a dynamic model of legislative bargaining

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A Dynamic Model of Legislative Bargaining

John Duggan Tasos Kalandrakis

February 21, 2006

Preliminary and Incomplete

Abstract

We prove existence of stationary Markov perfect equilibria in aninfinite-horizon model of legislative bargaining in which the policy out-come in one period determines the status quo in the next. We allow fora multidimensional policy space and arbitrary smooth stage utilities.

We prove that all such equilibria are essentially in pure strategies andthat proposal strategies are differentiable almost everywhere. Themodel is general enough to accommodate much of the institutionalstructure observed in real-world legislatures and parliaments.

1 Introduction

Political interaction in modern democracies qualifies as one of the most com-plex phenomena subjected to scientific inquiry. The need to accommodate

this complexity in formal political theory models stems not only from the de-sire to sate our intellectual curiosity, but seems also essential for the analysisof the effects of public policy and the design of constitutions. In this spirit,

Department of Political Science and Department of Economics, University ofRochester

Department of Political Science, University of Rochester

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we seek to develop a class of models of policy making that (i) accounts for the

multidimensional nature of public policy, (ii) captures the continuing natureof policy over time, (iii) is rich enough to reflect institutional structure ata fine level of detail, and (iv) allows for the kinds of random shocks (e.g.,on preferences and the social environment) to which political interaction issubjected over time. The political economy literature to date has had limitedsuccess in addressing these issues. At a formal level, the primary difficultythat arises is the existence of equilibria in which policy makers use relativelysimple and intuitive strategies. Beyond that is the problem of characterizingequilibria, once they are known to exist, and finally there is the task of ap-plying the model, i.e., developing useful special cases and, when the limits of

formal analysis are reached, bringing numerical techniques to bear.We lay the theoretical foundations for such future applications by es-

tablishing the existence of stationary Markov perfect equilibria in a classof models with the desiderata (i)(iv) identified above, and we show thatin every such model equilibria are essentially in pure strategies, and legisla-tors proposal strategies are differentiable almost everywhere. We consider aninfinite-horizon model of legislative bargaining where each period begins withthe random draw of a legislator, who proposes any feasible policy, which isthen subject to a majority vote. In this respect, our protocol is familiar fromthe Baron-Ferejohn (1989) model of distributive bargaining in the political

science literature. In their work, however, the game ends with the proposedallocation of surplus if a majority of legislators accept the proposal; other-wise, all agents receive a payoff of zero, and the bargaining game is repeatedin the next period. Thus, that model is appropriate for examining policychoices across legislative sessions only if policies remain in place for a singlesession, with an exogenously fixed default outcome in every future session inwhich a new agreement is not reached. This is often the case, for example, inbudgetary negotiations. The model is inadequate, however, for the analysisof continuing programs or legislation, where policy choices remain in placein the future, determining the status quo in future negotiations. In such

environments, policy-makers must not only consider the impact of a policyproposal on the present, but also the future policies that would follow thatchoice.

We consider a fully dynamic model of legislative bargaining, in whichevery period begins with a status quo policy; the policy outcome in that pe-riod is the proposed policy if it garners a majority, the status quo otherwise;

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and the status quo in the next period is determined by the outcome that

prevails in the current period. Thus, the path of play in this game generatesan infinite sequence of policies over time. The level of complexity of thesemodels is significantly greater than models with a fixed default outcome, asproposal strategies must now depend on the status quo in a non-trivial way.It is therefore natural to focus on stationary Markov perfect equilibria, whichdue to their simplicity minimize the difficulty of strategic calculations andmay possess a focal quality. From an econometric point of view, the Markovproperty of equilibrium strategies seems essential for estimation purposes. Inseminal work on the endogenous status quo model, Baron (1996) considersa one-dimensional policy space with single-peaked stage utilities and proves

that stationary equilibrium policy outcomes converge to the ideal point ofthe median voter over time. The model has been extended to special multidi-mensional settings by Kalandrakis (2004b,2005a), Fong (2005), Cho (2005),and Battaglini and Coate (2005), who give constructive proofs of equilibriumexistence relying on the particular structure of their models.

Our model is distinguished from other work on endogenous status quomodels in that we do not assume a specific set of policies or specific func-tional forms for legislators utility functions. Instead, we allow the set ofalternatives to be a very general subset of any finite-dimensional Euclideanspace defined by smooth feasibility constraints. We assume smooth stage

utility functions but do not impose any further conditions. Thus, we cap-ture standard models with resource and consumption constraints, such as theclassical spatial model of politics, economic environments, and distributivemodels in which a fixed surplus is allocated to the legislators. Furthermore,we incorporate uncertainty about future policy preferences and effects of pol-icy. Specifically, we assume that at the end of each period, (i) next periodsstatus quo is realized as the sum of the current periods policy outcome and a(possibly small) stochastic shock, and (ii) legislator preferences are subject to(possibly small) publicly observed stochastic shocks. These natural assump-tion smooth out the game sufficiently to allow us to deduce the existence of

stationary equilibrium. We also derive several technical properties of equi-librium that will facilitate applications of the models. Our existence result isnot covered in the abstract literature on stochastic games, because the tran-sition probabilities of our game violates a standard continuity assumptionused there.

In fact, we have thus far described a simplified version of our model, which

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we develop initially in Section 3. It is in this simplified framework where we

introduce strategies and our concept of stationary legislative equilibrium, andwhere we state our first theorem on existence of equilibrium. In Section 4,we give an elaboration of the benchmark model that is able to accommodatemuch of the institutional structure observed in real-world legislatures andparliaments.

[More on the full model to be inserted here.]

Proofs are collected in the appendix.

2 Literature Review

Before turning to the analysis, we first give a more in-depth review of theliterature on bargaining, as it relates to legislative modeling, and of theliterature on existence of Markov perfect equilibrium in stochastic games.

Bargaining Models Most of the existing work on bargaining considersan infinite-horizon game where in each period one agent makes a proposaland that proposal is either accepted, in which case the game ends with theproposed outcome, or rejected, in which case bargaining continues for at least

one more round. This literature begins with Rubinsteins (1982) work on two-person, alternating-offer bargaining, which is modified by Binmore (1987) toallow for a randomly determined proposer. This model was extended to coverlegislative politics by Baron and Ferejohn (1989), who allow for an arbitrarynumber of legislators and assume a simple majority is required for a proposalto pass. As with Rubinsteins and Binmores work, the subject of bargainingis the allocation of a fixed surplus, now interpreted as pork barrel spending.

A substantial literature cutting across economics and political science hasgrown from these papers. For example, Baron (1991) examines the case ofa two-dimensional set of alternatives, three or four voters with quadratic

preferences, and voting by majority rule. Merlo and Wilson (1995) proveuniqueness of stationary equilibrium, assuming unanimity rule and allowingthe amount of the surplus to vary stochastically over time. Eraslan (2002)proves uniqueness of stationary equilibrium in the original Baron-Ferejohnmodel. Banks and Duggan (2000) prove existence and examine connectionsto the core of the cooperative voting game in a version of the model with

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general set of alternatives, preferences, and voting rule. Kalandrakis (2004c)

gives a simplified proof of existence using a characterization of equilibriumin terms of the solutions to a finite number of equalities and inequalities.Kalandrakis (2006a) examines regularity of the general bargaining model forgeneric discount factors. While all of the previous work implicitly assumesthat delay is bad for the agents, Banks and Duggan (2006) allow for anarbitrary status quo, re-establish results from the earlier framework, andprovide a new analysis of the possibility of delay. Cho and Duggan (2003)prove uniqueness of stationary equilibrium in the one-dimensional model withquadratic utilities, and Cho and Duggan (2005) prove an asymptotic medianvoter theorem in the one-dimensional bargaining model without stationarity.

This class of models has found numerous applications to legislative policy-making,1 but while they capture some dynamic aspects of politics, but theyuniformly assume that the game ends once a proposal is accepted.

A small literature considers the effects of endogenizing the status quo:each period begins with a status quo, then one agent makes a proposal andthat proposal is either accepted, in which case it becomes the status quo forthe next period, or rejected, in which case the current status quo remains inplace. There are currently no general results for this model, though there areconstructions of stationary equilibria in special cases. Baron (1996) analyzesthe one-dimensional version of the model with single-peaked stage utilities.

Kalandrakis (2004a,2005a) establishes existence and continuity propertiesfor the constructed equilibrium strategies in the distributive model, obtainsa fully strategic version of McKelveys (1976,1979) dictatorial agenda settingin that setting, and studies the composition of equilibrium coalitions and theeffect of risk-aversion on equilibrium.2 Baron and Herron (2003) give a nu-merical calculation of equilibrium in a three-legislator, finite-horizon model.Fong (2005) considers a three-legislator model in which policies consist oflocations in a two-dimensional space and allocations of surplus. Cho (2005)analyzes policy outcomes in a similar environment but with a stage game em-ulating aspects of parliamentary government. Similar in spirit to the above,

Battaglini and Coate (2005) characterize stationary equilibria in a model ofpublic good provision and taxation with identical legislators and a stock of

1See, for example, Diermeier, Eraslan, and Merlo (2003), Jackson and Moselle (2002),Kalandrakis (2004a,2005b,2006b), McCarty (2000), and Merlo (1997).

2In contrast, Epple and Riordan (1987) prove folk theorem results in the distributivemodel.

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public goods that evolves over time. All of the above analyses of station-

ary equilibria consist of explicitly constructing equilibrium strategies, which,given the dependence of proposals on the status quo, can be extremely com-plex.

A number of related papers depart in various ways from the above liter-ature and our models. Penn (2005) considers a dynamic voting game withrandomly generated policy proposals and probabilistic voting on these pro-posals. Lagunoff (2005a,b) considers a class of stochastic games that in-corporate a social choice solution concept and analyzes endogenous politicalinstitutions. Finally, Gomez and Jehiel (2005) consider a class of stochas-tic games and characterize efficiency properties of equilibrium when players

are patient. Unlike our model, they assume a finite number of states andtransferable utility.

Stochastic Games Existence of stationary Markov perfect equilibriumis a central issue the literature on stochastic games, which analyzes dynamicgames at a more abstract level. It is well-known that existence in games withfinite state and action spaces follows from the straightforward application ofKakutanis fixed point theorem in finite dimensions (Rogers (1969) and Sobel(1971)). General results on existence have been elusive and have relied onthe imposition of relatively special structure or departures from the conceptof stationary equilibrium.3 All of the known results rely on fairly strongassumptions on the transition probability. Letting s denote a state and adenote a profile of actions, the transition probability is a measurable mappingt(|s, a) from state-action pairs to a probability measure on the set of states.Some assumptions used in the literature are, in increasing strength:

(A1) t is strongly continuous in a,4

(A2) t is norm-continuous in a

(A3) t is norm-continuous in a and absolutely continuous with respect tosome fixed probability measure t

(A4) t is norm-continuous in a and absolutely continuous with respect to afixed, non-atomic probability measure t

3Dutta and Sundaram (1998) provide a lucid review of much of the literature on stochas-tic games and the problem of existence of Markov Equilibrium.

4That is, for each measurable set Z of states, (Z|s, a) is continuous in a.

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(A5) t has a density f(s

|s, a) with respect to Lebesgue measure that is

continuous with respect to a.

It is well-known that even the weakest of the above assumptions, (A1), isinconsistent with deterministic transitions when action sets are uncountablyinfinite.

In finite-horizon stochastic games, Rieder (1979) (see also Chakrabarti(1999)) proves existence of Markov perfect equilibrium under (A1). By in-corporating time in the state variable of a finite-horizon game, we may infact view Rieders equilibrium as stationary. Under strong continuity as-sumptions on the transition probability, akin to (A5), Amir (1996, 2002)

and Curtat (1992) prove existence of stationary Markov perfect equilibriain games possessing strategic complementarities.5 Other results have beenobtained by weakening stationarity or considering weaker notions of equi-librium. Chackrabarti (1999) proves existence of (possibly non-stationary)Markov perfect equilibria in games satisfying (A3), and Mertens and Partha-sarathy (1987, 1991) drop the assumption of absolute continuity and ob-tain existence of equilibria that are nearly Markovian.6 Increasing (A3) to(A4), Chackrabarti (1999) proves existence of a stationary equilibrium, butnow in semi-Markov perfect strategies.7 Dutta and Sundaram (1998) givea simple proof of the existence of (possibly non-stationary) Markov perfect

-equilibria under (A1), whereas Nowak (1985) increases (A1) to (A4) andobtains a Markov perfect -equilibrium in stationary strategies. Himmel-berg, Parthasarathy, Raghavan, and van Vleck (1976) prove existence of p-equilibria assuming finite action sets.8 Finally, Nowak and Raghavan (1992)prove existence of stationary Markov perfect equilibria with public random-ization under (A4), and Duffie, Geanokoplos, Mas-Colell, and McLennan(1994) add mutual absolute continuity of transition probabilities and showthat the equilibrium induces an ergodic process.

A first difficulty in applying existing results on stochastic games to leg-

5Other work restricts the way in which players actions affect each others payoffs, e.g.,

Jovanovic and Rosenthal (1988), Bergin and Bernhardt (1992), and Horst (2005).6Players strategies in period t can depend not only on the current state st but the

previous state st1 as well.7That is, players may condition on the current state and the state in the previous

period, and the nature of that conditioning is constant over time.8Here, p is a probability measure on states, and a p-equilibrium is a strategy profile

such that players optimize at all but perhaps a set of states with p-measure zero.

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islative bargaining is that much of the above work uses weakenings of sta-

tionarity or the concept of equilibrium, whereas we seek truly stationarystrategies such that legislators optimize at all states, without the availabilityof a public randomization device. A second difficulty stems from the deter-ministic element inherent in structure of legislative procedure, which whenmodelled naturally violates (A1) and the other stronger conditions used ontransition probabilities. In describing legislative bargaining as a stochasticgame, the state variable must include all relevant details of the game whenlegislators take actions, whether proposing policy or voting over proposals.Thus, the state must specify when the legislature is in a proposal stage or avoting stage. In a proposal stage, the state must also specify the proposer

and status quo, and in a voting stage the stage must specify the proposedpolicy and the status quo. The transition from a voting stage to the subse-quent proposal stage is not problematic, for action sets in the voting stage arefinite, and the transition is trivially continuous following the votes of legisla-tors. The problem is the fact that the proposers action (the policy proposal)precisely determines the state in the subsequent voting stage, inevitably vio-lating (A1). Adding noise to the model in the form of uncertainty about thestatus quo and the policy preferences of legislators in future session does noteliminate this problem, which we take to be an inherent feature of legislativepolicy-making, yet it allows us to prove the existence of stationary equilibriasatisfying a number of desirable technical properties.

3 The Benchmark Model

We first present a simplified benchmark model that presents the most serioushurdles to establishing the existence of stationary Markov perfect equilibriaand omits extra structure that, while important in applications, is not criticalfor the development of our analytical techniques.

Framework We posit a finite set N of legislators, i = 1, . . . , n, who

must choose from a set X d of feasible policies. Legislative bargainingin each period t = 1, 2, . . . proceeds as follows. A status quo policy qt Xand a vector = (1, . . . , n) nd are taken as given. A legislator is drawnat random, with probabilities p1, . . . , pn, to propose a policy y X. Thelegislators vote simultaneously to accept y or reject it in favor of the status

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quo qt. The proposal passes if it at leastn+1

2legislators vote to accept, and it

fails otherwise. The policy for period t, denoted xt, is y if the proposal passesand is qt otherwise. Each legislator j receives utility uj(xt, j), where j dis the legislators utility shock. Finally, the status quo qt+1 for period t + 1is drawn from the density g(|xt), a new vector = (1, . . . , n) of shocks isdrawn from the density f() and publicly observed, and the above procedureis repeated in period t + 1. Payoffs in the dynamic game are given by theexpected discounted sum of stage utilities, as is standard, and we assume fornotational simplicity that legislators share the discount factor [0, 1).

We impose a number of regularity conditions on the model. We assumethat the set of feasible policies is cut out by a finite number of smooth

functions h : d , where varies over the index set K = {1, . . . , k}, i.e.,X = {x d | h(x) 0, K,

and that this set is compact. For technical reasons, we also impose the weakcondition that for all x d, {Dh(x) | L(y)} is linearly independent,where L(y) is the subset of K containing such that h(x) = 0. This allowsus to capture, for example, standard models with resource and consumptionconstraints, such as the classical spatial model of politics, economic envi-ronments, and distributive models in which an amount of surplus is to be

allocated among the legislators districts.We assume that the stage utilities take the form ui(x, i) = ui(x) + i x,

where ui : d is a smooth function, capturing the standard assumptionof strict quasi-concavity in the spatial modeling literature. Note that theutility shock enters stage utility in a linear fashion. In the special case ofquadratic utility, i.e., ui(x) = ||x xi||2 where xi X is a fixed ideal point,the linear functional form is equivalent to assuming a noise term added tothe ideal point of legislator i. To see this, note that

ui(x) + i

x =

(x

(xi +

1

2i))

(x

(xi +

1

2i)) + i

x +

1

4i

i,

which is just the sum of the constant term i x + 14 i i and the quadraticutility with ideal point xi +

12

i. In the special case of linear utility, theshock is simply a perturbation of the legislators gradients. More generally,the shock is a simple way to introduce well-behaved shifts of the indifferencecurves of legislators.

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We assume that each i is distributed iid and has support in some (possi-

bly small) open set , and we assume that for all x and all i, the expectation(ui(x)+i x)f()d is finite. It is therefore bounded as a function ofx, and

we let c denote a bound over x X and i N. We assume g : X ,with values g(q|x), is measurable in q and smooth in x, and that the supportof the density g(|x) lies in the set X of feasible policies. Furthermore, weassume that the the partial derivatives of all orders of g with respect to thecoordinates of x are uniformly bounded by some b.

Our approach to existence involves the addition of noise to policy out-comes and legislator utilities, but we emphasize that the status quo andthe utility shocks at the beginning of a period t are commonly known, and

therefore, given the strategies of others, a proposer knows whether any givenpolicy will pass or fail. Furthermore, once a vote is taken, the policy outcomeis pinned down for period t: the legislators know, conditional on the outcomeof voting, what the policy outcome in the current period will be, and a newstatus quo is drawn for period t+1 only after legislators receive their period tutilities from outcome xt. Thus, the addition of noise to the model amountsto the assumption that, while legislators are completely informed in the cur-rent period, there is at least some uncertainty about future policy preferencesand the effects of policy. We view these as natural modeling assumptions.In any case, the supports of f and g(, x) can be assumed arbitrarily small,so that the element of noise in the model can be made innocuous.

Strategies and Payoffs A strategy in the game consists of two compo-nents telling us the proposals of legislators when recognized to propose andthe votes of legislators after a proposal is made. While these choices can con-ceivably depend on histories arbitrarily, we seek subgame perfect equilibriain which legislators use simple strategies. We are therefore interested in pure,stationary strategies, which we denote i = (i, i). Here, i : X dis legislator is proposal strategy, where i(q, ) is the policy proposed byi given status quo x and utility shocks , and i : X X {a, r}is is voting strategy, where i(y, q, ) is is vote given proposal y, status

quo q, and shocks . We let = (1, . . . , n) denote a stationary strategyprofile. We may equivalently represent voting strategies by the set of pro-posals a legislator would vote for. We define this acceptance set for i asAi(q, ; ) = {y X | i(y, q, ) = a}. Letting C denote a coalition of

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legislators, we then define

AC(q, ; ) =iC

Ai(q, ; ) and A(q, ; ) =

C: #Cn+12

AC(q, ; )

as the coalitional acceptance set and social acceptance set, respectively. Thelatter consists of all policies that would pass, if proposed by legislator i.

Given strategies , we define legislator is induced preferences in the gameby

Ui(y, i; ) = (1 )(ui(y) + i y) + vi(y; ),

where vi(x; ) is is continuation value at the beginning of period t + 1 frompolicy outcome x in period t. We assume without loss of generality that whenindifferent, legislators vote for the policy proposed, and this then allows us tofocus on no-delay equilibria, in which no legislator ever proposes a policythat is rejected. (In lieu of that, the legislator can just as well propose thestatus quo.) For such equilibria, the continuation value vi satisfies

vi(x; ) =

q

jN

pjUi(j(q, ), i; )f()g(q|x)ddq

for all policies x.

Legislative Equilibrium With this formalism established, we can nowdefine a subset of stationary Markov perfect equilibria of special interest.Intuitively, we require that legislators always propose optimally and that theyalways vote in their best interest. It is well-known that the latter requirementis ambiguous in simultaneous voting games, as arbitrary outcomes can besupported by Nash equilibria in which no voter is pivotal. We follow thestandard approach of refining the set of Nash equilibria in voting subgamesby requiring that legislators delete votes that are dominated in the stagegame. Thus, we say is a stationary legislative equilibrium if

for all shocks , every status quo q, and every legislator i,

Ui(i(q, ), i; ) = sup{Ui(y, i; ) | y A(q, ; )}.

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for all shocks , every status quo q, every proposal y, and every legis-

lator i,

i(y, q, i) =

a if Ui(y, i; ) Ui(q, i; )r else.

Note that we build in the feature that voters defer to the proposer whenindifferent, and that we then without loss of generality restrict proposers tothe social acceptance set.

The main result of this section is that there is a stationary legislativeequilibrium satisfying a number of properties.

Theorem 1 There exists a stationary legislative equilibrium, , of thebenchmark model possessing the following properties.

1. Continuation values are smooth: for every legislatori, vi(q; ) is smoothas a function of q.

2. Coalitions are almost always minimum winning: for every status quo

q, almost all shocks , and every legislator i, if i(q, ) = q and thereexists k = i such that Uk(i(q, ), k; ) = Uk(q, k; ), then

|{j / N\ {i} | Uj(i(q, ), j) Uj(q, j)}| = n 12 .3. Proposals are almost always strictly best: for every status quo q, almost

all shocks , every legislator i, and every y A(q, ; ) distinct fromthe proposal i(q, ; ), we have Ui(i(q, ), i; ) > Ui(y, i; ).

4. The linear independence constraint qualification almost always holds

for the proposer: for every status quo q, almost all shocks , and everylegislator i, if i(q, ) = q, then the collection

{Dh(i(q, )), DyUj(i(q, ), i; ) | L, j C}is linearly independent, where L and C represent the feasibility andvoting constraints that bind at i(q, ).

5. Proposal strategies are almost always differentiable: for almost all shocks

and every legislator i, i(q, ) is differentiable as a function of (q, ).

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The fourth part of Theorem 1, which verifies that the linear independence

constraint qualification (LICQ) almost always holds, is particularly impor-tant, as it implies a characterization of optimal proposals by means of thefirst order conditions for a constrained maximum. In particular, for everystatus quo q and almost all shocks , the proposal i(q, ) is a critical pointof the Lagrangian and the complementary slackness conditions hold: thereexist , K, and j, j N such that

DyUi(i(q, ), i; ) +k

=1

Dyh(i(q, )) +n

j=1

jDyUj(i(q, ), j; ) = 0

j

0 and j(Uj(i(q, ), j ; )

Uj(q, j; )) = 0, j

N

0 and h(i(q, )) = 0, K.

In fact, we show in the proof of Theorem 1 that the complementary slacknessconditions hold strictly almost everywhere, i.e., j > 0 for all j such thatUj(i(q, ), j; ) = Uj(q, j; ), and > 0 for all such that h(i(q, )) = 0.This first order characterization of optimal proposals would seem necessaryfor a more detailed study of equilibria, either by analytical or numericalmeans.

We provide a sketch of the proof of Theorem 1, which is proved formally

in the appendix. As expected, the proof proceeds by defining a suitablemapping, establishing the existence of a fixed point, and then verifying thatit has the claimed properties. Let C(d, n) denote the space of smooth,bounded mappings from d to n (endowed with the topology ofC-uniformconvergence on compacta, as described in Mas-Colell (1985)). We defineUi(y, ; v) and Ai(q, ; v), in the obvious way, as the induced utilities andacceptance sets when continuation values are given by v. We proceed in anumber of steps

1. We note that for every status quo q and almost all shocks , the max-

imization problem maxy Ui(y, i; v)s.t. y A(q, ; v)

has a unique solution, which we denote i(q, ; v), almost everywhere.Elsewhere, we define the function i(; v) by arbitrarily selecting fromthe set of maximizers.

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2. We show that for every status quo q, almost all shocks , and every

legislator i, if is proposal is distinct from the status quo and makesany other legislator indifferent between accepting and rejecting, thenthe proposal garners a bare majority of votes, i.e., the coalition thatforms contains no redundant legislators.

3. By an application of the transversality theorem, we show that for everystatus quo q, almost all shocks , and every policy y A(q, ; v) distinctfrom q, the LICQ holds. This means that if the vector of continuationvalue functions is perturbed slightly, the proposer can find proposalsarbitrarily close to y that continue to satisfy the voting constraints.

4. For every status quo, almost all shocks , and every legislator i, LICQimplies that the necessary first and second order conditions for a con-strained maximum are satisfied. By an application of the transversalitytheorem, we show that the complementary slackness conditions actuallyhold strictly, so that binding constraints are associated with positivemultipliers.

5. We can then argue, by another application of the transversality theo-rem, that for every status quo and almost all shocks , every legislatorsproposal i(q, ; v) is differentiable in (q, ) whenever the legislator pro-poses the status quo.

6. We show by a maximum theorem-type argument that for every statusquo q, almost all shocks , and every legislator i, the optimal proposali(q, ; v) is continuous in v.

7. We define the subset V C(d, n) to consist of all mappings vthat are bounded by c and such that the partial derivatives of order1, 2, . . . share a particular bound. This set is nonempty, convex, andclosed. We show that the set of rth order derivatives of functions inV is equicontinuous for each r = 0, 1, 2, . . ., and it follows that V iscompact.

8. We define the mapping (v) = v, where

vi(x) =

q

jN

pjUi(j(q, ; v), i; v)f()g(q|x)ddq. (1)

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That is, (v) is the vector of continuation value functions generated

by optimal behavior when continuation values are given by v. By dif-ferentiating vi, we find that (V) V.

9. Using continuity of the proposal strategies i(q, ; v) almost every-where, we prove that the mapping is continuous.

10. Finally, we use standard fixed point arguments to show that the map-ping has a fixed point v.

From v, we define equilibrium proposal strategies i(q, ; v) and voting

strategies Ai(q, ; v). It follows immediately from (1) and the properties of

g that the continuation values vi are smooth, and the other properties inTheorem 1 follow from our equilibrium construction.

The operator constructed in the proof of Theorem 1 can be used to give asomewhat more general characterization result: since every legislative equi-librium continuation value v must be a fixed point of , it follows that theproperties described in Theorem 1 are necessarily satisfied by all stationarylegislative equilibria.

Theorem 2 Every stationary legislative equilibrium satisfies properties

15 of Theorem 1.

With existence of a stationary legislative equilibrium proved, it isof interest to consider the equilibrium dynamics of policy outcomes in themodel. In so doing, we define the transition probability on policy outcomesby

P(x, Y) =

q

iN

piIY(i (q, ))f()g(q|x)ddq,

which is the probability, conditional on policy outcome x this period, of apolicy outcome in the set Y X next period. We define the associatedMarkov operator T on the space of bounded, Borel measurable functions

: X by T (x) = (z)P(x,dz). The adjoint T operates on theBorel measures on X and is defined by T(Y) =

P(x, Y)(dx). This

describes, given a distribution of policy outcomes in the current period, thedistribution of outcomes in the next period. The iterates ofT, denoted Tm,give the distribution of policy outcomes m periods hence and are thereforekey in describing the long run policy outcomes of the model.

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It is straightforward to show that T maps continuous functions to con-

tinuous functions and, therefore, satisfies the Feller property. Furthermore,it is tight, so that it admits an invariant distribution such that = T.

[More on uniqueness to be inserted here.]

Our equilibrium construction also informs us of subgame perfect equi-libria (in stage-undominated voting strategies) in the finite-horizon versionof our model. Let v0 be the profile of zero functions. In the two-periodversion of the model, the continuation value from policy outcomes in thefirst period is given uniquely by (v0). This in turn generates essentiallyunique proposal strategies in the first period. By induction, the continuationvalue following the first period of the T-period version of the model is givenby (vT2). Thus, our construction yields the (essentially) unique subgameperfect equilibrium of the finite-horizon model.

4 The Full Model

[Much more on the full model to be inserted here.]

A Proofs of Theorems

Proof of Theorem 1 Let Cb (d, n) be the smooth, bounded func-tions from d into n with the topology ofC-uniform convergence on com-pacta, so that a sequence {m} of functions converges to if and only if forevery compact set Y d and each r = 0, 1, 2, . . ., all partial derivatives oforder r of m converge uniformly to zero on Y. Given v = (v1, . . . , vn) Cb (

d,

n), define the induced utility

Ui(y, i; v) = (1 )(ui(y) + i y) + vi(y),where future payoffs are assumed to be generated by v, and define the asso-ciated acceptance sets

Ai(q, ; v) = {y X | Ui(y, i; v) Ui(q, i; v)}.

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By Theorem I.3.1 of Mas-Colells (1985), for all i, this solution is uniquely

defined outside a measure zero set of is, so let Ci (i, q; v) denote themeasure zero set of i where is optimal proposal to C is not unique, giveni. Let i(i, q; v) denote the union of these sets, and let

i(q; v) = { | i i(i, q; v)},

which is measure zero in . Now take two distinct majority coalitions C andC, and take any . Suppose that Ci (q, ; v) =

C

i (q, ; v) = q and that thereexists j C\ C such that Uj(Ci (q, ; v), j; v) = Uj(q, j ; v). This equalitydefines a hyperplane in , which is lower-dimensional. If / (q; v), thenproposals are uniquely determined, so we may define

i (q; v) =

/ i(q; v)

there exist C, C and j C\ C such

that Ci (q, ; v) = C

i (q, ; v) = qand Uj(

Ci (q, ; v), j; v) = Uj(q, j; v)

,

which is measure zero by Fubinis theorem (see Aliprantis and Borders (1999)Theorem 11.26). By Step 1, outside the measure zero set i(q; v), player i hasa unique global maximizer i(q, ; v). Consider \ (i (q; v) i(q; v))and suppose that i(q, ; v) = q and Uk(i(q, ; v), k; v) = Uk(q, k; v) forsome k = i. Now let

C = {j N\ {i} | Uj(i(q, ; v), j ; v) Uj(q, j; v)},

and suppose, to obtain a contradiction, that |C| n+12

. Let C = C\ {k},

and note that C(q, ; v) = C

(q, ; v) = i(q, ; v) = q. This contradictsUk(i(q, ; v), k; v) = Uk(q, k; v), since / i (q; v). Thus, the conclusion ofthe claim holds outside the set of measure zero i (q; v) i(q; v).

We now give notation used in Steps 35. For arbitrary subsets L Kand C N, define the functions h : d |L| by h(y, ) = (h(y))Land U: d |C| by U(y, ; v) = (Ui(y, i; v) Ui(q, i; v))iC. Define

the mapping F

L,C

: (d

\ {q}) |L|+|C|

by

FL,C(y, ; v) =

h(y, )

U(y, ; v)

,

where here (and whenever relevant) we view vectors as column matrices,making FL,C(y, ; v) a (|L| + |C|) 1 matrix.

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3. Claim: for all q, almost all , and all y

A(q, ; v)

\ {q

}, LICQ is

satisfied at y. Fix q arbitrarily, and consider any L K and C N. ThenDFL,C(y, ; v) is the (|L| + |C|) (d + nd) matrix

Dh(y) 0 0DyU(y, ; v) (1 )(y q)T I|C| 0

,

where denotes Kronecker product. For all (y, ) such that FL,C(y, ; v) = 0,it follows that L is contained in the binding feasibility constraints at y,i.e., L L(y), and therefore rows 1, . . . , |L| are linearly independent byassumption. Since y = q, rows |L| + 1, . . . , |L| + |C| are linearly indepen-dent, and we see that DFL,C(y, ; v) has full row rank. We conclude that

FL,C is transversal to {0}. For each , define FL,C : d \ {q} |L|+|C|by FL,C (y; v) = F

L,C(y, ; v). By an application of the transversality the-orem (see Mas-Colells (1985) Theorem I.2.2), it follows that for almost all, FL,C {0}, i.e., 0 is a regular value of FL,C . Let L,C(q; v) be themeasure zero set of s where this does not hold. Let (q; v) be the finiteunion of these, which has measure zero. Now take any / (q; v) and anyy A(q, ; v) \ {q}, and let L and C represent the constraints satisfied withequality at y. Then y (FL,C )1({0}), so DFL,C (y) has full rank, i.e.,

{Dh(y), DyUi(y, i) | L, i C}is linearly independent, fulfilling LICQ.

Before we move to Step 4, we develop some necessary notation. For anylegislator i and any L K and C N, define the function LL,Ci : d |L|+|C| by

LL,Ci (y,,; v) = Ui(y, i; v) +L

h(y) +jC

j(Uj(y, j; v) Uj(q, j; v)).

Further, define GL,Ci : (d \ {q}) |L|+|C| d |L|+|C| by

GL,Ci (y,,; v) =

DyLL,C

i (y,,; v)FL,C(y, ; v)

.

where FL,C(y, ; v) is defined prior to Step 3.

4. Claim: for all q, all i, almost all , all L K, all C N, andevery (y, ) such that GL,Ci (y,,; v) = 0, we have m = 0 for all m

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L

C and D(y,)G

L,Ci (y,,; v) is non-singular. Fix q and i arbitrarily, and

consider any L K and C N. For m C L, define the mappingGL,C,mi : (d \ {q}) |L|+|C| d |L|+|C| by

GL,C,mi (y,,; v) =

GL,Ci (y,,; v)

m

.

Then the derivative DGL,C,m

i (y,,; v) is the (d + |L| + |C| + 1) (d + |L| +|C| + nd) matrix

DyyLL,C (y,,; v) DyFL,C(y, ; v)T (1 )Id ((1 )jId)jC

Dh(y) 0 0 0DyU(y, ; v) 0 0 (1 )(y q)T I|C|

0 0 1 0 0 0 .

(We omit zero-columns corresponding to derivatives with respect to j, j /C {i}.) For all (y,,) such that GL,C,mi (y,,; v) = 0, this derivativeevidently has maximal rank. Then, by the transversality theorem, for almostall , the mapping GL,C,mi, : (d \ {q}) |L|+|C| d |L|+|C| defined by G

L,C,m

i, (y, ; v) = GL,C,m

i (y,,; v) is transversal to {0}. Sincethe dimension of the domain of G

L,C,m

i, is smaller than that of the range,the preimage theorem (see Mas-Colells (1985) Theorem H.2.2) implies that

(GL,C,m )1({0}) is empty for almost all , i.e., outside a set L,C,mi (q; v) ofmeasure zero, G

L,C,m

i (y,,; v) = 0. Furthermore, defining GL,Ci, : (d\{q})|L|+|C| d |L|+|C| by GL,Ci, (y, ; v) = GL,Ci (y,,; v), an application ofthe transversality theorem to the mapping GL,Ci ensures that outside a setL,C(q; v) of measure zero, GL,Ci, (y, ; v) is also transversal to {0}. Repeatingthe above arguments for all L, C and all m LC, we conclude that outsidea set i(q; v) of measure zero in , every solution (y, ) to G

L,Ci (y,,; v) = 0

for any L and Cis such that m = 0 for all m LCand D(y,)GL,Ci (y,,; v)is non-singular.

5. Claim: for all q, all i, and almost all , if i(q, ; v) = q, then itis differentiable in (q, ). Fix q and i arbitrarily. Borrowing from Steps14, consider \ (i(q; v) (q; v) (q; v) i (q; v)), and assumei(q, ; v) = q. Since / i(q; v), i(q, ; v) is a unique global maximizer byStep 1. Suppose the binding constraints are given by L and C, which maybe empty. By Step 2, there exists C N such that i(q, ; v) is the solution

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to

maxyAC(q,;v) Ui(y, i; v), (2)

where C C, i / C, and |C| = n12

. Thus, borrowing from Step 2,

i(q, ; v) = C

i (q, ; v). We claim that C

i (; v) is continuous at (q, ). To seethis, consider any sequence {(qm, m)} converging to (q, ), and let y be anyaccumulation point of {Ci (qm, m; v)}. By closed graph of A, we have y A(q, ; v), and uniqueness of a global maximizer implies i(q, ; v) = y. SinceX is compact, this proves the claim. We now claim that there is an open setaround (q, ) such that is optimal proposal solves (2) over that set. Indeed,suppose the claim is false. Then there is a sequence {(qm, m)} convergingto (q, ) and a sequence {C

m

} of coalitions Cm

= C

, with i / Cm

and|Cm| n12

, such that Ui(

Cm

i (qm, m; v), mi ; v) > Ui(

C

i (qm, m; v), mi ; v)

for all m. By finiteness of N and compactness of X, we may suppose thatCm = C for all m and C

i (qm, m; v) y for some y X. By closed graph

of AC, we then have y A(q, ; v), and continuity implies Ui(y, i; v) Ui(

C

i (q, ; v), i; v). By uniqueness of the global maximizer, we then havey = C

i (q, ; v). Then continuity implies that

C {j N\ {i} | Uj(Ci (q, ; v), j; v) Uj(q, j; v)}.

If C

=

, then, by Step 2, C = C, a contradiction. If C =

, then for all

j = i, we have Uj(Ci (q, ; v), j; v) = Uj(q, j; v), and by continuity there areopen sets Y around C

i (q, ; v) and Z around (q, ) such that for all y Y,all (q, ) Z, and all j = i, we have Uj(y, j; v) > Uj(q, j; v) if and onlyif Uj(

C

i (q, ; v), j ; v) > Uj(q, j; v). This implies that for all (q, ) Z,we have Y AC(q, ; v) A(q, ; v). But for high enough m, we haveC

i (qm, m; v) Y and (qm, m) Z, and then Ui(Ci (qm, m; v), mi ; v)

Ui(C

i (qm, m; v), mi ; v), a contradiction that proves the claim. Now, as a

consequence, it suffices for Step 5 to show differentiability of the solutionC

i (q, ; v) to (2). By Theorems 2 and 3 in Fiacco and McCormick (1990),since LICQ is satisfied at C

i (q, ; v) by Step 3, the first and second order

necessary conditions must hold, i.e.,

(i) GL,C (C

i (q, ; v), ; v) = 0,

(ii) for all m L C, m 0,

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(iii) for all z such that DyFL,C(C

i (q, ; v), ; v)z = 0, we have

zTDyyLL,Ci (C

i (q, ; v), , ; v)z 0.

By Step 4, we have m = 0, and then (ii) implies that m > 0 for allm L C. By Step 4, D(y,)GL,Ci (Ci (q, ; v), , ; v) is non-singular, andtherefore the second order necessary condition implies that the second or-der sufficient condition is satisfied at C

i (q, ; v), i.e., for all z = 0 suchthat DyF

L,C(C

i (q, ; v), ; v)z = 0, zTDyyLL,Ci

C

i (q, ; v), ,

z < 0. For

if, instead, zTDyyLL,Ci

C

i (q, ; v), ,

z = 0 for some z = 0 such thatDyF

L,C(C

i (q, ; v), )z = 0, then

D(y,)GL,C

i (C

i (q, ; v), , ; v)

z0

=

00

,

which is impossible since D(y,)GL,C

(C

i (q, ; v), , ; v) is invertible and z =0. Thus, for almost all , C

i (q, ; v) = q satisfies all the conditions ofTheorem 6 of Fiacco and McCormick (1990), and we conclude that it isdifferentiable in an open set around q, .

6. Claim: for all q, all i, all {vm} converging to v, and almost all ,we have i(q, ; v

m) i(q, ; v). Fix q and i arbitrarily, and consider anysequence

{vm

}converging to v. Borrowing from Steps 1 and 3, let

i(q, {vm}) = i(q; v) (q; v)

m=1

(q; vm),

which has measure zero. Take any / i(q, {vm}), and suppose {i(q, ; vm)}does not converge to i(q, ; v). Since X is compact, we may go to a sub-sequence such that i(q, ; v

m) y = i(q, ; v) for some y X. By con-tinuity, A(q, ; v) has closed graph, so we know that y A(q, ; v). Sincei(q, ; v) is uniquely defined, by / i(q; v), and is distinct from y, we haveUi(i(q, ; v), i; v) > Ui(y, i; v). Case 1: i(q, ; v) = q. Since i(q, ; v) A(q, ; v) \ {q}, there exists some majority coalition C such that i(q, ; v) AC(q, ; v) \ {q}. By Step 3, we know that LICQ is satisfied at i(q, ; v), sothere exists a direction s such that Dh(y) s > 0 and DUi(q, i; v) s > 0 forall binding and i C. Choose > 0, small enough that i(q, ; v) + s AC(q, ; v) A(q, ; v) and Ui(i(q, ; v) + s,i; v) > Ui(y, i; v). Since noconstraints are binding at i(q, ; v) + s for v, we have i(q, ; v) + s

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AC(q, ; vm) for large enough m. By continuity, we have Ui(i(q, ; v) +

s, i; vm) Ui(i(q, ; v) + s, i; v) and Ui(i(q, ; vm), i; vm) Ui(y, i; v),but then Ui(i(q, ; v) + s, i; v

m) > Ui(i(q, ; vm), i; v

m) for large enoughm, contradicting optimality of i(q, i; v

m). Case 2: i(q, ; v) = q. ThenUi(q, i; v) > Ui(y, i; v), and, with the fact that q is always a feasible choice,continuity again leads to a contradiction with optimality of i(q, ; v

m).

7. The domain of continuation values. Define Vto consist of the functionsv Cb (d, n) that are bounded by c and such that for all i and all partialderivatives of order 1, 2, . . . ofvi are bounded by bc, where is the Lebesguemeasure ofX. Clearly, Vis a nonempty, convex, closed, and bounded subsetofCb (

d,

n). For each i and each v

V, let wi(x; v) be an arbitrary partial

derivative of order r = 0, 1, 2, . . . of vi. For every compact Y d, the setof functions

{(w1(; v)|Y, . . . , wn(; v))|Y | v V}is equicontinuous. Indeed, given > 0, set = /bc

nd. Suppose x, y Y

satisfy ||x y|| < . Since the partial derivatives of (; v) are bounded bybc, we have |wi(x; v) wi(y; v)| ||x y||bc

d < /

n. Therefore, by

Mas-Colells (1985) Theorem K.2.2, Vis compact.8. The fixed point mapping. We define the mapping : V Vas follows.

Let (v) = v, where

vi(x) =

q

jN

pjUi(j(q, ; v), i; v)f()g(q|x)ddq.

Let (q, x) be an arbitrary partial derivative of order r = 0, 1, 2, . . . of g(q|x)with respect to the coordinates of x, so that by Mas-Colells (1985) TheoremE.5.2, the corresponding partial derivative of vi(x) is

wi(x; v) =

q

jN

pjUi(j(q, ; v), i; v)f()(q, x)ddq.

Note that

|wi(x; v)| q

jN

pj|Ui(j(q, ; v), i; v)|f()d

|(q|x)|dq

q

c|(q|x)|dq.

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If r = 0, so that = g, then the latter integral is less than or equal to c; if

r 1, then it is less than or equal to bc. Thus, (V) V.9. Continuity of . Let {vm} be a sequence in V such that vm v.

We need to show that (vm) (v). Let (q, x) be an arbitrary partialderivative of order r = 0, 1, 2, . . . of g(q|x) with respect to the coordinates ofx, and define wi(x; v) as in Step 8. We must prove that wi(; vm) wi(; v)uniformly, i.e, for all > 0, there exists m such that for all m m and allx X, we have |wi(x; vm) wi(x; v)| < . If this does not hold, then thereexists a subsequence {vm}, still indexed by m, and a corresponding sequence{xm} in X such that for all m, |wi(xm; vm)wi(xm; v)| . By compactnessof X, we may further assume that xm

x for some x

X. Note that

wi(xm; vm) =

q,

jN

pjUi(j(q, ; vm), i; v

m)f()(q, xm)d(q, ).

LetZ = {(q, ) X | (q, {vm})}

denote the set of (q, ) pairs from Step 6 such that {(q, ; vm)} may notconverge to i(q, ; v), and note that this set has Lebesgue measure zero inX . Take any (q, ) / Z. Then j(q, ; vm) j(q, ; v) for all j, andjoint continuity ofUi implies that

Ui(j(q, ; vm), i; v

m)(q, xm) Ui(j(q, ; v), i; v)(q, x).

Therefore, by Lebesgues dominated convergence theorem (see Aliprantis andBorders (1999) Theorem 11.20), we have wi(x

m; vm) wi(x; v), a contra-diction. We conclude that is continuous.

10. Existence of equilibrium. By Mas-Colells (1985) Theorem K.1.1,C(d, n) is metrizable and therefore Hausdorff, and it is also locallyconvex. The space V inherits these properties, and therefore the Brouwer-Schauder-Tychonoff theorem (see Aliprantis and Borders (1999) Corollary

16.52) implies that has a fixed point, v (v). It is clear from our con-struction that the strategy profile defined by (i , A

i ) = (i(; v), Ai(; v))

for all i is a stationary legislative equilibrium that satisfies properties 15 ofthe theorem.

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