Coalition Governance with Incomplete Information Tiberiu Dragu1 and Michael Laver2 New York University

1 Department

of Politics, New York University, 19 West 4th St., New York, NY, 10012, Email: [email protected] 2 Department of Politics, New York University, 19 West 4th St., New York, NY, 10012, Email: [email protected]

Abstract How can coalition cabinets take policy decisions if the policy preferences of the coalition partners are private information? We use a mechanism design approach to study the important process of coalition governance in this setting. We show that, among all possible mechanisms that could structure decision-making within a coalition government, the mechanism that leads to the "best" policy compromise among coalition partners when members’ preferences are private information is "constrained ministerial government". This gives each particular cabinet minister considerable policy-making power within his or her policy jurisdiction, subject however to the key constraint that other cabinet members maintain some control by setting a bound on how far policy can be changed from the status quo. Our analysis provides a theoretical basis for a growing substantive literature on coalition governance, developing a systematic account of what is gained and lost by using any given decisionmaking procedure, and of what types of "compromise" are feasible and/or desirable when public policy is made by coalitions.

Keywords: cabinet decision making; coalition governance; principal-agent problems; incomplete information; mechanism design

Supplementary material for this article is available in the appendix in the online edition.

Parliamentary democracies aggregate the diverse preferences of individual citizens on issues to be decided into a single collective choice on these for society as a whole. This happens in three stages. First, preferences of a large number of citizens are represented in a parliament comprising a relatively small number of legislators, typically organized into more or less disciplined legislative parties. Second, these legislators choose a political executive comprising a much smaller number of "ministers", a cabinet. Third, these cabinet ministers choose policy outcomes, mindful of the need to maintain the support of a legislative majority. In a typical parliamentary democracy, proportional representation electoral rules have the effect that no single legislative party comprises a majority of legislators. This in turn means that the political executive must be supported by, and will typically itself comprise, a coalition of legislative parties. Our focus here is on the third stage of preference aggregation, in which the diverse preferences of a coalition of cabinet ministers are transformed into a single government position on key policy issues. We are concerned, in short, with coalition governance. In what follows, we study coalition governance when the policy preferences of coalition members are private information. This sets us apart from an immense scholarly literature on the politics of government coalitions, traceable in its current manifestation to the seminal work of William Riker (1962), nearly all of which assumes that politicians’ preferences on every relevant policy issue are common knowledge.1 While the assumption of complete information can be a good approximation in some circumstances, there are compelling reasons to study coalition governance with incomplete information. For one, politicians are usually reluctant to take clear stands on the issues of the day and often express their policy positions in ambiguous, vague and even contradictory terms when running for office (Downs 1957; Shepsle 1972; Page 1976; Bräuninger and Giger 2016). More importantly, however, even if for some reason all politicians did indeed reveal their true preferences at election time, many if not most important policy decisions facing any government were not, and could not 1

For surveys of this literature, see: Laver and Schofield 1998; Humphreys 2008; Strøm et al. 2008.

1

possibly have been, anticipated when the election was held and the government was formed. Coalition governance generically involves making decisions on vitally important but completely unanticipated problems. Think of the world financial crisis of 2008, for example, or the Syrian refugee crisis of 2016. Governments around the world faced crucial policy decisions that clearly had not been factored in to the politics of forming the government. Over and above these headline examples, much of the bread and butter of day-to-day governance involves policy decisions on unanticipated events. Any "contract" between members of any government is inevitably incomplete. Sophisticated politicians can anticipate at least some of the compromises they will need to make when in government, and can negotiate (more, or less, binding) resolutions of these as part of the government formation deal. But the world is stochastic. From their very first day in office, members of coalition cabinets are liable to face decisions on completely unexpected matters arising from random shocks to the political environment. To study coalition governance with incomplete information, we use a mechanism design approach.2 Theoretically, we develop a formal model of a multiparty government in which coalition members have preferences over a multidimensional policy space and must decide which (multidimensional) policy position to implement when they have incomplete information about each others’ policy preferences. We use this model to show that, among all possible mechanisms that could structure the decision-making process within a coalition government, the mechanism that leads to the "best" policy compromise (in ways we specify below) among coalition partners in such settings is one that we call "constrained ministerial government". This prescribes a decision-making environment in which some cabinet minister is given considerable policy-making autonomy within his or her policy domain, subject to the constraint that other cabinet members maintain some control by setting a bound on how far policy can be changed from the status quo. 2

This sets us apart from the existing theoretical literature on coalition formation and governance, which typically assumes a precise decision-making "protocol" that specifies: which types of agent can make which types of offer; to whom; in what sequence; the reversion point if offers are rejected; how agents can communicate with each other; and agents’ beliefs about other agents’ preferences.

2

"Constrained ministerial government" therefore offers a way to reconcile the (undisputed) substantial policy responsibility delegated to individual cabinet ministers with the (undisputed) final decision-making power of the full cabinet. This mechanism, furthermore, has several appealing properties: it is non-manipulable; it induces a genuine compromise among coalition members on every policy issue rather than concentrating decision-making power entirely in the hands of one party; it allows every coalition member to improve his or her payoff relative to the status-quo; and it leads to Pareto improvements on every policy issue for which all coalition members want a policy change from the status-quo in the same direction. Moreover, the "constrained ministerial government" mechanism can accommodate important empirical matters that might affect coalition governance, such as the robust empirical tendency for larger parties to be allocated more cabinet seats, or the particular desire of some "single issue" (for example environmentalist) party to be allocated some particular cabinet portfolio. Our suggested mechanism of constrained ministerial government is therefore consistent with a wide range of different models of coalition politics, including those that deal explicitly with how particular policy portfolios might be allocated to particular members of the government coalition. Substantively, our paper contributes to a growing literature on intra-coalition governance (Strøm et al. 2008; Martin and Vanberg 2015). Existing work does develop the intuition that, because coalition agreements are essentially incomplete contracts (Strøm and Müller 1999; Lupia and Strøm 2008; Martin and Vanberg 2004; Bowler et al. 2016), decision-making procedures that structure the bargaining process within a coalition government are essential for policy-making, especially when coalition members are uncertain about the precise policy preferences of their coalition partners. Our paper, however, is the first to develop a formal model of coalition governance in a multidimensional policy setting, designed to assess equilibrium coalition policies resulting under various decision-making procedures when preferences are private information. Our theoretical approach, mechanism design, is novel in the context of work on coalition governance and, more importantly, allows us to conduct a comprehensive 3

normative analysis without making any assumptions about the number of parties comprising the government coalition or about the beliefs and conjectures which coalition members might have about their partners’ policy preferences. Our analysis is therefore consistent with a range of settings where coalition governments make policy under incomplete information, including settings in which politicians express their policy views in ambiguous terms, settings in which coalition cabinets are likely to face decisions on completely unexpected issues, and those in which coalition governments make policy decisions in the (essentially uncertain) shadow of future elections (Lupia and Strøm 2008). Set in the larger literature on multiparty government, our mechanism design analysis provides a robust normative rationale for studying delegation within coalition cabinets, and the related issue of how to constrain ministerial drift. Several scholars have discussed, as a generic problem for coalition governance, the fact that individual cabinet ministers have obvious incentives to "shade" any policies agreed between coalition partners towards positions that favor their own private agenda and electoral fortunes.3 Coalition governance for this reason generates a classic principal-agent problem: coalition members must for practical reasons delegate governance to individual ministers, whose policy preferences may nonetheless diverge from those of other coalition partners (Indridason and Kristinsson 2013). Scholars discussing this agency problem have considered various institutional mechanisms that might address potential ministerial "drift", including: cabinet committees (Muller and Strøm 2003); "junior" ministers (Thies 2001); cabinet reshuffles (Indridason and Kam 2008); and legislative committees (Martin and Vanberg 2004; Carroll and Cox 2012). Implicitly or explicitly, this literature assumes that some compromise among coalition members is desirable (relative to allowing each minister absolute discretion over policy-making in his/her jurisdiction) and then proceeds to investigate, theoretically and empirically, how ministerial discretion might be constrained. Our mechanism design analysis provides a theoretical underpinning for this research tradition by developing a systematic account of what is gained, and what is lost, 3

For an excellent discussion of this important research topic, see Martin and Vanberg (2015).

4

by using some particular decision-making mechanism and therefore what kinds of "compromises" are desirable and feasible within a coalition government. In this context, our main theoretical result on "constrained ministerial government" suggests that substantive policing of ministerial discretion, along the lines explored in the literature we have just referred to, gives the full cabinet a practical way to enforce the bound on how far coalition policy can be moved on any issue by the delegated minister. This can not only mitigate a core principal-agent problem for coalition governance but can also lead to policy outcomes for the coalition that are normatively desirable.

Decision-Making Environment for Coalition Governance We describe coalition governance as interaction between leaders of two or more legislative parties (in general, N ≥ 2 parties) which comprise a government coalition.4 Party leaders have preferences over a multidimensional policy space RM where M ≥ 2. Each leader i’s preference over the multidimensional policy outcome p = (p1 , p2 , ..., pM ) ∈ RM is represented P j j j j j j by a utility function ui (p, θ i ) = M j=1 vi (p ; θi ), where vi (p ; θi ) is continuous and singlepeaked (about an ideal position θij ) and pj is the policy outcome implemented on issue j. Let Ui be the class of all such agent utility functions.5 There is an exogenous status-quo, q. Without loss of generality, we normalize the status quo policy to q = (0, ..., 0). Politicians’ policy preferences are private information. We describe politicians as having a variety of possible types regarding their most preferred policies. Let the type of party leader i be θ i = (θi1 , ..., θiM ) ∈ RM where θij ∈ R denotes the type of party leader i on issue j. We do not impose any assumption on the beliefs that party leaders might have about 4

We do not concern ourselves here with intra-party politics and think of leaders as controlling welldisciplined parties. This is without loss of generality since for "parties" we could substitute "legislators", each legislator being his or her own party, without any effect on the logic of our argument. 5 Examples the Euclidean qPof utility functions in this class are the utility function defined in terms of P M M j j j j 2 distance, − j=1 (θi − p ) ; the utility function defined in terms of the city block distance, − j=1 |θi −p |; PM the quadratic loss utility − j=1 αj (θij − pj )2 where αj > 0; and the utility function defined in terms of the P 1/t M j j s p-norm, − (θ − p ) for t ≥ 1. j=1 i

5

other coalition members’ policy preferences. It might be that in some instances coalition members might have an accurate assessment about their partners’ preferred policies or it might be that in other cases coalition members have less accurate information about policies they partners would like. Our subsequent analysis is comprehensive and applies to all these settings. The collective policy choice p(θ) depends on politicians’ ideal policies, and since the θ i ’s are private information, every coalition member must be relied upon to reveal his or her type truthfully in order for the coalition systematically to implement the policy p(θ). However, there is no reason to suppose this will happen. Indeed party leaders may well have incentives not to reveal this information truthfully if p(θ) is the implemented policy outcome. This means that the specific bargaining protocol according to which the coalition partners reach a policy decision is likely to shape their incentives to reveal their true preferences. In what follows, we deploy a mechanism design approach to identify policy outcomes which are incentive-compatible for all coalition partners, and which can therefore be implemented as equilibrium outcomes of a non-cooperative bargaining process when the coalition members’ preferred policies are private information. A mechanism in this setting can essentially be viewed as a bargaining protocol combined with a procedure for making a collective policy choice. In principle, such a mechanism can be any complex bargaining process (a set of possible actions and communication strategies) under which parties interact to choose a policy outcome.6 It includes any dynamic game in which the strategies of players comprise contingent plans for actions and messages. The task of identifying all policy outcomes that are incentive compatible for all coalition members might seem difficult if not impossible, since we might seem to be faced with the task of analyzing all possible mechanisms. However, we can actually attack this task in a relatively straightforward manner by exploiting the revelation principle, which tells us that we can restrict our analysis to a very simple 6

Formally, a mechanism in this particular setting, M = {S1 , S2 , ..., SN ; p(·)}, specifies a set of strategies available to each party leader and a rule p(·) stipulating the policy outcome implemented by the mechanism for a given strategy profile s = (s1 , s2 , ..., sN ).

6

class of mechanisms. These are direct revelation mechanisms, simple procedures in which the actions of agents are to reveal their types, with the policy outcome being a function of the reported types (Gibbard 1973; Myerson 1979). To analyze policy outcomes which can be implemented if coalition members’ ideal policies are private information, therefore, we need only consider direct revelation mechanisms subject to incentive compatibility constraints which require that truthful revelation of agents’ preferences is an equilibrium in the game of incomplete information induced by the mechanism in question. To this end, we require a policy mechanism p(θ) to be implemented in dominant strategies:7 Dominant-Strategy Incentive-Compatibility: A mechanism p(θ 1 , .., θ N ) is dominantstrategy incentive-compatible if and only if ui (p(θ i , θ˜−i ); θ i ) ≥ ui (p(θ˜i , θ˜−i ); θ i ) for all i, θ i , θ˜i , θ˜−i , and for all preference specifications ui ∈ Ui . Dominant-strategy incentive-compatibility requires that truthful revelation is an equilibrium in (weakly) dominant strategies in the game of incomplete information induced by the relevant mechanism.8 In other words, a coalition member has a (weakly) dominant strategy to reveal his or her preferred policy regardless of what other members do. The importance of this in the substantive context of coalition governance is that interactions governed by such a mechanism are straightforward, since one coalition member’s optimal action does not rely on his or her conjectures about other members’ preferences and conjectures. By requiring a mechanism to be dominant-strategy incentive-compatible, our analysis does not impose any common knowledge assumption or any other restriction on the beliefs of coalition members. This is substantively important since we are interested in outcomes that can be implemented 7

The revelation principle for dominant strategies states that every social choice function (i.e. a policy outcome that satisfies certain desiderata in our context) that can be implemented in dominant-strategies in some arbitrary bargaining game can also be implemented by a dominant-strategy incentive-compatible direct mechanism. 8 Notice that the incentive compatible direct mechanisms are required to operate on the basis of parties’ ideal points alone. In principle, these mechanisms could take into account all aspects of parties’ preferences, however, the existing literature has shown that allowing for the use of additional information does not enlarge the set of dominant-strategy incentive compatible mechanisms (Border and Jordan 1983); thus our restriction is without loss of generality and made for simplicity of exposition.

7

by coalition cabinets making decisions under incomplete information, taking account of any possible conjectures and beliefs politicians might have about each others’ preferred policies.9 Dominant-strategy incentive-compatible (non-manipulable) mechanisms, furthermore, have particular substantive attractions for analysts of coalition governance. In a multi-party setting with coalition cabinets, different parties each promote different policy programs to voters at election time, but are then forced to abandon some of their policy promises after the election if they are to agree a joint program of government with other parties. When legislators agree to policies in a coalition that differ, perhaps starkly, from promises they made to voters at election time, they are much more accountable to voters if the process according to which these policy compromises are made is non-manipulable and transparent. Over and above the foundational need for incentive compatibility, therefore, non-manipulability speaks directly to a common normative fear about coalition governments. This is that coalitions blur lines of public accountability because the policy compromises that are inevitably required take place, non-transparently, in strategic wheeling and dealing between politicians behind closed doors. Non-manipulable mechanisms, in contrast, are in this sense transparent.

Desirable Properties of Coalition Governance Coalition cabinets in parliamentary democracies aggregate the potentially diverse policy preferences of a set of public representatives into a unique policy profile that becomes government policy. As Arrow showed us many years ago, this process of preference aggregation has profound normative implications. We now describe and justify three normatively compelling desiderata for any system of coalition governance: non-dictatorship, individual rationality and efficiency, properties that the literature on coalition governance has mentioned as desirable features of policy-making in coalition governments. 9

Our analysis is in the vein of robust mechanism design (Bergemann and Morris 2012). Furthermore, if a mechanism is dominant-strategy incentive-compatible then it is also Bayesian-Nash incentive-compatible; this follows trivially from the fact that a (weakly) dominant strategy equilibrium is necessarily a BayesianNash equilibrium.

8

First, scholars have argued informally that the compromise between rival political parties that is the essence of coalition governance is an important desideratum in itself (Lijphart 1999; Powell 2000). We therefore require that a policy mechanism satisfies a minimal notion of compromise. This is non-dictatorship, which stipulates that, for any possible configuration of coalition members’ policy preferences, it is not the case that the implemented policy outcome is always the most preferred policy of one particular party leader. Otherwise, no matter what the profile of policy preferences of the other coalition members might be, the essence of coalition governance is undermined since the policy output takes no account whatsoever of these preferences. Formally: Non-Dictatorship: A mechanism p(θ 1 , .., θ N ) is non-dictatorial if @i ∈ N such that p(θ 1 , .., θ N ) = θ i for all (θ 1 , .., θ N ). That is, a policy mechanism is non-dictatorial if the policy outcome under that mechanism is not always the ideal policy of one particular coalition member, for any possible configuration of coalition members’ policy preferences. Second, we note a constitutional feature of coalition governance in parliamentary democracies that distinguishes this sharply from simple majority rule. While public policy decisions made by a legislature can be implemented despite the opposition of legislators on the "losing" side who prefer the status quo, members of a government coalition cannot be forced to stay in the coalition against their will and accept policy decisions with which they disagree (Austen-Smith and Banks 1990, Laver and Shepsle 1990, 1996). Substantively, this feature of coalition governance is deeply embedded in the constitutions of most parliamentary systems as a requirement for "collective cabinet responsibility". For any politician agreeing to continue in a government coalition, therefore, utility from the policy outcome resulting under that coalition must be at least as high as the utility arising should the status-quo policy remain in place. Another way to think of this is that coalition members may choose to veto shifts in coalition policy away from the status quo. This gives rise to our second substantive criterion for coalition cabinets making decisions when policy preferences are private information. This is an individual rationality constraint for each coalition member: the utility of 9

a coalition member i must be at least as high as i’s payoff from the status-quo. Formally: Individual Rationality: A mechanism p(θ 1 , .., θ N ) is individually rational if and only if ui (p(θ i , θ −i ); θ i ) ≥ ui (0; θ i ) for all i, θ i , θ −i , and for every preference specification ui ∈ Ui . Finally, we require that any mechanism for coalition governance is efficient in one of two senses: "global" Pareto efficiency and "issue-by-issue" Pareto efficiency. "Global Pareto efficiency" requires a mechanism to implement an outcome such that, taking all issues together, every coalition member either prefers this outcome, or is indifferent, when comparing it with any feasible alternative in the multidimensional space. Formally: Global Pareto Efficiency: A mechanism p(θ 1 , .., θ N ) is globally Pareto efficient if and only if, for any (θ 1 , θ 2 , .., θ N ), there is no other outcome p0 such that for some preference 0 N profile {ui }N i=1 ∈ Πi=1 Ui , ui (p ; θ i ) ≥ ui (p(θ i , θ −i ); θ i ) for all i, with strict preference for at

least one i = 1, ..., N . Global Pareto efficiency is also a desirable property of coalition governance since if a coalition policy outcome is globally Pareto efficient, then there can be no further improvement as a result of bargaining between coalition members. We also consider a weaker version of efficiency: "issue-by-issue Pareto efficiency." We note that Pareto efficiency in a one-dimensional issue space requires the policy outcome to be between the lowest and the highest of the coalition members’ preferred policies. This suggests an "issue-by-issue" notion of efficiency: outcomes should be Pareto efficient on each individual issue, considered without reference to other issues.10 Formally: Issue-by-issue Pareto efficiency: A mechanism p(θ 1 , .., θ N ) is issue-by-issue Pareto effij j } } ≤ pj (θ 1 , .., θ N ) ≤ max{θ1j , .., θN cient if and only if, for any (θ 1 , θ 2 , .., θ N ), min{θ1j , .., θN

for all j = 1, 2, ..., M . Issue-by-issue Pareto efficiency is desirable since it improves every coalition member’s payoff relative to the status-quo on any issue for which all coalition members want to change 10

Notice that when a mechanism satisfies global Pareto efficiency, it is also issue-by-issue Pareto efficient.

10

policy from the status-quo in the same direction.

Coalition Governance: A Mechanism Design Approach As mentioned, the literature on coalition governance has developed an informal argument that, because coalition agreements are essentially incomplete contracts, there are situations in which coalition members are uncertain about the precise policy preferences of their partners. Nevertheless, we lack general understanding of what policy outcomes can be implemented as equilibrium outcomes when preferences are private information. The reasons are both theoretical/computational and substantive. From a theoretical perspective, the standard approach to modeling coalition bargaining is to specify some particular non-cooperative bargaining protocol and then solve for the equilibrium policy. However, multilateral bargaining under incomplete information in a setting in which N > 2 agents engage in some sort of a back and forth negotiation process (for example according to the Baron-Ferejohn alternating offers bargaining process) is difficult, if not impossible, to solve analytically. From a substantive perspective, there can be a variety of empirically plausible multilateral bargaining protocols. Different protocols arise from changes in the timing of the game or from small variations the structure of a communication process between agents that stipulates, for example: what messages parties may send to each other; when they may send them; whether these messages are public or private. Since different communication protocols may give rise to different equilibrium policy outcomes, the question of which policy is chosen by a coalition government when preferences are private information is sensitive to the precise assumptions we make about the game of incomplete information that structures the incentives of parties to reveal their private information. This implies that even if we solve for the equilibrium policy of some multilateral bargaining game under incomplete information, we don’t know whether coalition members can achieve a better policy outcome under a different bargaining protocol.

11

A mechanism design approach allows us to overcome some of these theoretical and substantive problems. To this end, we begin our mechanism design analysis by first noting some properties of dominant-strategy incentive-compatible mechanisms. In a seminal paper, Moulin (1980) characterizes the class of all dominant-strategy incentive compatible mechanisms on single-peaked domains in a single dimension and shows that all such protocols can be understood as generalizations of the classical median voter rule. Subsequent work extended Moulin’s characterization to multidimensional environments for several (multidimensional) extensions of the class of single-peaked preferences (Border and Jordan 1983; Barbera, Gul, and Stacchetti 1993).11 These authors show that all mechanisms which are dominant-strategy incentive-compatible in a multidimensional policy space must be decomposable into a product of one-dimensional decision rules, in the sense that the implemented outcome on any given dimension depends only on the coordinates of agents’ ideal policies on that dimension. Existing formal results in this research program also show that, with more than two dimensions, the only dominant-strategy incentive-compatible mechanisms that are globally Pareto efficient are dictatorial (Border and Jordan 1983; Kim and Roush 1984; Nehring and Puppe 2007). The intuition is as follows. For a mechanism to induce agents to have a (weakly) dominant strategy to truthfully reveal their preferred policy it must implement some order statistic of coalition members’ ideal policies on each issue (for example the lowest, the highest, the median of agents’ ideal policies on issue j) or some constant policy. On the other hand, for a mechanism to be globally Pareto efficient, it must be inside the convex hull formed by the parties’ ideal policies. In three or more dimensions, only dictatorial mechanisms can be in the convex hull formed by the coalition members’ ideal policies and also satisfy the incentive compatibility requirement. In a two dimensional policy space, however, it is possible for non-dictatorial mechanisms to satisfy both global Pareto efficiency and the 11

These papers assume that preferences satisfy some multidimensional version of single-peakedness and show that separability of preferences across issues is an essential property to establish the existence of dominant-strategy incentive-compatible mechanisms in multidimensional environments.

12

incentive compatibility requirement. For example, consider the "median mechanism" that implements the median of coalition members’ (reported) ideal policies on each issue. The median mechanism in two dimensions, when N is odd, is both dominant-strategy incentivecompatible and globally Pareto efficient. To illustrate this, consider an example with three coalition members and two policy issues. Let the coalition members’ ideal policies be as follows: θ 1 = (θ11 , θ12 ) = (1, 2); θ 2 = (θ21 , θ22 ) = (4, 1), and θ 3 = (θ31 , θ32 ) = (6, 6). The direct mechanism that implements the median of the coalition members’ (reported) ideal policies on each issue dimension is p(θ˜1 , θ˜2 , θ˜3 ) = (med(θ˜11 , θ˜21 , θ˜31 ), med(θ˜12 , θ˜22 , θ˜32 )). The outcome under this mechanism, (4, 2), is in the convex hull of the parties’ ideal policies and thus the mechanism is globally Pareto efficient. This mechanism is also dominant-strategy incentive-compatible since no coalition member can do better by misrepresenting its ideal policy. In sum, in a space with three or more issue dimensions, the only mechanisms that are both globally Pareto efficient and dominant-strategy incentive-compatible are dictatorial mechanisms (Border and Jordan 1983; Kim and Roush 1984; Nehring and Puppe 2007). In a two-dimensional issue space, the only mechanisms that are both globally Pareto efficient and dominant-strategy incentive-compatible are dictatorial mechanisms and the median mechanism (Kim and Roush 1984; Nehring and Puppe 2007). In light of all this, a key question for us concerns whether the median mechanism satisfies the individual rationality condition? The answer is no. To show this consider the following example with two issues and three P j j 2 coalition members. If coalition members’ utility is represented by ui = − M j=1 (θi −p ) , and their ideal policies are θ 1 = (θ11 , θ12 ) = (1, 1); θ 2 = (θ21 , θ22 ) = (4, 3), and θ 3 = (θ31 , θ32 ) = (6, 2). The median mechanism implements the policy (4, 2). This mechanism is Pareto efficient and dominant-strategy incentive-compatible, but does not satisfy the individual rationality condition since member 1 is better off with the status-quo policy than with the outcome under this mechanism (4, 2).12 12 Dictatorial mechanisms also fail the individual rationality condition since we can always find a preference profile such that the individual rationality requirement is violated for some coalition member.

13

These arguments are synthesized in the following proposition. Proposition 1. No dominant-strategy incentive-compatible mechanism can simultaneously satisfy the non-dictatorship, individual rationality and global Pareto efficiency conditions. Dominant-strategy incentive-compatibility is a foundational property since we cannot assess whether the equilibrium outcome of the game of incomplete information induced by some mechanism is individually rational or globally Pareto efficient unless agents have incentives to reveal their private information truthfully. We can only assess whether any given decision-making protocol leads to individually rational or globally efficient (equilibrium) outcomes with respect to the reported preferences of coalition members. Since members act on the basis of their true policy preferences, the incentive-compatibility requirement is a prerequisite for investigating whether a certain decision-making protocol gives politicians incentives to act upon their private information so as to implement a desirable policy outcome. Non-dictatorship is a foundational property as well since we can’t really talk about coalition governance if it is always the case that the policy outcome is the preferred policy of a particular party, regardless of the policy preferences of any other coalition member. This highlights a fundamental tension between our desiderata since no non-dictatorial dominantstrategy incentive-compatible mechanism for coalition governance can satisfy both individual rationality and global Pareto efficiency. And, as we previously discussed, in three or more dimensions, no non-dictatorial dominant-strategy incentive-compatible mechanism for coalition governance can satisfy global Pareto efficiency.13 This leaves the question of whether there are other dominant-strategy incentive compatiblemechanisms that are non-dictatorial, individually rational and satisfy a weaker version of efficiency such as issue-by-issue Pareto efficiency. To investigate the question of whether is possible for any incentive compatible (direct) mechanism to simultaneously satisfy individual rationality and issue-by-issue Pareto efficiency in our setting, consider the following direct 13

Notice that, in this setting, there could be Bayesian-Nash incentive-compatible mechanisms that satisfy the non-dictatorial and global Pareto efficiency conditions since dominant-strategy incentive compatibility is a more demanding incentive compatibility requirement.

14

mechanism: 1 M Definition 1. The moderate mechanism is m(θ˜1 , ..., θ˜N ) = (m(θ˜11 , θ˜21 , ..., θ˜N ), .., m(θ˜1M , θ˜2M , ..., θ˜N ))

where for any j = 1, ..., M ,   j  min{θ˜1j , θ˜2j , ..., θ˜N }    j j )= m(θ˜1j , θ˜2j , ..., θ˜N max{θ˜1j , θ˜2j , ..., θ˜N }      0

if θ˜ij > 0 ∀i = 1, 2, .., N if θ˜ij < 0 ∀i = 1, 2, ..., N if θ˜ij ≤ 0 ≤ θ˜ij0 for some i, i0

This moderate mechanism implements the (reported) ideal policy that is closer to the status quo on those issue dimensions on which all players report an ideal policy on the same side of the status-quo, and implements the status-quo policy on those issue dimensions on which at least two players report ideal policies on the opposite side of the status-quo. We have the following result: Proposition 2. The unique incentive compatible (direct) mechanism that simultaneously satisfies the non-dictatorship, individual rationality and issue-by-issue Pareto efficiency conditions is the moderate mechanism. Proposition 2 tells us that there is no inevitable trade-off between non-dictatorship, individual rationality and issue-by-issue Pareto efficiency since there is an incentive compatible direct mechanism that can simultaneously satisfy these properties.14 To see that the moderate mechanism satisfies the incentive compatibility requirement, consider possible deviations on some issue j for some coalition member i. If the ideal policies of the parties on issue j are such that they disagree about the direction of policy change, then the implemented policy under the mechanism is 0. If some coalition member i deviates and declares an ideal policy 14

Dragu et al. (2014) have shown that, when all players prefer a policy change from the status-quo in the same direction, the moderate mechanism is a unique direct mechanism that can simultaneously satisfy the incentive compatibility requirement, issue-by-issue Pareto efficiency, and the individual rationality condition. Proposition 2 generalizes this result to a setting in which the coalition members’ ideal policies can also be on opposite sides of the status-quo on some issues. This setting presents novel theoretical challenges since a mechanism in this context has to extract information on both the magnitude and the direction on individual preferences; that is, intuitively, the mechanism has to sort out the issues on which players agree on the direction of policy change from the issues on which they don’t agree.

15

θˆij 6= θij , then i cannot improve its utility since such a deviation either makes no difference or results in a policy that is worse for i. And in the contingency in which all coalition members agree on the direction of policy change on issue j, the mechanism implements the ideal policy of the coalition member closest to the status-quo. If some coalition member i misrepresents its ideal policy, then this deviation either makes no difference or results in an implemented policy that is worse for i. To see that the moderate mechanism is individually rational notice that, for every issue where all coalition members want to move the outcome away from the status quo in the same direction, this mechanism implements the ideal policy of the coalition member closest to the status-quo. And when coalition members disagree on the direction of movement from the status quo, it implements the status quo policy. This means that, for every issue dimension, all coalition members are either better off with coalition policy than with the status-quo policy, or the status quo prevails. This mechanism cannot make any coalition member worse off than with the status quo, and so is individually rational. To illustrate this, consider a setting with two dimensions and three coalition members, where coalition member 1’s ideal policy is (9, 1), member 2’s ideal policy is (1, 9), and member 3’s ideal policy is (3, 3). The outcome under the moderate mechanism is (1, 1) and thus all three coalition members are better off with the policy (1, 1) than with the status-quo policy (0, 0). The moderate mechanism is issue-by-issue Pareto efficient since, on every issue dimension, it implements a policy outcome between the lowest and the highest of the coalition members’ preferred policies on that dimension. Finally, the moderate mechanism is non-dictatorial since, on every issue, it requires some sort of a compromise between two or more parties rather than concentrating decision-making power entirely in the hands of one coalition member. To illustrate this consider a setting with three dimensions and three coalition members, where coalition member 1’s ideal policy is (9, 1, 5), member 2’s ideal policy is (1, 9, 6), and member 3’s ideal policy is (3, 3, 1). The outcome under the moderate mechanism is (1, 1, 1), which, of

16

course is not the ideal policy of any of the three coalition member but rather a compromise.15 The outstanding question is whether the policy outcome implemented by the moderate mechanism can be obtained as the unique equilibrium outcome under some bargaining protocol. We provide a positive answer to this question in a subsequent section, but before doing so, notice that we can use our mechanism design analysis to evaluate any bargaining protocol under which coalition members decide what policy to implement. That is, proposition 2 can serve as a normative benchmark to assess the properties of various bargaining protocols. In principle, we can take any protocol under which coalition members decide on a policy outcome under incomplete information, formalize it as a mechanism, and analyze its resulting outcomes. If the outcome under that specific mechanism is the same as the outcome implemented under the moderate mechanism, then that respective bargaining protocol satisfies the non-dictatorship, individual rationality and issue-by-issue Pareto efficiency conditions. If not, then we know that that respective protocol fails one (or more) of these properties.

Protocols for Coalition Governance: A Normative Assessment In this section, we use our mechanism design results to evaluate some specific decision-making procedures for coalition governance that are commonly discussed in the existing literature, beginning with the "ministerial government" protocols. Under ministerial government, one or other specified coalition member is given exclusive jurisdiction over each policy issue to be decided. In the literature on government formation, "portfolio allocation" models proposed and analyzed by Laver and Shepsle (1990, 1996) and by Austen-Smith and Banks (1990), 15

Notice that since the outcome under the moderate mechanism is not globally Pareto optimal, then it is not necessarily stable with respect to renegotiation; however, if coalition members expect renegotiation to achieve a globally Pareto outcome, then the mechanism would not satisfy our incentive compatibility condition, as previously discussed.

17

leverage the observed departmentalization of cabinet decision making and posit a policymaking regime sometimes described as "ministerial government": each delegated coalition member has unbounded control over issues under its policy jurisdiction and therefore sets policy at its ideal point. We define a "ministerial government" protocol, MG, as follows: 1) every issue j = 1, 2, .., M , is assigned to the exclusive jurisdiction of some coalition member i; 2) for each issue j = 1, 2, ..., M , the coalition members state the policy they would like to implement; that is, each member i announces its preferred policy θ˜ij ; 3) there are at least two issues, j and j 0 , such that i(j) 6= i(j 0 ) where i(j) denotes the member who has the ministerial portfolio of issue j (that is, there is at least one pair of issues that are not controlled by the same j politician);16 4) the implemented policy outcome on issue j is pj = θ˜i(j) .

Since some coalition member i has exclusive jurisdiction on some issue j, the outcome j for any under the "ministerial government" protocol is p = (p1 , p2 , ..., pM ) where pj = θi(j)

possible combination of coalition members’ ideal policies, any beliefs, and any preference specification. Since this differs from the outcome under the moderate mechanism, this implies that the "ministerial government" protocol cannot simultaneously satisfy the incentivecompatibility requirement and the non-dictatorship, individual rationality and issue-by-issue Pareto efficiency conditions. More specifically, the MG protocol is dominant-strategy incentive-compatible. Member i cannot do better by misreporting his or her preferences since this has no effect on any policy issue over which another member has jurisdiction, whereas on policy issues over which member i has jurisdiction, the outcome is i’s preferred policy. The MG protocol is nondictatorial since, by definition, there are at least two policy dimensions such that the member with jurisdiction over one issue is not the same as the member with jurisdiction over the other issue. The "ministerial government" protocol is issue-by-issue Pareto efficient since, for every issue dimension, the outcome is one of the coalition members’ ideal policies. However, 16

Note that all conventional definitions of a coalition cabinet require that each cabinet member is allocated at least one ministerial portfolio, automatically satisfying condition (3)

18

MG protocol does not satisfy the individual rationality condition. To illustrate this, consider again a setting with two dimensions and two coalition members, where a coalition member’s P j j 2 preference is represented by the utility function ui = − M j=1 (θi − p ) . Now let member 1’s ideal policy be (9, 9), member 2’s ideal policy be (1, 1) and also let member 1 control issue 2 and member 2 control issue 1. The outcome under the MG protocol is (1, 9), which is clearly not individually rational since member 2 prefers the status-quo policy (0, 0) to the policy implemented under this mechanism. Next, consider the "median" protocol, which implements the dimension-by-dimension median of cabinet members’ ideal points for the set of new issues to be decided, "as if" cabinet members voted on each new issue as it came up. This protocol might well be seen as a plausible decision-making regime for coalition governance, perhaps as a "collective" alternative to ministerial government.17 Formally, the "median" protocol is defined as follows: 1) for each issue j, each coalition member i announces his or her preferred policy θ˜ij 2) the policy outcome on issue j is the median policy among the announced policies of coalition members. It is easy to see that the outcome under this protocol is the issue-by-issue median of coalition members’ ideal policies, for any possible combination of members’ ideal policies, any bej liefs, and any preference specification, i.e. p = (p1 , p2 , ..., pM ), where pj = med{θ1j , θ2j , ..., θN }

for all j = 1, 2, ..., M . Again, since this outcome differs from the policy outcome implemented by the moderate mechanism, this protocol cannot satisfy all properties at the same time. As we already mentioned, the "median" protocol is dominant-strategy incentive-compatible: for each coalition member, the outcome on every issue is either its own ideal policy or some policy lower or higher than its ideal policy. In the former case, a coalition member obviously has no incentive to deviate. In the latter case, the only way for a coalition member to change the outcome is to announce a policy that is lower than the reported median when its ideal policy is higher than the reported median or a policy that is higher than the reported median 17

Note that the dimension-by-dimension median is the equilibrium outcome arising from majority voting under Kadane’s (1972) "division of the question" setting.

19

when its ideal policy is lower. In both contingencies, the member would be worse off. The "median" protocol is issue-by-issue Pareto efficient and non-dictatorial. However, it is not individually rational as we have shown previously. Finally, a decision-making protocol for coalition governance that has implicitly or explicitly been discussed in published work as a plausible possibility is the "average" mechanism. This protocol implements the (weighted) average of coalition members’ reported ideal policies on every issue. Indeed many existing models of coalition formation under complete information assume that parties in a winning coalition will implement a policy that is some function of the coalition members’ ideal policies and legislative seat shares, for example the seat-weighted average of coalition members’ ideal policies (De Swaan 1973). Formally, the "average" decision-making procedure is defined as follows: 1) for each issue j, each member i announces its preferred policy θ˜ij ; 2) the policy outcome on issue j is (weighted) average of reported ideal policies on that issue. The equilibrium outcome of game of incomplete information induced by the "average" mechanism is p = (p1 , p2 , ..., pM ) where pj =

˜j ΣN i=1 θi N

for all j = 1, 2, ..., M . This mechanism is clearly not dominant-strategy

incentive-compatible: coalition members have self-evident incentives to mis-state their ideal policies and thereby drag the coalition mean closer to their own ideal points. For example, consider a scenario in which three equally-weighted parties have the following ideal policies on some issue j: (θ1j , θ2j , θ3j ) = (1, 2, 9). If the parties report their preferred policies on issue j truthfully, then the implemented policy would be pj = 4. However, party 3, for instance, is better off by reporting θˆ3j = 24 so that the implemented policy would be p = 9, party 3’s most preferred policy. The other two parties have similar incentives to misstate their ideal policies.

20

Constrained Ministerial Government In the previous section, we showed that the decision-making protocols discussed in the existing literature do not lead to policy outcomes which satisfy our desiderata. In this section, we turn to the substantively important question of whether there is some bargaining protocol under which the unique equilibrium outcome is the policy outcome implemented by the moderate mechanism. In other words, there is a critical implementation aspect of mechanism design since using the revelation principle to learn what is feasible in a certain setting doesn’t tell us whether there are plausible protocols that lead to desirable outcomes as unique equilibria. This is important because there can be substantive constraints on the types of bargaining protocol that can be considered plausible in a specific setting. For example, in our coalition governance setting, we will see that an important substantive constraint for any empirically plausible protocol is the observed departmentalization of cabinet decision-making. Almost invariably in practice, one or other coalition member is assigned jurisdiction over each policy dimension. Even if we were to show there is a specific bargaining protocol which implements the same policy as the moderate mechanism, then that protocol would be empirically implausible if, for example, it required two coalition members to have simultaneous jurisdiction over the same policy domain.18 We now specify a protocol, which we label the "constrained ministerial government" protocol, which not only leads to the same equilibrium outcome as the policy outcome implemented by the moderate mechanism, but is also consistent with robust empirical regularities about the departmentalization of government decision-making. Under the constrained ministerial government protocol, cabinet ministers have discretion to set policy in their own jurisdictions, but this discretion is subject to important constraints imposed by coalition partners. Under this protocol, one coalition member is designated to set policy on each relevant issue, but only after the other coalition members first set bounds on how far it 18

There is also an important theoretical point here since the direct revelation game in which all players simultaneously report their ideal policies and an outcome is implemented as a function of those reported ideal policies doesn’t have a unique equilibrium.

21

is acceptable to move policy away from the status-quo. Whenever all coalition members prefer to shift policy from the status quo in the same direction, the designated coalition member has discretion to choose any policy he or she wants, between the status-quo and the closest bound to the status-quo declared by the other coalition partners. If there is no agreement between coalition partners on the direction of policy change from the status quo, then the status quo policy prevails. In this way, each member of the government coalition has some potential effect on government policy on each salient policy dimension. The designated coalition member gets to set policy as close as possible to its ideal policy on the dimension concerned, subject to constraints set by each other coalition member, constraints which reflect their own ideal policies. Formally, define a "constrained ministerial government" protocol, CMG, with the following structure: • For each issue j, coalition members sequentially announce a preferred policy, θ˜ij .19 • For any issue j for which θ˜ij ≤ 0 ≤ θ˜kj for some pairs of coalition members i and k, the policy outcome on issue j is the status quo pj = 0. • For any issue j for which θ˜ij > 0 ∀i, let each of N − 1 coalition members first choose a policy bound Bkj ∈ R+ in a fixed sequential order, where k = 1, 2, ..., N − 1.20 The remaining n−th member then chooses a policy xj ∈ R+ to be implemented. The policy outcome on issue j is:    xj j p =   0

j −1 if xj ∈ [0, minN k=1 {Bk }] j −1 if xj ∈ / [0, minN k=1 {Bk }]

• For any issue j for which θ˜ij < 0 ∀i, let N − 1 coalition members first choose a policy bound Bkj ∈ R− in a fixed sequential order,where k = 1, 2, ..., N − 1.21 The remaining 19

The sequence in which coalition members announce a preferred policy is arbitrary. The identities of these N − 1 players can be arbitrary. 21 The identities of these N − 1 players can be arbitrary.

20

22

n−th member then chooses a policy xj ∈ R− . The policy outcome on issue j is:    xj j p =   0

j −1 if xj ∈ [maxN k=1 {Bk }, 0] j −1 if xj ∈ / [maxN k=1 {Bk }, 0]

The formal specification of the CMG mechanism may look complicated but the procedure itself is actually rather simple. First, coalition members announce in some (arbitrary) sequence the policy they would like to implement on each issue dimension. The status quo remains in place for every issue on which members disagree on the direction of policy change. For each issue on which the coalition members all agree on the direction of policy change, one member is designated to set policy on that issue. The other coalition members first set a bound on how far from the status quo the designated member can move policy. The designated member then sets policy in light of these bounds. The bound closest to the status quo is the one that "bites". The designated member can choose any policy between the status quo and this bound, otherwise the resulting outcome is the status quo.22 Let us define the policy outcome implemented by the moderate mechanism as the moderate policy. Recall that this is the policy outcome is the status-quo policy 0 for all issues on which coalition members disagree about the direction of policy change (relative to the status-quo policy) and, for each issue on which all members agree on the direction of policy change, this policy outcome is the preferred policy of the coalition member whose ideal point is closest to the status-quo on the issue concerned. Formally, 1 M Definition 2. The moderate policy is (m(θ11 , θ21 , ..., θN ), .., m(θ1M , θ2M , ..., θN )) where for any 22 Note that we analyze a general class of CMG protocols since we do not specify, for example, the identity of the party that sets the policy on issue j. Many different ways of selecting a particular coalition member in this example, may be consistent with this class of protocols.

23

j = 1, ..., M ,   j  min{θ1j , θ2j , ..., θN }    j j m(θ1j , θ2j , ..., θN )= } max{θ1j , θ2j , ..., θN      0

if θij > 0 ∀i = 1, 2, .., N if θij < 0 ∀i = 1, 2, ..., N if θij ≤ 0 ≤ θij0 for some i, i0

We have the following result: Proposition 3. The unique equilibrium outcome under the "constrained ministerial government" protocol is the moderate policy for any possible combination of parties’ ideal policies, any beliefs, and any preference specification ui ∈ Ui . To elaborate the intuition of this proposition, consider a scenario in which the coalition government consists of two members who must decide on a tax policy. The two coalition members, 1 and 2, respectively prefer income tax increases of three and ten percent. We want to show that the moderate policy (i.e., a tax increase of three percent) is the equilibrium outcome of the game induced by the "constrained ministerial government" protocol. To do this, we analyze first the outcome of the game in the contingency that both parties declare that they want to implement a tax increase. Let member 1 be the designated policy maker, choosing a tax policy, after member 2 has set the bound on the extent of policy change. In this context, member 2 will set the bound at its ideal policy of a ten percent raise and, in light of this, member 1 will then set a three percent tax increase. Neither coalition member can improve on this since the only way to change the outcome under this mechanism is for member 2 to set the bound below three percent, resulting in a worse outcome for member 2, as well as for member 1. Alternatively, let member 2 set the policy and member 1 set the bound. Member 1 will set the bound at a three percent raise and, in light of this, member 2 will then set a three percent tax increase, which is the best member 2 can now get. If member 2 proposes a tax increase above three percent, this is outside the bound and therefore results in the status-quo policy, a worse outcome for member 2. Therefore the 24

outcome is tax increase of three percent.23 Now let us analyze the first stage of the protocol in which the two members state the tax policy they would like to implement; that is, each member proposes a tax policy, θ˜i . Suppose, without loss of generality, that member 1 announce the tax increase she would like first and the member 2 makes an announcement about her preferred tax increase. It is easy to see that both parties have an optimal strategy to propose their preferred tax policy since misrepresenting their preference would either make no difference or result in a worse outcome. As a result, the unique equilibrium of the game of incomplete information induced by the "constrained ministerial government" protocol is the moderate policy, a tax increase of three percent. Also, notice that the outcome of the "constrained ministerial government" protocol does not depend on which particular coalition member has which particular policy preference, so that interchanging coalition members’ preferences results in the same outcome, a property referred to in the social choice literature as "anonymity".24 Another attractive feature of "constrained ministerial government" is that, as previewed briefly in the introduction, it can accommodate a range of important substantive factors that might affect coalition governance. For example, while details remain matters for debate, an extremely robust empirical finding in the government formation literature is that larger cabinet parties tend strongly to be allocated more seats at the cabinet table.25 Alternatively, some "single issue" party, say a green party, might only enter the governing coalition if it is allocated some particular cabinet portfolio, say the Department of the Environment. The "constrained ministerial government" mechanism can accommodate considerations such as this because it does not impose any restriction on the identity the coalition member allocated any particular policy portfolio, or on the number of policy portfolios allocated to any par23

The equilibrium outcome in the contingency that one of the parties or both parties state in the first stage of the mechanism to prefer a tax decrease is the status-quo policy. 24 Formally, a mechanism p(θ 1 , .., θ N ) satisfies anonymity if for any permutation σ : {1, ..., N } → {1, ..., N }, p(θ σ(1) , θ σ(2) , ..., θ σ(N ) ) = p(θ 1 , θ 2 , .., θ N ). 25 See for example: Snyder et al. (2005); Warwick and Druckman (2006); Laver et al. (2011).

25

ticular coalition member. As a result, "constrained ministerial government" mechanism can be consistent with a wide range of different models that deal with the formation of coalition cabinets in legislatures (Bassi 2013). While our focus in this section was to show there is a plausible protocol that leads to a unique equilibrium outcome satisfying our desiderata, the mechanism we proposed requires that the discretion of a particular minister be bounded. An important substantive question, therefore, concerns how can such policy bounds might be enforced in practice. First, notice that the issue of enforcing the bound might not appear at all under certain circumstances, such as when the coalition member whose ideal policy is closest to the status-quo on some policy dimension is also in charge of policy on that dimension. For example, in our tax example, suppose that the most conservative member, the member preferring a three percent tax increase, is also in charge of the tax policy. Since the other coalition member prefers a tax rate increase of 10 percent, our protocol requires that the conservative player to chooses some tax policy that does not exceed the bound of 10 percent, and given this player’s preference, the issue of enforcing the bound is not a problem in this scenario. However, there may well be situations in which the minister in charge of an issue has a preference for a policy change that is further from the status-quo than that of other coalition members. In such scenarios, the substantive issue of enforcing the bound is critical. The existing literature has addressed this matter by proposing and analyzing a variety of institutional devices by which ministerial discretion can be constrained. These include, among others: cabinet committees (Muller and Strøm 2003); "junior" ministers (Thies 2001); and legislative committees (Martin and Vanberg 2004; Carroll and Cox 2012). There is another, exogenous and powerful, institutional constraint that is not explicitly mentioned in the existing coalition governance literature but typically binds the discretion of cabinet ministers: this is the bureaucracy. Government ministers invariably depend on senior bureaucrats for practical policy implementation. If policy bounds arising from constrained ministerial government are public knowledge, or are at least known to senior bureaucrats, and if the 26

bureaucracy has a norm of allegiance to the government of the day rather than to a particular minister, then bureaucrats can serve as a constraint on the discretion of any minister by refusing to implement a policy that is outside the bounds set by the coalition government. Indeed, bureaucracies in many democratic societies have norms of refusing to implement an "illegal" order from a minister.26 In New Zealand, for example, the State Services Commission states that public servants should reject ministers’ directions if "it is reasonably held that instructions are unlawful because it would be unlawful for the minister to issue them ... where it would be unlawful for the officials to accept them ... where officials would have to break the law in order to carry out the directive."27

Conclusions We began this paper by asking fundamental questions about coalition governance. These concern how coalition cabinets can, and indeed should, take policy decisions in the very typical situation in which the policy preferences of the coalition partners are private information. This suggests an important new substantive focus for the analysis of coalition governance. Until now, much of the relevant literature has focused on the "making" and/or "breaking" of government coalitions, typically assuming high levels of information about the preferences of the politicians involved. The key questions have been, "assuming that agents know what can be known, how do coalition cabinets form, and how do they fail?" Returning to our motivating examples of the world financial or Syrian refugee crises, however, we see that a large and likely the most important part of coalition governance concerns how partners in government behave in relation to problems that, of their essence, cannot possibly be known at the moment of government formation. Since these problems cannot be known in advance, 26

For a formal analysis of how a legal bound can be self-enforcing when those making policy-decision want to implement an illegal policy (i.e., a policy outside the bound) but have to rely on administrators for implementation, see Dragu and Polborn (2013). 27 State Services Commission, The Senior Public Servant at 28, quoted in Kenneth Kernaghan, The Future Role of a Professional Non Partisan Public Service in Ontario (Panel on the Role of Government, Research Paper Series No. 13, 2003) at 313.

27

neither can the coalition partners’ preferences on these. This means that a fundamental and very typical problem for coalition governance is how, once a coalition cabinet has been formed, can its members make important policy decisions on issues for which the coalition partners’ preferences are private information. We analyze this problem using the tools of mechanism design, which allow us to provide a systematic account of what is gained, and what is lost, by using some particular decisionmaking mechanism and thereby to identify what types of "compromise" are desirable and feasible within a coalition government.28 We show that a regime of "constrained ministerial government" is the "best" mechanism (in the sense of our normative criteria) in this ubiquitous setting. This theoretical finding is substantively important. Our formal construct of "constrained ministerial government" in many ways resembles empirical accounts of how real coalition cabinets make decisions. It describes a substantive setting in which cabinet ministers do, for obvious practical reasons, have a lot of de facto power over policy outputs under their jurisdiction. But this is also a setting in which coalition partners are by no means irrelevant in relation to policies outside their own jurisdictions. It is a setting in which, for example, a Minister for Agriculture has a lot of power to affect the coalition’s agriculture policy. When a new Minister for Agriculture is appointed, therefore, everyone with a stake in agriculture policy pays very close attention. Nonetheless, the Minister for Agriculture is never in practice the absolute policy dictator on agriculture and, notwithstanding considerable practical power over agriculture policy, remains constrained by the policy preferences of coalition partners. This is a common theme in many critiques of the widely-cited portfolio allocation models, which assume that cabinet ministers are indeed policy dictators in their own jurisdictions. Many of these critiques, as we saw above, propose various practical ways in which ministerial power might be constrained in coalition cabinets. Our model and findings therefore provide a theoretical foundation for the ongoing re28

Our research also contributes to a literature that applies mechanism design to the study of political institutions and settings (Banks 1990; Baron 2000; Gailmard 2009, Dragu and Simpson 2017, among others) and also to a literature that studies delegation in different political settings (Gailmard 2002; Fox and Stuart 2011; Callander and Krehbiel 2014; Dewan et al. 2015, among others).

28

search program that focuses on practical empirical ways in which cabinet members might set bounds on the ability of their cabinet colleagues to specify government policy in domains over which they have jurisdiction. Our results show that such bounds can contribute in clear ways to normatively desirable outcomes from coalition governance, but we are silent on the practical empirical devices that might be used to enforce these bounds. Addressing this classic principal agent problem (Martin and Vanberg 2015; Indridason and Kristinsson 2013) recent authors have, as we saw above, investigated the role of: cabinet committees (Muller and Strøm 2003); "junior" ministers (Thies 2001); cabinet reshuffles (Indridason and Kam 2008); and legislative committees (Martin and Vanberg 2004; Carroll and Cox 2012). Our results set this empirical work in a normative context. Not only do these devices help coalition partners "keep tabs on" each other, in Thies’s terms, they enable cabinet members to set bounds on one another. These bounds in turn enable a "constrained ministerial government" regime for cabinet decision making. And this is important because, as we have shown here, constrained ministerial government allows members of coalition cabinets to make policy decisions in a way that is incentive compatible, individually rational, non-dictatorial and at least locally Pareto efficient, even when the coalition partners’ policy preferences are private information. Constrained ministerial government is above all coalition governance, taking account of the fact that it is never possible, at the moment a coalition cabinet is formed, to anticipate every key policy decision the new coalition will face. It is a non-manipulable mechanism for coalition governance that reconciles the inherently collective nature of decision making inside a coalition with the bandwidth required for effective decision making by any busy national government, which requires delegation of decision making to individual cabinet members.

29

Acknowledgements We thank Xiaochen Fan, Macartan Humphreys, Scott de Marchi, Mattias Polborn, Adam Przeworski, Ken Shepsle, and seminar participants at Duke University and New York University for helpful comments and suggestions. All errors are ours.

References [1] Austen-Smith, David, and Jeffrey Banks. 1990. “Stable governments and the allocation of policy portfolios." American Political Science Review 84 (3): 891-906. [2] Baron, David. 2000. “Legislative Organization with Information Committees." American Journal of Political Science 44 (3): 485-505. [3] Barbera, Salvador, Faruk Gul, and Ennio Stacchetti. 1993. “Generalized median voter schemes and committees." Journal of Economic Theory 61 (2): 262-289 [4] Bassi, Anna. 2013. “A Model of Endogenous Government Formation." American Journal of Political Science 57 (4): 777-793. [5] Bergemann, Dirk, and Stephen Morris. 2012. Robust mechanism design: The role of private information and higher order beliefs. Vol. 2. World Scientific. [6] Bowler, Shaun, Thomas Bräuninger, Marc Debus, and Indridi H. Indridason. 2016. “Let’s Just Agree to Disagree: Dispute Resolution Mechanisms in Coalition Agreements." The Journal of Politics 78 (4): 1264-1278.. [7] Border, Kim C., and James S. Jordan. 1983. “Straightforward elections, unanimity and phantom voters." The Review of Economic Studies 50 (1): 153-170. [8] Bräuninger, Thomas and Nathalie Giger. 2016. “Strategic Ambiguity of Party Positions in Multi-Party Competition." Political Science Research and Methods, in press. 30

[9] Callander, Steven, and Keith Krehbiel. 2014. “Gridlock and delegation in a changing world." American Journal of Political Science 58 (4): 819-834. [10] Carroll, Royce, and Gary W. Cox. 2012. "Shadowing Ministers Monitoring Partners in Coalition Governments." Comparative Political Studies 45 (2): 220-236. [11] De Swaan, Abram. 1973. “Coalition theories and cabinet formations." Elsevier Scientific Publishing Company, Amsterdam [12] Dewan, Torun, Andrea Galeotti, Christian Ghiglino, and Francesco Squintani. 2015. “Information aggregation and optimal structure of the executive." American Journal of Political Science 59 (2): 475-494. [13] Downs, Anthony. 1957. “An Economic Theory of Democracy." Harper, New York [14] Dragu, Tiberiu, and Mattias Polborn. 2013. “The administrative foundation of the rule of law." The Journal of Politics 75 (4): 1038-1050. [15] Dragu, Tiberiu, Xiaochen Fan, and James Kuklinski. 2014. “Designing Checks and Balances." Quarterly Journal of Political Science 9 (1): 45-86. [16] Dragu, Tiberiu and Hannah Simpson. 2017. ‘Veto Players, Institutional Design and Policy Change." Research and Politics 4 (3): 2053168017722704. [17] Fox, Justin, and Stuart V. Jordan. 2011. “Delegation and accountability." The Journal of Politics 73 (3): 831-844. [18] Gailmard, Sean. 2002. “Expertise, subversion, and bureaucratic discretion." Journal of Law, Economics, and Organization 18 (2): 536-555. [19] Gailmard, Sean. 2009. “Multiple principals and oversight of bureaucratic policymaking." Journal of Theoretical Politics 21 (2): 161-186.

31

[20] Gibbard, Allan. 1973. “Manipulation of voting schemes: a general result." Econometrica 41 (4): 587-601. [21] Kadane, Joseph B. 1972. “On division of the question." Public Choice 13 (1): 47-54. [22] Kim, Ki Hang, and Fred W. Roush. 1984. "Non-Manipulability in Two Dimensions." Mathematical Social Sciences 8 (1): 29-43. [23] Indridason, Indridi H., and Christopher Kam. 2008. “Cabinet reshuffles and ministerial drift." British Journal of Political Science 38 (4): 621-656. [24] Indridason, Indridi H., and Gunnar Helgi Kristinsson. 2013. “Making words count: Coalition agreements and cabinet management." European Journal of Political Research 52 (6): 822-846. [25] Laver, Michael, and Kenneth Shepsle. 1990. “Coalitions and Cabinet Government." American Political Science Review 84 (3): 873-890. [26] Laver, Michael, and Kenneth A. Shepsle. 1994. Cabinet ministers and parliamentary government. Cambridge University Press, New York, NY. [27] Laver, Michael, and Kenneth A. Shepsle. 1996. Making and breaking governments: Cabinets and legislatures in parliamentary democracies. Cambridge University Press, New York, NY. [28] Laver, Michael, and Norman Schofield. 1998. Multiparty government: the politics of coalition in Europe. Ann Arbor, Michigan: University of Michigan Press. [29] Laver, Michael, Scott de Marchi, and Hande Mutlu. 2011. “Negotiation in legislatures over government formation." Public Choice 147 (3): 285-304. [30] Lupia, Arthur and Kaare Strøm. 2008. “Bargaining, Transaction Costs, and Coalition Governance." In Kaare Strøm, Wolfgang C. Müller and Torbjörn Bergman (eds.), Cab32

inets and Coalition Bargaining: The Democratic Life Cycle in Western Europe. Oxford: Oxford University Press, 51-84. [31] Martin, Lanny W., and Georg Vanberg. 2004. “Policing the bargain: Coalition government and parliamentary scrutiny." American Journal of Political Science 48 (1): 13-27. [32] Martin, Lanny W, and Georg Vanberg. 2014. “Parties and policymaking in multiparty governments: The legislative median, ministerial autonomy, and the coalition compromise." American Journal of Political Science 58 (4): 979-96. [33] Martin, Lanny W. and Georg Vanberg. 2015. “Coalition Formation and Policymaking in Parliamentary Democracies." In Handbook of Comparative Political Institutions, ed. Jennifer Gandhi and Ruben Ruiz-Rufino. New York: Routledge Press. [34] Moulin, Herve. 1980. “On Strategy-Proofness and Single Peakedness." Public Choice 35 (4): 437-455 [35] Muller, Wolfgang C., and Kaare Strom. 2003.Coalition Governments in Western Europe. Oxford, Oxford University Press. [36] Nehring, Klaus, and Clemens Puppe. 2007. “Efficient and strategy-proof voting rules: A characterization." Games and Economic Behavior 59 (1): 132-153. [37] Page, Benjamin I. 1976. “The Theory of Political Ambiguity." American Political Science Review 70 (3): 742-752. [38] Riker, William H. 1982. Liberalism against Populism: A Confrontation between the Theory of Democracy and the Theory of Social Choice. San Francisco, CA: W.H. Freeman. [39] Snyder, James M., Michael Ting, and Stephen Ansolabehere. 2005. “Legislative bargaining under weighted voting." American Economic Review 95 (4): 981-1004. [40] Shepsle, Kenneth A. 1972. “The strategy of ambiguity: Uncertainty and electoral competition." American Political Science Review 66 (2): 555-568. 33

[41] Strøm, Kaare, and Müller, Wolfgang C. 1999. “The keys to togetherness: Coalition agreements in parliamentary democracies." Journal of Legislative Studies 5 (3/4): 25582. [42] Strøm, Kaare, Wolfgang C. Muller, and Torbjorn Bergman. 2006. Delegation and Accountability in Parliamentary Democracies. Oxford: Oxford University Press. [43] Strøm, Kaare, Wolfgang C. Müller, and Torbjörn Bergman (eds.). 2008. Cabinets and Coalition Bargaining: The Democratic Life Cycle in Western Europe. Oxford: Oxford University Press. [44] Thies, Michael F. 2001. “Keeping tabs on partners: The logic of delegation in coalition governments." American Journal of Political Science 45 (3): 580-598. [45] Warwick, Paul V., and James N. Druckman. 2006. “The portfolio allocation paradox: an investigation into the nature of a very strong but puzzling relationship." European Journal of Political Research 45 (4): 635-65.

34

Appendix In this appendix, we prove the propositions stated in the main text. Proof of Proposition 1. In text. Proof of Proposition 2. First, we show that the moderate mechanism is domiant strategy incentive compatible and satisfies the non-dictatorship, individual rationality and issue-byissue Pareto efficiency. To see that the moderate mechanism satisfies the dominant-strategy incentive-compatibility requirement, consider possible deviations on some issue j for some coalition member i. If the ideal policies of the parties on issue j are such that they disagree about the direction of policy change, then the implemented policy under the mechanism is the status-quo policy, 0. If some coalition member i deviates and declares an ideal policy θˆij = 6 θij , then i cannot improve her payoff since such a deviation either makes no difference or results in a policy that is worse for i. And in the contingency in which all coalition members agree on the direction of policy change on issue j, the mechanism implements the ideal policy of the coalition member closest to the status-quo. If some coalition member i misrepresents its ideal policy, then this deviation either makes no difference or results in an implemented policy that is worse for i. The "moderate" mechanism is issue-by-issue Pareto efficient since, by the definition, for j any preference profile (θ 1 , θ 2 , .., θ N ), min{θ1j , .., θN } j ≤ pj (θ1 , .., θN ) ≤ max{θ1j , .., θN } for all j = 1, 2, ..., M . Also, for any coalition member

i = 1, 2, ..., N , the resulting policy under this mechanism on some issue j is either pj = 0 or j j ) ≤ θij if θij > 0, or θij ≤ pj = m(θ1j , θ2j , ..., θN ) ≤ 0 if θij < 0. Recall 0 ≤ pj = m(θ1j , θ2j , ..., θN

that, for each coalition member i, the preference over the multidimensional policy outcome P j j j p = (p1 , p2 , ..., pM ) ∈ RM is represented by a utility function ui = M j=1 vi (p ; θi ), where vij (pj ; θij ) is continuous and single-peaked (about an ideal position θij ). For any such preference representation, the moderate mechanism satisfies the individual rationality condition 35

since on all issues on which the implemented policy is 0, the payoff for coalition member i is the same as implementing the status-quo policy, and on all issues on which the mechanism implements a policy different from the status-quo, the implemented policy is either i’s ideal policy or a policy closer to i’s ideal policy than the status-quo. Therefore, the "constrained ministerial discretion" mechanism is individual rational. Finally, the moderate mechanism is non-dictatorial since the implemented policy outcome under this mechanism is not the ideal policy of a certain party for all possible configurations of policy preferences. Second, we show that the moderate mechanism is the unique incentive compatible (direct) mechanism that simultaneously satisfies the non-dictatorship, individual rationality and issue-by-issue Pareto efficiency conditions. Recall that for a mechanism to be dominant strategy incentive compatible it must implement some order statistic of coalition members’ ideal policies on each issue (for example the lowest, the highest, the median of agents’ ideal policies on issue j) or some constant policy (Border and Jordan 1983; Kim and Roush 1984; Nehring and Puppe 2007). Moreover, the issue-by-issue Pareto efficiency requires that a dominant strategy incentive compatible mechanism implements a policy outcome on each issue dimension that is between the lowest and the highest of the coalition members’ preferred policies on that issue. Given these observations, suppose by contradiction that there exists some other mechanism p(θ 1 , .., θ N ) that is incentive compatible, satisfies the non-dictatorship, individual rationality and issue-by-issue Pareto efficiency conditions but is different than the moderate mechanism. For the mechanism p(θ 1 , .., θ N ) to be different than the moderate mechanism it must be that either this mechanism implements a policy that is different than the statusquo on those issues on which players disagree on the direction of changing the status-quo or implements a policy that is different than the ideal policy of player whose ideal policy is the closest to the status-quo on those issues on which all players prefer a change from the status-quo in the same direction, or both. Dragu et al. (2014) have shown that in a setting in which, for every issue j, the preference 36

profile is such that all players prefer a policy change in the same direction, the moderate mechanism is the unique incentive compatible mechanism that satisfies the individual rationality and issue-by-issue Pareto efficiency conditions.29 Since we require a mechanism to satisfy our desiderata for all preference profiles, this implies that the mechanism p(θ 1 , .., θ N ) cannot implement a different policy outcome than the moderate mechanism on issues on which all players prefer a change from the status-quo in the same direction. Let us assume then that the mechanism p(θ 1 , .., θ N ) implements a different policy than the moderate mechanism on issues on which players disagree on the direction of the change to the status-quo policy. Let us consider the following setting: two players and two issues, and let player 1’s ideal policies be to the left of the status-quo on both issues and player 2’s ideal policies be on the right of the status-quo on both issues. Under the moderate mechanism, the policy outcome is the status-quo on both issues. To satisfy the incentive compatibility and the issue-by-issue Pareto efficiency condition, the mechanism p(θ 1 , θ 2 ) has to implement one of the player’s ideal policy on each issue or some constant that lies between the players’ ideal policies on each issue, or a combination of a constant policy on an issue and a player’s ideal policy on the other issue. Whatever constant policy other than (0, 0) or order statistic of the players’ ideal policy is implemented by this mechanism (or combination of both), we can always find a preference profile such that the individual rationality condition is violated. To see this consider the situation in which the mechanism implements the lower of the players’ (reported) ideal policies on one issue and the higher of the players’ (reported) ideal policies on the other issue. Let coalition members’ preference P j j 2 be represented by the utility function ui = − M j=1 (θi − p ) ; also let member 1’s ideal policy be (−3, −3) and member 2’s ideal policy be (3, 3). The outcome under the above protocol is (−3, 3), which is clearly not individually rational since both players prefer the status-quo policy (0, 0) to the policy implemented under this mechanism. Similar violations can be shown if the highest of the players’ (reported) ideal policies is implemented on both issues 29

Proposition 5 in Dragu et al. (2014)

37

or if the lowest reported ideal policies is implemented on both issues or if a combination of a constant policy on one issue and the lowest (or the highest) reported ideal policies on the other issue. As a result, in this scenario, if any incentive compatible (direct) mechanism that satisfies the issue-by-issue Pareto condition implements a policy that is not the statusquo policy, then such mechanism fails to satisfy the individual rationality condition for all preference profiles and utility specifications. This implies that the mechanism p(θ 1 , .., θ N ) implements the status-quo policy on issues on which players disagree on the direction of the change to the status-quo policy in order to satisfy our specified desiderata for all preference profiles. Similarly, for any N>2, it can be shown analogously that whatever constant policy other than the status-quo or order statistic of the players’ ideal policy is implemented by this mechanism (or combination of both), we can always find a preference profile such that the individual rationality condition is violated. Taken together with the previous argument, this implies that the moderate mechanism is the unique incentive compatible (direct) mechanism that satisfies he non-dictatorship, individual rationality and issue-by-issue Pareto efficiency conditions.

Proof of Proposition 3. Let us solve for the equilibrium outcome of the "constrained ministerial government" protocol. First, consider some arbitrary issue j and let us look at what policy outcome results in the contingency that all coalition members state that the policy they want to implement on issue j is to the right of the status-quo, i.e., θ˜ij > 0 ∀i. Without loss of generality re-order coalition members such that the member with ideal policy θij is the i-th one to make a choice. That is, coalition members 1 through N − 1 choose a policy bound in the order 1, 2, ..., N − 1, and then member N (having observed all policy bounds chosen by the previous N − 1 players) chooses a policy on issue j. −1 j In the last stage of this game, the effective policy bound is minN i=1 Bi . If player N −1 j chooses a policy not exceeding minN i=1 Bi , then the outcome is the policy chosen by N ;

otherwise the outcome is the status quo. If the effective policy bound chosen by the pre38

N −1 j Bi , for any strategy of each of the previous N − 1 playvious N − 1 players is mini=1

ers, any preference specification and beliefs of player N , the optimal strategy for N is j j j j j N −1 j xj (B1j , ..., BN −1 , θN ) = min{mini=1 Bi , θN } if θN > 0 and Bi > 0 for all i = 1, 2, ..., N − 1 j j 30 and xj (B1j , ..., BN −1 , θN ) = 0 otherwise.

In the second-to-last stage, if the policy bounds chosen by the previous N − 2 players j are B1j , ..., BN −2 , for any strategy of each of the previous N − 2 players and any preference j specification and beliefs of player N −1, if θN −1 > 0, then setting the bound at its ideal policy j j j j j (i.e. BN −1 (B1 , ..., BN −2 , θN −1 ) = θN −1 ) is a (weakly) dominant strategy for player N −1, and j j j j j if θN −1 ≤ 0, player N − 1’s (weakly) dominant strategy is BN −1 (B1 , ..., BN −2 , θN −1 ) = 0. If j θN −1 > 0, setting the bound at its ideal policy is a (weakly) dominant strategy since, if player j N − 1 deviates to B 0 < θN −1 , then such deviation either does not change the outcome and

player N − 1 receives the same payoff, or it changes the outcome from somewhere between B 0 j 0 and θN −1 to B . Since players have single-peaked preferences on every issue, this change will

make player N − 1 worse off. A similar argument shows that this player also has no incentive j to deviate to B 0 > θN −1 . Iterating this argument back to the first stage, the strategy for any

other player that sets a policy bound is the same as the optimal strategy of player N − 1. As a result, the unique outcome in the contingency that all players declare that their preferred j policy is to the right of the status-quo policy, is pj = min(θ1j , θ2j , ..., θN ), if θij > 0 ∀i, and

pj = 0 otherwise, for any possible combination of coalition members’ ideal policies, any beliefs, and any preference specification ui ∈ Ui . The same argument applies to solving the outcome of the game in the event that θ˜ij < 0 ∀i for some issue j. That is, the unique outcome in the contingency that all players declare that the policy they would like to implement is to the left of the status-quo policy, j is pj = max(θ1j , θ2j , ..., θN ) if θij < 0 ∀i and pj = 0 otherwise, for any possible combination of

coalition members’ ideal policies, any beliefs, and any preference specification ui ∈ Ui . Next let us analyze the announcement stage of the game of incomplete information in30

N −1 j Or in this case, player N could also choose xj > mini=1 Bi , which will result in the same outcome 0.

39

duced by the constrained ministerial government protocol: a coalition member i’s decision to announce its preferred policy, θ˜ij for some issue j = 1, 2, ..., M . Without loss of generality let coalition member with ideal policy θij be the i-th one in the sequence in which players announce their ideal policies. In the ith stage of this announcement game, where i > 1, player i’s optimal strategy is as follows: 1) any announcement θ˜ij > 0 is optimal if θij > 0 and all previous players declare that the policy they would like to implement is to the right of the status-quo policy; 2) any announcement θ˜ij < 0 is optimal if θij < 0 and all previous players declare that the policy they would like to implement is to the left of the status-quo policy; 3) any policy announcement is optimal if the previous players’ announcements are such that at least one player declared a policy to the left of the status-quo and another player declared a policy to the right of the status-quo. This is the optimal strategy since if all previous players declared a policy to the right of the status-quo and player i’s ideal policy is to the right of the status-quo, then player i is better off announcing some policy to the right of the status-quo for otherwise the implemented policy on issue j is the status-quo whereas if all players have ideal policies to the right of the status-quo, such an announcement strategy will result a policy outcome that player i prefers over the status-quo. In the first stage of this announcement game, player 1’s optimal strategy is as follows: 1) any announcement θ˜ij > 0 is optimal if θij > 0 and 2) any announcement θ˜ij < 0 is optimal if θij < 0. This is the optimal announcement strategy for player 1 since if player 1’s ideal policy is to the right of the status-quo, an announcement of an ideal policy to the left of the status-quo will preclude any policy change to the right of the status-quo no matter what the other players do and player 1 would be worse off in the contingency in which all players’ ideal policies are to the right of the status-quo. A similar reasoning works for why it is optimal for player 1 to declare a policy to the left of the status-quo if player 1’s ideal policy is to the left of the status-quo. Given the optimal strategy of players in this announcement state of the game, all policy issues on which all players’ ideal policies are to the left or to the right of the status-quo 40

will be considered in the subsequent stage of the game (the stage starting after the players sequentially announce their ideal policies), and the implemented outcome on those issues will be the ideal policy of the player whose ideal policy is the closest to the status-quo, for any possible combination of coalition members’ ideal policies, any beliefs, and any preference specification ui ∈ Ui . For all policy issues on which players disagree on the direction of changing the status-quo, the resulting outcome is the status-quo policy, for any possible combination of coalition members’ ideal policies, any beliefs, and any preference specification ui ∈ Ui . As a result, the unique equilibrium outcome of the constrained ministerial governM 1 )) ), .., m(θ1M , θ2M , ..., θN ment protocol is the moderate policy m(θ 1 , ..., θ N ) = (m(θ11 , θ21 , ..., θN

where for any j = 1, ..., M ,   j  min{θ1j , θ2j , ..., θN }    j j m(θ1j , θ2j , ..., θN )= max{θ1j , θ2j , ..., θN }      0

41

if θij > 0 ∀i = 1, 2, ..., N if θij < 0 ∀i = 1, 2, ..., N if θij ≤ 0 ≤ θij0 for some i, i0