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Two-sided uncertainty in the monopoly agenda setter model
Journal
of Public
Economics
50 (1993) 429444.
North-Holland
Two-sided uncertainty agenda setter model
in the monopoly
Jeffrey S. Banks* Department of Economics and Department of Political Science, Harkness Hall, University of Rochester, Rochester, NY 14627-0156, USA Received
May 1990, revised version
received January
1992
We extend the Romer-Rosenthal model of representative democracy to a signaling environment, in which (i) only the representative knows the ‘status quo’ outcome resulting if her take-it-orleave-it policy proposal is rejected by the voters, while (ii) only the voters know their true preferences over policies. A separating sequential equilibrium is shown to exist, and to uniquely satisfy a common equilibrium refinement. Furthermore, this equilibrium has the property that, relative to the environment where the status quo is known to the voter, there is a downward bias in the setter’s proposal, and an associated upward bias in the probability of the proposal’s acceptance by the voter.
1. Introduction One of the more powerful results in the field of political economy concerns the fundamental interaction within a representative democracy in the determination of public policy, namely the interaction between a constituency and its chosen representative. The result, known as the ‘median voter theorem’ and attributable to Black (1958), says the following: if we have a unidimensional policy space, voters with single-peaked preferences, two offtcemotivated candidates selecting policy alternatives, majority-rule voting over the two alternatives, and common knowledge over all of the above specitications, then the equilibrium prediction is that the candidates’ policies will be identical and equal to the median of the voters’ ideal policies. The underlying rationale of this result, in common with the earlier location model of Hotelling (1929) in economics, is that the competition among the candidates for the role as the constituency’s representative will invariably lead to the attempted capture of the median voter’s vote, since if the median voter prefers candidate A over candidate B, then, by definition of the median voter Correspondence to: J.S. Banks, Department of Economics, Harkness Hall, University of Rochester, Rochester, NY 14627-0156, USA. *I wish to thank two referees for numerous constructive criticisms and suggestions, and the National Science Foundation and the Sloan Foundation for generous financial support. 004772727/93/$06.00
0
1993-Elsevier
Science Publishers
B.V. All rights reserved
430
J.S. Banks, Two-sided uncertainty
(and assuming single-peaked preferences), at least half the voters prefer A over B. Such a concise prediction of the policy outcome, varying in a simple fashion with the preferences of but a single member of the electorate, allows economists interested in issues such as, for example, local public finance, a convenient avenue for suppressing political factors in the determination of such outcomes. For instance, if a researcher wishes to estimate the demand for school expenditures within a district, she need simply apply the usual individual utility-maximizing assumption to the behavior of a single voter among the electorate in order to summarize the effective preferences of the entire constituency as well as their influence on the policy outcome.’ This straightforward characterization of the political decision-making process, sympathetic as it is in the familiar tools of microeconomic analysis, has led to the prominent use of the median voter theorem in a variety of economic studies, ranging beyond local public finance to include topics such as income redistribution, pollution control, and minimum wage legislation.’ While the median voter theorem provides a seductive means of sidestepping politics in certain economic analyses, Romer and Rosenthal (1979b) found the explanatory power of models employing the theorem to be wanting, due in part to their insensitivity to other institutional features of the typical policy environment. As Rosenthal (1990) mentions in a later survey, ‘The “median voter” studies in local public finance have provided substantial circumstantial evidence that agenda setting or some other form of institutional structure matters in local decision-making’ (p. 224). Indeed, Rosenthal (1990) goes on to note that in certain empirical studies, ‘the median voter model appears to apply better to municipalities with direct democracies than to municipalities with representative democracies’ (p. 224, emphasis added). With the latter type of decision-making being far more prevalent than the former, an enrichment of the representative-voter interaction seemed necessary in order to provide a more useful model of the political process. Romer and Rosenthal (1978, 1979a) developed one particularly insightful alternative to the median voter model, a model which forms the basis for the current paper’s investigations. Rather than focusing on the effect of competition on the policy outcome, Romer and Rosenthal assume the constituents’ representative has already been selected. Furthermore, this individual is endowed with the ability to ‘set’ the voting agenda by offering her constituency a choice between an exogenously determined status quo outcome, and an outcome selected by the representative. That is, the representative is allowed to make a ‘take-it-or-leave-it’ proposal to the voters. Assuming the preferences of the agenda setter, which are now over the policy ‘For estimation purposes, this median voter is typically assumed to be the member electorate with the median income. ‘See Romer and Rosenthal (1979b) for a lengthy list of such studies.
of the
J.S. Banks, Two-sided uncertainty
431
alternatives, are strictly increasing over the one-dimensional space, and hence are not coincident with those of the median voter, the relevant question then concerns the ability of the setter to bias the final policy outcome in her favor relative to the voters’ interests. Romer and Rosenthal (1978, 1979a) establish equilibrium predictions demonstrating how this bias can be quite severe depending on the location of the status quo outcome. In particular, the model often predicts a negatioe relationship between the status quo and the final outcome; that is, the worse the status quo situation the more the setter can extract from the voters. This follows since, if the status quo is less than the median voter’s ideal policy, the setter can capture the median voter’s vote by proposing an alternative greater than the latter’s ideal policy, and such that the median voter is indifferent between the status quo and the alternative. Obviously, in such instances the setter’s proposal, and hence (with complete information) the final policy, will be higher the lower is the status quo outcome. While the empirical support Romer and Rosenthal provide for their model nevertheless there are a number of theoretical issues appears intriguing,3 that do arise. Two such issues, which are the focus of the current paper, have to do with informational assumptions of the Romer-Rosenthal model. The first concerns the assumption that the setter possesses complete information about the voters’ preferences. From a realistic standpoint, it would appear more plausible to view the setter as being uncertain about the voters’ preferences, implying any proposal the setter offers generates a gamble over the status quo and the proposed alternative. Such uncertainty is addressed to a limited extent in Romer and Rosenthal (1979a), where they explore the effects of uncertain turnout, and is given a more comprehensive treatment in Morton ( 1988).4 A second issue concerns why the voters would permit the setter this degree of autonomy in the decision process in the first place, since they (in principle, at least) can alter this process if they so desire. One plausible answer is that once in office the representative gains some information concerning the relevant parameters of the policy decision, implying this degree of autonomy may be an eflicient means of distributing influence. The relevant question then concerns whether such efficiency gains offset (from the voters’ perspective) the bias in the outcome due to the setter’s agenda control. In Banks (1990) this informational asymmetry is modeled as the setter knowing with certainty the location of the status quo outcome (or, equivalently, the voters’ utility associated with the status quo outcome), information which the electorate lacks. Such an informational asymmetry may be though of as 3For more recent empirical support, see Rosenthal (1990). “Morton (1988) also investigates the effect of the setter having a number of attempts proposal passed, so if her first proposal fails to receive a majority, she proposes alternative, etc.
to have a a second
432
J.S. Banks,
Two-sided
uncertainty
resulting from the setter’s proximity to the qualitative features of the decision, e.g. the head of the school board will undoubtedly have better information than the voters concerning the current state of affairs in education. In the current paper we synthesize the earlier efforts of Morton (1988) and Banks (1990) as a means of better understanding the role of information in the original model of Romer and Rosenthal (1978, 1979a). In particular, we assume the setter is uncertain as to the location of the voters’ ideal policy outcomes, which as noted above is the principal determinant in the setter’s decision calculus in a complete information world. In addition, the voters are uncertain about the location of the status quo alternative, thereby treating the utility associated with voting against the setter’s proposal as stochastic. However, since the setter knows the location of the status quo, and selects her alternative with such knowledge, the voters may be able to infer this location through the setter’s choice. Hence we model the interaction between the setter and the voters as a signaling game, with one of the principal issues being the willingness and ability of the setter to signal the location of the status quo through her choice of proposals, when faced with an uncertain response by the voters. Theorem 1 below establishes the existence and uniqueness of a separating equilibrium in the model, thereby demonstrating the ability of the setter to credibly transmit all of the relevant information to the voters through the proposal process. Hence the decisions by the voters in choosing between the status quo and the proposed alternative will be equivalent to those they would make in a complete information environment. Furthermore, while other equilibria potentially exist, this separating equilibrium is the only equilibrium satisfying the refinement of universal divinity (Theorem 2). The existence of a separating equilibrium is in contrast to the main result in Banks (1990), which showed that when the setter knows the voters’ preferences only pooling equilibria exist, thereby limiting the amount of information the setter transmits to the voters and thus rendering some of the voters’ decisions ‘incorrect’ ex post. In addition, we find that the equilibrium behavior of the setter exhibits two important qualitative characteristics: (i) there will always exist a positioe relationship between the status quo and the setter’s proposal, contrary to the negative relationship exhibited in the model of Romer and Rosenthal (1978, 1979a), and (ii) the (separating) equilibrium proposal made by the setter will always be less than the proposal the setter would make if the status quo were known to the voters. The latter implies the existence of a downward pressure on the setter’s proposal, and an associated upward pressure on the acceptance rate of the proposals, relative to the model of Morton (1988). Hence the severity of the bias in the policy outcome due to the setter’s strategic advantage vis-a-vis the voters, emanating from her agenda control, is actually
433
J.S. Banks, Two-sided uncertainty
dampened by the presence of the setter’s informational advantage. This dampening therefore can be seen as a product of the very feature of the environment dictating the setter’s existence, namely the presence of an informational asymmetry between the setter and the voters.
2. The model The model describes the interaction between an agenda setter or bureaucrat, S, and a single voter, r/: in the determination of a policy outcome from [w.’ The setter’s preferences over R are represented by a continuously differentiable strictly increasing and (weakly) concave utility function us( .), &+/c~x >O, i3%,/13x~ SO. The voter’s preferences, represented by u,(.), are continuously differentiable, single-peaked about an ideal policy v, and (weakly) concave: &,/i3x$O as x $v, i32u,/ax2 SO. For expositional ease we also assume uy(.) is symmetric about v:Vx,YE R, Ix-v\ = ly-01 implies U”(X) =4(y). The sequence of actions is as follows: S makes a proposal of p E R, which V can either accept, in which case p becomes the policy outcome, or reject, in which case the status quo SE [w is the policy outcome. The informational assumptions are that only the setter knows the value of the status quo s, while only the voter knows her ideal policy v.~ Let [s, S] c R denote the set of possible values of the status quo, or equivalently the set of types of the setter, with the distribution F(.) describing the voter’s common knowledge prior belief about the status quo; F(.) is continuously differentiable, with aF/i?s-f(.), and f(s) >O iff SE [s,S]. The set of possible values for v, i.e. the set of types of the voter, is taken to be the entire real line; let G(.) be the twice continuously differentiable common knowledge distribution from which v is drawn, where aG/av =g(v) > 0, Vu E R. Given this informational structure a proposal strategy for S is a function
so n(s) is the proposal function
of a setter
of type s. A response
strategy
for V is a
r: R x R+(a,,a2},
‘The assumption of a single voter is not required for the results (see the discussion below) and is only adopted for simplicity. 6As noted in the Introduction, these informational assumptions differ from Romer and Rosenthal (1978, 1979a), where both s and v are common knowledge, Banks (1989) where u is common knowledge but s is private information, and Morton (1988) where s is common knowledge but v is private information.
434
J.S. Banks, Two-sided uncertainty
where r(p, u) =a1 denotes acceptance of the proposal p given ideal policy u, and r(p, u) = a2 denotes rejection. ’ Define X(p,r)= {UE R:r(p,~)=a,}, and let
(1) Thus A(p,r) is the probability the proposal p is accepted by the voter, given her response strategy r(.). The expected utility for the setter from proposing p, given type s and response strategy I( .) by the voter, defined os(p, s, r), is then a convex combination of the utility associated with the proposal p, and that associated with that status quo s:
os(P, s,4 = 4~~ rh(d + Clwhich is utility-equivalent lent. to
[Myerson
us(~, s, r) = 4~~ r)Cus(p) -
A(P, WS(4, (1985)],
(2)
and hence behaviorally
~~(41.
equiva-
(3)
Since U,( .) requires less notation than us(.), we will use U,( .) as our representation of the setter’s preferences. For the voter, let p(.,p) denote a belief about the setter’s type upon observing the proposal p. Her expected payoff from accepting a proposal of p, denoted U,(a,,p,p), is simply equal to u”(p), while if V rejects p she receives
U,(a,>P>P) = j uv(s)~(s> P) ds.
(4)
Definition. A sequential equilibrium [Kreps and Wilson (1982)] in the above game is a strategy pair (n*,r*) and a system of beliefs CL* such that, (i) VSE [s, a, n*(s) maximizes U,(p, s, r*), (ii) Vp E R, VUE R, r*(p, u) maximizes U,(a, p,p*), (iii) Vpc R, p*(p) is ‘consistent’ with the strategy rc(.) and the prior belief F(.). At a minimum, consistency requires ,U assign positive probability only to setter types in [s,s]. Consistency also requires the voter’s beliefs to be derived via Bayes’ Rule from the strategy n(.) and the prior belief F(.) where possible, i.e. for proposals p E R such that rt ‘(p) # 0. Hence, if according to ‘For simplicity we assume below indifferent voter types is of measure preferences are quadratic about v.
that the voter’s preferences are zero. This holds in equilibrium
such that the set of if, for example, V’s
J.S. Banks, Two-sided uncertainty
435
rc(.) a proposal p is sent by type s, and only by type s, then upon observing p the voter’s belief p( .) should be degenerate, placing probability one on the setter’s type being s. If on the other hand a subset of types send p, then the belief p(.) should place positive probability only on those types, and to the degree dictated by the prior belief F(.). In principle the belief ~1 might also be a non-trivial function of the voter’s type v as well; however, this will not be so in a sequential equilibrium. Formally, consistency requires the beliefs of the voter be the limit of a sequence of beliefs derived via Bayes’ Rule from a sequence of completely mixed strategies by the setter, where this sequence of strategies is independent of v (since v is unknown to S and is drawn independently from s) and converges to the equilibrium strategy rr(.). Bayes’ Rule is then applicable for all proposals in [w all along the sequence, thereby generating a sequence of beliefs, and a limit belief, which is independent of the voter’s ideal policy v. Before proceeding to characterize sequential equilibrium behavior, it is instructive to clarify in what respect the assumption of a single voter in the electorate is without loss of generality. Suppose instead the electorate consists of a set N={ l,,,,, n} of voters, with IZ finite and odd, and where majority rule determines the outcome. Suppose further that each voter’s preferences are single-peaked, with voter i’s ideal policy vi generated by a (smooth) distribution Gi with full support on [w. Then, using the same argument as employed above to show that the (single) voter’s beliefs will not be a function of her type, we see how consistency requires that in any sequential equilibrium all voters will possess the same belief concerning the status quo in and out of equilibrium. That is, since each voter’s beliefs are the limit beliefs generated by a sequence of completely mixed setter strategies, and these strategies are common among all the voters, the voters’ sequence of beliefs, and hence their limit beliefs, will coincide. Therefore, for all proposal-belief pairs, (p,p), we can identify the set of voter types who prefer the proposal over the (random) status quo outcome, and conversely. Assuming all voters vote according to their preferences,* we can interpret the distribution G(.) in the single-voter model as describing the probability at least half the voters have ideal points less than or equal to a given amount, where this probability is determined by the distributions Gr,. . . , G,.9 In this way the (admittedly stringent) sequential equilibrium requirement of common beliefs among the voters essentially reduces the model with n voters to one with a single voter.” *With only two alternatives this constitutes a voter’s weakly dominant strategy. ‘This follows since no restrictions, other than smoothness and full support on R, were placed on the prior G(.) in the single-voter model. “As a referee notes, such an equivalence fails to hold when there is a continuum of voters, all of whose ideal points are drawn from a common distribution, since in such an instance any electoral outcome is certain, rather than random.
436
J.S. Banks, Two-sided uncertainty
3. Results Let C denote the set of strategy pairs (n,r) such that there exists beliefs p, where (71,r, p) constitutes a sequential equilibrium, and for all (7t, r) E C define J(s) =A(Tc(s),Y) as the probability a type s setter’s equilibrium proposal is accepted. Our first result states some general properties of elements of the set C. Lemma 1. V(qr)EZ, (i) n( .) is weakly increasing in s, and T(S) > s, Vs; and (ii) A(.) is weakly decreasing in s, and A(s) > 0, Vs. As noted above, p(s,p)>O only if SE [s_,S]. An immediate implication Proof. of this is that, since I/ is (weakly) risk-averse, for any proposal p>S there will exist a subset of voter types, namely those close to and including p, that will accept this proposal regardless of their beliefs. Since S of type s can therefore guarantee herself an expected payoff strictly greater than zero by making such a proposal, for any (n,r)~C it must be that U,(x(s),s,r) >O as well. Thus, from (3) we must have X(S)> s, and 2(s) > 0, V’sE [s, Fl. To see the monotonicity results, let s>s’, and define p= n(s), p’= I, I.=A(s) and I,‘= %(s’). Then incentive compatibility [cf. d’Aspremont and Gerard-Varet (1979)], which is a necessary condition for (71,,I) to be part of an equilibrium, implies no setter type has a (strict) incentive to emulate the behavior of any other type, or
Al%(P)- us(s)1 2 ~‘C%(P’) - % (S)l, GUs(P’) -
(5)
~&‘)I 2 X%(P) - M)l.
Subtracting the RHS of (6) from the LHS of (5), and the LHS of (6) from the RHS of (5), we get A[u,(s’) - u,(s)] 2 i’[u,(s’) -us(s)]. Since us( .) is strictly increasing and s > s’, this implies Since i(.) >O, eqs. (5) and (6) also imply GP’) - 4s’) Us(P)- 4s’) Cross-multiplying
(ii).
> %(P’) - %(S) = Us(P) -us(s) .
and canceling
%(P)Cas(s) -
i’ 2 1, thus proving
W)l2
terms, we get
as(P’)C%(s) -
%Wl~
(9)
431
J.S. Banks, Two-sided uncertainty
implying
pzp’
and thus proving
(i).
Q.E.D.
The increasing nature of rr(.) is in contrast to the complete information result of Romer and Rosenthal (1978, 1979a), where the setter’s equilibrium proposals are strictly decreasing for status quo outcomes less than the voter’s (known) ideal point. The logic in Romer and Rosenthal (1978, 1979a) is based on the single-peaked nature of the voter’s preferences: if u = 0 and s < 0, then the voter will accept any proposal between s and -s;” therefore in equilibrium S proposes -s, which V accepts with probability one. With incomplete information, on the other hand, we get the opposite sort of monotonicity, where this result is now driven by the preferences and incentives of the setter; indeed, the single-peaked nature of the voter’s preferences is not at all consequential. Evidently, Lemma 1 constitutes a conclusion which holds for any sequential equilibrium of the model. To generate further results concerning behavior, we explore the characteristics of a special kind of equilibrium. Namely, we consider an equilibrium in which the setter’s proposal strategy is a strictly increasing function of the status quo, i.e. a separating equilibrium. While Banks (1990) has shown that if G(.) is degenerate (with a single mass point), no such equilibria exist, Theorem 1 below demonstrates the sensitivity of this result to the assumption of known voter preferences. Consider first the optimal proposal by the setter if the status quo s were known with certainty by the voter. l2 In such an environment, given a proposal p, all voter types less than (p+s)/2 prefer the status quo s to the proposal [by the symmetry of u,(.)], and conversely for u>(p+s)/2. Thus A(p, r) = [ 1 - G((p +s)/2)], and the optimal proposal by the setter, which we denote z’(s), solves max Cl - G((P + 4/2)X4(~)
-u&)1.
(10)
Suppose ke assume the hazard rate associated with the distribution g(u)/[l -G(u)], is increasing in u;13 this plus the concavity of us(.) insures solution to (10) is unique. Theorem 1. There exists a unique separating sequential z*(.), which satisfies the following differential equation: drr* ds
equilibrium
M*(s) N*(s) - M*(s)’
“Recall the symmetry assumption on uV( ,). “As noted above, this is the model of Morton (1988). 13This is a common assumption in private information McMillan (1987)].
G, the
strategy
(11)
models,
e.g. auctions
[cf. McAfee and
438
J.S. Banks, Two-sided uncertainty
where M*(s) =d(n*(s) + 4/N%(~*(s))
- %(SU~
N*(s) = 2[1- G((n*(s) +s)/2)](&+(7r*(s))/ap). Furthermore, n*(5) = n’(S), and for all s E [s_,S) Z*(S)c n’(s). Proof.
Let Ws, ~‘9PI = Cl - G((P + WYC~,(P)
- ~~(41
denote the setter’s expected utility from proposing p, given type s and given the voter believes S to be type s’ (with probability one), and let subscripts on W(.) denote partial derivatives. Given a strictly monotonic strategy n(.), if we substitute n(Y) into W(s,s’,p) we get the setter’s expected utility from imitating type s’ given true type s. Incentive compatibility implies that if (n, r) E 2, then ‘J’s,s’ E [Is,51, W(s, s, Z(S))2 W(s, s’, I). Since this holds with equality at s’ = s, we get the following ‘local’ incentive compatibility condition:
mqs,s’, 7c(s’)) as, -
=
W2(s,s, n(s)) + WJS, s, n(s)) g = 0,
(12)
sfEs
implying all WA%s, n(s)) as = - WJS, s, n(s))’
(13)
which is simply (11) when rc(.) is strictly monotonic, i.e. separating. To show that the model admits a strictly increasing solution to (13), and therefore this condition is sufficient for an equilibrium, we employ results due to Mailath (1987), who shows the following conditions to be sufficient for existence and uniqueness of a separating equilibrium: (i) W is twice continuously differentiable on [s_,S]” x R; in the current model this follows from the assumptions on G(.) and us(.). (ii) W, is never zero, and so is either positive or negative; here w = - CdP) - %(moP
f s)/2),
and so W,
(14) p>s.
J.S. Banks, Two-sided uncertainty
iv,,=;
=@I
i
-3-G((p+s’)/2)1
439
I
+ s’)/2) > 0.
(15)
(iv) W3(s, s, p) =0 has a unique solution in p, where W(s, s,p) is locally concave around this solution; as noted above this follows from us(.) concave and g(u)/[l - G(u)] increasing in u. (v) There exists k >O such that for all (s,p) E [g,S] x R, W,Js,s,p) 20 implies 1W3(s,s, p)I > k; this implies Ws(s, s, p) is bounded away from zero for p bounded away from X’(S). We can guarantee this condition is met in the current context by without loss of generality restricting the set of proposals to the interval [s_,+(S)]. Now W, ~0 implies the ‘worst’ belief, from the setter’s perspective, is where the voter is certain, s=S. Therefore in any separating sequential equilibrium it must be that n*(S) =rP(S), since otherwise S is separating at some other proposal; yet by switching to n”(S) the setter would receive a strictly higher payoff for any belief at Y(S). Furthermore, the following single-crossing property holds:
= -2(%/~4(~M~)C1-
G((P + s’)/2M(p + s’Y2)
U%(P) - @))d(P
+ W2)12
(16)
Then Mailath (1987) shows (Theorem 3) that the solution to the differential equation (13) along with the initial value condition n*(S) =rc’(S) generate a unique separating equilibrium in the above game given that the singlecrossing condition holds. The result that VSE [s, S) n*(s)O. Q.E.D. Recall that in the above description of the model minimal restrictions were placed on the prior distribution G(.); hence Theorem 1 shows how even a small amount of uncertainty concerning the voter’s preferences can translate into the existence of a separating equilibrium. The underlying rationale is that such uncertainty necessarily ‘smooths out’ the (expected) decision by the voter concerning any proposal, thereby allowing for the existence of a ‘smooth’ proposal strategy by the setter in response. All of this ‘smoothing’ is
J.S. Banks, Two-sided uncertainty
440
in contrast to the certainty case, where the setter knows with probability one whether a proposal will be accepted or rejected by the voter. Theorem 1 also states that the incentive constraints on the setter’s behavior place a downward pressure on the separating equilibrium proposals, since in general a setter of type s has an incentive to be thought of as a lower type, s’+ and some other type s’
Q(s,
p;
q
r) =
w;7L,4 .
(17)
a4 - us(s)’
thus if 8(s, p; n, I) E [0,11, then f3(.) gives the probability of acceptance making a type s setter indifferent between remaining along the equilibrium path [and receiving the payoff U(s; n, r)] and deviating to the proposal p. If p
In what follows we make use of the following
divine
equilibrium
7c- ‘(p) = 0,
if it is a only if
p(s’,p) >O
result.
14This downward pressure is in contrast to the upward pressure on sender choices found in other signaling models, e.g. Spence (1974). The difference is due to the fact that in the Spence model the sender (i.e. the worker) prefers to be thought of as a higher (more productive) type, ceteris paribus, while in the current model the sender prefers to be thought of as a lower type.
441
J.S. Banks, Two-sided uncertainty
2. V(n,r) E 1, U(s; z, r) is weakly decreasing and continuous in s; thus aU/as exists almost everywhere. Furthermore, where dUj8s exists it satisfies Lemma
au = as
i(s)%.
(18)
Proof. Let s > s’. Since us(.) is strictly increasing and enters negatively into 4s); U(.), s’ can achieve a payoff of at least U(s) by simply proposing therefore n(s’) and i(s’) must be such that U(s’)L U(s). Continuity of U(.) follows from a similar argument, as well as the continuity of us(.). Since U( .) is monotone, it is differentiable almost everywhere [Royden (1968)], so au/as exists for almost all s. Since (z,r) EZ, local incentive compatibility is satisfied:
$ IW)C%(W))- us(sm
S’=S
=~[us(z(s))-u~(s)]+i(s)~ g=o. Thus, where au/as
(19)
is defined,
~=g[us(~(s))-us(s)]+,‘.(s) @- !f!z [ @z ap as as] = -A(s) 2.
Q.E.D.
(20)
Incentive compatibility thus implies an ‘envelope theorem’-type of result; this follows since each type of setter is essentially optimizing over which type to behave as given the ‘suggested’ behavior from rc(.). In equilibrium it must be that each type prefers to behave ‘truthfully’, implying the indirect effect of increasing s on equilibrium utility, through changes in the equilibrium proposal Z(S) and the equilibrium probability of acceptance A(s), is zero. Theorem 2. The unique separating equilibrium the unique universally divine equilibrium.
defined
in Theorem
1 is also
Proof Let (rr, r) EC and let p be an out-of-equilibrium proposal; from the above definition we wish to characterize argmin, d(s, p; z, r). By Lemma 2, O(.) is differentiable almost everywhere and continuous everywhere; thus solving for M/as will assist in identifying the types most likely to defect. From Lemma 2,
a0 a u/as[us(p) -
as=
+ au,/as .U(S) us(s)1 2
t+(s)]
C&P) -
442
J.S. Banks, Two-sided uncertainty
= -
_
4s) ~usl~~cu,(P) - % (s)l+ au,/as~ W[Us(~(S))-us(s)] C%(P) - us( - Us(P)1 . C%(P) - % (S)l2
3s) w~~rus(m
Since A(s) >O and &,/&>O,
a6)/asso as
(21)
we have that
71(s) pp.
(22)
Now since rr( .) is monotone increasing (by Lemma 1) it is differentiable almost everywhere, and ‘locally’ is either pooling, &c/as= 0, or separating, &c/as>O. Suppose (qr) EC is such that rt has a pooling region (s’,s”), i.e. VS,~E(S’,S”), z(s) =rc(?)=p*. Then rr must have a jump discontinuity at s’, or else s’ =s otherwise s’ would be the supremum of a locally separating region, and the continuity of G(.) would imply a jump discontinuity in the probability of acceptance at s’ without a jump in 71, thereby violating U(., 71,r) continuous. Let p = p* --E for E> 0 and small; then from (22) we see that s’= argmin, @s,p). Thus universal divinity requires a jump discontinuity in the voter’s response at p*. Since the voter’s optimal response is strictly decreasing in ‘certainty’ beliefs, and V is (weakly) risk-averse, this is evidently a jump downward, or equivalently a jump up in the probability of acceptance, in going from p* to p. Thus s’ can achieve a strictly higher payoff from proposing p for E sufficiently small, thereby contradicting the assumption of an equilibrium. Therefore in any universally divine equilibrium there can be no pooling regions; implying the only such equilibria are separating. By Theorem 1, then, the unique universally divine equilibrium is the unique separating equilibrium. Q.E.D. Hence, although there may exist a plethora of sequential equilibria in the model, all but the unique separating equilibrium described in Theorem 1 will be discarded by the criterion of universal divinity. In this sense, then, focusing predominantly on the properties of the separating equilibrium is justified.
4. Conclusion As mentioned in the Introduction, the goal of this paper was to explore the various ramifications of the complete information assumptions inherent in the Romer and Rosenthal (1978, 1979a) model of agenda control. While Romer and Rosenthal identified certain weaknesses in the median voter
J.S. Banks,
Two-sided
uncertainty
443
model with respect to the political processes within which public policies are conceived, the model they themselves investigated was based on a number of seemingly unrealistic and/or restrictive assumptions concerning ‘who knows what’ (as is, of course, the median voter model).” And while restrictive assumptions do not necessarily subvert the importance of a model’s insights, relaxing such assumptions (while still maintaining a reasonably tractable model) enhances the analyst’s ability to identify which aspects of the behavioral predictions are due to the underlying incentives of the participants and structure of the interaction, and which are more specific to the details of the model employed. In this paper we have relaxed the complete information assumptions in the Romer-Rosenthal model on two fronts: the setter is now uncertain about the voters’ preferences, and the voters are now uncertain about the location of the status quo outcome. The principal results we find are threefold. First, the proposals offered by the agenda setter will be an increasing function of the status quo, rather than a decreasing function as deduced in the RomerRosenthal model. With uncertainty concerning the voters’ response, the setter is inevitably facing a gamble over final outcomes for whatever proposal she might offer; this plus the fact that the setter can condition her behavior on the actual location of the status quo while the voters cannot, leads to the reversal of the predicted setter behavior. Second, although the agenda setter is in sole possession of information relevant to the optimality of the voters’ decisions, namely the location of the status quo outcome, such information will be revealed by the setter through her choice of alternative. Hence the informational advantage the setter possesses ex ante will be alleviated through her own behavior. Finally, the setter’s proposals will always be less than that which she would propose if the voters also knew the status quo’s location. In this sense, then, the informational advantage the setter holds visa-vis the voters actually tends to diminish, rather than augment, the ability of the setter to implement preferred outcomes. Hence the current model also suggests that normative arguments against the presence of an agenda setter or ‘representative’ in the political system, with its concomitant differential influence of the setter in the determination of public policy outcomes, should be mitigated by the realization that informational effects can in theory act as a counterweight against such influence. ‘%f. Calvert
(1986) for a survey
of median
voter-type
models with imperfect
information
References d’Aspremont, C. and L.-A. Gerard-Varet, 1979, Incentives and incomplete of Public Economics 11, 2545. Banks, J., 1990, Monopoly agenda control with asymmetric information, Economics 105. 445464.
information, Quarterly
Journal Journal
of
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J.S. Banks, Two-sided uncertainty
Banks, J. and J. Sobel, 1987, Equilibrium selection in signaling games, Econometrica 55, 6477661. Black, D., 1958, The theory of committees and elections (Cambridge University Press, Cambridge). Calvert, R., 1986, Models of imperfect information in politics (Harwood Academic, Chur). Hotelling, H., 1929, Stability in competition, Economic Journal 39, 41-57. Kreps, D. and R. Wilson, 1982, Sequential equilibria, Econometrica 50, 8633894. Mailath, G., 1987, Incentive compatibility in signaling games with a continuum of types, Econometrica 55, 134991366. McAfee, P. and J. McMillan, 1987, Auctions and bidding, Journal of Economic Literature 25, 699-738. Morton, S., 1988, Strategic voting in repeated referenda, Social Choice and Welfare 5, 45568. Myerson, R., 1985, Bayesian equilibrium and incentive compatibility: An introduction, in: L. Hurwicz et al. eds., Social goals and social organizations: Essays in memory of Elisha Pazner (Cambridge University Press, Cambridge) 2288259. Romer, T. and H. Rosenthal, 1978, Political resource allocation, controlled agendas, and the status quo, Public Choice 33, 2744. Romer, T. and H. Rosenthal, 1979a, Bureaucrats vs. voters: On the political economy of resource allocation by direct democracy, Quarterly Journal of Economics 93, 5633588. Romer, T. and H. Rosenthal, 1979b, The elusive median voter, Journal of Public Economics 12, 1477170. Rosenthal, H., 1990, The setter model, in: J. Enelow and M. Hinich, eds., Advances in the spatial theory of voting (Cambridge University Press, Cambridge) 199-234. Royden, H., 1968, Real analysis, 2nd edn. (Macmillan, New York) 96. Spence, M., 1974, Market signaling (Harvard University Press, Cambridge, MA).
of Public
Economics
50 (1993) 429444.
North-Holland
Two-sided uncertainty agenda setter model
in the monopoly
Jeffrey S. Banks* Department of Economics and Department of Political Science, Harkness Hall, University of Rochester, Rochester, NY 14627-0156, USA Received
May 1990, revised version
received January
1992
We extend the Romer-Rosenthal model of representative democracy to a signaling environment, in which (i) only the representative knows the ‘status quo’ outcome resulting if her take-it-orleave-it policy proposal is rejected by the voters, while (ii) only the voters know their true preferences over policies. A separating sequential equilibrium is shown to exist, and to uniquely satisfy a common equilibrium refinement. Furthermore, this equilibrium has the property that, relative to the environment where the status quo is known to the voter, there is a downward bias in the setter’s proposal, and an associated upward bias in the probability of the proposal’s acceptance by the voter.
1. Introduction One of the more powerful results in the field of political economy concerns the fundamental interaction within a representative democracy in the determination of public policy, namely the interaction between a constituency and its chosen representative. The result, known as the ‘median voter theorem’ and attributable to Black (1958), says the following: if we have a unidimensional policy space, voters with single-peaked preferences, two offtcemotivated candidates selecting policy alternatives, majority-rule voting over the two alternatives, and common knowledge over all of the above specitications, then the equilibrium prediction is that the candidates’ policies will be identical and equal to the median of the voters’ ideal policies. The underlying rationale of this result, in common with the earlier location model of Hotelling (1929) in economics, is that the competition among the candidates for the role as the constituency’s representative will invariably lead to the attempted capture of the median voter’s vote, since if the median voter prefers candidate A over candidate B, then, by definition of the median voter Correspondence to: J.S. Banks, Department of Economics, Harkness Hall, University of Rochester, Rochester, NY 14627-0156, USA. *I wish to thank two referees for numerous constructive criticisms and suggestions, and the National Science Foundation and the Sloan Foundation for generous financial support. 004772727/93/$06.00
0
1993-Elsevier
Science Publishers
B.V. All rights reserved
430
J.S. Banks, Two-sided uncertainty
(and assuming single-peaked preferences), at least half the voters prefer A over B. Such a concise prediction of the policy outcome, varying in a simple fashion with the preferences of but a single member of the electorate, allows economists interested in issues such as, for example, local public finance, a convenient avenue for suppressing political factors in the determination of such outcomes. For instance, if a researcher wishes to estimate the demand for school expenditures within a district, she need simply apply the usual individual utility-maximizing assumption to the behavior of a single voter among the electorate in order to summarize the effective preferences of the entire constituency as well as their influence on the policy outcome.’ This straightforward characterization of the political decision-making process, sympathetic as it is in the familiar tools of microeconomic analysis, has led to the prominent use of the median voter theorem in a variety of economic studies, ranging beyond local public finance to include topics such as income redistribution, pollution control, and minimum wage legislation.’ While the median voter theorem provides a seductive means of sidestepping politics in certain economic analyses, Romer and Rosenthal (1979b) found the explanatory power of models employing the theorem to be wanting, due in part to their insensitivity to other institutional features of the typical policy environment. As Rosenthal (1990) mentions in a later survey, ‘The “median voter” studies in local public finance have provided substantial circumstantial evidence that agenda setting or some other form of institutional structure matters in local decision-making’ (p. 224). Indeed, Rosenthal (1990) goes on to note that in certain empirical studies, ‘the median voter model appears to apply better to municipalities with direct democracies than to municipalities with representative democracies’ (p. 224, emphasis added). With the latter type of decision-making being far more prevalent than the former, an enrichment of the representative-voter interaction seemed necessary in order to provide a more useful model of the political process. Romer and Rosenthal (1978, 1979a) developed one particularly insightful alternative to the median voter model, a model which forms the basis for the current paper’s investigations. Rather than focusing on the effect of competition on the policy outcome, Romer and Rosenthal assume the constituents’ representative has already been selected. Furthermore, this individual is endowed with the ability to ‘set’ the voting agenda by offering her constituency a choice between an exogenously determined status quo outcome, and an outcome selected by the representative. That is, the representative is allowed to make a ‘take-it-or-leave-it’ proposal to the voters. Assuming the preferences of the agenda setter, which are now over the policy ‘For estimation purposes, this median voter is typically assumed to be the member electorate with the median income. ‘See Romer and Rosenthal (1979b) for a lengthy list of such studies.
of the
J.S. Banks, Two-sided uncertainty
431
alternatives, are strictly increasing over the one-dimensional space, and hence are not coincident with those of the median voter, the relevant question then concerns the ability of the setter to bias the final policy outcome in her favor relative to the voters’ interests. Romer and Rosenthal (1978, 1979a) establish equilibrium predictions demonstrating how this bias can be quite severe depending on the location of the status quo outcome. In particular, the model often predicts a negatioe relationship between the status quo and the final outcome; that is, the worse the status quo situation the more the setter can extract from the voters. This follows since, if the status quo is less than the median voter’s ideal policy, the setter can capture the median voter’s vote by proposing an alternative greater than the latter’s ideal policy, and such that the median voter is indifferent between the status quo and the alternative. Obviously, in such instances the setter’s proposal, and hence (with complete information) the final policy, will be higher the lower is the status quo outcome. While the empirical support Romer and Rosenthal provide for their model nevertheless there are a number of theoretical issues appears intriguing,3 that do arise. Two such issues, which are the focus of the current paper, have to do with informational assumptions of the Romer-Rosenthal model. The first concerns the assumption that the setter possesses complete information about the voters’ preferences. From a realistic standpoint, it would appear more plausible to view the setter as being uncertain about the voters’ preferences, implying any proposal the setter offers generates a gamble over the status quo and the proposed alternative. Such uncertainty is addressed to a limited extent in Romer and Rosenthal (1979a), where they explore the effects of uncertain turnout, and is given a more comprehensive treatment in Morton ( 1988).4 A second issue concerns why the voters would permit the setter this degree of autonomy in the decision process in the first place, since they (in principle, at least) can alter this process if they so desire. One plausible answer is that once in office the representative gains some information concerning the relevant parameters of the policy decision, implying this degree of autonomy may be an eflicient means of distributing influence. The relevant question then concerns whether such efficiency gains offset (from the voters’ perspective) the bias in the outcome due to the setter’s agenda control. In Banks (1990) this informational asymmetry is modeled as the setter knowing with certainty the location of the status quo outcome (or, equivalently, the voters’ utility associated with the status quo outcome), information which the electorate lacks. Such an informational asymmetry may be though of as 3For more recent empirical support, see Rosenthal (1990). “Morton (1988) also investigates the effect of the setter having a number of attempts proposal passed, so if her first proposal fails to receive a majority, she proposes alternative, etc.
to have a a second
432
J.S. Banks,
Two-sided
uncertainty
resulting from the setter’s proximity to the qualitative features of the decision, e.g. the head of the school board will undoubtedly have better information than the voters concerning the current state of affairs in education. In the current paper we synthesize the earlier efforts of Morton (1988) and Banks (1990) as a means of better understanding the role of information in the original model of Romer and Rosenthal (1978, 1979a). In particular, we assume the setter is uncertain as to the location of the voters’ ideal policy outcomes, which as noted above is the principal determinant in the setter’s decision calculus in a complete information world. In addition, the voters are uncertain about the location of the status quo alternative, thereby treating the utility associated with voting against the setter’s proposal as stochastic. However, since the setter knows the location of the status quo, and selects her alternative with such knowledge, the voters may be able to infer this location through the setter’s choice. Hence we model the interaction between the setter and the voters as a signaling game, with one of the principal issues being the willingness and ability of the setter to signal the location of the status quo through her choice of proposals, when faced with an uncertain response by the voters. Theorem 1 below establishes the existence and uniqueness of a separating equilibrium in the model, thereby demonstrating the ability of the setter to credibly transmit all of the relevant information to the voters through the proposal process. Hence the decisions by the voters in choosing between the status quo and the proposed alternative will be equivalent to those they would make in a complete information environment. Furthermore, while other equilibria potentially exist, this separating equilibrium is the only equilibrium satisfying the refinement of universal divinity (Theorem 2). The existence of a separating equilibrium is in contrast to the main result in Banks (1990), which showed that when the setter knows the voters’ preferences only pooling equilibria exist, thereby limiting the amount of information the setter transmits to the voters and thus rendering some of the voters’ decisions ‘incorrect’ ex post. In addition, we find that the equilibrium behavior of the setter exhibits two important qualitative characteristics: (i) there will always exist a positioe relationship between the status quo and the setter’s proposal, contrary to the negative relationship exhibited in the model of Romer and Rosenthal (1978, 1979a), and (ii) the (separating) equilibrium proposal made by the setter will always be less than the proposal the setter would make if the status quo were known to the voters. The latter implies the existence of a downward pressure on the setter’s proposal, and an associated upward pressure on the acceptance rate of the proposals, relative to the model of Morton (1988). Hence the severity of the bias in the policy outcome due to the setter’s strategic advantage vis-a-vis the voters, emanating from her agenda control, is actually
433
J.S. Banks, Two-sided uncertainty
dampened by the presence of the setter’s informational advantage. This dampening therefore can be seen as a product of the very feature of the environment dictating the setter’s existence, namely the presence of an informational asymmetry between the setter and the voters.
2. The model The model describes the interaction between an agenda setter or bureaucrat, S, and a single voter, r/: in the determination of a policy outcome from [w.’ The setter’s preferences over R are represented by a continuously differentiable strictly increasing and (weakly) concave utility function us( .), &+/c~x >O, i3%,/13x~ SO. The voter’s preferences, represented by u,(.), are continuously differentiable, single-peaked about an ideal policy v, and (weakly) concave: &,/i3x$O as x $v, i32u,/ax2 SO. For expositional ease we also assume uy(.) is symmetric about v:Vx,YE R, Ix-v\ = ly-01 implies U”(X) =4(y). The sequence of actions is as follows: S makes a proposal of p E R, which V can either accept, in which case p becomes the policy outcome, or reject, in which case the status quo SE [w is the policy outcome. The informational assumptions are that only the setter knows the value of the status quo s, while only the voter knows her ideal policy v.~ Let [s, S] c R denote the set of possible values of the status quo, or equivalently the set of types of the setter, with the distribution F(.) describing the voter’s common knowledge prior belief about the status quo; F(.) is continuously differentiable, with aF/i?s-f(.), and f(s) >O iff SE [s,S]. The set of possible values for v, i.e. the set of types of the voter, is taken to be the entire real line; let G(.) be the twice continuously differentiable common knowledge distribution from which v is drawn, where aG/av =g(v) > 0, Vu E R. Given this informational structure a proposal strategy for S is a function
so n(s) is the proposal function
of a setter
of type s. A response
strategy
for V is a
r: R x R+(a,,a2},
‘The assumption of a single voter is not required for the results (see the discussion below) and is only adopted for simplicity. 6As noted in the Introduction, these informational assumptions differ from Romer and Rosenthal (1978, 1979a), where both s and v are common knowledge, Banks (1989) where u is common knowledge but s is private information, and Morton (1988) where s is common knowledge but v is private information.
434
J.S. Banks, Two-sided uncertainty
where r(p, u) =a1 denotes acceptance of the proposal p given ideal policy u, and r(p, u) = a2 denotes rejection. ’ Define X(p,r)= {UE R:r(p,~)=a,}, and let
(1) Thus A(p,r) is the probability the proposal p is accepted by the voter, given her response strategy r(.). The expected utility for the setter from proposing p, given type s and response strategy I( .) by the voter, defined os(p, s, r), is then a convex combination of the utility associated with the proposal p, and that associated with that status quo s:
os(P, s,4 = 4~~ rh(d + Clwhich is utility-equivalent lent. to
[Myerson
us(~, s, r) = 4~~ r)Cus(p) -
A(P, WS(4, (1985)],
(2)
and hence behaviorally
~~(41.
equiva-
(3)
Since U,( .) requires less notation than us(.), we will use U,( .) as our representation of the setter’s preferences. For the voter, let p(.,p) denote a belief about the setter’s type upon observing the proposal p. Her expected payoff from accepting a proposal of p, denoted U,(a,,p,p), is simply equal to u”(p), while if V rejects p she receives
U,(a,>P>P) = j uv(s)~(s> P) ds.
(4)
Definition. A sequential equilibrium [Kreps and Wilson (1982)] in the above game is a strategy pair (n*,r*) and a system of beliefs CL* such that, (i) VSE [s, a, n*(s) maximizes U,(p, s, r*), (ii) Vp E R, VUE R, r*(p, u) maximizes U,(a, p,p*), (iii) Vpc R, p*(p) is ‘consistent’ with the strategy rc(.) and the prior belief F(.). At a minimum, consistency requires ,U assign positive probability only to setter types in [s,s]. Consistency also requires the voter’s beliefs to be derived via Bayes’ Rule from the strategy n(.) and the prior belief F(.) where possible, i.e. for proposals p E R such that rt ‘(p) # 0. Hence, if according to ‘For simplicity we assume below indifferent voter types is of measure preferences are quadratic about v.
that the voter’s preferences are zero. This holds in equilibrium
such that the set of if, for example, V’s
J.S. Banks, Two-sided uncertainty
435
rc(.) a proposal p is sent by type s, and only by type s, then upon observing p the voter’s belief p( .) should be degenerate, placing probability one on the setter’s type being s. If on the other hand a subset of types send p, then the belief p(.) should place positive probability only on those types, and to the degree dictated by the prior belief F(.). In principle the belief ~1 might also be a non-trivial function of the voter’s type v as well; however, this will not be so in a sequential equilibrium. Formally, consistency requires the beliefs of the voter be the limit of a sequence of beliefs derived via Bayes’ Rule from a sequence of completely mixed strategies by the setter, where this sequence of strategies is independent of v (since v is unknown to S and is drawn independently from s) and converges to the equilibrium strategy rr(.). Bayes’ Rule is then applicable for all proposals in [w all along the sequence, thereby generating a sequence of beliefs, and a limit belief, which is independent of the voter’s ideal policy v. Before proceeding to characterize sequential equilibrium behavior, it is instructive to clarify in what respect the assumption of a single voter in the electorate is without loss of generality. Suppose instead the electorate consists of a set N={ l,,,,, n} of voters, with IZ finite and odd, and where majority rule determines the outcome. Suppose further that each voter’s preferences are single-peaked, with voter i’s ideal policy vi generated by a (smooth) distribution Gi with full support on [w. Then, using the same argument as employed above to show that the (single) voter’s beliefs will not be a function of her type, we see how consistency requires that in any sequential equilibrium all voters will possess the same belief concerning the status quo in and out of equilibrium. That is, since each voter’s beliefs are the limit beliefs generated by a sequence of completely mixed setter strategies, and these strategies are common among all the voters, the voters’ sequence of beliefs, and hence their limit beliefs, will coincide. Therefore, for all proposal-belief pairs, (p,p), we can identify the set of voter types who prefer the proposal over the (random) status quo outcome, and conversely. Assuming all voters vote according to their preferences,* we can interpret the distribution G(.) in the single-voter model as describing the probability at least half the voters have ideal points less than or equal to a given amount, where this probability is determined by the distributions Gr,. . . , G,.9 In this way the (admittedly stringent) sequential equilibrium requirement of common beliefs among the voters essentially reduces the model with n voters to one with a single voter.” *With only two alternatives this constitutes a voter’s weakly dominant strategy. ‘This follows since no restrictions, other than smoothness and full support on R, were placed on the prior G(.) in the single-voter model. “As a referee notes, such an equivalence fails to hold when there is a continuum of voters, all of whose ideal points are drawn from a common distribution, since in such an instance any electoral outcome is certain, rather than random.
436
J.S. Banks, Two-sided uncertainty
3. Results Let C denote the set of strategy pairs (n,r) such that there exists beliefs p, where (71,r, p) constitutes a sequential equilibrium, and for all (7t, r) E C define J(s) =A(Tc(s),Y) as the probability a type s setter’s equilibrium proposal is accepted. Our first result states some general properties of elements of the set C. Lemma 1. V(qr)EZ, (i) n( .) is weakly increasing in s, and T(S) > s, Vs; and (ii) A(.) is weakly decreasing in s, and A(s) > 0, Vs. As noted above, p(s,p)>O only if SE [s_,S]. An immediate implication Proof. of this is that, since I/ is (weakly) risk-averse, for any proposal p>S there will exist a subset of voter types, namely those close to and including p, that will accept this proposal regardless of their beliefs. Since S of type s can therefore guarantee herself an expected payoff strictly greater than zero by making such a proposal, for any (n,r)~C it must be that U,(x(s),s,r) >O as well. Thus, from (3) we must have X(S)> s, and 2(s) > 0, V’sE [s, Fl. To see the monotonicity results, let s>s’, and define p= n(s), p’= I, I.=A(s) and I,‘= %(s’). Then incentive compatibility [cf. d’Aspremont and Gerard-Varet (1979)], which is a necessary condition for (71,,I) to be part of an equilibrium, implies no setter type has a (strict) incentive to emulate the behavior of any other type, or
Al%(P)- us(s)1 2 ~‘C%(P’) - % (S)l, GUs(P’) -
(5)
~&‘)I 2 X%(P) - M)l.
Subtracting the RHS of (6) from the LHS of (5), and the LHS of (6) from the RHS of (5), we get A[u,(s’) - u,(s)] 2 i’[u,(s’) -us(s)]. Since us( .) is strictly increasing and s > s’, this implies Since i(.) >O, eqs. (5) and (6) also imply GP’) - 4s’) Us(P)- 4s’) Cross-multiplying
(ii).
> %(P’) - %(S) = Us(P) -us(s) .
and canceling
%(P)Cas(s) -
i’ 2 1, thus proving
W)l2
terms, we get
as(P’)C%(s) -
%Wl~
(9)
431
J.S. Banks, Two-sided uncertainty
implying
pzp’
and thus proving
(i).
Q.E.D.
The increasing nature of rr(.) is in contrast to the complete information result of Romer and Rosenthal (1978, 1979a), where the setter’s equilibrium proposals are strictly decreasing for status quo outcomes less than the voter’s (known) ideal point. The logic in Romer and Rosenthal (1978, 1979a) is based on the single-peaked nature of the voter’s preferences: if u = 0 and s < 0, then the voter will accept any proposal between s and -s;” therefore in equilibrium S proposes -s, which V accepts with probability one. With incomplete information, on the other hand, we get the opposite sort of monotonicity, where this result is now driven by the preferences and incentives of the setter; indeed, the single-peaked nature of the voter’s preferences is not at all consequential. Evidently, Lemma 1 constitutes a conclusion which holds for any sequential equilibrium of the model. To generate further results concerning behavior, we explore the characteristics of a special kind of equilibrium. Namely, we consider an equilibrium in which the setter’s proposal strategy is a strictly increasing function of the status quo, i.e. a separating equilibrium. While Banks (1990) has shown that if G(.) is degenerate (with a single mass point), no such equilibria exist, Theorem 1 below demonstrates the sensitivity of this result to the assumption of known voter preferences. Consider first the optimal proposal by the setter if the status quo s were known with certainty by the voter. l2 In such an environment, given a proposal p, all voter types less than (p+s)/2 prefer the status quo s to the proposal [by the symmetry of u,(.)], and conversely for u>(p+s)/2. Thus A(p, r) = [ 1 - G((p +s)/2)], and the optimal proposal by the setter, which we denote z’(s), solves max Cl - G((P + 4/2)X4(~)
-u&)1.
(10)
Suppose ke assume the hazard rate associated with the distribution g(u)/[l -G(u)], is increasing in u;13 this plus the concavity of us(.) insures solution to (10) is unique. Theorem 1. There exists a unique separating sequential z*(.), which satisfies the following differential equation: drr* ds
equilibrium
M*(s) N*(s) - M*(s)’
“Recall the symmetry assumption on uV( ,). “As noted above, this is the model of Morton (1988). 13This is a common assumption in private information McMillan (1987)].
G, the
strategy
(11)
models,
e.g. auctions
[cf. McAfee and
438
J.S. Banks, Two-sided uncertainty
where M*(s) =d(n*(s) + 4/N%(~*(s))
- %(SU~
N*(s) = 2[1- G((n*(s) +s)/2)](&+(7r*(s))/ap). Furthermore, n*(5) = n’(S), and for all s E [s_,S) Z*(S)c n’(s). Proof.
Let Ws, ~‘9PI = Cl - G((P + WYC~,(P)
- ~~(41
denote the setter’s expected utility from proposing p, given type s and given the voter believes S to be type s’ (with probability one), and let subscripts on W(.) denote partial derivatives. Given a strictly monotonic strategy n(.), if we substitute n(Y) into W(s,s’,p) we get the setter’s expected utility from imitating type s’ given true type s. Incentive compatibility implies that if (n, r) E 2, then ‘J’s,s’ E [Is,51, W(s, s, Z(S))2 W(s, s’, I). Since this holds with equality at s’ = s, we get the following ‘local’ incentive compatibility condition:
mqs,s’, 7c(s’)) as, -
=
W2(s,s, n(s)) + WJS, s, n(s)) g = 0,
(12)
sfEs
implying all WA%s, n(s)) as = - WJS, s, n(s))’
(13)
which is simply (11) when rc(.) is strictly monotonic, i.e. separating. To show that the model admits a strictly increasing solution to (13), and therefore this condition is sufficient for an equilibrium, we employ results due to Mailath (1987), who shows the following conditions to be sufficient for existence and uniqueness of a separating equilibrium: (i) W is twice continuously differentiable on [s_,S]” x R; in the current model this follows from the assumptions on G(.) and us(.). (ii) W, is never zero, and so is either positive or negative; here w = - CdP) - %(moP
f s)/2),
and so W,
(14) p>s.
J.S. Banks, Two-sided uncertainty
iv,,=;
=@I
i
-3-G((p+s’)/2)1
439
I
+ s’)/2) > 0.
(15)
(iv) W3(s, s, p) =0 has a unique solution in p, where W(s, s,p) is locally concave around this solution; as noted above this follows from us(.) concave and g(u)/[l - G(u)] increasing in u. (v) There exists k >O such that for all (s,p) E [g,S] x R, W,Js,s,p) 20 implies 1W3(s,s, p)I > k; this implies Ws(s, s, p) is bounded away from zero for p bounded away from X’(S). We can guarantee this condition is met in the current context by without loss of generality restricting the set of proposals to the interval [s_,+(S)]. Now W, ~0 implies the ‘worst’ belief, from the setter’s perspective, is where the voter is certain, s=S. Therefore in any separating sequential equilibrium it must be that n*(S) =rP(S), since otherwise S is separating at some other proposal; yet by switching to n”(S) the setter would receive a strictly higher payoff for any belief at Y(S). Furthermore, the following single-crossing property holds:
= -2(%/~4(~M~)C1-
G((P + s’)/2M(p + s’Y2)
U%(P) - @))d(P
+ W2)12
(16)
Then Mailath (1987) shows (Theorem 3) that the solution to the differential equation (13) along with the initial value condition n*(S) =rc’(S) generate a unique separating equilibrium in the above game given that the singlecrossing condition holds. The result that VSE [s, S) n*(s)
J.S. Banks, Two-sided uncertainty
440
in contrast to the certainty case, where the setter knows with probability one whether a proposal will be accepted or rejected by the voter. Theorem 1 also states that the incentive constraints on the setter’s behavior place a downward pressure on the separating equilibrium proposals, since in general a setter of type s has an incentive to be thought of as a lower type, s’
Q(s,
p;
q
r) =
w;7L,4 .
(17)
a4 - us(s)’
thus if 8(s, p; n, I) E [0,11, then f3(.) gives the probability of acceptance making a type s setter indifferent between remaining along the equilibrium path [and receiving the payoff U(s; n, r)] and deviating to the proposal p. If p
In what follows we make use of the following
divine
equilibrium
7c- ‘(p) = 0,
if it is a only if
p(s’,p) >O
result.
14This downward pressure is in contrast to the upward pressure on sender choices found in other signaling models, e.g. Spence (1974). The difference is due to the fact that in the Spence model the sender (i.e. the worker) prefers to be thought of as a higher (more productive) type, ceteris paribus, while in the current model the sender prefers to be thought of as a lower type.
441
J.S. Banks, Two-sided uncertainty
2. V(n,r) E 1, U(s; z, r) is weakly decreasing and continuous in s; thus aU/as exists almost everywhere. Furthermore, where dUj8s exists it satisfies Lemma
au = as
i(s)%.
(18)
Proof. Let s > s’. Since us(.) is strictly increasing and enters negatively into 4s); U(.), s’ can achieve a payoff of at least U(s) by simply proposing therefore n(s’) and i(s’) must be such that U(s’)L U(s). Continuity of U(.) follows from a similar argument, as well as the continuity of us(.). Since U( .) is monotone, it is differentiable almost everywhere [Royden (1968)], so au/as exists for almost all s. Since (z,r) EZ, local incentive compatibility is satisfied:
$ IW)C%(W))- us(sm
S’=S
=~[us(z(s))-u~(s)]+i(s)~ g=o. Thus, where au/as
(19)
is defined,
~=g[us(~(s))-us(s)]+,‘.(s) @- !f!z [ @z ap as as] = -A(s) 2.
Q.E.D.
(20)
Incentive compatibility thus implies an ‘envelope theorem’-type of result; this follows since each type of setter is essentially optimizing over which type to behave as given the ‘suggested’ behavior from rc(.). In equilibrium it must be that each type prefers to behave ‘truthfully’, implying the indirect effect of increasing s on equilibrium utility, through changes in the equilibrium proposal Z(S) and the equilibrium probability of acceptance A(s), is zero. Theorem 2. The unique separating equilibrium the unique universally divine equilibrium.
defined
in Theorem
1 is also
Proof Let (rr, r) EC and let p be an out-of-equilibrium proposal; from the above definition we wish to characterize argmin, d(s, p; z, r). By Lemma 2, O(.) is differentiable almost everywhere and continuous everywhere; thus solving for M/as will assist in identifying the types most likely to defect. From Lemma 2,
a0 a u/as[us(p) -
as=
+ au,/as .U(S) us(s)1 2
t+(s)]
C&P) -
442
J.S. Banks, Two-sided uncertainty
= -
_
4s) ~usl~~cu,(P) - % (s)l+ au,/as~ W[Us(~(S))-us(s)] C%(P) - us( - Us(P)1 . C%(P) - % (S)l2
3s) w~~rus(m
Since A(s) >O and &,/&>O,
a6)/asso as
(21)
we have that
71(s) pp.
(22)
Now since rr( .) is monotone increasing (by Lemma 1) it is differentiable almost everywhere, and ‘locally’ is either pooling, &c/as= 0, or separating, &c/as>O. Suppose (qr) EC is such that rt has a pooling region (s’,s”), i.e. VS,~E(S’,S”), z(s) =rc(?)=p*. Then rr must have a jump discontinuity at s’, or else s’ =s otherwise s’ would be the supremum of a locally separating region, and the continuity of G(.) would imply a jump discontinuity in the probability of acceptance at s’ without a jump in 71, thereby violating U(., 71,r) continuous. Let p = p* --E for E> 0 and small; then from (22) we see that s’= argmin, @s,p). Thus universal divinity requires a jump discontinuity in the voter’s response at p*. Since the voter’s optimal response is strictly decreasing in ‘certainty’ beliefs, and V is (weakly) risk-averse, this is evidently a jump downward, or equivalently a jump up in the probability of acceptance, in going from p* to p. Thus s’ can achieve a strictly higher payoff from proposing p for E sufficiently small, thereby contradicting the assumption of an equilibrium. Therefore in any universally divine equilibrium there can be no pooling regions; implying the only such equilibria are separating. By Theorem 1, then, the unique universally divine equilibrium is the unique separating equilibrium. Q.E.D. Hence, although there may exist a plethora of sequential equilibria in the model, all but the unique separating equilibrium described in Theorem 1 will be discarded by the criterion of universal divinity. In this sense, then, focusing predominantly on the properties of the separating equilibrium is justified.
4. Conclusion As mentioned in the Introduction, the goal of this paper was to explore the various ramifications of the complete information assumptions inherent in the Romer and Rosenthal (1978, 1979a) model of agenda control. While Romer and Rosenthal identified certain weaknesses in the median voter
J.S. Banks,
Two-sided
uncertainty
443
model with respect to the political processes within which public policies are conceived, the model they themselves investigated was based on a number of seemingly unrealistic and/or restrictive assumptions concerning ‘who knows what’ (as is, of course, the median voter model).” And while restrictive assumptions do not necessarily subvert the importance of a model’s insights, relaxing such assumptions (while still maintaining a reasonably tractable model) enhances the analyst’s ability to identify which aspects of the behavioral predictions are due to the underlying incentives of the participants and structure of the interaction, and which are more specific to the details of the model employed. In this paper we have relaxed the complete information assumptions in the Romer-Rosenthal model on two fronts: the setter is now uncertain about the voters’ preferences, and the voters are now uncertain about the location of the status quo outcome. The principal results we find are threefold. First, the proposals offered by the agenda setter will be an increasing function of the status quo, rather than a decreasing function as deduced in the RomerRosenthal model. With uncertainty concerning the voters’ response, the setter is inevitably facing a gamble over final outcomes for whatever proposal she might offer; this plus the fact that the setter can condition her behavior on the actual location of the status quo while the voters cannot, leads to the reversal of the predicted setter behavior. Second, although the agenda setter is in sole possession of information relevant to the optimality of the voters’ decisions, namely the location of the status quo outcome, such information will be revealed by the setter through her choice of alternative. Hence the informational advantage the setter possesses ex ante will be alleviated through her own behavior. Finally, the setter’s proposals will always be less than that which she would propose if the voters also knew the status quo’s location. In this sense, then, the informational advantage the setter holds visa-vis the voters actually tends to diminish, rather than augment, the ability of the setter to implement preferred outcomes. Hence the current model also suggests that normative arguments against the presence of an agenda setter or ‘representative’ in the political system, with its concomitant differential influence of the setter in the determination of public policy outcomes, should be mitigated by the realization that informational effects can in theory act as a counterweight against such influence. ‘%f. Calvert
(1986) for a survey
of median
voter-type
models with imperfect
information
References d’Aspremont, C. and L.-A. Gerard-Varet, 1979, Incentives and incomplete of Public Economics 11, 2545. Banks, J., 1990, Monopoly agenda control with asymmetric information, Economics 105. 445464.
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