Institutional Foundations of Coalition Governance Tiberiu Dragu and Michael Laver New York University

Abstract Almost as soon as they have formed, all governments face the need to make important policy decisions on problems that were not foreseen, and could not possibly have been foreseen, during the process of government formation. This problem is particularly acute for coalition cabinets, whose members almost inevitably have diverse policy preferences, preferences which are essentially private with respect to new and unexpected matters. An important substantive feature of coalition governance, therefore, concerns how members of coalition cabinets take decisions on unanticipated new matters. In this paper we use a mechanism design approach to analyze the properties of some commonly discussed mechanisms for coalition governance, before extending the analysis to all possible mechanisms. Stipulating two normatively desirable properties of any mechanism for coalition governance (efficiency and stability), we show that it is impossible for a policy mechanism to satisfy these properties while at the same time inducing coalition members to reveal their policy preferences truthfully. Nonetheless, one possible mechanism, which we call "constrained ministerial government", offers what we argue is an empirically plausible and normatively appealing compromise in the face of the inescapable trade-offs involved.

As soon as any government takes office, its key players face the need to make collective decisions. Some of these decisions will have been anticipated. Explicitly or implicitly, they will have been baked into the understanding between government members that emerged during the government formation process. Many if not most of these decisions, however, will concern important matters that were not anticipated, and could never have been anticipated, when the government formed. Think of the world financial crisis of 2008, or the Greek debt and Syrian refugee crises of 2015. These are high-profile situations in which governments around the world, be they democracies or autocracies, faced crucial policy decisions that could not possibly have 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, as opposed to routine public administration, involves policy decisions on unanticipated problems. Any "contract" between members of any government is inevitably incomplete. This is particularly evident for coalition cabinets, which of their essence comprise members with potentially conflicting policy objectives. 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 problems arising from random shocks to the political environment. If we want to understand coalition governance, therefore, we need to understand how members of coalition cabinets make decisions on the stream of new problems that were not part of the deal when their cabinet was formed and for which, furthermore, the preferences of their coalition partners are essentially private information. We first locate our argument in the extensive literature on the making and breaking of (coalition) governments in parliamentary systems with minority legislatures. We move on to a formal description of the decision-making environment for coalition governance, in a setting where coalition members’ preferences on unforeseen problems are private information. We specify simple normative desiderata for decision-making in such settings. These concern the stability and efficiency of the decision making process. With these tools in hand, we turn to an evaluation of different mechanisms that cabinet members might use for decision-making on unanticipated problems. Although our analysis can be applied to any such mechanism, we evaluate three types of mechanism, which we call "ministerial", "constrained ministerial" and "prime ministerial" government. We identify an inescapable trade-off at the heart of any possible system for coalition governance, not just the three mechanisms we consider. Simply put, no non-manipulable mechanism is both stable and globally efficient. When coalition governments must make decisions in the face of new and unexpected events, as will inevitably be the case, our framework allows us to specify what is gained, and what is lost, by using 1

some particular decision-making mechanism. There is a vast literature on the "making and breaking" of government coalitions in parliamentary democracies. This ranges from cross-national empirical analyses to rigorous formal models.1 The plain fact that coalition cabinets, once formed, must survive in a stream of "critical events" has informed a body of work on cabinet stability. This manifested first in the use of event history models as the standard statistical framework for analyzing empirical covariates of government survival.2 Substantively, this approach impounds a generic assumption that any critical event is likely to be somewhat destabilizing for any government and that more stable governments are those more likely to survive more critical events. Crudely speaking, this literature treats coalition cabinets as black boxes, some more some less resilient, that must survive intact while being pummeled by a stream of random events. Lupia and Strom (1995) specified an explicit formal model linking the survival of a twoparty coalition government in a three-party system to what goes on inside the black box of cabinet governance. This modeled the effect of what Laver and Shepsle (1998) subsequently called "public opinion shocks", which perturb cabinet members’ expectations about changes in party seat shares in the event of an immediate election, and thereby change their incentives to precipitate such an election and bring down the government.3 The Lupia-Strom model was analyzed empirically within the event history framework by Diermeier and Stevenson (1999, 2000). Laver and Shepsle (1995) extended this approach to look at "policy shocks", unanticipated events which force politicians to take positions on issues they had not considered before, and "agenda shocks", which unexpectedly raise the salience of particular issues. All of this work takes explicit account of the fact that coalition cabinets are subject to a stream of generic shocks from the moment they form, though only the Lupia-Strom model offers an explicit account of cabinet politics inside the black box. Lupia-Strom shocks, however, are shocks to expectations about the next election result, not important new matters for decision that unexpectedly pop up on the cabinet agenda. Their model does not therefore address our core question, which concerns how members of coalition cabinets make collective policy decisions when faced with unexpected new problems on which their policy preferences are private information. Martin and Vanberg (2014) do address the question of how coalition cabinets make policy decisions, with an empirical analysis of changes to government bills in three countries 1

The relevant literature is much too big to attempt a review here. A widely-cited "mid-term" review can be found in Laver and Schofield (1998); an excellent synthesis of the formal literature on coalitions can be found in Humphreys (2008); a very helpful recent review can be found in Martin and Vanberg (2015). 2 See Browne, Frendries and Gleiber (1984, 1986, 1988); King et al (1990); Warwick (1994). 3 Although in equilibrium coalition members may renegotiate their deal to avoid the costs of holding an election.

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as these progress through the legislature, designed to assess rival assumptions about cabinet decision-making. These assumptions are: "legislative median" (policy preferences of members of the government are irrelevant, all that matters are preferences of the median legislator); "coalition compromise" (privileging the seat-weighted mean policy positions of the coalition partners); "ministerial autonomy" (privileging the policy position of the minister with jurisdiction over the relevant policy area). Their empirical findings are clear, and are "inconsistent with the notion that ministers can unilaterally determine policy in areas under their jurisdiction. We also find strong evidence that policy lies closer to the coalition compromise than to the ideal point of the legislative median ... our evidence suggests that policies that are adopted in parliamentary systems reflect a compromise among the policy positions of coalition partners" Martin and Vanberg (2014: 994). This striking and relevant finding deals only with the process of passing legislation, not with cabinet policy-making more generally; nor does it address the problem of cabinet decision-making on unanticipated problems for which cabinet members’ and relevant minister’s preferences may well be private information. But it does provide an empirical benchmark for such an analysis, to which we now turn. While our motivation here is to analyze properties of decision-making protocols for coalition governance when cabinet members face unexpected problems and events, the mechanism design analysis we develop below has broader substantive relevance. It applies to any situation in which members of a coalition cabinet must aggregate their announced policy preferences into a collective policy decision, notwithstanding the fact that their actual policy preferences are private information.

Decision-Making Environment for Coalition Governance We now set out to open the black box and analyze policy making inside incumbent coalition cabinets. We describe coalition governance as interaction between leaders of two or more well-disciplined legislative parties (in general, N ≥ 2 parties) which comprise a government coalition.4 Each party leader has preferences over a multidimensional policy space RM where M ≥ 2. These preferences are continuous, separable across the M issues, and single-peaked on each issue dimension. That is, party leader i’s preference for policy outcomes on each issue j is given by a continuous single-peaked preference about an ideal position θij ∈ R. This single-peaked preference over the multidimensional policy outcome p = (p1 , p2 , ..., pM ) ∈ RM 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.

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can be represented by a utility function ui for each party leader i. Let Ui be the class of all such separable, single-peaked 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). While some information about party leaders’ preferences on matters that can be anticipated may be revealed during election campaigns and subsequent government formation negotiations, their preferences concerning completely unanticipated new problems are essentially private information. We describe party leaders 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. The leaders’ types, denoted by θ = (θ 1 , ..., θ N ), are drawn according to a probability distribution function Φ ∈ ∆H, where H = RM × RM × ... × RM , and ∆H is the set of all probability distribution functions over the set of party leaders’ types. In a slight abuse of terminology, we henceforth refer to ∆H as the set of all beliefs party leaders have about each others’ preferred policies. Coalition members may make any statement they like about which particular policy outcome they prefer. Clearly, however, they may well not find it in their best interests to reveal their "true" preferences about this; in many circumstances they can hope to benefit by mis-stating these. Since coalition members’ preferences are private information, revealed only through actions taken and statements made while governing, we cannot be certain that any given protocol for coalition governance (possible actions open to agents and timing of their interactions) will yield a particular desired outcome unless this protocol gives members incentives to interact in such a way that this outcome is realized. Simply put, for the same set of coalition members’ sincere preferences, different protocols for coalition governance give the members different incentives to reveal these preferences, and therefore different incentives to interact in such a way as to achieve some particular outcome of interest. There is an effectively unlimited number of different protocols that set out how coalition members make and implement collective policy decisions in the face of new and unexpected events. One such protocol, for example, is for a designated coalition member to propose a policy on some issue and for coalition members to decide collectively, using simple majority rule, whether to implement the proposal or maintain the status-quo. Another protocol is that policy on each issue is set unilaterally by a single preeminent member of the government, 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 |; j=1 (θi − p ) ; the utility function defined in terms of the city block distance, − P 1/t M j j s the utility function defined in terms of the p-norm, − for t ≥ 1; the quadratic loss utility j=1 (θi − p ) PM j j 2 − j=1 αj (θi − p ) where αj > 0. In general, utilities from the class of separable, single-peaked utilities PM can be characterized by j=1 vij (pj ; θij ), where vij (pj ; θij ) is continuous and single-peaked (about an ideal position θij ) and pj is the policy outcome implemented on issue j.

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the "prime minister". Since different protocols give parties different incentives to act upon their private information when choosing a policy outcome and since some outcomes may well be more normatively desirable than others, it is important to investigate whether there are particular decision-making protocols that lead to policy outcomes that are desirable. Our goal is to analyze the properties of decision-making protocols for coalition governance in the face of a stream of unexpected events, and we use a mechanism design approach to do this. 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 ). To illustrate this, consider the decision-making protocol according to which some coalition member i proposes a policy on some issue j and then coalition members decide collectively, using simple majority rule, whether to implement the proposal or maintain the status-quo. In this mechanism, the strategy for party i is a policy proposal x ∈ R and a vote {yes; no} at the voting stage; the strategy for parties other than i is a vote {yes, no} at the stage where coalition members decide whether to accept or not i’s proposal. The outcome rule is to implement i’s proposed policy, pj = x, if a majority of members vote "yes" and to maintain the status quo, x = 0, otherwise. This particular mechanism has a simple structure but, in principle, a mechanism can be any complex dynamic process under which parties interact to choose a policy outcome. Notice a crucial difference between a mechanism and a non-cooperative game. In a mechanism design setting, the consequence of a profile of strategies is an outcome rather than a vector of payoffs, as in a non-cooperative game. This is because, in a mechanism design analysis, agents’ preferences vary and the analysis is carried for all possible preference specifications. Once preferences of the agents are specified, say, each coalition member’s policy preference is represented by a specific utility function ui (together with assumptions on the possible types of agents and their beliefs), then a mechanism induces a game of incomplete information among agents. Mechanism design thus allows us to investigate the properties of decision-making protocols for all contingencies, which is particularly relevant for assessing the institutional arrangements under which coalition governments make policy in the face of new and unexpected events. We don’t want to analyze some particular game that models cabinet decision-making in a stream of random events, which was for example the approach adopted by Lupia and Strom (1995) and indeed by many if not most extant theories of coalition governance. We want to say something about all possible cabinet decision-making regimes, all possible coalition games in such a setting. Although this task might appear intractable, we are helped by the revelation principle. Consider some mechanism M = {S1 , S2 , ..., SN ; p(·)}. The revelation principle states that, 5

for any equilibrium of a game of incomplete information induced by this mechanism, there exists a truth-revealing direct mechanism that is payoff-equivalent with such equilibrium (Myerson 1979). In a truth-revealing direct mechanism, the strategy spaces are precisely the type spaces, and in equilibrium all parties reveal their true types.6 Formally, a truth-revealing direct mechanism p(θ 1 , θ 2 , ..., θ N ) specifies a policy outcome p ∈ RM as a function of the agents’ true preferred policies (θ 1 , θ 2 , ..., θ N ). To analyze properties of all possible decisionmaking protocols, therefore, it is sufficient to consider only truth-revealing direct mechanisms subject to incentive compatibility constraints requiring that truthful revelation be an equilibrium in the game of incomplete information induced by the respective mechanism. To this end, there are two notions of incentive compatibility we could require a mechanism to satisfy: dominant-strategy incentive-compatibility and Bayesian-Nash incentive-compatibility. In the context of our analysis, we require that a mechanism p(θ 1 , θ 2 , ..., θ N ) is dominantstrategy incentive-compatible: 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 is the most robust notion of incentive compatibility since it requires that truthful revelation is an equilibrium in (weakly) dominant strategies in the game of incomplete information induced by the mechanism p(θ 1 , θ 2 , ..., θ N ). In other words, a coalition member has a (weakly) dominant strategy to reveal its preferred policy regardless of what other players 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. In contrast, if we were to require a mechanism to be Bayesian-Nash incentive-compatible, then predicting whether it will yield some desired outcome depends critically on a range of detailed assumptions about the various party leaders’ conjectures about each others’ 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 normative desiderata that can be implemented by coalition cabinets making decisions in the face of unanticipated new prob6

A direct revelation mechanism is a simple procedure in which the actions of agents are to reveal their types and an outcome results as a function of the reported types; a direct revelation mechanism is a special case of a general mechanism M = {S1 , S2 , ..., SN ; p(·)} with Si = RM ∀i. Mechanisms that are not direct revelation mechanisms are typically referred to as indirect mechanisms (the three types of mechanism that we will investigate in detail, the "ministerial", "constrained ministerial" and "prime ministerial" mechanisms, are examples of indirect mechanisms).

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lems, taking account of any possible conjectures and beliefs party leaders might have about each others’ preferred policies. Furthermore, if a mechanism is dominant-strategy incentivecompatible then it is also Bayesian-Nash incentive-compatible; this follows trivially from the fact that a (weakly) dominant strategy equilibrium is necessarily a Bayesian-Nash equilibrium. Dominant-strategy incentive-compatible (non-manipulable) mechanisms have further advantages 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 in 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 these blur lines of public accountability because the policy compromises that are inevitably required take place, non-transparently, in strategic wheeling and dealing between party leaders behind closed doors. Non-manipulable mechanisms, in contrast, are in this sense transparent. To illustrate our incentive compatibility requirement, consider one mechanism that satisfies it and one that does not. The first mechanism sets the outcome on issue j as the median of the coalition members’ reported preferred policies on that issue.7 This mechanism 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 on issue j 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 when its ideal policy is lower than that. In both contingencies, the member would be worse off. Now consider a second mechanism whereby coalition members announce their ideal policies on each issue, and coalition policy is the mean of the players’ announced policies on each issue.8 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 M 1 )), Formally, p(θ 1 , .., θ N ) = (p1 (θ11 , θ21 , ..., θN ), ..., pM (θ1M , θ2M , ..., θN j j j med{θ1 , θ2 , ..., θN } for all j = 1, 2, ..., M . 7

j where pj (θ1j , θ2j , ..., θN )

j 1 M Formally, p(θ 1 , .., θ N ) = (p1 (θ11 , θ21 , ..., θN ), ..., pM (θ1M , θ2M , ..., θN )), where pj (θ1j , θ2j , ..., θN )= all j = 1, 2, ..., M . 8

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j ΣN i=1 θi N

= for

closer to their own ideal points.

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 two normatively compelling desiderata for any system of coalition governance: stability and efficiency. First, 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 first substantive property for coalition cabinets making decisions in the face of unexpected events. A mechanism for coalition governance, p(θ 1 , .., θ N ), must satisfy an individual rationality constraint we call coalition stability. This requires a coalition’s utility for any member be at least as high as the member’s payoff from the status-quo. Coalition Stability: A mechanism p(θ 1 , .., θ N ) is coalition stable 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 . Second, we require that a mechanism p(θ 1 , .., θ N ) for coalition governance is efficient. We consider two different notions of efficiency: "issue-by-issue" and "global" Pareto efficiency. First, notice that Pareto efficiency in a one-dimensional issue space requires the policy outcome 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. This property, which we call "issue-by-issue Pareto efficiency", is desirable since it improves each coalition member’s payoff relative to the status-quo policy on those issues on which all coalition members want 8

a policy change from the status-quo in the same direction. 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 . Second, we consider a stronger version of efficiency: "global Pareto efficiency." This requires that a mechanism implements an outcome such that, taking all issues together, all coalition members either prefer it, or are indifferent, when comparing it with any feasible alternative in the full multidimensional space. 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 ui (p0 ; θ i ) ≥ ui (p(θ i ; θ −i ), θ i ) for all i and for all preference specifications ui ∈ Ui , and uk (p0 ; θ k ) > uk (p(θ k ; θ −k ), θ k ) for some k. Global Pareto efficiency is also a desirable property of coalition governance since if a coalition policy outcome is globally Pareto efficient, then it is "renegotiation proof" in the sense that there can be no further improvement as a result of bargaining between coalition members. This is because, if the outcome is globally Pareto efficient, then by definition any feasible alternative is either preferred by no coalition member, or is less preferred by at least one coalition member. This means that attempts to bargain away from the globally efficient outcome within the coalition will be unsuccessful. To illustrate global Pareto efficiency for coalition government outcomes, consider two mechanisms, one that satisfies the condition and one that doesn’t. The first mechanism chooses the coalition member’s reported ideal policy which is closest to the status quo in the multidimensional policy space.9 This mechanism is globally Pareto efficient: it always implements the ideal policy of one of the coalition members and any move away from the chosen policy would make this member worse off. Second, consider the mechanism which sets the outcome on each issue j at the minimum of coalition members’ preferred policies on that issue.10 This mechanism is not globally Pareto efficient. To see this, consider a setting with two dimensions and two coalition members, where a coalition member’s preference is P j j 2 represented by the utility function ui = − M j=1 (θi − p ) . Let member 1’s ideal policy be (9, 1) and member 2’s ideal policy be (1, 9). The outcome of this mechanism is (1, 1). This is clearly not globally Pareto efficient since many other policies, say (5, 5), improve the utilities of both coalition members. However, this mechanism is issue-by-issue Pareto efficient since Formally, this is the mechanism p(θ 1 , .., θ N ) = θ i∗ , where θ i∗ is the ideal policy of player i∗ ≡ argminN i=1 d(0, θ i ), with d representing the Euclidean distance function. j j 10 Formally, this is the mechanism p(θ 1 , .., θ N ), where pj (θ1j , θ2j , ..., θN ) = min{θ1j , θ2j , ..., θN } for all j = 1, 2, ..., M . 9

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the implemented outcome on each dimension is one coalition member’s ideal policy on that dimension. More generally, these examples illustrate that global Pareto efficiency implies issue-byissue Pareto efficiency: when a mechanism satisfies global Pareto efficiency, it is also issueby-issue Pareto efficient.

Coalition Governance: A Mechanism Design Approach In order to analyze coalition governance in the most general setting, we develop our mechanism design analysis in two steps. First, we evaluate the properties of three specific classes of mechanisms for coalition governance: "ministerial government", "constrained ministerial government", and "prime-ministerial government". These mechanisms vary in the extent to which decision-making power is shifted away from individual ministers. We show that the latter two mechanisms better satisfy at least one of our desiderata than the "ministerial government" mechanism, suggesting that some centralization of cabinet decision-making may be desirable when coalition cabinets make policy in the face of new and unexpected events. This analysis also shows that, while the "constrained ministerial government" and "primeministerial government" mechanisms are both dominant-strategy incentive-compatible, the former is coalition stable but not globally Pareto efficient while the latter is globally Pareto efficient but not coalition stable. Following our analysis of these classes of mechanism, we undertake a more general mechanism design analysis to investigate whether there is any possible mechanism that is dominant-strategy incentive-compatible and can simultaneously satisfy stability and global Pareto efficiency conditions. We identify an inescapable contradiction between desiderata at the heart of coalition governance. No dominant-strategy incentive-compatible mechanism can simultaneously satisfy coalition stability and global Pareto efficiency. An inevitable tension exists between coalition stability and global Pareto efficiency when coalition governments make policy in the face of new and unexpected events. Which particular decision-making protocol for coalition governance is seen as better in such situations depends, therefore, on whether greater importance is attached to stability or efficiency. To summarize, mechanism design allows us to investigate which decision-making protocols satisfy particular desired properties, in the very general setting where coalition cabinets must aggregate the announced policy preferences of coalition members into a single collective policy outcome, despite the fact that their actual policy preferences are private information. Our analysis of the three classes of mechanisms mentioned above proceeds as follows. First, we investigate whether truthful revelation of policy preferences is an equilibrium in (weakly) 10

dominant strategies in the game of incomplete information induced by the mechanism under consideration. This incentive compatibility requirement is a prerequisite for the next step of our analysis: if a mechanism is dominant-strategy incentive-compatible, then we can assess whether the equilibrium outcome of the game of incomplete information induced by that mechanism satisfies our two normative criteria: stability and efficiency.

Ministerial Government Our first mechanism for coalition governance is informed by a striking and pervasive regularity in real word government decision making. All governments, whether or not these comprise coalitions of different parties, operate according to a strong and often constitutionally embedded institutional structuring of governance that "departmentalizes" policy making and implementation. This imposes a formal division of labor according to which particular issues fall under the "jurisdiction" of particular government departments responsible for policy development and implementation in the relevant domain. Thus the Department of Education is formally responsible for developing and implementing policies on education issues, the Department of the Environment on environmental issues, and so on. Political control of each government department is given to a government "minister", almost always a politician associated with one of the government parties, as a "portfolio" of issues for which this person is responsible. This amounts to formal delegation of responsibility for particular portfolio of issues to particular members of the cabinet.11 This institutionally embedded departmental governance structure sets up what Kadane (1972) called a "division of the question". Decision-making bodies may be confronted with bundles of issues, but institutional division of responsibility for each issue, for example in a legislative committee system (Shepsle 1979) or a set of government departments, means that decisions are processed on an issue-by-issue basis. In the context of the literature on government formation, the "portfolio allocation" models proposed and analyzed by Laver and Shepsle (1990, 1996) and by Austen-Smith and Banks (1990), leverage the empirical departmentalization of cabinet decision making referred to above and posit a policy-making 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.12 Accordingly, we define a "ministerial government" mechanism, MG, as follows: 11 For empirical surveys of the delegation of policy making and implementation in coalition governance, see Laver and Shepsle (1994); Strom, Muller and Bergman (2006). 12 Substantively, control over policy in some domain is achieved in "portfolio allocation" models by giving one coalition member the ministerial portfolio with exclusive jurisdiction over this domain. This is the decision-making regime referred to by Martin and Vanberg (2014) as "ministerial autonomy".

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• Every issue j = 1, 2, .., M , is assigned to the exclusive jurisdiction of some coalition member i. • The coalition member with jurisdiction over issue j proposes some policy xj . • The policy outcome on issue j is pj = xj . The "ministerial government" mechanism gives rise straightforwardly to the following proposition. Proposition 1. The "ministerial government" mechanism is dominant-strategy incentivecompatible. Let the coalition member who has the ministerial portfolio of issue j be i(j). The equilibrium outcome of the game of incomplete information induced by the "ministerial j government" mechanism is p = (p1 , p2 , ..., pM ) where pj = θi(j) for any possible combination of coalition members’ ideal policies, any beliefs Φ ∈ ∆H, and any preference specification ui ∈ Ui . The "ministerial government" mechanism MG is dominant-strategy incentive-compatible since, for any possible contingency, the outcome is the preferred policy of the coalition member with jurisdiction over the issue under consideration. No coalition member can do better by misrepresenting his or her preferred policy. For an issue over which member i has jurisdiction, the policy outcome is i’s preferred policy. For an issue on which some other member k has jurisdiction, the outcome is k’s ideal policy. Member i cannot do better by misreporting his or her preferences since this has no effect on those policy issues on which another member has jurisdiction, whereas on those policy issues over which member i has jurisdiction, the outcome is i’s preferred policy. Given that the "ministerial government" mechanism is dominant-strategy incentivecompatible, we next assess the normative properties of this mechanism. We have the following proposition: Proposition 2. The "ministerial government" mechanism is issue-by-issue Pareto efficient but is neither coalition stable nor globally Pareto efficient. The "ministerial government" mechanism is issue-by-issue Pareto efficient since, for every issue dimension, the outcome is one of the coalition members’ ideal policies. However, MG is not globally Pareto efficient.13 To see this, consider a setting with two dimensions and For the "ministerial government" mechanism we assume that there are at least two issue j and j 0 such that i(j) 6= i(j 0 ), i.e., the party that has jurisdiction over issue j is not the same as the party that has jurisdiction over issue j 0 . The scenario when the same party controls all issues is captured by the "primeministerial" mechanism, which we analyze below. 13

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two coalition members, where a coalition member’s preference is represented by the utility P j j 2 function ui = − M j=1 (θi − p ) . Let member 1’s ideal policy be (9, 1) and member 2’s ideal policy be (1, 9). Let member 1 control issue 2 and member 2 control issue 1. The outcome under the MG mechanism is (1, 1). This is clearly not globally Pareto efficient since many other policies, say (5, 5), improve the utilities of both coalition members. Moreover, the "ministerial government" mechanism does not satisfy the coalition stability condition. To illustrate this, consider again a setting with two dimensions and two coalition members, where a coalition member’s preference is represented by the utility function ui = P j j 2 − M j=1 (θi − p ) . 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 mechanism is (1, 9), which is clearly not stable since member 2 prefers the status-quo policy (0, 0) to the policy implemented under this mechanism. This analysis leads us to investigate a different mechanism, which imposes some cabinet control over the ministerial discretion.

Constrained Ministerial Government A common critique of portfolio allocation models in the literature on government formation is that the assumption of unbounded ministerial discretion over policy is unrealistically extreme.14 The argument is that, while cabinet ministers do have substantial discretion to set policy in their own jurisdictions, this discretion is subject to important constraints imposed by coalition partners. We therefore turn to mechanisms in which the policy choices of ministers are subject to such constraints, specifying a simple mechanism we describe as "constrained ministerial government". Under this, 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 is acceptable to move policy away from the status-quo. The designated coalition member has discretion to choose any policy he or she wants, between the status-quo policy and the closest bound to the status-quo declared by each of the other coalition partners on those issues on which the coalition prefers a change over the status-quo. In this way, each member of the government coalition has some potential effect on government policy on each salient policy dimension, without the need for negotiations or agreements between coalition partners. 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" mechanism, CMG, with the 14

For an excellent synthesis of this argument, see Martin and Vanberg (2015).

13

following structure: • For each issue j = 1, 2, ..., M , the coalition members state the policy they would like to implement; that is, each member i proposes a policy, θ˜ij . • For any issue j for which θ˜ij ≤ 0 ≤ θ˜kj for some coalition member 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 − 1.15 The remaining n−th member then chooses a policy xj ∈ R+ to be implemented. The policy outcome on issue j is: ( j

p =

xj 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 − 1.16 The remaining n−th member then chooses a policy xj ∈ R− . The policy outcome on issue j is: ( j

p =

xj 0

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

The specification of the CMG mechanism may look complicated but the procedure itself is actually rather simple. First, coalition members declare what policy they would like to implement. 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.17 15

The identities of these N − 1 players can be arbitrary. The identities of these N − 1 players can be arbitrary. 17 Note that we analyze a general class of CMG mechanisms 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 mechanisms. 16

14

We now define a particular type of policy outcome that will be useful to us: the moderate policy. This is the policy outcome that implements 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, implements the preferred policy of the coalition member whose ideal point is closest to the status-quo on the issue concerned. Formally, 1 M Definition 1. The moderate policy is m(x1 , ..., xN ) = (m(θ11 , θ21 , ..., θN ), .., m(θ1M , θ2M , ..., θN )) where for any j = 1, ..., M ,

 j j j   min{θ1 , θ2 , ..., 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 "constrained ministerial government" mechanism is dominant-strategy incentive-compatible. The equilibrium outcome of the game of incomplete information induced by the "constrained ministerial government" mechanism is the moderate policy for any possible combination of parties’ ideal policies, any beliefs Φ ∈ ∆H, and any preference specification ui ∈ Ui . To elaborate the intuition of this proposition, consider the 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" mechanism. To do this, we analyze first the outcome of the subgame 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 15

the bound and therefore results in the status-quo policy, a worse outcome for member 2. Therefore the equilibrium outcome of this subgame is tax increase of three percent.18 Now let us analyze the first stage of the mechanism in which the two members state the tax policy they would like to implement; that is, each member proposes a tax policy, θ˜i . Both parties have a weakly dominant 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 equilibrium of the game of incomplete information induced by the "constrained ministerial government" mechanism is the moderate policy, a tax increase of three percent. Furthermore, notice that the CMG mechanism is dominant-strategy incentive-compatible. We have already seen that the outcome under "constrained ministerial government" is the moderate policy for all contingencies, so no coalition member has an incentive to misrepresent its preferred policy. If some member were indeed to misrepresent its preferred policy, then this would either have no effect on the implemented outcome or would result in outcome that is worse for that member. Given Proposition 3, we next investigate the properties of the "constrained ministerial government" mechanism. We have the following result: Proposition 4. The "constrained ministerial government" mechanism is coalition stable and issue-by-issue Pareto efficient but not globally Pareto efficient. To unpack the intuition of this proposition, first note that the "constrained ministerial government" mechanism is coalition stable. For every issue which 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 coalition stable. 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 CMG protocol 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). Also, the "constrained ministerial government" mechanism is issue-by-issue Pareto efficient since on every issue dimension it implements a policy outcome that is between the lowest and the highest of the coalition members’ preferred policies on that dimension. However, 18

The equilibrium outcome of the subgame 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.

16

CMG is not globally Pareto efficient. To see this, consider a setting with two dimensions and two coalition members, where a coalition member’s preference is represented by the utility P j j 2 function ui = − M j=1 (θi − p ) . Let coalition member 1’s ideal policy be (9, 1), member 2’s ideal policy be (1, 9). The outcome under the CMG mechanism is (1, 1) and this is not globally Pareto efficient as there other policy outcomes (for example, (5, 5)) that both members would prefer to the policy (1, 1).

Prime-Ministerial Government Constrained ministerial government "centralizes" decision making away from individual cabinet members towards the cabinet as a whole. We now analyze the properties of an extremely centralized mechanism, "prime-ministerial government". A single cabinet member is designated to hold a particularly privileged position, prime-minister, and has full authority to implement a policy outcome on any issue. Formally, the PG mechanism is as follows: • Some coalition member k is assigned the position of prime-minister. • Party k proposes some policy x. • The policy outcome is p = x. The outcome of the "prime-ministerial government" mechanism, for all preference specifications ui ∈ Ui , is the ideal policy of the coalition member who holds the position of prime-minister.19 This gives rise to the following result: Proposition 5. The "prime-ministerial government" mechanism is dominant-strategy incentivecompatible. Let k denote the coalition member holding the prime-ministerial position. The equilibrium outcome of game of incomplete information induced by the "prime-ministerial government" mechanism is p = θ k for any possible combination of members’ ideal policies, any beliefs Φ ∈ ∆H, and any preference specification ui ∈ Ui . It is easy to see that the "prime-ministerial government" mechanism is dominant-strategy incentive-compatible. The equilibrium outcome of the game of incomplete information induced by this mechanism is the ideal policy of the coalition member holding the position of prime minister. This member has no incentive to misrepresent its preferences; no other coalition member can change the outcome by misrepresenting their preferences and thus 19

Note that we could also add a stage in this mechanism (as well as in the "ministerial discretion" mechanism) in which parties state what policy they would like to implement, θ˜ij , just as in the "constrained ministerial discretion" mechanism. Including such a stage is inconsequential to the outcome implemented by this mechanism, since one party has full control over what policy to implement on some issue j.

17

none have an incentive to do so. Given Proposition 5, we investigate the properties of the PG mechanism and have the following result: Proposition 6. The "prime-ministerial government" mechanism is globally Pareto efficient but does not satisfy the coalition stability condition. The PG mechanism is globally Pareto efficient, since it implements the ideal policy of one party on every single issue dimension.20 The "prime-ministerial government" mechanism is not coalition stable, however. To see this, consider a setting with three dimensions and three coalition members, where a coalition member’s preference is represented by the utility P j j 2 function ui = − M j=1 (θi − p ) . Let member 1’s ideal policy be (1, 1, 1), member 2’s ideal policy be (2, 2, 2), and member 3’s ideal policy be (7, 7, 7). Let member 3 be the party holding the prime-ministerial position. The outcome under PG is (7, 7, 7). This is not coalition stable since members 1 and 2 are both worse off with the policy (7, 7, 7) than with the status-quo policy. More generally, it is clear that putative coalition partners have no incentive to stay inside a government coalition if the coalition’s decision protocol is that some other coalition member has the unfettered power to set policy outcomes, completely regardless of what that policy might be.

A Tension between Stability and Efficiency Our analysis of three particular policy mechanisms suggests that some form of cabinet control over ministerial discretion may be desirable since both "constrained ministerial government" and "prime-ministerial government" are each better than "ministerial government" on some dimension. Our analysis also shows that both CMG and PG mechanisms are dominantstrategy incentive-compatible. The former is coalition stable but not globally Pareto efficient, however, while the latter is globally Pareto efficient but not coalition stable. The outstanding question is whether there is some other dominant-strategy incentive-compatible mechanism that is both globally Pareto optimal and coalition stable. We now investigate this question using a more general axiomatic analysis, showing that these desiderata cannot be satisfied simultaneously by any possible policy mechanism. We first note 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 20

By implication, it is also issue-by-issue Pareto efficient.

18

several (multidimensional) extensions of the class of single-peaked preferences (Border and Jordan 1983; Barbera, Gul, and Stacchetti 1993).21 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 (such as the "prime-ministerial government" mechanism) 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 incentive compatibility requirement. For example, consider another potential mechanism for coalition governance, which we call the "median mechanism". This implements the median of coalition members’ ideal policies on each issue, and is both dominant-strategy incentive-compatible and globally Pareto efficient.22 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 mechanism that implements the median of the coalition members’ ideal policies on each issue dimension is p(θ 1 , θ 2 , θ 3 ) = (med(θ11 , θ21 , θ31 ), med(θ12 , θ22 , θ32 )). 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 dominantstrategy incentive-compatible since no coalition member can do better by misrepresenting its ideal policy. Since the median of an odd number of coalition members’ ideal points in two dimensions will always be in the convex hull of these, this illustrates a more general result: given two policy dimensions and odd number of parties, in addition to "prime-ministerial government" mechanisms, the "median mechanism" is also both dominant-strategy incentive21

These papers assume that preferences satisfy some multidimensional version of single-peakedness and show that separability of preferences is an essential property to establish the existence of dominant-strategy incentive-compatible mechanisms in multidimensional environments. 22 When N is odd.

19

compatible and globally Pareto efficient. In sum we know from existing formal results that, in a space with three or more issue dimensions, the only mechanisms that are both globally Pareto efficient and dominantstrategy incentive-compatible are dictatorial mechanisms, "prime-ministerial government" mechanisms in our context (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 (Nehring and Puppe 2007). In light of all this, a key question for us concerns whether any of these mechanisms satisfies the coalition stability condition? The answer is no. Neither a dictatorial mechanism nor the median mechanism satisfies the coalition stability condition for every preference specification ui ∈ Ui . Some simple examples demonstrate this impossibility result. Consider a setting with three P j coalition members and three issues where a coalition members’ utility is ui = − M j=1 (θi − pj )2 . The dictatorial mechanism that implements member 1’s ideal policy, p(θ 1 , θ 2 , θ 3 ) = θ 1 , is dominant-strategy incentive-compatible and globally Pareto efficient. But it clearly fails to satisfy the coalition stability condition. If member 1’s ideal policy is θ 1 = (θ11 , θ12 , θ13 ) = (10, 10, 10) and member 2’s ideal policy is θ 2 = (θ21 , θ22 , θ23 ) = (1, 1, 1), then member 2 is better off with the status-quo policy than with the policy implemented under this mechanism: (10, 10, 10). A different example with two issues and three coalition members shows that the median mechanism does not necessarily satisfy the coalition stability condition. If coalition P j j 2 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 coalition stability condition since member 1 is better off with the status-quo policy than with the outcome under this mechanism (4, 2). This latter result is also substantively important since the median mechanism might well be seen as a plausible decision-making regime for coalition governance, perhaps as a "collective" alternative to ministerial government. This implements the dimension-bydimension 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.23 The previous example has just illustrated, however, that the median mechanism is not coalition stable. Intuitively, it 23

Note that the dimension-by-dimension median is the equilibrium outcome arising from majority voting under Kadane’s (1972) "division of the question" setting. Note also that, in portfolio allocation models, (Laver and Shepsle 1990, 1996; Austen Smith and Banks 1990), the selection of cabinet ministers who each have ideal points at the median of members’ (common knowledge) ideal points on the dimension over which they have jurisdiction is always one equilibrium.

20

is very easy to think of settings in which a coalition member with an ideal point close to the status quo would prefer the status quo to the median of all coalition member’s ideal points. Another mechanism that has implicitly or explicitly appeared in existing works as plausible decision-making procedure for how coalitions reach decisions is the "average" mechanism, the mechanism that, on every issue, implements the (weighted) average of the reported ideal policies on that issue. As mentioned, this mechanism is doesn’t satisfy the incentive compatibility requirement as parties have obvious incentives to mis-state their preferred policy in order to bring the policy implemented on some issue domain closer to what they really want. To summarize, we know from previously published formal results that the only mechanisms satisfying global Pareto efficiency and our incentive compatibility requirement in multidimensional settings are either the "prime-ministerial" mechanism or the "median" mechanism (the latter in just two dimensions). We have just seen from simple examples that neither "prime-ministerial" nor the "median" mechanism satisfies the coalition stability condition in the general case, since each is liable to leave some coalition member preferring the status quo. These arguments are synthesized in the following proposition. Proposition 7. No dominant-strategy incentive-compatible mechanism can simultaneously satisfy the coalition stability and global Pareto efficiency conditions. In a nutshell, Proposition 7 shows that no dominant-strategy incentive-compatible mechanism can be both coalition stable and globally Pareto efficient when coalition cabinets must reach decisions in the face of events that could not possibly have been anticipated during negotiations to form the incumbent coalition. 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 under consideration is coalition stable or globally Pareto efficiency unless agents have incentives to reveal their private information truthfully. We can only assess whether any given decision-making protocol leads to stable or globally efficient (equilibrium) outcomes, for example, 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 parties incentives to act upon their private information so as to implement a desirable policy outcome. There is therefore a fundamental tension between coalition stability and global Pareto efficiency, since no dominant-strategy incentive-compatible mechanism can satisfy both desiderata. This inescapable contradiction between desiderata lies at the heart of coalition governance when coalition governments must reach decisions in the face of new and unexpected events. 21

Taken together with the previous analysis, Proposition 7 implies that we cannot find an unequivocally "better" dominant-strategy incentive-compatible policy mechanism than "constrained ministerial" or "prime-ministerial" government. We might then ask which of these policy mechanisms is normatively the more appealing? The answer depends on whether we attach more weight to stability or efficiency. At the minimum, knowledge of such a tradeoff between desiderata can help scholars and practitioners make informed assessments about what must be sacrificed by using some particular decision-making protocol when coalition governments must make policy in the face of new and unexpected events. More specifically, however, we now sketch an informal argument about why the "constrained ministerial government" mechanism, and particular decision-making protocols consistent with this, are normatively more desirable. Some 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). On this basis, "constrained ministerial government" is appealing since it requires compromise between two or more parties rather than concentrating decision-making power entirely in the hands of the prime minister. Another attractive feature of "constrained ministerial government" is that 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.24 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 substantive 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 particular 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 minority legislatures, including those that deal explicitly with how particular policy portfolios might be allocated to particular members of the government coalition. More generally, the outcome of the "constrained ministerial government" mechanism 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".25 In contrast, the "prime-ministerial 24

See for example: Gamson (1961); Browne and Franklin (1973); Snyder et al. (2005); Warwick and Druckman (2006); Laver et al. (2011); Cutler et al. (2014). 25 Formally, a mechanism p(θ 1 , .., θ N ) satisfies anonymity if for any permutation σ : {1, ..., N } → {1, ..., N },

22

government" mechanism does not satisfy the anonymity requirement, since implemented policy depends fundamentally on the identity of the coalition member designated as prime minister. In the context of multi-party elections, anonymity is perhaps an appealing property as it requires that policy decisions agreed by a particular coalition should be affected by the policies of the coalition’s members, but not by the label of the coalition member promoting some particular policy. Third, the "constrained ministerial government" mechanism can be seen as a practical response to real world problems of coalition governance. Given the strong empirical regularity of departmentalized decision-making that we referred to above, this mechanism prescribes a decision-making environment in which the relevant minister does indeed have considerable policy-making autonomy within his or her policy domain, while at the same time the cabinet does indeed maintain some control over individual ministers in that it prescribes a bound on how far policy can be changed. "Constrained ministerial government" essentially offers a manner of reconciling the (undisputed) substantial policy responsibility delegated to individual cabinet ministers with the (undisputed) final decision-making power of the full cabinet. In contrast, the "prime-ministerial government" mechanism requires, notwithstanding the pervasive departmentalization of decision making, that each individual minister implements exactly the policy mandated by the prime minister. Unless the prime minister can install clones of him- or herself in all cabinet portfolios, prime ministerial government requires a very strong, and in our view impractical, set of institutions for controlling ministerial discretion. The "constrained ministerial government" mechanism, on the other hand, requires only that a certain bound on how far an individual minister can move policy be enforced and monitored by the cabinet. Overall, we have shown that "ministerial government" is non-manipulable but neither coalition stable nor globally efficient. We have shown that "constrained ministerial government" is non-manipulable, issue-by-issue Pareto efficient and coalition stable but not globally efficient. And we have shown that "prime-ministerial government" is dominant-strategy incentive-compatible and globally efficient but not coalition stable. This leaves the question of whether there are other dominant-strategy incentive compatible-mechanisms that are stable and issue-by-issue Pareto efficient (like constrained ministerial government) or other dominant strategy incentive compatible mechanisms that are globally Pareto efficient (like prime ministerial government). We already answered the latter question when showing that, for three or more issues, the only dominant-strategy incentive-compatible mechanisms that are globally Pareto efficient are "prime-ministerial government" mechanisms. In relation to the former question, existing work showed that only mechanisms that implements the moderp(θ σ(1) , θ σ(2) , ..., θ σ(N ) ) = p(θ 1 , θ 2 , .., θ N )

23

ate policy can simultaneously satisfy the incentive compatibility requirement, issue-by-issue Pareto efficiency, and an individual rationality constraint similar to our stability condition when all players want a change from the status-quo (Dragu, Fan and Kuklinski 2014). Since the moderate policy is the outcome under the "constrained ministerial government" mechanism, this suggests that this is representative of what can normatively be accomplished using decision-making protocols that require some cabinet control over individual ministers but do not fully centralize decision-making authority in the hands of the prime minister.

Conclusions Substantively, our mechanism design analysis provides a robust normative rationale for studying delegation within multiparty coalition governments, 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" policies agreed between the coalition partners towards positions that favor their own private agenda and electoral fortunes.26 Coalition governance for this reason generates a classic principal-agent problem: coalition members must in practice delegate governance to individual ministers, whose policy preferences may well 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 Strom 2003); "junior" ministers (Thies 2001); cabinet reshuffles (Indridason and Kam 2008); and legislative committees (Martin and Vanberg 2004; Caroll and Cox 2012). Implicitly or explicitly, this literature assumes that some compromise among coalition members is desirable (relative to allowing each minister complete discretion over policy-making in his/her jurisdiction) and then proceeds to investigate theoretically and empirically how ministerial discretion can be mitigated. Our mechanism design analysis provides a systematic analysis of what is gained, and what is lost, by using some particular decision-making mechanism and therefore what kinds of "compromises" are desirable and feasible within a coalition government. In this context, the "constrained ministerial government" mechanism suggests that effective policing of ministerial discretion (giving the full cabinet a practical way to enforce the bound on how far coalition policy can be moved by the delegated minister on a any given policy issue) 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. Theoretically, we propose a novel mechanism, the "constrained ministerial government" 26

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

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mechanism, that is normatively appealing in the face of the inescapable trade-offs involved in coalition governance and, more generally, in settings in which multiple agents need to agree to effect a policy change. Above all, however, our argument suggests an important new substantive focus for the analysis of coalition governance. Up until now, a large part 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 key agents. 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, Greek debt and 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, neither can the coalition partners’ preferences on these. This means that a fundamental problem for coalition governance is how to make important policy decisions on unexpected new issues on which the coalition partners’ preferences are private information. This is the problem we have addressed, deploying the tools of mechanism design. We find that no non-manipulable mechanism can simultaneously satisfy two straightforward desiderata, but that a regime of "constrained ministerial government" seems the "best" compromise in this ubiquitous setting.

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Appendix In this Appendix, we proof the propositions stated in the main text. Proof of Proposition 1. Given this mechanism, party i that has jurisdiction over issue j is going to propose its ideal policy on issue j and therefore the equilibrium outcome of the game of incomplete information induced by the "ministerial government" mechanism is j p = (p1 , p2 , ..., pM ) where pj = θi(j) for any possible combination of coalition members’ ideal policies, any beliefs Φ ∈ ∆H, and any preference specification ui ∈ Ui . The "ministerial government" mechanism is dominant-strategy incentive-compatible since no coalition member can benefit by misrepresenting its preferences. To see this, note that on a policy issue on which party i does not have jurisdiction, the policy outcome on that issue dimension is independent of whatever policy preferences party i states. And on a policy issue on which party i has jurisdiction, party i cannot benefit by misrepresenting its policy preference since the outcome is its ideal policy on that dimension. Proof of Proposition 2. The "ministerial government" mechanism is issue-by-issue Pareto efficient since the outcome on each issue is the ideal policy of some party. Finally, the examples in the main text show that the "ministerial government" mechanism is not globally Pareto efficient nor coalition stable. Proof of Proposition 3. Let us solve the subgame of the game of incomplete information induced by the "constrained ministerial discretion" mechanism that begins after the contingency that all coalition members state that the policy the 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 subgame, the effective policy bound is minN i=1 Bi . If player −1 j N 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 N −1 j previous N − 1 players is mini=1 Bi , for any strategy of each of the previous N − 1 players, any preference specification and beliefs of player N , the unique 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 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 30

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 j 0 B 0 and θN −1 to either B or 0. Since players have single-peaked preferences, 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 of the subgame that begins after players declare j that their preferred policy is to the right of the status-quo policy, is pj = min(θ1j , θ2j , ..., θN ), j j if θi > 0 ∀i, and p = 0 otherwise, for any possible combination of coalition members’ ideal policies, any beliefs Φ ∈ ∆H, and any preference specification ui ∈ Ui . The same argument applies to solving the subgame that starts after the event that θ˜ij < 0 ∀i for some issue j. That is, the unique outcome of the subgame that begins after 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 Φ ∈ ∆H, and any preference specification ui ∈ Ui . Next let us analyze the first stage of the game of incomplete information induced by the mechanism: a coalition member’s i’s decision to state its preferred policy, θ˜ij for some issue j = 1, 2, ..., M . We argue that each coalition member has a (weakly) dominant strategy to choose θ˜ij = θij . Suppose that coalition member i’s ideal policy is θij > 0; that is, i prefers a change to the right of the status-quo. If coalition member i misrepresents its preferred policy, then such strategy can only be worse than truthfully revealing its desired policy. This is because coalition member i’s decision can only make a difference when all other coalition members’ statements are θ˜kj > 0 or θ˜kj < 0 for k = 1, 2, .., i − 1, i + 1, ..., N . (In any other contingency, the resulting policy is the status-quo policy, so it doesn’t matter what coalition member i does). Now, if all players other than i state θ˜kj > 0 and i states θ˜ij < 0, then the resulting policy would be 0, which is worse than the outcome that results if i truthfully reports its preferred policy. On the other hand, if all players other than i choose θ˜kj < 0 and i states θ˜ij < 0, then the outcome is pj = 0, which is the same as the outcome that results if i truthfully states its preferred policy. Thus, player i has a (weakly) dominant strategy to state its true preferred policy, θ˜ij = θij . A similar argument shows that players i has a (weakly) dominant strategy to state θ˜ij = θij if θij < 0 or if θij = 0. Therefore, each

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player has a (weakly) dominant strategy to reveal their true preferred policy for any possible combination of coalition members’ ideal policies, any beliefs Φ ∈ ∆H, and any preference specification ui ∈ Ui . As a result, the equilibrium outcome of the game of incomplete information induced by 1 M this mechanism is the moderate policy m(θ 1 , ..., θ N ) = (m(θ11 , θ21 , ..., θN ), .., m(θ1M , θ2M , ..., θN )) where for any j = 1, ..., M ,  j j j   min{θ1 , θ2 , ..., θN } j j )= m(θ1j , xj2 , ..., 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

The "constrained ministerial discretion" mechanism is dominant-strategy incentive-compatible since, for any issue j = 1, 2, ..., M , it implements the moderate policy. To see that the mechanism satisfies this incentive compatibility requirement, consider possible deviations on some issue j for some coalition member i. If the ideal policies of the party leaders on issue j are such that they disagree about the direction of policy change, the implemented policy under the mechanism is 0. If some coalition member i deviates and declares an ideal policy 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. Proof of Proposition 4. By the definition of the moderate policy, for any preference profile j (θ 1 , θ 2 , .., θ N ), min{θ1j , .., θN } j j ≤ pj (θ1 , .., θN ) ≤ max{θ1 , .., θN } for all j = 1, 2, ..., M . Therefore, the "constrained ministerial discretion" mechanism is issue-by-issue Pareto efficient. Also, for any coalition member i = 1, 2, ..., N , the resulting policy under this mechanism j ) ≤ θij if θij > 0, or θij ≤ pj = on some issue j is either pj = 0 or 0 ≤ pj = m(θ1j , θ2j , ..., θN j ) ≤ 0 if θij < 0. Recall that, for each coalition member, the preference over m(θ1j , θ2j , ..., θN the multidimensional policy x = (x1 , x2 , ..., xM ) ∈ RM is given by single-peaked preference that is separable across the M issues. For any such preference representation, the moderate policy satisfies the coalition stability condition 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 32

policy than the status-quo. Therefore, the "constrained ministerial discretion" mechanism is coalition stable. Finally, the example in the text shows that the "constrained ministerial discretion" mechanism is not globally Pareto efficient. Proof of Proposition 5. Given that under the "prime-ministerial government" mechanism the outcome is party k’s policy choice (i.e., the policy chosen by the party assigned the prime minister position), in the game of incomplete information induced by the "primeministerial government" mechanism, party k chooses its ideal policy p = θ k for any possible combination of members’ ideal policies, any beliefs Φ ∈ ∆H, and any preference specifications ui ∈ Ui . The policy outcome under the "prime-ministerial government" mechanism is θ k where k is the party assigned the prime minister position. Thus no coalition member can profit by misreporting its policy preferences since a party that doesn’t hold the prime-minister position cannot affect the outcome no matter what it reports and the party that holds the prime-ministerial position can implement its ideal policy under this mechanism. As a result, the "prime-ministerial government" mechanism is dominant-strategy incentive-compatible. Proof of Proposition 6. The "prime-ministerial government" mechanism is globally Pareto efficient since it implements the ideal policy of one of the parties (therefore, it is also issueby-issue Pareto efficient). The example in the main text shows that this mechanism is not coalition stable.

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