Dynamic Collective Choice with Endogenous Status Quo∗

Dynamic Collective Choice With Endogenous Status Quo∗

Wioletta Dziuda† Antoine Loeper‡

August 2014 First version: December 2009

Abstract

This paper analyzes an ongoing bargaining situation in which preferences evolve over time and the previous agreement becomes the next status quo, determining the payoffs until a new agreement is reached. We show that the endogeneity of the status quo induces perverse incentives that exacerbate the players’ conflict of interest: Players disagree more often than they would if the status quo was exogenous. This leads to ineffi ciencies and status quo inertia. When players are suffi ciently patient, the endogenous status quo can lead the negotiations to a complete gridlock in which players never reach an agreement even though their preferences agree arbitrarily often. When applied to the legislative setting, our model predicts partisan behavior, and explains why oftentimes legislators fail to react in a timely fashion to economic shocks. JEL Code: D72.

∗Previously circulated under the title “Ongoing Negotiation with an Endogenous Status Quo.”We thank David Austen-Smith, David Baron, Daniel Diermeier, Bard Harstad, and seminar participants at Bonn University, Leuven University, CUNEF, Northwestern University, Nottingham University, Rice University, Simon Fraser University, Universidad de Alicante, Universidad Carlos III, Universidad Complutense, the University of Chicago, and the Paris Game Theory Seminar. Yuta Takahashi proﬁded an excellent research assistance. †Corresponding author. Kellogg School of Management, Northwestern University, 2001 Sheridan Road, Evanston, 60208, IL. Email: [email protected] ‡Universidad Carlos III de Madrid. Email: [email protected]

1 1 Introduction

The legislative gridlock that paralyzed the 112th U.S. Congress and led to a fiscal cliff highlighted the inability of legislators to agree on much needed fiscal policy reforms, in particular concerning the various entitlement programs such as medicare and social security. This gridlock puzzled the popular press and public, who bemoaned the partisan behavior of the legislators and interpreted it as blind allegiance to their parties and ideologies. The economic literature has proposed several ra- tional explanations for the inability of legislators to reach seemingly beneficial agreements: electoral calculus, vested interests, uncertainty over the distributional impact of reforms, lags in the effects of reforms, or communication failures.1 This paper shows that partisan behavior and legislative inertia can be the strategic consequence of two features. First, many bargaining decisions such as the laws that determine entitlement programs are continuing in nature. That is, they remain in effect until a new agreement is reached. Second, a constantly evolving environment requires these decisions to be periodically revisited.

To understand how the combination of the continuing nature of policies and the evolving environment leads to excessive disagreement, consider the case of legislators negotiating over the level of fiscal spending. During a contraction, generous deficit spending may be favored by all parties to stimulate short-term economic growth. During a boom, all parties may agree to use the extra tax revenues to bring the public debt under control. In normal times, however, legislators may genuinely disagree on the optimal level of public spending. If today’s policy becomes the default for future negotiations, fiscal conservatives may be reluctant to increase public spending during a recession, out of fear that their liberal counterparts will veto a return to fiscal discipline when the economy improves. Similarly, anticipating future disagreement, liberals may refuse to lower spending in times of economic growth, out of fear that conservatives will oppose a fiscal expansion when the boom is over.

We ﬁrst formalize this insight in a basic model in which two players engage in an inﬁnite sequence of choices over two alternatives, called L and R. At the beginning of each period, one alternative serves as the status quo. If both players agree to move away from the status quo, the new agreement is implemented. Otherwise, the status quo stays in place. In both cases, the

1 See Section 2 and Drazen (2002) for a detailed review of this literature.

2 implemented agreement determines the players’payoﬀs in this period and becomes the status quo for the next; that is, players operate under the endogenous status quo protocol. Players’preferences change over time in a stochastic fashion. We assume that players preferences over the alternatives sometimes disagree, and whenever they do, one player (the rightist) prefers R and the other (the leftist) prefers L.

We show that indeed the endogeneity of the status quo exacerbates the players’ conﬂict of interest and leads to equilibrium behavior that resembles partisanship: in all equilibria, the leftist player votes for L and the rightist player votes for R more often than they would if the policies were not continuing in nature or if there were no shocks to the preferences. As a result, players may fail to reach an agreement even when the status quo is Pareto dominated. Hence, the status quo stays in place too often, and the bargaining outcome is less responsive to the environment.

Our analysis further shows that players’ equilibrium behavior is driven by a vicious cycle in which partisanship and disagreement feed on each other: more partisan players disagree more often; more frequent disagreement increases the importance of securing the favorable status quo; and this further increases partisanship. As a result, the polarizing eﬀect of the endogenous status quo can be quite dramatic. In particular, we show that the negotiations can come to a complete gridlock: even if their preferences agree arbitrarily often, suﬃ ciently patient players never reach an agreement, and the bargaining outcome is completely unresponsive to the evolution of preferences.

When players bargain over more than two alternatives in each period, the polarizing effect of the endogenous status quo can be accompanied by other equilibrium effects which depend on the fine details of the model. Their complex interaction makes it hard to isolate analytically the impact of the endogenous status quo on equilibrium behavior. However, our analysis delivers two important insights. First, these additional equilibrium effects arise only for a particular allocation of bargaining power, while the polarization effect is independent of this allocation. Consistently, we show that for a class of bargaining procedures– which has been found to be of particular empirical relevance in Riboni and Ruge-Murcia (2010)– only the polarization effect occurs, and the endogenous status quo generates partisanship and status quo inertia in all equilibria. Second, for the allocation of bargaining power to matter, and thus for the other equilibrium effects to arise, the environment must be able to change drastically before players have an opportunity to react. Consistently, we show that irrespective of the details of the bargaining procedure used in each period, when players

3 can revise the status quo as soon as the environment makes it inadequate, the polarization eﬀect dominates, and equilibria exhibit status quo inertia.

Our model is institutionally sparse, but can shed light on a vast array of dynamic bargaining situations which feature an endogenous status quo and shocks to the environments. For example, about two-thirds of the U.S. federal budget– called mandatory spending– continues year after year by default, and legislators’preferences over it are affected by shocks such as business cycles, changes in the country’scredit rating, or the vagaries of public opinion. Recurrent delays in political action have led many macroeconomists to consider fiscal policy as a very imperfect stabilization tool if left to politician’sdiscretion. Our results provide a strategic rationale for the inability of ideologically polarized legislators to react in a timely fashion to a changing environment. Our model also applies to other legislative policy domains. For instance, the laws regulating civil liberties and privacies, are typically continuing in nature. At the same time, citizens’and legislators’preferences are affected by shocks such as terrorist attacks, national security threats, and technological innovation.

Our results also have implications for monetary policy making. In some countries, monetary policy is set by a committee with heterogeneous preferences and beliefs, and the interest rate stays the same until the committee agrees to change it according to its internal voting rule (see Riboni and Ruge-Murcia 2008). Our results show that the endogeneity of the status quo, and the voting rule used in monetary policy making, can aﬀect the ability of the committee to respond to economic shocks.

Similarly, the renegotiations of ongoing contracts such as trade agreements, international treaties

(e.g., for the World Trade Organization or the European Union), and financial or labor contracts are usually conducted in the presence of a binding previous agreement.2 The impossibility of making these contracts fully contingent on all economic shocks and political events creates the need for frequent renegotiation. In the case of labor contracts, the intuition highlighted in this model suggests a possible explanation for wage stickiness: During a recession, lower wages may leave the workers and the employers better off, thanks to a lower risk of bankruptcy and layoffs.

Yet, the workers maybe reluctant to agree on lower wages out of fear that the employers will refuse to increase them when the economy improves. Conversely, during a boom, higher wages

2 In the case of international agreements, each party can always unilaterarily violate the terms of the agreement. The consequences of that, however, are oftentimes severe enough that in practice the relevant default is the prevailing international agreement. Hence, in eﬀect, countries negotiate under the endogenous status quo protocol.

4 may be necessary to retain skilled workers and improve morale. However, the employers may be reluctant to increase wages out of fear that the workers will refuse to reduce them when the situation deteriorates.

The paper is organized as follows. Section 2 reviews the related literature, Section 3 analyzes the basic model with two alternatives, and Section 4 generalizes the model to more than two alternatives. Section 5 concludes and discusses several extensions.

2 Related literature

2.1 Ineﬃ ciency and inertia in dynamic policy making models

Policy makers’ incentives (not) to adopt socially beneficial policy changes have been the study of numerous papers. Fernandez and Rodrik (1991) show that if there is uncertainty over who benefits from the policy change, the change may not be undertaken, even if it is known to benefit the majority ex post. Similarly, Strulovici (2010) shows in a dynamic collective experimentation setting that such distributional uncertainty can discourage players from policy experimentation.

In our model, it is not the uncertainty over future preferences that matters but the fact that they evolve.

Status quo inertia can arise also if the costs of policy changes are borne disproportionately by those who initiate these changes. Alesina and Drazen (1991) show that in such cases, players engage in a war of attrition, which results in delays. Glomm and Ravikumar (1995), Krusell and Rios-Rull

(1996), Coate and Morris (1999) show that policy persistence may also occur if citizens respond to the current policy by undertaking private actions that make them more likely to oppose a policy change in the future. Persson and Svensson (1989), Alesina and Tabellini (1990), Krusell and

Rios-Rull (1999), Azzimonti (2005), and Battaglini and Coate (2007, 2008) build dynamic political economy models of public finance in which the policy maker/voters in a given period distorts her decision in order to affect the incentives of future policy makers/voters. In contrast, in this model, players’ actions do not affect future preferences, and the dynamic linkage comes solely from the endogeneity of the status quo.

Besley and Coate (1998), Acemoglu and Robinson (2001), Cho (2005), Fong (2006), Acemoglu,

Egorov and Sonin (2008, 2012, 2013), Bai and Lagunoﬀ (2011), Baron, Diermeier and Fong (2012),

5 and Duggan and Forand (2013) analyze dynamic games in which the policy implemented in a given period affects the allocation of decision power in the next period. They show that this dynamic linkage can lead to ineffi cient decisions. In particular, ineffi cient status quo can persist because the winners from an effi cient reform cannot commit to use their future political power to compensate the losers. In this paper, the allocation of decision power is assumed to be exogenous so as to isolate the effect of the endogenous status quo on the evolution of policies.

Casella (2005) and Jackson and Sonnenschein (2007) show that linking voting decisions across time allows voters to express their preference intensity, which can be socially beneficial. Our results suggest that the endogenous status quo protocol, despite the pervasiveness of this institution, is not an effi cient way to elicit preference intensity. Barbera and Jackson (2010) let ex-ante identical voters choose the group decision rule after having learned their first-period preferences. Similarly to our paper, in their framework bundling the current and future decision rules generates ineffi ciencies.

But since the dynamic linkage is only between the ﬁrst period and the subsequent ones, suﬃ ciently patient players always select the optimal voting rule.

2.2 Bargaining in a static environment with an endogenous status quo

Our paper complements the literature on dynamic bargaining with an endogenous status quo. Al- though most of this literature considers environments in which preferences are static, it is instructive to understand the relation with this paper. In these models, the bargaining outcome changes over time because the identity of the proposer changes, or because the same proposer seeks the support of a different coalition. In our basic model with two alternatives and unanimity rule, these two channels do not play any role. Instead, the impetus for policy change comes solely from the evolution of preferences. In environments with common interest policies, the literature with static preferences (Baron, 1996; Baron and Herron, 2003; and Zapal, 2011a) finds that the endogenous status quo has as a moderating effect: players vote for policies that are more moderate than their instantaneous preferences would imply. In contrast, this paper shows that in environments with evolving preferences, the endogenous status quo has a polarizing effect. Hence, the endogenous status quo has a diametrically different impact on equilibrium behavior in these two environments.

The moderating eﬀect highlighted by the aforementioned literature is similar to the one that occurs in this model in the case of more than two alternatives. This eﬀect is closely related to the

6 observation initially made by Romer and Rosenthal (1978) that an extreme status quo increases the leverage of the proposer, and is thus detrimental to non-proposers. Therefore, under the endogenous status quo protocol, implementing a moderate agreement today can constrain the bargaining power of the next proposer. Diermeier and Fong (2011) show that this effect can limit the bargaining power of a monopolistic proposer, and Bowen et al. (2014a) show that it can mitigate the dynamic ineffi ciency due to a change of proposer. We contribute to that literature by showing that in a stochastic environment, under reasonable conditions, the moderation effect is dominated by the polarization effect.

Since Kalandrakis (2004), a large literature has looked at the case of purely distributive policies.3

These papers focus on the equitability of the division of the pie, on whether proposals are approved by minimal coalitions, and on whether these coalitions are stable. By virtue of analyzing common interest policies under the unanimity rule, our paper has nothing to say on these issues. Instead, we focus on the responsiveness of policies to the changing environment, and on the phenomenon of partisanship, on which the aforementioned papers are mute, because in the distributive setting, players’preferences do not change over time and are always diametrically opposed.

2.3 Bargaining in an evolving environment with an endogenous status quo

Even though dynamic bargaining with an endogenous status quo in a stochastic environment is at the center of many economically relevant situations, the existing literature on this topic is scarce.

This may be a consequence of the intractability of these games. As Romer and Rosenthal (1978) showed in a static setup with single-peaked preferences, the induced preferences over the status quo are typically not convex, which makes the multi-period extension technically hard to analyze.

To the best of our knowledge, only Riboni and Ruge-Murcia (2008), Zapal (2011b), Duggan and

Kalandrakis (2012), and Bowen et al. (2014b) make progress on this front. Riboni and Ruge-Murcia

(2008) were the ﬁrst to analyze the endogenous status quo in a stochastic environment. They solve a two-period, two-state example with quadratic preferences, but use numerical solutions for the general model. Adding noise to the status quo, Duggan and Kalandrakis (2012) establish the existence of an equilibrium in a very general setting. We consider instead a more stylized environment

3 See, among others, Kalandrakis (2010), Bernheim et al (2006), Anesi (2010), Diermeier and Fong (2011), Bowen and Zahran (2012), Baron and Bowen (2013), Richter (2013), and Anesi and Seidmann (2014).

7 and characterize the equilibria for a large class of stochastic processes.4 Zapal (2011b) characterizes an equilibrium in a two-state environment with quadratic preferences and compares it to a different dynamic bargaining protocol. In his setting, a constant policy is optimal; hence, questions of the responsiveness of policies to shocks are mute. Our paper differs from these contributions in that our equilibrium characterization allows us to isolate the effect of the endogenous status quo in a transparent way across all equilibria. Bowen et al. (2014b) show that effi ciency is restored when the endogenous status quo can be contingent on future payoffs.

Montagnes (2010) looks at a two-period financial contracting environment in which the current contract serves as the default option in future negotiations. He shows that ineffi ciencies may arise, and hence both contracting parties may prefer to commit ex ante to ceding a future decision power to avoid these ineffi ciencies.

3 The 2 alternative model − 3.1 The model

Two players, l and r, are in a relationship that lasts for inﬁnitely many periods. There are two alternatives, X = L, R . In each period t, a status quo q (t) L, R is in place, and players vote { } ∈ { } simultaneously on which alternative to adopt. If both players vote for the same alternative, this alternative is implemented. If they disagree, this period’s status quo q (t) is implemented. The implemented alternative x (t), be it the new agreement or the status quo, determines the players’ payoﬀ in t and becomes the status quo for the next period t + 1.

The payoﬀ of player k l, r in period t from alternative x X is denoted by Uk (θ (t) , x) , ∈ { } ∈ where θ (t) denotes the state of nature. To capture the dynamic nature of the environment, we assume that θ (t) follows a stationary Markov process. That is, the state θ (t) depends on the past only through last period’s state θ (t 1), and its distribution is independent of t. Note that this − assumption does not rule out deterministic processes. For all θ Θ, Pθ denotes the probability ∈ distribution of θ (t + 1) conditional on θ (t) = θ. The state θ (t) is observed in t by both players before they cast their votes. We assume that Uk (θ, x) is bounded over Θ and X, and for all x X ∈ 4 Unlike in Duggan and Kalandrakis (2012), in our model the status quo is a deterministic function of players’ decisions, but due to other simpliﬁcations we make, equilibrium existence is not an issue.

8 and all θ Θ, θ0 Uk θ0, x is Pθ measurable. Players maximize the expected discounted sum of ∈ → payoffs, and δ (0, 1) is the discount factor. ∈ We denote the above game by Γen. In this game, the dynamic linkage across periods comes solely from the continuing nature of the policies. To characterize the impact of the endogenous status quo on the equilibrium behavior and outcomes, we will compare Γen to the game Γex, which differs from Γen only in that in every period t, the status quo is exogenously fixed at some q (t) X, ∈ irrespective of players’past actions.

As is customary in dynamic voting games with an inﬁnite horizon, we look for Markov perfect equilibria in stage-undominated strategies (henceforth, equilibria) as deﬁned in Baron and Kalai

5 ex en (1993). A Markov strategy for player k in Γ or Γ , denoted by σk, maps in each period t the current state θ (t) and status quo q (t) into a probability distribution over votes. Stage undomination amounts to assuming that in each period, each player votes for the alternative that gives her the greater continuation payoﬀ. It rules out pathological equilibria such as both players always voting for the status quo. When indiﬀerent, we assume that players vote for R.

Additional assumptions and notations

Definition 1 A payoff function U :Θ X R is more leftist than another payoff function U 0 (or × → equivalently, U is more rightist than U) if for all θ Θ, 0 ∈

U (θ, R) U (θ, L) U 0 (θ, R) U 0 (θ, L) . − ≤ −

Section 3, we assume that Ur is more rightist than Ul, and we occasionally refer to player r and l as the rightist and the leftist player, respectively. This assumption has a natural interpretation in political economy applications: players can be unambiguously ranked on the ideological spectrum.

Note, however, that it imposes no restrictions on the preference distribution of a single player, nor on the severity of the conﬂict of interest between players: both players might prefer the same

5 Stage-undominated Markov perfect equilibria, or variants thereof, are used in almost all of the infinite-horizon models cited in this paper. The only exceptions that we are aware of are Epple and Riordan (1987), and Baron and Ferejohn (1989). Both papers prove results that have the flavor of the folk theorem in repeated games. As shown in Baron and Kalai (1993), stage-undominated Markov perfect equilibria have a focal point property that derives from their simplicity. Markov perfection in the legislative sphere can be justified on the grounds that the game is played by a sequence of legislators who are never certain to be reelected. In such cases, the institutional memory required for more sophisticated nonstationary equilibria involving infinitely nested punishment strategies may be inappropriate. See Baskhar et al (2013) for a formalization of this argument, and see Krehbiel (1991) for a critical discussion of the prevalence of folk-theorem like cooperation among legislators.

9 agreement for an arbitrarily large subset of Θ.

For all θ Θ and all x X, deﬁne Uk (θ) Uk (θ, R) Uk (θ, L) . The expression Uk (θ) measures ∈ ∈ $ − the relative gain in expected payoﬀ that player k receives from alternative R as compared to L in a period in which the state is θ. We call Uk (θ) player k0s current preference, as the sign of Uk (θ) determines her preference for the current period without taking into account the consequences of today’sbargaining outcome on future periods.

To illustrate our results, sometimes we will use the following family of payoff profiles in which each player has single-peaked preferences with a peak equal to the state plus a player-specific shift.

This shift can be interpreted as her ideological bias.

Definition 2 When Θ R and X R, a profile of payoff functions (Ul,Ur) is state-dependent ⊆ ⊂ 2 single-peaked if for all k l, r , θ Θ, and x X, Uk (θ, x) = (x (θ + bk)) for some ∈ { } ∈ ∈ − − 6 bl, br R such that bl < br. ∈

Comments

A few comments about the model are in order. First, we analyze a two-player game which requires unanimity for changing the status quo, but in Dziuda and Loeper (2014), we show that our results easily extend to an arbitrary number of players with a large class of voting rules. For example, with a qualiﬁed majority rule such as the ﬁlibuster rule in the U.S. Senate, if players can be ranked from a most leftist to a most rightist, the same two players are pivotal in every decision, so the situation is strategically equivalent to the two-player model analyzed in this paper.

Even when simple majority rule is used within each decision body, bicameralism, judicial review by a constitutional court, presidential veto power, or public initiatives imply the existence of two distinct pivotal voters.

Second, we assume that in each period the outcome is decided by a simultaneous vote, but with two alternatives and two players, most bargaining protocols are equivalent.7 This assumption will be relaxed when we allow for more alternatives. 6 The careful reader will note that when Θ is unbounded, this family of payoff profiles violates the assumption that U is bounded over Θ. However, the results stated for this class of payoff profiles are derived without using this assumption. 7 For instance, using standard equilibrium concepts, equilibrium outcomes are the same when players vote simultaneously, sequentially, when they make take-it-or-leave-it offers, or when they make several alternating offers at each bargaining time.

10 Third, to assess the role of the endogenous status quo, we compare Γen to the game with the exogenous status quo protocol Γex. The exogenous status quo protocol is the simplest dynamic bargaining protocol which severs the link between today’s agreement and tomorrow’s status quo.

Hence, this benchmark protocol allows us to show in a transparent way that it is the continuing nature of policies that leads to polarization. The comparison between Γen and Γex is also of an independent interest, as the exogenous status quo protocol is probably the most commonly observed alternative to the endogenous status quo in actual bargaining environments. For instance, about one-third of the U.S. federal budget– called discretionary spending– requires annual appropriation bills. This means that for all such programs, the status quo is exogenously ﬁxed at 0.

Finally, in this model, today’sbargaining outcome affects the future only via tomorrow’sstatus quo. In a more general model, it could also affect tomorrow’s states θ. We abstract from this dependence to isolate the effect of the endogenous status quo in a transparent way. In Section 5, we discuss how making the evolution of the state dependent on the implemented policies would affect our results.

3.2 Equilibrium analysis

As a benchmark, we ﬁrst look at the game with an exogenous status quo Γex. Consider player k in period t in which the state is θ (t) = θ. Since the bargaining outcome implemented in t has no impact on the subgame starting in the next period t + 1, player k votes for R if and only if

Uk (θ) 0. (1) ≥

Hence, Γex has a unique equilibrium in which players simply vote for the best alternative according to their current preferences.

Consider now the same situation in the game with an endogenous status quo Γen. The agreement x implemented in t aﬀects player k’spayoﬀ in t, and since x becomes the status quo in t + 1, it also

σ aﬀects her continuation value from the subgame that starts in t + 1. Let Wk (θ, x) be the expected value of that continuation game, conditional on θ (t) = θ and on strategy proﬁle σ = (σl, σr) being played after t. Let W σ (θ) W σ (θ, R) W σ (θ, L). One can interpret W σ as the relative expected k $ k − k k gain for player k from having R instead of L as the next period’s status quo. With a slight abuse

11 σ of notation, we call Wk the continuation value of player k. When no ambiguity arises, we drop the superscript σ.

Stage-undomination requires that in each period, players vote as if they were pivotal. Hence, in Γen, if the current state is θ, player k votes for R if and only if

ρδ Uk (θ) + e− Wk (θ) 0. (2) ≥

Comparing (1) and (2), one can see that the eﬀect of the endogenous status quo on equilibrium behavior is completely captured by Wk. If Wk (θ) = 0 for all θ, then player k votes according to her

ex current preference as in Γ . If Wk (θ) is positive (negative), then player k votes for R more (less) often in Γen than in Γex.

The ﬁrst proposition shows that the endogenous status quo has a polarizing eﬀect on players’ behavior: it makes the leftist player vote for L and the rightist player vote for R more often than in Γex.

en Proposition 1 The game Γ has an equilibrium. In all equilibria, Wl (θ) 0 Wr (θ). ≤ ≤

Proof. All proofs are in the Appendix.

The intuition for Proposition 1 is as follows. The alternative implemented today aﬀects future bargaining outcomes only if players disagree in the next period. Since r is relatively more rightist than l, when players disagree, r prefers R while l prefers L. Hence, status quo L gives more option value to player l than status quo R. As a result, player l is willing to vote for L to secure L as the next period’s status quo even when alternative R would give her a greater payoﬀ in the current period.

The intuition behind Proposition 1 reveals that the direction in which the endogenous status quo biases players’behavior depends only on players’relative ideological positions. It depends on which alternative each player prefers when players disagree but not on which alternative they prefer on average.

To understand the implications of Proposition 1 for the equilibrium outcomes, note that under our assumptions, for all θ Θ,Ul (θ) Ur (θ) . Therefore, from (1), when the status quo is ∈ ≤

12 exogenous, players do not reach an agreement when

Ul (θ) < 0 Ur (θ) , ≤ while from (2), when the status quo is endogenous, players do not reach an agreement when

Ul (θ) + δWl (θ) < 0 Ur (θ) + δWr (θ) . ≤

These inequalities together with Proposition 1 imply the following.

Corollary 1 (Status Quo Inertia) In any equilibrium of Γen, the set of states in which players do not reach an agreement is greater in the inclusion sense than in the equilibrium of Γex. In particular, in any period t, the probability that the status quo stays in place in t + 1 is higher in Γen than in Γex.

The behavior of the players described in Proposition 1 and Corollary 1 resembles what is commonly referred to as partisanship. One dictionary definition for partisanship is “a prejudice in favor of a particular cause; a bias.”In multiparty systems, this term carries a negative connotation: it refers to those who wholly support their party’s policies and are reluctant to acknowledge any common ground with their political opponents. This definition resonates with the players’behavior in our model: players favor policies that are in line with their relative ideology rather than policies that are optimal given their current preferences, which leads to more disagreement and ineffi cient policies. This model hence shows that when the status quo is endogenous, partisanship can be generated by strategic considerations without assuming any obedience to a doctrine or a group.

Proposition 1 and Corollary 1 do not characterize the conditions under which the endogenous status quo strictly increases the probability that players fail to reach an agreement. However, for the equilibrium of Γex to be also an equilibrium of Γen, the process θ (t): t 0 must satisfy quite { ≥ } restrictive conditions. To see this, consider a sequence of states in which both players’ current preferences favor R, but l0s preferences for R are vanishingly weak. Then if the probability that players’ preferences disagree in the next period is not vanishingly small in the limit (e.g., if the state is independently and identically drawn in every period), l will ﬁnd it proﬁtable to vote for

L instead of R. In section 3.4, we analyze a class of environments in which such states arise with

13 positive probability in every period, and thus status quo inertia is strictly greater in all periods in

Γen than in Γex.

To fully understand the strategic underpinning of partisanship in our model, it is important to note that the equilibrium degree of partisanship is the result of a vicious cycle in which players’ behaviors feed on themselves. To see this, observe that as player l becomes more partisan for L

(i.e., as she votes for L for a larger set of θ), players are less likely to reach an agreement in each period, which in turns makes player r more willing to defend her preferred status quo R, and thus makes her more partisan for R.

Because of this strategic complementarity, there can be multiple equilibria. Proposition 2 below shows that the equilibria can be ranked in terms of their degree of partisanship, and the more partisan players are, the worse-oﬀ they are.

Proposition 2 The set of equilibria is a complete lattice in the partisan order deﬁned as follows: for any two strategy proﬁles σ and σ , σ is more partisan than σ if for all θ Θ,W σ0 (θ) 0 0 ∈ l ≤ W σ (θ) 0 W σ (θ) W σ0 (θ) . Moreover, if σ and σ are two equilibria such that σ is more l ≤ ≤ r ≤ r 0 0 partisan than σ, then σ Pareto dominates σ0.

To conclude this section, let us note that in this simple model, the ineffi ciency generated by the endogenous status quo arises only if preferences evolve over time. To see this, observe that if players’ preferences were fixed, then in Γen, each player would vote according to her current preferences. The status quo could be replaced only in the first period, and the constant bargaining outcome would be trivially Pareto optimal. Hence, it is the combination of the endogenous status quo and the evolving environment that generates partisanship and status quo inertia.

3.3 The determinants of partisanship and gridlock eﬀect

In this section, we investigate the main drivers of the magnitude of partisanship.

Since partisanship is driven by the anticipation of future disagreement, it should be affected by the degree of players’conflict of interest and by their patience. Players whose current preferences are more likely to disagree, and who care more about the future should be more willing to sacrifice today’spayoff to obtain their preferred status quo for tomorrow’snegotiations. The next proposition formalizes this intuition.

14 Proposition 3 Let (Ul,Ur) and (Ul0,Ur0 ) be two profiles of payoff functions such that Ul0 is more leftist than Ul and Ur0 is more rightist than Ul (in the sense of Definition 1), and let δ and δ0 be two discount factors such that δ0 > δ. If σ and σ0 denote the Pareto best equilibria for the parameters

Ul,Ur, and δ, and for Ul0,Ur0 and δ0, respectively, then σ0 is more partisan than σ (in the sense of Proposition 2). The same comparative statics hold for the Pareto worst equilibrium.

Since partisanship implies status quo inertia, Proposition 3 implies that more patient and more ideologically polarized players are less likely to agree on Pareto improving policies after a shock to the environment has made the status quo suboptimal.

The next proposition shows that the impact of the endogenous status quo on equilibrium behavior can be quite dramatic: under some conditions, there exists a gridlock equilibrium in which players always vote for opposite alternatives, and hence the bargaining outcome is totally unresponsive to the evolution of the environment.

Proposition 4 Let σg be the strategy proﬁle in which player l always votes for L and player r always votes for R. Then σg is an equilibrium if and only if for any initial state θ (0) Θ, ∈

∞ t ∞ t E δ Ul (θ (t)) < 0 E δ Ur (θ (t)) . (3) θ(0) ≤ θ(0) "t=0 # "t=0 # X X

Condition (3) means that player r0s (l0s) expected payoff from implementing R in every period is higher (strictly lower) than her expected payoff from implementing L in every period for any initial state. In the appendix, we show that this condition is satisfied if and only if players are suffi ciently patient and their current preferences are suffi ciently polarized (see Proposition 10). Gridlock, however, can occur for arbitrarily mild conflict of interests. For example, if the profile of payoff functions (Ul,Ur) is state-dependent single-peaked (see Definition 2), if θ (t): t 0 is a regular { ≥ } Markov chain on a finite Θ, and if ¯θ denotes the mean of its stationary distribution, then Condition L+R ¯ (3) is satisfied whenever players’ideological biases bl and br are such that bl < θ < br and δ 2 − is suffi ciently large (see Proposition 11 for a formal proof). Hence, as bl and br become arbitrarily close to L+R ¯θ, players’current preferences agree arbitrarily often in all periods. Yet, Proposition 2 − 4 implies that if players are suffi ciently patient, there exists an equilibrium in which they always vote for opposite alternatives.

15 3.4 An illustration with single-peaked preferences

In this section, we illustrate our results in the following simple environment: the profile of payoff functions U is state-dependent single-peaked (see Definition 2) with bl < br, Θ = R, and the process

θ (t): t N is a random walk: for all t N, θ (t + 1) = θ (t) + ν (t), where the random variables { ∈ } ∈ (ν (t)) are i.i.d. We further assume that each ν (t) admits a probability density function f that t N ∈ is single-peaked with a peak at 0. We denote the games Γen and Γex by Γen (X, b) and Γex (X, b) , respectively, to emphasize their dependence on the set of alternatives X = L, R and on players’ { } ideological biases b = (bl, br). In this environment, condition (1) can be rewritten as follows: in Γex (X, b), player k votes for

L+R L+R R if and only if θ bk. Since bl < br, this implies that L is implemented when θ < br, ≥ 2 − 2 − L+R R is implemented when θ bl, and the status quo stays in place when ≥ 2 −

L + R L + R θ br, bl . (4) ∈ 2 − 2 −

The following proposition shows that in this environment the impact of the endogenous status quo on the equilibrium behavior takes a particularly simple form.

ex Proposition 5 There exist pl < 0 and pr > 0 such that for all X, the equilibrium of Γ (X, b + p)

en is also an equilibrium of Γ (X, b) . Moreover, for any b such that b b = br bl, the equilibrium 0 r0 − l0 − ex en of Γ (X, b0 + p) is also an equilibrium of Γ (X, b0) .

Proposition 5 states that there exists an equilibrium of Γen (X, b) in which players behave as if the status quo were exogenous but their ideological biases were b + p instead of b. Since p depends on b only through br bl, Proposition 5 implies that the impact of the endogenous status − on equilibrium behavior depends only on players’ ideological diﬀerences, and since pl < 0 and pr > 0, the endogenous status quo magniﬁes them. Hence, pk can be interpreted as the degree | | of partisanship of player k. Note that Proposition 5 also implies that in this environment, players’ degree of partisanship does not depend on X; in particular it does not depend on how close the two alternatives L and R are in the policy space.

In the equilibrium of Γen (X, b) characterized in Proposition 5, the status quo stays in place

L+R L+R when θ br pr, bl pl . By comparing this interval with (4), we see that the ∈ 2 − − 2 − − 16 inertial eﬀect of the endogenous status quo is driven by players’degree of partisanship pl and pr . | | | | In particular, when ν (t) has full support, status quo inertia is strictly greater in Γen (X, b) than in

Γex (X, b) in all periods with positive probability.

Figure 1 below represents the equilibrium of Γex (X, b) (dashed lines) and of Γen (X, b) (solid lines). The horizontal lines represent the set of states in which a given status quo stays in place, and the arrows represent the cutoﬀ states at which the policy switches.

Figure 1: The evolution of the policy.

For ν (t) N (0, 0.1) and br bl = 1, we calculate pr numerically (by symmetry, pl = pr). ∼ − − Figure 2 shows pr as a function of patience. As we see, as δ 1, partisanship pr converges → | | to br bl = 1. Hence, when players are very patient, their equilibrium behavior is three times − more polarized in Γen (X, b) than in Γex (X, b). For instance, if θ (t) is in the middle of the static disagreement region (4), then the probability that players agree in the next period in Γex (X, b) is

0.31, while in Γen (X, b) it is only 0.07.8

Figure 2: Partisanship of player r

8 All simulation results are uvailable from the authors upon request.

17 4 The N alternative model − In the 2 alternative model analyzed in Section 3, the implemented alternative is selected by a − simultaneous vote, but any other reasonable within-period bargaining procedure would lead to the same results. With more than two alternatives, the bargaining outcome is no longer independent of the bargaining procedure used within each period. Moreover, characterizing all equilibria in stochastic bargaining games with an endogenous status quo is a task that is known to be elusive, even for a particular within-period bargaining procedure (see, e.g., Riboni and Ruge-Murcia 2008, or Bowen et al 2014b), mainly because of the richness of the strategic eﬀects.9 Nevertheless, in this section, we show that under reasonable assumptions, the partisanship and status quo inertia highlighted in the 2 alternative model are still the dominant equilibrium eﬀects. −

4.1 Polarization and moderation: Intuition

With an arbitrary number of alternatives, it is still true that the status quo matters only when players disagree, so players’equilibrium behavior still depends on which policies they prefer when they disagree. However, players’preferences conditional on disagreement are harder to determine, because there is more than one way to disagree. As in the 2 alternative model, players might − disagree about which policies are better than the status quo. For example, in the context of legislative bargaining, the conservatives may want to decrease public spending while the liberals may want to increase it, and no common ground can be found. In such disagreement states, the status quo stays in place irrespective of how the bargaining procedure allocates the bargaining power. Clearly, the anticipation of such disagreement biases the conservatives in favor of smaller and the liberals in favor of bigger budgets. This is the same polarizing eﬀect as in the case of two alternatives.

However, with more than two alternatives, players may also agree that a certain subset of policies is better than the status quo, but disagree about the ranking of these alternatives. Such disagreements can generate incentives that are qualitatively different from the ones generated by the polarizing effect. To see this, consider again the case of legislators bargaining over fiscal policies,

9 For instance, in the redistributive setting with majoritarian bargaining, the qualitative impact of the endogenous status quo has been found to be quite sensitive to the parameters of the models (e.g., risk aversion, time horizon) and the choice of the equilibrium. Compare for instance Bernheim et al. (2006) and Diemeier and Fong (2011), or Kalandrakis (2004), Bowen and Zahran (2012), Richter (2014), and Anesi and Seidman (2014).

18 and suppose that a shock severely contracts the economy. In that case, all legislators may want to expand the budget to stimulate growth, but the liberals will want to expand it more. If the conservatives have all bargaining power, they can increase the budget size at will, as all increases favored by them will be better for the liberals than the status quo. However, as ﬁrst pointed out by

Romer and Rosenthal (1978), if the liberals have all bargaining power, how much they can leverage it depends on how inadequate the current status quo is. If the status quo budget is excessively small given the economic conditions, they can propose a bigger expansion than the conservatives would like, and the conservatives will have no choice but to accept it. Anticipating this scenario in normal times, the conservatives may prefer a level of public spending that is larger than their bliss point, so as to be able to constrain the liberals with the threat of a veto in a case of a severe downturn. Hence, this equilibrium eﬀect tends to bias the conservatives in favor of more liberal policies, and vice versa. Thus, the endogenous status quo can also have a moderating eﬀect.

Note that these two effects have opposite implications in terms of the dynamics of the bargaining outcome. Polarization means that players disagree more often than their current preferences do, and thus leads to status quo inertia. In contrast, the moderation effect biases players’ behavior in favor of alternatives that are typically favored by the opponent. Therefore, if this effect is suffi ciently strong, the liberals (conservatives) may agree to replace the status quo with a more conservative (liberal) policy before their current preferences favor such a change, and thus before the status quo becomes Pareto dominated. That is, moderation may lead to status quo instability

The above discussion suggests two conclusions. First, the polarization effect occurs irrespective of the within-period bargaining procedure. The moderation effect, however, occurs only if the within-period bargaining procedure favors the more drastic policy change whenever players agree on the direction of the change but not its magnitude. Second, to generate the type of disagreement that leads to moderation, the preferences must change drastically from one period to the next: players must agree on a quite conservative policy at some point in time, but prefer suffi ciently liberal policies when the next chance to renegotiate arises.

In the next two sections, we formalize these observations. In Section 4.2, we consider a particular within-period bargaining procedure that favors moderate policy changes. This procedure has been shown to be empirically relevant by Riboni and Ruge-Murcia (2010). We show that the results

19 from Section 3 extend to this environment.10 In Section 4.3, we allow for a general class of within- period bargaining procedures, but consider instead an environment in which preferences change continuously over time. We show that when players can revise the policy agreement as soon as the environment prompts it, the polarizing eﬀect prevails, and status quo inertia occurs in equilibrium. We believe that the assumption that players are not restricted to bargain at ﬁxed times is empirically relevant because in actual bargaining situations, we rarely observe sharp constraints on when bargaining parties can propose changes to the status quo. In volatile times, the negotiating parties typically do not wait for the next scheduled meeting to revise the previous agreement.11

4.2 Bargaining with the consensus procedure

The set of alternatives X = x1, x2, ..., xN is a subset of R, with x1 < ... < xN . As in the { }

2 alternative model, we assume that the proﬁle of payoﬀ functions (Uk (θ, x))k l,r ,θ Θ,x X is − ∈{ } ∈ ∈ bounded, and that player r has more rightist preferences than player l:

Assumption 1 For all θ Θ and all x, y X such that x < y, Ul (θ, y) Ul (θ, x) Ur (θ, y) ∈ ∈ − ≤ − Ur (θ, x) .

In each period, the bargaining outcome is decided using the consensus procedure introduced by

Riboni and Ruge-Murcia (2010). Hence, players play the following stochastic game. Let θ and xn denote the state and status quo in a given period. In the first stage of that period, the proposer p l, r (whose identity can depend on the state θ) can play left or right. If she plays right ∈ { } (left), then in the subsequent stages of that period, players vote via unanimity rule on how far to the right (left) they should incrementally move the status quo. Formally, the first temporary agreement is the status quo xn, and players vote on whether to move the temporary agreement to xn+1 (xn 1). If one of them votes no, then the game stops. If both vote yes, then xn+1 (xn 1) − − 10 An effect similar to the moderation effect arises in a related model by Acemoglu, Egorov, and Sonin (2013). They rule it out by assuming that in each period the status quo can be replaced only by one of the adjacent alternatives or that the set of quasi-median voters in different states has at most one player in common. Neither of these assumptions seem appropriate in our setting. 11 Suppose for instance that a legislature implements a fiscal stimulus after an economic contraction, but quickly thereafter, the markets lose faith in the ability of the government to repay its debts. In that scenario, the legislators are unlikely to wait for the next ordinary legislative session to start negotiating over how to react to the new situation. Similarly, since 1981 the Federal Open Market Committee have held eight regularly scheduled meetings each year at intervals of five to eight weeks. If circumstances require, however, members may be called on to participate in a special meeting. For example, during the financial crisis the committee met eleven times in 2007 and fourteen in 2008. Source http://www.federalreserve.gov/monetarypolicy/fomchistorical2008.htm

20 becomes the temporary agreement for the next stage. In the next stage, players vote on whether to move the temporary agreement to xn+2 (xn 2), and so on. Bargaining continues until one player − plays no, or until the temporary agreement is xN (x1). The last temporary agreement x becomes the bargaining outcome for that period, players obtain payoﬀs (Uk (θ, x))k l,r , and the game ∈{ } moves to the next period. The next period starts with the status quo x and a new state θ0 drawn

en from Pθ. The discounting δ (0, 1) occurs only between periods. We call this game Γ . ∈ c ex en Let Γc denote the game which differs from Γc in that the status quo in each period t N ∈ is exogenously fixed at some q (t). As in the 2 alternative model, the effect of the endogenous − status quo on equilibrium behavior is captured by the shape of the continuous value function

σ q W (θ, q) . To see this, suppose that the status quo is xn. Under the exogenous status quo, → k player k prefers to replace xn with xn+1 in a given period only if Uk (θ, xn+1) Uk (θ, xn), while ≥ under the endogenous status quo, she may do so only if

σ σ Uk (θ, xn+1) + δW (θ, xn+1) Uk (θ, xn) + δW (θ, xn) . k ≥ k

σ Hence, if Wk (θ, x) is increasing (decreasing) in x, player k is in favor of replacing xn with xn+1 more (less) often than under the exogenous status quo. The following deﬁnition extends the notion of partisanship to this environment.

Deﬁnition 3 A Markov strategy proﬁle σ of Γen is partisan if for all θ Θ, W σ (θ, q) is weakly c ∈ r σ 12 increasing in q and Wl (θ, q) is weakly decreasing in q.

In most bargaining environments, the rules describing who and when can make proposals are not binding or not explicitly speciﬁed. So it is hard to tell whether the consensus procedure is a better model of a given decision making body than other bargaining games. Nevertheless, Riboni and Ruge-Murcia (2010) show empirically that it describes the decisions of monetary committees better than other standard procedures.

12 Note that in the case of X = 2, Section 3 defines a partisan strategy profile as a strategy profile σ such that σ | |σ σ σ σ for all θ Θ, Wr (θ) 0 and Wl (θ) 0. Since W (θ) = W (θ, R) W (θ, L) , this definition is consistent with Definition∈ 3. ≥ ≤ −

21 4.2.1 The general result

Independent of its empirical relevance, the consensus procedure is of particular interest for our purpose because it favors more moderate policy changes. To see this, note that if in some period, the status quo q is replaced by some x > q, then each player can unilaterally impose a more moderate policy change x [q, x], while deviating to a more radical change x > x requires the approval of 0 ∈ 00 both players. As argued in Section 4.1, this property precludes the kind of disagreement that leads to the moderation eﬀect, and thus players’equilibrium behavior should be partisan. Proposition 6 conﬁrms that intuition.

Proposition 6 If Θ is ﬁnite, there exists a partisan equilibrium.

We cannot ascertain that all equilibria are partisan, because if the proposer is indifferent between choosing left and right for some status quos, she can condition her equilibrium behavior on the status quo in arbitrary ways. This degree of freedom can be used to distort the other player’s induced preferences over the status quo, which complicates the analysis.13 Below we define a large class of environments in which such indifferences occur with negligible probability, and show that in these environments all equilibria are partisan.

Assumption 2 In all periods t, the state can be decomposed into two coordinates θ (t) = (θ1 (t) , θ2 (t)) such that

(i) conditionally on θ1 (t) and θ1 (t + 1) , (Uk (θ1 (t + 1) , θ2 (t + 1) , x))k l,r ,x X has an absolutely ∈{ } ∈ l,r X continuous distribution on R{ }× ;

(ii) conditionally on θ1 (t) , θ1 (t + 1) and θ2 (t + 1) are independent of θ2 (t) .

Roughly speaking, Assumption 2 requires that players’payoﬀs depend on a stochastic parameter

θ1 (t) that is arbitrarily correlated across time, and on a shock θ2 (t) that smoothly perturbs the payoffs (condition (i)) but does not affect the distribution of future payoffs (condition (ii)). Note

13 To see this, consider the following one-period environment: X = 1, 0, 1 , player r is the proposer and is indiﬀerent among all three alternatives, while l strictly prefers 1 to 0 to{−1. In} this environment it is optimal for r to play right when the initial status quo q is 0, and left when−q is 1 or 1, and it is optimal for both players to vote no after proposal right and yes after proposal left. In this case, q−= 0 leads to outcome 0 while q = 1 leads to outcome 1, which is less preferred by l than 0. Hence, the induced preferences of player l over the status quo are not partisan:− she strictly prefers q = 1 to q = 0.

22 that conditional on θ1 (t) and θ1 (t + 1) , the distribution of (Uk (θ1 (t + 1) , θ2 (t + 1) , x))k l,r ,x X ∈{ } ∈ can have an arbitrarily small support. So the environment described in Assumption 2 can be viewed as an inﬁnitesimal perturbation of the environment in which the (arbitrary) Markov process

θ1 (t): t 0 is the only payoﬀ relevant state. Alternatively, if θ1 (t): t 0 is deterministic, { ≥ } { ≥ } Assumption 2 encompasses the case in which players’payoﬀs are independently distributed over time.

Proposition 7 If Assumptions 1 and 2 are satisﬁed, then all equilibria are partisan. If furthermore

Θ1 is ﬁnite, an equilibrium exists.

4.2.2 An illustration with single-peaked preferences

Propositions 6 and 7 show that players’equilibrium behavior exhibits partisanship, but are silent on whether the moderating effect makes the partisanship weaker than in the 2 alternative model. − en In this section, we show that in the environment considered in Section 3.4, partisanship in Γc (X, b) is exactly the same as in the 2 alternative model. This confirms the intuition that by favoring − moderate policy changes, the consensus procedure precludes any moderation effect.

en ex In line with the notations introduced in Section 3.4, Γc (X, b) and Γc (X, b) refer to the games en ex Γc and Γc , respectively, when the set of alternatives is X, players’ payoﬀ functions are state- dependent single-peaked with ideological biases b = (bl, br), and θ (t): t 0 is a random walk as { ≥ } described in Section 3.4.

ex It is instructive to ﬁrst describe players’behavior in the equilibrium of the game Γc (X, b) . As shown in Riboni and Ruge-Murcia (2010), in a given period with state θ and status quo xn, if the peaks of both players are on the same side of the status quo, then the bargaining outcome is the peak that is closer to the status quo. If their peaks are on the opposite sides of the status quo xn, that is, when xn + xn 1 xn+1 + xn θ − br, bl , (5) ∈ 2 − 2 − the status quo xn stays in place. The following proposition characterizes the equilibrium under the endogenous status quo.

Proposition 8 There exist pl < 0 and pr > 0 such that for all ﬁnite X R, the equilibrium of ⊂ ex en Γc (X, b + p) is an equilibrium of Γc (X, b).

23 Note that pl and pr are the same for all X, and are thus the same as those identiﬁed in Propo- sition 5. This means, perhaps somewhat surprisingly, that adding policy compromises between

L and R does not mitigate the polarizing eﬀect of the endogenous status quo identiﬁed in the

2 alternative model. From (5), in the equilibrium of Γen (X, b) characterized in Proposition 8, the − c xn+xn 1 xn+1+xn status quo xn stays in place when θ − br pr, bl pl . By comparing this ∈ 2 − − 2 − − interval with (5), we see that the partisanshiph induced by the endogenous status quo generates the same status quo inertia as in the 2 alternative model. − en ex Figure 3 depicts the equilibrium in Γc (solid lines) and Γc (dashed lines) using the same conventions as in Figure 1 in Section 3.4. Comparing Figure 3 to Figure 1, one can see how the evolution of the policy changes when a third alternative M (L, R) becomes available. ∈

Figure 3: The evolution of the policy

4.3 Bargaining in a continuously changing environment

In this section, we formalize the intuition laid out in Section 4.1 that when preferences evolve continuously and players can bargain as soon as the status quo becomes obsolete, the moderation eﬀect becomes negligible, and thus status quo inertia prevails.

The set of alternatives X is finite. Time is continuous, but players bargain only every ∆ > 0 units of time. As before, at each bargaining time t 0, ∆, 2∆, .. , a status quo q (t) is in place, ∈ { } players observe the current state θ (t) , and they determine the bargaining outcome by playing some (possibly state-dependent and stochastic) within-period bargaining procedure. Its outcome determines players’flow payoff for the period [t, t + ∆), and the status quo for the next bargaining

24 time q (t + ∆). The within-period bargaining procedure used in state θ Θ and status quo q X ∈ ∈ is denoted by ℘ (θ, q). The state θ (t) follows a continuous-time stochastic Markov process on Θ.

The flow payoff of player k l, r in state θ from alternative x X is denoted by uk (θ, x), which ∈ { } ∈ we assume is uniformly bounded. Players maximize the expected discounted sum of flow payoffs,

en and ρ > 0 is the discount factor. We denote this game by Γ℘ (∆). Note that the models analyzed in Sections 3 and 4.2 are special cases of this model, where ∆ is

ρ∆ ﬁxed, Pθ is the distribution of θ (t + ∆) conditional on θ (t) = θ, δ = e− , the payoﬀ function is given by ∆ ρt Uk (θ, x) = Eθ(0)=θ (uk (θ (t) , x)) ρe− dt , (6) Z0 and ℘ is the voting protocol or the consensus protocol, respectively.

4.3.1 A numerical illustration with single-peaked preferences

Suppose X = 1, 0, 1 , θ (t): t N is a standard Brownian motion on Θ = R, and the profile {− } { ∈ } of flowpayoffs is state-dependent single-peaked (see Definition 2). Note that conditional on θ (t),

θ (t + ∆) is distributed according to N θ (t) , √∆ ; hence, if players can update the policy frequently, the probability that their preferences change drastically between periods is small. Note also that the proﬁle of payoﬀ functions as given by (6) is state-dependent single-peaked (modulo a

ex constant). Hence, when ℘ is the consensus protocol (denoted by ℘ = c), then the games Γc (∆) en ex en and Γc (∆) are equivalent to Γc (X, b) and Γc (X, b) as deﬁned in Section 4.2.2 respectively. Therefore, for a given ∆, the equilibria described in Section 4.2.2 are also equilibria of the games

ex en ρ∆ Γc (∆) and Γc (∆) with e− = δ and √∆ = σ. To illustrate the eﬀect of the bargaining friction ∆ and of the within-bargaining procedure ℘ on the relative magnitude of the polarization and moderation eﬀects, we will compare the equilibria

ex en en of Γc (∆) and Γc (∆) to the equilibrium of Γ℘ (∆) for diﬀerent values of ∆, where ℘ denotes the within-period bargaining game in which r is a monopolistic proposer. That is, r makes a take-it- or-leave-it proposal to l, and l can either accept it, in which case the proposal is implemented, or reject it, in which case the status quo stays in place. In what follows, we denote the corresponding

en dynamic bargaining game Γr (∆). en We postulate the following cutoﬀ equilibrium in Γr (∆): there exists four cutoﬀs cLM , cMR, cRM ,

25 Figure 4: The evolution of the policy cML R with cLM < cMR and cRM < cML. When q = L and θ increases, then r proposes a change ∈ from L to M when θ [cLM , cMR), and when q = M and θ increases, r proposes a change from M ∈ to R when θ cMR. When q = R and θ decreases, then r proposes a change from R to M when ≥ θ [cML, cRM ), and when q = M and θ decreases, then r proposes a change from M to L when ∈ 14 θ < cML. All proposals of r are accepted by l.

We numerically solve for cML, cLM , cRM , and cMR, and obtain that for all parameters we used, the equilibrium exhibits partisanship and status quo inertia, but less so than the equilibrium of

en ex en Γc (∆) . To see this, consider Figure 4. For Γc (∆) (dashed lines) and Γc (∆) (solid lines), the en evolution of the policy is as in Figure 3 from Section 4.2.2. The policy evolution in Γr (∆) is depicted using dotted lines. The dotted arrows are always farther apart than the dashed arrows of

Γex (∆) ; hence, there is the status quo inertia. The moderation eﬀect can be seen by comparing the

en en equilibria of Γr (∆) and Γc (∆) . The equilibrium cutoffs between R and M are the same in both 15 en games. However, the equilibrium cutoffs between M and L are closer to each other in Γr (∆) en than in Γc (∆) . This means that players’behavior is more moderate and there is less status quo en inertia than in Γc (∆) , which is consistent with the intuition provided in Section 4.1. en Recall that in Γc (∆) , partisanship and status quo inertia were measured by (pl, pr) . Similarly, 14 The full equilibrium requires specifying also what happens for drastic changes of θ (t) , but this is not relevant for the point made in this example; hence, it is omitted. 15 This should not be surprising in light of the intuition laid out in Section 4.1. With a rightist status quo q M,R , the moderation effect can bias players if they expect to disagree on how far to left to move the status quo,∈ { i.e., to} M or to L, and the player who wants the most drastic change of policy, i.e., to L, has the bargaining en power. But in such a disagreement, r prefers the more moderate change of policy to M, and in the game Γr (∆) , she has all the bargaining power.

26 en ex in Γr (∆) they can be measured by the difference between the equilibrium cutoffs in Γ (∆) and en 16 the corresponding cutoffs in Γr (∆). We denote these differences pLM and pML (see Figure 4).

Figure 5 plots (pl, pr) (solid) and (pLM , pML) (dotted) as a function of ∆ for ρ = 0.001. Interestingly, in both games, partisanship and status quo inertia increase as ∆ 0. This is because when ∆ is → small, defending the preferred status quo for ∆ amount of time is less costly. Moreover, players behave in the same way in Γen (∆) and in Γen (∆) as ∆ 0. Hence, these numerical results show r c → that as the bargaining friction ∆ vanishes, the moderation eﬀect vanishes but the polarization eﬀect does not.

Figure 5: Status quo inertia

The intuition from Section 4.1 suggests that status quo instability could arise if preferences were allowed to change drastically between bargaining periods. To illustrate this phenomenon,

en we consider the same bargaining game as Γr (∆) but with a diﬀerent continuous time process θ (t): t 0 with discontinuous shocks: at each instance, θ (t) either stays constant or is redrawn { ≥ } from a uniform distribution over [ 2, 2] . The occurrence of draws is distributed as a Poisson process − with some parameter λ > 0. Numerical simulations show that in equilibrium, pLM > 0 and pML < 0 even as ∆ 0. That is, M replaces L before M becomes Pareto optimal and L replaces M before → L becomes Pareto optimal, which means that the moderation eﬀect is strong enough to generate status quo instability.

16 L+M L+M Formally, pLM = bl cLM and pML = br cML. 2 − − 2 − −

27 4.3.2 The general result

We generalize the qualitative findings from Section 4.3.1 to a large class of within bargaining procedures ℘ and stochastic processes θ (t): t 0 . { ≥ } The only restriction we impose on ℘ is that either player can unilaterally block any change away from the status quo. Formally, for all θ Θ and q X, in any subgame perfect equilibrium ∈ ∈ of ℘ (θ, q), with probability 1, both players weakly prefer the equilibrium outcome to the status quo q. Hence, ℘ can be a game form in which a state-dependent, randomly drawn proposer makes a take-it-or-leave-it offer to the other player, or the voting procedure used in Section 3.17 It is worth noting that our conclusions also hold in the case in which the bargaining outcome within each period is determined by some cooperative bargaining solution such as the Nash bargaining solution, where players’payoffs are their continuation value from implementing a given alternative.

To capture the requirement that preferences evolve smoothly over time, we assume that the state space Θ is endowed with a topology such that for all x X, uk (θ, x) is continuous in θ, and ∈ the process θ (t): t 0 satisﬁes the following condition: { ≥ }

Assumption 3 For all θ Θ, and all neighborhoods B of θ, if tˆB denotes the ﬁrst exit time of B, ∈ then Pr tˆB t = o (t) as t 0. θ(0)=θ ≤ → Assumption 3 is satisﬁed for all standard continuous Markov processes such as the Brownian motion used in Section 4.3.1, and is violated for all standard discontinuous Markov processes such as the Poisson process used in 4.3.1.18

en Let σ be a Markov strategy profile of Γ℘ (∆) . For every (θ, q), the strategy profile σ induces a probability distribution over the possible outcomes of the bargaining stage-game, and we denote by φσ (θ, q; x) the probability that x is implemented in that state. We call φσ the outcome function induced by the strategy profile σ.

In what follows, to characterize the equilibrium behavior as the bargaining friction ∆ vanishes, we fix the within-period bargaining procedure ℘, and we consider a sequence (∆ ) such that n n N ∈ 17 It is easy to see that the consensus procedure outlined in Section 4.2 satisfies this restriction in any pure strategy equilibrium: If the implemented policy left a player worse off than the status quo, this player would veto the first move away from the status quo. Under Assumption 2, our restriction is also satisfied for mixed-strategy equilibria. 18 See Remark 1 in the appendix for a formal proof of these claims. Note also that if θ (t): t 0 is deterministic, then Assumption 3 is satisfied if and only if θ (t): t 0 is continuous. When Θ is a{ metric space,≥ } it is satisfied if θ (t): t 0 is Lipschitz continuous in t with{ probability≥ } 1. { ≥ }

28 ∆ 0, and a sequence of strategy profiles (σ ) such that for all n , σ is a Markov perfect n n n N N n → ∈ ∈ en equilibrium of Γ℘ (∆n). This sequence is fixed throughout, and we denote the corresponding sequence of outcome functions φ . σn n N ∈ Definition 4 An alternative x is stable at (q, θ) in the limit if there exists a neighborhood B of θ, some ε > 0, and some N > 0 such that for all ζ B and all n N, ∈ ≥

(i) there exists a sequence of alternatives x1, ..., xM with x1 = q and xM = x such that φσn (ζ, xm; xm+1) > ε for all m = 1, ..., M 1; −

(ii) φσn (ζ, x; x) = 1.

Hence, if x is stable at (q, θ), then for ∆n suﬃ ciently small, with nonvanishing probability, the bargaining outcome moves from q to x with positive probability, possibly in several steps (condition

(i) in Deﬁnition 4), and stays at x as long as θ (t) stays suﬃ ciently close to θ (condition (ii) in

Definition 4). The notion of stability is not a demanding one. To see this, note that for all (θ, q) , conditions (i) and (ii) are always satisfied at ζ = θ : since X is finite and the bargaining outcome in state θ must always weakly improve players’intertemporal payoff relative to the status quo, if the state were to stay at θ for several periods, players would eventually reach a bargaining agreement

19 that they would not want to revise. Since X is finite, the path x1, ..., xM must be the same for some subsequence of (σ ) . Definition 4 further requires that this path is the same for all states ζ n n N ∈ suffi ciently close to θ. Roughly speaking, this requirement is satisfied whenever σn does not change discontinuously in a neighborhood of θ for all n suffi ciently large. For example, it is met for almost all θ if the equilibria (σ ) are in cutoff strategies as the equilibria analyzed in Sections 4.2.2 n n N ∈ and 4.3.1.20

We are ready to state our result.

19 To be more precise, there might exists an equilibrium which prescribes players to cycle indefinitely between some subset of alternatives X0 X in some state θ. However, in that case, both players must be indifferent between ⊂ all alternatives in X0, and one can avoid such cycles by requiring that for all q, x X0, whenever the equilibrium prescribes to move from q to x, one player blocks such a change and imposes to stay∈ at q. The resulting strategy profile induces the same continuation value, and is thus still an equilibrium. 20 en To see this, note that these equilibria have the following properties: at each action node of the game Γ℘ , players are prescribed pure action, and the set of states at which a given action is prescribed is an interval of Θ = R. Hence, for any sequence σn of such equilibria, there exists a subsequence in which these intervals converge, and in any state θ that does not coincide with the limit of the boundaries of these intervals, the actions prescribed by σn are locally constant in a neighborhood of θ for any status quo q.

29 Proposition 9 For all θ Θ and all q, x X, if x is stable at (q, θ) in the limit, then for all ∈ ∈ k l, r , uk (θ, x) uk (θ, q) . ∈ { } ≥

Proposition 9 states that when the negotiating parties can revise the bargaining agreement suﬃ ciently frequently, the status quo q can be replaced by x only if x Pareto improves on q in terms of the current preferences. Note that with an exogenous status quo, the status quo q is replaced exactly when an alternative x Pareto improves on q in terms of the current preferences. Hence,

Proposition 9 implies that if the endogenous status quo distorts the dynamics of the bargaining agreement relative to the exogenous status quo, this distortion must take the form of status quo inertia.

The intuition for Proposition 9 is fairly simple. Consider player k deciding whether to agree to replace q with x at some bargaining time t. If the other player prefers such a change, then since preferences evolve smoothly, a small instant later she is still likely to prefer it. Hence, if q gives player k a higher current payoﬀ than x, player k prefers to oppose the change in t and reconsider it an instant latter.

Since Proposition 9 imposes minimal assumptions on the environment, the inertial effect of the endogenous status quo cannot be quantified in this general setup. However, as the numerical example in Section 4.3.1 illustrates, this effect typically does not disappear as ∆ tends to 0. To see why, note that as ∆ tends to 0, it is very unlikely that in a given period, players agree to implement some alternative x, and ∆ unit of time latter, more than one alternative Pareto dominates x. Hence, the second type of disagreement is unlikely. However, players are still likely to disagree on what alternatives are better than the status quo. Using the intuition provided in Section 4.1, this means that when players are allowed to revise the status quo whenever they want, the kind of disagreement that leads to moderation becomes unlikely, while the kind of disagreement that leads to polarization does not.21 21 It is easy to show that when X = 2, then Proposition 4 applies to this environment, and by substituting (6) into (3), we obtain that a gridlock| equilibrium| exists if and only if for all θ (0) Θ, ∈

∞ ρt ∞ ρt Eθ(0) e− (ul (θ (t) ,R) ul (θ (t) ,L)) < 0 < Eθ(0) e− (ur (θ (t) ,R) ur (θ (t) ,L)) . (7) − − Z0 Z0 Since ∆ does not appear in the above expression, the conditions under which gridlock and hence extreme status quo inertia occur in the two-alternative model are independent of ∆.

30 5 Concluding remarks and directions for further research

Negotiations in a changing environment with an endogenous status quo are at the center of many economically relevant situations. They present the negotiating parties with a fundamental trade- off between responding adequately to the current environment and securing a favorable bargaining position for the future. In this paper, we show that this trade-off generates perverse incentives that have a detrimental impact on the effi ciency of bargaining outcomes and their responsiveness to political and economic shocks.

The natural and most common alternative to the endogenous status quo protocol is the exogenous status quo protocol.22 For example, discretionary spending is legislated using the latter procedure. In the legislative sphere, the exogenous status quo is also implemented in the form of automatic sunset provisions: clauses that specify a duration after which an act expires, unless further legislative action is taken.23 The partisanship and status quo inertia generated by the endogenous status quo may suggest that the exogenous status quo and sunset provisions are better institutions.

However, such a conclusion would be premature. Automatic sunset provisions and the exogenous status quo protocols are certain to be beneﬁcial only if their ﬁxed reversion point is socially optimal.

This may not be the case if the optimal reversion point depends on the current state of nature.

Since under the endogenous status quo, the status quo is determined by the equilibrium behavior, and hence evolves with the state, the endogenous status quo may dominate.24

In light of the above, it would be interesting to endogenize the occurrence of sunset provisions.

One can shed light on this issue by considering an extension in which in every bargaining round,

22 For example, the permanent provisions of the Agricultural Adjustment Act of 1938 and the Agriculture Act of 1949 serve as a fixed status quo for U.S. farm bills (Kwan 2009). Also, bilateral international agreements implicitely have an exogenous status quo of no agreement because either country can unilaterally opt out. See Lowi (1969), Weaver (1985, 1988), Hird (1991), and Gersen (2007) for more detailed studies of the ongoing and temporary nature of the laws enacted by the U.S. congress. 23 One example of automatic sunset provisions is the sunset legislation in twenty-four U.S. states that requires automatic termination of a state agency, board, commission, or committee (see The Book of the States, 2011, Council of State Governments). See Kearney (1990) for more on the use of sunset provisions by U.S. state legislatures. In the U.S., automatic sunset clauses are less common at the federal level, although there have been attempts to introduce them systematically in Congress (the Federal Sunset Act). In the budget process, the Byrd rule is equivalent to imposing an automatic sunset clause on any provision that increases the deficit and that does not garner a filibuster- proof majority. In a similar spirit, in 2007, the Liberal Democratic Party in Australia proposed an automatic sunset provision in all legislation that does not get the support of a 75 percent parliamentary supermajority. In Canada, any law that overrides the Canadian Charter of Rights and Freedoms (section 33) has an automatic 5-year sunset. In Germany, all emergency legislations have an automatic sunset of six months. 24 An interested reader is referred to Example 1 in Dziuda and Loeper (2012) which shows an environment in which the endogenous status quo may dominate.

31 decision makers bargain both on the new policy and on whether and when it should sunset. We leave this for future research.

Given the inertia of political action, some macroeconomists have argued that automatic stabilizers could help public policy to react in a timely manner to economic shocks. An automatic stabilizer is a clause attached to a continuing policy that triggers its change in response to a change in certain observable indicators (e.g., price levels, unemployment rate). Hence, an automatic stabilizer changes how today’s policy aﬀects tomorrow’s status quo. By weakening the link between today’s policy and tomorrow’s status quo, automatic stabilizers could mitigate players’partisanship and status quo inertia without foregoing completely the responsiveness of the status quo to the environment. One could investigate the impact of automatic stabilizers by considering an extension of this model in which today’s status quo depends on yesterday’s policy and on the realization of a (veriﬁable) random variable that is correlated with the state of nature.

Finally, in many applications, the evolution of the state is likely to depend on the implemented policies. For example, increasing public spending may not only alleviate the pain from economic recession, but also speed up the recovery. In our model, we abstract from this dependence to isolate the effect of the endogenous status quo in a transparent way. However, the polarizing effect we have identified is likely to be robust to the introduction of this dependence. Additional equilibrium effects may arise, but their nature will depend on the fine details of the model. To see this, note that if today’spolicy affects tomorrow’sstate, it does so in the same way under the exogenous and the endogenous status quo. Hence, the difference in players’behavior under these two protocols will come only from two sources. First, as in our model, under the endogenous status quo the policy implemented today will affect tomorrow’spolicy in case players disagree tomorrow. This dynamic linkeage biases the players in favor of the policy that they prefer in disagreement states; hence, will lead to polarization. Second, since today’s policy affects tomorrow’s policy differently under the two protocols, today’s policy also differently affects the distribution of the state two periods from now. The anticipation of this difference biases players in favor of policies that deliver better future states, which may be different than the policies they prefer conditional on disagreement. Whether the second effect reinforces or mitigates the first one depends on particular applications. We leave them for future research.

32 6 Appendix A

Section 6.1.1 introduces necessary notation, and Section 6.1.2 proves lemmas that will be used throughout the appendix.

6.1 Proofs in the 2 alternative model − 6.1.1 Notations

We denote by m the bound of Uk (θ, x) over all k l, r , θ Θ, and x X, and by Pθ the | | ∈ { } ∈ ∈ probability distribution of the random variable θ (t) , conditional on θ (t 1) = θ. Let be the − F m m set of mappings fk from Θ into 1 δ , 1 δ such that for all θ Θ, fk is Pθ measurable, and let − − − ∈ − h i f = (fl, fr) be an arbitrary pair of such mappings. Denote

D (f) = θ Θ: fl (θ) < 0 and fr (θ) 0 { ∈ ≥ }  . (8)  D0 (f) = θ Θ: fl (θ) 0 and fr (θ) < 0  { ∈ ≥ } We deﬁne the mappings V and Ω from 2 into itself as follows: for all f 2, all k l, r , and F ∈ F ∈ { } all θ Θ, ∈ Vk (fk, θ) $ Uk (θ) + δfk (θ) , (9) and

Ωk (f)(θ) = fk (ζ) dPθ (ζ) . (10) ζ D(f) D (f) Z ∈ ∪ 0 σ σ When fk is equal to some continuation value Wk for some strategy proﬁle σ, then Vk (Wk , θ) is simply the relative gain for player k of implementing alternative R instead of L in period t in the game Γen, conditional on θ (t) = θ and conditional on σ being played thereafter. Then,

Ωk (V (f)) (θ) can be interpreted as the expectation of that relative gain in period t + 1 conditional on θ (t) = θ and on players disagreeing in t + 1.

We denote by ( , ) the partial order on 2 defined as follows: for all f and f , f ( , ) f if ≤ ≥ F 0 0 ≤ ≥ for all θ Θ, f (θ) fl (θ) and f (θ) fr (θ). This order on payoff profiles corresponds to the ∈ l0 ≤ r0 ≥ partisanship order on strategy profiles defined in Proposition 2.

33 6.1.2 Preliminary Lemmas

Lemma 1 Let T be a set that is Pθ measurable for all θ Θ, and let fk be such that for all − ∈ ∈ F θ Θ, using the notation introduced in (9), ∈

fk (θ) Vk (fk, ζ) dPθ (ζ) . ≥ ζ T Z ∈

If Uk (θ) 0 for all θ T, then for all θ Θ, fk (θ) 0. The symmetric implication holds with the ≥ ∈ ∈ ≥ opposite inequalities.

Proof. Let f = inf f (θ) and let (θ ) be a sequence of states in Θ such that f (θ ) ∗ θ Θ k n n N k n ∈ ∈ → f ∗. Under our assumptions, using (9), for all n N, ∈

fk(θn) Vk (fk, ζ) dPθn (ζ) Uk (ζ) dPθn (ζ) + δf ∗Pθn (T ) . (11) ≥ ζ T ≥ ζ T Z ∈ Z ∈

Since ζ T Uk (ζ) dPθn (ζ) m and Pθn (T ) 1, we can assume w.l.o.g. that these two sequences ∈ ≤ | | ≤ converge R to some u and p, respectively. Necessarily, p 1, and since Uk (θ) 0, u 0. Taking the ≤ ≥ ≥ limit in (11) and rearranging terms, we obtain f u/ (1 pδ) 0, as needed. The second part is ∗ ≥ − ≥ proved analogously by using a sequence (θ ) such that f (θ ) sup f (θ) . n n N k n θ Θ k ∈ → ∈ Lemma 2 The map Ω is isotone for the order ( , ). ≤ ≥ Proof. For all f, f 2, 0 ∈ F

Ωr f 0, θ Ωr (f, θ) = f 0 (ζ) dPθ (ζ) fr (ζ) dPθ (ζ) − r − ZD(f 0) ZD(f) + f 0 (ζ) dPθ (ζ) fr (ζ) dPθ (ζ) r − ZD0(f 0) ZD0(f)

Let Ar (f, f 0, θ) and Br (f, f 0, θ) denote the expression on the ﬁrst and second line, respectively, of the right-hand side of the above equation.

Suppose that f ( , ) f. Then from (8), D (f) D (f ), so 0 ≤ ≥ ⊂ 0

Ar f, f 0, θ = f 0 (ζ) dPθ (ζ) f 0 (ζ) dPθ (ζ) + f 0 (ζ) dPθ (ζ) fr (ζ) dPθ (ζ) r − r r − ZD(f 0) ZD(f) ZD(f) ZD(f) = f 0 (ζ) dPθ (ζ) + f 0 (ζ) fr (ζ) dPθ (ζ) . r r − ZD(f 0) D(f) ZD(f) \ 34 From (8), fr0 is positive on D (f 0); hence, D0 (f ) D(f) fr0 (ζ) dPθ (ζ) is positive. Since fr0 fr, 0 \ ≥ (f (ζ) fr (ζ)) dPθ (ζ) is positive. Therefore,R Ar (f, f , θ) is positive. D(f) r0 − 0 R From (8), if f ( , ) f, D (f ) D (f), so 0 ≤ ≥ 0 0 ⊂ 0

Br f, f 0, θ = f 0 (ζ) dPθ (ζ) fr (ζ) dPθ (ζ) + fr (ζ) dPθ (ζ) fr (ζ) dPθ (ζ) r − − ZD0(f 0) ZD0(f 0) ZD0(f 0) ZD0(f) = f 0 (ζ) fr (ζ) dPθ (ζ) fr (ζ) dPθ (ζ) . r − − ZD0(f 0) ZD0(f) D0(f 0) \

From (8), fr is negative on D0 (f), so D (f) D (f ) fr (ζ) dPθ (ζ) is negative. Since fr0 fr, then 0 \ 0 0 ≥ (fr0 (ζ) fr (ζ)) dPθ (ζ) is positive.R Therefore, Br (f, f 0, θ) is positive, and Ωr (f 0, θ) Ωr (f, θ). D0(f 0) − ≥ AR symmetric argument for player l completes the proof.

Lemma 3 For all W 2,W is the continuation value of some equilibrium σ if and only if it is ∈ F a ﬁxed point of the map f Ω(V (f)). →

Proof. Let σ be an equilibrium profile of strategies, let W σ be the corresponding continuation value as defined in Section 3.2, and let x (t) be the alternative implemented in t. This alternative becomes the status quo in the next period q (t + 1) . Note that q (t + 1) affects the outcome in t + 1 only when players vote for different alternatives in that period, in which case q (t + 1) is implemented in t + 1. Using Condition (2) and the notations introduced in (8), players vote for different alternatives in t + 1 when θ (t + 1) D (V (W σ)) D (V (W σ)). From the preceding ∈ ∪ 0 discussion, for any such state θ (t + 1), the difference in continuation value from t + 1 onwards for player k between x (t) = R and x (t) = L is simply

σ Uk (θ (t + 1)) + δWk (θ (t + 1)) . (12)

By deﬁnition, W σ is simply the expectation of (12) over all disagreement states θ (t + 1) conditional on θ (t), so

σ σ Wk (θ (t)) = [Uk (ζ) + δWk (ζ)] dPθ(t) (ζ) . (13) ζ D(V (W σ)) D (V (W σ)) Z ∈ ∪ 0 σ Using the notations introduced in (9) and (10), the above equation can be rewritten as Wk (θ (t)) = σ Ωk (V (W ) , θ (t)), as needed.

o 2 o o Reciprocally, let W be such that W = Ωk (V (W )). Consider the strategy profile σ such ∈ F 35 o σ that each player k votes for R if and only if Uk (θ) + δW (θ) 0. By definition, W is given by k ≥ the expectation of (12) over all disagreement states θ (t + 1) conditional on θ (t) . By construction of σ, players disagree in state θ (t + 1) if and only if θ D (V (W o)) D (V (W o)) . Therefore, W σ ∈ ∪ 0 k satisfies

σ σ Wk (θ) = [Uk (ζ) + δWk (ζ)] dPθ (ζ) (14) ζ D(V (W o)) D (V (W o)) Z ∈ ∪ 0 Equation (14) can be seen as a functional equation in W σ, and since W o = Ω (V (W o)) ,W o is also a solution to (14). From (9), the right hand-side of (14) is an δ-contraction in W σ for the sup norm.

Therefore, (14) has a unique solution, which implies that W o = W σ. Together with the deﬁnition

σ of σ, this shows that under σ, player k votes for R if and only if Uk (θ) + δW (θ) 0. This means k ≥ that player k use stage undominated strategies, so σ is an equilibrium.

6.1.3 Proofs for Section 3.2

Proof of Proposition 1. Equilibrium existence follows from Proposition 2 and the fact that a complete lattice is non empty. Let W be an equilibrium continuation function. From Lemma 3,

W = Ω (V (W )) , so Wk (θ) = ζ D(V (W )) D (V (W )) Vk (Wk, ζ) dPθ (ζ) . Hence, we can write ∈ ∪ 0 R

Wr (θ) Wl (θ) = (Vr (Wr, ζ) Vl (Wl, ζ)) dPθ (ζ) . − ζ D(V (W )) D (V (W )) − Z ∈ ∪ 0

Setting fk = Wr Wl in Lemma 1 and using the fact that Ur (θ) Ul (θ) 0 for all θ Θ, − − ≥ ∈ we obtain that Wr (θ) Wl (θ) 0. Substituting this inequality and Ur (θ) Ul (θ) in (9), − ≥ ≥ we obtain Vr (Wr, θ) Vl (Wl, θ). From (8), this implies that D (V (W )) = . Hence, Wk = ≥ 0 ∅ Vk (Wk, ζ) dPθ (ζ) , which implies that W ( , ) 0, as needed. ζ:Vl(Wl,ζ)<0 and Vr(Wr,ζ) 0 ≥ ≤ ≥ R Proof of Proposition 2. Note that 2 is a complete lattice for ( , ). One can easily see F ≤ ≥ from (9) that the map f V (f) is isotone for the order ( , ). Together with Lemma 2, this → ≤ ≥ shows that the map f Ω(V (f)) is isotone. Tarski’sfixed point theorem implies then that the set → of fixed points of f Ω(V (f)) is a complete lattice for the order ( , ). The result follows then → ≤ ≥ from Lemma 3, and the fact that the order ( , ) on continuation value functions corresponds to ≤ ≥ the partisan order on strategy profiles as defined in Proposition 2.

To show the Pareto ranking properties, let σ and σˆ be two equilibria such that W σˆ ( , ) W σ. ≤ ≥

36 We will show that σ is Pareto superior to σˆ by constructing a series of “one-bargaining time deviations”that lead from σ to σˆ such that all deviations are payoﬀ reducing for both players. Suppose that both players "deviate" from σ to σˆ only in t = 0. Since W σˆ ( , ) W σ, from Proposition 1, ≤ ≥ players disagree in t = 0 more often under σˆ than under σ, so the only way in which this deviation can change the outcome at t = 0 is if under σ players agree to change the status quo while under σˆ, the status quo remains. Suppose w.l.o.g. that q (0) = R, and hence L is implemented under σ while R stays in place under σ.ˆ Since players play their equilibrium strategy σ in the subgame starting in t = 1, the net eﬀect of the deviation from σ to σˆ in period 0 for player k

σ is Uk (θ (0)) + δWk (θ (0)). By assumption, both players vote for L under σ, and since σ is an σ equilibrium, Uk (θ (0)) + δW (θ (0)) must be negative for both k l, r , which implies that the k ∈ { } deviation decreases the payoﬀ of both players.

To conclude the argument, consider the strategy proﬁle in which players play σˆ in t = 0 and σ afterwards, and let players deviate from that strategy by playing according to σˆ also in t = 1. The same reasoning as above shows that this deviation decreases the payoﬀs of both players irrespective of the status quo distribution in t = 1. By induction on the number of periods in which players deviate from σ to σˆ, the proposition follows.

6.1.4 Proofs for Section 3.3

Proof of Proposition 3. From Lemma 3, W is an equilibrium continuation value if and only if W is a fixed point of the mapping f Ω(V (f)), and from Proposition 1, we can restrict → attention to the fixed points of that mapping in Π+ = f 2 : f ( , ) 0 . Let V and V ∈ F ≤ ≥ 0 refer to the mappings defined in (9) corresponding to (U, δ) and to U 0, δ0 , respectively. From

+ (9), U ( , ) U and δ0 > δ imply that for all f Π , V (f)( , )V (f), and from Lemma 2, 0 ≤ ≥ ∈ 0 ≤ ≥ Ω(V (f)) ( , )Ω(V (f)). Since f Ω(V (f)) and f Ω(V (f)) are isotone in f for ( , ) (see 0 ≤ ≥ → → 0 ≤ ≥ the proof of Proposition 2), Corollary 1 in Villas-Boas (1997) implies that their respective smallest

(or greatest) fixed points W and W for the order ( , ) are such that W ( , ) W . 0 ≤ ≥ 0 ≤ ≥ g g Proof of Proposition 4. Suppose σl is not a best response for player l to σr. From the one-step-deviation principle, this is true if and only if for some θ Θ, conditional on θ (0) = θ, ∈ player l is better-off voting for R than for L at t = 0, given that she expects σg to be played in all future periods. By definition of σg, this is equivalent to saying that for θ (0) = θ player l is

37 better-oﬀ with alternative R in all periods, than with alternative L in all periods. Hence, we have

g g shown that σl is not a best response for player l to σr if and only if Condition (3) is not satisﬁed for player l. A symmetric argument for player r completes the proof.

Proposition 10 If Condition (3) is satisﬁed for some (Ul,Ur) and some δ, then it is also satisﬁed any (U ,U )( , )(Ul,Ur) and for any δ0 δ. l0 r0 ≤ ≥ ≥

Proof. The comparative statics with respect to Uk is straightforward and omitted for brevity. t j δ0 δ0 δ t δ0 Let δ0 δ. Using = 1 + − and the law of iterated expectation, we have ≥ δ δ j=0 δ P t s ∞ t ∞ t δ0 δ δ0 E δ0 U (θ (t)) = E U (θ (t)) δ 1 + − θ(0) l θ(0) l δ δ "t=0 # "t=0 s=0 !# X X X t s ∞ δ0 δ ∞ δ0 = E δtU (θ (t)) + − δtU (θ (t)) θ(0) l δ l δ "t=0 t=0 s=0 # X X s X ∞ δ0 δ ∞ δ0 ∞ = E δtU (θ (t)) + − E δtU (θ (t)) . θ(0) l δ δ θ(s) l "t=0 s=0 " t=s ## X X X

If Condition (3) is satisﬁed for player l for δ, then since θ (t): t N is stationary, for all s N and { ∈ } ∈ t all realizations of θ (s), E ∞ δ Ul (θ (t)) 0. Since δ0 δ, this means that the right-hand θ(s) t=s ≤ ≥ side of the above equation is negative,P and hence Condition (3) is satisﬁed for player l for δ0. The proof for r is analogous.

Proposition 11 Suppose that the proﬁle of payoﬀ functions U is state-dependent single-peaked and

θ (t): t 0 is a regular Markov chain on a finite Θ whose stationary distribution has mean ¯θ. { ≥ } L+R ¯ Then Condition (3) is satisfied whenever bl < θ < br and δ is suffi ciently large. 2 −

Proof. Since θ (t): t N is a regular Markov process on a ﬁnite Θ, it has a unique invariant { ∈ } t distribution, and one can easily show that for all θ (0) Θ,E (1 δ) ∞ θ (t) δ ¯θ as ∈ θ(0) − t=0 → δ 1. Hence, P →

∞ t ∞ t L + R (1 δ) E δ Uk (θ (t)) = (1 δ) E 2 (R L) δ θ (t) + bk − θ(0) − θ(0) − − 2 "t=0 # " t=0 # X X ∞ t L + R = 2 (R L) (1 δ) E δ θ (t) + bk − − θ(0) − 2 "t=0 # ! X ¯ L + R δ 12 (R L) θ + bk . → → − − 2 38 L+R ¯ Therefore, if bl < θ < br, then for δ suﬃ ciently large, for all θ (0) Θ, the left-hand side of 2 − ∈ the above equation is strictly positive for k = r and strictly negative for k = l, as needed.

6.1.5 Proofs for Section 3.4

Proof of Proposition 5. Consider the game Γen ( L, R , b) with L = 1/4 and R = 1/4. In { } − what follows, we say players play cutoff strategies c = (cl, cr) if the strategy profile σ prescribes player k to vote for R if θ ck and for L if θ < ck. If players play cutoff strategies c, then from ≥ (13) the continuation value W c is given by

cl c 2 2 c W (θ) = (1/4 (ζ + br)) + ( 1/4 (ζ + br)) + δW (ζ) f (ζ θ) dζ (15) k − − − − k − Zcr cl c = (ζ + bk + δW (ζ)) f (ζ θ) dζ. k − Zcr

c Given Wr , the stage undominated strategy for player r in each period is to vote for +1/4 in all states 2 c 2 c θ such that (1/4 (θ + br)) + δW (θ, 1/4) ( 1/4 (θ + br)) + δW (θ, 1/4) . Therefore, − − r ≥ − − − r − the cutoﬀ cr is a stage-undominated best response for player r to the cutoﬀ cl if and only if

c θ + br + δW (θ) 0 if and only if θ cr. (16) r ≥ ≥

For all cl bl, deﬁne ≥ −

c Cr (cl) = cr R : cr br & for all θ cr, θ + br + δWr (θ) 0 . (17) { ∈ ≤ − ≥ ≥ }

Together with (16), steps 0-4 below establish that if player l plays a cutoff strategy cl bl, then ≥ − there exists a best response of player r that is in cutoff strategy, and the best response cutoff is given by cr∗ (cl) = inf Cr (cl) . c Step 0: Wk (θ) defined in (15) is jointly continuous in c and θ and differentiable in c. c c c Note that (15) can be viewed as a linear functional equation in Wk of the form Wk = M (ζ + bk)ζ Θ + ∈ c c c δM (Wk ) , where M is a bounded linear operator. Since δ < 1, this equation has a unique solution k c k given by k∞=0 δ (M ) (ζ + bk)ζ Θ . Step 0 follows from the fact that each of the term of this ∈ series is differentiable,P and the sum of the derivatives converge uniformly. The details of the proof

39 are standard and are omitted for brevity.

Step 1: cr∗ (cl) exists and is ﬁnite.

If cr = br, then cr < bl cl, and for all θ [cr, cl], θ + br 0. So Lemma 1 applied to (15) − − ≤ ∈ ≥ cl, br implies that Wr − ( br) 0. Combined with (17), this implies that br Cr (cl), so Cr (cl) = . − ≥ − ∈ 6 ∅ c Clearly, limcr Wk (cr) = , so Cr (cl) is bounded below. →−∞ −∞ cl,cr∗(cl) Step 2: If θ c (cl), then θ + br + δWr (θ) 0. ≥ r∗ ≥ cl,cr From Step 0, Wr (cr) is continuous in cr, so Cr (cl) is a closed subset of R, and cr∗ (cl) Cr (cl). ∈ Step 2 follow then from the deﬁnition of Cr (cl).

cl,cr∗(cl) Step 3: c (cl) + br + δWr (c (cl)) = 0, and c (cl) < br. r∗ r∗ r∗ − c From Step 0, W (θ) is diﬀerentiable w.r.t. cr, and from (15), for all θ Θ, r ∈

c cl c ∂Wr ∂Wr c (θ) = δ (ζ) f (ζ θ) dζ (cr + bk + δW (cr)) f (cr θ) . (18) ∂c ∂c − − r − r Zcr r

Suppose

cl,cr∗(cl) cr∗ (cl) + bk + δWk (cr∗ (cl)) > 0. (19)

From Step 0, the left-hand side of (19) is continuous in c (cl), so (19) must hold for all cr r∗ ∈ cl,cr ∂Wk [c∗ (cl) ε, c∗ (cl)] for some ε > 0. Lemma 1 applied to (18) implies that for all θ Θ, (θ) r − r ∈ ∂cr evaluated at all cr [c (cl) ε, c (cl)] is weakly negative. Hence, c (cl) ε Cr (cl), which ∈ r∗ − r∗ r∗ − ∈ contradicts the definition of cr∗ (cl). This proves the first point of Step 3. For the second point, observe that if c (cl) = br, then for all θ (c (cl) , cl), θ + br > 0, so Lemma 1 implies that for r∗ − ∈ r∗ c all ζ (c (cl) , cl), ζ + br + δW (ζ) > 0. Substituting the latter inequality in (15), we see that ∈ r∗ r c cl,cr∗(cl) Wr (cr∗ (cl)) > 0, so cr∗ (cl) + br + δW (cr∗ (cl)) > 0, which contradicts the first point.

cl,cr Step 4: If θ < cr∗ (cl), then θ + br + δWr (θ) < 0.

For all θ θ0 c (cl), ≤ ≤ r∗

cl cl,c∗(cl) cl,c∗(cl) cl,cr∗(cl) W r θ0 W r (θ) = ζ + bk + δW (ζ) f ζ θ0 f (ζ θ) dζ. (20) r − r k − − − Zc∗r (cl)

Since f is single-peaked with a peak at 0, and since θ θ0 c (cl), the second term in parenthesis ≤ ≤ r∗ inside the integral in (20) is weakly positive. By deﬁnition of cr∗ (cl), the ﬁrst parenthesis is weakly

cl,cr∗(cl) positive. So (20) implies that Wr (θ) is weakly increasing in θ on θ c (cl). Therefore, ≤ r∗

40 cl,cr∗(cl) θ + br + δWr (θ) is strictly increasing in θ on θ c (cl). Step 4 follow then from step 3. ≤ r∗ Steps 5 and 6 below show that an equilibrium in cutoﬀ strategies exists.

Step 5: For all cl bl, c (cl) is weakly decreasing in cl. ≥ − r∗ c From Step 0, W (θ) is diﬀerentiable w.r.t. cl, and from (15), for all θ Θ, k ∈

cl,cr∗(cl) c cl,cr∗(cl) ∂Wr l ∂Wr cl,cr∗(cl) (θ) = δ (ζ) f (ζ θ) dζ + cl + bk + δWr (cl) f (cl θ) . ∂cl cr ∂cl − − Z h i

From deﬁnition of cr∗ (cl), the term inside the brackets on the right-hand side of the above equation cl,cr∗(cl) is weakly positive. Lemma 1 implies then that for all θ Θ, ∂Wr (θ) is weakly positive. From ∈ ∂cl (17), this implies that Cr (cl) is weakly increasing in cl in the inclusion sense, and thus that cr∗ (cl) is weakly decreasing in cl.

Step 6: There exists an equilibrium in cutoﬀ strategies such that cr < br and cl > bl. − −

Construct cl∗ (cr) analogously to cr∗ (cl). A symmetric argument shows that cl∗ (cr) is weakly decreasing in cr. The mapping (cl, cr) (c (cr) , c (cl)) maps [ bl, + ) ( , br] into itself, and → l∗ r∗ − ∞ × −∞ − it is isotone for the order ( , ). As such, it has a ﬁxed point. ≤ ≥ We have established the existence of an equilibrium of Γen ( 1/4, 1/4 , b) in which each player {− } ex k votes for 1/4 if and only if θ ck. In the equilibrium of Γ ( 1/4, 1/4 , b + p) , k votes for 1/4 if ≥ {− } ex and only if θ bk pk. Hence if we set p = b c, then the equilibrium of Γ ( 1/4, 1/4 , b + p) ≥ − − − − {− } en is also an equilibrium of Γ ( 1/4, 1/4 , b) . The inequalities pl < 0 and pr > 0 follow from step {− } 3.

We now show that for all L < R, the equilibrium of Γex ( L, R , b + p) is also an equilibrium of { } Γen ( L, R , b). Note first that in the equilibrium of Γex ( L, R , b + p) , each player k votes for R { } { } L+R L+R if and only if θ bk pk, so she uses a cutoff c = bk pk. Suppose now that players ≥ 2 − − k0 2 − − play the cutoff strategy profile c in the game Γen ( L, R , b) . Then from (13), the corresponding 0 { } value function W is given by: for all k l, r and all θ Θ, 0 ∈ { } ∈

c l0 L + R W 0 (θ) = W 0 (θ, R) W 0 (θ, L) = 2 (R L) ζ + bk + δW 0 (ζ) f (ζ θ) dζ. k k − k − − 2 k − Zc0r

41 L+R The change of variable ζ0 = ζ yields − 2

L+R L+R cl0 =cl Wk0 (θ) − 2 Wk0 ζ0 + 2 L + R = ζ0 + bk + δ f ζ0 + θ dζ, 2 (R L) L+R 2 (R L) 2 c0r =cr ! − − Z − 2 −

L+R Wk0 (θ+ 2 ) so θ 2(R L) is the fixed point of the contraction defined in (15). Therefore, for all θ Θ, → − ∈ W (θ) = 2 (R L) W c θ L+R , and c = L+R + c. Voting for R is a stage-undominated action 0 − − 2 0 2 for player k in the gameΓen ( L, R , b) when the state is θ and continuation play is given by cutoffs { } c0 if and only if

2 2 (R (θ + bk)) + δW 0 (θ, R) (L (θ + bk)) + δW 0 (θ, L) − − k ≥ − − k L + R c L + R 2 (R L) θ + bk + δ2 (R L) W θ 0 ⇔ − − 2 − k − 2 ≥ L + R c L + R θ + bk + δW θ 0. ⇔ − 2 k − 2 ≥

L+R From (16), we know that the last inequality is satisfied if and only if θ ck, or equivalently, − 2 ≥ L+R θ + ck = c . Hence, we have shown that the profile of cutoffs c are stage-undominated best ≥ 2 k0 0 response to each other in the game Γen ( L, R , b) , as needed. { } To complete the proof it remains to show that for any b such that b b = br bl, the 0 r0 − l0 − ex en equilibrium of Γ (X, b0 + p) is also an equilibrium of Γ (X, b0) . To see this, note that if the equilibrium of Γex ( L, R , b + p) is an equilibrium of Γen ( L, R , b) , then if we change the ori- { } { } gin on the state and policy space from 0 to some s R, we obtain that the equilibrium of ∈ Γex ( L s, R s , b (s, s) + p) is an equilibrium of Γen ( L s, R s , b (s, s)) . This change { − − } − { − − } − of origin does not affect the properties of the process that governs the evolution of the state. Thus, the equilibrium of Γex ( L, R , b (s, s) + p) is an equilibrium of Γen ( L, R , b (s, s)) . { } − { } −

6.2 Proofs for Section 4.2

Throughout this section, we use the following notations.

en Notation 1 The state space of the stochastic game Γc , denoted by S, is comprised of three kinds of states: direction states, in which the proposer must choose a direction, voting states, in which players vote simultaneously, and terminal states, in which players have no actions to choose, the

42 current agreement is implemented, and players receive the corresponding payoff. A typical direction state and terminal state of Γen are denoted by θ, q and θ, q , respectively, where θ Θ is c h i hh ii ∈ the period-specific state which determines the payoffs for the corresponding period, and q X is ∈ the temporary agreement or the final agreement, respectively. A typical voting state is denoted by

θ, q, q , where θ and q have the same interpretation as in θ, q , and q is the proposal under h 0i h i 0 consideration.

For all strategy proﬁles σ of Γen and all states s S, V σ (s) denotes the continuation value of c ∈ en σ the game Γc in state s when players play σ, φ (s) denotes the (possibly stochastic) bargaining outcome in a given period starting from state s when players play σ, and as before, W σ (θ, q) denote the expected continuation value at the beginning of period t + 1 conditional on θ (t) = θ, q (t + 1) = q, and continuation play σ. Note that by deﬁnition V σ ( θ, q ) V σ ( θ, q, q ) for the k h i ≥ k h 0i proposer.

For each θ Θ, let Γc (θ) denote a one-period consensus bargaining game played in state θ that ∈ starts with k being the proposer and ends if a policy is implemented. Players obtain payoffs only from terminal nodes, and we denote this payoff by (V ( θ, x ))x X . In what follows, we omit θ hh ii ∈ whenever it is fixed.

Sketch of the proof of Proposition 6.

Because of the possible indifferences mentioned in the text, we cannot prove that all equilibria are partisan, which complicates the proof considerably. Instead, we build an auxiliary game in which all equilibria are partisan, and show that some equilibria of this game are the equilibria of the original game. This part is quite long and not very illuminating. Hence, for the sake of brevity, we relegate it to a not-for-publication appendix B and only provide a detailed sketch here. ên en Consider an auxiliary game Γc δ0 that differs from Γc in two respects: First, the proposer can choose direction in every stage of each period, not only in the first stage. Second, there is discounting between the stages within a given period, with the discount rate δ0 [0, 1). The first ∈ modification assures that in any Markov equilibrium the proposer resolves her indifferences in a way that makes the bargaining outcome from any status quo q and state θ weakly increasing in q, irrespective of players’induced preferences over terminal states (to see that, note that if some xn+m is implemented when players start from θ, xn , this means that the proposer said right in h i

43 all θ, xs such that xs [xn, xn+m), and hence the Markov property assures that starting from h i ∈ any θ, xs , xn+m must be implemented as well). Using the same argument as in the proof of h i Proposition 7, this implies that the continuation value W σˆ satisfies Assumption 1, so σˆ is partisan. ên Note that Γc δ0 can potentially have infinitely many stages within a given period, if players’ action prescribes them to cycle indefinitely within a subset of temporary agreements. However, ên because Γc δ0 features discounting between stages, in any equilibrium, such cycles never occur. ên This together with the finiteness of Θ and X assures that Γc δ0 has an equilibrium, and that there exists a sequence of stage-discounting rates δ0 and a sequence of strategy profiles (ˆσn) n n N n N ∈ ∈ ên such that for all n N, σˆn is an equilibrium of Γ c δn0 , and as δn0 1, σˆn converges to some σˆ ∈ → which is also partisan. Moreover, no cycling also assures that σˆ can be mapped into an equilibrium

en σ of the original game Γc that has the same path of play as σ,ˆ so it is partisan as well.

Lemma 4 Using the conventions in Notation 1, in the game Γc, the following holds:

1. If the proﬁle of ordinal preferences induced by (V ( x ))x X is strict, then any two stage- hh ii ∈ undominated Markov perfect equilibria of Γc induce the same outcome function φ. φ ( q ) is h i deterministic and increasing in q;

2. Suppose (V ( x ))x X satisﬁes Assumption 1 (i.e., for all x, y X such that x < y, hh ii ∈ ∈ Vl ( y ) Vl ( x ) Vr ( y ) Vr x ). If σ is a stage-undominated Markov perfect hh ii − hh ii ≤ hh ii − hh ii σ σ equilibrium of Γc, and if φ ( q ) is increasing in the F.O.S.D. sense in q, then V ( q ) is h i l h i weakly decreasing in q and V σ ( q ) is weakly increasing in q. r h i

Proof. Throughout this proof, to ﬁx ideas, we assume that the proposer is player r.

To prove point 1, note that Γc has a ﬁnite number of action nodes. Therefore, uniqueness and determinism of φ follow from backward induction and the assumption that players have strict preferences over the bargaining outcomes. To show that φ ( q ) is weakly increasing in q, suppose h i by contradiction that for some n < N, φ ( xn ) > φ ( xn+1 ). Necessarily, φ ( xn ) xn+1 or h i h i h i ≥ φ ( xn+1 ) xn. Suppose the former, the proof in the latter case is identical. If r plays right h i ≤ in state xn+1 , then since players use Markov strategies and since φ ( xn ) xn+1, we must h i h i ≥ have φ ( xn ) = φ ( xn+1 ), which is a contradiction. Therefore, r plays left in xn+1 , and hence h i h i h i φ ( xn+1 ) xn+1. Note that r plays left in xn+1 only if she weakly prefers φ ( xn+1 ) to φ ( xn ) , h i ≤ h i h i h i

44 and since the ordinal preferences induced by V are strict, r strictly prefers φ ( xn+1 ) to φ ( xn ) . h i h i But she could secure φ ( xn+1 ) in state xn , either by playing left in xn if φ ( xn+1 ) xn, or h i h i h i h i ≤ by playing right in xn and voting no in xn+1, xn+2 if φ ( xn+1 ) = xn+1, which is a proﬁtable h i h i h i deviation.

σ To prove point 2, let us first show that for all n 1, ..., N 1 , it cannot be that V ( xn ) > ∈ { − } k h i σ σ σ V ( xn+1 ) for both k l, r , or that V ( xn ) < V ( xn+1 ) for both k l, r . Suppose k h i ∈ { } k h i k h i ∈ { } σ σ by contradiction that V ( xn ) < V ( xn+1 ) for both k l, r ; the proof in the other case k h i k h i ∈ { } is identical. Then there exists an action d left, right that r plays with positive probabil- ∈ { } ity in state xn+1 such that the continuation payoff from that action is strictly higher for both h i σ σ σ players than V ( xn ). Suppose first that d is right, that is V ( xn+1, xn+2 ) > V ( xn ) k h i r h i r h i ≥ σ V ( xn, xn+1 ). Since σ is stage undominated, this implies that in xn, xn+1 strategy profile σ r h i h i prescribes r to vote yes with probability 1. Also, l must vote no in xn, xn+1 as otherwise r h i σ σ would be better of playing right in xn in which case we would have V ( xn ) = V ( xn+1 ) h i k h i k h i σ σ and not V ( xn ) < V ( xn+1 ) as we assumed. So since l plays no in xn, xn+1 , it must be k h i k h i h i σ σ that V ( xn ) V ( xn+1, xn+2 ) , as yes would have lead to state xn+1, xn+2 . But since l hh ii ≥ l h i h i each player can unilaterally impose xn after the state xn by voting no, we always have hh ii h i σ σ σ V ( xn ) V ( xn ) , the previous inequality contradicts the initial assumption that V ( xn ) < l h i ≥ l hh ii l h i σ σ V ( xn+1, xn+2 ) = V ( xn+1 ) . Suppose now that the aforementioned direction d is left, so l h i l h i σ σ σ σ Vr ( xn+1, xn ) > Vr ( xn ) Vr ( xn, xn 1 ) . The nature of Γc implies that Vk ( xn+1, xn ) is a h i h i ≥ h − i h i σ σ convex combination of Vk ( xn+1 ) and Vk ( xn, xn 1 ) , so the previous inequality implies that hh ii h − i σ σ Vr ( xn+1 ) > Vk ( xn, xn 1 ). This implies that a stage-undominated σ must prescribe r to vote hh ii h − i σ σ σ no with probability 1 in xn+1, xn , so V ( xn+1 ) = V ( xn+1, xn ) = V ( xn+1 ) . Therefore, h i k h i k h i k hh ii σ σ the initial assumption implies that V ( xn+1 ) > V ( xn ) for all k l, r , and since we always k hh ii k h i ∈ { } σ σ σ σ have V ( xn ) V ( xn ) , it also implies that V ( xn+1 ) > V ( xn ) . This in turns im- k h i ≥ k hh ii k hh ii k hh ii plies that in a stage undominated σ, both players must vote yes in xn, xn+1 . This in turn implies h i that playing right in xn is a strictly profitable deviation for r, which is a contradiction. h i To complete the proof of point 2, note that since one player must weakly prefer φ ( xn+1 ) h i to φ ( xn ), since φ ( xn+1 ) first order stochastically dominates φ ( xn ), and since (V ( x ))x X h i h i h i hh ii ∈ satisfies Assumption 1, r must weakly prefer φ ( xn+1 ) to φ ( xn ) . Likewise, since one player must h i h i weakly prefer φ ( xn ) to φ ( xn+1 ) , l must weakly prefer φ ( xn ) to φ ( xn+1 ) , as needed. h i h i h i h i

45 Proof of the ﬁrst claim in Proposition 7. We show that under Assumptions 1 and 2, if

en σ σ is an equilibrium of Γc , then σ is partisan. The deﬁnition of W and Assumption 2(ii) imply that for all (θ1, θ2) Θ1 Θ2 and all q X, ∈ × ∈

σ σ σ W (θ1, θ2, q) = E [V ( θ (t + 1) , φ ( θ (t + 1) , q ) )] (21) θ(t)=θ hh h i ii = E [U (θ (t + 1) , φσ ( θ (t + 1) , q )) + δW (θ (t + 1) , φσ ( θ (t + 1) , q ))] . θ1(t)=θ1 h i h i

σ so W (θ, q) depends on θ only through θ1. Let

σ σ σ σ ∆ = inf [Wr (θ1, y) Wr (θ1, x) (Wl (θ1, y) Wl (θ1, x))] , θ1 Θ1,x,y X:y x − − − ∈ ∈ ≥ and let x , y X and a sequence (θm) ΘN be such that ∆ = lim [W σ (θm, y ) ∗ ∗ 1 m N 1 m r 1 ∗ ∈ ∈ ∈ →∞ − W σ (θm, x ) (W σ (θm, y ) W σ (θm, x ))]. Using (21) and the law of iterated expectation, we r 1 ∗ − l 1 ∗ − l 1 ∗ obtain

σ σ Ur (θ (t + 1) , φ ( θ (t + 1) , y∗ )) Ur (θ (t + 1) , φ ( θ (t + 1) , x∗ )) ∆ = lim E E h i − h i (22) m m    θ1(t)=θ1 θ1(t+1) σ σ →∞ (Ul (θ (t + 1) , φ ( θ (t + 1) , y∗ )) Ul (θ (t + 1) , φ ( θ (t + 1) , x∗ )))   − h i − h i     σ σ σ σ Wr (θ1 (t + 1) , φ ( θ (t + 1) , y∗ )) Wr (θ1 (t + 1) , φ ( θ (t + 1) , x∗ )) +δ lim E E h i − h i m m    θ1(t)=θ1 θ1(t+1) σ σ σ σ →∞ (Wl (θ1 (t + 1) , φ ( θ (t + 1) , y∗ )) Wl (θ1 (t + 1) , φ ( θ (t + 1) , x∗ )))   − h i − h i    

Assumption 2(i) implies that conditional on θ1 (t) and θ1 (t + 1), almost surely, the proﬁle of ordinal

σ preferences induced by (U (θ (t + 1) , x) + δW (θ1 (t + 1) , x))x X is strict. Hence, from part 1 of ∈ Lemma 4, φσ ( θ (t + 1) , q ) is deterministic and weakly increasing in q almost surely. Therefore, h i Assumption 1 implies that for all θ (t + 1) Θ, the ﬁrst bracketed term on the right hand-side of ∈ (22) is weakly positive. By the same token, Lemma 4 part 1 and the deﬁnition of ∆ imply that for all θ (t + 1), the second bracketed term on the right hand-side of (22) is greater than ∆. Therefore,

(22) implies that ∆ δ∆, and thus ∆ 0. The definition of ∆ then implies that W σ satisfies ≥ ≥ Assumption 1. Since U also satisfies Assumption 1, so does U + δW σ, which by definition is equal

σ to (V ( θ, x ))θ Θ,x X . To complete the proof, note that the actions prescribed by σ in a period hh ii ∈ ∈ with state θ must be an equilibrium of the game Γc in which players’payoﬀ in the terminal states

σ en are (V ( θ, x ))x X and the proposer is the same as in Γc in the state θ. Lemma 4 part 2 implies hh ii ∈

46 then that V σ ( θ, x ) is weakly decreasing and V σ ( θ, x ) is weakly increasing in x. But then l hh ii r hh ii σ σ (21) implies that Wl (θ, x) is weakly decreasing and Wr (θ, x) is weakly increasing in x; hence, σ is partisan.

en Proof of the second claim in Proposition 7. We show that when Θ1 is ﬁnite, Γc admits l,r Θ1 X m m { }× × an equilibrium. Let W 1 δ , 1 δ , where m is the uniform bound on U. Let σ (W ) ∈ − − − be a strategy proﬁle that forh each θ isi equal to a stage-undominated Markov perfect equilibrium of

en Γc (θ) with the terminal payoﬀs V ( θ, x ) = U (x, θ)+δW (x, θ1) and the same proposer as in Γ hh ii c in the state θ. Such equilibrium always exists and by construction, σ (W ) is Markovian. Let W σ(W )

en be the corresponding continuation value in Γc . In what follows, we show that irrespective of how Θ X σ(W ) m m × σ (W ) is picked for each W , the map W W is continuous. Since it maps 1− δ , 1 δ → − − h i into itself, Brower’sﬁxed point theorem implies that it has a ﬁxed point W ∗. By construction, the

en corresponding strategy proﬁle σ (W ∗) is an equilibrium of Γc .

Let us ﬁrst show that conditional for all θ1 (t + 1) , for almost all θ2 (t + 1) , the mapping W → l,r Θ1 X σ(W ) m m { }× × φ ( θ (t + 1) , q ) is locally constant in W . Fix θ1 (t + 1) . For a given W 1 δ , 1 δ , h i ∈ − − − h i θ2 (t + 1) Θ2, and k l, r , the proﬁle of continuation values (Uk (θ (t + 1) , x) + δWk (θ1 (t + 1) , x))x X ∈ ∈ { } ∈ induces an ordinal preferences over X, which we denote W,θ2(t+1) . From Assumption 2(i), for al- k W,θ2(t+1) most all θ2 (t + 1) , is strict. Since Uk + δWk is continuous in Wk, at any θ2 (t + 1) at k which W,θ2(t+1) is strict for k l, r , W,θ2(t+1) is locally constant in W for k l, r . For all k ∈ { } k ∈ { } σ(W ) such θ2 (t + 1) , Lemma 4 part 1 implies then that φ ( θ (t + 1) , q ) is locally constant in W, as h i needed.

The law of iterated expectations yields

σ(W ) W (θ1, q) (23)

σ(W ) σ(W ) = E E Uk θ (t + 1) , φ ( θ (t + 1) , q ) + δWk θ1 (t + 1) , φ ( θ (t + 1) , q ) . θ1(t) θ1(t+1) h i h i h i

σ(W ) Fix θ1 (t) and θ1 (t + 1) . Since W φ ( θ (t + 1) , q ) is locally constant, and thus locally → h i continuous, the term inside the bracket in (23) is continuous in W for almost all θ2 (t + 1) . Since it is uniformly bounded, a standard application of Lebesgue’sdominated convergence theorem implies

σ(W ) that its expectation over θ2 (t + 1) is continuous in W, and thus W W is continuous. →

47 Proof of Proposition 8. Let p be the biases corresponding to the equilibrium characterized

p 2 in the proof of Proposition 5. For all k l, r and all θ Θ, let U (θ, x) = (x θ bk pk) ∈ { } ∈ k − − − − p and let nk (θ) be such that xnk(θ) is the greatest alternative that maximizes Uk (θ, x) in X. Note that xnr(θ) > xnl(θ). Let σ be the strategy profile defined as follows: in any direction state θ, q , the proposer k h i plays right if x q and left otherwise. In any voting state θ, q, q , each player k votes yes nk(θ) ≥ h 0i if q0 is closer to her ideal point xnk(θ) than q, and no otherwise. By construction, σ is Markov. If players play σ, then for almost all θ Θ and all q X, the outcome in a given period is x if ∈ ∈ nr(θ) x q, x if q x , and q otherwise. nr(θ) ≤ nl(θ) ≤ nl(θ) p Since for almost all θ R, x U (θ, x) is strictly single-peaked over X, from Riboni and ∈ → k ex Ruge-Murcia (2010) we know that σ is an equilibrium of Γc (X, b + p) irrespective of the proposer ex recognition rule. It is easy to see that σ is also an equilibrium of Γc (X, b + p) for any profile of payoff functions U 0 as long as in all states θ R, for all k l, r ,U 0 (θ, x) induces the same ordinal ∈ ∈ { } k p en preferences over xn, xn+1 as U (θ, x). So, to show that σ is also an equilibrium of Γ (X, b) , { } k c en it suffi ces to show that the continuation value in Γc (X, b) when players play σ is such that for σ all k l, r and all θ R, Uk (θ, x) + δW (θ, x) induces the same ordinal preferences over all ∈ { } ∈ p xn, xn+1 as U (θ, x). { } k en To show the latter property, consider a state θ, xn and a state θ, xn+1 in Γ (X, b). By h i h i c σ σ construction of σ, φ ( θ, xn ) differs from φ ( θ, xn+1 ) if and only if nl (θ) n and n+1 nr (θ) , h i h i ≤ ≤ in which case the status quo stays in place in both cases. By definition of nl (θ) and nr (θ), we have nl (θ) n and n + 1 nr (θ) if and only if ≤ ≤

2 2 2 2 (xn θ bl pl) > (xn+1 θ bl pl) and (xn θ br pr) (xn+1 θ br pr) , − − − − − − − − − − − − ≤ − − − −

xn+xn+1 xn+xn+1 or equivalently, θ br pr, bl pl . Therefore, for all k l, r and θ Θ, ∈ 2 − − 2 − − ∈ { } ∈ h

σ σ W (θ, xn+1) W (θ, xn) k − k xn+xn+1 bl pl 2 − − 2 2 σ σ = (xn+1 ζ bk) + (xn ζ bk) + δ (W (ζ, xn+1) W (ζ, xn)) f (ζ θ) dζ. xn+xn+1 br pr − − − − − − − Z 2 − − h i

48 σ σ As argued in the proof of Proposition 5, (W (θ, xn+1) W (θ, xn))θ Θ that solves the above − ∈ equation is equal to the continuation value of the cutoff strategy profile with cutoffs c = ( xn+xn+1 2 − xn+xn+1 en en bl pl, br pr) in the 2 alternative game Γ ( xn, xn+1 , b) . In Γ ( xn, xn+1 , b) , − 2 − − − { } { } σ σ the equilibrium cutoff is such that Uk (θ, xn+1) + δW (θ, xn+1) Uk (θ, xn) + δW (θ, xn) if and k ≥ k xn+xn+1 p p only if θ ck = bk pk, which in turn is equivalent to U (θ, xn+1) U (θ, xn). Hence, ≥ 2 − − k ≥ k σ p x Uk (θ, x) + W (θ, x) is ordinally equivalent to x U (θ, x) over xn, xn+1 , as needed. → k → k { }

6.3 Proofs for Section 4.3

For all θ Θ, x X, and n N, let V n (θ, q) be the continuation value for player k in the ∈ ∈ ∈ k game Γen of implementing alternative q X at some bargaining time t such that θ (t) = θ, and ℘,∆n ∈ conditional on players playing σn from t + ∆n onwards:

∆n n ρt ρ∆n n V (θ, q) = E (uk (θ (t) , q)) ρe− dt +e− E V (θ (∆n) , y) φ (θ (∆n) , q; y) . k θ(0)=θ θ(0)=θ  k σn  0 y X Z X∈  

Since σn is a Markov perfect equilibrium, our assumption on the bargaining stage game ℘ implies that for all θ Θ, q, x X, and k l, r , ∈ ∈ ∈ { }

φ (θ, q; x) > 0 V n (θ, x) V n (θ, q) . (24) σn ⇒ k ≥ k

For all n, let en (t): t 0 be the continuous-time process equal to the path of the bargaining q ≥ outcome in the game Γen when players play σ and the initial status quo is q. For all p > 0, let ℘,∆n n n eˆ (θ, q, p) denote the random variable on X that is equal to the bargaining outcome under σn at a bargaining time p∆n if the state and status quo at that time are θ and q.

Deﬁnition 5 For all n N and all q X, a continuous time stochastic process dn (t): t 0 is ∈ ∈ { ≥ } a deviation from en (t): t 0 if q ≥ n (i) for all integer p 0, d (t) is constant on t [p∆n, (p + 1) ∆n), ≥ ∈ (ii) dn (0) = q,

n n n n (iii) for all integer p > 0, d (p∆n) is such that d (p∆n) is either equal to eˆ (θ (p∆n) , d ((p 1) ∆n) , p) − n or to the status quo d ((p 1) ∆n). − 49 Hence, dn (t): t 0 is a deviation of en (t): t 0 if d departs from the equilibrium transi- { ≥ } q ≥ tion eˆn only by staying at the status quo. If dn (t): t 0 is an continuous time processes on X { ≥ } that is measurable with probability 1, we denote the (random) discounted payoﬀ of player k from dn as follows:

∞ n ρt Πk (d) = uk (θ (t) , d (t)) ρe− dt. Z0 n n Note that if m denotes a uniform bound on ﬂow payoﬀs, for any two such processes d and d 0,

n n Πk (d ) Πk d 0 2m. (25) − ≤

n n Note also that Eθ(0)=θ Πk eq = Vk (θ, q). Lemma 5 If x is stable at (q, θ) in the limit, then there exists a neighborhood Bθ of θ such that for all ζ Bθ, ∈ n n Vk (ζ, q) Vk (ζ, x) n 0. (26) − → →∞

Proof. Property (i) in Deﬁnition 4 implies that there exists a neighborhood Bθ of θ, some

ε > 0, some N > 0, and a sequence of alternatives x1, ..., xM with x1 = q and xM = x such that for all m = 1, ..., M 1, all ζ Bθ, and all n N, φσ (ζ, xm; xm+1) > ε. For all n N, Let − ∈ ≥ n ∈ dn (t): t 0 be a deviation from en (t): t 0 such that dn either stays at status quo or moves q ≥ q ≥ q n from xm to xm +1 whenever the equilibrium transition eˆ does (point (i) below), and after reaching

n n x, dq follows the equilibrium transition eˆ (point (ii) below). Formally, for all p > 0, n n n (i) if for all p < p, d (p ∆n) = x, and if d ((p 1) ∆n) = xm for some m < M, then d (p∆n) = 0 q 0 6 q − q n n n n xm+1 if eˆ (θ (p∆n) , d ((p 1) ∆n) , p) = xm+1 and d (p∆n) = d ((p 1) ∆n) otherwise, − q q − n n n n (ii) if for some p < p, d (p ∆n) = x, then d (p∆n) =e ˆ θ (p∆n) , d ((p 1) ∆n) , p . 0 q 0 q q − n n n Since dq is a deviation of eq , then at all bargaining times dq either follows the equilibrium path or deviates by staying at the status quo. Therefore, individual rationality implies that for all k and all ζ Θ, ∈ n n n E Πk d E Πk e = V (ζ, q) . θ(0)=ζ q ≤ θ(0)=ζ q k Property (i) in Deﬁnition 4 together with (24) imply that for all k l, r and all ζ Bθ, ∈ { } ∈

n n n n n V (ζ, q) = V (ζ, x1) V (ζ, x2) ... V (ζ, xM ) = V (ζ, x) . k k ≤ k ≤ ≤ k k 50 The two inequalities above imply that

n n n n E Πk d V (ζ, q) V (ζ, x) = E [Πk (e )] . (27) θ(0)=ζ q ≤ k ≤ k θ(0)=ζ x n n In the rest of the proof, we show that lim infn Eθ(0)=ζ Πk dq Πk (ex) 0, which, to- →∞ − ≥ gether with (27), implies (26). Let tˆB be the first exit time of θ(t): t 0 from Bθ. Property θ { ≥ } n n (ii) in Definition 4 implies that for all n N and all t tˆB , e (t) = x. Let pˆ be the first ≥ ≤ θ x n n period p such that dq (p∆n) = x. If the deviation dq reaches x before the state exits Bθ, that is, n n n if ∆npˆ tˆB , then at any time t < ∆npˆ , the flow payoff from e (t) is uk (θ (t) , x) while by ≤ θ x n definition of m, the flow payoff from d (t) is no smaller than uk (θ (t) , x) 2m, and at any time q − n n t ∆npˆ , by point (ii) in the definition of d , the deviation follows the equilibrium path, that is, ≥ q n n n n ˆ dq (t) = ex (t). If the state exits Bθ before the deviation dq reaches x, that is, if ∆npˆ > tB, then n n from (25), we have Πk d Πk (e ) 2m. Therefore, q ≥ x − n n E Πk d Πk (e ) (28) θ(0)=ζ q − x n n ρ∆npˆ ρ∆npˆ n E 1 n 1 e ( 2m) + e 0 + Pr ∆ pˆ > tˆ ( 2m) . θ(0)=ζ ∆npˆ tˆB − − n Bθ ≥ ≤ θ − − × θ(0)=ζ − h i n By definition of dq and from property (i) in Definition 4, as long as θ stays in Bθ, the probability n n that dq moves from some xm to some xm+1 is at least ε. Therefore, conditional on dq reaching x n before tˆB (i.e., conditional on ∆npˆ tˆB ), the probability that it does so in more than p rounds θ ≤ θ is smaller than the probability of having less than M successes out of p draws in a Bernoulli trial

n n p m p m with success probability ε . Hence, Pr pˆ p ∆npˆ tˆB ε (1 ε) − . So, for ≥ | ≤ θ ≤ m

n ρ∆npˆ ρ∆np E 1 n 1 e E 1 n 1 e 1 n + 1 n 1 n θ(0)=ζ ∆npˆ tˆB − θ(0)=ζ ∆npˆ tˆB − pˆ

51 Likewise, for all p N+, ∈

ˆ n ˆ ˆ n Pr tBθ < ∆npˆ Pr tBθ < p∆n + Pr tBθ p∆n &p ˆ > p θ(0)=ζ ≤ θ(0)=ζ θ(0)=ζ ≥ ˆ p m p m Pr tBθ < p∆n + ε (1 ε) − . ≤ θ(0)=ζ m − m

p m p m From Assumption 3, for all ζ Bθ, Pr tˆB < p∆n tends to 0 as p∆n 0. Also, ε (1 ε) − ∈ θ(0)=ζ θ → m − 1/2 tends to 0 as p . Therefore, if we take p = ∆n− in the two inequalities above, we obtain → ∞ n ρ∆npˆ j k n that E 1 n 1 e and Pr tˆ < ∆ pˆ both tend to 0 as n . This θ(0)=ζ ∆npˆ tˆB − θ(0)=ζ B n ≤ − → ∞ shows that theh right-hand side of (28)i also tends to 0 as n , which completes the proof. → ∞ n,s Proof of Proposition 9. For all s > 0 and all n N, let dq (t): t 0 be a deviation ∈ { ≥ } n n,s from e (t): t 0 (see Deﬁnition 5) such that dq (t) stays at the status quo before s/∆n (point q ≥ n,s n (i) below) and dq follows the equilibrium transition eˆ after s/∆n (point (ii) below). Formally, for all p N+, ∈ n,s (i) if p s/∆n, dq (p∆n) = q, ≤ n,s n n,s (ii) if p > s/∆n, dq (p∆n) =e ˆ (θ (p∆n) , d ((p 1) ∆n) , p). − Using the same argument as in the proof of Lemma 5, property (i) in Deﬁnition 4 together with

(24) imply that

n,s n n n E Πk d V (θ, q) V (θ, x) = E [Πk (e )] (29) θ(0)=θ q ≤ k ≤ k θ(0)=θ x

In the rest of the proof, we show that if by contradiction of Proposition 9, uk (θ, q) > uk (θ, x) , then for n suﬃ ciently large,

n,s n Eθ(0)=θ Πk dq > Eθ(0)=θ [Πk (ex)] , (30) which contradicts (29).

Suppose that uk (θ, q) > uk (θ, x). Since uk is continuous in θ, there exists a neighborhood Bθ0 of θ such that for all ζ B , uk (ζ, q) uk (ζ, x) + for some > 0. Property (ii) in Deﬁnition 4 ∈ θ0 ≥ implies that there exists N > 0 and a neighborhood B of θ such that for all n N and all ζ B , θ00 ≥ ∈ θ00 φ (ζ, x; x) = 1. Let Bθ = B B and let tˆB be the ﬁrst exit time of θ (t): t 0 from Bθ. For ∆n θ0 ∩ θ00 θ { ≥ }

52 n all n N and all t tˆB , θ (t) B so e (t) = x. Therefore, conditional on tˆB > s, for all t s, ≥ ≤ θ ∈ θ00 x θ ≤ n n,s the flow payoff from ex is uk (θ (t) , x) while the flow payoff from dq is uk (θ (t) , q). Conditional n,s n n on tˆB s, inequality (25) implies that Πk (dq ) Πk (e ) 2m. Therefore, if s = ∆n s/∆n θ ≤ ≥ x − b c denotes the last bargaining time before s,

n,s n E Πk d Πk (e ) (31) θ(0)=θ q − x sn 2m Pr tˆ s + E 1 (u (θ (t) , q) u (θ (t) , x)) ρ exp ( ρt) dt Bθ θ(0)=θ tˆB >s k k ≥ − θ(0)=θ ≤ θ 0 − − Z ρsn n n n n +Eθ(0)=θ 1tˆ >se− (Vk (θ (s ) , q) Vk (θ (s ) , x)) Bθ − h i

n n n Since s s, conditional on tˆB > s, θ (s ) B . Moreover, V is normalized to the units ≤ θ ∈ θ00 k of the ﬂow payoﬀs and is thus bounded by m. Therefore, Lemma 5 together with the bounded convergence theorem imply that the last term on the right-hand side of (31) tend to 0 as ∆n tends to 0. Therefore, letting n in (31) we obtain → ∞

n,s n lim inf Eθ(0)=θ Πk d Πk (e ) (32) n q x →∞ − s 2m Pr tˆ s + E 1 (u (θ (t) , q) u (θ (t) , x)) ρ exp ( ρt) dt Bθ θ(0)=θ tˆB >s k k ≥ − θ(0)=θ ≤ θ 0 − − Z ˆ ρs 2m Pr tB s + Eθ(0)=θ 1tˆ >s 1 e− ≥ − θ(0)=θ θ ≤ Bθ − h i ρs (2m + ε) Pr tˆB s + 1 e− ≥ − θ(0)=θ θ ≤ −

ρs From Assumption 3, Prθ(0)=θ tˆB s = o (s), and since 1 e− ρs as s 0, the right-hand θ ≤ − ∼ → side of the above inequality is strictly positive for some s > 0. This shows that for some s > 0, and for n suﬃ ciently large, (30) must hold.

Remark 1 Assumption 3 is satisﬁed for Brownian processes. It is violated for Poisson processes.

Proof. Let θ (t): t 0 is a standard Brownian motion, and let M (t) = maxs [0,t] (θ (s) θ (0)) { ≥ } ∈ − denote its running maximum. As is well known, M (t) /√t has a normal distribution truncated at

0. By symmetry, m (t) = mins [0,t] (θ (s) θ (0)) has the same distribution (more precisely, m (t) ∈ − − has the same distribution). If D (t) = maxs [0,t] θ (s) θ (0) denotes the running maximal devia- ∈ | − | tion, D(t) is equal to max M(t) , m(t) , so D (t) is ﬁrst order stochastically dominated by twice √t √t − √t

53 a truncated normal. If f denotes the p.d.f. of the normal distribution, and if τ B denotes the ﬁrst exit time from [θ b, θ + b] conditional on θ (0) = θ, −

2 D (t) b ∞ t ∞ b t ∞ 2 Pr (τ B t) = Pr 2f (µ) dµ = 2 f (µ) dµ 2 µ f (µ) dµ. ≤ √t ≥ √t ≤ b b2 b t ≤ b2 b Z √t Z √t Z √t

2 Since the normal distribution has a finite second moment, ∞b µ f (µ) dµ 0 as t 0. So the √t → → above inequality shows that Pr (τ B t) = o (t) as t 0. R ≤ → Let θ (t): t 0 be a process in which at any t, θ (t) either stays constant or is redrawn { ≥ } according to some given distribution, and the occurrence of such draws is Poisson distributed with parameter λ. Note that for B suffi ciently small, the process exits B as soon as a Poisson shock occurs, so τ B is approximately the occurrence of the first shock, and Pr (τ B t) λt as θ(0)=θ ≤ ∼ t 0. →

7 Appendix B: Proof of Proposition 6 (not for publication)

To prove Proposition 6, we deﬁne an auxiliary game, construct an equilibrium of that game that has convenient properties, and then show that this equilibrium can be mapped into a partisan

en equilibrium of Γc . Throughout the proof, we will use the conventions in Notation 1. en en For all δ0 [0, 1) , consider the game Γˆ δ0 that diﬀers from Γ in two respects. First, unlike ∈ c c in Γen, when both players vote yes in a voting state θ, q, q , the game moves to the direction c h 0i state θ, q in which the same proposer again chooses the direction again. Second, the discounting h 0i en between the terminal state of a given period and the beginning of the next period is δ as in Γc , but the discounting between two states within a given period is δ0, instead of 1. We also deﬁne a

en en en game Γˆ as follows: Γˆ has the same game form as Γˆ δ0 for δ0 (0, 1) , and the payoff from a c c c ∈ en en strategy profile σˆ in Γˆ is the limit of the payoff of σˆ inΓˆ δ0 as δ0 1. Note that the set of c c → en en en states of Γˆ δ0 for all δ0 [0, 1] is the same as the set of states S of Γˆ and Γ . For all s S, c ∈ c c ∈ σ ên ˆ σ,δ0 ˆ ˆ σ and for any Markov strategy profile σ of Γc δ0 , we define V (s) , φ (s) , and W analogously σ σ σ en to V (s) , φ (s) , and W for the game Γc , and we omit the superscript δ0 when δ0 = 1.

ˆen Lemma 6 There exists a Markov strategy proﬁle σˆ of Γc with the following properties:

ˆen 1. σˆ is stage-undominated in Γc .

54 2. For all n 2, ..., N 1 and all θ, σˆ does not prescribe the proposer to play right with ∈ { − } positive probability in θ, xn , left with positive probability in θ, xn+1 , and yes with positive h i h i probability in θ, xn, xn+1 and in θ, xn+1, xn . h i h i 3. For all n 2, ..., N 1 , if σˆ prescribes the proposer to play a mixed strategy in some ∈ { − } direction state θ, xn , then it prescribes both players to vote yes with positive probability in h i θ, xn, xn+1 if and only if it does so also in θ, xn, xn 1 . h i h − i σˆ 4. For all n 2, ..., N 1 , if φˆ ( θ, xn ) = xn with probability 1, then σˆ prescribes a mixed ∈ { − } h i action at θ, xn , and it prescribes some player to vote no with probability 1 in θ, xn, xn+1 h i h i and in θ, xn, xn 1 (possibly a distinct voter for each voting state). h − i

en Proof. Note that for all δ0 (0, 1) , Γˆ δ0 is a standard discounted stochastic game. Since ∈ c the state space S is finite, and since each player has at most two actions in each state, the agent- normal form game associated with this game (as defined in Selten 1975) has a finite set of players and strategies. Therefore, it has a trembling-hand perfect Nash equilibrium σ˜ δ0 . Since σ˜ δ0 is ên a Nash equilibrium, the corresponding strategy profile of Γc δ0 , which we denote by σˆ δ0 , is a

Markov perfect equilibrium, and since σ˜ δ0 is trembling-hand perfect, σˆ δ0 is stage-undominated.

We now show that σˆ δ0 satisﬁes the property 2 of the Lemma. Suppose that σˆ δ0 prescribes the proposer, say r, to play right with positive probability in θ, xn and to vote yes with h i positive probability in θ, xn, xn+1 . In that case, stage undomination of σˆ δ0 in θ, xn, xn+1 im- h i h i σˆ(δ0),δ0 σˆ(δ0),δ0 en plies that Vˆr ( θ, xn ) Vˆr ( θ, xn+1 ) . The nature of the game Γˆ δ0 implies that hh ii ≤ h i c σˆ(δ0),δ0 σˆ(δ0),δ0 σˆ(δ0),δ0 Vˆr ( θ, xn, xn+1 ) is a convex combination of δ0Vˆr ( θ, xn ) and δ0Vˆr ( θ, xn+1 ), h i hh ii h i σˆ(δ0),δ0 σˆ(δ0),δ0 so the previous inequality implies that Vˆr ( θ, xn, xn+1 ) δ0Vˆr ( θ, xn+1 ) . Since σˆ δ0 h i ≤ h i prescribes r to play right with positive probability in θ, xn , stage undomination of σˆ δ0 re- h i σˆ(δ0),δ0 σˆ(δ0),δ0 quires that Vˆr ( θ, xn ) = δ0Vˆr ( θ, xn, xn+1 ) , so the previous inequality implies that h i h i σˆ(δ0),δ0 2 σˆ(δ0),δ0 Vˆr ( θ, xn ) δ0 Vˆr ( θ, xn+1 ) . By the same token, if σˆ δ0 prescribes r to play left h i ≤ h i with positive probability in θ, xn+1 and to vote yes with positive probability in θ, xn+1, xn , h i h i σˆ(δ0),δ0 2 σˆ(δ0),δ0 necessarily, Vˆr ( θ, xn+1 ) δ0 Vˆr ( θ, xn ) . Since δ0 < 1, the two conditions cannot h i ≤ h i be met at the same time.

The ﬁniteness of S and of the action space implies the existence of a sequence δ0 1 such n → σ,δ that σˆ δ0 converges to some σˆ. Since for all s S, Vˆ 0 (s) is continuous in σ, δ0 on the space n ∈ 55 of Markov strategy proﬁles, Vˆ σˆ(δn0 ),δn0 tends to Vˆ σˆ (recall that Vˆ σˆ denotes Vˆ σ,ˆ 1). Therefore, if

Vˆ σˆ (s) > Vˆ σˆ (s ) for some k l, r and some s, s S, necessarily, Vˆ σˆ(δn0 ),δn0 (s) > Vˆ σˆ(δn0 ),δn0 (s ) for k k 0 ∈ { } 0 ∈ k k 0 ˆen some n N. Since σˆ δn0 is stage-undominated in Γc δn0 , this implies that σˆ is stage-undominated ∈ ˆen in Γc . Clearly, σˆ is Markov, and it inherits property 2 by continuity (because property 2 simply states that the product of four probabilities is 0, and since each of the four probabilities converge, the product of the limit must also be 0).

To prove property 3, suppose that σˆ prescribes the proposer to play right with a probability pr ( θ, xn ) (0, 1) in some direction state θ, xn , both players to vote yes with positive probability h i ∈ h i in θ, xn, xn+1 , and some player to vote no with probability 1 in θ, xn, xn 1 ; the proof in the h i h − i opposite case is identical. We show that we can change σ by making her prescribe a pure action in

θ, xn without changing the other properties of σ.ˆ Let pyes ( θ, xn, xn+1 ) denote the probability h i h i that the proposer votes yes in θ, xn, xn+1 . Consider the strategy profile σˆ0 that differs from σˆ h i only in that σˆ0 prescribes the proposer to play right with probability 1 in θ, xn and yes with h i probability pr ( θ, xn ) pyes ( θ, xn, xn+1 ) in θ, xn, xn 1 , the actions of the other player being h i h i h − i unchanged. Then starting from θ, xn , the probability that the state becomes θ, xn, xn+1 and h i h i that the proposer votes yes in that state is the same with σˆ and σˆ0. So σˆ and σˆ0 induce the same distribution over bargaining outcomes starting from state θ, xn , and thus in all direction states. h i As a result, σˆ0 inherits stage-undomination from σ.ˆ Moreover, σˆ0 still satisfies property 2 because if σˆ0 prescribes the proposer to vote yes with positive probability at θ, xn, xn+1 , so does σ,ˆ and h i if σˆ0 prescribes a given direction with positive probability at θ, xn , so does σˆ. Repeating this h i procedure for all such states θ, xn , we obtain a stage undominated strategy profile which satisfies h i properties 1, 2, and 3. In what follows, the resulting strategy profile is denoted σˆ0.

σˆ0 To prove property 4, suppose that φˆ ( θ, xn ) = xn with probability 1, and suppose that σˆ0 h i prescribes both players to vote yes with positive probability in a voting state that follows θ, xn , h i say θ, xn, xn 1 . For this, σˆ0 must prescribes her to play right with probability 1 and some player h − i to vote no with probability 1 in θ, xn, xn+1 . Therefore (assuming that r is the proposer; the other h i case is analogous),

ˆ σˆ0 ˆ σˆ0 ˆ σˆ0 Vr ( θ, xn ) = Vr ( θ, xn, xn+1 ) Vr ( θ, xn, xn 1 ) . hh ii h i ≥ h − i

56 ˆ σˆ0 ˆ σˆ0 ˆ σˆ0 Moreover, Vr ( θ, xn, xn 1 ) is a convex combination of Vr ( θ, xn ) and Vr ( θ, xn 1 ) , with a h − i hh ii h − i ˆ σˆ0 positive weight on Vr ( θ, xn 1 ) since both players vote yes with positive probability in θ, xn, xn+1 . h − i h i ˆ σˆ0 ˆ σˆ0 Therefore, the above inequality implies that Vr ( θ, xn ) Vr ( θ, xn 1 ). So it is stage- hh ii ≥ h − i undominated for player r to vote no with probability 1 in θ, xn, xn 1 . Consider the strategy profile h − i σˆ00 that differs from σˆ0 only in that σˆ00 prescribes r to vote no with probability 1 in θ, xn, xn 1 and h − i randomize between left and right at θ, xn . Note that σˆ0 and σˆ00 induce the same distribution over h i bargaining outcomes starting from state θ, xn , and thus in all direction states. As a result, σˆ00 h i inherits stage-undomination from σˆ0. Clearly, σˆ00 still satisfies property 2, and since σˆ00 prescribes some player to vote no with probability 1 in θ, xn, xn 1 and in θ, xn, xn+1 , it inherits property h − i h i 3 from σˆ0. Repeating this procedure for all such states θ, xn , we obtain a stage undominated h i strategy profile which satisfies properties 1, 2, 3, and 4. ên Proof of Proposition 6. Let σˆ be an equilibrium of Γc that satisfies the properties of Lemma 6. In what follows, we say that a state s is on the path of σˆ if it is reached with positive probability from some direction state when players play σˆ in Γên (including the original θ, q ). c h i Hence, all direction states are on the path of σ,ˆ but not necessarily all voting states.

en Step 1: There exists a Markov strategy proﬁle σ of Γc that satisﬁes:

(i) For any state s that is on the path of σˆ, σ prescribes the same action as σˆ in state s, and σˆ φσ (s) = φˆ (s) . Moreover, V σ and Vˆ σˆ coincides on any states on the path of σˆ, and also on

any terminal states.

(ii) In any state θ, x, x that is not on the path of σˆ, σ prescribes each voter k to vote yes with h 0i probability 1 if V σ ( θ, x ) > V σ ( θ, x ) , and no with probability 1 otherwise. k h 0i k hh ii Set σ to prescribe the same actions as σˆ in any state that is on the path of σˆ, consistently with property (i) . Irrespective of how we specify the actions prescribed by σ in the other states, φσ and σˆ φˆ must coincide on all states on the path of σ.ˆ To see why, note ﬁrst that for any direction state

θ, xn , since σ and σˆ coincide in that state, the probabilities that the game moves from θ, xn to h i h i ˆen θ, xn, xn 1 , θ, xn, xn+1 , and θ, xn are the same when players play σˆ in the game Γc as when h − i h i hh ii en player play σ in the game Γ . Consider now a voting state on the path of σˆ, say θ, xn, xn+1 . c h i By construction of σ, the probability that some player votes no, and thus that the game moves to θ, xn is the same in both situations as well. However, when both players are prescribed by hh ii 57 en σˆ and σ to vote yes in such a state, the game moves to θ, xn+1 in Γˆ and to θ, xn+1, xn+2 in h i c h i en Γ . When σˆ prescribes the proposer to play right with probability 1 in θ, xn+1 , then the state c h i en moves to θ, xn+1, xn+2 in Γˆ as well. Consider then the case in which σˆ prescribes the proposer h i c to play left with positive probability π in θ, xn+1 . In that case, Lemma 6 part 2 implies that σˆ h i prescribes the proposer to vote no with probability 1 in θ, xn+1, xn . So either π = 1, and hence h i σˆ φˆ ( θ, xn+1 ) = xn+1 with probability 1, or π < 1, in which case Lemma 6 part 3 implies that h i σˆ φˆ ( θ, xn+1 ) = xn+1 with probability 1 as well. Lemma 6 part 4 implies then that θ, xn+1, xn+2 h i h i is on the path of σ,ˆ and that σˆ prescribes some player to vote no with probability 1 in that state.

Therefore, so does σ. So in that case, the game moves to θ, xn+1 with probability 1 in both hh ii situations.

The actions prescribed by σ in the states that are not on the path of σˆ are speciﬁed as follows. Let

θ, xn, xn+1 be a voting state that is not on the path of σˆ. Then there exists m such that for all h i s = n, ..., m 1, θ, xs, xs+1 is not on the path of σˆ, and either m = N or θ, xm, xm+1 is on the − h i h i path of σˆ. Then in both cases, the continuation payoﬀs in case of acceptance or rejection of xm

en σ σ σˆ in the game Γc at θ, xm 1, xm ,V ( θ, xm ) and V ( θ, xm 1 ) , are equal to V ( θ, xm ) h − i hh ii hh − ii hh ii σˆ and V ( θ, xm 1 ) respectively. We set σ in θ, xm, xm+1 to prescribe each player to vote yes hh − ii h i if and only if her continuation payoff of acceptance is strictly higher than that of rejection. We reiterate the procedure until all states of the form θ, xn, xn+1 that are not on the path of σˆ have h i been assigned an action, and we do the same procedure for all states of the form θ, xn, xn 1 that h − i σ are not on the path of σ.ˆ At each step, V ( θ, xn, xn 1 ) is uniquely defined. A simple induction h − i shows that σ satisfies property (ii) .

σ Step 2: If σ satisfies property (i) and (ii) from step 1, if n + 1 < N, and if V ( θ, xn ) < k h i σ V ( θ, xn+1, xn+2 ) for the proposer, then either θ, xn+1, xn+2 is on the path of σˆ or σ prescribes k h i h i some player to vote no with probability 1 at θ, xn+1, xn+2 . h i σ To fix ideas let r be the proposer. Suppose that V ( θ, xn ) < Vr ( θ, xn+1, xn+2 ) and θ, xn+1, xn+2 r h i h i h i is not on the path of σ.ˆ Consider the path of play starting from state θ, xn+1, xn+2 when players h i en play σ in the game Γ . Let p be such that θ, xp, xp+1 is either the first state on the path of σˆ, c h i or the first state at which σ prescribes some player to vote no with positive probability, whichever

σ σ comes ﬁrst. By deﬁnition of p and from property (ii) ,V ( θ, xn+1, xn+2 ) V ( θ, xp, xp+1 ) . So h i ≤ h i

58 σ the assumption that V ( θ, xn ) < Vr ( θ, xn+1, xn+2 ) together with property (i) imply that r h i h i

σˆ σ σ Vˆ ( θ, xn ) = V ( θ, xn ) < V ( θ, xp, xp+1 ) , (33) r h i r h i r h i

By deﬁnition of p, for all s n + 1, ..., p 1 , θ, xs, xs+1 is not on the path of σˆ but σ prescribes ∈ { − } h i both players to vote yes in that state. Using successively properties (i) and (ii) , this implies that for all k l, r , ∈ { }

σˆ σ σ Vˆ ( θ, xs ) = V ( θ, xs ) < V ( θ, xp, xp+1 ) for all s n + 1, ..., p 1 . (34) k hh ii k hh ii k h i ∈ { − }

Since by deﬁnition of p, θ, xp 1, xp is not on the equilibrium path, it must be that σˆ prescribes r h − i to play left with probability 1 in θ, xp 1 . From Lemma 6 part 2 and the fact that σˆ is Markov, h − i ˆσˆ ˆ σˆ this implies that the support of φr ( θ, xp 1, xp 2 ) is to the left of xp 1, so Vr ( θ, xp 1 ) is a h − − i − h − i ˆ σˆ ˆ σˆ convex combination of Vr ( θ, xs ) and Vr ( θ, xn ) , which from (33), and (34) are hh ii s=n+1,...,p 1 h i − σˆ all are strictly smaller than Vˆ ( θ, xp, xp+1 ) , so r h i

ˆ σˆ ˆ σˆ Vr ( θ, xp 1 ) < Vr ( θ, xp, xp+1 ) . (35) h − i h i

From (35), for right not to be a one-step proﬁtable deviation from σˆ for r in θ, xp 1 , it must be h − i that σˆ prescribes l to vote no with positive probability in θ, xp 1, xp , so h − i

ˆ σˆ ˆ σˆ Vl ( θ, xp 1 ) Vl ( θ, xp ) . (36) hh − ii ≥ h i

Suppose θ, xp, xp+1 is not on the path of σˆ. This means that σˆ prescribes r to play left with h i probability 1 at θ, xp . Then since σˆ is stage undominated, (35) implies that σˆ must prescribe r to h i σˆ vote no with probability 1 in θ, xp, xp 1 , so φ ( θ, xp ) = xp with probability 1. Lemma 6 part 3 h − i h i implies then that θ, xp, xp+1 is on the path of σ,ˆ which is a contradiction. So θ, xp, xp+1 is on h i h i σˆ σˆ σ the path of σˆ. Property (i) implies then Vˆ ( θ, xp ) = Vˆ ( θ, xp, xp+1 ) = V ( θ, xp, xp+1 ) . This h i h i h i ˆ σˆ ˆ σˆ σ equality together with (36) implies that Vl ( θ, xp 1 ) Vl ( θ, xp ) = Vl ( θ, xp, xp+1 ), which hh − ii ≥ h i h i contradicts (34).

Step 3: If σ satisﬁes property (i) and (ii) from Step 1, then σ prescribes stage undominated

59 en actions in any direction state in Γc .

Let θ, xn be a direction state for some n 2, ..., N 1 , and to ﬁx ideas let r be the proposer. h i ∈ { − } For the sake of clarity, we divide the proof into several cases.

Case 1: σ prescribes a mixed action at θ, xn . h i From property (i), so does σˆ. From Lemma 6 part 1, σˆ is stage-undominated, so mixing requires ˆ σˆ ˆ σˆ that Vr ( θ, xn, xn+1 ) = Vr ( θ, xn, xn 1 ). Since θ, xn, xn+1 and θ, xn, xn 1 are on the path of h i h − i h i h − i σ σ σ,ˆ property (i) implies then that Vr ( θ, xn, xn+1 ) = Vr ( θ, xn, xn 1 ) , so no profitable deviation h i h − i exists at θ, xn . h i Case 2: σ prescribes a pure action in θ, xn , say left; hence, so does σ.ˆ h i Consider the one-step deviation in which r plays right in that state, and both players play σ afterwards. Assume by contradiction that this deviation is profitable. Since by saying left and then no, r can assure herself θ, xn , for her right to be profitable, σ must prescribe both players hh ii to vote yes with positive probability at θ, xn, xn+1 . If n + 1 = N, then it is easy to see that the h i ên same deviation is also a profitable deviation from σˆ in the game Γc , which contradicts Lemma 6 part 1. In what follows, we assume n < N 1. Since θ, xn, xn+1 is not on the path of σ,ˆ property − h i (ii) implies then that

σˆ σ σ for all k l, r , Vˆ ( θ, xn ) = V ( θ, xn ) < V ( θ, xn+1, xn+2 ) , (37) ∈ { } k h i k h i k h i where the ﬁrst equality comes from property (i) .

Case 2.1: σˆ prescribes r to play right with probability 1 in θ, xn+1 . h i σ Then θ, xn+1, xn+2 is on the path of σ,ˆ and then from property (i) ,V ( θ, xn+1, xn+2 ) = h i h i σˆ σˆ Vˆ ( θ, xn+1, xn+2 ) = Vˆ ( θ, xn+1 ) , so (37) implies that for all k l, r , h i h i ∈ { }

σˆ σˆ Vˆ ( θ, xn ) < Vˆ ( θ, xn+1 ) . (38) k h i k h i

Therefore, stage-undomination of σˆ requires that both players vote yes in θ, xn, xn+1 under σ.ˆ So h i en playing right in θ, xn in the game Γˆ is a proﬁtable one-step deviation from σ,ˆ which contradicts h i c σˆ0s stage undomination.

Case 2.2: σˆ prescribes a mixed action in θ, xn+1 . h i

60 σ σˆ From property (i) ,V ( θ, xn+1, xn+2 ) = Vˆ ( θ, xn+1, xn+2 ) , and since σˆ is stage-undominated, h i h i σˆ σˆ Vˆ ( θ, xn+1, xn+2 ) = Vˆ ( θ, xn+1 ). Substituting this equality in (37), we obtain that (38) holds r h i r h i for k = r. Since σˆ is stage undominated in θ, xn , the inequality (38) for k = r implies that h i σˆ prescribes r to vote no with probability 1 in θ, xn+1, xn . Lemma 6 part 3 implies then that h i σˆ prescribes some player to vote no with probability 1 also in θ, xn+1, xn+2 . Therefore, using h i property (i) ,

σ σ σˆ V ( θ, xn+1, xn+2 ) = V ( θ, xn+1 ) = Vˆ ( θ, xn+1 ) . h i h i k h i

Substituting the above equality in (37), we obtain that (38) also holds for k = l. The same argument as in the previous case leads to the same contradiction.

Case 2.3: σˆ prescribes r to play left with probability 1 in θ, xn+1 , so θ, xn+1, xn+2 is not on h i h i the path of σ.ˆ

Then, step 2 together with (37) imply then σ prescribes some player to vote no with probability 1 at θ, xn+1, xn+2 . This implies that for all k l, r , h i ∈ { }

σˆ σ σ Vˆ ( θ, xn+1 ) = V ( θ, xn+1 ) V ( θ, xn+1, xn+2 ) , k h i k h i ≥ k h i where the ﬁrst equality comes from property (i) . Substituting the above inequality in (37), we obtain that (38) holds for all k l, r , and thus the same contradiction as in the previous case. ∈ { } Step 4: There exists a stage-undominated σ0 which generates the same transition between states (and thus the same continuation value) as σ.

Let σ be the strategy constructed in step 1. From step 3, it is stage-undominated in any direction state. It remains to show that σ can be made stage undominated in all voting states without changing the transition between states. Consider a voting stage s, and suppose to ﬁx ideas that s = θ, xn, xn+1 the proposer is r (the proofs in the other cases are identical). If θ, xn, xn+1 is h i h i not on the path of σˆ, then property (ii) implies that σ is stage undominated in s. If θ, xn, xn+1 h i is on the path of σˆ, then by (i) , σ and σˆ prescribe the same action. Moreover, the continuation

σ σˆ value from saying no is the same in both cases: V ( θ, xn ) = V ( θ, xn ). The cases below hh ii hh ii deal with diﬀerent possible plays after yes is played.

Case 1: σˆ prescribes r to play right with probability 1 in θ, xn+1 or n + 1 = N. h i

61 en Property (i) implies that the continuation value from saying yes the same under σ in Γc and under ˆen σˆ in Γc . Hence, stage-undomination of σ comes from stage-undomination of σˆ

Case 2: n + 1 < N and σˆ prescribes r to play left with positive probability in θ, xn+1 . h i Case 2.1: σˆ prescribes r to vote no with probability 1 in θ, xn+1, xn . h i σˆ If r plays left at θ, xn+1 with probability 1, then φ ( θ, xn+1 ) = xn+1 with probability 1. If r h i h i randomizes on direction at θ, xn+1 , then Part 3 of Lemma 6 says that she must veto the change so h i σˆ again φ ( θ, xn+1 ) = xn+1 with probability 1. Hence, part 4 of Lemma 6 implies that σˆ prescribes h i a mixed strategy in θ, xn+1 (so θ, xn+1, xn+2 is on the path of σˆ) and some voter to vote no h i h i with probability 1 in θ, xn+1, xn+2 . This together with property (i) implies that h i

σ σˆ σˆ V ( θ, xn+1, xn+2 ) = Vˆ ( θ, xn+1, xn+2 ) = Vˆ ( θ, xn+1 ) , h i h i h i

en σˆ so the continuation value from yes at θ, xn, xn+1 in Γˆ , Vˆ ( θ, xn+1 ) , is the same as the con- h i c h i en σ tinuation value from yes in Γ ,V ( θ, xn+1, xn+2 ) . Hence, stage-undomination of σ comes from c h i stage-undomination of σˆ

Case 2.2: σˆ prescribes r to vote yes with positive probability in θ, xn+1, xn , and θ, xn+1, xn+2 h i h i is not on the path of σ.ˆ

Part 2 of Lemma 6 implies that σˆ prescribes r to vote no with probability 1 in θ, xn, xn+1 . Then h i from the fact that θ, xn, xn+1 is on the path of σ,ˆ σ also prescribes r to vote no; hence it must be h i σ σ σˆ that V ( θ, xn ) = Vˆ ( θ, xn ) = Vˆ ( θ, xn ). r h i r hh ii r hh ii Case 2.2.1 Suppose

σˆ σ σ Vˆ ( θ, xn ) = V ( θ, xn ) < V ( θ, xn+1, xn+2 ) . (39) r hh ii r h i r h i

Then step 2 implies that σ prescribes some player to vote no with probability 1 at θ, xn+1, xn+2 . h i σ σ σˆ σ Therefore, V ( θ, xn+1, xn+2 ) = V ( θ, xn+1 ) . Thus, (39) implies that Vˆ ( θ, xn ) < V ( θ, xn+1 ) = h i hh ii r hh ii r hh ii σˆ V ( θ, xn+1 ) , where the last equality comes from property (i) . This in turn contradicts the fact r hh ii that a stage-undominated σˆ prescribes r to vote no with probability 1 in θ, xn, xn+1 . h i σˆ σ Case 2.2.2. If (39) does not hold; that is, Vˆ ( θ, xn ) V ( θ, xn+1, xn+2 ), voting no in r hh ii ≥ r h i en θ, xn, xn+1 is stage undominated for r in the game Γ with continuation play σ. If the action h i c

62 prescribed by σ to l in θ, xn, xn+1 is not stage undominated, the change σl at θ, xn, xn+1 to h i h i make it stage undominated. Since r votes no with probability 1, this change will have no impact on the transition between states induced by σ; and hence, is still an equilibrium.

Case 2.3 σˆ prescribes r to vote yes with positive probability in θ, xn+1, xn , and θ, xn+1, xn+2 is h i h i on the path of σ.ˆ

By assumption, θ, xn, xn+1 and θ, xn+1, xn are on the path of σ,ˆ and σˆ prescribes r to vote h i h i yes with positive probability in θ, xn+1, xn , so part 2 of Lemma 6 implies that σˆ prescribes h i σˆ σˆ r to vote no with probability 1 in θ, xn, xn+1 : hence, V ( θ, xn ) = V ( θ, xn ) . Since σˆ h i r h i r hh ii σˆ σˆ is stage-undominated, this implies that Vˆ ( θ, xn ) Vˆ ( θ, xn+1 ) . Moreover, we always have r hh ii ≥ r h i σˆ σˆ σˆ σˆ σˆ Vˆ ( θ, xn+1 ) Vˆ ( θ, xn+1, xn+2 ) . Therefore, V ( θ, xn ) = Vˆ ( θ, xn ) Vˆ ( θ, xn+1, xn+2 ) , r h i ≥ r h i r h i r hh ii ≥ r h i σˆ and since θ, xn+1, xn+2 is on the path of σ,ˆ property (i) implies then that Vˆ ( θ, xn+1, xn+2 ) = h i r h i σ σˆ σ Vˆ ( θ, xn+1, xn+2 ) so Vˆ ( θ, xn ) V ( θ, xn+1, xn+2 ). The rest follows from case 2.2.2. r h i r hh ii ≥ r h i Step 5: The strategy proﬁle σ0 constructed in step 4 is a partisan, stage-undominated Markov

en perfect equilibrium of Γc . Note that since each state has only one player moving or two players voting over two alternatives, stage undomination implies Markov perfection. So from step 4, σ0 is a stage-undominated Markov en ên perfect equilibrium of Γc . We now show that it is partisan. The nature of the game Γc together σˆ with the fact that σˆ is Markov implies that for all θ Θ, φˆ ( θ, q ) is monotonic in q for the first ∈ h i order stochastic dominance order. Therefore, one can easily adapt the argument in the proof of σˆ Proposition 7 to show that Wˆ σˆ satisfies Assumption 1. From step 4, φˆ and φσ0 coincide on all direction states, so Wˆ σˆ = W σ0 , and since U also satisfies Assumption 1, so does U + δW σ0 . Since

φσ0 ( θ, q ) is monotonic in q for the ﬁrst order stochastic dominance order, Lemma 4 part 2 implies h i σ0 σ σ0 that for all θ Θ, Uk θ, φ ( θ, q ) + δW θ, φ ( θ, q ) is weakly increasing for k = r and ∈ h i k h i weakly decreasing for k= l. Taking expectation over θ conditional on the previous period’s state, we obtain that σ0 is partisan.

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