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Massachusetts Institute of Technology Department of Economics Working Paper Series Dynamics and Stability of Constitutions, Coalitions, and Clubs Daron Acemoglu Georgy Egorov Konstantin Sonin Working Paper 08-1 August 2008 4, Room E52-251 50 Memorial Drive MA Cambridge, 021 42 This paper can be downloaded without charge from the Social Science Research Network Paper Collection at http://ssrn.com/abstract=1201982 Digitized by the Internet Archive in 2011 with funding from Boston Library Consortium IVIember Libraries http://www.archive.org/details/dynamicsstabilitOOacem Dynamics and Stability of Constitutions, Coalition.s, and Clubs* Daron Acemoglu Georgy Egorov Konstantin Sonin MIT Harvard New Economic School August 2008 Abstract A central feature of dynamic collective decision-making is that the rules that govern the procedures forfuturedecision-making andthe distributionofpolitical power across players are determined by current decisions. For example, current constitutional change must take into account how the new constitution may pave the way for further changes in laws and regulations. We develop a general framework for the analysis of this class of dynamic problems. Under relatively natural acyclicity assumptions, we provide a complete characterization of dynamically stable states as functions of the initial state and determine conditions for their uniqueness. We show how this framework can be applied in political economy, coalition formation, and the analysis of the dynamics of clubs. The explicit characterization we provide highlights two intuitive features of dynamic collective decision-making: (1) a social arrangement is made stable by the instability of alternative arrangements that are preferred by sufficiently many members of the society; (2) efficiency-enhancing changes are often resisted because offurther social changes that they will engender. Keywords: commitment, constitutions, dynamic coalition formation, political economy, stability, voting. JEL Classification: D71, D74, C71. *Daron Acemoglu gratefully acknowledges financial support from the National Science Foundation and the Canadian Institute for Advanced Research. We thank John Duggan, Matthew Jackson, Robert Powell, and conference and seminar participants at the ASSA 2008 (New Orleans), the 2008 Game Theory World Congress, Berkeley, California Institute ofTechnology, Harvard and Princeton for useful comments. 1 Introduction Consider the problem of a society choosing its constitution. Naturally, the current rewards from adopting a specific constitution will influence this decision. But, as long as the members of the society are forward-looking and patient, the future implications of the constitution may be even more important. For example, a constitution that encourages economic activity and benefits the majority of the population may nonetheless create future instability or leave room — — for a minority to seize political control. Ifso, the society or the majority of its members may rationally shy away from adopting such a constitution. Many problems in political economy, club theory, coalition formation, organizational economics, and industrial organization have a structure resembling this example of constitutional choice. We develop a general framework for the analysis of dynamic group-decision-making over constitutions, coalitions, and clubs. Formally, we consider a society consisting ofa finite number of infinitely-lived individuals. The society starts in a particular state, which can be thought of as the constitution of the society, regulating how economic and political decisions are made. It determines stage payoffs and also how the society can determine its future states (constitutions), for example, which subsets of individuals can change the constitution. Our focus is on (Markov perfect) equilibriaofthis dynamic game when individuals are sufficiently forward-looking. Under natural acyclicity assumptions, which rule out Condorcet-type cycles, we prove the existence and characterize the structure of [dynanucally) stable states, which are defined as states that arise and persist. An equihbrium is represented by a mapping which designates the dynamically (f), stable state </>(so) as a function of the initial state sq. We show that the set of dynamically stable states is largely independent of the details of agenda-setting and voting protocols. Although our main focus is the analysis of the noncooperative game outlined in the previous paragraph, we first start with an axiomatic characterization of stable states. This character- ization relies on the observation that sufficiently forward-looking individuals will not wish to support change towards a state (constitution) that might ultimately lead to another, less pre- ferred state. This observation is encapsulated in a simple stability axiom. We also introduce two other natural axioms ensuring that individuals do not support changes that will give them lower utihty. We characterize the set of mappings, <I>, that are consistent with these three axioms and provide conditions under which there exists a unique </> € $ (Theorem 1). Even when <& is not a singleton, the sets of stable states according to any two c^ij, 02 € $ are identical. Ourmain results aregiven in Theorem 2. This theoremshows thatfor any agenda-setting and voting protocol the equilibria of the dynamic game we outline can be represented by some <^ g $ and that for any ^ G $, there exists a protocol such that the equilibrium will be represented by 4>.^ This means that starting with initial state sq, any equihbrium leads to a dynamically stable state cp (sq) for some (^ £ $. Naturally, when such is unique, all equilibria result in the uniquely-defined dynamically stable 0(so). An attractive feature of this analysis is that the set of dynamically stable states can be characterized recursively. This characterization is not only simple (the set ofdynamically stable states can be computed using induction), but it also emphasizes a fundamental insight: a par- ticular state is dynamically stable only if there does not exist another state that is dynamically stable and is preferred by a set ofplayers that form a winning coalition within the current state. At the center ofour approach is the natural lack of commitment in dynamic decision-making — problems those that gain additional decision-making power as a result of a reform cannot commit to refraining from further choices that would hurt the initial set of decision-makers. This lack of commitment, together with forward-looking behavior, is at the root of the general characterization result provided in these theorems. It also leads to two simple intuitive results: 1. A particular social arrangement (constitution, coalition, or club) is made stable not by the absence of a powerful set of players that prefer another alternative, but because of the absence of an alternative stable arrangement that is preferred by a sufficiently powerful constituency. To understand why certain social arrangements are stable, we must thus study the instabilities that changes away from these arrangements would unleash.^ — 2. Dynamically stable states can be inefficient in the sense that they may be Pareto domi- nated by the payoffs in another state. ' - . - • ' -. Our final general result, Theorem 3, provides sufficient conditions for the acyclicity assump- tions in Theorems 1 and 2 to hold when states belong to an ordered set (e.g., when they are a subset of M). In particular, it shows that these results apply when (static) preferences satisfy a single-crossing property or are single peaked (and some minimal assumptions on the structure of winning coalitions are satisfied). This theorem makes our main results easy to apply in a variety of environments. We also show that Theorems 1 and 2 apply in a range of situations in which states do not belong to an ordered set. The next two examples provide simple illustrations of the dynamic trade-offs emphasized by our approach. An additional assumption in the analysis of this game is that there is a transaction cost incurred by all individuals every time there is a change in the state. This assumption is used to prove the existence of a pure-strategy equihbrium and to rule out cycles without imposing stronger assumptions on preferences (see also Examples 3 and 4 in Appendix C). 'Thisresultalsoshowsthat, incontrasttoRiker's (1962) emphasisinthecontextofpoliticalcoalitionformation games, the equilibrium will typically not involve a "minimum winning coalition," because the state corresponding to this coalition is generally unstable. . Example 1 (Inefficient Inertia) Consider a society consisting of two individuals, E and M. M E represents the elite and initially holds power, and corresponds to the middle class. There are three states: (1) absolutist monarchy a, in which E rules, with no checks and no political M rights for M; (2) constitutional monarchy c, in which has greater security and is willing to M invest; (3) democracy d, where becomes more influential and the privileges of E disappear. ^ Suppose that stage payoffs satisfy - We {d) < We (a) < We (c) and w^.j (a) < wm (c) < w^ {d) , In particular, we (a) < we (c) means that E has higher payoff under constitutional monarchy M than under absolutist monarchy, for example, because greater investments by increase tax M revenues. clearly prefers democracy to constitutional monarchy and is least well-off under absolutist monarchy. Both parties discount the stage payoffs with discount factor /3 G (0, 1). As described above, "states" not only determine payoffs but also specify decision rules. In absolutist monarchy, E decides which regime will prevail tomorrow. To simplify the discussion, suppose M that starting in both regimes c and d, decides next period's regime. In terms of the notation introduced above, this implies that d is a dynamically stable state, and cp (d) = d. In contrast, c is not a dynamically stable state, since starting from c, there will be a transition to d and thus 4>{c) = d. Therefore, if, starting in regime a, E chooses a reform towards c, this will lead to d in the following period, and thus give E a discounted payoff of t/E (reform) = u;£ (c) + /3^^^. In contrast, if E decides to stay in a forever, its payoff is Ue (no reform) = we (a) / (1 — /3). If /3 is sufficiently small, then Ue (no reform) < Ue (reform), and reform will take place. However, when players are forward looking and /? is large, then (/f; (no reform) > C/g (reform). In this — case, the unique dynamically stable state starting with a is a that is, 4> (a) = a. This example, when players are sufficiently forward-looking, illustrates both of the intuitive results mentioned above. First, state a is made stable by the instability of another state, c, that M is preferred by those who are powerful in a. Second, both E and would be strictly better off in c than in a, so the stable state starting from a is Pareto inefficient. It also illustrates another general insight: the set of stable states is larger when players are forward-looking (when (3 is small, only d is stable, whereas when j3 is large, both a and d are stable). A similar game can be used to model the implications ofconcessions in wars. For example, a concession that increases the payoffs to both warring parties may not take place because it will change the future balance of power. It could also be used to illustrate how organizations might act "conservatively" andresist efficiency-enhancing restructuring. For instance, the appointment ofa CEO who would increase the value ofthe firm may not be favored by the board ofdirectors if they forecast that the CEO would then become powerful and reduce their privileges.'^ Example 2 (Voting in Clubs) Considerthe problemofvotingin clubs. The society consisting of A'' individuals. A club is a subset of the society. Each individual i receives a stage payoff Wi (st), which is as a function ofthe current club st, and current club members decide (according to some voting rule) tomorrow's club st+i. The seminal unpublished paper by Roberts (1999) studies a special case of this environment, where individuals are ordered, i = 1,2,..., A^, any club St must take the form Xk = {!,...,/c} for some k = l,2,..,N, and decisions are made by majoritarian voting. Under a range of additional assumptions Roberts estabhshes the existence ofmixed-strategy (Markovian) equilibria and characterizes some of their properties. Our model nests a more general version of this environment and enables us to establish the existence of a unique dynamically stable club (and a pure-strategy equilibrium) under weaker conditions. In addition, our approach allows a complete characterization of dynamically stable states and clarifies the reasons for potential Pareto inefficiency. • - These examples illustrate some of the possible applications of our approach. We view the rich set ofenvironments that are covered by our model and the relative simplicity ofthe resulting dynamic stable states as its major advantages. Both our specific results and the general ideas can be applied to a range of problems in political economy, organizational economics, club theory, and other areas. Some of these additional examples are discussed in Section 6. On the theoretical side, Roberts (1999) and Barbera, Maschler, and Shalev (2001) can be viewed as the most important precursors to our paper. Barbera, Maschler, and Shalev (2001) study a dynamic game ofclub formation in which any member ofthe club can unilaterally admit a new agent. The recent ambitious paper by Lagunoff (2006), which constructs a general model ofpolitical reform and relates reform to the time-inconsistency ofinduced social rules, is another precm-sor. Acemoglu and Robinson's (2000, 2006a) and Lizzeri and Persico's (2004) analyses of franchise extension and Barbera and Jackson's (2004) model of constitutional stability are on related themes aswell. How these papers can be viewed as applications ofour general framework is discussed in Section 6. ' The two papers most closely related to our work are Chwe (1994) and Gomes and Jehiel ' Ideas related to this example have been discussed in a number of different contexts. Robinson (1997) and Bourguignon and Verdier (2000) discuss how a dictator or an oligarchy may refrain from providing productive public goods or from educational investments, because they may be afraid of losing power. Rajan and Zingales (2000) also emphasize similar ideas and apply them in the context of organizations. Acemoglu and Robinson (2006a) construct a dynamic model in which the elite may block technological improvements or institutional reforms, because they will destabilize the existing regime. Fearon (1996, 2004) and Powell (1998) discuss similar ideas in the context ofcivil wars and international wars, respectively. (2005). Chwe studies a model where payoffs are determined by states and there are exogenous rules governing transitions from one state to another. Chwe demonstrates the relationship between two distinct notions from cooperative game theory, the consistent and stable sets. However, in Chwe's setup, neither a noncooperative analysis nor characterization results are possible, while such results are at the heart of our paper. The link between Chwe's consistent sets and our dynamically stable states is discussed further below. Gomes and Jehiel study a related environment with side payments. They show that aplayer maysacrifice his instantaneous payoff to improve his' bargaining position for the future, which is related to the unwillingness of winning coalitions to make transitions to non-stable states in our paper. They also show that equilibrium may be inefficient when the discount factor is small. In contrast, in our game Pareto dominated outcomes are not only possible in general, but they may emerge as unique equilibria and are more Hkely when discount factors are large (as illustrated by Example 1). More generally, we also provide a full set of characterization (and uniqueness) results, which are not present in Gomes and Jehiel (and in fact, with side payments, we suspect that such restilts are not possible). Finally, in our paper a dynamically stable state depends on the initial state, while in Gomes and Jehiel, as the discount factor tends to 1, there is "ergodicity" in the sense that the ultimate distribution of states does not depend on the initial state. Finally, our work is also related to the literature on club theory (see, for example, Buchanan, 1956, Ellickson et al., 1999, Scotchmer, 2001). While early work in this areawas static, anumber of recent papers have investigated the dynamics of club formation. In addition to Roberts (1999) and Barbera, Maschler, and Shalev (2001), which were discussed above, some of the important papers in this area include Burkart and Wallner (2000), who develop an incomplete contracts theory of club enlargement, and Jehiel and Scotchmer (2001), who show that the requirement of a majority consent for admission to a jurisdiction may not be more restrictive than an unrestricted right to migrate. Alesina, Angeloni, and Etro (2005) apply a simplified version of Roberts's model to the enlargement of the EU and Bordignon and Brusco (2003) study the role of "enhanced cooperation agreements" in the dynamics of EU enlargement. The rest of the paper is organized as follows. Section 2 introduces the general environment. Section 3 motivates and presents our axiomatic analysis, which also acts as a preparation for our noncooperative analysis. In Section 4, we prove the existence of a (pure-strategy) Markov perfect equilibrium of the dynamic game for any agenda setting and voting protocol and estab- lish the equivalence between these equilibria and the axiomatic characterization in Section 3. Section 5 shows how the results of Sections 3-4 can be applied when states belong to an ordered set. Section 6 discusses a range of applications of our framework, including the two examples presented above. Section 7 concludes. Appendix A presents the main proofs omitted from the . text. Appendix B and C, which are not for publication, contain a number of generalizations, additional results, examples, and some omitted proofs. 2 Environment There is a finite set of players X. Time is discrete and infinite, indexed by i (i > 1). Tliere is a finite set of states which we denote by S. Throughout the paper, |X| denotes the number of elements of set X, so \J\ and |<5| denote the number of individuals and states, respectively. States represent both different institutions affecting players' payoffs and procedures for decision- making (e.g., the identity of the ruling coalition in power, the degree of supermajority, or the weights or powers of different agents). Although our game is one of non-transferable utility, a limited amount of transfers can also be incorporated by allowing multiple (but a finite set of) states that have the same procedure for decision-making but different payoffs across players. The initial state is denoted by sq G iS. This state can be thought of as being determined as part of the description of the game or as chosen by Nature according to a given probability distribution. For any t > 1, the state st & S is determined endogenously. A nonempty set X c I is called coalition, and we denote the set of coalitions by C (that is, C is the set of nonempty subsets ofI). Each state s E S is characterized by a pair ({lo, (s)}j£j Ws) Here, for each state , s G S,Wi (s) is a (strictly) positive stage payoffassigned to each individual i G I. The restriction that Wj (s) > is a normalization, making zero payoff the worst outcome. W^ is a (possibly empty) subset of C representing the set of winning coalitions in state s. We use Ws to model pohtical institutions in state s. This allows us to summarize different political procedures, such as weighted majority or supermajority rules, in an economical fashion. For example, if in state s a majority is required for decision-making, Ws includes all subsets ofI that form a majority; if in state s individual i is a dictator, Ws includes all coalitions that include i^ Since Ws is a function of the state, the procedure for decision-making can vary across states. ; Throughout the paper, we maintain the following assumption. . • , Assumption 1 (Winning Coalitions) For any state s G S, Ws C C satisfies: (a) IfX,Y e C, X c y, and X eWs then Y e Ws. , : • (b) IfX,Ye Ws, then X r\Y =^ 0. X Part (a) simply states that ifsome coalition is winning in state s, then increasing the size of the coalition will not reverse this. This is a natural assmuption for almost any decision rule. Notice that Ws or the pohtical rules do not specify certain institutional details, such as who makes proposals, how voting takes place and so on. These are specified by the agenda-setting and voting protocols ofour dynamic game. We will show that these only have a limited effect on equilibrium outcomes, justifying our focus on W,, as a representation of "political rules"