NBER WORKING PAPER SERIES POLITICAL ECONOMY IN A CHANGING WORLD. Daron Acemoglu Georgy Egorov Konstantin Sonin

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1 NBER WORKING PAPER SERIES POLITICAL ECONOMY IN A CHANGING WORLD Daron Acemoglu Georgy Egorov Konstantn Sonn Workng Paper NATIONAL BUREAU OF ECONOMIC RESEARCH 1050 Massachusetts Avenue Cambrdge, MA June 2013 An earler draft was crculated under the ttle Markov Votng Equlbra: Theory and Applcatons. We thank partcpants of Walls Insttute Annual Conference, CIFAR meetng n Toronto, and of semnars at Georgetown, ITAM, Northwestern, London School of Economcs, Stanford, UPenn, Warwck and Zurch for helpful comments. Acemoglu gratefully acknowledges fnancal support from Army Research Offce. The vews expressed heren are those of the authors and do not necessarly reflect the vews of the Natonal Bureau of Economc Research. NBER workng papers are crculated for dscusson and comment purposes. They have not been peerrevewed or been subject to the revew by the NBER Board of Drectors that accompanes offcal NBER publcatons by Daron Acemoglu, Georgy Egorov, and Konstantn Sonn. All rghts reserved. Short sectons of text, not to exceed two paragraphs, may be quoted wthout explct permsson provded that full credt, ncludng notce, s gven to the source.

2 Poltcal Economy n a Changng World Daron Acemoglu, Georgy Egorov, and Konstantn Sonn NBER Workng Paper No June 2013 JEL No. C71,D71,D74 ABSTRACT We provde a general framework for the analyss of the dynamcs of nsttutonal change (e.g., democratzaton, extenson of poltcal rghts or represson of dfferent groups), and how these dynamcs nteract wth (antcpated and unantcpated) changes n the dstrbuton of poltcal power and n economc structure. We focus on the Markov Votng Equlbra, whch requre that economc and poltcal changes should take place f there exsts a subset of players wth the power to mplement such changes and who wll obtan hgher expected dscounted utlty by dong so. Assumng that economc and poltcal nsttutons as well as ndvdual types can be ordered, and preferences and the dstrbuton of poltcal power satsfy natural sngle crossng (ncreasng dfferences) condtons, we prove the exstence of a pure-strategy equlbrum, provde condtons for ts unqueness, and present a number of comparatve statc results that apply at ths level of generalty. We then use ths framework to study the dynamcs of poltcal rghts and represson n the presence of radcal groups that can stochastcally grab power. We characterze the condtons under whch the presence of radcals leads to represson (of less radcal groups), show a type of path dependence n poltcs resultng from radcals comng to power, and dentfy a novel strategc complementarty n represson. Daron Acemoglu Department of Economcs MIT, E52-380B 50 Memoral Drve Cambrdge, MA and CIFAR and also NBER daron@mt.edu Konstantn Sonn New Economc School 47 Nakhmovsky prosp. Moscow, RUSSIA ksonn@nes.ru Georgy Egorov Kellogg School of Management Northwestern Unversty 2001 Sherdan Road Evanston, IL and NBER g-egorov@kellogg.northwestern.edu An onlne appendx s avalable at:

3 1 Introducton Poltcal change often takes place n the mdst of uncertanty and turmol, whch sometmes brngs to power or paves the road for the rse of the most radcal factons, such as the mltant Jacobns durng the Regn of Terror n the French Revoluton or the Nazs durng the crss of the Wemar Republc. The possblty of extreme outcomes s of nterest not only because the resultng regmes have caused much human sufferng and powerfully shaped the course of hstory, but also because, n many epsodes, the fear of such radcal extremst regmes has been one of the drvers of represson aganst a whole gamut of opposton groups. The events leadng up to the October Revoluton of 1917 n Russa llustrate both how an extremst frnge group can ascend to power, and the dynamcs of represson partly motvated by the desre of rulng eltes to prevent the empowerment of extremst and sometmes also of more moderate elements. Russa entered the 20 th century as an absolute monarchy, but started a process of lmted poltcal reforms n response to labor strkes and cvlan unrest n the aftermath of ts defeat n the Russo-Japanese war of Despte the formaton of poltcal partes (for the frst tme n Russan hstory) and an electon wth a wde franchse, the represson aganst the regme s opponents contnued, and the parlament, the Duma, had lmted powers and was consdered by the tsar as an advsory rather than legslatve body (Ppes, 1995). The tsar stll retaned control, n part relyng on represson aganst the leftst groups, hs veto power, the rght to dssolve the Duma, full control of the mltary and cabnet appontments, and hs ablty to rule by decree when the Duma was not n sesson. Ths may have been partly motvated by the fear of further strengthenng the two major leftst partes, Socal Revolutonares and Socal Democrats (correspondng to communsts, consstng of the Bolshevks and the Menshevks), whch together controlled about 2/5 of the 1906 Duma and explctly targeted a revoluton. 1 World War I, whch became very unpopular followng large casualtes and terrtoral losses, created the openng for the Bolshevks, brngng to power the Provsonal Government n the February Revoluton of 1917, and then later, the moderate Socal Revolutonary Alexander 1 Lenn, the leader of the Bolshevk wng of the Socal Democrats, recognzed that a revoluton was possble only by explotng turmol. In the context of the 1906 Duma, he stated: Our task s [... ] to use the conflcts wthn ths Duma, or connected wth t, for choosng the rght moment to attack the enemy, the rght moment for an nsurrecton aganst the autocracy. Later, he argued: [... ] the Duma should be used for the purposes of the revoluton, should be used manly for promulgatng the Party s poltcal and socalst vews and not for legslatve reforms, whch, n any case, would mean supportng the counter-revoluton and curtalng democracy n every way. 1

4 Kerensky. Addtonal mltary defeats of the Russan army n the summer of 1917, the destructon of the mltary chan of command by Bolshevk-led solder commttees, and Kerensky s wllngness to enter nto an allance wth Socal Democrats to defeat the attempted coup by the army durng the Kornlov affar strengthened the Bolshevks further. Though n the electons to the Consttuent Assembly n November 1917, they had only a small fracton of the vote, the Bolshevks successfully exploted ther control of Petrograd Sovets to outmaneuver the more popular Socal Revolutonares, frst enterng nto an allance wth so-called Left Socal Revolutonares, and then coercng them to leave the government so as to form ther own one-party dctatorshp. Ths epsode thus llustrates both the possblty of a seres of transtons brngng to power some of the most radcal groups and the potental mplcatons of the concerns of moderate poltcal transtons further empowerng radcal groups. Despte a growng lterature on poltcal transtons, the ssues we have just llustrated n the context of the Bolshevk Revoluton cannot be studed wth exstng models, 2 because they necesstate a dynamc model where several groups can form temporary coaltons and a rch set of stochastc shocks creates a changng envronment, potentally leadng to a sequence of poltcal transtons away from current powerholders. Such a model could also shed further lght on key questons n the lterature on regme transtons, ncludng those concernng poltcal transtons wth several heterogeneous groups, gradual enfranchsement, and the nteractons between regme dynamcs and coalton formaton. In ths paper, we develop a framework for the study of dynamc poltcal economy n the presence of stochastc shocks and changng envronments, whch we then apply to an analyss of the mplcatons of potental shfts of power to radcal groups durng tumultuous tmes. The next example provdes a frst glmpse of the type of abstracton we wll use. Example 1 Consder a socety consstng of n groups, spannng from l < 0 (left-wng) to r > 0 (rght-wng), wth group 0 normalzed to contan the medan voter. For concreteness, suppose that n = 3, and that the rghtmost player corresponds to the Russan tsar, the mddle player to moderate groups, and the leftmost group to Bolshevks. The stage payoff of each group depends on current polces, whch are determned by the poltcally powerful coalton n the current poltcal state. Suppose that there are 2n 1 poltcal states, each state specfyng 2 These types of poltcal dynamcs are not confned to epsodes n whch extreme left groups mght come to power. The power struggles between secularsts and relgous groups n Turkey and more recently n the Mddle East and North Afrca are also partly motvated by concerns on both sdes that poltcal power wll rrevocably or at least persstently shft to the other sde. 2

5 whch of the extreme players are repressed and excluded from poltcal decson-makng. Wth n = 3, the fve states are s = 2 (both moderates and Bolshevks are repressed and the tsar s the dctator), 1 (Bolshevks are repressed), 0 (nobody s repressed and power les wth moderates), 1 (the tsar s repressed or elmnated), and fnally 2 (the tsar and moderates are repressed,.e. a Bolshevk dctatorshp). Snce current polces depend on the poltcal state, we can drectly defne stage payoffs as a functon of the current state for each player, u (s) (whch s nclusve of represson costs, f any). Suppose that startng n any state s 2, a stochastc shock can brng the Bolshevks to power and ths shock s more lkely when s s lower. In addton to provng the exstence and characterzng the structure of pure-strategy equlbra, our framework enables us to establsh the followng types of results. Frst, n the absence of stochastc shocks brngng Bolshevks to power, s = 0 (no represson or democracy) s stable n the sense that moderates would not lke to ntate represson, but s > 0 may also be stable, because the tsar may prefer to ncur the costs of represson to mplement polces more n lne wth hs preferences. Second, and more nterestngly, moderates may also ntate represson startng wth s = 0 f there s the possblty of a swtch of power to Bolshevks. Thrd, and paradoxcally, the tsar may be more wllng to grant poltcal rghts to moderates when Bolshevks are stronger, because ths mght make a coalton between the latter two groups less lkely (ths s an llustraton of what we refer to as slppery slope consderatons and shows the general non-monotonctes n our model: when Bolshevks are stronger, the tsar has less to fear from the slppery slope). Fourth, there s hstory dependence n the sense that once Bolshevks come to power and leave power, a new (dfferent) stable state may emerge. Fnally, there s strategc complementarty n represson: the antcpaton of represson by Bolshevks encourages represson by moderates and the tsar. 3 Though stylzed, ths example communcates the complex strategc nteractons nvolved n dynamc poltcal transtons n the presence of stochastc shocks and changng envronments. Aganst ths background, the framework we develop wll show that, under natural assumptons, we can characterze the equlbra of ths class of envronments farly tghtly and perform comparatve statcs, sheddng lght on these and a varety of other dynamc strategc nteractons. 3 Ths result s also nterestng as t provdes a new perspectve on why represson may dffer markedly across socetes. For example, Russa before the Bolshevk Revoluton repressed the leftsts, and after the Bolshevk Revoluton systematcally repressed the rghtsts and centrsts, whle the extent of represson of ether extreme has been more lmted n the Unted Kngdom. Such dfferences are often ascrbed to dfferences n poltcal culture. Our result nstead suggests that (small) dfferences n economc nterests or poltcal costs of represson can lead to sgnfcantly dfferent represson outcomes. 3

6 Formally, we consder a generalzaton of the envronment dscussed n the example. Socety conssts of = 1, 2,..., n players (groups or ndvduals) and s = 1, 2,..., m states, whch represent both dfferent economc arrangements wth varyng payoffs for dfferent players, and dfferent poltcal arrangements and nsttutonal choces. Stochastc shocks are modeled as stochastc changes n envronments, whch contan nformaton on preferences of all players over states and the dstrbuton of poltcal power wthn states. Ths approach s general enough to capture a rch set of permanent and transtory (as well as both antcpated or unantcpated) stochastc shocks dependng on the current state and envronment. Players care about the expected dscounted sum of ther utlty, and they make jont choces among feasble poltcal transtons, based on ther poltcal power. Our key assumpton s that both preferences and the dstrbuton of poltcal power satsfy a natural sngle-crossng (ncreasng dfferences) property: we assume that players and states are ordered, and hgher-ndexed players relatvely prefer hgher-ndexed states and also tend to have greater poltcal power n such states. (Changes n envronments shft these preferences and dstrbuton of poltcal power, but mantan ncreasng dfferences). Our noton of equlbrum s Markov Votng Equlbrum (MVE), whch comprses two natural requrements: (1) that changes n states should take place f there exsts a subset of players wth the power to mplement them and who wll obtan hgher contnuaton utlty (along the equlbrum path) by dong so; (2) that strateges and contnuaton utltes should only depend on payoff-relevant varables and states. Under these assumptons, we establsh the exstence of pure-strategy equlbra. Furthermore, we show that the stochastc path of states n any MVE ultmately converges to a lmt state.e., to a state that does not nduce further changes once reached, though ths lmt state may depend on the exact tmng and sequence of shocks (Theorems 1 and 3). 4 Although MVE are not always unque, we also provde suff cent condtons that ensure unqueness (Theorems 2 and 4). We further demonstrate a close correspondence between these MVE and the pure-strategy Markov perfect equlbra of our envronment (Theorem 5). Despte the generalty of the framework descrbed here and the potental countervalng forces hghlghted by our example above, we also establsh a number of comparatve statc results. Here we only menton one of them. Consder a change n envronment whch leaves preferences or the allocaton of poltcal power n any of the states s = 1,...s unchanged, but potentally changes them n states s = s + 1,..., m. The result s that f the steady state of equlbrum 4 Ths last result also mples that, n contrast to many other models of nsttutonal persstence, ours features true path dependence as defned, for example, by Page (2006), who crtczes many exstng models of path dependence for beng nvarant to the sequencng of shocks. 4

7 dynamcs descrbed above, x, dd not experence change (.e., x s ), then the new steady state emergng after the change n envronment can be no smaller than ths steady state (Theorem 6). Intutvely, before the change, a transton to any of the smaller states s x could have been chosen, but was not. Now, gven that preferences and poltcal power dd not change for these states, they have not become more attractve. 5 An nterestng and novel mplcaton of ths result s that n some envronments, there may exst crtcal states, such as a suff cently democratc consttuton, and f these crtcal states are reached before the arrval of certan major shocks or changes (whch mght have otherwse led to ther collapse), there wll be no turnng back (see Corollary 1). Ths result provdes a dfferent nterpretaton of the durablty of certan democratcs regmes than the approaches based on democratc captal (e.g., Persson and Tabelln, 2009): a democracy wll survve forever f t s not shocked or challenged severely whle stll progressng towards the suff cently democratc consttuton/state, but wll fall f there s a shock before ths state s reached. The second part of the paper apples our framework to the emergence and mplcatons of radcal poltcs. After establshng that our framework and comparatve statcs can be drectly appled to the class of problems descrbed n Example 1, we derve a number of addtonal results for ths applcaton, some of whch were outlned above. Our paper s related to a large poltcal economy lterature. Frst, our prevous work, n partcular Acemoglu, Egorov, and Sonn (2012), takes one step n ths drecton by ntroducng a model for the analyss of the dynamcs and stablty of dfferent poltcal rules and consttutons. However, that approach not only heavly reles on determnstc and statonary envronments (thus rulng out changes n poltcal power or preferences) but also focuses on envronments n whch the dscount factor s suff cently close to 1 so that all agents just care about the payoff from a stable state (that wll emerge and perssts) f such a state exsts. Here, n contrast, t s crucal that poltcal change and choces are motvated by the entre path of payoffs. 6 Second, several papers on dynamc poltcal economy and on dynamcs of clubs emerge as 5 In contrast, some of the hgher-ranked states may have become more attractve, whch may nduce a transton to a hgher state. In fact, perhaps somewhat surprsngly, transton to a state s s + 1 can take place even f all states s = s + 1,..., m become less attractve for all agents n socety. 6 In Acemoglu, Egorov and Sonn (2010), we studed poltcal selecton and government formaton n a populaton wth heterogeneous abltes and allowed stochastc changes n the competences of poltcans. Nevertheless, ths was done under two assumptons, whch sgnfcantly smplfed the analyss and made t much less applcable. In partcular, stochastc shocks were assumed to be very nfrequent and the dscount factor was taken to be close to 1. Acemoglu, Egorov and Sonn (2011) took a frst step towards ntroducng stochastc shocks, but only confned to the exogenous emergence of new extreme states (and wthout any of the general characterzaton or comparatve statc results presented here). 5

8 specal cases of our paper. Among these, Roberts (1999) deserves specal menton as an mportant precursor of our analyss. Roberts studes a dynamc model of club formaton n whch current members of the club vote about whether to admt new members or exclude some of the exstng ones. Roberts focuses on a lmted set of transtons, also makes sngle-crossng type assumptons and only consders non-stochastc envronments and majortaran votng (see also Barberà, Maschler, and Shalev, 2001, for a related setup). Both our framework and characterzaton results are more general, not only because they ncorporate stochastc elements and more general dstrbutons of poltcal power, but also because we provde condtons for unqueness, convergence to steady states, and general comparatve statc results. In addton, Gomes and Jehel s (2005) paper, whch studes dynamcs n a related envronment wth sde transfers, s also noteworthy. Ths paper, however, does not nclude stochastc elements or smlar general characterzaton results ether. Strulovc (2010), who studes a votng model wth stochastc arrval of new nformaton, s also related, but hs focus s on nformaton leadng to neff cent dynamcs, whle changes n poltcal nsttutons or votng rules are not part of the model. Thrd, our motvaton s also related to the lterature on poltcal transtons. Acemoglu and Robnson (2000a, 2001) consder envronments n whch nsttutonal change s partly motvated by a desre to reallocate poltcal power n the future to match the current dstrbuton of power. 7 Acemoglu and Robnson s analyss s smplfed by focusng on a socety consstng of two socal groups (and n Acemoglu and Robnson, 2006, wth three socal groups). In Acemoglu and Robnson (2001), Fearon (2004), Powell (2006), Hrshlefer, Boldrn and Levne (2009), and Acemoglu, Tcch, and Vndgn (2010), antcpaton of future changes n poltcal power leads to neff cent polces, cvl war, or collapse of democracy. There s a growng lterature that focuses on stuatons where decsons of the current polcy makers affect the future allocaton of poltcal power (see also, Besley and Coate, 1998). Fourth, there s a small lterature on strategc use of represson, whch ncludes Acemoglu and Robnson (2000b), Gregory, Schroeder, and Sonn (2011) and Woltzky (2011). In Woltzky (2011), dfferent poltcal postons (rather than dfferent types of players) are repressed n order to shft the poltcal equlbrum n the context of a two-perod model of poltcal economy. In Acemoglu and Robnson (2000b), represson arses because poltcal concessons can be nterpreted as a sgn of weakness. None of the papers dscussed n the prevous three paragraphs 7 Other related contrbutons here nclude Alesna, Angelon, and Etro (2005), Barberà and Jackson (2004), Messner and Polborn (2004), Bourgugnon and Verder (2000), Burkart and Wallner (2000), Jack and Lagunoff (2008), Lagunoff (2006), and Lzzer and Persco (2004). 6

9 study the ssues we focus on or make progress towards a general framework of the sort presented here. The rest of the paper s organzed as follows. In Secton 2, we present our general framework and ntroduce the concept of MVE. Secton 3 contans the analyss of MVE. We start wth the statonary case (wthout shocks), then extend the analyss to the general case where shocks are possble, and then compare the concepts of MVE to Markov Perfect Equlbrum n a properly defned dynamc game. We also establsh several comparatve statc results that hold even at ths level of generalty; ths allows us to study the socety s reactons to shocks n appled models. Secton 4 apples our framework to the study of radcal poltcs. Secton 5 dscusses a number of extensons. Secton 6 concludes. 2 General Framework Tme s dscrete and nfnte, ndexed by t 1. The socety conssts of n players (representng ndvduals or groups), N = {1,..., n}. The set of players s ordered, and the order reflects the ntal dstrbuton of some varable of nterest. For example, hgher-ndexed players may be rcher, or more pro-authortaran, or more rght-wng on socal ssues. In each perod, the socety s n one of the h envronments E = {E 1,..., E h }, whch determne preferences and the dstrbuton of poltcal power n socety (as descrbed below). We model stochastc elements by assumng that, at each date, the socety transtons from envronment E to envronment E wth probablty π (E, E ). Naturally, E E π (E, E ) = 1. We assume: Assumpton 1 (Ordered Transtons) If 1 x < y h, then π (E y, E x ) = 0. Assumpton 1 mples that there can only be at most a fnte number of shocks. It also stpulates that envronments are numbered so that only transtons to hgher-numbered envronments are possble. 8 Though ths s wthout loss of generalty, t enables us to use the conventon that once the last envronment, E h, has been reached, there wll be no further stochastc shocks. 9 We model preferences and the dstrbuton of poltcal power by means of states, belongng to a fnte set S = {1,..., m}. 10 The set of states s ordered: loosely speakng, ths wll generally 8 Assumpton 1 does not preclude the possblty that the same envronment wll recur several tmes. For example, the possblty of q transtons between E 1 and E 2 can be modeled by settng E 3 = E 1, E 4 = E 2, etc. 9 Ths does not mean that the socety must reach E h on every path: for example, t s permssble to have three envronments wth π (E 1, E 2) = π (E 1, E 3) > 0, and all other transton probabltes equal to zero. 10 The mplct assumpton that the set of states s the same for all envronments s wthout any loss of generalty. 7

10 mply that hgher-ndexed states provde both greater economc payoffs and more poltcal power to hgher-ndexed players. An example would be a stuaton n whch hgher-ndexed states correspond to more non-democratc arrangements, whch are both economcally and poltcally better for rcher, more elte groups. The payoff of player N n state s S and envronment E E s u E, (s). To capture relatve preferences and power of players n dfferent states, we wll frequently make use of the followng defnton: Defnton 1 (Increasng Dff erences) Vector {w (s)} s B A, where A, B R, satsfes the ncreasng dfferences condton f for any agents, j A such that > j and any states x, y B such that x > y, w (x) w (y) w j (x) w j (y). The followng s one of our key assumptons: Assumpton 2 (Increasng Dff erences n Payoff s) In every envronment E E, the vector of utlty functons, {u E, (s)} s S N, satsfes the ncreasng dfferences condton. Note that payoffs {u E, (s)} are drectly assgned to combnatons of states and envronments. An alternatve would be to assgn payoffs to some other actons, e.g., polces, whch are then determned endogenously by the same poltcal process that determnes transtons between states. Ths s what we do n Secton 4, and as our analyss there shows, under farly weak condtons, the current state wll determne the choce of acton (polcy), so payoffs wll then be ndrectly defned over states and envronments. Here we are thus reducng notaton by drectly wrtng them as {u E, (s)}. We model the dstrbuton of poltcal power n a state usng the noton of wnnng coaltons. Ths captures nformaton on whch subsets of agents have the (poltcal) power to mplement economc or poltcal change, here correspondng to a transton from one state to another. We denote the set of wnnng coaltons n state s and envronment E by W E,s, and mpose the followng standard assumpton: Assumpton 3 (Wnnng Coaltons) For envronment E E and state s S, the set of wnnng coaltons W E,s satsfes: 1. (monotoncty) f X Y N and X W E,s, then Y W E,s ; 8

11 2. (properness) f X W E,s, then N \ X / W E,s ; 3. (decsveness) W E,s. The frst part of Assumpton 3 states that f some coalton has the capacty to mplement change, then a larger coalton also does. The second part ensures that f some coalton has the capacty to mplement change, then the coalton of the remanng players (ts complement) does not (effectvely rulng out submajorty rule ). Fnally, the thrd part, n the lght of monotoncty propery, s equvalent to N W E,s, and thus states that f all players want to mplement a change, they can do so. Several common models of poltcal power are specal cases. For example, f a player s a dctator n some state, then the wnnng coaltons n that state are all those that nclude hm; f we need unanmty for transtons, then the only wnnng coalton s N; f there s majortaran votng n some state, then the set of wnnng coaltons conssts of all coaltons wth an absolute majorty of the players. Assumpton 3 puts mnmal and natural restrctons on the set of wnnng coaltons W E,s n each gven state s S. Our man restrcton on the dstrbuton of poltcal power wll be, as dscussed n the Introducton, the requrement of some monotoncty of poltcal power that hgher-ndexed players have no less poltcal power n hgher-ndexed states. To formally formulate ths restrcton, we need the noton of a quas-medan voter (see Acemoglu, Egorov, and Sonn, 2012). Defnton 2 (Quas-Medan Voter) Player ranked s a quas-medan voter (QMV) n state s (n envronment E) f for any wnnng coalton X W E,s, mn X max X. Let M E,s denote the set of QMVs n state s n envronment E. Then by Assumpton 3, M E,s for any s S and E E; moreover, the set M E,s s connected: whenever < j < k and, k M E,s, j M E,s. In many cases, the set of quas-medan voters s a sngleton, M E,s = 1. Examples nclude: one player s the dctator,.e., X W E,s f and only f X (and then M E,s = {}), or majortaran votng among sets contanng odd numbers of players, or there s a weghted majorty n votng wth generc weghts (see Secton 4). An example where M E,s s not a sngleton s the unanmty rule. The followng assumpton ensures that the dstrbuton of poltcal power s monotone over states. 9

12 Assumpton 4 (Monotone Quas-Medan Voter Property, MQMV) In any envronment E E, the sequences {mn M E,s } s S and {max M E,s } s S are nondecreasng n s. The essence of Assumpton 4 s that poltcal power (weakly) shfts towards hgher-ndexed players n hgher-ndexed states. For example, f a certan number of hgher-ndexed players are powerful enough to mplement a transton n some state, then they are also suff cently powerful to do so n a hgher-ndexed state. Ths would hold n a varety of applcatons, ncludng the one we present n Secton 4 and Roberts s (1999) model. Trvally, f M E,s s a sngleton n every state, t s equvalent to M E,s beng nondecreasng (where M E,s s treated as the sngle element). For some applcatons, one mght want to restrct feasble transtons between states that the socety may mplement; for example, t mght be realstc to assume that only transtons to adjacent states are possble. To ncorporate such possbltes, we ntroduce the mappng F = F E : S 2 S, whch maps every x S nto the set of states to whch socety may transton. In other words, y F E (x) means that the socety may transton from x to y n envronment E. We do not assume that y F E (x) mples x F E (y), so certan transtons may be rreversble. We mpose: Assumpton 5 (Feasble Transtons) For each envronment E E, F E satsfes: 1. For any x S, x F E (x); 2. For any states x, y, z S such that x < y < z or x > y > z: If z F E (x), then y F E (x) and z F E (y). The key requrement, encapsulated n the second part, s that f a transton between two states s feasble, then any transtons (n the same drecton) between ntermedate states are also feasble. Specal cases of ths assumpton nclude: (a) any transton s possble: F E (x) = S for any x and E; (b) one-step transtons: y F E (x) f and only f x y 1; (c) one drectonal transtons: y F E (x) f and only f x y. 11 Fnally, we assume that the dscount factor, β [0, 1), s the same for all players and across all envronments. To recap, the full descrpton of each envronment E E s gven by a tuple ( N, S, β, {u E, (s)} s S N, {W E,s} s S, {F E (s)} s S ). 11 In an earler verson, we also allowed for costs of transtons between states, whch we now omt to smplfy the exposton. 10

13 Each perod t starts wth envronment E t 1 E and wth state s t 1 nherted from the prevous perod; Nature determnes E t wth probablty dstrbuton π (E t 1, E t ), and then the players decde on the transton to any feasble s t as we descrbe next. We take E 0 E and s 0 S as gven. At the end of perod t, each player receves the stage payoff v t = u Et, (s t ). (1) Denotng the expectaton at tme t by E t, the expected dscounted payoff of player by the end of perod t can be wrtten as The tmng of events wthn each perod s: V t = E t k=0 βk u Et+k, (s t+k ). 1. The envronment E t 1 and state s t 1 are nherted from perod t There s a change n envronment from E t 1 to E t E wth probablty π (E t 1, E t ). 3. Socety (collectvely) decdes on state s t, subject to s t F E (s t 1 ). 4. Each player gets stage payoff gven by (1). We omt the exact sequence of moves determnng transtons across states (n step 3) as ths s not requred for the Markov Votng Equlbrum (MVE) concept. The exact game form s ntroduced when we study the noncooperatve foundatons of MVE. 12 MVE wll be characterzed by a collecton of transton mappngs φ = {φ E : S S} E E. We let φ k E be the kth teraton of φ E (wth φ 0 E (s) = s). Wth φ, we can assocate contnuaton payoffs V φ E, (s) for player n state s and envronment E, whch are recursvely gven by V φ E, (s) = u E, (s) + β π ( E, E ) V φ E, (φ E (s)). (2) E E As 0 β < 1, the values V φ E, (s) are unquely defned by (2). Defnton 3 (Markov Votng Equlbrum, MVE) A collecton of transton mappngs φ = {φ E : S S} E E s a Markov Votng Equlbrum f the followng three propertes hold: 1. (feasblty) for any envronment E E and for any state x S, φ E (x) F E (x); 12 In what follows, we use MVE both for the sngular (Markov Votng Equlbrum) and plural (Markov Votng Equlbra). 11

14 2. (core) for any envronment E E and for any states x, y S such that y F E (x), { } N : V φ E, (y) > V φ E, (φ E (x)) / W E,x ; (3) 3. (status quo persstence) for any envronment E E and for any state x S, } { N : V φ E, (φ E (x)) V φ E, (x) W E,x. Property 1 requres that MVE nvolves only feasble transtons (n the current envronment). Property 2 s satsfed f no (feasble) alternatve y φ (x) s supported by a wnnng coalton n x over φ E (x) prescrbed by the transton mappng φ E. Ths s analogous to a core property: no alternatve should be preferred to the proposed transton by some suff cently powerful coalton of players; otherwse, the proposed transton would be blocked. Of course, n ths comparson, players should focus on contnuaton utltes, whch s what (3) mposes. Property 3 requres that t takes a wnnng coalton to move from any state to some alternatve.e., to move away from the status quo. Ths requrement sngles out the status quo f there s no alternatve strctly preferred by some wnnng coalton. In addton, we say φ E s monotone f for all x, y S such that x y, we have φ E (x) φ E (y) (φ s monotone f each of the φ E s s monotone). For now, we focus on monotone MVE,.e., MVE wth monotone transton mappngs for each E E. In many cases ths s wthout loss of generalty, and Theorem 9 states mld suff cent condtons for when all MVE are (genercally) monotone. We also refer to any state x such that φ E (x) = x as a steady state or stable n E. In what follows, wth some abuse of notaton, we wll often suppress the reference to the envronment and use, e.g., u (s) nstead of u E, (s) or φ nstead of φ E, when ths causes no confuson. 3 Analyss In ths secton, we analyze the structure of MVE. We frst prove exstence of monotone MVE n a statonary (determnstc) envronment. We then extend these results to stuatons n whch there are stochastc shocks and nonstatonary elements. After establshng the relatonshp between MVE and Markov Perfect Equlbra (MPE) of a dynamc game representng the framework of Secton 2, we present a number of comparatve statc results for our general model. 12

15 3.1 Nonstochastc envronment We frst study the case wthout any stochastc shocks, or equvalently the case of only one envronment ( E = 1) and thus suppress the subscrpt E. For any mappng φ : S S, the contnuaton utlty of player after a transton to s has taken place s gven by V φ (s) = u (s) + ( ) k=1 βk u φ k (s). (4) We start our analyss wth several lemmas whch wll form the bass of our man results. The next one emphaszes the role that the noton of quas-medan voters (QMV) plays n our theory. Lemma 1 Suppose that vector {w (s)} satsfes ncreasng dfferences for S S. Take x, y S, s S and N and let P = { N : w (y) > w (x)}. Then P W s f and only f M s P. A smlar statement s true for relatons, <,. Lemma 1 s a consequence of the followng reasonng: From ncreasng dfferences n payoffs, f w (y) > w (x) for members of W s, then ths holds for all max M s f y < x and for all mn M s f y > x. In ether case, ths establshes the f part of the lemma. The only f part also follows from ncreasng dfferences: w (y) > w (x) must hold for a connected coalton, and therefore t holds for all members of M s (from Defnton 2). For each s S, let us ntroduce the bnary relaton > s on the set of n-dmensonal vectors to desgnate that there exsts a wnnng coalton n s strctly preferrng one payoff vector to another. Formally: w 1 > s w 2 { N : w 1 > w 2 } Ws. The relaton s s defned smlarly. Lemma 1 now mples that f a vector {w (x)} satsfes ncreasng dfferences, then for any s S, the relatons > s and s are transtve on {w (x)} x S. Our next result s crtcal for the rest of our analyss, establshng that, under Assumpton 2 { } s S and 5, when φ s monotone, then contnuaton utltes (s) satsfy ncreasng dfferences. { Lemma 2 For a mappng φ : S S, the vector dff erences f V φ V φ } s S (s) N N, gven by (4), satsfes ncreasng 1. φ s monotone; or 13

16 2. for all x S, φ (x) x 1. Ths result s at the root of the central role of QMVs n our model. As s well known, medan voter type results do not generally apply wth multdmensonal polcy choces. Snce our players are effectvely votng over nfnte dmensonal choces (a sequence of polces), a natural conjecture would have been that such results would not apply n our settng ether. The reason they do has a smlar ntuton to why votng sequentally over two dmensons of polcy, over each of whch preferences satsfy sngle crossng (ncreasng dfferences) or sngle peakedness, does lead to the medan voter type outcomes. By backward nducton, the second vote has a well-defned medan voter, and then gven ths choce, the medan voter over the frst one can be determned. Loosely speakng, our recursve formulaton of today s value enables us to apply ths reasonng between the vote today and the vote tomorrow, and the fact that contnuaton utltes satsfy ncreasng dfferences s the crtcal step n ths argument. For mappng φ to consttute a MVE, t must satsfy the three propertes of Defnton 3. Of these, the core property s the most substantve one. The next lemma smplfes the analyss consderably by provng that f for a monotone mappng φ the core property s volated (.e., there s a devaton that makes all members of some wnnng coalton n the current state better off), then one can fnd a monotone devaton.e., a vald devaton such that the resultng mappng after the devaton s also monotone. We call ths result the Monotone Devaton Prncple wth analogy to the One-Stage Devaton Prncple n extensve form games, whch also smplfes the set of devatons one has to consder (because f some devaton makes a player better off, then there s a one-stage devaton whch also does so). Lemma 3 (Monotone Devaton Prncple) Suppose that φ : S S s feasble (part 1 of Defnton 3) and monotone but the core property s volated n the sense that for some x, y S (such that y F (x)), V φ (y) > x V φ (φ (x)). (5) Then there exst x, y S such that y F (x), (5) stll holds, and the mappng φ : S S gven by { φ φ (s) f s x (s) = (6) y f s = x s monotone. Wth the help of the Monotone Devaton Prncple, we can prove the followng result, whch wll be used to establsh the exstence of MVE. 14

17 Lemma 4 (No Double Devaton) Let a [1, m 1], and let φ 1 : [1, a] [1, a] and φ 2 : [a + 1, m] [a + 1, m] be two monotone mappngs whch are MVE on ther respectve domans. Let φ : S S be defned by { φ1 (s) f s a φ (s) = (7) φ 2 (s) f s > a Then exactly one of the followng s true: 1. φ s a MVE on S; 2. there s z [a + 1, φ (a + 1)] such that z F (a) and V φ (z) > a V φ (φ (a)); 3. there s z [φ (a), a] such that z F (a + 1) and V φ (z) > a+1 V φ (φ (a + 1)). Intutvely, ths lemma states that f we splt the set of states nto two subsets, [1, a] and [a + 1, m], and fnd (by nducton) the MVE on these respectve domans, then the combned mappng may fal to be an MVE only f ether a wnnng coalton n a prefers to move to some (feasble) state n [a + 1, m], or a wnnng coalton n a + 1 prefers to move to some state n [1, a]. But crucally, these two possbltes are mutually exclusve a result whch we use to prove our next theorem, establshng the exstence of MVE. Theorem 1 (Exstence) There exsts a monotone MVE. Moreover, f φ s a monotone MVE, then the equlbrum path s 0, s 1 = φ (s 1 ), s 2 = φ (s 2 ),... s monotone, and there exsts a lmt state s τ = s τ+1 =... = s. We now provde a bref sketch of the proof of ths theorem whch s by nducton on the number of states (here we assume for smplcty that all transtons are feasble). If m = 1, then φ : S S gven by φ (1) = 1 s an MVE for trval reasons. For m > 1, we assume, to obtan a contradcton, that there s no MVE. Take any of m 1 possble splts of S nto nonempty C a = {1,..., a} and D a = {a + 1,..., m}, where a {1,..., m 1}, and then take MVE φ a 1 on C a and MVE φ a 2 on D a (assume for smplcty that they are unque; the Appendx descrbes the way we select φ a 1 and φ a 2 n the general case). Lemma 4 mples that ether there s a devaton from a to [a + 1, φ a 2 (a + 1)] or a devaton from a + 1 to [φ a 1 (a), a], but not both. Denote g (a) = r (for rght ) n the former case, and g (a) = l n the latter. Then g s a well-defned sngle-valued functon. We then have the followng possbltes. If g (1) = r, we can extend the MVE φ 1 2 onto the entre doman by assgnng φ (1) [2, m] approprately; smlarly, f g (m 1) = l, we can extend φ m 1 1 by choosng φ (m) [1, m 1] 15

18 approprately (the detals are provded n the Appendx). It remans to consder the case where g (1) = l and g (m 1) = r. Then there must exst a {2,..., m 1} such that g (a 1) = l and g (a) = r. We take equlbra φ a 1 1 on [1, a 1] and φ a 2 on [a + 1, m], and consder φ : S S gven by φ (s) = φ a 1 1 (s) f s < a b f s = a φ a 2 (s) f s > a where b [ φ a 1 1 (a 1), a 1 ] [a + 1, φ a 2 (a + 1)] s pcked so that V φ (b) s maxmzed for some M a (and b F (a)). Suppose, wthout loss of generalty, that b < a, then φ [1,a] s a MVE on [1, a]. By Lemma 3, to show that the core property s satsfed, t suff ces to check that there s no devaton from a + 1 to [b, a]; ths follows from g (a) = r. The other two propertes, feasblty and perstence, hold by constructon, and thus φ s MVE. The Appendx flls n the detals of ths argument. We next study the unqueness of monotone MVE. We frst ntroduce the followng defntons. Defnton 4 (Sngle-Peaked Preferences) Indvdual preferences are sngle-peaked f for every N there exsts x S such that whenever, for states y, z S, z < y x or z > y x, u (z) < u (y). Defnton 5 (One-Step Transtons) We say that only one-step transtons are possble f for any x, y S wth x y > 1, y / F (x). The next examples shows that a monotone MVE s not always unque., Example 2 (Example wth two MVE) Suppose that there are three states A, B, C, and two players 1 and 2. The decson-makng rule s unanmty n all states. Payoffs are gven by d A B C Then, wth β suff cently close to 1 (e.g., β = 0.9), there are two MVE. In one, φ 1 (A) = φ 2 (B) = A and φ 1 (C) = C. In another, φ 2 (A) = A, φ 2 (B) = φ 2 (C) = C. Ths s possble because preferences are not sngle-peaked, and there s more than one QMV n all states. Example 6 n the Appendx shows that makng preferences sngle peaked s by tself nsuff cent to restore unqueness. The next theorem provdes suff cent condtons for generc unqueness of monotone MVE. 16

19 Theorem 2 (Unqueness) The monotone MVE s (genercally) unque f 1. for every s S, M s s a sngleton; and/or 2. only one-step transtons are possble and preferences are sngle-peaked. Though somewhat restrctve, several nterestng appled problems satsfy one or the other parts of the condtons of ths theorem. In addton, Theorem 9 below shows that under essentally the same assumptons any MVE s monotone. 3.2 Stochastc envronments We now extend our analyss to the case n whch there are stochastc shocks, whch wll also enable us to deal wth nonstatonary n the economc envronment, for example, because the dstrbuton of poltcal power or economc preferences wll change n a specfc drecton n the future. By Assumpton 1, envronments are ordered as E 1, E 2,..., E h so that π (E x, E y ) = 0 f x > y. Ths means that when (and f) we reach envronment E h, there wll be no further shocks, and the analyss from Secton 3.1 s applcable from then on. In partcular, we get the same condtons for exstence and unqueness of MVE. We can now use backward nducton from envronment E h to characterze equlbrum transton mappngs n lower-ndexed envronments, essentally usng Lemma 2, whch establshed that when φ s monotone, contnuaton utltes satsfy ncreasng dfferences. Here we outlne ths backward nducton argument. Take an MVE φ Eh n envronment E h (ts exstence s guaranteed by Theorem 1). Suppose that we have characterzed an MVE {φ E } E {Ek,...,E h } for some k = 1,..., h 1; let us construct φ E k whch would make {φ E } E {Ek,...,E h } an MVE n {E k,..., E h }. Contnuaton utltes n envronment E k are: V φ E k, (s) = u E k, (s) + β π ( E k, E ) V φ E, (φ E (s)) = u Ek, (s) + β E {E k,...,e h } E {E k+1,...e h } + βπ (E k, E k ) V φ E k, ( φek (s) ). π ( E k, E ) V φ E, (φ E (s)) (8) By nducton, we know φ E and V φ E (φ E (s)) for E {E k+1,..., E h }. We next show that there } exsts φ Ek that s an MVE gven contnuaton values {V φ Ek, (s) from (8). Denote s S ũ Ek, (s) = u Ek, (s) + β π ( E k, E ) V φ E,j (φ E (s)), β = βπ (E k, E k ) E {E k+1,...,e h } 17

20 Then rearrangng equaton (8): V φ E k, (s) = ũ E k, (s) + βv φ E k, ( φek (s) ). Snce {ũ Ek, (s)} s S N satsfy ncreasng dfferences, we can smply apply Theorem 1 to the modfed ( envronment E = N, S, β, ) {ũ Ek, (s)} s S N, {W E k,s} s S, {F Ek (s)} s S to characterze φ Ek. Then by defnton of MVE, snce {φ E } E {Ek,...,E h } was an MVE, we have that {φ E} E {Ek,...E h } s an MVE n {E k,..., E h }, provng the desred result. Proceedng nductvely we characterze an entre MVE φ = {φ E } E {E1,...E h }. Ths argument establshes: Theorem 3 (Exstence) There exsts an MVE φ = {φ E } E E. Furthermore, there exsts a lmt state s τ = s τ+1 =... = s (wth probablty 1) but ths lmt state depends on the tmng and realzaton of stochastc shocks and the path to a lmt state need not be monotone. Establshng the unqueness of MVE s more challengng because sngle peakedness s not necessarly nherted by contnuaton utltes (ths s shown, for nstance, by Example 7 n the Appendx). Nevertheless, the followng theorem provdes straghtforward suff cent condtons for unqueness. Theorem 4 (Unqueness) The monotone MVE s (genercally) unque f at least one of the followng condtons holds: 1. for every envronment E E and any state s S, M E,s s a sngleton; 2. n each envronment, only one-step transtons are possble; each player s preferences are sngle-peaked; and, moreover, for each state s there s a player such that M E,s for all E E and the peaks (for all E E) of s preferences do not le on dfferent sdes of s. The frst suff cent condton s the same as n Theorem 2, whle the second strengthens ts equvalent: t would be satsfed, for example, f players blss ponts and the dstrbuton of poltcal power do not change much as a result of shocks. 3.3 Noncooperatve game We have so far presented the concept of MVE wthout ntroducng an explct noncooperatve game. Ths s partly motvated by the fact that several plausble noncooperatve games would 18

21 underpn the notonal MVE. In ths secton, we provde one plausble and transparent noncooperatve game and formally establsh the relatonshp between the Markov Perfect Equlbra (MPE) of ths game and the set of MVE. For each envronment E E and state s S, let us ntroduce a protocol θ E,s, whch s a fnte sequence of all states n F s \ {s} capturng the order n whch dfferent transtons are consdered wthn the perod. Then the exact sequence of events n ths noncooperatve game s gven as follows: 1. The envronment E t 1 and state s t 1 are nherted from perod t Envronment transtons are realzed: E t = E E wth probablty π (E t 1, E). 3. The frst alternatve, θ Et,st 1 (j) for j = 1, s voted aganst the status quo s. That s, all players are ordered n a sequence and must support ether the current proposal θ Et,st 1 (j) or the status quo s. 13 If the set of those who supported θ Et,st 1 (j) s a wnnng coalton.e., t s n W Et,st 1 then s t = θ Et,st 1 (j); otherwse, ths step repeats for the next j. If all alternatves have been voted and rejected for j = 1,..., F s 1, then the new state s s t = s t Each player gets stage payoff gven by (1). We study (pure-strategy) MPE of ths game. Naturally, each MPE nduces an equlbrum behavor whch can be represented by a set of transton mappngs φ = {φ E } E E. In partcular, here φ E (s) s the state to whch the equlbrum play transtons startng wth state s n envronment E. Then we have: Theorem 5 (MVE vs. MPE) 1. For any MVE φ, there exsts a set of protocols {θ E,s } s S E E whch nduces φ. 2. Conversely, f for some set of protocols {θ E,s } s S E E transton mappng φ = {φ E } E E s monotone, then t s an MVE. such that there exsts a MPE and some MPE σ, the correspondng 13 To avod the usual problems wth equlbra n votng games, we assume sequental votng for some fxed sequence of players. See Acemoglu, Egorov, and Sonn (2009) for a soluton concept whch would refne out unnatural equlbra n votng games wth smultaneous votng. 19

22 Ths theorem thus establshes the close connecton between MVE and MPE. Essentally, any MVE corresponds to an MPE (for some protocol) and, conversely, any MPE corresponds to an MVE, provded that ths MPE nduces monotone transtons. 3.4 Comparatve statcs In ths secton, we present general comparatve statc results. We assume that parameter values are generc. We say that envronments E 1 and E 2, defned for the same set of players and set of states, concde on S S, f for each N and for any state x S, u E1, (x) = u E2, (x), W E1,x = W E2,x, and also F E1 S = F E2 S (n the sense that for x, y S, y F E1 (x) y F E2 (x)). Our next result shows that n two envronments E 1 and E 2 that concde on a subset of states (and dffer arbtrarly on other states), there s a smple way of characterzng the transton mappng of one envronment at the steady state of the other. We also say that the MVE s unque on S S f there exsts a unque equlbrum when (transtons are) restrcted to the set of states S. For the results n ths secton, we assume that there exsts a unque MVE (e.g., ether set of condtons of Theorem 4 hold). 14 Theorem 6 (General Comparatve Statcs I) Suppose that envronments E 1 and E 2 concde on S = [1, s] S and that there s a unque MVE n both envronments. For MVE φ 1 n E 1, suppose that φ 1 (x) = x for some x S. Then for MVE φ 2 n E 2 we have φ 2 (x) x. The theorem says that f x s a steady state (lmt state) n envronment E 1 and envronments E 1 and E 2 concde on a subset of states [1, s] that ncludes x, then the MVE n E 2 wll ether stay at x or nduce a transton to a greater state than x. Of course, the two envronments can be swapped: f y S s such that φ 2 (y) = y, then φ 1 (y) y. Moreover, snce the orderng of states can be reversed, a smlar result apples when S = [s, m] rather than [1, s]. The ntuton for Theorem 6 s nstructve. The fact that φ 1 (x) = x mples that n envronment E 1, there s no wnnng coalton wshng to move from x to y < x. But when restrcted to S, economc payoffs and the dstrbuton of poltcal power are the same n envronment E 2 as n E 1, so n envronment E 2 there wll also be no wnnng coalton supportng the move to y < x. Ths mples φ 2 (x) x. Note, however, that φ 2 (x) > x s possble even though φ 1 (x) = x, snce 14 A smlar result can be establshed wthout unqueness. For example, one can show that f for some x S, for each MVE φ 1 n E 1, φ 1 (x) x, wth at least one MVE φ 1 such that φ 1 (x) = x, then all MVE φ 2 n E 2 satsfy φ 2 (x) x. Because both the statements of these results and the proofs are more nvolved, we focus here on stuatons n whch MVE are unque. 20