l) £ f(R
Furthermore, by the symmetry of the above argument,
i and all
)
).
Now, by construction, there exists a coalition C
such that
eft™"
a(q-l)
1
-q
e
q
V R
R
(R..
%<~s\ s
-
..
a(q-l) aP a(q-l)
Thus, g(s
1
ea?"
A
-q
But now choose R
^ q.
q
with
.
be a strong equilibrium for g with respect to
s
if g(i
Va
„i
R
n
..=...=
m+1
1
Let
I
coalitions C such that
|c|
= m-1.
(1)
c>
R_
)
Q
holds for all
Now because m
>_
3,
the
'
cardinality of C is at least
C = {n-m+2,
.
.
Construct
,n}.
.
V(R.,...,R
)
e(R""
n-m+29
A
1
nH" 2
,
From symmetry, we may take
2.
f*
:
(^
Furthermore, taking
4.9
n-m+2
= S
-,
n-m+2
x
x
S
= S
S
i
g*
and defining
,
f (R
, .
.
.
,R
n-m+2
1
R
.,,...,&
.,)
n-m+2
n-m+2
does and because the coalition C
f
J
S
so that
f*(R 1 ,...,R
=
)
n-m+2
1
Now f* satisfies no veto power because
has no veto power.
A
:£
for j=l,
n-m+2
S
II
:
,
-*
.
.
.
,
n-m+2 and
A where
j-l
= 8^S
1'*
"
n-m+2^'
one may easil y verlf y that g*
implements f* for strong equilibrium.
But we have now succeeded in
g*(s*
S
n-m+2^
S
Continuing iteratively, one
reducing the number of players by m-2.
can reduce the number of players to the number of alternatives.
At this
point, the argument from the beginning of the proof applies.
Q.E.D.
Theorem
is false when the number of alternatives is exactly two,
2
as the following simple example shows.
&±
= Gl
is,
let
a
b
=
2
b
a
'
a
Let
a
b
MAJT
b
=
MAJ
a
MAJ
= f
a
MAJ
b
9
a
=
9
a
b
b^
a
f.
MAJ
b
=
{a}
a
b
b
f
aba
»
MAJ
a
b
(a
MAJ
a
9
b
>
>
b
b
a
b
a
»
MAJ
9
MAJ
%
SCC.
aba
a
abb
f.
b
b
be tn e majority
J
J rule
f„.
a
Let n=2, A = {a,b} and take
b
bl
baa
9
a
9
=
{b}
That
The following game form Implements
player
1
for strong equilibrium, where
f
chooses rows, 2, columns, and 3, matrices:
a
a
a
b
a
b
b
b
Individual Rationality
3.
No veto power, although appealing, may not always make sense.
In some circumstances, one may wish to guarantee players payoffs which
leave them no worse off then their initial welfare levels; i.e.,
one may require the SCC to be individually rational
In such cases,
.
players must be able to veto alternatives which entail net losses.
In
this section I investigate the set of individually rational SCC's which
can be implemented for strong equilibria.
It will turn out that this
set is a singleton, consisting only of the SCC which for any preference
profile, selects all individually rational Pareto optima.
I
first show
that this SCC can indeed be implemented.
Theorem 3:
f
:
y
Q^ °
a
Let a
t A
,
-L
Q
1
>
•
•
•
»R
respect to
f
V(R
be the SCC such that
f«(R-,
Then
A be identified as the status quo, and let
e
o
n
)
(R.
1
= {a e A
,...,R )}
may be implemented.
n
aR.a
i o
.
.
.
.
,R
n
)
e
(R ?
.
Pi.
Vi, a Pareto optimal with
.
10
Proof
The proof is constructive.
:
In
With each a
cribed in the hypotheses.
Take
s(a).
=
s1
.
.
.
Suppose that
= s
= {s(a)
I
is the SCC des-
f
In
A associate a strategy
e
Define g
a e A}
.
if s
= s
x
Sn
:
.
.
x
.
S
->
A
so that
fa,
/
g(s
,...,s
1
\
n
)
J
<.
=
a
^
Now if a
1
o
,
= ... = s
'
otherwise
In
In
f(R, ,...,R ), then I claim that s = (s.,...,s
e
Clearly g(s) =
is a strong equilibrium.
= s(a)
n
= (s(a)
)
,
o
.
.
.
,s(a))
No coalition C smaller than
a.
since deviating
the grand coalition can improve itself by deviating from s
yields a
,
which, by individual rationality, no one prefers to a.
On
the other hand, the grand coalition cannot improve itself because a is
Therefore f(R, ,...,R )
1
n
Pareto efficient.
is a strong equilibrium with respect to
s
Pareto optimal.
Therefore, if g(s) = a
,
—
(R
,
.
a
.
o
,
o
i o
In
f (R n
,
.
.
then a
o
,R ).
n
&
g(s) is obviously
f(R,,...,R ).
1
n
If
for all i since anyone can force the outcome
then aR.a
Therefore a e
.
1
o
g(s) = a $ a
Now suppose
SE (R, ,...,R ).
n
1
g
<^-
.
.
,R
)
,
and SE
gin O In
(R,
,
.
.
.
,R
)
-
f (R ,
,
.
.
.
,R
)
Q.E.D.
Next
I
show that f
vidually rational.
is the only implementable SCC which is indi-
11
Theorem 4
f:(ft*...x<3,±A
A
A
If
:
is Individually rational
and implementable for strong equilibrium, then
Proof
implies that there exist (R
,
...,R
x
n
)
e
CR
is Pareto optimal for (R, ,...,R ), aR.a
V
,
...,R
e
A 3
8
e
IIS.
:
a z A
f
^ f
This
.
such that a
for all i, and yet
i o
-*
A implement
f for
strong equilibrium.
for which
IIS
(1)
Vj
(2)
g(s
bP a
C
e
s_
,
c
c
Cho ose (R.
,
.
1
(3)
.
such that g(s) = a, there exists a coalition C such that
s e IIS.
3 b
Let g
).
and
.
a
n
1
a i f(R
f
Then there exists an implementable
Suppose not.
:
f =
.
.
,R )
n
(R......R
1
n
e
(J,
)
:
aP>
i
\
A\
({a
o
}
U
D)
ln\o
=
(R.
R
)
:
v
A\({a
5/
} \J
D)
bl a for all i}
|
a P.b.
(4)
Vb
(5)
Vi aP.b for all b
=>
= b
)
such that
,
A
\
where D = {b
;
Suppose that
oil
s
e
e
D
gin
SE (R, ,...,R ).
If g(s) = a,
then from the
above argument there exist a coalition C, alternative b, and deviation
s
ellS. such that
being an equilibrium.
(1)
and (2) are satisfied, a contradiction of s's
Suppose g(s) = b
j*
a, b i D.
Because a is Pareto
12
optimal for (R
,
...,R ), there exists
contradicting the hypotheses on
a Pareto inefficient outcome for
be a strong equilibrium.
g
1
n
)
=
{l,...,n} such that aP.b.
Therefore b is not an individually rational outcome,
From (4), a P.b.
SE (R, ,...,R
i e
4>
.
If g(s)
f.
(R.,...,R
1
n
e
),
D,
then from (5), g(s) is
and therefore
s
cannot
Thus, in all cases, contradictions arise.
So
13
Footnotes
1.
and
Throughout, P
I
shall denote, respectively, the strong and
indifference relations associated with R
gls
,
S
)
— g(,S.,...,S., S
.
g(s n
,
s
)
= g(s) where s. =
J
(
3.
If p = 1, let a(p-l) = a(n)
5.
The notation
R
:
T
s
±
I
4.
S
,
s
±
.
._,...
,
it
C
,
i/
C
,
S
J
denotes "R restricted to the set T."
14
Reference
[1]
Maskin, E. S.
,
"Nash Equilibrium and Welfare Optimality,"
forthcoming in Mathematics of Operations Research
.
;
%
?
'
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