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COMPACTNESS AND CONVEXITY OF STATIC CONTRACTUAL SETS
by
Murat
433 - 69
R.
Sertel
December, 1969
MASSACHUSETTS
INSTITUTE OF TECHNOLOGY
50 MEMORIAL DRIVE
CAMBRIDGE, MASSACHUSETTS 02139
COMPACTNESS AND CONVEXITY OF STATIC CONTRACTUAL SETS
by
Murat
433 - 69
R.
Sertel
December, 1969
no.
cjECt'^'^'^
FEc
^33-^9
A
COMPACTNESS AND CONVEXITY OF STATIC CONTRACTUAL SETS
Murat R. Sertel
This is a sequel of an earlier paper (Sertel, 1969), where, among
other things, a static social contract is defined for a collectivity of
and shown to exist
(sub jectlvely-)expected-utility maximizing behavors
if behavors' impressions of social states are continuous in the latter.
Here it is shown that the static contractual set is compact and that, if
behavors' impression functions are linear on it, the set is then convex
also. Some implications of these results are briefly discussed.
,
Introduction
;
This brief paper is meant to be intelligible after having read
Sertel (1969). There it was shown that the static contractual set
,
which is the briefer term we adopt here to designate the set of static
social contracts, is non-empty. Here we demonstrate a compactness and
n
certain convexity property for that set.
For the sake of continuity, the notation of the mentioned paper is
adhered to. For the same reason, we also continue with the numbering of
results, etc., from where we left it there. First we take up the compactness, and then the convexity result to be proved. Remarks follow.
Compactness
:
Without delay, we prove the following fact.
32.
COR
.
The static contractual set C of a collectivity N is compact.
Proof: Since C is
a
subset of the compact social state space P, C is
534
'.).";
a
bounded. Therefore, we need only prove that C is closed. Let {c
sequence of points in C converging to
so that
=
c^
<c,
exists an open ball
C.
^
Then
c
be a
S(c),
i
Since
(Sertel, 1969)], this implies that there
LEM.
6)
B(c^;
c
the graph of the social evolution S.
c> i G(S),
C(S) is closed [by 21.
Suppose
c.
}
with radius
fl
B(c; 6)
'
about
c_,
such that
G(S) = 0.
'
Let K be large enough so that, for all
c*^,
6
greater than or equal to K,
C
<
c|
6
I
Then, for all
dieting that
Convexity
C
greater than or equal to
c
c
e
C.
Hence,
c
C,
e
K,
= <c
c_
,
c
>
s^
G(S), contra-
so that C is closed.
:
Now we prove the following fact.
33.
C(Jt\
.
if,
for all
j
the impression function I.
in N,
is linear on C,
Chen C is convex.
Proof
A
e
:
[0,
Let c, c' be static social contracts, i.e., elements of C; let
1], X'
= 1 - A,
and correspondingly for
c
c'.
= Ac
,
c
+ A'c'. Denote
,
and for c,,
c.
c
.
= f.(c),
Denote also
V.(p) = U.(I.(p)) = U.(p. * I^(f^(p))).
3
IJ
J
J
c-^
=
f-"
(c)
,
Since
I.
is linear on C, and since U.
J
J
is
linear [by
8.
ASS.
(Sertel,
(A2)
1969)], V. is linear on C. Hence,
V.(C) = AV.(c) + A'V.(c'),
J
J
J
But
* I^(cJ))
V,(c) = U,(c.
>
U.(p. * I^(c^)),
/
p'.
(p.,
3
V.(c') = U,(c'.
J
Also,
c.
= Ac,
2
2
+
2
2
REM
.
..., c
-22
U.(p'.
>
3
I^(c'^)),
2
2
= V
J
3 A.
2
*
since f. is linear. Hence, denoting p. = Ap. +
X'c',5
V.(?!)
Thus, y = (c„,
* I^Cc'^))
(Z.
Jl
)
c
* I^(c^))
>.
U.(p. * I^(e^)),
J
J
Y(c), implying
c e
J
P.)
e
1
A
'pi
2
(p.
e
2
P.),
J
2(Y(c)) = S(c). Hence,
J
c
e
C.
Without following them through very far, we may briefly indicate
some implications of the last two results. Apart from giving some motiv-
ation for these results, the remarks below are hoped to be suggestive and
inviting to other researchers.
As a result of 32. COR., every continuous functional defined on C
will attain a
maxiraura
(mininiu-m)
on that set. As a result of 33. COR.
,
allows us to define concave or convex functionals on itself. Thus, one
may now go ahead and consider continuous objective functionals on C,
specify the risk aversion reflected by their curvatures, and speak of
maximizing (minimizing)
then-i
on C,
C
The relevance of all this is clearer when one begins to consider
"social leaders" or "controllers" who may be willing and able to influence
the equilibrium outcomes, i.e., social contracts, obtaining for collect-
ivities such as N, by control over the environment, impression functions,
or tastes (utility vectors) involved. Such controllers may be imagined
as
forming a "higher level" "controllers' collectivity" of their own,
which may be analyzed quite the same as N. In that case, it will be useful to be able to specify well-defined and tractable optimization prob-
lems for the controller-type behavors of this new sort of collectivity,
which is precisely where the last two results should be welcome indeed.
It is hoped that these remarks may be justified by an undelayed
excursion in the indicated direction.
NOTES
*
This research is part of an ongoing effort inthe area of Management
Information and Controls at the Sloan School of Management of M.I.T.,
and was partially supported by a NASA grant under the direction of
Prof. Zenon S. Zannetos.
REFERENCES
(Sertel, 1969)
Murat
R.
:
Sertsl^^ "A Formal Framework and Fundamental Results for
Social Analysis", SSM Working Paper 429-69, October, 1969.
5:^^KBiate- Due
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