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EQUILIBRIUM POLICY PROPOSALS WITH ABSTENTIONS

by

Peter J. Coughlin

- I'

Technical Report No. 333

May 1981

A REPORT OF THECENTER FOR RESEARCH ON ORGANIZATIONAL EFFICIENCY

STANFORD UNIVERSITY

Contract ONR-NO0014-79-C-0685, United States Office of Naval Research

THE ECONOMICS SERIES

INSTITUTE FOR MATHEMATICAL STUDIES IN THE SOCIAL SCIENCESFourth Floor, Encina Hall

Stanford UniversityStanford, California

94305

I-.M

2- ~~.* -

- - -

ABSTRACT

This paper analyzes spatial models of electoral competitions with

abstentions in which candidates have directional or local strategy sets.

It includes, as a special case, situations in which incumbents must defend

the status quo. The results derived here provide necessary and sufficient

conditions for directional, convergent stationary and convergent local

electoral equilibria for these spatial models. These conditions provide a

method for finding such equilibria. They also provide existence results

for directional, stationary and local electoral equilibria for societies

with abstentions. The results on stationary and local electoral equilibria

are obtained by analyzing cumulative plurality (or plurality potential)

functions.

L D11 T AB

I I I

EQUILIBRIUM POLICY PROPOSALS WITH ABSTENTIONS*

by

Peter Coughlin**

1. Introduction

Spatial analyses of economic policy formation in elections when voters

may choose to abstain have been developed by Hinich and Ordeshook [19691

[1970], Hinich, Ledyard and Ordeshook [1972], Slutsky [1975], McKelvey [1975],

Denzau and Kats [1977] and Hinich [1978]. One of the primary concerns of

these investigations has been the determination of conditions under which

there exist pure strategy equilibria for vote seeking candidates (and hence

predictable outcomes). The only societies for which such equilibria have

been shown to exist, thus far, have been ones in which the distribution of

voters' ideal points is radially symmetric or special concavity conditions

are satisfied. However, these special assumptions are highly restrictive

1!and have additionally been criticized for being empirically ad hoc.- These

(and related) analyses have also been criticized for assuming that candidates

have perfect mobility (or global strategy sets).-2/

This paper, alternatively, considers the nature of electoral equilibria

without including any special radial symmetry or concavity conditions.

Additionally, it studies societies in which candidates have directional or

local strategy sets- / and includes in its analysis the important case in

which incumbents must defend the status quo. The spatial voting model

*This paper has benefited from helpful comments and suggestions which have

been provided by Professors Kenneth Arrow, Melvin Hinich and Shmuel Nitzan.Financial support for this research was provided by Office of Naval ResearchGrant No. ONR-NO001-79-C-0685.

**Institute of Economics and Statistics, Oxford University. I

-2-

analyzed in this paper also includes probabilistic voter choices between

candidates L/ as well as between voting and abstaining. This analysis pro-

vides necessary and sufficient conditions for directional, stationary and

local electoral equilibria. These conditions, in turn, provide general

existence results for electoral equilibria when the society has an oppor-

tunity set which satisfies assumptions which are standard in microeconomics.

All proofs are in the appendix.

2. Electoral Competitions with Abstentions-/

The set of social alternatives is given by a non-empty, open, convex

Euclidean policy space, X C Rn. At any given time, the social opportunity

set will be a feasible compact subset, S C X. In an electoral competition

with perfect mobility, candidates compete for votes by proposing any of the

feasible policies for the society. Their global or basic strategy set is

therefore the set S.

C = {1,21 will be an index set for the two candidates. Whenever we

are considering a pair of proposals made by the candidates, they will always

be ordered according to the candidates' indices. In particular, we will let

ie S denote a basic strategy for candidate i. Then ( ,1,%) will be

a pair of policies proposed by the candidates.

Individual voters will be indexed by the elements, a , of a set

A C R n Their choice behavior will be summarized in (aggregate) probability

functions,

(1) Pi: X X X [0,1]Ia

-3-

i

for i = 0,1,2 and a E A. P (i,2 ) with i E C is used to denote the

robability that an individual who is randomly drawn from the citizens

lexed by a will vote for candidate i when the pair (i,p 2) is pro-

osed. P0 (, 2) = i - P1(II2) - P (, ) is the probability that sucha 12 al'2 CL12

an individual chooses to abstain.

We will additionally assume that (for i = 1,2),

(2) P 0p1 ,p 2 = P'(G( 1 ),Ga4 2 ))

where G (4) is a scalar-valued function on X. This enables us to include

both the utility-based probabilistic voting and abstentions of Hinich, Ledyard,

Ordeshook, et al and the metric symmetry of McKelvey.-/ We will also take

both the Pi and G to be twice continuously differentiable functions.-a a

This follows from (but does not require) the assumptions about aggregate

voting behavior in Hinich, Ledyard and Ordeshook (e.g. see [1972], pp. 147-148).

We will also assume that there is policy-related voting, by which we

mean P(G(x),G (y)) = P2 (G (y),G(x)) for all x,y E X. The policy relateda a a a a a

voting in McKelvey [19751 implies this assumption for the voting behavior

which he has studied (when Ga(x) = U(x)). It also follows from the assumptions

in Hinich, Ledyard and Ordeshook [1972], [1973] and Denzau and Kats [1977].

The population of individuals (and, hence, their aggregated choice

probabilities) will be summarized by a probability measure space (A,A,p).

For technical reasons (and with essentially no restriction) we assume that

iG , and P and their first and second partial derivatives are integrablea a

with respect to this measure space. When the candidates have incomplete

information about the distribution of individual characteristics in the

population, this p must be estimated. However, since candidates usually

. .. . * - .. L

have access to the same polls, past election data and other sources of infor-

mation, we are implicitly assuming that in this case they have a common

estimator- (as in Coughlin and Nitzan [1981]).

Finally, we will assume that candidates are interested in maximizing

their expected pluralities.-- Pei(1, 2 ) will be the notation used for

10 /the expected plurality- for candidate i at the pair of proposals

1 x2

3. Directional and Stationary Electoral Equilibria

In this section we will be concerned with situations in which candidates

can (at most) marginally vary previously established positions. As is

standard in microeconomic analyses, we will examine the consequences of

candidates being concerned with the marginal changes in their respective

expected pluralities which can result from their strategic choices.

At any basic strategy, ip E- S, the directional strategy set for

candidate i, T(4W), consists of all the feasible directions in S1/

together with the zero vector in Rn (i.e. together with "no change").

We will use u E T( *) and v E T( *) to denote directions which may be

selected by candidates 1 and 2, respectively.

The payoff function for candidate i, i E C, (on the directional

strategy sets) when the candidates are at the basic strategy pair (*,**) E

is given by the directional derivative (equation (15) in the appendix)

(3) Pi(u,v) = D (uv)Pi l( 12) at (p1,42) =

for every (u~v) E T(**) X T(**). This is simply the net effect on the candi-

dates' plurality of the simultaneous variations in position by both the candidates.ii

Therefore, a pair of directions, (u*,v*) e T(**) X T(**), is a direc-

tional electoral equilibrium (in pure strategies) at the basic pair of

policies (**,*) E S2 if and only if

(4) Pl(u,v*) : Pl(u*,v*) , V u E T(**) , and

P2 (u*,v) : P2 (u*,v*) , V v E T(V*)

Theorem 1: (u*,v*) E T(**) x T(*) is a directional electoral equili-2

brium at ) • S2 if and only if u* and v* maximize the directional

derivatives DuPt(I and DvPtI(, 2 ) at 1= and '2 =

respectively.

This implies a general existence result for such equilibria:

Corollary 1.1: There is a directional electoral equilibrium (in pureS2"

strategies) at every (i1 , 2) E S

The strategic maneuvering of the candidates is in a state of rest if,

and only if, both of them choose to remain at their current basic strategies.

Therefore, to say that there is a stationary electoral equilibrium at the

basic strategy pair (*i, 2) E S2 means that (0,0) E R2n is a directional

electoral equilibrium at

The remainder of this section will be concerned with existence questions

for stationary electoral equilibria. The analysis will answer these questions

in the affirmative by showing, even more specifically, how to find convergent

pairs of basic strategies (i.e., ones at which both candidates propose the

same basic policies) where there are stationary electoral equilibria.

It should be observed that, thus far, the discussion has assumed that

each of the candidates can vary his or her basic position in S in any feas-

ible direction. However, quite often, an incumbent does not have this mobility

12/and must, instead, defend the status quo.-/ Furthermore, challengers to

a status quo or an incumbent might be restricted to feasible directions away

13/from the status quo.-'

Hence, we could alternatively study electoral games in which there is

a fixed incumbent and a challenger whose objective is to maximize his expected

plurality by appropriately varying his position in a feasible direction away

from the status quo. However, we will show, this is equivalent to studying

stationary electoral equilibria at convergent pairs of basic strategies.

Consequently, this important case will be included in our analysis.

More formally, the possible payoffs available to the challenger at

the status quo * E X are given by the directional derivative,

(5) DuPtl(pl,*) at i =

for the u E T(*). An optimal strategy for a challenger is therefore any

u* which maximizes DuP41 (*I,) at 'l = € " Consequently, he'll be willing

to not vary his position away from the status quo if and only if

(6) DuPL(l,) DoP1 (*.4 ) at i =

for every u E T(p).

Therefore we say that there is a stationary equilibrium at * E S when

the incumbent must defend the status quo if and only if (6) is satisfied.

Theorem 2: There is a stationary electoral equilibrium at , E S2

if and only if there is a stationary equilibrium at * when the incumbent

must defend the status quo.

We will now turn the discussion to some preliminary considerations which

will enable us to define a third social choice mechanism. This third mech-

anism will be of special interest since it will provide us with necessary and

sufficient conditions and general existence results for the stationary equilib-

ria of Theorem 2.

Given an institutional setting in which only directional or infinitesimal

changes are possible, a society may be concerned with the consequences of

following different possible C1 (or at least piecewise C ) paths in its

policy space. For instance, alternative paths could be compared on the

cumulative total of the expected plurality (positive or negative) for changes

along the paths.

To calculate this accumulated plurality, we will use the following

notation for the gradient of marginal expected pluralities for changes from

a status quo x,

(7) VlPf[(x'x) , x

where h - 1,... ,n. Using this notation, the cumulative (marginal expected)

1plurality for the changes along a (piecewise C) path K from x tox0 1x

is the line integral=--

4

-8-

Xl=K(b)

(a)) 1 tl(K(T),K(-r))dKXo=K(a)

bfV 1Z (t(K(T),K(T)). K'(T)dTa

(where • means inner product)..

It should be observed that this expected plurality would never be accumu-

lated by a candidate who was competing against a rival (since both of them

can make the same calculations and changes). However, this integral can be

used as a measure of social approval or dissatisfaction with a path (i.e.

with making all of the successive changes along K).

Since X is convex, there are many piecewise smooth paths from any

x0 E X to any x1 e X. Therefore, for any particular x0 we can specify

a function

(9) K = K(x0,x) = O(x; xO0

which assigns a piecewise smooth curve from x0 to x for each x E X.

For each specified x and 0 we then have a cumulative plurality function,

x=K(b)

(10) G(x) =G(x; x0,4) = f VIPtI(y,y)dKxo=K(a)

defined on X (see (8)).

Any vector-valued function from Rn into Rn is called a vector

field. Therefore, F(x) = VIPt(x,x) is clearly a vector field defined on

X. Additionally, if F(x) is a vector field and there exists a function

t..

f(x) for which Vf(x) = F(x) for all x E X, then F is called a gradient

field and f is called a potential function for the gradient field.

For the class of electoral games being studied in this paper we have

the following important result:

Theorem 3: F(x) = V P 1(x,x) is a gradient field.

One consequence of this theorem is

xCorolary3-1:Thevalue of the line integral fl ViPtl(y,y)dK is the

1~ x 0same for every piecewise CI path K from x0 to xI .

This says that, given x0 E X, the function G(x) =G(x; x0 ) = G(x; x04 )

is independent of the function €. We are therefore justified in calling

G(x; x 0 ) (see (10)) the cumulative plurality function at x0.

Theorem 3 also implies

Corollary 3.2: The cumulative plurality function at any given x0 E X

is a potential function for F(x) = V P1 (xx).

The cumulative plurality function at a given x0 E X will consequently

also be referred to (interchangeably) as a plurality potential function.

This brings us to the third social choice mechanism which we will examine

in this paper. Many procedures which have been suggested for making social

choices involve the maximization of a social objective function (for instance,

social welfare functions and Borda scores). When a society can choose among

only feasible directions at a status quo it can, analogously, be concerned with

maximizing the marginal change in its objective function. We will therefore

refer to any ' E X as a stationary outcome for the society's plurality

potential function at x0 E X if and only if

- 4A

-10-

(ii) D G(x; x0) _ D G(x; x) at x =U 0 0 '0

for every feasible direction, a E T(*).

This enables us to provide necessary and sufficient conditions for the

stationary equilibria in Theorem 2 which specify their locations:

Theorem 4: There is a stationary electoral equilibrium at the basic

strategy pair (ip , ) E S2 if and only if P is a stationary outcome for the

society's cumulative plurality function.

Our development of the cumulative plurality function also now provides

us with the following general existence results:

Corollary 4.1: There exists a basic strategy pair (i,$2) E S2 at

which there is a stationary electoral equilibrium.

And even more specifically,

Corollary 4.2: There exists a convergent pair of basic strategies

2(4,w) E S at which there is a stationary electoral equilibrium, and

Corollary 4.3: There exists a status quo, P E S, at which there is

a stationary equilibrium when the incumbent must defend the status quo.

4. Local Electoral Equilibria

Stationary electoral equilibria which occur at critical points of a

cumulative plurality function may have the undesirable property that each

candidate is minimizing his objective function (while taking his rival's

current policy position as given). Then, if the candidates obtain more local

information than just marginal pluralities, they will not remain at their

positions. Alternatively, even if they know just their marginal pluralities

but are perturbed to slightly different basic positions they may select

directions for their positions which point away from (rather than back toward)

the nearby stationary electoral equilibrium point. I.e., such equilibria

can be locally unstable.

We therefore now consider existence questions for local equilibria. In

particular, a local electoral equilibrium is a pair of basic strategies

(i1"2' such that

(12) P 2(*I,€p) Pll(4('ip) , and

2 1 2 2 1(2,

for every *1 E N l(p*) and 2 C N2(*) for some c1, 2 > 0 (i.e., in

some pair of e-neighborhoods of and **). Additionally, there is a

local equilibrium at 4 when the incumbent must defend the status quo if

and only if

Pt I ] Pq€,

for all i N () for some C 0. These definitions are based on the

concept of a local voting equilibrium in Kramer and Klevorick [1974].

We will study these local equilibria under assumptions on the social

opportunity set, S, which are standard for such sets in microeconomic analyses.

Specifically, we will assume that S is a compact subset of X which is

defined by m(<n) C2 equations of the form

(13) gk(X) =0 (k 1,...,m)

-12-

This means that the candidates and the society have Lagrangean maximization

problems.

Additionally, in order to study this problem for C2 payoff functions

we will analyze those situations in which G(x) (see (10)) and Pt (x,y)

are both non-degenerate functions relative to the constraints given by (13).12/

It should be remarked that this imposes essentially no further restriction

on the class of electoral competitions we are analyzing since, generically,

every C2 function is non-degenerate. 16/

Theorem 5: There is a local electoral equilibrium at the basic strategy

pair (*,'p) ( S2 if and only if there is a local equilibrium at P when the

incumbent must defend the status quo.

With this equivalence result in mind, we can also specify the locations

of these local equilibria with a theorem which is analogous to Theorem 4:

Theorem 6: There is a local electoral equilibrium at the basic strategy

pair ($,*) E S2 if and only if 4 ib a local maximum of the society's

cumulative plurality function at 4.

It should be noted that this characterization of the convergent local

electoral equilibria for the candidates is similar to the results of Arrow,

Gould and Howe [1973] for local optimization problems. Here, however, we

have found a constrained optimization problem which will solve our game's

local saddle point problem, instead of the other way around.

Again, this gives us general existence results:

Corollary 6.1: There exists a basic strategy pair, (i2) E S2,

which is a local electoral equilibrium.

-13-

Furthermore,

Corollary 6.2: There exists a convergent pair of basic strategies,

(€,i) E S 2 , which is a local electoral equilibrium, and

Corollary 6.3: There exists a social alternative in S at which there

is a local equilibrium when the incumbent must defend the status quo.

5. Applications to Related Spatial Voting Models

This work has extended the earlier research of Hinich, Ledyard and

Ordeshook [1972],11973] (among others). In their papers they included special

concavity assumptions. These additional assumptions are sufficient for the

candidates' expected plurality functions to be strictly concave in their own

strategies. It then followed that there was a unique global electoral equili-

brium. What this does is to convert any stationary electoral equilibrium

into a global electoral equilibrium. Their existence result, therefore,

follows, alternatively, from Corollary 4.1 in this paper. The strictness

of the concavity assures that the electoral equilibrium is unique. Convergence

to the equilibrium then follows as in the original papers of Hinich, Ledyard

and Ordeshook.

This work has also extended the research of Coughlin and Nitzan [1981]

for electorates with probabilistic voting and no abstentions. This earlier

paper derived a specific functional form for the P1 (see equation (1)) from

assumptions in the Mathematical Psychology literature for binary choices. The

resulting model is a special case for the analysis in this paper. Consequently,

the existence theorems which were previously derived by Counghlin and Nitzan

[1981] follow, alternative, from Corollaries 5.1 and 6.1 in this paper.

" . . . .1. . t -

6. Conclusion

This paper has analyzed electoral competitions with abstentions and

probabilistic voting when candidates have directional, local or no mobility.

It provides necessary and sufficient conditions for directional, convergent

stationary and convergent local electoral equilibria. These conditions specify

the locations of all of the convergent stationary and local electoral equili-

bria. They additionally provide general existence results for these equilibria.

1

-15-

Footnotes

l/ For example, see Slutsky 11975], Kramer [1977], and Rubenstein [19791.

2/ For example, see Matthews [1979] and Kramer and Klevorick [1974].

3/ Electoral competitions and majority rule with directional or localstrategy sets or choice sets have previously been studied for societieswith no abstentions in Plott [1967], Kramer and Klevorick [1974],Schofield 11978], Matthews [1979], Cohen and Matthews [1980], andCoughlin and Nitzan [1981].

4/ This incorporates the recent work of Hinich [1977], Kramer [1978]and Coughlin and Nitzan [1981] with the work on abstentions referredto in the first paragraph.

5/ The notations and assumptions in this paper are from the referencescited in the first paragraph and in footnotes i/-4/.

6/ For the utility-based probabilistic voting and abstentions of Hinich,Ledyard and Ordeshook [1972] and Denzau and Kats [1977], letG,(x) = Ua(x). For the metric symmetry of McKelvey [1975] (assump-tion 3.3), let G (x) = lix - xi1A, P1:(x,y) = W(G,(x),G (y)) and

aa5 Ca

7/ It should be observed that the Pi will not be in C2 when individualswith a common label have the same utility function, everyone makesa deterministic choice of the candidate whose policy proposal hasthe greater utility for him and the candidates know the voters' behavior.However, it is easily satisfied in the aggregate when individualschoose probabilistically and/or candidates have smooth estimatorsfor the voters' choice behavior.

8/ Of course, in directional and local electoral competitions the can-didates will only obtain the portions of this estimator or of thefuictions which we derive from it that pertain to their possiblestrategies.

9/ When there is no chance eiement in voters' decisions, the maximizationof expected plurality is simply the familiar maximization of plurality.When some or all of the voters' choices are probabilistic (or estimatedas such), candidates could be concerned with their expected pluralitiesor their probabilities of winning. However, Hinich [1977] has shownthat these two objectives are equivalent whenever there is a largepopulation with probabilistic voting. Hence we consider only thefirst objective.

a

-16-

Footnotes Continued

10/ See Section I in the Appendix.

i_/ The feasible directions in S at *t are the n-vectors of unitIlength, w, for which there is some positive real number A suchthat + A • w E S for every X E (0,Xw).

12/ This is the structure in Kramer [1977], for instance.

13/ This is the structure in Plott [19671, Schofield [1978] and elsewhere.

14/ See Section IV in the Appendix.

15/ See Section V in the Appendix.

16/ E.g., see Hirsch [1976], Theorem 6.1.2.

i

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Appendix

Section I. The expected pluralities for the candidates are given by:

(14) Ptl(xy) = f(p(x,y) - P2(xy)) di( ) , andA

APt2(~y) f(PQ(x'y) - P,(x,y)) dla(Q) -PZ1(x,y)

2for every (x,y) E X

Section II. Proof of Theorem 1:

(15) Pi(u,v) D(u v) PIi(,1,2)

n apl . (3P 1' i 2 ) n apt. (*,19 2)h=l li •h Vhh~l a Ihh=l a2h

Therefore, Pi(u,v) is additively separable in u and v for both i E C.

Hence any u* and v* which maximize

n t(16) DuPt(* '*) - M " h

h=l Th

and

n tD n (P§2(,),2)i~~~~~~ DPt2% '2 W "v h=l a3 2 h h

at l = and = respectively, are dominant strategies.1 2 2

r -18-

Conversely if u* (respectively, v*) does not maximize DuPt

(respectively, DvP2(¢pi2)) at ( i2 = (i1,2) then it is dominatedv 2 12 12 1' 2

by some other direction. Q.E.D.

Proof of Corollary 1.1:

First, since DuPt1 (p1 ,*2 ) and DvPt2(I,4 2 ) are linear in u and v,

they are also continuous in u and v, respectively. Secondly, since S

is compact, T(**) and '.'. compact. Therefore maxima exist. Q.E.D.

Section III. Proof of . , 2 "

If tnere is a s< i.oi.Ij electoral equilibrium at (4,*) E S , then

Theorem 1 directly impli~s that (6) is satisfied.

Conversely, if (6) is satisfied then we immediately know that u* = o

maximizes D uPt 1( 414) at i= i " Therefore, we need only show that

v* = o is a dominant strategy for candidate 2.

By (14) and (2),

(17) Ptl(x,y) = fPI(G (x),G (y)) * diJ(a)A

- A• x,~() I~t

Therefore, since there is policy-related voting (p. 3),

(18) Pt (x,y) f P 2(G (y),G (x)) • dP(ct)A

SfP 1(G (y),%(x)) d(a)

Pk P~(yx)

-. - V . . .2

-19-

Consequently,

(19) -

at i = *2 = i" Therefore,

n apt(20) DvPt2 (pi 2 ) = • vh

h=l 2h

n aPZ (1i,)

h=l 3 lh h

at l= 2 =

Hence, since u = o maximizes DuP (t,1i) at i = \b, it must also

be true that v = o maximizes DvPt 2 (*' 2 ) at *2 = 4. Q.E.D.

Section IV. A curve K in Rn is a continuous function mapping a closed

interval [a,b] of real numbers into Rn. T E [a,b] is the parameter

of the curve K. The derivative of the curve K(T) = (KI (T),...,K n(T))

with respect to the parameter T is K'(T) = (dKI(T)/dT,...,dKn(-t)/dT).

If the derivative K'(T) is continuous for all values of T, then a

is called a smooth curve. If the interval [a,b] may be partitioned into

subintervals, a = T 0 < T1 < ... < Tn = b, such that K is smooth on each

of the subintervals, then K is said to be piecewise smooth.

The line integral of a function h from Rn into Rn along a piece-

wise smooth curve K is defined to be

II* - '.4 *t

-20-

X,=K(b)

(21) fh = f h(CK(r))dK(1 ic x = (a)

b= fhCK(,)) • K'( )dT

a

(where * is inner product). (See Curtis [1972] or Friedman [1971] for a

thorough discussion of these integrals.)

2 nSection V. Let f(x) be a C function on X C R and let gk(x) = 0

(k = 1,...,m < n) be C2 constraints on X which define a feasible set,

S C X. Then a point x* is called a critical point of f relative to these

constraints if and only if it satisfies these constraints and has associated

with it a Lagrange function

(22) L = f + Xg1 + ... + X mgm

such that VF(x* ) = 0.

f is said to be non-degenerate relative to these constraints if and

only if the determinant of the Bordered Hessian

WLx) Dgk W)

axhaxZI 3

(23)

agk.(X)

axt

(where the dimensions of the sub-matrices are as follows:

-21-

is (n n) , [-- is (n m)

Ikx] is (mx n) and [01 is (mX M)

is non-zero at every critical point of f relative to these constraints

(see Hestenes [1975, p. 153).

Section VI. Proof of Theorem 3:

By Theorem 9.4 in Curtis [1972], if F'(s) is a symmetric matrix

for each s E X then it follows that F(s) = V iPt (ss) is a gradient

field. We therefore consider the entries in the matrixr2 21a2Pe.l(S S) a2Pe.1(s,s) j

(2 4 ) F ' (s ) = P lh1 Plk + a t1 a*2s )l0h *ik a* lh alp2k

with h,k = 1,...,n.

First,

2 t (s,s) a 2Pt (s,s)(25) alh a*lk a* lk0 lh

by Young's Theorem (e.g., see Chiang [1974]) and Corollary 5.9 in Bartle

4 [1965].

Secondly,

-22-

(26) aplxy

f)lh G,, (X),,,(Y)) dIP(a) -P A a(x),Ga (y)) dII(a]

=---'(G (x),G(y)) -dP.(a) -ag 2( *X () dlI(a)

A~la a a A lh

A ah

fO a a a d (a)

A aGx X

by (14), (2), the Chain Rule (e.g., see Chiang [1974]) and Corollary 5.9

in Bartle [19651.

Therefore,

21

t a a a -I . - di'(a)

- a a - - ' dl'(a)

9(x)aG (Y) * Xh

by Corollary 5.9 in Bartle [19751 and the Chain Rule.

-23-

Similarly,

2(28) a Pl(xy)

(2) -*lk '2h

a 2P (G (x),G (y)) 3G (y) 3G (x)

=f a aa -A a (x)3G (y) kh a dI(a)

a 2 (G (x),G (y)) 3G (y) 3G (x)aa a a a *dPa

-A aG a(x)DG ay•) da(a)

At x y = s we have

(29 G(y) 3(x) G (x) 3G (y)

ayk ax h y=s ax k yh x=s

X=s y=s

Therefore,

a 2Pt1 (Ss) a 2Pt1 (Ss)(30) - - = i 2hlh 32k l *2

Finally, (25) and (30) imply that t'E(s) is symmetric.

Therefore, F(s) = Vl Pt (s,s) is a gradient field on X. Q.E.D.

Proof of Corollary 3.1:

This follows directly from Theorem 3 (above) and Theorem 9.4.1 in

Friedman [19711.

z

-2--

-24-

Proof of Corollary 3.2:

This follows directly from Theorem 3 (above) and Theorem 9.3 in

Curtis [1972].

Section VI. Proof of Theorem 4:

By Corollary 3.2 and Theorem 9.3 in Curtis [1972],

(31) VG(y) = V1PZ1 (y,y)

at every y e S. Recall that

(32) DuPZ1 (y,y) = V1Pt1 (y,y) • u

(where • is inner product). Therefore,

(33) D G(*) = VG(*) - u

= V Pt m1 1~~

= D Pt ~ ~)u 1

Hence, Theorem 4 follows by an argument analogous to the proof of Theorem 2.

Proof of Corollary 4.1:

Follows directly from Corollary 4.2.

Proof of Corollary 4.2:

Since each G(x) is continuously differentiable, G(x) is a continuous

function of x. Therefore, since S is compact, G must achieve a maximum

* I- ,.. ~ -.. ~--

-25-

at some # E S. Any such 0 must satisfy (10). Therefore, there must

also be a stationary electoral equilibrium at (i ,) by Theorem 4.

Proof of Corollary 4.3:

By Corollary 4.2 and Theorem 2.

Section VII. Proof of Theorem 5:

This follows by a second-order argument analogous to the first-order

argument in the proof of Theorem 2.

Proof of Theorem 6:

(34) [3yhaY] VVG(y)

= V(V1Ptl(y,y))

6_ a2Pel (YY) 3 2P l (y,y)]

By the proof of Theorem 2, Pt1 (x,y) = -P'l1 (Y,x). Therefore,

a 2Pt 1 (x,y) -a 2PL 1 (y,x)(35) -- 3alh 2k '1 2halk

But, by the proof of Theorem 3, when x y = we have

a 2Pti(xy) a2Pel(x,y)(36) lh2k '2h'lk

.-.

- -26-

Therefore,

~lh ~2k

at every ~,for h,k = 1,... ,n.

Hence,

E38 2 2G() 1 a 2 Pt (Y 1(38)h ahYk J alh ~lkJ

(with h ,k =1.. .,n).

Now, suppose that there is a local electoral equilibrium at EJ S.

Then, since Pt1(4p,14) is non-degenerate with respect to the constraints

given by (13), there exist unique multipliers m1 .. , such that, if

we set L(x) = Pt 1 (X,4) + X g(x) + .. + Xmgm(x), then VL,(*) = 0 and

n n DLip(39) L.(i i w l*lawtx

for every non-zero vector w E R n which satisfies the equation Vg~1I k w = 0

(k =l..m. (E.g., see the proof of Theorem 3.3.2 in Hestenes [1975]).

Now consider G(x). Since VG(x) = VPt (x,4*) at x = ,we must have

L() G(x) + Xg(x) + .. + Xmgm(x) for the samne X1 *..,~ implies

VL2 x W 0. Additionally, (38) and (39) imply

* 2

ip -27-

n n DL2( )

(40) L"(i - w, . . > 02 =i h=lchX t h W

for every non-zero vector w G Rn which satisfies Vgk(L) k w = 0

(k = 1,...m). Therefore, since G(x) is non-degenerate with respect to

the constraints given by (15), * must be a strict local maximum of L(x)

(e.g., see Theorem 3.3.2 in Hestenes).

The converse follows similarly (as in Theorems 2 and 4). Q.E.D.

Proof of Corollary 6.1:

By Corollary 6.2.

Proof of Corollary 6.2:

By the proof of Corollary 4.2, G(x) must achieve a local maximum at

some * E S. By Theorem 6, there is a local equilibrium whenever both candi-

dates choose the strategy * (i.e., at (',i) E 2).

Proof of Corollary 6.3:

By Corollary 6.2 and Theorem 2.

*- q

-- -28-

References

Arrow, K., F. Gould and S. Howe [1973], "A General Saddle Point Result for

Constrained Optimization," Mathematical Programming, 5, 225-23h.

Bartle, R. [1965], The Elements of Integration, New York: Wiley.

Chiang, A. [1974], Fundamental Methods of Mathematical Economics, New York:McGraw-Hill.

Cohen, L. and S. Matthews [1980], "Constrained Plott Equilibria, DirectionalEquilibria, and Global Cycling Sets," Review of Economic Studies, h6,975-986.

Coughlin, P. and S. Nitzan [1981], "Directional and Local Electoral Competi-tions with Probabilistic Voting," Journal of Economic Theory, forth-coming.

Curtis, P. [1972], Multivariate Calculus, New York: Wiley.

Denzau, A. and A. Kats [1977], "Expected Plurality Voting Equilibrium andSocial Choice Functions," Review of Economic Studies, hh, 227-233.

Friedman, A. [1971], Advanced Calculus, New York: Holt, Rinehart and Winston.

Hestenes, M. [19751, Optimization Theoy, New York: Wiley.

Hinich, M. [1977], "Equilibrium in Spatial Voting: The Median Voter Resultis an Artifact," Journal of Economic Theory, 16, 208-219.

Hinich, M. [1978], "Some Evidence on Non-voting Models in the Spatial Theory

, of Electoral Competition," Public Choice, 33, 83-102.

Hinich, M. and P. Ordeshook [19691, "Abstentions and Equilibrium in theElectoral Process," Public Choice, 7, 81-106.

Hinich, M., J. Ledyard and P. Ordeshook [1972], "Non-voting and the Existence* of Equilibrium Under Majority Rule," Journal of Economic Theory, h,

144-153.

Hirsch, M. [19761, Differential Topology, New York: Springer-Verlag.

Kramer, G. [1977], "A Dynamical Model of Political Equilibrium," Journal ofEconomic Theory, 16, 310-33h.

Kramer, G. [1978], "Robustness of the Median Voter Result," Journal ofEconomic Theory, 19, 565-567.

-29-

References Continued

Kramer, G. and A. Klevorick [1974], "Existence of a 'Local' CooperativeEquilibrium in a Class of Voting Games," Review of Economic Studies,41, 539-547.

Matthews, S. [1979], "A Simple Direction Model of Electoral Competition,"Public Choice, 34, 141-156.

McKelvey, R. [1975], "Policy Related Voting and Electoral Equilibrium," Ecn-ometrica, 43, 815-844.

Plott, C. [1967], "A Notion of Equilibrium and Its Possibility Under MajorityRule," American Economic Review, 57, 787-806.

Rubenstein, A. [1979], "A Note About the 'Nowhere Denseness' of SocietiesHaving an Equilibrium Under Majority Rule," Econometrica, 47, 511-514.

Schofield, N. [19783, "Instability of Simple Dynamic Games," Review of EconomicStudies, 45, 575-594.

Slutsky, S. [1975], "Abstentions and Majority Equilibrium," Journal of EconomicTheory, 11, 292-304.

-'4

REPORTS IN THIS SERIES

160. "The Structure and Stability ofCoulpetilive Dynamical Systems." by David (ass and Karl Shell.161. "Monopolistic Competition and the Capital Market," by J. E. Stglit/.I (2. "The Corporation Tax," by J. E. Stiglitl,163. "Measuring Returns to Scale in tle Aggregate and tie Scale Effect of Public Goods," by David A. Starrett.

1o4. "'Monopoly. Quality, and Regulation," by Michael Spence.165. "A Note on tihe Budget Constraint in a Model of Borrowing," by Duncan K. Foley and Martin F. llellwig,1(it). "Incentitives, Risk, and Information: Notes Towards a Theory of Ilierarchy," by Joseph 1. Stiglitz.I 7. "Asymptotic Expansions of the Distributions of Estimates in Simultaneous Equations for Alternative Parameter

Sequences," by T. W. Anderson.168. "Estimation of Linear Functional Relationships: Approximate Distributions and Connections with Simultaneous

Equations in Econometrics," by T. W. Anderson.I6). "Monopoly and the Rate of Extraction of Exhaustible Resources," by Joseph E. Stiglitiz.170. "Equilibrium in Competitive Insurance Markets: An Essay ott the Economics of Imperfect Information," by

Michael Rothschild and Joseph Stiglitz.171. "Strong Consistency of Least Squares Estimates in Normal Linear Regression," by T. W. Anderson and John B. Taylor.172. "Incentive Schemes under Differential Information Structures: An Application to Trade Policy," by Partha Dasgupta

and Joseph Stiglitz.173. "The Incidence and Efficiency Effects of Taxes otn Income from Capital," by John B. Shoven.174. "Distribution of a Maximum Likelihood Estimate of a Slope Coefficient: The LIML Estimate for Known Covariance

Matrix," by T. W. Anderson and Takamitsu Sawa.175. "A Commtent on the [est of Overidentifying Restrictions." by Joseph B. Kadane and T. W. Anderson.176. "An Asymptotic Expansion of tile )istribution of the Maxinmum Likelihood Estimate of the Slope (oefflcient In a

Linear Functional Relationship," by T. W. Anderson.177. "Sotue Experimental Results on the Statistical Properties of Least Squares Estimates in Control Problems," by

T. W. Anderson and John B. Taylor.178. "A Note on "Fulfilled Expectations" Equilibria," by David M. Kreps.

179. "Uncertainty and the Rate of Extraction under Alternative Institutional Arrangements." by Partha Dasgupta andJoseph 1. Stiglitz.

180. "Budget Displacement Effects of Inflationary Finance." by Jerry Green and E. Sheshinski.181. "Towards a Marxist Theory of Money," by Duncan K. Foley.182. "The Existence of Futures Markets, Noisy Rational Expectations and Informational Externalities." by Sanford Grossman.183. "On time Efficiency of Competitive Stock Markets where Traders have Diverse Information." b Sanford Grossman.184. "A Bidding Model of Perfect Competition," by Robert Wilson.185. "A Bayesian Approach to the Production of Information and Learning by Doing." by Sanford J. Grossman. Richard 1.

Kihlstrom and Leonard J. Mirman.186. "DiseqUilibritm Allocations and Recontracting." by Jean-Michel Grandmont. GuI Laroque and Yves Younmes.

187. "Agreeing to Disagree." by Robert J. Aumann.188. "Tite Maximum Likelihood and the Nonlinear Three Stage Least Squares Estimator in the General Nonlinear Simultaneous

Equation Model," by Takeshi Amemiya.189. "Tile Modified Second Round Estimator in the General Qualitative Response Model," by Takeshi Amemiya.190. "Some Theorems in the Linear Probability Model," by Takeshi Amemiya.191. "The Bilinear Complementarity Problem and Competitive Equilibria of Linear Economic Models." by Robert Wilson.192. "Noncooperative Equilibrium Concepts for Oligopoly Theory," b5 .. A. Gerard-Varet.193. "Inflation and Costs of Price Adjustment." by Eytan Sheshinski and Yoram Weiss.194. "Power and Taxes in a Multi-Commodity Economy," by R. J. Aumann and M. KI,z.195. "Distortion of Preferences, Income Distirbution and the Case for a Linear Income Tax." by Mordecai Kium.196. "'Search Strategies for Nonrenewable Resource Deposits," by Richard J. Gilbert,197. "Demand for Fixed Factors, Inflation and Adjustment Costs," by Eytan Sheshinski and Yoram Weiss.

198. "Bargains and Ripoffs: A Model of Monopolistically Competitive Price Dispersions." b, Steve Salop and Joseph Stiglit/.199. "Tle Design of Tax Structure: Direct Versus Indirect Taxation by A. B. Atkinson and J. F-. Stiglitz.200. "Market Allocations of Location Choice in a Model with Free Mobility," by David Starrett.201. "Efficiency in the Optimum Supply of Public Goods," by Lawrence J. Lau, Eytan Sheshinski and Joseph 13. Stiglit,.

202. "Risk Sharing, Sharecropping and Uncertain Labor Markets," by David M. G. Newberr,.

203. "On Non-Walrasian Equilibria," by Frank Hahn.204. "A Note on E.lasticity of Substitution Functions." by Lawrence J. Lati.205. "Quantity Constraints as Substitutes for Inoperative Markets: The Case of the Credit Markets." by Mordecai Kuri,

206. "Incremental Consumer's Surplus and tledonic Price AdjustrmenI," by Robert 1). Willig.

RI PORtS IN rMIS SERIES10, -Op iii I)CVICT TI'rio ill I II~cijiT Stock. hx Richaiid G;ilbert.

- hal~ l. ' ll .l\C' III .1W C i)lIjllMtii oII puliimmIli lmTRI I t qli \ Iaiiiicl,-h I S cin ii J. 6townii d I.Sielj k.-ILL I IC l~ ~p~T .i~ I ll j TI.IT I omm \IIiliI\, Dir l i lii NI I'I I 1111C."i hiici\ i \1 a.LS011 ie n L h su s

-1 1- 'AheC speIti'l i ) plj~ T I il ut III jlC ''I I \ IklllAll' I T JL. CTI ll .aeh Stighi,.- I ". .i d L~ II I ,,II 'Ivi ', D.c~i --\ It i ( mycTN iiiC ike I C M ikoN. L.Ti Jsl'li IC. S iht.I.'4.idii ai iiiiI.fa l.

- I,. I C~i~.TTiI ~c. t~TITI q)I I.wlli\ ,I d I t'llili. I T'i.C IClS .I . )11 .i I dII. \I . W.~i Aiidersm a nd1. o hn BCT . al

- i4. IcjCi. i l IlljTI latIti I qll Ii. he Iiii (T Ic IT 1 11 l Mic il~I Ilir . 1spiI.Sih

I .2 ITTTITT ii lc l''CTT .01 11111 l. 1 I111'O iii .111 11 it'. MOII2. I C CI II 1 (,i cqkiiil. ii ",-. ! s ll

I4 IC hI N lkIITI VIl i I J)ikii S I'IlT I itioi il %% tii i~I ws imnioT~ I' iiiIId pI. -h MiiSiia el 1ii 1d .piV tg

I, \\ ItTc \i. II tijI-l tw of I lieII1,'iicmIi& I I:mIc. I a livid 1 . ,Trcl C. l

I LL I h l 'iI- it.'I I vT'II \I.'III \ Il Ii ll I 'IT I.T 1.i C S lITTd h\ii~ (I T iT i TIRCTil i 1&Ik TTtn.sd/.ilai

CoplI~ Moo by \Kw 'IO~l

- TN i ( ,i'.l IICT III. jIi\dtheIIN Milmk id Iii.IC % T J a lihel (tindiilTmo t C. byI A ilraad .Sn

1\T I i.1 III llI~- I i I' Sim c i .\1 T 1 TI\ItI (i I-mp[Iilie h\iiijiT Tij. ~I r -isaeL s S u e siia

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24 . sToTll.' ItiTTIIII \ I IIITdol T II'.' ITj 1, cilMo e. m I'. TITTikiTIe '.. i~ A II ~ avjd J iri

Vi4 ' h ' I L A' I ICLT.111 C.. I 1111 ihl' lII I RT'',cIiiiiii ji~~ iciiIII'I~iTijiii y KnltiJArw

44 '(CIII T JIC T ,,,.III, I'. TI 1.11 2ll I ITIC \ I DCTT kaIldT i Sart.ei .(T~Tadha lehis

.4 "I .lcIilpk .1i II> ( I I illil I 111 IT. C III i' v I peL'l.' I% k\ i1) CS Ivc and e R .M. im Iaiii .

-C' I \ i. lT 11 I I qoiTCT.ITC Il.iT -i TT iI~ii'.: Ili CIS qw le, 1 SjIii d IJd . bysiml L arn c I . .1r

- 41 \ppITI .TI.. I l,I I CII. MoIT l \ Ic I'' I~Itei lCWeIi Id~' IM'e POMid ofm tilei Ineies. b Lo .'nce h\ Lsad arin

.51 Ild B<CI.'.l T 7111ThI, Cill [i \11\11 e~i~ I T11) lI h I pITII ~Sl a ige gac at. y('in Isai

Il: \I'4 , -11h ]I I TiSTim I I liiilihi o (T ii liiT. inl Isill (it (lie S lpe I'artlia Ieasup a d ricmi askns.

-hITTilict'.I i I q)1 C ll C i l I"? STlIw~ ITT iIIT. IT of iiIeliiv Stckiir,."b\S (

2 I he R I.' I T11 Im ,I I IICTITI m id T 'lI tCTITT.I T C T.11I'it .\ Taesi A. Ixrep .

o ' I IT-T nl , "I I it'!1 .i \Ummu T'TIil' TTIT.ITC IN li l ibia iilllblic1 Mo~del. b T. eodore arv and Sel

-1 "(W I lieTTC lT I k po t l , ! ill C TiTTloe Tin l-1 -u lli i re \ Darvd laf.

14, T 11ic II~ tT1i~I of T the ITT.TTIoiT i. - iclo tj M "'ii mikCTS7 I Salidi.Grs n a l lan o eh b. tiLwrenc .Lu

REPORTS IN THIS SERIES

26o). "Values for Gaines ~ithout Sidepaynietits: Some' 1)111kul ties With Current C'oncepts," by Alvin I .Roth.

2b1. "Niartingle.s and the Valuation of Redundant Assets," by J. Michael Harrison and David M. Kreps.

262. 'Autoregressive Modelling and Money Income (ajusality Detection." by (heng lisiao.

263. "Measurement IError in a Dynamiic Simultaneous Fqutimns Model without Stationary Disturbances," by (heng Ilsiao.

r264. "1 lie Measurement ol Ileadweiglii Loss Res tsited," by %. F-. LDiewert.

265. "1 lie 1lhisticity of D~erived Net Supply and a Generalized Le (hatelher Principle," by W. 1%.. 1)iewert.

2bb. "Income D~istribution and Distortion of Pref'erences. the Q Commodity Case," by Mordecai Kurz.267. "in- Order Mean Squared Errors of the Maminuin Likelihood and the Mininuii Logit Chi-Square I-siiiator," by lakeshi Arnemiya.

268. " letporal Von Neuioann-Morgensiern and lndu.-d Prectce~' b David M. Kreps and Lvan L. Porteus.

269. "Trake-Over Bids and the I heory of the Corporation." by Stanford Grossman and Oliver D. Hart.27o. "11w Nuiterical Values of Sonie Key Parameters in Economnetric Models." by 1. W. Anderson, Kutnio Morintune and Takatnitsu Sawa.

271. "-1 %o Representations of Inforination Structures and their Comparisons," by Jerry Green and Nancy Stokey.

272. "Asymtotic Expansions of the Distributions of Estinatmrs in a Linear 1-unctional Relationship when the Sample Size is Larye." byNaoi o Kunitoino.

273. "Public Goods and Power," by R. J. Auinann. M. Kurz and A. Neynran.

274. "An Axiomatic Approach to the Efficiency of' Non-Cooperative Equilibrium in Economies with a Coninuum of ITraders," by A. Mas'('olell.

275. "alsof'the UExact Distribution F-unction of the Linmited Information Maximumi Likelihood E-stimator w hen the Covariance Matrix inKnown," by 1. W. Anderson and Fakaimitsu Sawa.

276. "Autoregressive Modeling of"Canadian Money and Income Data," by Cheng Ilsjao.

277. "We Can't Disagree IForever," by John 1). Geanakoplos and Heraklis Pilenarchakis.

278. "Constrained Excess Deitatnd Functions," by Hlerklis M. Polemnarchakis.

279. "On the Bayesian Selection of' Nash Equilibrium," by Akira Iotnioka.

280. "Disequilibrium Fcm'notncmrics in Simultaneous Equations Syseti, b) C. Giourierous. J. J. Laftont and A. Monfort.

281. "Duality Approache.s to Microeconomic Fheory." by %. L.. Diewert.

282. "ATime Series Analysis of the Impact of ('anadian Wage and Price Controls," b, "heng IHsiao and Oluwjaayo I-akiyesi.

283. "A Strategic Theoiry oft Inflation," by Mordecai Kurz.

284. "A Characterization of Vector Meastire Gamecs in pN.A," by Yair T auntati.

285. "On the Method of Taxation and the Provision of Local Public Goods," by D)avid A. Starrett.

286. "An Optimization Problem Arising in Economics; Approximate Solutions, Linearity, and a Law of Large Numbers." by Sergiu Hart.

287. "Asymptotic Expansions of the Distributions of ihe Estimates of Coefficients in mt Simultaneous Equation System," by Yasunoriimjjikosbi, Xitmic, Moriiune, Naoto Kunitoino and Masanobu Taniguchi.

288. "*Optimal & Voluntary Income Distribution," by K. J. Arrow.

289. "'Asymptotic Values mif Mixed Gaime,.," by Abraham Neymnan.

290. "Tinie Series Modelling and ('ausal Ordering mif ('anadian Motney, Income and Interest Rate," by Chieng Hsiao.

291. "An Analysis of Power in Exchange Economies," by Martin J. Osborne.

292. "Estinatiomn of the Reciprocal of a Normal Mean,"* by Asad Zamnan.

* - 293. "Imprioving thme Maximum Likelihood Estimate in Linear Functional Relationships for Alternative Parameter Sequences." by KimioMorimune and Naoto Kunitomo.

294. "C'alculation oft Bivartate Norial Integrals by the Use oif Incomplete Negative-Order Moments," by Kei Takeuchi and Akitnichi Taketnura.

295. "On Partitioning mif a Sample with Binary-Type Questions in Lieu of Collecting Observations." by Kenneth 1. Arrow. Leon Pesotchminsk%

296.

297. "Thme FIwo Stage Least Absolute Deviations Estimlators," by Takeshi Atnetniya.

298. "'1hree Essays urn C'apital Markets," by David NM. Kreps.

299. "Ininite Horizon Programts." by Michael J. P. Magill.

30(o. "Electoral Outcoeots and Social Log-Likelihood Maxima," by) Peter C'oughlin and Shmuel Nitzan.

301. "Nortes on Social Choice and Voting,'' by Peter Coughlin.302. "Overlapping Expectations and Hart's ('onditions f~or Equilibrium in a Securities Model," by Peter J. Hammond.30)3. "D~irectional and Local Electorate Competitions with Probabilistic Voting," by Peter Coughlin and Shmnuel Nitzan.

3(04. "Asymptotic Expansions of the Distributions of ihe Trest Statistics for Overidentifying Restrictions in a System oif SimultaneousEquations," by Kunisomo, Morinmune, and Isukuda.

305. "Incomplete Markets and the Observability of' Risk Preference Properties," by H. H. Polemarchakis and L. Selden.

306. "Multiperiod Securities and the IFfficient Allocation of Risk: A C'omnment on the Black-Scholes Option Pricing Model," byD~avid M. Kreps.

307. "Asymptotic Expansion% if thme Distributions of k-C'lass Estimators when the Disturbances are Small," by Naoto Kunitomo,Kimio Morimune, and Yoshihiko Fsukuda.

3518, "Arbitrage and Equilibrium in -cumimimies with Infinitely Many C'ommodities," by David M. Kreps.

1309. "Unemployment Equilibrium in an Economy with Linked Prices," by Mordecai Kur.

310. "Pareto Optimal Nash Equilibria are Competitive in a Repeated Economy," by Mordecai Kurz and Sergiu Hart.

311. "Identification," by Cheng Hsiao.

312. "An Introduction tom I~kt-Person Zero Sumi Repeated Gaimes with Incomplete Information," by Sylvain Sorin.

Reports in this Series

313. "Estimation of Dynamic Models With Error Components," by T. W.Anderson and Cheng Hsiao.

314. "On Robust Estimation in Certainty Equivalence Control," byAnders H. Westlund and Hans Stenlund.

315. "On Industry Equilibrium Under Uncertainty," by J. Drbze andE. Sheshinski.

316. "Cost Benefit Analysis and Project Evaluation From the Viewpointof Productive Efficiency" by W. E. Diewert.

317. "On the Chain-Store Paradox and Predation: Reputation forToughness," by David M. Kreps and Robert Wilson.

318. "On the Number of Cormr.odities Required to Represent a MarketGames," Sergiu Hart.

319. "Evaluation of the Distribution Function of the LimitedInformation Maximum Likelihood Estimator," by T. W. Anderson,Naoto Kunitomo, and Takamitsu Sawa.

320. "A Comparison of the Logit Model and Normal DiscriminantAnalysis When the Independent Variables Are Binary," by TakeshiAmemiya and James L. Powell.

321. "Efficiency of Resource Allocation by Uninformed Demand," byTheodore Groves and Sergiu Hart.

322. "A Comparison of the Box-Cox Maximum Likelihood Estimator andthe Nonlinear Two Stage Least Squares Estimator," by TakeshiAmemiya and James L. Powell.

323. "Comparison of the Densities of the TSLS and LIMLK Estimatorsfor Simultaneous Equations," by T. W. Anderson, Naoto Kunitomo,and Takamitsu Sawa.

324. "Admissibility of the Bayes Procedure Corresponding to the UniformPrior Distribution for the Control Problem in Four Dimensions butNot in Five," by Charles Stein and Asad Zanan.

325. "Some Recent Developments on the Distributions of Single-EquationEstimators," by T. W. Anderson.

326. "On Inflation", by Frank Hahn

327. Two Papers on Majority Rule: "Continuity Properties of MajorityRule with Intermediate Preferences," by Peter Coughlin and Kuan-PinLin; and, "Electoral Outcomes with Probabilistic Voting and NashSocial Welfare Maxima," by Peter Coughlin and Shmuel Nitzan.

328. "On the Endogenous Formation of Coalitions," by Sergiu Hart andMordecai Kurz.

329. "Controliability, Pecuniary Externalities and Optimal Taxation,"by David Starrett.

330. "Nonlinear Regression Models," by Takeshi Amemiya.

Reports in this Series

331. "Paradoxical Results From Inada's Conditions for Majority Rule,"by Herve Raynaud.

332. "On Welfare Economics vith Incomplete Information and the SocialValue of Public Information," by Peter J. Hammond.

333. "Equilibrium Policy Proposals With Abstentions," by Peter J.Coughlin.

ILMEI