Designing Checks and Balances

Designing Checks and Balances Tiberiu Dragu, Xiaochen Fan, and James Kuklinski1 University of Illinois at Urbana-Champaign September 30, 2011 1 We t...
1 downloads 0 Views 233KB Size
Designing Checks and Balances Tiberiu Dragu, Xiaochen Fan, and James Kuklinski1 University of Illinois at Urbana-Champaign September 30, 2011

1 We

thank Doug Bernheim, Josh Cohen, Jim Fearon, John Ferejohn, Matt Jackson, Leslie Johns, Terry Moe, Mattias Polborn, and Richard Van Weelden for helpful comments and suggestions. All errors are ours.

Abstract Notions of checks and balances pervade thinking about how to structure governments and organizations. We use a mechanism design approach to study the design of checks and balances institutions when two (or more) parties, in environments with single-peaked preferences, need to agree on changing an existing policy. We characterize a class of checks and balances rules that satisfy strategy proofness and responsiveness and identify a unique rule that satisfies these properties. It consists of implementing the ideal policy of the more moderate player, that is, the player that prefers the less aggressive change from the policy status-quo. We also show that the moderate rule can be implemented as a unique equilibrium by means of a simple institutional arrangement. Our analysis can serve as a benchmark to assess all checks and balances institutions; we consider, in detail, its implications for a particularly important institution, constitutional review.

1

Introduction

Notions of checks and balances pervade thinking about how to structure governments and organizations. The U.S. Constitution, rooted in the writings of Montesquieu and Madison, is the best-known statement. In it, the founders created three distinct branches –legislative, executive, and judiciary– and assigned specific functions to each. To ensure that government would not abridge the rights and liberties of its citizens, they established means by which each branch could check the abuses of either of the others. Thus, for example, the President can veto statutes passed by the Senate and House of Representatives; Congress can overturn international treaties signed by the president; and the Supreme Court can review the constitutionality of legislation passed by Congress and approved by the president. In Federalist No. 51, James Madison, the most prominent and persuasive of the Founding Fathers, argues compellingly that checks and balances are necessary to stemming the abuse of power and to ensuring that government as a whole works in citizens’ interests. However, there is more than one way to design how the players in any particular checks and balances setting interact. Scholars as prominent as Thomas Hobbes, Adam Smith, and Frederick Hayek have observed that the rules and structural arrangements designed to coordinate their interactions will shape the outcomes that emerge. Even when the agents’ information and preferences remain constant, different sets of rules will generate different outcomes, with different implications for social welfare. It follows that understanding how best to design checks and balances institutions is a challenging task of utmost importance. In this paper, we study the design of checks and balances institutions in situations in which two (or more) parties need to agree on changing an existing policy. Although our analysis applies generally to any two (or more) interacting parties, we examine the interaction between two players, G and C, in environments with single-peaked preferences. We characterize a class of checks and balances rules that are strategy-proof and responsive. Strategy proofness requires that a player has no incentive to misreport its preference, regardless of what other players do. Responsiveness requires that the policy outcome be positively responsive to changes in the preferences of the players whenever both agents prefer a change from the status-quo policy. We find a unique rule that meets the two criteria. It consists of implementing the ideal policy of the more moderate player, that is, the player that prefers the less aggressive change from the policy status-quo. The moderate choice rule is also the unique checks and balances rule that is strategy proof and (ex-post) Pareto efficient. The moderate choice rule, we show, can be implemented as the unique equilibrium outcome by means of a simple institutional arrangement: C specifies a legal limit and G chooses a policy; the outcome is G’s chosen

1

policy if that policy falls within the C-determined legal limit, otherwise it is the status quo policy. This institutional arrangement has two notable features. First, there is an asymmetry with respect to the players’ available strategies: C sets a legal limit on policy, and G chooses a policy; second, it gives the first-move advantage to C. Our analysis can serve as a benchmark to assess existing checks and balances institutions; we consider its normative and policy implications within the context of one particularly important institution, constitutional review. This checks and balances institution is a critical pillar of established democracies, and, increasingly, of developing democracies. To promote the rule of law in countries undergoing transitions from autocratic to democratic regimes, the World Bank and other influential international institutions have unfailingly pressured them to adopt constitutional review. During the past two decades alone, more than 80 such countries have written bills of rights and established constitutional review as the means by which to protect them (Hirschl 2004). Today, the terms “rule of law” and “constitutional review” are almost inseparable (Ackerman 1997). Many private and international organizations employ the defining feature of systems of governmental checks and balances, namely, the use of multiple players whose agreement is required to effect policy change. In the corporate realm, the board of directors needs the approval of shareholders for mergers, dissolutions, sales of assets, and charter amendments. International organizations that emphasize the sovereignties of the member states and thus want to afford their members a capacity to block undesirable outcomes also employ unanimity rules. For example, the UN Security Council requires that all members approve specified courses of action before undertaking them; and the European Union Council of Minister requires unanimity on policy proposals that touch on vital interests of the member states, such as asylum, taxation, immigration, and foreign and security affairs. Our analysis not only speaks to such corporations and organizations, it sheds light on the voting procedures of organizations that require the endorsements of all their members before a decision can be undertaken. We study specifically how to design a voting procedure that implements the moderate outcome as an equilibrium outcome in organizations requiring unanimous consent. By extending the application of mechanism design to the analysis of checks and balances, this paper contributes to a literature that uses mechanism design to study the organization of various types of markets, including labor (Roth and Xing 1994; Bulow and Levin 2006; Kojima and Pathak 2009), auctions (Bulow and Roberts 1989; Klemperer 2002; DeMarzo, Kremer, and Skrzypacz 2005), health care (Roth and Peranson 1999; Roth, Sonmez, and Unver 2004; Roth, Sonmez, and Unver 2007), and school assignment (Abdulkadiroglu and Snmez 2003; Abdulkadiroglu, Pathak, and Roth 2009), among other topics.1 Our work 1

The literature on market design and auctions is extensive. The above citations represent only some of

2

builds on and contributes to the social choice literature on the domain of single-peaked preferences. Moulin (1980) and Barbera and Jackson (1994) have characterized strategy-proof social choice functions, and shown that all such functions are given by generalized majority voting rules.2 We show that the moderate choice rule is the only checks and balances rule that satisfies strategy proofness and responsiveness. We also propose a simple institutional arrangement that implements the moderate choice rule as the unique equilibrium outcome, and discuss the implications of this arrangement in the context of constitutional review. Finally, this paper also contributes to a political economy literature on rules and institutional design. Topics include, amongst others, the effect of veto players on economic development and policy making (Tsebelis 1995, Cox and McCubbins 2001, North and Weingast 1989), the choice of institutional rules (Buchanan and Tullock 1962; Barbera and Jackson 2004; Messner and Polborn 2004), and the internal organization of government (Diermeier and Myerson 1999; Laffont 2005; Dixit 1997; Tirole and Maskin 2004). More directly relevant here, there is a small but growing game theoretic literature on separation of powers and checks and balances. This work finds that if checks and balances are organized in a certain way, the government, as a whole, reveals to citizens the true conditions under which it operates (Persson, Roland, and Tabellini 1997); and that checks and balances are less likely to emerge as equilibrium institutions whenever rents from holding office are low, elites are well organized to influence politicians, or social inequality and potential taxes are high (Acemoglu, Robinson, and Torvik 2011). We undertake a mechanism design approach that complements these game theoretic studies and provides a normative benchmark to assess any checks and balances institution, regardless of its specific design.3 Our discussion proceeds as follows. Section 2 describes our setup model, and section 3 characterizes the class of mechanisms that satisfy the strategy proofness and responsive criteria. Section 4 proposes a simple institutional arrangement that implements the moderate choice rule as the unique equilibrium, section 5 discusses the implications of our analysis for the institutional design of constitutional review, and section 6 generalizes our results to situations with multiple decision-makers. the works in this field. 2 Barbera (2001) and Jackson (2001) present a thorough review of existing results regarding strategy-proof social choice functions on the domain of single-peaked preferences. 3 Manin, Przeworski, and Stokes (1999) observe the lack of a systematic way to assess checks and balances institutions because of considerable variation in constitutional rules. Our work attempts to overcome this problem.

3

2

The Model

There are two players, which we label G and C. Both have preferences over a one-dimensional policy space, X = R. An exogenous status-quo, q, exists. It can be viewed as the policy resulting from inaction. G’s preference is represented by a continuous and single-peaked (about an ideal position g) utility function UG (·, g) that satisfies the single-crossing property.4 C’s preference is represented by a continuous and single-peaked (about an ideal position c) utility function UC (·, c) that satisfies the single-crossing property.5 A mechanism or allocation rule x(c, g) specifies an outcome x ∈ X as a function of C’s and G’s reported types.6 As mentioned, we are interested in environments where both players must agree to effect policy change. This requirement translates into two (ex-post) individual rationality constraints: UG (x(c, g), g) ≥ UG (q, g) and UC (x(c, g), c) ≥ UC (q, c) for any (c, g).7 We refer to rules that satisfy these individual rationality constraints as checks and balances rules. The situation in which the players have ideal points on opposite sides of q is trivial to analyze, then, the parties never agree on a policy change, and the outcome of that interaction is the status-quo policy q, regardless of the institutional arrangement.8 We therefore focus on situations in which in which the rules of the game matter; that is, the players have ideal points on the same side of the status-quo. Thus let g ≥ q, and c ≥ q; i.e., both players’ ideal policies are higher than or equal to the status-quo policy.9 Without loss of generality, we normalize q = 0. G’s ideal position g is private information and given by a distribution FG (g) with full support on [0, LG ]. C’s ideal position c is private information and given by a distribution FC (c) with full support on [0, LC ].10 Both LC and LG can be infinity.11 We want to characterize checks and balances rules that are strategy proof and responsive 4

Formally, UG (x, g) satisfies the single-crossing property if for any x1 > x2 and g1 > g2 , UG (x1 , g2 ) ≥ UG (x2 , g2 ) implies UG (x1 , g1 ) > UG (x2 , g1 ), and UG (x2 , g1 ) ≥ UG (x1 , g1 ) implies UG (x2 , g2 ) > UG (x1 , g2 ). 5 Formally, UC (x, c) satisfies the single-crossing property if for any x1 > x2 and c1 > c2 , UC (x1 , c2 ) ≥ UC (x2 , c2 ) implies UC (x1 , c1 ) > UC (x2 , c1 ) and UC (x2 , c1 ) ≥ UC (x1 , c1 ) implies UC (x2 , c2 ) > UC (x1 , c2 ). 6 The Revelation Principle allows a focus on direct mechanisms without loss of generality. 7 The ex-post individual rationality constraints ensure that our mechanisms are robust in the sense that they do not depend on players’ conjectures about the other players’ behaviors. For a discussion of expost individual rationality conditions as the appropriate concepts when the implementation of the proposed mechanisms should not rely on conjectures about players’ high order beliefs see Chung and Ely (2006). 8 Moreover, if c < q < g or g < q < c, the checks and balances rule x(c, g) = q is also strategy proof and (trivially) responsive. 9 The case in which the players’ ideal points are lower or equal to the status-quo is similar. 10 The support of the distributions is R+ given that q = 0, c ≥ 0, and g ≥ 0. 11 We make no assumption about whether the types are independent or not.

4

to players’ preferences. Strategy Proofness: A rule x(c, g) is strategy proof if and only if UC (x(c, g), c) ≥ UC (x(e c, g), c) and UG (x(c, g), g) ≥ UG (x(c, ge), g) for all c, e c, g, ge. Strategy proofness, or non-manipulability, is a desirable property of institutions. If an institution is non-manipulable, all players have a dominant strategy to tell the truth regardless of what other players do. The interactions within such an institution are straightforward, since one player’s optimal action does not rely on conjectures about the other players’ behaviors. More importantly, designing the structure of government to increase the capacity of citizens to assess the behaviors of elected politicians would seem a paramount goal of democratic governments.12 If the constitutional rulers prescribing the interactions among governmental agencies are manipulable, there is room for politicians to act strategically within such institutions. Such manipulation, when it occurs, will blur their policy records, making it difficult for citizens to ascertain whether their representatives acted in their interests. Conversely, when the interaction within the constitutional rules is strategy-proof, representatives do not benefit from manipulation; they have a dominant strategy to reveal their true preferences regardless of what other governmental actors do. Because there is no discrepancy between the representatives’ true and stated preferences, it is easier for citizens to assess their policy records. Responsiveness: A rule x(c, g) is (strictly) responsive to players’ preferences if and only if for any c0 > c ≥ 0 and any g 0 > g ≥ 0, x(c0 , g 0 ) > x(c, g). Responsiveness requires that the outcome rule x(c, g) respond positively to an increase in the players’ preferences when both players want a change from the policy status-quo. The responsiveness criterion requires no more than absence of a situation where both agents want a policy change and yet the policy does not change.13 We also investigate the set of checks and balances rules that are strategy-proof and expost Pareto efficient. That is, we impose (ex-post) efficiency rather than responsiveness as a desirable property in our characterization. In fact, we show, the set of checks and 12

During the constitutional convention of 1787, the Federalists and the Anti-federalists debated, how best to design the operation of the government to make it comprehensible to ordinary citizens (Manin 1994). 13 Note that our analysis can be applied to situations in which there is an upper bound on the policies that can be chosen, that is x(c, g) ≤ B for all c, g. For example, it may be the case that a constitution absolutely prohibits certain policies, such as torture. In this instance, we can define responsiveness as follows: A rule x(c, g) is (strictly) responsive to players’ preferences if and only if for any c0 > c ≥ 0 and any g 0 > g ≥ 0, either x(c0 , g 0 ) > x(c, g) or x(c, g) = B.

5

balances that satisfies strategy proofness and responsiveness and the set that satisfies strategy proofness and ex-post efficiency are equivalent. Pareto Efficiency: A rule x(c, g) is (ex-post) Pareto efficient if and only if for any (c, g), there is not another rule x0 (·, ·) such that UC (x0 (c, g), c) ≥ UC (x(c, g), c) and UG (x0 (c, g), g) ≥ UG (x(c, g), g), and at least one of the above inequalities holds with strict inequality. The Pareto efficiency criterion requires that the chosen outcome is such that there is no other feasible outcome that is preferred by one player, and not less preferred by the other. Pareto efficiency is a desirable property of institutions. It requires that the structure of interactions allows players to chose their own actions while avoiding outcomes that none would have chosen. When the interactions of two or more people lead to a result that is not Pareto optimal, such undesirable outcomes are coordination failures. In other words, ex-post efficiency requires the absence of coordination failures in checks and balances institutions. Moreover, if the outcome x(c, g) is (ex-post) efficient, then the outcome is renegotiation proof in that there cannot be further bargaining improvements. We do not propose that these criteria are the only desirable properties of checks and balances institutions. Rather, we view our analysis of the set of checks and balances rules that satisfies these properties as a necessary first step in investigating the optimal checksand-balances designs. Someone seeking to consider the effects of additional properties can assess, for example, whether any rule within the set of strategy proof and efficient checks and balances rules satisfies the additional properties. A negative answer implies a trade-off among different, desirable criteria. This knowledge can help scholars and practitioners to make informed assessments regarding which properties are more suitable for the institutional setting under investigation.

3

The Moderate Choice Rule

We first characterize some properties of strategy-proof rules. The revelation principle states that, for any direct revelation mechanism, there exists a direct mechanism that is payoffequivalent and for which truthful revelation is an equilibrium. Thus it is sufficient to consider only truth-revealing direct mechanisms.14 In a truth-revealing direct mechanism, the message spaces are precisely the type spaces –in our case, e c ∈ [0, L] and ge ∈ [0, LG ]– and in equilibrium all individuals reveal their true types. 14

The revelation principle was first formulated in dominant strategy equilibria by Gibbard (1973), and then extended by Green and Laffont (1977), Dasgupta et al. (1979), and Myerson (1979).

6

For strategy-proof rules, the strategies proposed to the players must be optimal for each type, independently of what other types do. Because each player type is required to have a dominant strategy, strategy proof rules produce the desired outcomes independently of what players think about each other. Rules that are strategy proof for C, given an arbitrary type of G, are as follows: Lemma 1. For any g, any rule x(c, g) that is strategy proof for player C is weakly increasing in c; if x(c, g) is strictly increasing in c on an open interval (c1 , c2 ), then x(c, g) = c on (c1 , c2 ). Proof. See Appendix. This result is well known in the mechanism design literature. It suggests that strategy proof rules consist either of regions where the outcome is the same regardless of player C 0 s type or of regions where the outcome is C’s ideal policy. Rules that are strategy proof for G, given an arbitrary type of C, are as follows: Lemma 2. For any c, any rule x(c, g) that is strategy proof for player G is weakly increasing in g; if x(c, g) is strictly increasing in g on an open interval (g1 , g2 ), x(c, g) = g on (g1 , g2 ). Proof. See Appendix. Thus strategy proof rules implement either the player’s ideal policy or a constant policy. For example, the rules x(c, g) = max{c, g} and x(c, g) = min{c, g} are both strategy proof. The second rule is strategy proof because, for each player, the outcome is either its own ideal policy or some policy lower than its ideal policy. In the former case, a player has no incentive to deviate; in the latter case, the only way to change the outcome is to announce and implement an even lower policy, which would make the player worse off. The first rule is not a checks and balances rule because it violates individual rationality; for (c, g) such that max{c, g} > 2min{c, g}, this rule yields a lower payoff than the status quo for the player with the more moderate preference. Before proceeding to our main characterization, consider the following examples of checks and balances rules that violate either strategy proofness or responsiveness. } satisfies responFirst, the checks and balances rule x(c, g) = min{2min{c, g}, c+g 2 15 siveness and not strategy proofness. This rule is responsive because for any c0 > c ≥ 0 0 > c+g . Therefore x(c0 , g 0 ) = 0 and g 0 > g ≥ 0, min{c0 , g 0 } > min{c, g} and c +g 2 2 0 0 } > x(c, g) = min{2min{c, g}, c+g }. However, it is not strategy min{2min{c0 , g 0 }, c +g 2 2 15

c+g It is a checks and balances rule because |min{2min{c, g}, c+g 2 }−c| ≤ c and |min{2min{c, g}, 2 }−g| ≤

g.

7

proof, as the following counter example illustrates. Let g = 100 and c = 120. Then the = 110. But C has an incentive to deviate and announce c0 = 140 because outcome is c+g 2 0 the outcome is c +g = 120 and C is better off. 2 Second, the checks and balance rule x(c, g) = 0 for all c and g satisfies strategy proofness and not responsiveness. This rule is trivially strategy proof because it does not depend on the announced values of the players’ ideal points. Moreover, the outcome is always the status quo. However, the rule is not responsive. A non-degenerate checks and balances rule that is strategy proof and not responsive is x(c, g) = min{a, c, g}, in which a > 0 is any positive constant. This rule is strategy proof because for any player, the outcome is either her ideal policy or is lower than her ideal policy when telling the truth. If a player misrepresents her ideal point, the player can only change the outcome to a lower policy outcome, which makes her worse off. This rule is not responsive because for any c0 > c > a and g 0 > g > a, x(c0 , g 0 ) = x(c, g) = a. There is a unique check and balances rule that satisfies both strategy proofness and responsiveness: the players report their ideal points and the outcome consists of implementing the more moderate of the two ideal points (e.g., the ideal point closer to the status-quo policy). The result is as follows: Proposition 1. The unique checks and balances rule that satisfies the strategy proofness and responsiveness conditions is the moderate choice rule, x(c, g) = min{c, g}. Proof. See Appendix. Proposition 1 can serve as a benchmark to assess designs of existing checks and balances institutions, including the interactions between two chambers of bicameral legislatures, between legislatures and executives, and between courts and governments. It suggests that if the equilibrium outcome in any checks and balances institution is not the moderate rule, then the institution in question is either manipulable or non responsive to the preferences of the players.16 As mentioned, we also investigate the set of checks and balances rules that are strategyproof and ex-post Pareto efficient. That is, we impose (ex-post) efficiency rather than responsiveness as a desirable property in our axiomatic characterization. Ex-post efficiency of x(c, g) is equivalent to saying that for any c and g, x(c, g) lies in the closed interval between c and g. The result is as follows: 16

Proposition 1 indicates that the moderate choice rule is the unique rule that simultaneously satisfies the following three properties: (1) individual rationality, (2) strategy proofness, and (3) responsiveness. We can obtain this result with a (much) weaker notion of individual rationality: x(c, g) = 0 whenever c = 0 or g = 0, which we label individual rationality at the status-quo. Thus we can restate our result as follows: the moderate rule is the unique rule that simultaneously satisfies three properties: (1) individual rationality at status quo, (2) strategy proofness, and (3) responsiveness.

8

Proposition 2. The unique checks and balances rule that satisfies the strategy proofness and Pareto efficiency conditions is the moderate choice rule, x(c, g) = min{c, g}. Proof. See Appendix. The set of checks and balances institutions that are strategy-proof and responsive and those that are strategy-proof and efficient are equivalent. Proposition 2 thus suggests that a checks and balances institution in which the outcome is not the moderate player’s ideal policy is manipulable or (ex-post) inefficient. Because the moderate rule is (ex-post) efficient, whenever the outcome is the moderate rule, the outcome is renegotiation proof in that there is no room for further bargaining improvements.17 Note that the moderate choice rule is also anonymous; thus the moderate choice rule is the only checks and balances rule that satisfies strategy proofness, responsiveness, ex-post efficiency, and anonymity. Note that for continuous and single-peaked utilities that do not satisfy the single-crossing property, the moderate choice rule x(c, g) = min{c, g} is a checks and balances rule that satisfies both strategy proofness and responsiveness or strategy proofness and ex-post Pareto efficiency. In sum, the moderate choice rule is the unique checks and balances rule that satisfies strategy proofness and responsiveness or strategy proofness and ex-post Pareto efficiency conditions for all continuous and single-peaked utilities. The moderate choice rule is also the unique checks and balances rule that is strategy proof and maximizes the players’ expected utilities under certain assumptions on their beliefs and utility functions.18 These assumptions are as follows: Let G’s preference be represented by a twice continuously differentiable, symmetric single-peaked (about an ideal position g) and (weakly) concave utility function UG (·, g); and C’s preference be represented by a symmetric single-peaked (about an ideal position c) and (weakly) concave utility function UC (·, c). Also, let g and c, the ideal positions of G and C, be independently distributed according to a uniform distribution [0, LG ] and [0, LC ], respectively. Then, we get the following result: Proposition 3. The unique strategy proof checks and balances rule that maximizes G’s and C’s expected payoffs is the moderate choice rule, x(c, g) = min{c, g}. Proof. See Appendix. 17

We can also (re)state the result of Proposition 2 using a weaker condition, individual rationality at status quo. Then the moderate rule is the unique rule that satisfies three properties: (1) individual rationality at status quo, (2) strategy proofness, and (3) efficiency. 18 It should be noted that maximizing the governmental agents’ utilities is secondary to designing checks and balances in light of principles that increase the citizenry’s capacity to influence governmental policy. Our point here is simply to show that the moderate choice rule, the unique rule that satisfies our proposed criteria, need not be in conflict with governmental agents’ welfare.

9

Proposition 3 suggests, for example, that when c ∈ [0, g], G’s optimal rule is x(c, g) = c. To provide some intuition for this result, we compare G0 s expected utility when the rule is x(c, g) = c with G0 s utility when the rule is x(c, g) = g. In the first mechanism, the resulting policy outcome is c since C accepts the policy. In the second, the resulting outcome is x(c, g) = g if c ∈ [ g2 , g] and x(c, g) = 0 if c ∈ [0, g2 ]. In other words, the expected outcome is 0 with probability 1/2 and g with probability 1/2 in the latter mechanism, whereas the outcome is c for c ∈ [0, g] in the former mechanism. Now consider two distributions: one in which the outcome is c for c ∈ [ g2 , g] and one for which the outcome is 0 if c ∈ [0, g2 ] and g if c ∈ [ g2 , g], given that c is uniformly distributed on [0, g]. The two distributions have the same mean g2 , but the second distribution has a bigger variance. Thus the first distribution secondorder stochastically dominates the second, and, because, G has weakly-concave preferences, it is better off under the first distribution. As a result, G prefers the mechanism x(c, g) = c for c ∈ [0, g] to the mechanism x(c, g) = g for c ∈ [ g2 , g] and x(c, g) = 0 for c ∈ [0, g2 ]. The same rationale applies when comparing the mechanism x(c, g) = c for c ∈ [0, t] with the mechanism x(c, g) = t for c ∈ [ 2t , t] and x(c, g) = 0 for c ∈ [0, 2t ] where t ∈ [0, g].

4

Institutional Design and the Moderate Choice Rule

We showed that the moderate choice rule is the unique checks and balances rule that is strategy proof and responsive, and strategy proof and efficient. Are there game forms for which the strategic interaction always induces the players to choose actions that lead to the desired outcome? That is, can one design an institution that ensures the moderate rule to be the unique equilibrium outcome? As our next result shows, the moderate rule can be obtained as the unique equilibrium by means of a very simple and natural institutional arrangement. The players’ strategies are as follows: Player C chooses a legal limit ` ∈ R+ , and for each observed ` player G chooses a policy x ∈ R+ . The players’ choices translate into the outcome of the game as follows: if G’s policy x is consistent with the legal limit `, i.e. if x ≤ `, then the outcome is x; and if G’s chosen policy is inconsistent with the legal limit `, i.e. if x > `, then its policy is illegal and the outcome is the status-quo policy, q = 0. The timing of the interaction is as follows: C defines the legal limit ` first and then G chooses a policy x. The unique equilibrium outcome in this institution is the same as the mechanism defined in Proposition 1. The strategy profile `(c) = c and x(`, g) = min{g, `} is a Bayesian Nash equilibrium strategy, resulting in the unique perfect Bayesian Nash equilibrium outcome x(`) = min{c, g}. We prove this result by backward induction. In the second stage, if the legal limit chosen 10

by C in the first stage is ` for any C’s strategy and any G’s beliefs, then x(`, g) = min{g, `} is the unique optimal strategy for G. In the first stage, if C deviates to `0 < c, then for all g ≤ `0 , C receives the same payoff; for all g ∈ (`0 , c], C is worse off because the outcome is g if it chooses c and `0 < g ≤ c if it chooses `0 ; and for all g > c, C is worse off because the outcome is c if it chooses c and `0 < c if it chooses `0 . A similar argument shows that C also has no incentive to deviate to `0 > c. Thus the following proposition: Proposition 4. The moderate player’s ideal policy is the equilibrium outcome in the institution where C defines a legal limit ` ∈ R+ before G implements a policy, G then chooses a policy x ∈ R+ , and the outcome is x if x ≤ ` and q if x > `. Proof. In Text. Note the simplicity of the institutional arrangement defined in Proposition 4, which can be a desirable property from the perspective of enforcing the rules of the game. Leonid Hurwicz (2008) argues that a Nash equilibrium is self-enforcing only when the players have incentives to respect the rules of the game so that they do not resort to strategies outside the prescribed rules of the interaction. Of course, the issue of how to enforce the rules of the game is a problem for any institution, and the practical solution will likely be context dependent. In the context of governmental checks and balances, for example, stipulating a set of rules in a constitution will not ensure that governmental actors comply with them. The maintenance of any constitution requires the vigilance of the citizens. To this end, institutional designs that facilitate citizens’ policing of governmental actors’ unconstitutional behaviors will be especially valuable. For the institution that Proposition 4 defines, the stakeholders face a relatively simple task: to observe whether C establishes a legal limit on the actions of G and whether G0 s chosen policy falls within that limit. Proposition 4 proposes an asymmetry in the players’ strategies: C is to choose a legal limit ` on policy and G is to choose a policy x. Within the context of governmental checks and balances, scholars have in fact hinted at such an asymmetry of strategies, especially when it comes to interactions between governments and courts with respect to constitutionallyguaranteed rights. In this context, the court protects constitutionally-guaranteed rights by answering (binary) questions of law (Kornhauser 1992), such as: Do suspected terrorists have a constitutional right to habeas corpus? Are certain enhanced interrogation techniques legal in the context of terrorism prevention? In providing answers to such constitutional questions, the court’s influence on policy typically consists of stating a bound on what the government can and cannot do in a specific situation.19 19 Suppose, for example, the issue is to decide on a new legal policy regarding the detention rights of suspected terrorists. More specifically, suppose that the legal question the court needs to answer is how

11

Proposition 4 also proposes that in order to implement the moderate outcome, given the asymmetry in the players’ strategies, C, the limit-setter, should move first. In the case of government-court interactions, constitutional rules vary in setting the timing of them. Our analysis thus can be used to evaluate existing designs of constitutional review.

5

Implications for the Design of Constitutional Review

Liberal societies share a normative commitment to constitutional review, as most recently evidenced by its widespread adaption after the Second World War. Constitutional review, in turn, means that governments cannot trample on constitutionally protected rights, including even when they enjoy the support of a majority of citizens; and that the constitutional court holds the authority to make sure they do not. However, the specific designs of constitutional review vary. Scholars routinely distinguish the American from the European model of constitutional review (Stone Sweet 2000; Ferejohn and Pasquino 2002). In the United States, constitutional review was not explicitly designed at a constitutional convention. Rather, a Supreme Court ruling, Madison v. Marbury, established the practice of constitutional review in the United States (Monaghan 1973). European constitutional review, in contrast, was explicitly designed at the constitutional table (Stone Sweet 2000; Ferejohn and Pasquino 2002). For example, Germany and Italy after 1945, Spain and Portugal after the collapse of their authoritarian governments, and Central and Eastern European countries after the collapse of Soviet hegemony, all adopted new democratic constitutions and established a review mechanism to protect the constitutionally-enshrined limits on power (Schwartz 2000; Ferejohn and Pasquino 2004). Although there are exceptions to the rule, the American model confines constitutional review to legal controversies following governmental implementation of a policy. It requires that an individual bring suit on the grounds that he or she was unduly harmed by a particular policy. The European model, in contrast, allows for constitutional review prior to implementation. Political actors –president, prime minister, members of parliament, and long a government can keep a suspected terrorist in prison without giving him access to some form of legal representation. This has been a prominent question in several liberal democracies, including the United States and the United Kingdom, in the aftermath of the 9/11 terrorist attacks. The status-quo legal policy before 9/11 was to give suspected criminals access to legal representation within 48 hours of detention. Suppose the court’s most preferred policy is to keep a suspected terrorist for five days in detention without legal representation. The court itself cannot implement this policy even if it states that it is legal to keep a suspected terrorist for 5 days without legal rights. Such a ruling only puts a bound on what the government can and cannot do. After all, the government can choose to give the right to legal representation after only three days if it wishes.

12

members of state governments– can ask the constitutional court for early review, which is commonly known as abstract review (Stone Sweet 2000). Roughly speaking, the timing of constitutional review is ex-post in the American case and ex-ante in the European. Given this difference in the timing of review, we can represent the two models of constitutional review as the following games. Let the court’s strategy be to choose a legal limit ` ∈ R+ and the government’s strategy be to choose a policy x ∈ R+ . The outcome of the game is x if x ≤ ` and q = 0 if x > `. In ex-ante constitutional review (the European model), the court defines the legal limit ` first and then the government chooses to implement a policy x. In the ex-post constitutional review (the American model), the government chooses to implement a policy first x and then the court defines the legal limit `. Proposition 4 suggests that the equilibrium outcome under ex-ante review is the moderate choice rule, and thus this institution is strategy proof and responsive (or strategy proof and efficient). On the other hand, the equilibrium outcome under ex-post constitutional review cannot always be the moderate’s most preferred policy,20 and thus this institution can be either manipulable or inefficient, which has implications for scholarly and policy debates about constitutional review.21 First, James Thayer (1983) advanced an important criticism of constitutional review more than a century ago. Very possibly, he argued, constitutional review will cause members of elected branches of government strategically to allow unconstitutional policies because they know they can rely on the court to address such concerns (Tushnet 1999). Our analysis suggests that Thayer’s criticism applies to a specific institutional design, American (ex-post) constitutional review, rather than to constitutional review more generally. Ex-ante review does not create “judicial overhang” because it is strategy-proof, implying that members of elected branches cannot benefit from acting strategically. Should the strategic behavior that might arise under ex-post judicial review reach the point of diluting elected officials’ sense of constitutional responsibility, as Thayer feared, adopting ex-ante constitutional review can, in principle, overcome the problem. Second, ex-post constitutional review can sometimes produce Pareto inefficient policies. Both the government and court might prefer a change from the policy status quo but because 20

The same conclusion holds if we consider a variation of the American model of constitutional review: expost constitutional review with the possibility of communication. That is, the court can send the government legal advice (a cheap talk message) before the government implements a policy, but the court can review the constitutionality of the policy only after implementation. For example, state constitutions in Colorado, Florida, Maine, Massachusetts, Michigan, New Hampshire, Rhode Island, and South Dakota authorize the judiciary to give advice when the legislature or the governor requests it. 21 The analysis here thus provides some normative justifications for the European model of constitutional review.

13

the government implements a policy under uncertainty, the court might reject the policy on grounds of falling outside its range of acceptability. When such policy coordination failure actually occurs, scholars sometimes direct their criticism to the particular agents involved in the policy process: perhaps the President or Congress were too aggressive in their policy choices; perhaps the Supreme Court justices were out of tune with the majority preferences of the day; and so forth. Although such criticisms might sometimes be warranted, our analysis suggests that policy coordination failures could easily be caused by the design of constitutional review. Then, assigning responsibility to the agents acting under the particular institution would be misleading. Proposition 6 indicates that (ex-ante) constitutional review consistently avoids policy coordination failures and thus promotes efficiency; ex-post review does not. Finally, constitutions cannot possibly enumerate an exhaustive set of provisions specifying how the government should behave under all possible circumstances.22 The ambiguities and lacunae arising from constitutional incompleteness become most evident and dramatic when unforeseen circumstances such as the 9/11 terrorist attacks arise. In such situations of legal uncertainty, systems of checks and balances generally compensate for constitutional incompleteness. The government is responsible of protecting the safety and welfare of its citizens in the here-and-now. On the other hand, although it cannot ignore the immediate safety and welfare of citizens, the court is responsible of protecting the long-term rights and values of the society. Proposition 3 suggests that allowing the court to define the legal limits before the government implements a policy is optimal, with respect to maximizing expected payoffs for both government, the agent of today’s majority, and court, the agent of long-term constitutional rights and values.23 22 Politically, any attempt to do so would impede compromise among constitutional convention parties with different and often conflicting political views. Even if the parties tried, the inherent ambiguities of language, combined with unforeseen contingencies arising from societal and technological changes, would ensure failure. 23 In the United States, the court’s capacity to annul governmental acts was fragile at the time that Marshall wrote the opinion that established the institution of constitutional review. Not unexpectedly, in retrospect, the Court formulated a flexible range of doctrines–ripeness, mootness, standing, political question, abstention, and exhaustion of remedies–all of which provided vehicles for limiting the exercise of constitutional review to cases and controversies (Monaghan 1973). Over the long-haul, citizens came to equate a proper exercise of judicial review with specific, ex-post litigation. Following Weingast (1997), we can construe the interaction between the court and the government as embedded in a larger game where citizens potentially might take action against governmental encroachment of judicial power. If citizens define a proper exercise of judicial review to include the court’s assessment of the legality of government action in the context of specific litigation, then ex-post review will emerge as an equilibrium outcome in this bigger game in which the government has the option of ignoring judicial rulings and citizens have the choice of taking action against the government.

14

6

Checks and Balances with Multiple Decision-Makers

Our main results can be extended to a situation in which there are n > 2 decision-makers who must agree on changing an existing policy. Similar to the previous section, each of these n players has veto power in the sense that each has a reservation payoff given by the status quo policy. Formally, for i = 1, 2, ..., n, we denote the ideal points of player i by pi . A mechanism x(p1 , p2 , ..., pn ) specifies an outcome x ∈ X as a function of all members’ reported types. A rule x(p1 , p2 , ..., pn ) is a checks and balances rule if Ui (x(p1 , p2 , ..., pn ), pi ) ≥ Ui (q, pi ) for all i. We restate the following properties for the case of n > 2 players: Strategy Proofness: A rule x(p1 , p2 , ..., pn ) is strategy proof if and only if Ui (x(pi , p−i ), pi ) ≥ Ui (x(˜ pi , p−i ), pi ) for all i, pi , p˜i and p−i . Responsiveness: A rule x(p1 , p2 , ..., pn ) is (strictly) responsive to players’ preferences if and only if for any (p01 , p02 , ..., p0n ) and (p1 , p2 , ..., pn ), where p0i > pi ≥ 0 for all i, then x(p01 , p02 , ..., p0n ) > x(p1 , p2 , ..., pn ). Pareto Efficiency: A rule x(p1 , p2 , ..., pn ) is (ex-post) Pareto efficient if and only if for any (p1 , p2 , ..., pn ), there is not another rule x0 (p1 , p2 , ..., pn ) such that Ui (x0 (p1 , p2 , ..., pn ), pi ) ≥ Ui (x(p1 , p2 , ..., pn ), pi ) for all i, and Uj (x0 (p1 , p2 , ..., pn ), pj ) ≥ Uj (x(p1 , p2 , ..., pn ), pj ) for some j. We have the following results: Proposition 5. The unique checks and balances rule that satisfies strategy proofness and responsiveness is the moderate choice rule x(p1 , p2 , ..., pn ) = minni=1 pi . Proof. See Appendix. Proposition 6. The unique checks and balances rule that satisfies strategy proofness and efficiency is the moderate choice rule x(p1 , p2 , ..., pn ) = minni=1 pi . Proof. See Appendix. Similar to what we found in the previous section, the moderate choice rule can be implemented as the unique equilibrium outcome by means of a simple institutional arrangement. In this institution, n − 1 players choose a legal limit `i ∈ R+ in a fixed sequential order,24 and the remaining player chooses a policy x ∈ R+ . A policy x is legal if and only if x does not exceed the lowest of the n − 1 legal limits. In other words, a policy is legal if and only if it is within all the legal bounds set by the previous n − 1 players. The outcome is x if x is legal and the status quo, q = 0, otherwise. We have the following proposition: 24

The identities of these n − 1 players can be arbitrary.

15

Proposition 7. The unique equilibrium outcome in the institution in which n − 1 players define the legal limits `i ∈ R+ (in a fixed sequential order), and then the remaining player chooses a policy x ∈ R+ , is minni=1 pi . Proof. See Appendix. Our analysis can also be used to understand the design of voting procedures in organizations that require the endorsements of all their members before a decision can be undertaken. International organizations that emphasize the sovereignties of the member states and thus want to afford their members a capacity to block undesirable outcomes employ unanimity rules. For example, the UN Security Council requires that all members approve specified courses of action before undertaking them; and the European Union Council of Minister requires unanimity on policy proposals that touch on vital interests of the member states, such as taxation, asylum and immigration, and foreign and security affairs. In each of these cases, and others, each member essentially possesses veto power. The moderate choice rule in such a setting can be implemented as the unique equilibrium outcome by the following voting procedure: 1) every member makes a policy proposal; 2) if any policy proposal is less than the status quo, the voting game ends with the initial status quo as the policy outcome; 3) if all policy proposals are higher than the status-quo, then they are ranked according to their distance from the status-quo; 4) the members vote in a pairwise contest between the status-quo and the proposed policy that is closest to the statusquo; if the latter wins, under the unanimity rule, it becomes the new status-quo; 5) the new status-quo is pitted against the proposed policy that is closest to the (new) status-quo, from the set of proposed policies defined in step 3, and so on; 6) if in any pairwise voting round, at least one member votes in favor of the status-quo, that is, the proposed amendment fails to gather unanimous agreement, the game ends with the status-quo of that voting round as the policy outcome. We can formalize this voting game as follows. An organization consists of n members, and for i = 1, 2, ..., n denote player i0 s ideal point by pi ∈ R+ . For each pi , member i0 s strategy consists of two components: a policy proposal and a voting choice. Denote member i0 s policy proposal by pˆi ; also denote pˆ(0) = 0.25 For any proposal profile (ˆ p1 , pˆ2 , ..., pˆn ), re-order the policy proposals (ˆ p(1) , pˆ(2) , ..., pˆ(n) ) such that pˆ(1) ≤ pˆ(2) ≤ ... ≤ pˆ(n) .26 For any proposal profile (ˆ p1 , pˆ2 , ..., pˆn ), member i0 s voting strategy is a collection of (binary) choice from {ˆ p(j−1) , pˆ(j) } for every j = 1, 2, ..., n. More exactly, i0 s policy proposal is pˆi (pi ) but for simplicity we omit the argument pi in both proposal and voting strategies. 26 That is, pˆ(1) = min{ˆ p1 , pˆ2 , ..., pˆn }, and for any j ≥ 1, pˆ(j+1) = min({ˆ p1 , pˆ2 , ..., pˆn }\{ˆ p(1) , ..., pˆ(j) }). 25

16

The game ends when any of the following conditions obtains: when pˆi < 0 for some i, in which case the outcome is the status-quo 0; when for all i, the choice between {ˆ p(0) , pˆ(1) } is pˆ(0) = 0, in which case the outcome is 0; when there exists 1 ≤ k < n for all i the choice between {ˆ p(j−1) , pˆ(j) } is pˆ(j) for all j = 1, ..., k, and for some i the choice between {ˆ p(k) , pˆ(k+1) } is pˆ(k) , in which case the outcome is pˆ(k) ; or when for all i the choice between {ˆ p(j−1) , pˆ(j) } is pˆ(j) for all j = 1, ..., n, in which case the outcome is pˆ(n) . We characterize the equilibrium of such a voting procedure in the next proposition. Proposition 8. The moderate choice rule, x(p1 , p2 , ..., pn ) = min{p1 , p2 , ..., pn } is an equilibrium outcome of the previously defined voting game. Proof. See Appendix.

7

Conclusions

In this paper, we have analyzed the institutional design of the checks and balances institutions. We showed that there is a unique checks and balances rule that is strategy proof and responsive, strategy proof and Pareto efficient, and strategy proof and utility maximizing under certain conditions. It entails implementing the ideal policy of the moderate player (moderate relative to the status-quo policy). The moderate choice rule can be implemented as the unique equilibrium outcome through a very simple institutional arrangement: C specifies a legal limit on the actions of G and G chooses a policy. The outcome is G0 s chosen policy if that policy is within C-determined legal limits; otherwise it is the status-quo policy. These results can serve as a benchmark to assess various checks-and-balances institutions.

Appendix Proof of Lemma 1. First, we show that a strategy proof rule x(c, g) is weakly increasing in c for all g. Let us assume that for some g, x(c, g) is not weakly increasing; that is, without loss of generality, let c2 > c1 and x(c1 , g) > x(c2 , g). Given this assumption, we consider the following four cases, which exhaust all possibilities: i) c1 ≥ x(c1 , g) > x(c2 , g). Then c2 > x(c1 , g) > x(c2 , g); the court type c2 prefers x(c1 , g) to x(c2 , g) and will have incentives to misrepresent its type. Thus x(c, g) is not strategy proof. ii) x(c1 , g) > x(c2 , g) ≥ c1 . Then by the single-peakedness of UC (·), the court type c1 prefers x(c2 , g) to x(c1 , g) and will have incentives to misrepresent its type. Thus x(c, g) is not strategy proof. 17

iii) c2 ≥ x(c1 , g) > c1 > x(c2 , g). Then by the single-peakedness of UC (·), the court type c2 prefers x(c1 , g) to x(c2 , g) and will have incentives to misrepresent its type. Thus x(c, g) is not strategy proof. iv) x(c1 , g) > c2 > c1 > x(c2 , g). Incentive compatibility at c1 implies that UC (x(c1 , g), c1 ) ≥ UC (x(c2 , g), c1 ). Single-crossing property further implies that UC (x(c1 , g), c2 ) > UC (x(c2 , g), c2 ). As a result, the court type c2 will prefer x(c1 , g) to x(c2 , g); thus x(c, g) is not strategy proof. Second, we show that for arbitrary g, if x(c, g) strictly increasing in c on an open interval (c1 , c2 ), then x(c, g) = c on (c1 , c2 ). Suppose not. Without loss of generality consider the case in which x(c∗ , g) < c∗ for some c∗ ∈ (c1 , c2 ). If x(c, g) is continuous at c∗ , there exists an  > 0 such that x(c∗ , g) < x(c∗ + , g) < c∗ and thus UC (x(c∗ + , g), c∗ ) > UC (x(c∗ , g), c∗ ) so x(·, g) is not strategy proof. If x(c, g) is not continuous at c∗ , since x(c, g) is strictly increasing in c on (c1 , c2 ), then x(c, g) can only have jump discontinuities and there are at most countable discontinuity points. If limc→c∗+ x(c, g) < c∗ , we can use the preceding argument to show a contradiction (i.e, x(·, g) is not strategy proof). If limc→c∗+ x(c, g) = c∗ , since UC is continuous there exists  > 0 such that UC (x(c∗ + , g), c∗ ) > UC (x(c∗ , g), c∗ ) and thus x(·, g) is not strategy proof. If limc→c∗+ x(c, g) > c∗ , let us denote limc→c∗+ x(c, g) ≡ c∗ + ∆ where ∆ > 0. Then for any c ∈ (c∗ , min{c∗ +∆, c2 }), we have x(c, g) > limc→c∗+ x(c, g) = c∗ +∆ > c. Since x(c, g) is strictly increasing on (c∗ , min{c∗ + ∆, c2 }), there exists c∗∗ ∈ (c∗ , min{c∗ + ∆, c2 }) such that x(c, g) is continuous at c∗∗ , and from the previous line we also have x(c∗∗ , g) > c∗∗ . Then there exists  > 0 such that x(c∗∗ , g) > x(c∗∗ −, g) > c∗∗ , and therefore UC (x(c∗∗ −, g), c∗∗ ) > UC (x(c∗∗ , g), c∗∗ ). As a result, x(·, g) is not strategy proof. A similar argument holds if x(c∗ , g) > c∗ for some c∗ ∈ (c1 , c2 ). As a result, for any g, any rule x(c, g) that is strategy proof for player C is weakly increasing in c, and if x(c, g) strictly increasing in c on an open interval (c1 , c2 ), x(c, g) = c on (c1 , c2 ). Proof of Lemma 2. Similar to the proof of Lemma 1. Before proceeding to proof our main result, we first prove the following lemmas. Lemma 3. Let x(c, g) be a strategy proof rule. Then for any g, if x(c, g) = c on (c1 , c2 ), then x(c, g) is continuous at both c1 and c2 . Similarly, for any c, if x(c, g) = g on (g1 , g2 ), then x(c, g) is continuous at both g1 and g2 . Proof. We prove the first half of the lemma, and the proof for the second half is completely analogous. Without loss of generality we prove that x(c, g) is continuous at c1 . The case 18

of c2 is analogous. Suppose to the contrary that x(c, g) is discontinuous at c1 . Since x(c, g) is (weakly) increasing in c, limc→c1 − x(c, g) < limc→c1 + x(c, g) = c1 . Since UC is continuous there exists small enough  > 0 such that x(c1 − , g) ≤ limc→c1 − x(c, g) < c1 −  and UC (x(c1 + , g), c1 − ) > UC (x(c1 − , g), c1 − ). That is, court type c1 −  can do better by reporting c1 +  and thus a contradiction to x(c, g) being strategy proof. In other words, the only discontinuity points of a strategy proof rule x(c, g) are the points that connect two flat segments. The next lemma further characterizes the discontinuity points of x(c, g): Lemma 4. Let x(c, g) be a strategy proof rule. Then for any g, if cˆ is a discontinuity point of x(c, g), then cˆ − limc→ˆc− x(c, g) > 0 and limc→ˆc+ x(c, g) − cˆ > 0. Similarly, for any c, if gˆ is a discontinuity point of x(c, g), then gˆ − limg→ˆg− x(c, g) > 0 and limg→ˆg+ x(c, g) − gˆ > 0. Proof. Again we prove the first half of the lemma. Let us show that both terms are positive. There are only three other possibilities: (i) cˆ − limc→ˆc− x(c, g) ≤ 0 and limc→ˆc+ x(c, g) − cˆ ≤ 0, but this is impossible because x(c, g) is increasing and discontinuous at cˆ; (ii) cˆ − limc→ˆc− x(c, g) ≤ 0 and limc→ˆc+ x(c, g) − cˆ > 0; and (iii) cˆ − limc→ˆc− x(c, g) > 0 and limc→ˆc+ x(c, g) − cˆ ≤ 0. We show a contradiction for case (ii), and the argument for case (iii) is analogous. Suppose cˆ−limc→ˆc− x(c, g) ≤ 0 and limc→ˆc+ x(c, g)− cˆ > 0. Since x(c, g) is increasing and is discontinuous at cˆ, we have limc→ˆc+ x(c, g) > limc→ˆc− x(c, g) ≥ cˆ. Since UC is continuous, there exists  > 0 such that UC (x(ˆ c − , g), cˆ + ) > UC (x(ˆ c + , g), cˆ + ), so the court type cˆ +  will have an incentive to misreport his type as cˆ −  and thus x(c, g) is not strategy proof. Lemma 5. Let x(c, g) be a strategy proof rule. Then for any g, if for some cˆ x(ˆ c, g) = a 6= cˆ, then x(c, g) = a for all c ∈ (min{a, cˆ}, max{a, cˆ}). Similarly, for any c, if for some gˆ x(c, gˆ) = a 6= gˆ, then x(c, g) = a for all g ∈ (min{a, gˆ}, max{a, gˆ}). Proof. We prove the first half of the lemma for the case a < cˆ. The proof for a > cˆ and the proof for the second half are analogous. Suppose there exists c˜ ∈ (a, cˆ) such that x(˜ c, g) 6= a, then since x(c, g) is (weakly) increasing, x(˜ c, g) < a. Thus x(˜ c, g) < x(ˆ c, g) = a < c˜, and since UC (˙,c˜) is single-peaked with peak c˜, we have UC (x(ˆ c, g), c˜) > UC (x(˜ c, g), c˜), a contradiction to x(c, g) being strategy proof. Proof of Proposition 1. First, it is easy to check that the rule x(c, g) = min{c, g} satisfies strategy-proofness and individual-rationality constraints of both C and G, and is responsive. Next, we show that any rule that is strategy-proof and responsive must be that x(c, g) ∈ {c, g} for all (c, g). Suppose not, then there exist (ˆ c, gˆ) and a ∈ / {ˆ c, gˆ} such that x(ˆ c, gˆ) = a. 19

Then min{a, cˆ} = 6 max{a, cˆ} and min{a, gˆ} = 6 max{a, gˆ}. Lemma 5 implies that x(c, gˆ) = a for all c ∈ [min{a, cˆ}, max{a, cˆ}], and x(ˆ c, g) = a for all g ∈ [min{a, gˆ}, max{a, gˆ}]. This further implies that x(c, g) = a for all c ∈ [min{a, cˆ}, max{a, cˆ}] and g ∈ [min{a, gˆ}, max{a, gˆ}]. Therefore x(min{a, cˆ}, min{a, gˆ}) = x(max{a, cˆ}, max{a, gˆ}) = a but max{a, cˆ} > min{a, cˆ} and max{a, gˆ} > min{a, gˆ}, a contradiction to that x(c, g) is responsive. Therefore, if x(c, g) is strategy-proof and responsive, x(c, g) ∈ {c, g} for all (c, g). We next prove that the unique strategy-proof and responsive checks and balances rule is x(c, g) = min{c, g}. Suppose there exists another strategy-proof and responsive checks and balances rule x˜(c, g). From what we show above, since x˜(c, g) is strategy-proof and responsive, x˜(c, g) ∈ {c, g}. For any c = g, x˜(c, g) ∈ {c, g} implies x˜(c, g) = min{c, g}, so there exists c˜ 6= g˜ such that x˜(˜ c, g˜) 6= min{˜ c, g˜}. With loss of generality let c˜ < g˜, then x˜(˜ c, g˜) = g˜. Since x˜(c, g) is a checks and balances rule, x˜(c, g) is individually rational. In particular, we have x˜(0, g˜) = 0. By Lemma 1 and 4, x˜(0, g˜) = 0 and x˜(˜ c, g˜) = g˜ > c˜ imply that there exists c1 ∈ (0, c˜] such that x˜(c, g˜) is discontinuous at c1 , and there exists c2 < c1 such that x˜(c2 , g˜) < c2 . Since c2 < c1 ≤ c˜ < g˜, x˜(c2 , g˜) ∈ / {c2 , g˜}, a contradiction. Therefore the unique checks and balances rule that is strategy-proof and responsive is x(c, g) = min{c, g}.

Proof of Proposition 2. First, it is easy to check that the rule x(c, g) = min{c, g} satisfies strategy-proofness and individual-rationality constraints of both C and G. Next note that ex-post efficiency is equivalent to stating that for any (c, g), x(c, g) ∈ [min{c, g}, max{c, g}]. Therefore the rule x(c, g) = min{c, g} is ex-post efficient and if c = g, the unique ex-post efficient outcome is x(c, g) = c = g = min{c, g}. Now suppose that there exist another strategy-proof and efficient checks and balances rule x˜(c, g), such that x˜(˜ c, g˜) 6= min{˜ c, g˜} for some c˜ 6= g˜ and without loss of generality let c˜ < g˜. Ex-post efficiency implies that x˜(˜ c, g˜) > c˜. Individual rationality for player C with ideal point 0 implies that x˜(0, g˜) = 0. Lemma 1 then implies that x˜(c, g˜) has at least one discontinuous point on [0, c˜]. Now if x˜(c, g˜) is discontinuous at c = 0, i.e. if limc→0+ x˜(c, g˜) > 0, then there exists  > 0 such that UC (0, ) > UC (˜ x(, g˜), ). But then player C with ideal point  has an incentive to deviate and report type 0, a contradiction to x˜(c, g) being strategy-proof. Therefore, x˜(c, g˜) is continuous at c = 0 and therefore x˜(c, g˜) has at least one discontinuous point on (0, c˜]. Let cˆ ∈ (0, c˜] be a discontinuous point of x˜(c, g˜). From Lemma 4, we have cˆ−limc→ˆc− x˜(c, g˜) > 0. Therefore, there exists 0 < c1 ≤ cˆ such that x˜(c1 , g˜) < c1 . But since c1 ≤ cˆ ≤ c˜ < g˜, x˜(c1 , g˜) is not ex-post efficient. We have a contradiction. 20

Therefore the unique checks and balances rule that is strategy-proof and efficient is x(c, g) = min{c, g}. Proof of Proposition 3. First, it’s easy to see that x(c, g) = min{c, g} is a strategy proof checks and balances rule. We show that x(c, g) = min{c, g} is the unique strategy proof checks and balances rule that maximizes the expected payoff of G. The proof for x(c, g) = min{c, g} being also the unique strategy proof checks and balances rule that maximizes the expected payoff of C is analogous. That is, we show that for any g0 ∈ [0, LG ], x(c, g) = min{c, g} solves the following maximization problem LC

Z max x(c,g)

UG (x(c, g0 ), g0 )fC (c)dc,

(1)

0

subject to the incentive compatibility constraints (i.e. the rule is strategy proof) UC (x(c, g), c) ≥ UC (x(e c, g), c), ∀c, e c, g

(2)

UG (x(c, g), g) ≥ UG (x(c, ge), g), ∀g, ge, c

(3)

and the individual rationality constraints (i.e. the rule is a checks and balances rule) UC (x(c, g), c) ≥ UC (q, c), ∀c, g

(4)

UG (x(c, g), g) ≥ UG (q, g).∀c, g

(5)

It suffices to show that x(c, g0 ) = min{c, g0 } solves the following maximization problem Z max x(c,g0 )

LC

UG (x(c, g0 ), g0 )fC (c)dc,

(6)

0

subject to incentive compatibility and individual rationality constraints of C when G is at g0 UC (x(c, g0 ), c) ≥ UC (x(e c, g0 ), c), ∀c, e c

(7)

UC (x(c, g0 ), c) ≥ UC (q, c), ∀c

(8)

.

21

This is because the second problem maximizes the same objective function from a larger pool of rules than the first problem. Therefore if x(c, g0 ) = min{c, g0 } solves the second problem, then x(c, g) = min{c, g} for all c and g, which satisfies all the constraints of the first problem, is the solution to the first problem. Therefore, denote the solution to the second problem by x∗ (c, g0 ), it suffices to show that x∗ (c, g0 ) = min{c, g0 }. First, we show that x∗ (c, g0 ) is continuous in c. Suppose not, and let cˆ be a discontinuity point of x∗ (c, g0 ), then by Lemma 4, cˆ − limc→ˆc− x∗ (c, g0 ) > 0 and limc→ˆc+ x∗ (c, g0 ) − cˆ > 0. Furthermore, since UC is symmetric, we have cˆ − limc→ˆc− x∗ (c, g0 ) = limc→ˆc+ x∗ (c, g0 ) − cˆ. We show this by contradiction. Suppose not, and with loss of generality suppose that cˆ − limc→ˆc− x∗ (c, g0 ) > limc→ˆc+ x∗ (c, g0 ) − cˆ(> 0). From Lemma 1 and Lemma 3 we know that any discontinuity point must be one that connects two flat segments, then there exists  > 0 such that x∗ (ˆ c − , g0 ) = limc→ˆc− x∗ (c, g0 ) and x∗ (ˆ c + , g0 ) = limc→ˆc+ x∗ (c, g0 ). Since (ˆ c − ) − x∗ (ˆ c − , g0 ) > x∗ (ˆ c + , g0 ) − (ˆ c − ), the court type cˆ −  will have an incentive to ∗ misreport his type as cˆ +  and thus x (c, g0 ) violates the incentive compatibility constraint for c = cˆ − . Denote a ≡ limc→ˆc− x∗ (c, g0 ) and b ≡ limc→ˆc+ x∗ (c, g0 ), from the above we have cˆ − a = . By Lemma 5, x∗ (c, g0 ) = a for all c ∈ [a, a+b ), and x∗ (c, g0 ) = b b − cˆ > 0, that is cˆ = a+b 2 2 for all c ∈ ( a+b , b]. 2 Now consider x0 (c, g0 ) such that x0 (c, g0 ) = c for c ∈ [a, b] and x0 (c, g0 ) = x∗ (c, g0 ) for all other c. Since x∗ (c, g0 ) satisfies the incentive compatibility and individual rationality constraints for all c, it is easy to check that x0 (c, g0 ) also satisfies the incentive compatibility and individual rationality constraints for all c. We now show that x0 (c, g0 ) yields a higher expected payoff for G than x∗ (c, g0 ). Since x0 (c, g0 ) and x∗ (c, g0 ) differ only on [a, b], it Rb Rb R a+b suffices to show that a UG (c, g0 )fC (c)dc > a 2 UG (a, g0 )fC (c)dc + a+b UG (b, g0 )fC (c)dc. 2 The outcome of the left hand side is a random variable, denoted by x, which is uniformly distributed on [a, b]; while the outcome of the right hand side is a random variable, denoted by y, such that y = a with probability 12 and y = b with probability 21 . We have Ex = Ey = a+b . 2 2 2 Varx = (b−a) . Vary = 12 (a− a+b )2 + 12 (b− a+b )2 = (b−a) . Since Ex = Ey and Varx < Vary, 12 2 2 4 y is a mean-preserving spread of x. Since UG (·, g0 ) is (weakly) concave, G prefers x to y, that is, the left hand side is greater than the right hand side, a contradiction to x∗ (c, g0 ) being the solution to the above maximization problem. Therefore, the solution x∗ (c, g0 ) is continuous in c. Individual rationality constraint at c = 0 yields x∗ (0, g0 ) = 0. This together with Lemma 1 and continuity imply that x∗ (c, g0 ) can only take the following form: for some d ∈ [0, LC ], x∗ (c, g0 ) = c for c ∈ [0, d] and x∗ (c, g0 ) = d for c > d. Therefore x∗ (c, g0 ) is equivalent to the 22

solution to the following problem Z max d∈[0,LC ]

d

Z UG (c, g0 )fC (c)dc +

0

LC

UG (d, g0 )fC (c)dc

(9)

d

. Rd We claim the solution to this problem is d∗ = g0 . To see this, denote Φ(d) ≡ 0 UG (c, g0 )fC (c)dc+ R LC UG (d, g0 )fC (c)dc. d Rd Rg RL For any d < g0 , Φ(g0 ) = 0 UG (c, g0 )fC (c)dc+ d 0 UG (c, g0 )fC (c)dc+ g0 C UG (g0 , g0 )fC (c)dc, RL Rg Rd and Φ(d) = 0 UG (c, g0 )fC (c)dc + d 0 UG (d, g0 )fC (c)dc + g0 C UG (d, g0 )fC (c)dc. Φ(g0 ) − Rg RL Φ(d) = d 0 [UG (c, g0 ) − UG (d, g0 )]fC (c)dc + g0 C [UG (g0 , g0 ) − UG (d, g0 )]fC (c)dc > 0, because UG (c, g0 ) − UG (d, g0 ) > 0 for any c ∈ (d, g0 ) due to single-peakedness of UG (·, g0 ), and UG (g0 , g0 ) − UG (d, g0 ) > 0 since g0 is the peak of UG (·, g0 ). Rg Rd RL For any d > g0 , Φ(g0 ) = 0 0 UG (c, g0 )fC (c)dc+ g0 UG (g0 , g0 )fC (c)dc+ d C UG (g0 , g0 )fC (c)dc, Rg Rd RL and Φ(d) = 0 0 UG (c, g0 )fC (c)dc + g0 UG (c, g0 )fC (c)dc + d C UG (d, g0 )fC (c)dc. Φ(g0 ) − Rd RL Φ(d) = g0 [UG (g0 , g0 ) − UG (c, g0 )]fC (c)dc + d C [UG (g0 , g0 ) − UG (d, g0 )]fC (c)dc > 0, because UG (g0 , g0 ) − UG (c, g0 ) > 0 and UG (g0 , g0 ) − UG (d, g0 ) > 0 since g0 is the peak of UG (·, g0 ). Therefore, Φ(g0 ) > Φ(d) for any d 6= g0 , and therefore d∗ = g0 . That is, x∗ (c, g0 ) = c for c ∈ [0, g0 ] and x∗ (c, g0 ) = g0 for c > g0 , namely x∗ (c, g0 ) = min{c, g0 }.

Before proving the results with multiple decision-makers, we generalize Lemma 1 through Lemma 5 to the model with n players. Lemma 6. For any i and any p−i , any rule x(pi , p−i ) that is strategy proof for player i is weakly increasing in pi , and if x(pi , p−i ) strictly increasing in pi on an open interval (pi 1 , pi 2 ), x(pi , p−i ) = pi on (pi 1 , pi 2 ). Lemma 7. Let x(p1 , p2 , ..., pn ) be a strategy proof rule. Then for any i and any p−i , if x(pi , p−i ) = pi on (pi 1 , pi 2 ), then x(pi , p−i ) is continuous at both p1i and p2i . Lemma 8. Let x(p1 , p2 , ..., pn ) be a strategy proof rule. Then for any i and any p−i , if pˆi is a discontinuity point of x(pi , p−i ), then pˆi −limpi →ˆp−i x(pi , p−i ) > 0 and limpi →ˆp+i x(pi , p−i )−pˆi > 0. Lemma 9. Let x(p1 , p2 , ..., pn ) be a strategy proof rule. Then for any i and any p−i , if for some pˆi x(ˆ pi , p−i ) = a 6= pˆi , then x(pi , p−i ) = a for all pi ∈ (min{a, pˆi }, max{a, pˆi }). The proof of the above lemmas are exactly the same as the proof of Lemma 1, 3, 4 and 5.

23

Proof of Proposition 5. First, it is easy to check that the rule x(p1 , p2 , ..., pn ) = minni=1 pi satisfies strategy-proofness and individual-rationality constraints of all members, and is responsive. Next, we show that any rule that is strategy-proof and responsive must be that x(p1 , p2 , ..., pn ) ∈ {p1 , p2 , ..., pn } for all (p1 , p2 , ..., pn ). Suppose not, then there exist (ˆ p1 , pˆ2 , ..., pˆn ) and a ∈ / {ˆ p1 , pˆ2 , ..., pˆn } such that x(ˆ p1 , pˆ2 , ..., pˆn ) = a. Then min{a, pˆi } = 6 max{a, pˆi } for all i. Lemma 9 implies that x(pi , pˆ−i ) = a for all i and pi ∈ [min{a, pˆi }, max{a, pˆi }]. This further implies that x(p1 , p2 , ..., pn ) = a for all pi ∈ [min{a, pˆi }, max{a, pˆi }]. In particular, x(min{a, pˆ1 }, min{a, pˆ2 }, ..., min{a, pˆn }) = x(max{a, pˆ1 }, max{a, pˆ2 }, ..., max{a, pˆn }) = a but max{a, pˆi } > min{a, pˆi } for all i, a contradiction to x(p1 , p2 , ..., pn ) being responsive. Therefore, if x(p1 , p2 , ..., pn ) is strategy-proof and responsive, x(p1 , p2 , ..., pn ) ∈ {p1 , p2 , ..., pn } for all (p1 , p2 , ..., pn ). We next prove that the unique strategy-proof and responsive checks and balances rule is x(p1 , p2 , ..., pn ) = minni=1 pi . Suppose there exists another strategyproof and responsive checks and balances rule x˜(p1 , p2 , ..., pn ). From what we show above, since x˜(p1 , p2 , ..., pn ) is strategy-proof and responsive, x˜(p1 , p2 , ..., pn ) ∈ {p1 , p2 , ..., pn } for all (p1 , p2 , ..., pn ). For any (p1 , p2 , ..., pn ) such that pi = pj for all i 6= j, x˜(p1 , p2 , ..., pn ) ∈ {p1 , p2 , ..., pn } implies x˜(p1 , p2 , ..., pn ) = minni=1 pi . Therefore there exists (˜ p1 , p˜2 , ..., p˜n ) n with p˜i 6= p˜j for some i 6= j, such that x˜(˜ p1 , p˜2 , ..., p˜n ) 6= mini=1 p˜i . Let i∗ be such that p˜i∗ = minni=1 p˜i , then x˜(˜ pi∗ , p˜−i∗ ) > p˜i∗ . Since x˜(p1 , p2 , ..., pn ) is a checks and balances rule, x˜(pi∗ , p˜−i∗ ) is individually rational for i∗ . In particular, we have x˜(0, p˜−i∗ ) = 0. By Lemma 6 and 8, x˜(0, p˜−i∗ ) = 0 and x˜(˜ pi∗ , p˜−i∗ ) > p˜i∗ imply that there exists p1i∗ ∈ (0, p˜i∗ ] such that x˜(pi∗ , p˜−i∗ ) is discontinuous at p1i∗ , and there exists p2i∗ < p1i∗ such that x˜(p2i∗ , p˜−i∗ ) < p2i∗ . Since p2i∗ < p1i∗ ≤ p˜i∗ ≤ p˜j for all j 6= i∗ , x˜(p2i∗ , p˜−i∗ ) ∈ / {p2i∗ , p˜−i∗ }, a contradiction. Therefore the unique checks and balances rule that is strategy-proof and responsive is x(p1 , p2 , ..., pn ) = minni=1 pi .

Proof of Proposition 6. In our model with single-peaked preferences, an outcome x is efficient if and only if minni=1 pi ≤ x ≤ maxni=1 pi . First, it is easy to check that the mechanism x(p1 , p2 , ..., pn ) = minni=1 pi satisfies strategyproofness and individual-rationality constraints of all players and ex-post efficiency. If pi = pj for all i 6= j, the unique ex-post efficient outcome is x(p1 , p2 , ..., pn ) = minni=1 pi . Now suppose that there exist another strategy-proof and efficient checks and balances rule x˜(p1 , p2 , ..., pn ). That is, for some (˜ p1 , p˜2 , ..., p˜n ) such that p˜i 6= p˜j for some i 6= j, n x˜(˜ p1 , p˜2 , ..., p˜n ) 6= mini=1 p˜i . Ex-post efficiency implies that x˜(˜ p1 , p˜2 , ..., p˜n ) > minni=1 p˜i . Let i∗ ∈ {1, 2, ..., n} be such that p˜i∗ = minni=1 p˜i .

24

Individual rationality for player i∗ with ideal point 0 implies that x˜(0, p˜−i∗ ) = 0. Lemma 6 then implies that x˜(pi∗ , p˜−i∗ ) has at least one discontinuous point on [0, p˜i∗ ]. Now if x˜(pi∗ , p˜−i∗ ) is discontinuous at pi∗ = 0, i.e. if limpi∗ →0+ x˜(pi∗ , p˜−i∗ ) > 0, then there exists  > 0 such that Ui∗ (0, ) > Ui∗ (˜ x(, p˜−i∗ ), ). But then the member i∗ with ideal point  has an incentive to deviate and report type 0, a contradiction to x˜(p1 , p2 , ..., pn ) being strategyproof. Therefore, x˜(pi∗ , p˜−i∗ ) is continuous at pi∗ = 0 and therefore x˜(pi∗ , p˜−i∗ ) has at least one discontinuous point on (0, p˜i∗ ]. Let pˆi∗ ∈ (0, p˜i∗ ] be a discontinuous point of x˜(pi∗ , p˜−i∗ ). From Lemma 8, we have pˆi∗ − limpi∗ →ˆp−∗ x˜(pi∗ , p˜−i∗ ) > 0. i Therefore, there exists 0 < p1i∗ ≤ pˆi∗ such that x˜(p1i∗ , p˜−i∗ ) < p1i∗ . But since p1i∗ ≤ pˆi∗ ≤ p˜i∗ ≤ p˜j for all j 6= i∗ , x˜(p1i∗ , p˜−i∗ ) < p1i∗ ≤ min{p1i∗ , p˜−i∗ } and therefore is not ex-post efficient. We have a contradiction. Therefore the unique checks and balances rule that is strategy-proof and efficient is x(p1 , p2 , ..., pn ) = minni=1 pi .

Proof of Proposition 7. Without loss of generality re-order the players such that the player with ideal points pi is the i-th one to make a choice. That is, player 1 through n − 1 each chooses a legal limit in the order 1, 2, ..., n − 1, and then player n (observing all the legal limits chosen by the previous n − 1 players) chooses a policy. We claim that the strategy profile `i (`1 , ...`i−1 , pi ) = pi for i = 1, 2, ..., n − 1 and xn (`1 , ..., `n−1 , pn ) = min{minn−1 i=1 `i , pn } is a Bayesian Nash equilibrium strategy and it gives as the unique Bayesian Nash equilibrium outcome the ideal policy of the most moderate player. We prove this claim by backward induction. n−1 In the last stage, the effective legal limit is mini=1 `i , i.e. if player n chooses a policy n−1 not exceeding mini=1 `i the outcome is this chosen policy, otherwise the outcome is the n−1 status quo. If the effective legal limit chosen by the previous n − 1 players is mini=1 `i , for any strategy of the previous n − 1 players and any beliefs of player n, xn (`1 , ..., `n−1 , pn ) = min{minn−1 i=1 `i , pn } is the unique optimal strategy for player n. In the second-to-last stage, if the legal limits chosen by the previous n − 2 players are `1 , ..., `n−2 , for any strategy of the previous n − 2 players and any beliefs of player n − 1, truth-telling (i.e. `n−1 (`1 , ..., `n−2 , pn−1 ) = pn−1 is optimal for player n − 1. This is the case since if player n − 1 deviates to `0 < pn−1 , he either does not change the outcome and thus receives the same payoff, or changes the outcome from somewhere in between `0 and pn−1 to `0 . Since players have single-peaked preferences, this change will make player n − 1 worse off. A similar argument shows that the player has no incentive to deviates to `0 > pn−1 either. Iterating this argument back to the first stage proves our claim. 25

Proof of Proposition 8. We solve the game by backward induction. For any proposal profile (ˆ p1 , pˆ2 , ..., pˆn ), consider the voting stage. If the members ever get to choose between pˆ(n−1) and pˆ(n) , it is dominant strategy for all to vote truthfully since there are only two alternatives. Now consider the choice between pˆ(n−2) and pˆ(n−1) , for any member i, her vote is pivotal only if all other members vote for pˆ(n−1) (because otherwise the outcome is pˆ(n−2) no matter how i votes). In such case in which i’s vote is pivotal, voting for pˆ(n−2) generates (immediate) outcome of pˆ(n−2) , while voting for pˆ(n−1) moves the game to the next-round voting between pˆ(n−1) and pˆ(n) . Since Ui is single-peaked and pˆ(n−2) ≤ pˆ(n−1) ≤ pˆ(n) , there are 4 possible rankings of the three alternative by i: (1) Ui (ˆ p(n−2) ) ≥ Ui (ˆ p(n−1) ) ≥ Ui (ˆ p(n) ).27 In this case, in the voting between pˆ(n−1) and pˆ(n) i will vote for pˆ(n−1) and therefore the outcome of the next-round voting is pˆ(n−1) , that is, the current vote between pˆ(n−2) and pˆ(n−1) is equivalent to the choice between two outcomes pˆ(n−2) and pˆ(n−1) , and i will vote for pˆ(n−2) ; (2) Ui (ˆ p(n−1) ) ≥ Ui (ˆ p(n−2) ) ≥ Ui (ˆ p(n) ). In this case, in the voting between pˆ(n−1) and pˆ(n) i will vote for pˆ(n−1) and therefore the outcome of the next-round voting is pˆ(n−1) , that is, the current vote between pˆ(n−2) and pˆ(n−1) is equivalent to the choice between two outcomes pˆ(n−2) and pˆ(n−1) , and i will vote for pˆ(n−1) ; (3) Ui (ˆ p(n−1) ) ≥ Ui (ˆ p(n) ) ≥ Ui (ˆ p(n−2) ). In this case, in the voting between pˆ(n−1) and pˆ(n) i will vote for pˆ(n−1) and therefore the outcome of the next-round voting is pˆ(n−1) , that is, the current vote between pˆ(n−2) and pˆ(n−1) is equivalent to the choice between two outcomes pˆ(n−2) and pˆ(n−1) , and i will vote for pˆ(n−1) ; (4) Ui (ˆ p(n) ) ≥ Ui (ˆ p(n−1) ) ≥ Ui (ˆ p(n−2) ). In this case, both outcomes from the next-round voting is preferred by i to pˆ(n−2) , so in the current voting i will vote for pˆ(n−1) . Therefore, for any i and any pi , it is dominant strategy for i to vote truthfully between pˆ(n−2) and pˆ(n−1) . Iterating this argument backwards we can show that it is dominant strategy for all members to vote truthfully in any voting round that they reach. Now consider the proposal-making stage. For any i and any pi , given that all other members report truthfully, if i report her ideal point truthfully, i.e. pˆi = pi , the outcome of the game is minni=1 pi . For any p−i , either pi 6= minni=1 pi so there is no deviation for i that can change the outcome of the game; or pi = minni=1 pi so the outcome is i’s ideal point so there is no profitable deviations for i. Therefore, for i reporting truthfully is the 27

For simplicity we omit the parameter pi in the utility function Ui .

26

best response to all others reporting truthfully, and thus truthful reporting is an equilibrium proposal strategy.

References [1] Acemoglu, Daron, James A. Robinson and Ragnar Torvik. 2011. Equilibrium Checks and Balances. mimeo. [2] Ackerman, Bruce. 1997. “The Rise of World Constitutionalism.” Virginia Law Review 83(4): 771-797. [3] Atila Abdulkadiroglu and Tayfun Snmez (2003). School Choice: A Mechanism Design Approach. American Economic Review, 93, 729-747. [4] Abdulkadiroglu, Atila, Parag A. Pathak, and Alvin E. Roth. 2009. Strategy- proofness versus Efficiency in Matching with Indifferences: Redesigning the NYC High School Match. American Economic Review. [5] Aghion, Phillipe and Jean Tirole. 1997. “Real and Formal Authority in Organizations.“ Journal of Political Economy 105 (1): 1-29. [6] Barbera, Salvador and Matthew Jackson. 1994. “A characterization of strategy-proof social choice functions for economies with pure public goods.” Social Choice and Welfare 11:241-252 [7] Barbera, Salvador and Matthew O. Jackson. 2004. “Choosing How to Choose: SelfStable Majority Rules and Constitutions.” The Quarterly Journal of Economics 119 (3): 1011-1048. [8] Buchanan, James and Gordon Tullock. 1962. “The Calculus of the Consent. Logical Foundations of Constitutional Democracy.” Ann Arbor: Michigan University Press. [9] Bulow, Jeremy and John Roberts. 1989. The Simple Economics of Optimal Auctions,” Journal of Political Economy 97(5): 1060-1090. [10] Bulow, Jeremy and Jonathan Levin. 2006. Matching and Price Competition. American Economic Review 96(3).

27

[11] Cox, Gary and Mathew McCubbins. 2001. “The Institutional Determinants of Economic Policy Outcomes ” in Presidents, Parliaments and Policy, Stephan Haggard and Mathew McCubbins, editors. New York: Cambridge University Press. [12] Dasgupta, Partha, Peter Hammond, and Eric Maskin. 1979. “The Implementation of Social Choice Rules.” Review of Economic Studies 46: 185-216 [13] DeMarzo, P., Kremer, I. and Skrzypacz, A. 2005. Bidding with Securities: Auctions and Security Design. American Economic Review 95 (4): 936-959. [14] Diermeier, Daniel and Roger B. Myerson. 1999. “Bicameralism and Its Consequences for the Internal Organization of Legislatures.” The American Economic Review 89(5): 1182-1196. [15] Dixit, Avinash. 1997. “Power of Incentives in Private versus Public Organizations.” The American Economic Review 87(2): 378-382. [16] Ferejohn, John and Pasquale Pasquino. 2002. “Constitutional Courts as Deliberative Institutions.” in Constitutional Justice, East and West, W. Sadurski ed., Kluwer Law International: The Hague. [17] Ferejohn, John and Pasquale Pasquino. 2004. “Constitutional Adjudication: Lessons from Europe.” Texas Law Review 82: 1671 [18] Gibbard, A. 1973. “Manipulation for Voting Schemes.” Econometrica 41: 587-601 [19] Green, J., and J.-J. Laffont. 1977. “Characterization of Satisfactory Mechanisms for the Revelation of Preferences for Public Goods. ” Econometrica 45: 427-438 [20] Hirschl, Ran. 2004. “Towards Juristocracy. The Origins and Consequences of the New Constitutionalism. ” Cambridge, MA: Harvard Univ. Press [21] Hurwicz, Leonid. 2008. “But Who Will Guard the Guardians?” American Economic Review 98: 577585. [22] Jackson, Matthew. 2001. “Mechanism Theory. ” Encyclopedia of Life Support Systems. [23] Klemperer, Paul. 2002. What Really Matters in Auction Design? Journal of Economic Perspectives 16(1): 169-189. [24] Kojima, Fuhito; Pathak, Parag A. 2009. Incentives and Stability in Large Two-Sided Matching Markets. The American Economic Review 99 (3): 608-627 28

[25] Kornhauser, Lewis. 1992. “Modeling Collegial Courts. II. Legal Doctrine.” Journal of Law, Economics, & Organization 8 (3): 441-470. [26] Manin, Bernard. 1994. Checks, balances and boundaries: the separation of powers in the constitutional debate of 1787. in Biancamaria Fontana (ed.), The Invention of the Modern Republic, Cambridge University Press. [27] Manin, Bernad, Adam Przeworski, and Susan C. Stokes. 1999. Elections and Representation. in Adam Przeworski, Susan C. Stokes, and Bernard Manin (eds.), Democracy, Accountability, and Representation, Cambridge University Press. [28] Maskin, Eric and Jean Tirole. 2004. “The Politician and the Judge: Accountability in Government. ” The American Economic Review 94(4): 1034-1054. [29] Messner, Matthias and Mattias Polborn. 2004. “Voting on Majority Rules.” Review of Economic Studies 71: 115-132. [30] Monaghan, Henry. 1973. “To Whom and When.” Yale Law Journal 82: 1363-1397. [31] Moulin, Herve. 1980. “On strategy-proofness and single peakedness.” Public Choice 354: 437-455 [32] Myerson, Roger. 1979. “Incentive Compatibility and the Bargaining Problem.” Econometrica 47: 61-73 [33] Myerson, Roger. 1982. “Optimal Coordination Mechanisms in Generalized PrincipalAgent Problems.” Journal of Mathematical Economics 10: 67-81. [34] North, Douglass and Barry Weingast 1989. “Constitutions and Commitment: The Evolution of Institutions Governing Public Choice in Seventeenth-Century England. ” Journal of Economic History 49: 803-832. [35] Persson, Torsten, Gerard Roland and Guido Tabellini. 1997. “Separation of Powers and Political Accountability.” The Quarterly Journal of Economics 112 (4): 1163-1202 [36] Roth, Alvin E. and X. Xing. 1994. Jumping the Gun: Imperfections and Institutions Related to the Timing of Market Transactions. American Economic Review 84: 9921044. [37] Roth, Alvin. E. and Elliott Peranson.1999. The Redesign of the Matching Market for American Physicians: Some Engineering Aspects of Economic Design American Economic Review 89 (4): 748-780. 29

[38] Roth, Alvin E., Tayfun Sonmez, and M. Utku Unver. 2004. Kidney Exchange. Quarterly Journal of Economics 119 (2): 457-488. [39] Roth, Alvin E., Tayfun Sonmez , and M. Utku Unver. 2007. Efficient Kidney Exchange: Coincidence of Wants in Markets with Compatibility-Based Preferences. American Economic Review 97 (3): 828-851. [40] Stone Sweet, Alec. 2000. “Governing with Judges: Constitutional Politics in Europe.” Oxford: Oxford University Press. [41] Thayer, James Bradley. 1893. “The Origin and Scope of the American Doctrine of Constitutional Law.” Harvard Law Review 7(3):129156. [42] Tushnet, Mark. 1999. “Taking the Constitution Away from the Courts.” Princeton: Princeton University Press. [43] Tsebelis, George. 2002. “Veto Players: How Political Institutions Work. ” Princeton: Princeton University Press. [44] Weingast, Barry. 1997. “The Political Foundations of Democracy and the Rule of Law.” American Political Science Review 91: 245-63.

30