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Legislative coalitions with incomplete information

  1. Michael Lavera,1
  1. aDepartment of Politics, New York University, New York, NY 10012
  1. Edited by Torun Dewan, London School of Economics, London, United Kingdom, and accepted by Editorial Board Member Mary C. Waters January 17, 2017 (received for review May 26, 2016)

Significance

It is rare in parliamentary democracies for one party to win more than half the votes. Choosing new policies or new governments depends on coalitions of legislative parties. We study the formation of these coalitions under the assumption of incomplete information. Therefore, rather than assuming that each politician’s private preferences are known to everybody, we make the more plausible and general assumption that politicians can never be certain about the preferences of others, especially because those others may have strategic incentives to misstate these preferences. This approach has implications that are both distinctive and plausible. Policy coalitions should typically be either of the center left or the center right, but should not comprise parties from both left and right of center.

Abstract

In most parliamentary democracies, proportional representation electoral rules mean that no single party controls a majority of seats in the legislature. This in turn means that the formation of majority legislative coalitions in such settings is of critical political importance. Conventional approaches to modeling the formation of such legislative coalitions typically make the “common knowledge” assumption that the preferences of all politicians are public information. In this paper, we develop a theoretical framework to investigate which legislative coalitions form when politicians’ policy preferences are private information, not known with certainty by the other politicians with whom they are negotiating over what policies to implement. The model we develop has distinctive implications. It suggests that legislative coalitions should typically be either of the center left or the center right. In other words our model, distinctively, predicts only center-left or center-right policy coalitions, not coalitions comprising the median party plus parties both to its left and to its right.

In most parliamentary democracies, proportional representation electoral rules mean that no single party controls a majority of seats in the legislature. The formation of majority legislative coalitions in such settings is therefore of critical political importance. Such coalitions are required before the legislature can pass any policy proposal. The binding constitutional constraint in parliamentary democracies, furthermore, is that governments must both secure and maintain the support of a majority of legislators, invariably organized into more or less disciplined political parties. Whether enacting new legislation, investing a new government, or supporting an incumbent, a successful proposal requires (at least implicit) support from a coalition of legislative parties. Different parties, representing different sections of the citizenry and having made different policy promises to voters during the election campaign, must come together in a single legislative coalition that aggregates all of these diverse preferences into a single outcome for any given proposal. The outcome may be continuation of the status quo or replacement of this by some alternative policy or government. To understand this crucial process of preference aggregation at the heart of parliamentary democracy, therefore, we need to understand how policy coalitions form in legislatures. To keep things simple, the bulk of our argument concerns the formation of legislative coalitions in support of particular policy proposals. We do, however, return in our conclusions to consider government formation in settings where the policy preferences of all politicians are well described in terms of a single latent dimension of ideology.

In what follows, we study the formation of policy coalitions in legislatures when politicians’ policy preferences are private information. This sets our study apart from almost all existing theoretical work on coalition formation, which analyzes policy coalitions in legislatures under complete information, assuming that the “true” policy preferences of all politicians are common knowledge (1??4). (For a review of the existing literature, see refs. 5 and 6.) Although the assumption of complete information may sometimes be a reasonable approximation, there are compelling reasons to study the formation of legislative coalitions assuming incomplete information. First, politicians are often reluctant to take clear stands on issues of the day, expressing their views in vague terms when running for office (7??10). Second, even after an election in which politicians made clear public statements about their issue positions, others can never know whether these statements reflect strategic position taking or sincere statements of true policy preferences. Third, legislators face decisions on completely unexpected problems arising from random events that were never anticipated at the time of the previous election. Think of the world financial crisis of 2008 or the Syrian refugee crisis of 2016. Forced to deal with new problems such as these, politicians may make public statements but everyone else has no way of knowing whether these are truthful revelations of genuine preferences or strategic position taking designed to achieve some private objective. Seeking a deeper understanding of coalition formation in legislatures, therefore, we need to understand how parties form policy coalitions on issues for which their true policy preferences are private information.

In what follows, we model legislative coalition formation assuming incomplete information about politicians’ policy preferences and derive a distinctive implication. This is that we should observe only policy coalitions of the center left or of the center right; that is, coalitions should include the median party and only parties to the left or to the right of it. We should not observe “centrist” policy coalitions, including parties on both sides the legislative median. Our analysis also implies that whether the policy coalition is center left or center right depends on whether the election result generates a median party with a policy position to the left or right of the status quo.

To study coalition formation with incomplete information, we use a mechanism design approach. Legislative coalitions aggregate the potentially diverse policy preferences of a set of representatives into a unique policy profile that becomes government policy. As the social choice literature showed us many years ago, this process of preference aggregation has profound normative implications. However, when preferences are private information, every party must be relied upon to reveal preferred policies truthfully. Mechanism design provides a systematic framework for analyzing which legislative coalitions are desirable and feasible when we need to consider the incentives of parties to truthfully reveal their preferences. The mechanism design problem of preference revelation is canonical and arises in many other important settings, including the design of trading and regulatory schemes, the structuring of auctions, the design of electoral systems, and environmental agreements.

The Model

We study a coalition formation game in a legislature comprising a set of <mml:math><mml:mrow><mml:mpadded width="+1.7pt"><mml:mi mathvariant="script">N</mml:mi></mml:mpadded><mml:mo>=</mml:mo><mml:mrow><mml:mo stretchy="false">{</mml:mo><mml:mn>1</mml:mn><mml:mo>,</mml:mo><mml:mn>2</mml:mn><mml:mo>,</mml:mo><mml:mn>3</mml:mn><mml:mrow><mml:mi mathvariant="normal">…</mml:mi><mml:mi>n</mml:mi></mml:mrow><mml:mo stretchy="false">}</mml:mo></mml:mrow></mml:mrow></mml:math>N={1,2,3…n} parties. Each party has preferences over a unidimensional policy space <mml:math><mml:mi>?</mml:mi></mml:math>?. Party <mml:math><mml:mi>i</mml:mi></mml:math>i’s preference for policy outcomes is represented by a utility function <mml:math><mml:mrow><mml:msub><mml:mi>u</mml:mi><mml:mi>i</mml:mi></mml:msub><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>p</mml:mi><mml:mo>,</mml:mo><mml:msub><mml:mi>θ</mml:mi><mml:mi>i</mml:mi></mml:msub><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mrow></mml:math>ui(p,θi), which is continuous and single peaked about an ideal position <mml:math><mml:mrow><mml:mpadded width="+1.7pt"><mml:msub><mml:mi>θ</mml:mi><mml:mi>i</mml:mi></mml:msub></mml:mpadded><mml:mo>∈</mml:mo><mml:mi>?</mml:mi></mml:mrow></mml:math>θi∈?, where <mml:math><mml:mi>p</mml:mi></mml:math>p is the implemented policy. There is an exogenous status quo, <mml:math><mml:mi>q</mml:mi></mml:math>q; without loss of generality, we normalize the status quo policy to <mml:math><mml:mn>0</mml:mn></mml:math>0.

The policy preferences of parties are private information. We describe parties as having a variety of possible types regarding their most preferred policies. (In effect we treat legislative parties as disciplined groups of legislators who behave “as if” they all have the same type and set aside intraparty politics as something outside the model.) That is, let <mml:math><mml:mrow><mml:mpadded width="+1.7pt"><mml:msub><mml:mi>θ</mml:mi><mml:mi>i</mml:mi></mml:msub></mml:mpadded><mml:mo>∈</mml:mo><mml:mi>?</mml:mi></mml:mrow></mml:math>θi∈? denote the type of party <mml:math><mml:mi>i</mml:mi></mml:math>i. Let <mml:math><mml:mrow><mml:mi>P</mml:mi><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>??</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mrow></mml:math>P(??) denote a joint probability distribution over party types <mml:math><mml:mrow><mml:mpadded width="+1.7pt"><mml:mi>??</mml:mi></mml:mpadded><mml:mo>=</mml:mo><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:msub><mml:mi>θ</mml:mi><mml:mn>1</mml:mn></mml:msub><mml:mo>,</mml:mo><mml:mi mathvariant="normal">…</mml:mi><mml:mo>,</mml:mo><mml:msub><mml:mi>θ</mml:mi><mml:mi>n</mml:mi></mml:msub><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mrow></mml:math>??=(θ1,…,θn) and also let <mml:math><mml:mrow><mml:mi>G</mml:mi><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:msub><mml:mi>??</mml:mi><mml:mrow><mml:mo>?</mml:mo><mml:mi>??</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">|</mml:mo><mml:msub><mml:mi>θ</mml:mi><mml:mi>i</mml:mi></mml:msub><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mrow></mml:math>G(?????|θi) be the conditional distribution of party types <mml:math><mml:mrow><mml:mpadded width="+1.7pt"><mml:msub><mml:mi>??</mml:mi><mml:mrow><mml:mo>?</mml:mo><mml:mi>??</mml:mi></mml:mrow></mml:msub></mml:mpadded><mml:mo>=</mml:mo><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:msub><mml:mi>θ</mml:mi><mml:mn>1</mml:mn></mml:msub><mml:mo>,</mml:mo><mml:mi mathvariant="normal">…</mml:mi><mml:mo>,</mml:mo><mml:msub><mml:mi>θ</mml:mi><mml:mrow><mml:mi>i</mml:mi><mml:mo>?</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:mo>,</mml:mo><mml:msub><mml:mi>θ</mml:mi><mml:mrow><mml:mi>i</mml:mi><mml:mo>+</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:mo>,</mml:mo><mml:mi mathvariant="normal">…</mml:mi><mml:mo>,</mml:mo><mml:msub><mml:mi>θ</mml:mi><mml:mi>n</mml:mi></mml:msub><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mrow></mml:math>?????=(θ1,…,θi?1,θi+1,…,θn) given that party <mml:math><mml:mi>i</mml:mi></mml:math>i knows that its own ideal policy is <mml:math><mml:msub><mml:mi>θ</mml:mi><mml:mi>i</mml:mi></mml:msub></mml:math>θi.

No party has a majority of seats in the legislature. Parties must therefore form nonsingleton coalitions to jointly implement some policy <mml:math><mml:mrow><mml:mi>p</mml:mi><mml:mo>∈</mml:mo><mml:mi>?</mml:mi></mml:mrow></mml:math>p∈?. Each party <mml:math><mml:mi>i</mml:mi></mml:math>i has a voting weight <mml:math><mml:msub><mml:mi>w</mml:mi><mml:mi>i</mml:mi></mml:msub></mml:math>wi that represents its share of seats in the legislature, where <mml:math><mml:mrow><mml:mpadded width="+1.7pt"><mml:msub><mml:mi>w</mml:mi><mml:mi>i</mml:mi></mml:msub></mml:mpadded><mml:mo><</mml:mo><mml:mrow><mml:mn>1</mml:mn><mml:mo>/</mml:mo><mml:mn>2</mml:mn></mml:mrow></mml:mrow></mml:math>wi<1/2 for all <mml:math><mml:mrow><mml:mpadded width="+1.7pt"><mml:mi>i</mml:mi></mml:mpadded><mml:mo>∈</mml:mo><mml:mi mathvariant="script">N</mml:mi></mml:mrow></mml:math>i∈N. Let <mml:math><mml:mrow><mml:mi mathvariant="script">W</mml:mi><mml:mo>=</mml:mo><mml:mrow><mml:mo stretchy="false">{</mml:mo><mml:mrow><mml:mpadded width="+1.7pt"><mml:mi>T</mml:mi></mml:mpadded><mml:mo>∈</mml:mo><mml:msup><mml:mn>2</mml:mn><mml:mi>N</mml:mi></mml:msup></mml:mrow><mml:mo stretchy="false">|</mml:mo><mml:mrow><mml:mrow><mml:mpadded width="+5pt"><mml:msub><mml:mi mathvariant="normal">Σ</mml:mi><mml:mrow><mml:mpadded width="+1.7pt"><mml:mi>j</mml:mi></mml:mpadded><mml:mo>∈</mml:mo><mml:mi>T</mml:mi></mml:mrow></mml:msub></mml:mpadded><mml:mpadded width="+1.7pt"><mml:msub><mml:mi>w</mml:mi><mml:mi>j</mml:mi></mml:msub></mml:mpadded></mml:mrow><mml:mo>></mml:mo><mml:mrow><mml:mn>1</mml:mn><mml:mo>/</mml:mo><mml:mn>2</mml:mn></mml:mrow></mml:mrow><mml:mo stretchy="false">}</mml:mo></mml:mrow></mml:mrow></mml:math>W={T∈2N|Σj∈Twj>1/2} denote the set of all winning coalitions (coalitions for which the aggregate voting weight of parties in the coalition is higher than <mml:math><mml:mrow><mml:mn>1</mml:mn><mml:mo>/</mml:mo><mml:mn>2</mml:mn></mml:mrow></mml:math>1/2), where <mml:math><mml:mi>j</mml:mi></mml:math>j denotes a (generic) party belonging to a winning coalition.

For any winning coalition <mml:math><mml:mrow><mml:mpadded width="+1.7pt"><mml:mi>T</mml:mi></mml:mpadded><mml:mo>∈</mml:mo><mml:mi mathvariant="script">W</mml:mi></mml:mrow></mml:math>T∈W that might form, the parties that compose the coalition <mml:math><mml:mi>T</mml:mi></mml:math>T make a collective choice <mml:math><mml:mrow><mml:mpadded width="+1.7pt"><mml:mi>p</mml:mi></mml:mpadded><mml:mo>∈</mml:mo><mml:mi>?</mml:mi></mml:mrow></mml:math>p∈?. Because each party has preferences over policy outcomes, the parties’ induced preferences over the set of legislative coalitions depend on which policy is expected to be chosen by each possible winning coalition. Existing models of coalition formation under complete information typically assume that parties in a winning coalition will implement a policy that is some function of the coalition members’ ideal policies, for example the seat-weighted average of coalition members’ ideal policies (originally posited by ref. 2). If these ideal policies are private information, however, this approach is problematic. If the collective policy choice <mml:math><mml:mi>p</mml:mi></mml:math>p depends on <mml:math><mml:msub><mml:mi>??</mml:mi><mml:mi>T</mml:mi></mml:msub></mml:math>??T, where <mml:math><mml:msub><mml:mi>??</mml:mi><mml:mi>T</mml:mi></mml:msub></mml:math>??T is the vector of ideal policies of parties belonging to coalition <mml:math><mml:mi>T</mml:mi></mml:math>T, and if the <mml:math><mml:msub><mml:mi>θ</mml:mi><mml:mi>i</mml:mi></mml:msub></mml:math>θis are private information, then every party must be relied upon to reveal its type truthfully for a coalition <mml:math><mml:mi>T</mml:mi></mml:math>T to implement the policy <mml:math><mml:mrow><mml:mi>p</mml:mi><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:msub><mml:mi>??</mml:mi><mml:mi>T</mml:mi></mml:msub><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mrow></mml:math>p(??T). However, there is no reason to suppose this will happen. Indeed a party may well have incentives not to reveal this information truthfully if coalition <mml:math><mml:mi>T</mml:mi></mml:math>T would implement the policy <mml:math><mml:mrow><mml:mi>p</mml:mi><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:msub><mml:mi>??</mml:mi><mml:mi>T</mml:mi></mml:msub><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mrow></mml:math>p(??T).

This means that the specific bargaining protocol according to which parties form a winning coalition to reach a policy decision is likely to shape their incentives to reveal their true preferences. This in turn means that, to analyze which particular policy is likely to be implemented by some legislative coalition <mml:math><mml:mrow><mml:mpadded width="+1.7pt"><mml:mi>T</mml:mi></mml:mpadded><mml:mo>∈</mml:mo><mml:mi mathvariant="script">W</mml:mi></mml:mrow></mml:math>T∈W when policy preferences are private information, we must specify some particular noncooperative bargaining protocol that structures these incentives and then solve for the equilibrium policy of such bargaining under incomplete information. This raises serious theoretical/computational and substantive problems. First, multilateral bargaining under incomplete information in a setting in which agents engage in some sort of a back and forth negotiation process [for example, according to the Baron–Ferejohn alternating-offers bargaining process (11)] is difficult, if not impossible, to solve analytically. Second, and more important from a substantive perspective, there is a huge variety of empirically plausible multilateral bargaining protocols. Different protocols arise from changes in the timing of the game or from small variations in 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, and whether these messages are public or private. However, the formation of legislative coalitions is arguably the biggest game in politics, and there is no exogenously enforceable rule stipulating how politicians can act or communicate with each other when they try to form a winning coalition to reach a policy compromise. Any assumption about the specifics of some communication protocol is essentially arbitrary. But, because different communication protocols may give rise to different equilibrium policy outcomes, the question of which policy is chosen by some legislative coalition when preferences are private information is deeply sensitive to the precise assumptions we make about the game of incomplete information that structures the incentives of parties to reveal their private information.

To overcome these theoretical and substantive problems, we use a mechanism design approach to identify policy outcomes that are incentive compatible and that can therefore be implemented as equilibrium outcomes of a noncooperative bargaining process when the parties’ ideal 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 form a winning coalition and choose a policy outcome. It includes any dynamic game in which the strategies of players comprise contingent plans for actions and messages.

To identify all policy outcomes that are incentive compatible if parties’ ideal policies are private information, the revelation principle tells us that we need only consider direct revelation mechanisms subject to incentive compatibility constraints that require that truthful revelation of agents’ preferences is an equilibrium in the game of incomplete information induced by the mechanism in question (12). In this context, we require that, for any <mml:math><mml:mrow><mml:mpadded width="+1.7pt"><mml:mi>T</mml:mi></mml:mpadded><mml:mo>∈</mml:mo><mml:mi>W</mml:mi></mml:mrow></mml:math>T∈W that might form, the policy mechanism <mml:math><mml:mrow><mml:mi>p</mml:mi><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:msub><mml:mi>??</mml:mi><mml:mi>T</mml:mi></mml:msub><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mrow></mml:math>p(??T) be implemented in dominant strategies.

Incentive Compatibility.

For any coalition <mml:math><mml:mrow><mml:mpadded width="+1.7pt"><mml:mi>T</mml:mi></mml:mpadded><mml:mo>∈</mml:mo><mml:mi mathvariant="script">W</mml:mi></mml:mrow></mml:math>T∈W, a policy mechanism <mml:math><mml:mrow><mml:mi>p</mml:mi><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:msub><mml:mi>??</mml:mi><mml:mi>T</mml:mi></mml:msub><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mrow></mml:math>p(??T) is dominant-strategy incentive compatible if and only if <mml:math><mml:mrow><mml:mrow><mml:msub><mml:mi>u</mml:mi><mml:mi>j</mml:mi></mml:msub><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mrow><mml:mi>p</mml:mi><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:msub><mml:mi>θ</mml:mi><mml:mi>j</mml:mi></mml:msub><mml:mo>;</mml:mo><mml:msub><mml:mi>??</mml:mi><mml:mrow><mml:mo>?</mml:mo><mml:mi>??</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mrow><mml:mo>,</mml:mo><mml:msub><mml:mi>θ</mml:mi><mml:mi>j</mml:mi></mml:msub><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mrow><mml:mo>≥</mml:mo><mml:mrow><mml:msub><mml:mi>u</mml:mi><mml:mi>j</mml:mi></mml:msub><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mrow><mml:mi>p</mml:mi><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:msub><mml:mrow><mml:mover accent="true"><mml:mi>θ</mml:mi><mml:mo stretchy="false">~</mml:mo></mml:mover></mml:mrow><mml:mi>j</mml:mi></mml:msub><mml:mo>;</mml:mo><mml:msub><mml:mi>??</mml:mi><mml:mrow><mml:mo>?</mml:mo><mml:mi>??</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mrow><mml:mo>,</mml:mo><mml:msub><mml:mi>θ</mml:mi><mml:mi>j</mml:mi></mml:msub><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mrow></mml:mrow></mml:math>uj(p(θj;?????),θj)≥uj(p(θ~j;?????),θj) for all <mml:math><mml:mrow><mml:mpadded width="+1.7pt"><mml:mi>j</mml:mi></mml:mpadded><mml:mo>∈</mml:mo><mml:mi>T</mml:mi></mml:mrow></mml:math>j∈T and all <mml:math><mml:msub><mml:mi>θ</mml:mi><mml:mi>j</mml:mi></mml:msub></mml:math>θj, <mml:math><mml:msub><mml:mrow><mml:mover accent="true"><mml:mi>θ</mml:mi><mml:mo stretchy="false">~</mml:mo></mml:mover></mml:mrow><mml:mi>j</mml:mi></mml:msub></mml:math>θ~j, <mml:math><mml:msub><mml:mi>??</mml:mi><mml:mrow><mml:mo>?</mml:mo><mml:mi>??</mml:mi></mml:mrow></mml:msub></mml:math>?????.

Dominant-strategy incentive compatibility requires that truthful revelation is a (weakly) dominant strategy in the game of incomplete information induced by the mechanism <mml:math><mml:mrow><mml:mi>p</mml:mi><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:msub><mml:mi>??</mml:mi><mml:mi>T</mml:mi></mml:msub><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mrow></mml:math>p(??T). [Because the outcome of the mechanism <mml:math><mml:mrow><mml:mi>p</mml:mi><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:msub><mml:mi>??</mml:mi><mml:mi>T</mml:mi></mml:msub><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mrow></mml:math>p(??T) depends only on the announced preferred policies of the parties in the winning coalition <mml:math><mml:mi>T</mml:mi></mml:math>T, the incentive compatibility requirement is trivially satisfied for any party <mml:math><mml:mrow><mml:mpadded width="+1.7pt"><mml:mi>k</mml:mi></mml:mpadded><mml:mo>?</mml:mo><mml:mi>T</mml:mi></mml:mrow></mml:math>k?T.] In other words, each party has a (weakly) dominant strategy to reveal his or her preferred policy regardless of what other players do given that the legislative coalition <mml:math><mml:mi>T</mml:mi></mml:math>T will form to implement the policy <mml:math><mml:mrow><mml:mi>p</mml:mi><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:msub><mml:mi>??</mml:mi><mml:mi>T</mml:mi></mml:msub><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mrow></mml:math>p(??T). This has the crucial implication that interactions governed by such a mechanism are straightforward and transparent, in the sense that any given coalition member’s optimal action does not rely in any way on his or her conjectures about other coalition members’ preferences and conjectures.

We also require that, for any legislative coalition <mml:math><mml:mrow><mml:mpadded width="+1.7pt"><mml:mi>T</mml:mi></mml:mpadded><mml:mo>∈</mml:mo><mml:mi>W</mml:mi></mml:mrow></mml:math>T∈W that might form, the policy outcome implemented by this coalition satisfies two criteria. The first one is that the outcome is such that all members of coalition <mml:math><mml:mi>T</mml:mi></mml:math>T are better off with that policy outcome than with the status quo policy. The second one is that the outcome is Pareto efficient for parties in coalition <mml:math><mml:mi>T</mml:mi></mml:math>T. First, therefore, we require that the policy outcome must satisfy an individual rationality constraint for members of a coalition <mml:math><mml:mi>T</mml:mi></mml:math>T: The utility of a party <mml:math><mml:mrow><mml:mpadded width="+1.7pt"><mml:mi>j</mml:mi></mml:mpadded><mml:mo>∈</mml:mo><mml:mi>T</mml:mi></mml:mrow></mml:math>j∈T must be at least as high as that party’s payoff from the status quo.

Individual Rationality.

For any coalition <mml:math><mml:mrow><mml:mpadded width="+1.7pt"><mml:mi>T</mml:mi></mml:mpadded><mml:mo>∈</mml:mo><mml:mi mathvariant="script">W</mml:mi></mml:mrow></mml:math>T∈W, a policy mechanism <mml:math><mml:mrow><mml:mi>p</mml:mi><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:msub><mml:mi>??</mml:mi><mml:mi>T</mml:mi></mml:msub><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mrow></mml:math>p(??T) is individually rational if and only if <mml:math><mml:mrow><mml:mrow><mml:msub><mml:mi>u</mml:mi><mml:mi>j</mml:mi></mml:msub><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mrow><mml:mi>p</mml:mi><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:msub><mml:mi>θ</mml:mi><mml:mi>j</mml:mi></mml:msub><mml:mo>,</mml:mo><mml:msub><mml:mi>??</mml:mi><mml:mrow><mml:mo>?</mml:mo><mml:mi>??</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mrow><mml:mo>,</mml:mo><mml:msub><mml:mi>θ</mml:mi><mml:mi>j</mml:mi></mml:msub><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mrow><mml:mo>≥</mml:mo><mml:mrow><mml:msub><mml:mi>u</mml:mi><mml:mi>j</mml:mi></mml:msub><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mn>0</mml:mn><mml:mo>,</mml:mo><mml:msub><mml:mi>θ</mml:mi><mml:mi>j</mml:mi></mml:msub><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mrow></mml:mrow></mml:math>uj(p(θj,?????),θj)≥uj(0,θj) for all <mml:math><mml:mrow><mml:mpadded width="+1.7pt"><mml:mi>j</mml:mi></mml:mpadded><mml:mo>∈</mml:mo><mml:mi>T</mml:mi></mml:mrow></mml:math>j∈T and all <mml:math><mml:msub><mml:mi>θ</mml:mi><mml:mi>j</mml:mi></mml:msub></mml:math>θj, <mml:math><mml:msub><mml:mi>??</mml:mi><mml:mrow><mml:mo>?</mml:mo><mml:mi>??</mml:mi></mml:mrow></mml:msub></mml:math>?????.

Second, we require that a mechanism <mml:math><mml:mrow><mml:mi>p</mml:mi><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:msub><mml:mi>??</mml:mi><mml:mi>T</mml:mi></mml:msub><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mrow></mml:math>p(??T) is Pareto efficient for all <mml:math><mml:mrow><mml:mpadded width="+1.7pt"><mml:mi>j</mml:mi></mml:mpadded><mml:mo>∈</mml:mo><mml:mi>T</mml:mi></mml:mrow></mml:math>j∈T, which means that there can be no further improvement as a result of bargaining between coalition members in <mml:math><mml:mi>T</mml:mi></mml:math>T. Note that Pareto efficiency requires the policy outcome to be between the lowest and the highest of the coalition members’ preferred policies.

Pareto Efficiency.

For any coalition <mml:math><mml:mrow><mml:mpadded width="+1.7pt"><mml:mi>T</mml:mi></mml:mpadded><mml:mo>∈</mml:mo><mml:mi mathvariant="script">W</mml:mi></mml:mrow></mml:math>T∈W, a policy mechanism <mml:math><mml:mrow><mml:mi>p</mml:mi><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:msub><mml:mi>??</mml:mi><mml:mi>T</mml:mi></mml:msub><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mrow></mml:math>p(??T) is Pareto efficient if and only if <mml:math><mml:mrow><mml:mrow><mml:munder><mml:mi>min</mml:mi><mml:mrow><mml:mpadded width="+1.7pt"><mml:mi>j</mml:mi></mml:mpadded><mml:mo>∈</mml:mo><mml:mi>T</mml:mi></mml:mrow></mml:munder><mml:mrow><mml:mo stretchy="false">{</mml:mo><mml:msub><mml:mi>θ</mml:mi><mml:mi>j</mml:mi></mml:msub><mml:mo stretchy="false">}</mml:mo></mml:mrow></mml:mrow><mml:mo>≤</mml:mo><mml:mrow><mml:mi>p</mml:mi><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:msub><mml:mi>??</mml:mi><mml:mi>T</mml:mi></mml:msub><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mrow><mml:mo>≤</mml:mo><mml:mrow><mml:munder><mml:mi>max</mml:mi><mml:mrow><mml:mpadded width="+1.7pt"><mml:mi>j</mml:mi></mml:mpadded><mml:mo>∈</mml:mo><mml:mi>T</mml:mi></mml:mrow></mml:munder><mml:mrow><mml:mo stretchy="false">{</mml:mo><mml:msub><mml:mi>θ</mml:mi><mml:mi>j</mml:mi></mml:msub><mml:mo stretchy="false">}</mml:mo></mml:mrow></mml:mrow></mml:mrow></mml:math>minj∈T{θj}≤p(??T)≤maxj∈T{θj} for all <mml:math><mml:msub><mml:mi>??</mml:mi><mml:mi>T</mml:mi></mml:msub></mml:math>??T.

For any coalition <mml:math><mml:mrow><mml:mi>T</mml:mi><mml:mo>∈</mml:mo><mml:mi mathvariant="script">W</mml:mi></mml:mrow></mml:math>T∈W and any <mml:math><mml:msub><mml:mi>??</mml:mi><mml:mi>T</mml:mi></mml:msub></mml:math>??T, let <mml:math><mml:mrow><mml:mi mathvariant="script">P</mml:mi><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:msub><mml:mi>??</mml:mi><mml:mi>T</mml:mi></mml:msub><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mrow></mml:math>P(??T) be the set of all equilibrium policy outcomes that are induced by a policy mechanism that satisfies dominant-strategy incentive compatibility, individual rationality, and Pareto efficiency. For simplicity of exposition, let <mml:math><mml:msub><mml:mi>p</mml:mi><mml:mi>T</mml:mi></mml:msub></mml:math>pT denote a policy in <mml:math><mml:mrow><mml:mi mathvariant="script">P</mml:mi><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:msub><mml:mi>??</mml:mi><mml:mi>T</mml:mi></mml:msub><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mrow></mml:math>P(??T).

Given this definition, we require that any legislative coalition <mml:math><mml:mrow><mml:mpadded width="+1.7pt"><mml:mi>T</mml:mi></mml:mpadded><mml:mo>∈</mml:mo><mml:mi mathvariant="script">W</mml:mi></mml:mrow></mml:math>T∈W that might form be stable in the sense that there is no other winning coalition <mml:math><mml:mi>T</mml:mi><mml:mo>′</mml:mo></mml:math>T′ that can block <mml:math><mml:mi>T</mml:mi></mml:math>T in the sense that each member of coalition <mml:math><mml:mi>T</mml:mi><mml:mo>′</mml:mo></mml:math>T′ is better off with any policy outcome <mml:math><mml:msub><mml:mi>p</mml:mi><mml:mrow><mml:mi>T</mml:mi><mml:mo>′</mml:mo></mml:mrow></mml:msub></mml:math>pT′ implemented by coalition <mml:math><mml:mi>T</mml:mi><mml:mo>′</mml:mo></mml:math>T′ than with any outcome <mml:math><mml:msub><mml:mi>p</mml:mi><mml:mi>T</mml:mi></mml:msub></mml:math>pT implemented by coalition <mml:math><mml:mi>T</mml:mi></mml:math>T, where <mml:math><mml:mrow><mml:mpadded width="+1.7pt"><mml:msub><mml:mi>p</mml:mi><mml:mi>T</mml:mi></mml:msub></mml:mpadded><mml:mo>∈</mml:mo><mml:mrow><mml:mi mathvariant="script">P</mml:mi><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:msub><mml:mi>??</mml:mi><mml:mi>T</mml:mi></mml:msub><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mrow></mml:mrow></mml:math>pT∈P(??T) and <mml:math><mml:mrow><mml:mpadded width="+1.7pt"><mml:msub><mml:mi>p</mml:mi><mml:mrow><mml:mi>T</mml:mi><mml:mo>′</mml:mo></mml:mrow></mml:msub></mml:mpadded><mml:mo>∈</mml:mo><mml:mrow><mml:mi mathvariant="script">P</mml:mi><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:msub><mml:mi>??</mml:mi><mml:mrow><mml:mi>T</mml:mi><mml:mo>′</mml:mo></mml:mrow></mml:msub><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mrow></mml:mrow></mml:math>pT′∈P(??T′). This is more formally stated as follows.

Coalition Stability.

A coalition <mml:math><mml:mrow><mml:mpadded width="+1.7pt"><mml:mi>T</mml:mi><mml:mo>′</mml:mo></mml:mpadded><mml:mo>∈</mml:mo><mml:mi mathvariant="script">W</mml:mi></mml:mrow></mml:math>T′∈W is a profitable deviation from a winning coalition <mml:math><mml:mi>T</mml:mi></mml:math>T if <mml:math><mml:mrow><mml:mrow><mml:msub><mml:mi>u</mml:mi><mml:mi>j</mml:mi></mml:msub><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:msub><mml:mi>p</mml:mi><mml:mrow><mml:mi>T</mml:mi><mml:mo>′</mml:mo></mml:mrow></mml:msub><mml:mo>,</mml:mo><mml:msub><mml:mi>θ</mml:mi><mml:mi>j</mml:mi></mml:msub><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mrow><mml:mo>></mml:mo><mml:mrow><mml:msub><mml:mi>u</mml:mi><mml:mi>j</mml:mi></mml:msub><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:msub><mml:mi>p</mml:mi><mml:mi>T</mml:mi></mml:msub><mml:mo>,</mml:mo><mml:msub><mml:mi>θ</mml:mi><mml:mi>j</mml:mi></mml:msub><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mrow></mml:mrow></mml:math>uj(pT′,θj)>uj(pT,θj) for all <mml:math><mml:mrow><mml:mpadded width="+1.7pt"><mml:mi>j</mml:mi></mml:mpadded><mml:mo>∈</mml:mo><mml:mi>T</mml:mi><mml:mo>′</mml:mo></mml:mrow></mml:math>j∈T′, for all <mml:math><mml:mrow><mml:mpadded width="+1.7pt"><mml:msub><mml:mi>p</mml:mi><mml:mi>T</mml:mi></mml:msub></mml:mpadded><mml:mo>∈</mml:mo><mml:mrow><mml:mi mathvariant="script">P</mml:mi><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:msub><mml:mi>??</mml:mi><mml:mi>T</mml:mi></mml:msub><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mrow></mml:mrow></mml:math>pT∈P(??T) and for all <mml:math><mml:mrow><mml:mpadded width="+1.7pt"><mml:msub><mml:mi>p</mml:mi><mml:mrow><mml:mi>T</mml:mi><mml:mo>′</mml:mo></mml:mrow></mml:msub></mml:mpadded><mml:mo>∈</mml:mo><mml:mrow><mml:mi mathvariant="script">P</mml:mi><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:msub><mml:mi>??</mml:mi><mml:mrow><mml:mi>T</mml:mi><mml:mo>′</mml:mo></mml:mrow></mml:msub><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mrow></mml:mrow></mml:math>pT′∈P(??T′). A winning coalition <mml:math><mml:mrow><mml:mpadded width="+1.7pt"><mml:mi>T</mml:mi></mml:mpadded><mml:mo>∈</mml:mo><mml:mi mathvariant="script">W</mml:mi></mml:mrow></mml:math>T∈W is stable if there is no profitable deviation from it.

We now develop our analysis of coalition formation with incomplete information in two steps. For any winning coalition <mml:math><mml:mrow><mml:mpadded width="+1.7pt"><mml:mi>T</mml:mi></mml:mpadded><mml:mo>∈</mml:mo><mml:mi mathvariant="script">W</mml:mi></mml:mrow></mml:math>T∈W, we determine the set of equilibrium outcomes induced by policy mechanisms that are dominant-strategy incentive compatible, individually rational, and Pareto efficient. We then identify which legislative coalitions are stable if, for any winning coalition <mml:math><mml:mrow><mml:mpadded width="+1.7pt"><mml:mi>T</mml:mi></mml:mpadded><mml:mo>∈</mml:mo><mml:mi mathvariant="script">W</mml:mi></mml:mrow></mml:math>T∈W, the implemented policy is some policy from the set <mml:math><mml:mrow><mml:mi mathvariant="script">P</mml:mi><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:msub><mml:mi>??</mml:mi><mml:mi>T</mml:mi></mml:msub><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mrow></mml:math>P(??T).

Analysis

To begin our analysis, consider the following policy mechanism. If some coalition members disagree about the direction of policy change relative to the status quo, then the mechanism implements the status quo policy. If all members agree on the direction of policy change, then the mechanism implements the preferred policy of the coalition member whose ideal point is closest to the status quo. Formally we have the following:

Definition 1: For any <mml:math><mml:mi>T</mml:mi></mml:math>T and <mml:math><mml:msub><mml:mi>??</mml:mi><mml:mi>T</mml:mi></mml:msub></mml:math>??T let<mml:math display="block"><mml:mrow><mml:mrow><mml:msup><mml:mi>p</mml:mi><mml:mi>m</mml:mi></mml:msup><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:msub><mml:mi>??</mml:mi><mml:mi>T</mml:mi></mml:msub><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mrow><mml:mo>≡</mml:mo><mml:mrow><mml:mrow><mml:mo>{</mml:mo><mml:mtable displaystyle="true"><mml:mtr><mml:mtd columnalign="left"><mml:mrow><mml:mpadded width="+5pt"><mml:msub><mml:mi>min</mml:mi><mml:mrow><mml:mi>j</mml:mi><mml:mo>∈</mml:mo><mml:mi>T</mml:mi></mml:mrow></mml:msub></mml:mpadded><mml:msub><mml:mi>θ</mml:mi><mml:mi>j</mml:mi></mml:msub></mml:mrow></mml:mtd><mml:mtd></mml:mtd></mml:mtr><mml:mtr><mml:mtd columnalign="left"><mml:mrow><mml:mpadded width="+5pt"><mml:msub><mml:mi>max</mml:mi><mml:mrow><mml:mi>j</mml:mi><mml:mo>∈</mml:mo><mml:mi>T</mml:mi></mml:mrow></mml:msub></mml:mpadded><mml:msub><mml:mi>θ</mml:mi><mml:mi>j</mml:mi></mml:msub></mml:mrow></mml:mtd><mml:mtd></mml:mtd></mml:mtr><mml:mtr><mml:mtd columnalign="left"><mml:mn>0</mml:mn></mml:mtd><mml:mtd></mml:mtd></mml:mtr></mml:mtable></mml:mrow><mml:mrow><mml:mo>{</mml:mo><mml:mtable displaystyle="true"><mml:mtr><mml:mtd columnalign="left"><mml:mrow><mml:mrow><mml:mpadded width="+1.7pt"><mml:mtext>if</mml:mtext></mml:mpadded><mml:mo>?</mml:mo><mml:msub><mml:mi>θ</mml:mi><mml:mi>j</mml:mi></mml:msub></mml:mrow><mml:mo>></mml:mo><mml:mrow><mml:mpadded width="+5pt"><mml:mn>0</mml:mn></mml:mpadded><mml:mrow><mml:mo>?</mml:mo><mml:mi>j</mml:mi></mml:mrow></mml:mrow><mml:mo>∈</mml:mo><mml:mi>T</mml:mi></mml:mrow></mml:mtd></mml:mtr><mml:mtr><mml:mtd columnalign="left"><mml:mrow><mml:mrow><mml:mpadded width="+1.7pt"><mml:mtext>if</mml:mtext></mml:mpadded><mml:mo>?</mml:mo><mml:msub><mml:mi>θ</mml:mi><mml:mi>j</mml:mi></mml:msub></mml:mrow><mml:mo><</mml:mo><mml:mrow><mml:mpadded width="+5pt"><mml:mn>0</mml:mn></mml:mpadded><mml:mrow><mml:mo>?</mml:mo><mml:mi>j</mml:mi></mml:mrow></mml:mrow><mml:mo>∈</mml:mo><mml:mi>T</mml:mi></mml:mrow></mml:mtd></mml:mtr><mml:mtr><mml:mtd columnalign="left"><mml:mrow><mml:mrow><mml:mrow><mml:mrow><mml:mpadded width="+1.7pt"><mml:mtext>if</mml:mtext></mml:mpadded><mml:mo>?</mml:mo><mml:msub><mml:mi>θ</mml:mi><mml:mi>j</mml:mi></mml:msub></mml:mrow><mml:mo>≤</mml:mo><mml:mn>0</mml:mn><mml:mo>≤</mml:mo><mml:mrow><mml:mpadded width="+5pt"><mml:msub><mml:mi>θ</mml:mi><mml:mrow><mml:mi>j</mml:mi><mml:mo>′</mml:mo></mml:mrow></mml:msub></mml:mpadded><mml:mpadded width="+1.7pt"><mml:mtext>for</mml:mtext></mml:mpadded><mml:mo>?</mml:mo><mml:mpadded width="+5pt"><mml:mtext>some</mml:mtext></mml:mpadded><mml:mo>?</mml:mo><mml:mi>j</mml:mi></mml:mrow></mml:mrow><mml:mo>,</mml:mo><mml:mrow><mml:mi>j</mml:mi><mml:mo>′</mml:mo><mml:mo>∈</mml:mo><mml:mi>T</mml:mi></mml:mrow></mml:mrow><mml:mo>.</mml:mo></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:mrow></mml:mrow></mml:math>pm(??T)≡{minj∈Tθjmaxj∈Tθj0{if?θj>0?j∈Tif?θj<0?j∈Tif?θj≤0≤θj′for?some?j,j′∈T.

Dragu et al. (13) have shown that the mechanism <mml:math><mml:mrow><mml:msup><mml:mi>p</mml:mi><mml:mi>m</mml:mi></mml:msup><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:msub><mml:mi>??</mml:mi><mml:mi>T</mml:mi></mml:msub><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mrow></mml:math>pm(??T) is the unique mechanism that satisfies the dominant-strategy incentive compatibility, individual rationality, and Pareto efficiency conditions (13). This mechanism is dominant-strategy incentive compatible, because by misrepresenting its preferences, a party has no effect on the outcome or induces a worse outcome for that party. It is Pareto efficient because for any possible combination of players’ ideal policies, the resulting outcome is always a policy between the lowest and the highest of coalition parties’ ideal policies. And it satisfies the individual rationality criterion because for any combination of parties’ ideal policies, the resulting outcome under this mechanism is either a policy that is preferred by all parties to the status quo policy or the status quo policy. [The mechanism <mml:math><mml:mrow><mml:msup><mml:mi>p</mml:mi><mml:mi>m</mml:mi></mml:msup><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:msub><mml:mi>??</mml:mi><mml:mi>T</mml:mi></mml:msub><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mrow></mml:math>pm(??T) is the unique mechanism satisfying these criteria because, intuitively speaking, any other mechanism that is incentive compatible and Pareto efficient cannot satisfy the individual rationality constraint for all possible combinations of parties’ ideal policies.] Denote the policy outcome induced by the mechanism <mml:math><mml:mrow><mml:msup><mml:mi>p</mml:mi><mml:mi>m</mml:mi></mml:msup><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:msub><mml:mi>??</mml:mi><mml:mi>T</mml:mi></mml:msub><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mrow></mml:math>pm(??T) by <mml:math><mml:msubsup><mml:mi>p</mml:mi><mml:mi>T</mml:mi><mml:mi>m</mml:mi></mml:msubsup></mml:math>pTm. We have the following:

Proposition 1.

For any coalition <mml:math><mml:mrow><mml:mpadded width="+1.7pt"><mml:mi>T</mml:mi></mml:mpadded><mml:mo>∈</mml:mo><mml:mi mathvariant="script">W</mml:mi></mml:mrow></mml:math>T∈W, the set of equilibrium policy outcomes that are induced by a dominant-strategy incentive compatible, individually rational, and Pareto efficient mechanism is <mml:math><mml:mrow><mml:mrow><mml:mi mathvariant="script">P</mml:mi><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:msub><mml:mi>??</mml:mi><mml:mi>T</mml:mi></mml:msub><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mrow><mml:mo>=</mml:mo><mml:mrow><mml:mo stretchy="false">{</mml:mo><mml:msubsup><mml:mi>p</mml:mi><mml:mi>T</mml:mi><mml:mi>m</mml:mi></mml:msubsup><mml:mo stretchy="false">}</mml:mo></mml:mrow></mml:mrow></mml:math>P(??T)={pTm}.

Proposition 1 tells us that for any coalition <mml:math><mml:mrow><mml:mpadded width="+1.7pt"><mml:mi>T</mml:mi></mml:mpadded><mml:mo>∈</mml:mo><mml:mi mathvariant="script">W</mml:mi></mml:mrow></mml:math>T∈W that might form, the set of equilibrium policy outcomes that are induced by an incentive compatible, individual rational, and Pareto efficient mechanism is singleton. This implements the status quo policy if the ideal policies of at least two members of coalition <mml:math><mml:mi>T</mml:mi></mml:math>T are on the opposite side of the status quo; otherwise, it implements the ideal policy of the coalition member whose ideal policy is the closest to the status quo. Given this result, we next investigate which coalitions <mml:math><mml:mrow><mml:mpadded width="+1.7pt"><mml:mi>T</mml:mi></mml:mpadded><mml:mo>∈</mml:mo><mml:mi mathvariant="script">W</mml:mi></mml:mrow></mml:math>T∈W satisfy our stability criterion. Without loss of generality we index parties <mml:math><mml:mrow><mml:mpadded width="+1.7pt"><mml:mi>i</mml:mi></mml:mpadded><mml:mo>∈</mml:mo><mml:mi mathvariant="script">N</mml:mi></mml:mrow></mml:math>i∈N by their ideal policies such that <mml:math><mml:mrow><mml:mpadded width="+1.7pt"><mml:mi>i</mml:mi></mml:mpadded><mml:mo><</mml:mo><mml:mi>j</mml:mi><mml:mo>?</mml:mo><mml:mpadded width="+1.7pt"><mml:msub><mml:mi>θ</mml:mi><mml:mi>i</mml:mi></mml:msub></mml:mpadded><mml:mo>≤</mml:mo><mml:msub><mml:mi>θ</mml:mi><mml:mi>j</mml:mi></mml:msub></mml:mrow></mml:math>i<j?θi≤θj. The median party is then defined, absolutely conventionally, as follows:

Definition 2:

The median party, <mml:math><mml:mi>m</mml:mi></mml:math>m, is the party with ideal policy <mml:math><mml:msub><mml:mi>θ</mml:mi><mml:mi>m</mml:mi></mml:msub></mml:math>θm such that <mml:math><mml:mrow><mml:mrow><mml:msub><mml:mi mathvariant="normal">Σ</mml:mi><mml:mrow><mml:mi>i</mml:mi><mml:mo>≤</mml:mo><mml:mi>m</mml:mi></mml:mrow></mml:msub><mml:mpadded width="+1.7pt"><mml:msub><mml:mi>w</mml:mi><mml:mi>i</mml:mi></mml:msub></mml:mpadded></mml:mrow><mml:mo>≥</mml:mo><mml:mrow><mml:mn>1</mml:mn><mml:mo>/</mml:mo><mml:mn>2</mml:mn></mml:mrow></mml:mrow></mml:math>Σi≤mwi≥1/2 and <mml:math><mml:mrow><mml:mrow><mml:msub><mml:mi mathvariant="normal">Σ</mml:mi><mml:mrow><mml:mpadded width="+1.7pt"><mml:mi>i</mml:mi></mml:mpadded><mml:mo>≥</mml:mo><mml:mi>m</mml:mi></mml:mrow></mml:msub><mml:mpadded width="+1.7pt"><mml:msub><mml:mi>w</mml:mi><mml:mi>i</mml:mi></mml:msub></mml:mpadded></mml:mrow><mml:mo>≥</mml:mo><mml:mrow><mml:mn>1</mml:mn><mml:mo>/</mml:mo><mml:mn>2</mml:mn></mml:mrow></mml:mrow></mml:math>Σi≥mwi≥1/2. [We assume that <mml:math><mml:mrow><mml:mrow><mml:msub><mml:mi mathvariant="normal">Σ</mml:mi><mml:mrow><mml:mi>i</mml:mi><mml:mo>∈</mml:mo><mml:mi>M</mml:mi></mml:mrow></mml:msub><mml:msub><mml:mi>w</mml:mi><mml:mi>i</mml:mi></mml:msub></mml:mrow><mml:mo>≠</mml:mo><mml:mrow><mml:mpadded width="+5pt"><mml:mfrac><mml:mn>1</mml:mn><mml:mn>2</mml:mn></mml:mfrac></mml:mpadded><mml:mrow><mml:mo>?</mml:mo><mml:mpadded width="+1.7pt"><mml:mi>M</mml:mi></mml:mpadded></mml:mrow></mml:mrow><mml:mo>?</mml:mo><mml:mi mathvariant="script">N</mml:mi></mml:mrow></mml:math>Σi∈Mwi≠12?M?N and <mml:math><mml:mrow><mml:msub><mml:mi>θ</mml:mi><mml:mi>i</mml:mi></mml:msub><mml:mo>≠</mml:mo><mml:msub><mml:mi>θ</mml:mi><mml:mi>j</mml:mi></mml:msub></mml:mrow></mml:math>θi≠θj for <mml:math><mml:mrow><mml:mi>i</mml:mi><mml:mo>≠</mml:mo><mml:mi>j</mml:mi></mml:mrow></mml:math>i≠j for the median party to be unique (this is equivalent to the assumption of having an odd number of legislators with equal weights to ensure there is a unique median legislator).]

In words, the median party is the party with a policy position such that there is a winning coalition comprising the median party and all parties with policy positions to the left of it and another winning coalition comprising the median party and all parties with a policy positions to the right of it. Now, and critically for our argument, we define the set <mml:math><mml:mi>A</mml:mi></mml:math>A of parties that prefer a change from the status quo in the same direction as the median party, but prefer the same or a greater policy change from the status quo than that preferred by the median party. That is, let <mml:math><mml:mrow><mml:mpadded width="+1.7pt"><mml:mi>A</mml:mi></mml:mpadded><mml:mo>=</mml:mo><mml:mrow><mml:mo stretchy="false">{</mml:mo><mml:mrow><mml:mi>k</mml:mi><mml:mo>∈</mml:mo><mml:mi mathvariant="script">N</mml:mi></mml:mrow><mml:mo stretchy="false">|</mml:mo><mml:mrow><mml:mrow><mml:mtext>sign</mml:mtext><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:msub><mml:mi>θ</mml:mi><mml:mi>k</mml:mi></mml:msub><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mrow><mml:mo>=</mml:mo><mml:mrow><mml:mtext>sign</mml:mtext><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:msub><mml:mi>θ</mml:mi><mml:mi>m</mml:mi></mml:msub><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mpadded width="+5pt"><mml:mtext>and</mml:mtext></mml:mpadded><mml:mrow><mml:mo stretchy="false">|</mml:mo><mml:msub><mml:mi>θ</mml:mi><mml:mi>k</mml:mi></mml:msub><mml:mo stretchy="false">|</mml:mo></mml:mrow></mml:mrow><mml:mo>≥</mml:mo><mml:mrow><mml:mo stretchy="false">|</mml:mo><mml:msub><mml:mi>θ</mml:mi><mml:mi>m</mml:mi></mml:msub><mml:mo stretchy="false">|</mml:mo></mml:mrow></mml:mrow><mml:mo stretchy="false">}</mml:mo></mml:mrow></mml:mrow></mml:math>A={k∈N|sign(θk)=sign(θm)and|θk|≥|θm|}. Now let <mml:math><mml:mrow><mml:mpadded width="+1.7pt"><mml:msub><mml:mi mathvariant="script">W</mml:mi><mml:mi>A</mml:mi></mml:msub></mml:mpadded><mml:mo>=</mml:mo><mml:mrow><mml:mo stretchy="false">{</mml:mo><mml:mrow><mml:mpadded width="+1.7pt"><mml:mi>T</mml:mi></mml:mpadded><mml:mo>∈</mml:mo><mml:msup><mml:mn>2</mml:mn><mml:mi>A</mml:mi></mml:msup></mml:mrow><mml:mo stretchy="false">|</mml:mo><mml:mrow><mml:mrow><mml:msub><mml:mi mathvariant="normal">Σ</mml:mi><mml:mrow><mml:mi>k</mml:mi><mml:mo>∈</mml:mo><mml:mi>T</mml:mi></mml:mrow></mml:msub><mml:mpadded width="+1.7pt"><mml:msub><mml:mi>w</mml:mi><mml:mi>k</mml:mi></mml:msub></mml:mpadded></mml:mrow><mml:mo>></mml:mo><mml:mrow><mml:mn>1</mml:mn><mml:mo>/</mml:mo><mml:mn>2</mml:mn></mml:mrow></mml:mrow><mml:mo stretchy="false">}</mml:mo></mml:mrow></mml:mrow></mml:math>WA={T∈2A|Σk∈Twk>1/2} be the set of winning coalitions that comprise only parties from <mml:math><mml:mi>A</mml:mi></mml:math>A. We now state our main result:

Proposition 2.

The set of stable coalitions contains all winning coalitions comprising only parties from <mml:math><mml:mi>A</mml:mi></mml:math>A; that is, the set of stable coalitions is <mml:math><mml:msub><mml:mi mathvariant="script">W</mml:mi><mml:mi>A</mml:mi></mml:msub></mml:math>WA.

Proposition 2 suggests that the only legislative coalitions that are stable include the median party and parties that want to change policy in the same direction as does the median party, but prefer a greater departure from the status quo than the median party. Note that there can be multiple winning coalitions that are stable, depending on the distribution of legislative weights; however, all such coalitions are payoff equivalent because the implemented policy under any such coalition is <mml:math><mml:msub><mml:mi>θ</mml:mi><mml:mi>m</mml:mi></mml:msub></mml:math>θm. If all parties in <mml:math><mml:mi>A</mml:mi></mml:math>A are pivotal for forming a winning coalition, then the set of stable coalitions is a singleton. Note also, however, that the set of stable legislative coalitions might contain “surplus majority” coalitions, comprising more members than are needed to control a legislative majority.

Finally, note that the set of stable coalitions identified by Proposition 2 can be obtained as the equilibrium outcome of the following direct mechanism: (i) Parties simultaneously announce their preferred policies, <mml:math><mml:msub><mml:mrow><mml:mover accent="true"><mml:mi>θ</mml:mi><mml:mo stretchy="false">^</mml:mo></mml:mover></mml:mrow><mml:mi>i</mml:mi></mml:msub></mml:math>θ^i for <mml:math><mml:mrow><mml:mpadded width="+1.7pt"><mml:mi>i</mml:mi></mml:mpadded><mml:mo>∈</mml:mo><mml:mi mathvariant="script">N</mml:mi></mml:mrow></mml:math>i∈N, and (ii) a legislative coalition comprising all parties announcing a preferred policy that is the same as or farther from the status quo in the same direction as the announced policy of the median party forms to implement the policy <mml:math><mml:mrow><mml:mpadded width="+1.7pt"><mml:msubsup><mml:mrow><mml:mover accent="true"><mml:mi>p</mml:mi><mml:mo stretchy="false">^</mml:mo></mml:mover></mml:mrow><mml:mi>T</mml:mi><mml:mi>m</mml:mi></mml:msubsup></mml:mpadded><mml:mo>=</mml:mo><mml:msub><mml:mrow><mml:mover accent="true"><mml:mi>θ</mml:mi><mml:mo stretchy="false">^</mml:mo></mml:mover></mml:mrow><mml:mi>m</mml:mi></mml:msub></mml:mrow></mml:math>p^Tm=θ^m. Given the game of incomplete information induced by this mechanism, we can see that each party has a (weakly) dominant strategy to truthfully state its preferred policy. This is because, if a party misrepresents its preferred policy, then this can lead only to a worse equilibrium outcome for that party. Whether or not a party is in the winning coalition, misreporting its ideal policy either has no effect on the equilibrium outcome or induces a policy outcome that is farther away from that party’s preferred policy. Because each party has a weakly dominant strategy to tell the truth, a legislative coalition in <mml:math><mml:msub><mml:mi mathvariant="script">W</mml:mi><mml:mi>A</mml:mi></mml:msub></mml:math>WA forms to implement <mml:math><mml:msub><mml:mi>θ</mml:mi><mml:mi>m</mml:mi></mml:msub></mml:math>θm.

Some examples might be helpful to illustrate the implications of Proposition 2. Consider two examples with the same five legislative parties <mml:math><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mn>1</mml:mn><mml:mo>,</mml:mo><mml:mn>2</mml:mn><mml:mo>,</mml:mo><mml:mn>3</mml:mn><mml:mo>,</mml:mo><mml:mn>4</mml:mn><mml:mo>,</mml:mo><mml:mn>5</mml:mn><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:math>(1,2,3,4,5) and legislative seat shares <mml:math><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mn>0.2</mml:mn><mml:mo>,</mml:mo><mml:mn>0.2</mml:mn><mml:mo>,</mml:mo><mml:mn>0.35</mml:mn><mml:mo>,</mml:mo><mml:mn>0.05</mml:mn><mml:mo>,</mml:mo><mml:mn>0.2</mml:mn><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:math>(0.2,0.2,0.35,0.05,0.2) as shown in Fig. 1. Party 3 is the median party in both scenarios.

In the first example, the parties’ ideal policies are <mml:math><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mrow><mml:mo>?</mml:mo><mml:mn>3</mml:mn></mml:mrow><mml:mo>,</mml:mo><mml:mrow><mml:mo>?</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mo>,</mml:mo><mml:mn>1</mml:mn><mml:mo>,</mml:mo><mml:mn>3</mml:mn><mml:mo>,</mml:mo><mml:mn>5</mml:mn><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:math>(?3,?1,1,3,5). Note that these policies straddle the status quo and that the ideal policy of the median party is to the right of the status quo. The set of parties who prefer either the position of the median party (<mml:math><mml:mrow><mml:mpadded width="+1.7pt"><mml:msub><mml:mi>θ</mml:mi><mml:mi>m</mml:mi></mml:msub></mml:mpadded><mml:mo>=</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:math>θm=1) or some position farther to the right of this is <mml:math><mml:mrow><mml:mpadded width="+1.7pt"><mml:mi>A</mml:mi></mml:mpadded><mml:mo>=</mml:mo><mml:mrow><mml:mo stretchy="false">{</mml:mo><mml:mn>3</mml:mn><mml:mo>,</mml:mo><mml:mn>4</mml:mn><mml:mo>,</mml:mo><mml:mn>5</mml:mn><mml:mo stretchy="false">}</mml:mo></mml:mrow></mml:mrow></mml:math>A={3,4,5}. The set of stable coalitions is <mml:math><mml:mrow><mml:mo stretchy="false">{</mml:mo><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mn>3</mml:mn><mml:mo>,</mml:mo><mml:mn>4</mml:mn><mml:mo>,</mml:mo><mml:mn>5</mml:mn><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mo>,</mml:mo><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mn>3</mml:mn><mml:mo>,</mml:mo><mml:mn>5</mml:mn><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mo stretchy="false">}</mml:mo></mml:mrow></mml:math>{(3,4,5),(3,5)}. Note that coalitions in the stable set may or may not contain the “surplus” member, party 4, whose legislative seat share is never needed for a majority. The stable set may or may not be “connected,” in the sense of comprising parties with adjacent policy positions; coalition <mml:math><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mn>3</mml:mn><mml:mo>,</mml:mo><mml:mn>5</mml:mn><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:math>(3,5) can add or drop party 4 without affecting its winning status or implemented policy position.

The second example offers a more striking illustration of how predictions from our model differ from naive intuition. In this scenario, the parties’ ideal policies are <mml:math><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mn>1</mml:mn><mml:mo>,</mml:mo><mml:mn>3</mml:mn><mml:mo>,</mml:mo><mml:mn>5</mml:mn><mml:mo>,</mml:mo><mml:mn>7</mml:mn><mml:mo>,</mml:mo><mml:mn>9</mml:mn><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:math>(1,3,5,7,9). The status quo policy is now to the left of the ideal policies of every party. This might seem to imply that the legislative coalition that forms might be any winning coalition that includes the median party, with a policy outcome at the ideal policy of the median party. However, the individual rationality constraint binds to make our model more precise and rules out any coalition containing any party with a policy to the left of the median party. We know that the policy of each possible winning coalition will be the policy position of the coalition member that is closest to the status quo. In this example, the median party will therefore block any coalition that includes members to the left of it. The set of stable coalitions in this scenario is thus the same as that in the previous example: <mml:math><mml:mrow><mml:mo stretchy="false">{</mml:mo><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mn>3</mml:mn><mml:mo>,</mml:mo><mml:mn>4</mml:mn><mml:mo>,</mml:mo><mml:mn>5</mml:mn><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mo>,</mml:mo><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mn>3</mml:mn><mml:mo>,</mml:mo><mml:mn>5</mml:mn><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mo stretchy="false">}</mml:mo></mml:mrow></mml:math>{(3,4,5),(3,5)}. Indeed, this is the set of stable coalitions for any scenario in which the status quo is to the left of the position of the median party.

Note that the policy outcome of the coalition formation game is the median party’s ideal policy, <mml:math><mml:msub><mml:mi>θ</mml:mi><mml:mi>m</mml:mi></mml:msub></mml:math>θm, which is the unique core policy outcome and is also a prediction that might well be made by many alternative models of coalition formation (14). Our prime concern here, however, is not to predict policy outcomes but to identify which legislative coalitions might form in a setting where politicians’ policy preferences are private information. It is the individual rationality constraint that plays the determining role in specifying the set of stable coalitions identified in Proposition 2. To see the importance of the individual rationality constraint for the results in Proposition 2, consider the question of which legislative coalitions form if we drop this constraint and require only that the implemented policy for any winning coalition satisfies dominant-strategy incentive compatibility and Pareto efficiency.

We have already seen that the median of the coalition members’ ideal policies is dominant-strategy incentive compatible and Pareto efficient and therefore can investigate which coalitions form if, for any winning coalition, the implemented policy is the median ideal policy of the parties in that coalition. [Any order statistic of parties’ ideal policies other than the median ideal policy, such as max or min, would also be incentive compatible and Pareto efficient (15).] Given this, the set of stable coalitions comprises all winning coalitions for which there are some parties to the left and some to the right of the median party, such that the median party’s ideal policy is also the median policy of the coalition. (Note that the median party’s ideal policy is not necessarily the median ideal policy among the parties’ ideal policies. The definition of the median party accounts for the legislative weight of each party whereas the notion of median ideal policy of parties in some coalition does not.) To illustrate this, consider another example with five parties, <mml:math><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mn>1</mml:mn><mml:mo>,</mml:mo><mml:mn>2</mml:mn><mml:mo>,</mml:mo><mml:mn>3</mml:mn><mml:mo>,</mml:mo><mml:mn>4</mml:mn><mml:mo>,</mml:mo><mml:mn>5</mml:mn><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:math>(1,2,3,4,5), with ideal policies <mml:math><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mrow><mml:mo>?</mml:mo><mml:mn>5</mml:mn></mml:mrow><mml:mo>,</mml:mo><mml:mrow><mml:mo>?</mml:mo><mml:mn>4</mml:mn></mml:mrow><mml:mo>,</mml:mo><mml:mrow><mml:mo>?</mml:mo><mml:mn>3</mml:mn></mml:mrow><mml:mo>,</mml:mo><mml:mrow><mml:mo>?</mml:mo><mml:mn>2</mml:mn></mml:mrow><mml:mo>,</mml:mo><mml:mrow><mml:mo>?</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:math>(?5,?4,?3,?2,?1) and legislative weights <mml:math><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mn>0.2</mml:mn><mml:mo>,</mml:mo><mml:mn>0.2</mml:mn><mml:mo>,</mml:mo><mml:mn>0.2</mml:mn><mml:mo>,</mml:mo><mml:mn>0.2</mml:mn><mml:mo>,</mml:mo><mml:mn>0.2</mml:mn><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:math>(0.2,0.2,0.2,0.2,0.2). The set of stable coalitions when we require the implemented policy to be the median ideal policy of coalition members is <mml:math><mml:mrow><mml:mo stretchy="false">{</mml:mo><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mn>1</mml:mn><mml:mo>,</mml:mo><mml:mn>3</mml:mn><mml:mo>,</mml:mo><mml:mn>5</mml:mn><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mo>,</mml:mo><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mn>1</mml:mn><mml:mo>,</mml:mo><mml:mn>3</mml:mn><mml:mo>,</mml:mo><mml:mn>4</mml:mn><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mo>,</mml:mo><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mn>2</mml:mn><mml:mo>,</mml:mo><mml:mn>3</mml:mn><mml:mo>,</mml:mo><mml:mn>5</mml:mn><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mo>,</mml:mo><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mn>2</mml:mn><mml:mo>,</mml:mo><mml:mn>3</mml:mn><mml:mo>,</mml:mo><mml:mn>4</mml:mn><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mo>,</mml:mo><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mn>1</mml:mn><mml:mo>,</mml:mo><mml:mn>2</mml:mn><mml:mo>,</mml:mo><mml:mn>3</mml:mn><mml:mo>,</mml:mo><mml:mn>4</mml:mn><mml:mo>,</mml:mo><mml:mn>5</mml:mn><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mo stretchy="false">}</mml:mo></mml:mrow></mml:math>{(1,3,5),(1,3,4),(2,3,5),(2,3,4),(1,2,3,4,5)}. Note that all these coalitions are centered around the median party, party <mml:math><mml:mn>3</mml:mn></mml:math>3, and that the implemented outcome in each case is the median party’s ideal policy. The equilibrium policy outcome is the same as that generated by our model, the median party’s ideal policy, but predicted legislative coalitions are very different if we require the implemented policy also to be individually rational. Indeed, in this scenario Proposition 2 implies the quite different prediction that the only stable coalition is the center-left coalition <mml:math><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mn>1</mml:mn><mml:mo>,</mml:mo><mml:mn>2</mml:mn><mml:mo>,</mml:mo><mml:mn>3</mml:mn><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:math>(1,2,3).

Substantively, therefore, our model has the distinctive (and empirically plausible) implication that we should observe only policy coalitions of the center left or of the center right, but not centrist coalitions comprising parties on both sides of the legislative median. Furthermore, our model implies that whether the policy coalition is center left or center right depends on whether the election result generates a median party with a policy position to the left or right of the status quo.

Our core result is contained in Proposition 2, which identifies the set of stable policy coalitions. It is also of interest perhaps to establish whether the formation of coalitions in this stable set might actually be achieved by using some well-specified bargaining procedure. In SI Appendix, we show this by setting out a simple bargaining protocol that generates the set of stable policy coalitions as the equilibrium of a noncooperative game of incomplete information.

SI Appendix

In SI Appendix, we provide the proofs of the propositions stated in the main text in Proof of Propositions. We also show in Implementation of Stable Coalitions how the formation of coalitions in the stable set identified in Proposition 2 can be achieved as an equilibrium outcome by using some well-specified bargaining procedure.

Discussion

Our main interest in this paper has been to investigate the set of stable legislative coalitions that form when policy preferences are private information. Our core conclusion is summarized in Proposition 2, which tells us about the set of stable legislative coalitions. This comprises the median party and other parties either exclusively to the left or to the right of the median. In other words our model, distinctively, predicts only center-left or center-right policy coalitions, not coalitions comprising the median party plus parties both to its left and to its right.

Thus far, we have discussed legislative coalitions in general, whether these form to pass some particular policy proposal, to invest a new government, or to support an incumbent government under challenge. Although the executive coalition and the legislative coalition that supports it are theoretically and constitutionally distinct, our conclusions about stable legislative coalitions have clear implications for government formation (14, 16?18). Any stable government coalition will comprise either all of the parties in some stable legislative support coalition or some subset of these. We have already shown that stable legislative coalitions may contain surplus members, who may leave the coalition yet leave it still winning. “Minority governments,” executive coalitions whose members do not themselves control a majority of legislative seats, may also be stable if they maintain the support of some majority legislative coalition. What our results highlight in relation to government formation in parliamentary democracies, therefore, is that the set of parties that are members of any stable executive coalition is the set of parties in <mml:math><mml:mi>A</mml:mi></mml:math>A. Subject to this restriction, stable executive coalitions may take any of the forms we frequently observe in the real world; they may be minimum winning, minority, or surplus majority administrations. Furthermore, if the policy preferences of politicians can be described in terms of a single latent dimension of ideology, then our model implies that the coalition cabinets that form should comprise parties from the center left or the center right; they should not be centrist coalitions comprising parties from both right and left.

Proof of Propositions

In this section, we provide the proofs for Propositions 1 and 2.

Proof of Proposition 1:

See proposition 7 in appendix <mml:math><mml:mi mathvariant="script">B</mml:mi></mml:math>B in Dragu et al. (13).

Proof of Proposition 2:

First, we show that any coalition <mml:math><mml:mrow><mml:mi>T</mml:mi><mml:mo>?</mml:mo><mml:msub><mml:mi mathvariant="script">W</mml:mi><mml:mi>A</mml:mi></mml:msub></mml:mrow></mml:math>T?WA is not stable. Note that any winning coalition <mml:math><mml:mrow><mml:mpadded width="+1.7pt"><mml:mi>T</mml:mi></mml:mpadded><mml:mo>?</mml:mo><mml:msub><mml:mi mathvariant="script">W</mml:mi><mml:mi>A</mml:mi></mml:msub></mml:mrow></mml:math>T?WA must contain some party with ideal policy <mml:math><mml:msub><mml:mi>θ</mml:mi><mml:mi>k</mml:mi></mml:msub></mml:math>θk such that <mml:math><mml:mrow><mml:mrow><mml:mo stretchy="false">|</mml:mo><mml:msub><mml:mi>θ</mml:mi><mml:mi>m</mml:mi></mml:msub><mml:mo stretchy="false">|</mml:mo></mml:mrow><mml:mo>></mml:mo><mml:mrow><mml:mo stretchy="false">|</mml:mo><mml:msub><mml:mi>θ</mml:mi><mml:mi>k</mml:mi></mml:msub><mml:mo stretchy="false">|</mml:mo></mml:mrow></mml:mrow></mml:math>|θm|>|θk| or <mml:math><mml:mrow><mml:mrow><mml:mtext>sign</mml:mtext><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:msub><mml:mi>θ</mml:mi><mml:mi>k</mml:mi></mml:msub><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mrow><mml:mo>≠</mml:mo><mml:mrow><mml:mtext>sign</mml:mtext><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:msub><mml:mi>θ</mml:mi><mml:mi>m</mml:mi></mml:msub><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mrow></mml:mrow></mml:math>sign(θk)≠sign(θm). That is, the ideal policy of party <mml:math><mml:mi>k</mml:mi></mml:math>k is closer to the status quo than the ideal policy of the median party or the ideal policies of the median party and party <mml:math><mml:mi>k</mml:mi></mml:math>k are on the opposite sides of the status quo. The policy implemented under the coalition <mml:math><mml:mi>T</mml:mi></mml:math>T is some policy closer to the status quo than <mml:math><mml:msub><mml:mi>θ</mml:mi><mml:mi>m</mml:mi></mml:msub></mml:math>θm or the status quo policy. In either situation, there is a profitable deviation by forming another coalition <mml:math><mml:msub><mml:mi mathvariant="script">W</mml:mi><mml:mi>A</mml:mi></mml:msub></mml:math>WA because the policy outcome under any coalition <mml:math><mml:msub><mml:mi mathvariant="script">W</mml:mi><mml:mi>A</mml:mi></mml:msub></mml:math>WA is <mml:math><mml:msub><mml:mi>θ</mml:mi><mml:mi>m</mml:mi></mml:msub></mml:math>θm and all parties in <mml:math><mml:msub><mml:mi mathvariant="script">W</mml:mi><mml:mi>A</mml:mi></mml:msub></mml:math>WA are better off with the policy <mml:math><mml:msub><mml:mi>θ</mml:mi><mml:mi>m</mml:mi></mml:msub></mml:math>θm than with the status quo policy or some policy closer to the status quo than <mml:math><mml:msub><mml:mi>θ</mml:mi><mml:mi>m</mml:mi></mml:msub></mml:math>θm.

Second, we show that any winning coalition <mml:math><mml:mrow><mml:mpadded width="+1.7pt"><mml:mi>T</mml:mi></mml:mpadded><mml:mo>∈</mml:mo><mml:msub><mml:mi mathvariant="script">W</mml:mi><mml:mi>A</mml:mi></mml:msub></mml:mrow></mml:math>T∈WA is stable. The implemented policy outcome for any coalition <mml:math><mml:mrow><mml:mpadded width="+1.7pt"><mml:mi>T</mml:mi></mml:mpadded><mml:mo>∈</mml:mo><mml:msub><mml:mi mathvariant="script">W</mml:mi><mml:mi>A</mml:mi></mml:msub></mml:mrow></mml:math>T∈WA is <mml:math><mml:msub><mml:mi>θ</mml:mi><mml:mi>m</mml:mi></mml:msub></mml:math>θm. Suppose that <mml:math><mml:mi>T</mml:mi><mml:mo>′</mml:mo></mml:math>T′ is a profitable deviation, which implies that <mml:math><mml:mrow><mml:mi>m</mml:mi><mml:mo>?</mml:mo><mml:mi>T</mml:mi><mml:mo>′</mml:mo></mml:mrow></mml:math>m?T′. This implies that <mml:math><mml:mrow><mml:mrow><mml:mrow><mml:mo>?</mml:mo><mml:mpadded width="+1.7pt"><mml:mi>j</mml:mi></mml:mpadded></mml:mrow><mml:mo>&</mml:mo><mml:mi>k</mml:mi></mml:mrow><mml:mo>∈</mml:mo><mml:mi>T</mml:mi><mml:mo>′</mml:mo></mml:mrow></mml:math>?j&k∈T′ such that <mml:math><mml:mrow><mml:mrow><mml:mi>min</mml:mi><mml:mrow><mml:mo stretchy="false">{</mml:mo><mml:msub><mml:mi>θ</mml:mi><mml:mi>j</mml:mi></mml:msub><mml:mo>,</mml:mo><mml:msub><mml:mi>θ</mml:mi><mml:mi>k</mml:mi></mml:msub><mml:mo stretchy="false">}</mml:mo></mml:mrow></mml:mrow><mml:mo><</mml:mo><mml:mpadded width="+1.7pt"><mml:msub><mml:mi>θ</mml:mi><mml:mi>m</mml:mi></mml:msub></mml:mpadded><mml:mo><</mml:mo><mml:mrow><mml:mi>max</mml:mi><mml:mrow><mml:mo stretchy="false">{</mml:mo><mml:msub><mml:mi>θ</mml:mi><mml:mi>j</mml:mi></mml:msub><mml:mo>,</mml:mo><mml:msub><mml:mi>θ</mml:mi><mml:mi>k</mml:mi></mml:msub><mml:mo stretchy="false">}</mml:mo></mml:mrow></mml:mrow></mml:mrow></mml:math>min{θj,θk}<θm<max{θj,θk}. This in turn implies that the policy implemented under coalition <mml:math><mml:mi>T</mml:mi></mml:math>T is closer to the status quo than <mml:math><mml:msub><mml:mi>θ</mml:mi><mml:mi>m</mml:mi></mml:msub></mml:math>θm. But then one of parties <mml:math><mml:mi>j</mml:mi></mml:math>j and <mml:math><mml:mi>k</mml:mi></mml:math>k is worse off with the policy outcome implemented by coalition <mml:math><mml:mi>T</mml:mi><mml:mo>′</mml:mo></mml:math>T′ than by coalition <mml:math><mml:mi>T</mml:mi></mml:math>T. Thus, <mml:math><mml:mi>T</mml:mi><mml:mo>′</mml:mo></mml:math>T′ cannot be a profitable deviation from <mml:math><mml:mi>T</mml:mi></mml:math>T.

Implementation of Stable Coalitions

In this section, we illustrate how the formation of coalitions in the stable set identified in Proposition 2 can be achieved by using some well-specified bargaining procedure. We show this by setting out a simple bargaining protocol that generates the set of stable coalitions as an equilibrium of a noncooperative game of incomplete information. This bargaining protocol <mml:math><mml:mi mathvariant="script">B</mml:mi></mml:math>B is as follows:

  • ? Parties simultaneously announce their preferred policies, <mml:math><mml:msub><mml:mrow><mml:mover accent="true"><mml:mi>θ</mml:mi><mml:mo stretchy="false">^</mml:mo></mml:mover></mml:mrow><mml:mi>i</mml:mi></mml:msub></mml:math>θ^i for <mml:math><mml:mrow><mml:mpadded width="+1.7pt"><mml:mi>i</mml:mi></mml:mpadded><mml:mo>∈</mml:mo><mml:mi mathvariant="script">N</mml:mi></mml:mrow></mml:math>i∈N.

  • ? These announcements identify the median party, denoted by <mml:math><mml:msub><mml:mrow><mml:mover accent="true"><mml:mi>θ</mml:mi><mml:mo stretchy="false">^</mml:mo></mml:mover></mml:mrow><mml:mi>m</mml:mi></mml:msub></mml:math>θ^m, as well as a proto-coalition comprising all parties announcing a preferred policy that is the same as or farther from the status quo in the same direction as the announced policy of the median party.

  • ? All members of this proto-coalition simultaneously announce whether or not they will join a legislative coalition implementing the announced policy of the coalition members that is closest to the status quo.

  • ? If the aggregate legislative weight of parties announcing they will join the legislative coalition is more than <mml:math><mml:mrow><mml:mn>1</mml:mn><mml:mo>/</mml:mo><mml:mn>2</mml:mn></mml:mrow></mml:math>1/2, the legislative coalition consisting of only these parties forms; the implemented policy is <mml:math><mml:msub><mml:mrow><mml:mover accent="true"><mml:mi>θ</mml:mi><mml:mo stretchy="false">^</mml:mo></mml:mover></mml:mrow><mml:mi>m</mml:mi></mml:msub></mml:math>θ^m.

  • ? If the aggregate legislative weight of parties announcing they will join the legislative coalition is not more than <mml:math><mml:mrow><mml:mn>1</mml:mn><mml:mo>/</mml:mo><mml:mn>2</mml:mn></mml:mrow></mml:math>1/2 and if the aggregate legislative weight of these parties plus all parties with announced ideal policies between the status quo and <mml:math><mml:msub><mml:mrow><mml:mover accent="true"><mml:mi>θ</mml:mi><mml:mo stretchy="false">^</mml:mo></mml:mover></mml:mrow><mml:mi>m</mml:mi></mml:msub></mml:math>θ^m is still no more than <mml:math><mml:mrow><mml:mn>1</mml:mn><mml:mo>/</mml:mo><mml:mn>2</mml:mn></mml:mrow></mml:math>1/2, the legislative coalition consisting of all other parties forms; the implemented policy is the status quo.

  • ? If the aggregate legislative weight of parties announcing they will join the legislative coalition is not more than <mml:math><mml:mrow><mml:mn>1</mml:mn><mml:mo>/</mml:mo><mml:mn>2</mml:mn></mml:mrow></mml:math>1/2, but if the aggregate legislative weight of these parties plus all parties with announced ideal policies between the status quo and <mml:math><mml:msub><mml:mrow><mml:mover accent="true"><mml:mi>θ</mml:mi><mml:mo stretchy="false">^</mml:mo></mml:mover></mml:mrow><mml:mi>m</mml:mi></mml:msub></mml:math>θ^m is more than <mml:math><mml:mrow><mml:mn>1</mml:mn><mml:mo>/</mml:mo><mml:mn>2</mml:mn></mml:mrow></mml:math>1/2, a new proto-coalition is identified. This adds, to parties who have already agreed to join the legislative coalition, parties announcing policies between the status quo and <mml:math><mml:msub><mml:mrow><mml:mover accent="true"><mml:mi>θ</mml:mi><mml:mo stretchy="false">^</mml:mo></mml:mover></mml:mrow><mml:mi>m</mml:mi></mml:msub></mml:math>θ^m, starting with the party announcing a policy closest to <mml:math><mml:msub><mml:mrow><mml:mover accent="true"><mml:mi>θ</mml:mi><mml:mo stretchy="false">^</mml:mo></mml:mover></mml:mrow><mml:mi>m</mml:mi></mml:msub></mml:math>θ^m and adding parties increasingly close to the status quo until the aggregate weight of parties in the proto-coalition is more than <mml:math><mml:mrow><mml:mn>1</mml:mn><mml:mo>/</mml:mo><mml:mn>2</mml:mn></mml:mrow></mml:math>1/2.

  • ? All members of this new proto-coalition simultaneously announce whether or not they will join a legislative coalition implementing the announced policy closest to the status quo among the coalition members.

  • ? If the aggregate legislative weight of these parties is more than <mml:math><mml:mrow><mml:mn>1</mml:mn><mml:mo>/</mml:mo><mml:mn>2</mml:mn></mml:mrow></mml:math>1/2, the legislative coalition consisting of only these parties forms; the implemented policy is the announced policy closest to the status quo among the coalition members.

  • ? If the aggregate legislative weight of all parties announcing they will join the latter proto-coalition is not more than <mml:math><mml:mrow><mml:mn>1</mml:mn><mml:mo>/</mml:mo><mml:mn>2</mml:mn></mml:mrow></mml:math>1/2, the legislative coalition consisting of all other parties forms; the implemented policy is the status quo.

The bargaining protocol <mml:math><mml:mi mathvariant="script">B</mml:mi></mml:math>B looks complicated when written down but is actually rather simple. It takes account of the fact that, regardless of legislative majorities, every party makes an autonomous decision, based on its own private preference, about whether or not to join any legislative coalition. The protocol encodes a situation in which all parties announce their preferred policy positions and the only way to change the status quo policy in a setting where politicians’ policy preferences are private information is to form a majority legislative coalition whose members all privately prefer some alternative policy to the status quo, knowing that the only available alternative policy is a policy publicly announced by the median party (given the announced policies) or some other policy between the policy announced by the median party and the status quo. If such a coalition exists, it forms and the policy outcome is the policy closest to the status quo among all members of that coalition. Otherwise there is in effect a bargaining failure and an implicit ragtag coalition of all other parties forms, with the policy outcome being the status quo.

A strategy for a party in this game of incomplete information is a statement of its preferred policy (as a function of its true ideal policy) and, in the contingency that a party is included in any proto-coalition, a decision whether to announce it will join or not that legislative coalition to implement the announced policy of the coalition members that is closest to the status quo.

Next, we show that any stable coalition <mml:math><mml:mrow><mml:mpadded width="+1.7pt"><mml:mi>T</mml:mi></mml:mpadded><mml:mo>∈</mml:mo><mml:msub><mml:mi mathvariant="script">W</mml:mi><mml:mi>A</mml:mi></mml:msub></mml:mrow></mml:math>T∈WA identified in Proposition 2 can result as the equilibrium coalition in a perfect Bayesian Nash equilibrium of this game. Consider some stable coalition <mml:math><mml:mrow><mml:mpadded width="+1.7pt"><mml:mi>T</mml:mi></mml:mpadded><mml:mo>∈</mml:mo><mml:msub><mml:mi mathvariant="script">W</mml:mi><mml:mi>A</mml:mi></mml:msub></mml:mrow></mml:math>T∈WA and let <mml:math><mml:mi>j</mml:mi></mml:math>j denote a party that is a member of this stable coalition <mml:math><mml:mi>T</mml:mi></mml:math>T. The next proposition states that the bargaining protocol <mml:math><mml:mi mathvariant="script">B</mml:mi></mml:math>B has a perfect Bayesian equilibrium in which all parties truthfully announce their preferred policy <mml:math><mml:mrow><mml:mpadded width="+1.7pt"><mml:msub><mml:mrow><mml:mover accent="true"><mml:mi>θ</mml:mi><mml:mo stretchy="false">^</mml:mo></mml:mover></mml:mrow><mml:mi>i</mml:mi></mml:msub></mml:mpadded><mml:mo>=</mml:mo><mml:msub><mml:mi>θ</mml:mi><mml:mi>i</mml:mi></mml:msub></mml:mrow></mml:math>θ^i=θi, a party <mml:math><mml:mrow><mml:mpadded width="+1.7pt"><mml:mi>j</mml:mi></mml:mpadded><mml:mo>∈</mml:mo><mml:mi>T</mml:mi></mml:mrow></mml:math>j∈T will announce it will join any legislative coalition (in the contingency that it is included in any proto-coalition), and a party <mml:math><mml:mrow><mml:mpadded width="+1.7pt"><mml:mi>k</mml:mi></mml:mpadded><mml:mo>?</mml:mo><mml:mi>T</mml:mi></mml:mrow></mml:math>k?T will announce that it will not join any legislative coalition (in the contingency that it is included in any proto-coalition). The proof of Proposition 3 is provided at the end of this section.

Proposition 3.

Consider a stable coalition <mml:math><mml:mrow><mml:mpadded width="+1.7pt"><mml:mi>T</mml:mi></mml:mpadded><mml:mo>∈</mml:mo><mml:msub><mml:mi mathvariant="script">W</mml:mi><mml:mi>A</mml:mi></mml:msub></mml:mrow></mml:math>T∈WA. The bargaining protocol <mml:math><mml:mi mathvariant="script">B</mml:mi></mml:math>B has a perfect Bayesian Nash equilibrium in which every party truthfully reveals its preferred policy, <mml:math><mml:mrow><mml:mrow><mml:msub><mml:mrow><mml:mover accent="true"><mml:mi>θ</mml:mi><mml:mo stretchy="false">^</mml:mo></mml:mover></mml:mrow><mml:mi>i</mml:mi></mml:msub><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:msub><mml:mi>θ</mml:mi><mml:mi>i</mml:mi></mml:msub><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mrow><mml:mo>=</mml:mo><mml:msub><mml:mi>θ</mml:mi><mml:mi>i</mml:mi></mml:msub></mml:mrow></mml:math>θ^i(θi)=θi, a party <mml:math><mml:mrow><mml:mpadded width="+1.7pt"><mml:mi>j</mml:mi></mml:mpadded><mml:mo>∈</mml:mo><mml:mi>T</mml:mi></mml:mrow></mml:math>j∈T will announce it will join any legislative coalition (in the contingency that it is included in any proto-coalition), and a party <mml:math><mml:mrow><mml:mpadded width="+1.7pt"><mml:mi>k</mml:mi></mml:mpadded><mml:mo>?</mml:mo><mml:mi>T</mml:mi></mml:mrow></mml:math>k?T will announce that it will not join any legislative coalition (in the contingency that it is included in any proto-coalition). The policy outcome of this perfect Bayesian Nash equilibrium of the game is <mml:math><mml:mrow><mml:mpadded width="+1.7pt"><mml:msup><mml:mi>p</mml:mi><mml:mo>?</mml:mo></mml:msup></mml:mpadded><mml:mo>=</mml:mo><mml:msub><mml:mi>θ</mml:mi><mml:mi>m</mml:mi></mml:msub></mml:mrow></mml:math>p?=θm and the stable coalition <mml:math><mml:mrow><mml:mpadded width="+1.7pt"><mml:mi>T</mml:mi></mml:mpadded><mml:mo>∈</mml:mo><mml:msub><mml:mi mathvariant="script">W</mml:mi><mml:mi>A</mml:mi></mml:msub></mml:mrow></mml:math>T∈WA is the equilibrium legislative coalition in this perfect Bayesian Nash equilibrium.

The intuition for Proposition 3 is simple. If all parties truthfully state their preferred policies, parties <mml:math><mml:mrow><mml:mpadded width="+1.7pt"><mml:mi>j</mml:mi></mml:mpadded><mml:mo>∈</mml:mo><mml:mi>T</mml:mi></mml:mrow></mml:math>j∈T will be included in the initial proto-coalition and will announce they will join a legislative coalition for which the implemented policy is the median party’s ideal policy. No party in this stable coalition <mml:math><mml:mi>T</mml:mi></mml:math>T has an incentive to deviate and misreport its preferred policy because it cannot improve upon the policy outcome <mml:math><mml:msub><mml:mi>θ</mml:mi><mml:mi>m</mml:mi></mml:msub></mml:math>θm. For a party that is not in <mml:math><mml:mi>T</mml:mi></mml:math>T, if that party misreports its preferred policy, such a strategy cannot change the outcome. By misreporting its ideal policy, a party might change the composition of the initial proto-coalition at the expense of some party (parties) from the coalition <mml:math><mml:mi>T</mml:mi></mml:math>T. That party can announce that it will not join the initial legislative coalition but, because parties in the coalition <mml:math><mml:mi>T</mml:mi></mml:math>T can form a winning legislative coalition in the next round, misrepresenting its ideal policy has no consequence for the outcome.

The next example illustrates the equilibrium described in Proposition 3. Consider a scenario with five legislative parties, <mml:math><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mn>1</mml:mn><mml:mo>,</mml:mo><mml:mn>2</mml:mn><mml:mo>,</mml:mo><mml:mn>3</mml:mn><mml:mo>,</mml:mo><mml:mn>4</mml:mn><mml:mo>,</mml:mo><mml:mn>5</mml:mn><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:math>(1,2,3,4,5), with ideal policies <mml:math><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mrow><mml:mo>?</mml:mo><mml:mn>2</mml:mn></mml:mrow><mml:mo>,</mml:mo><mml:mn>1</mml:mn><mml:mo>,</mml:mo><mml:mn>3</mml:mn><mml:mo>,</mml:mo><mml:mn>4</mml:mn><mml:mo>,</mml:mo><mml:mn>5</mml:mn><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:math>(?2,1,3,4,5) and legislative weights <mml:math><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mn>0.2</mml:mn><mml:mo>,</mml:mo><mml:mn>0.2</mml:mn><mml:mo>,</mml:mo><mml:mn>0.2</mml:mn><mml:mo>,</mml:mo><mml:mn>0.2</mml:mn><mml:mo>,</mml:mo><mml:mn>0.2</mml:mn><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:math>(0.2,0.2,0.2,0.2,0.2). In this setting the set of stable coalitions is singleton and consists of parties from the set <mml:math><mml:mrow><mml:mpadded width="+1.7pt"><mml:mi>A</mml:mi></mml:mpadded><mml:mo>=</mml:mo><mml:mrow><mml:mo stretchy="false">{</mml:mo><mml:mn>3</mml:mn><mml:mo>,</mml:mo><mml:mn>4</mml:mn><mml:mo>,</mml:mo><mml:mn>5</mml:mn><mml:mo stretchy="false">}</mml:mo></mml:mrow></mml:mrow></mml:math>A={3,4,5}. If all parties truthfully state their ideal policies in the first stage, <mml:math><mml:mrow><mml:mrow><mml:msub><mml:mrow><mml:mover accent="true"><mml:mi>θ</mml:mi><mml:mo stretchy="false">^</mml:mo></mml:mover></mml:mrow><mml:mi>i</mml:mi></mml:msub><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:msub><mml:mi>θ</mml:mi><mml:mi>i</mml:mi></mml:msub><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mrow><mml:mo>=</mml:mo><mml:msub><mml:mi>θ</mml:mi><mml:mi>i</mml:mi></mml:msub></mml:mrow></mml:math>θ^i(θi)=θi, the proto-coalition after the announcement of ideal policy consists of parties from the set <mml:math><mml:mrow><mml:mpadded width="+1.7pt"><mml:mi>A</mml:mi></mml:mpadded><mml:mo>=</mml:mo><mml:mrow><mml:mo stretchy="false">{</mml:mo><mml:mn>3</mml:mn><mml:mo>,</mml:mo><mml:mn>4</mml:mn><mml:mo>,</mml:mo><mml:mn>5</mml:mn><mml:mo stretchy="false">}</mml:mo></mml:mrow></mml:mrow></mml:math>A={3,4,5}. All these parties will announce they will join this coalition because the implemented outcome is <mml:math><mml:mrow><mml:mpadded width="+1.7pt"><mml:msub><mml:mi>θ</mml:mi><mml:mi>m</mml:mi></mml:msub></mml:mpadded><mml:mo>=</mml:mo><mml:mn>3</mml:mn></mml:mrow></mml:math>θm=3 if the coalition forms whereas a policy closer to the status quo than <mml:math><mml:msub><mml:mi>θ</mml:mi><mml:mi>m</mml:mi></mml:msub></mml:math>θm or the status quo will result if the coalition fails to form. No party from the set <mml:math><mml:mi>A</mml:mi></mml:math>A can benefit by misreporting its ideal policy because such a strategy can only lead to a worse policy outcome than <mml:math><mml:msub><mml:mi>θ</mml:mi><mml:mn>3</mml:mn></mml:msub></mml:math>θ3. On the other hand, party 1 would be better off if the status quo (or a policy closer to the status quo than <mml:math><mml:msub><mml:mi>θ</mml:mi><mml:mn>3</mml:mn></mml:msub></mml:math>θ3) is implemented. However, if party 1 misreports <mml:math><mml:mrow><mml:mpadded width="+1.7pt"><mml:msub><mml:mrow><mml:mover accent="true"><mml:mi>θ</mml:mi><mml:mo stretchy="false">^</mml:mo></mml:mover></mml:mrow><mml:mn>1</mml:mn></mml:msub></mml:mpadded><mml:mo>≤</mml:mo><mml:msub><mml:mi>θ</mml:mi><mml:mn>3</mml:mn></mml:msub></mml:mrow></mml:math>θ^1≤θ3 and the other parties truthfully state their preferred policies, such a strategy has no effect on the outcome because all parties from <mml:math><mml:mi>A</mml:mi></mml:math>A are still included in the initial proto-coalition. On the other hand, if party 1 announces a policy <mml:math><mml:mrow><mml:mpadded width="+1.7pt"><mml:msub><mml:mrow><mml:mover accent="true"><mml:mi>θ</mml:mi><mml:mo stretchy="false">^</mml:mo></mml:mover></mml:mrow><mml:mn>1</mml:mn></mml:msub></mml:mpadded><mml:mo>></mml:mo><mml:msub><mml:mi>θ</mml:mi><mml:mn>3</mml:mn></mml:msub></mml:mrow></mml:math>θ^1>θ3 and the other parties truthfully state their preferred policies, such a strategy changes the composition of the initial proto-coalition to <mml:math><mml:mrow><mml:mpadded width="+1.7pt"><mml:mrow><mml:mover accent="true"><mml:mi>A</mml:mi><mml:mo stretchy="false">^</mml:mo></mml:mover></mml:mrow></mml:mpadded><mml:mo>=</mml:mo><mml:mrow><mml:mo stretchy="false">{</mml:mo><mml:mn>1</mml:mn><mml:mo>,</mml:mo><mml:mn>4</mml:mn><mml:mo>,</mml:mo><mml:mn>5</mml:mn><mml:mo stretchy="false">}</mml:mo></mml:mrow></mml:mrow></mml:math>A^={1,4,5}. Because the policy implemented if this legislative coalition forms is <mml:math><mml:mrow><mml:mpadded width="+1.7pt"><mml:msub><mml:mrow><mml:mover accent="true"><mml:mi>θ</mml:mi><mml:mo stretchy="false">^</mml:mo></mml:mover></mml:mrow><mml:mi>m</mml:mi></mml:msub></mml:mpadded><mml:mo>=</mml:mo><mml:mrow><mml:mi>min</mml:mi><mml:mrow><mml:mo stretchy="false">{</mml:mo><mml:msub><mml:mrow><mml:mover accent="true"><mml:mi>θ</mml:mi><mml:mo stretchy="false">^</mml:mo></mml:mover></mml:mrow><mml:mn>1</mml:mn></mml:msub><mml:mo>,</mml:mo><mml:mn>4</mml:mn><mml:mo>,</mml:mo><mml:mn>5</mml:mn><mml:mo stretchy="false">}</mml:mo></mml:mrow></mml:mrow></mml:mrow></mml:math>θ^m=min{θ^1,4,5} and because the policy outcome should this coalition fail to form can only be lower than <mml:math><mml:msub><mml:mrow><mml:mover accent="true"><mml:mi>θ</mml:mi><mml:mo stretchy="false">^</mml:mo></mml:mover></mml:mrow><mml:mi>m</mml:mi></mml:msub></mml:math>θ^m, parties <mml:math><mml:mn>4</mml:mn></mml:math>4 and 5 will announce that they will join this coalition whereas party <mml:math><mml:mn>1</mml:mn></mml:math>1 will announce that it will not. Because this legislative coalition fails to form and because the total weight of parties <mml:math><mml:mn>4</mml:mn></mml:math>4 and <mml:math><mml:mn>5</mml:mn></mml:math>5 (i.e., parties that announced they would join the previous coalition) plus the weight of parties <mml:math><mml:mn>2</mml:mn></mml:math>2 and <mml:math><mml:mn>3</mml:mn></mml:math>3 [i.e., parties whose announced policies are between the status quo and the (announced) median’s ideal policy, <mml:math><mml:msub><mml:mrow><mml:mover accent="true"><mml:mi>θ</mml:mi><mml:mo stretchy="false">^</mml:mo></mml:mover></mml:mrow><mml:mi>m</mml:mi></mml:msub></mml:math>θ^m] is higher than <mml:math><mml:mrow><mml:mn>1</mml:mn><mml:mo>/</mml:mo><mml:mn>2</mml:mn></mml:mrow></mml:math>1/2, a new proto-coalition is identified. It consists of parties <mml:math><mml:mn>3</mml:mn></mml:math>3, <mml:math><mml:mn>4</mml:mn></mml:math>4, and <mml:math><mml:mn>5</mml:mn></mml:math>5; note that party <mml:math><mml:mn>2</mml:mn></mml:math>2 is not included because party <mml:math><mml:mn>3</mml:mn></mml:math>3 is the party with the closest announced policy to <mml:math><mml:msub><mml:mrow><mml:mover accent="true"><mml:mi>θ</mml:mi><mml:mo stretchy="false">^</mml:mo></mml:mover></mml:mrow><mml:mi>m</mml:mi></mml:msub></mml:math>θ^m and the aggregate legislative weight of parties <mml:math><mml:mn>3</mml:mn></mml:math>3, <mml:math><mml:mn>4</mml:mn></mml:math>4, and <mml:math><mml:mn>5</mml:mn></mml:math>5 is more than <mml:math><mml:mrow><mml:mn>1</mml:mn><mml:mo>/</mml:mo><mml:mn>2</mml:mn></mml:mrow></mml:math>1/2. All these parties will announce they will join a legislative coalition consisting of parties <mml:math><mml:mn>3</mml:mn></mml:math>3, <mml:math><mml:mn>4</mml:mn></mml:math>4, and <mml:math><mml:mn>5</mml:mn></mml:math>5 to implement the policy <mml:math><mml:msub><mml:mi>θ</mml:mi><mml:mn>3</mml:mn></mml:msub></mml:math>θ3. So by misreporting its ideal policy, party <mml:math><mml:mn>1</mml:mn></mml:math>1 cannot change the outcome to a more advantageous policy than <mml:math><mml:msub><mml:mi>θ</mml:mi><mml:mn>3</mml:mn></mml:msub></mml:math>θ3. The same argument applies to understanding the incentives of party <mml:math><mml:mn>2</mml:mn></mml:math>2.

Proposition 3 describes a perfect Bayesian equilibrium in which a stable coalition <mml:math><mml:mi>T</mml:mi></mml:math>T can be obtained as the equilibrium coalition of the game. Note that when the set of stable coalitions is not singleton, any legislative coalitions from the set of stable coalitions defined in Proposition 2 can be obtained as the equilibrium coalition in a perfect Bayesian equilibrium of the game as the one described in Proposition 3 by having parties in that respective coalition announcing that they will join any legislative coalition (in the contingency that it is included in any proto-coalition) and having parties that are not in that stable coalition announcing that they will not join any legislative coalition (in the contingency that it is included in any proto-coalition). All perfect Bayesian equilibria (of the sort described in Proposition 3) that implement a stable coalition as an equilibrium coalition are payoff equivalent in that the implemented policy is <mml:math><mml:msub><mml:mi>θ</mml:mi><mml:mi>m</mml:mi></mml:msub></mml:math>θm for any of these coalitions.

Below, we provide the proof of Proposition 3.

Proof of Proposition 3:

Consider a stable coalition <mml:math><mml:mrow><mml:mpadded width="+1.7pt"><mml:mi>T</mml:mi></mml:mpadded><mml:mo>∈</mml:mo><mml:msub><mml:mi mathvariant="script">W</mml:mi><mml:mi>A</mml:mi></mml:msub></mml:mrow></mml:math>T∈WA. We want to show that the bargaining protocol <mml:math><mml:mi mathvariant="script">B</mml:mi></mml:math>B has a perfect Bayesian Nash equilibrium in which every party truthfully reveals its preferred policy, <mml:math><mml:mrow><mml:mrow><mml:msub><mml:mrow><mml:mover accent="true"><mml:mi>θ</mml:mi><mml:mo stretchy="false">^</mml:mo></mml:mover></mml:mrow><mml:mi>i</mml:mi></mml:msub><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:msub><mml:mi>θ</mml:mi><mml:mi>i</mml:mi></mml:msub><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mrow><mml:mo>=</mml:mo><mml:msub><mml:mi>θ</mml:mi><mml:mi>i</mml:mi></mml:msub></mml:mrow></mml:math>θ^i(θi)=θi, a party <mml:math><mml:mrow><mml:mpadded width="+1.7pt"><mml:mi>j</mml:mi></mml:mpadded><mml:mo>∈</mml:mo><mml:mi>T</mml:mi></mml:mrow></mml:math>j∈T announces it will join any legislative coalition (in the contingency that it is included in any proto-coalition), and a party <mml:math><mml:mrow><mml:mpadded width="+1.7pt"><mml:mi>k</mml:mi></mml:mpadded><mml:mo>?</mml:mo><mml:mi>T</mml:mi></mml:mrow></mml:math>k?T announces it will not join any legislative coalition (in the contingency that it is included in any proto-coalition).

Suppose that <mml:math><mml:mrow><mml:mpadded width="+1.7pt"><mml:mn>0</mml:mn></mml:mpadded><mml:mo><</mml:mo><mml:msub><mml:mi>θ</mml:mi><mml:mi>m</mml:mi></mml:msub></mml:mrow></mml:math>0<θm (the case in which <mml:math><mml:mrow><mml:mpadded width="+1.7pt"><mml:msub><mml:mi>θ</mml:mi><mml:mi>m</mml:mi></mml:msub></mml:mpadded><mml:mo><</mml:mo><mml:mn>0</mml:mn></mml:mrow></mml:math>θm<0 is analogous) and suppose that all parties truthfully reveal their ideal policy in the first stage; i.e., <mml:math><mml:mrow><mml:mrow><mml:msub><mml:mrow><mml:mover accent="true"><mml:mi>θ</mml:mi><mml:mo stretchy="false">^</mml:mo></mml:mover></mml:mrow><mml:mi>i</mml:mi></mml:msub><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:msub><mml:mi>θ</mml:mi><mml:mi>i</mml:mi></mml:msub><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mrow><mml:mo>=</mml:mo><mml:msub><mml:mi>θ</mml:mi><mml:mi>i</mml:mi></mml:msub></mml:mrow></mml:math>θ^i(θi)=θi. Given the bargaining protocol <mml:math><mml:mi mathvariant="script">B</mml:mi></mml:math>B, all parties <mml:math><mml:mrow><mml:mpadded width="+1.7pt"><mml:mi>j</mml:mi></mml:mpadded><mml:mo>∈</mml:mo><mml:mi>T</mml:mi></mml:mrow></mml:math>j∈T will be included in the initial proto-coalition and will simultaneously announce whether to form or not form a legislative coalition. Because for any party <mml:math><mml:mrow><mml:mpadded width="+1.7pt"><mml:mi>j</mml:mi></mml:mpadded><mml:mo>∈</mml:mo><mml:mi>T</mml:mi></mml:mrow></mml:math>j∈T we have <mml:math><mml:mrow><mml:mpadded width="+1.7pt"><mml:msub><mml:mi>θ</mml:mi><mml:mi>j</mml:mi></mml:msub></mml:mpadded><mml:mo>≥</mml:mo><mml:msub><mml:mi>θ</mml:mi><mml:mi>m</mml:mi></mml:msub></mml:mrow></mml:math>θj≥θm and because if this coalition fails to form the implemented policy is some policy closer or equal to the status quo than <mml:math><mml:msub><mml:mi>θ</mml:mi><mml:mi>m</mml:mi></mml:msub></mml:math>θm, any party <mml:math><mml:mrow><mml:mpadded width="+1.7pt"><mml:mi>j</mml:mi></mml:mpadded><mml:mo>∈</mml:mo><mml:mi>T</mml:mi></mml:mrow></mml:math>j∈T with ideal policy <mml:math><mml:mrow><mml:mpadded width="+1.7pt"><mml:msub><mml:mi>θ</mml:mi><mml:mi>j</mml:mi></mml:msub></mml:mpadded><mml:mo>≥</mml:mo><mml:msub><mml:mi>θ</mml:mi><mml:mi>m</mml:mi></mml:msub></mml:mrow></mml:math>θj≥θm will announce it will join the legislative coalition regardless of its beliefs about the other parties’ (true) ideal policies. If the set of stable coalitions is not singleton, the set of parties included in the initial proto-coalition includes some parties <mml:math><mml:mrow><mml:mpadded width="+1.7pt"><mml:mi>k</mml:mi></mml:mpadded><mml:mo>∈</mml:mo><mml:mi>A</mml:mi></mml:mrow></mml:math>k∈A but <mml:math><mml:mrow><mml:mpadded width="+1.7pt"><mml:mi>k</mml:mi></mml:mpadded><mml:mo>?</mml:mo><mml:mi>T</mml:mi></mml:mrow></mml:math>k?T. Any such party <mml:math><mml:mi>k</mml:mi></mml:math>k is indifferent between announcing it will or will not join the legislative coalition because either way the implemented policy outcome is <mml:math><mml:msub><mml:mi>θ</mml:mi><mml:mi>m</mml:mi></mml:msub></mml:math>θm.

Given these strategies of parties in <mml:math><mml:mi>T</mml:mi></mml:math>T and parties not in <mml:math><mml:mi>T</mml:mi></mml:math>T, no party has a profitable deviation to misreport its ideal policy in the first stage given that the other parties report their ideal policies truthfully. If a party <mml:math><mml:mrow><mml:mpadded width="+1.7pt"><mml:mi>j</mml:mi></mml:mpadded><mml:mo>∈</mml:mo><mml:mi>T</mml:mi></mml:mrow></mml:math>j∈T misreports its ideal policy, such a strategy either has no effect on the outcome or results in a policy outcome worse than <mml:math><mml:msub><mml:mi>θ</mml:mi><mml:mi>m</mml:mi></mml:msub></mml:math>θm for party <mml:math><mml:mi>j</mml:mi></mml:math>j. If a party <mml:math><mml:mrow><mml:mpadded width="+1.7pt"><mml:mi>k</mml:mi></mml:mpadded><mml:mo>?</mml:mo><mml:mi>T</mml:mi></mml:mrow></mml:math>k?T misreports its ideal policy such that <mml:math><mml:mrow><mml:mpadded width="+1.7pt"><mml:msub><mml:mrow><mml:mover accent="true"><mml:mi>θ</mml:mi><mml:mo stretchy="false">^</mml:mo></mml:mover></mml:mrow><mml:mi>k</mml:mi></mml:msub></mml:mpadded><mml:mo>≤</mml:mo><mml:msub><mml:mi>θ</mml:mi><mml:mi>m</mml:mi></mml:msub></mml:mrow></mml:math>θ^k≤θm, such a strategy has no effect on the equilibrium outcome of the game. And if a party <mml:math><mml:mrow><mml:mpadded width="+1.7pt"><mml:mi>k</mml:mi></mml:mpadded><mml:mo>?</mml:mo><mml:mi>T</mml:mi></mml:mrow></mml:math>k?T with true ideal policy <mml:math><mml:mrow><mml:mpadded width="+1.7pt"><mml:msub><mml:mi>θ</mml:mi><mml:mi>k</mml:mi></mml:msub></mml:mpadded><mml:mo><</mml:mo><mml:msub><mml:mi>θ</mml:mi><mml:mi>m</mml:mi></mml:msub></mml:mrow></mml:math>θk<θm misreports its ideal policy such that <mml:math><mml:mrow><mml:mpadded width="+1.7pt"><mml:msub><mml:mrow><mml:mover accent="true"><mml:mi>θ</mml:mi><mml:mo stretchy="false">^</mml:mo></mml:mover></mml:mrow><mml:mi>k</mml:mi></mml:msub></mml:mpadded><mml:mo>></mml:mo><mml:msub><mml:mi>θ</mml:mi><mml:mi>m</mml:mi></mml:msub></mml:mrow></mml:math>θ^k>θm, such a strategy might change the identity of the initial proto-coalition in that party <mml:math><mml:mi>k</mml:mi></mml:math>k might be included in the initial proto-coalition, one or more parties from the stable coalition <mml:math><mml:mi>T</mml:mi></mml:math>T are excluded from the initial proto-coalition, and party <mml:math><mml:mi>k</mml:mi></mml:math>k’s positive announcement is necessary to form the initial legislative coalition. In this contingency, any party <mml:math><mml:mrow><mml:mpadded width="+1.7pt"><mml:mi>j</mml:mi></mml:mpadded><mml:mo>∈</mml:mo><mml:mi>T</mml:mi></mml:mrow></mml:math>j∈T (still included in the initial proto-coalition) will announce it will join this proto-coalition and a party <mml:math><mml:mi>k</mml:mi></mml:math>k’s optimal strategy is to announce it will not. Because this legislative coalition fails to form, in the next round (given the rules of the bargaining protocol <mml:math><mml:mi mathvariant="script">B</mml:mi></mml:math>B) a new proto-coalition comprising all parties from the stable coalition <mml:math><mml:mi>T</mml:mi></mml:math>T forms and any party <mml:math><mml:mrow><mml:mpadded width="+1.7pt"><mml:mi>j</mml:mi></mml:mpadded><mml:mo>∈</mml:mo><mml:mi>T</mml:mi></mml:mrow></mml:math>j∈T will announce it will join this legislative coalition to implement <mml:math><mml:msub><mml:mi>θ</mml:mi><mml:mi>m</mml:mi></mml:msub></mml:math>θm. Thus, by misreporting its ideal policy, party <mml:math><mml:mi>k</mml:mi></mml:math>k cannot change the policy outcome <mml:math><mml:msub><mml:mi>θ</mml:mi><mml:mi>m</mml:mi></mml:msub></mml:math>θm.

Note that in this type of perfect Bayesian Nash equilibrium, the belief of party <mml:math><mml:mi>i</mml:mi></mml:math>i regarding party <mml:math><mml:mi>j</mml:mi></mml:math>j’s type after observing?<mml:math><mml:mi>j</mml:mi></mml:math>j’s reported ideal policy is <mml:math><mml:mrow><mml:mi>P</mml:mi><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mpadded width="+1.7pt"><mml:msub><mml:mi>θ</mml:mi><mml:mi>j</mml:mi></mml:msub></mml:mpadded><mml:mo>=</mml:mo><mml:msub><mml:mrow><mml:mover accent="true"><mml:mi>θ</mml:mi><mml:mo stretchy="false">^</mml:mo></mml:mover></mml:mrow><mml:mi>j</mml:mi></mml:msub><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mo>=</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:math>P(θj=θ^j)=1 for any <mml:math><mml:mi>i</mml:mi></mml:math>i and <mml:math><mml:mi>j</mml:mi></mml:math>j. This belief is consistent with the reporting strategies and given this belief, a party’s strategy regarding whether to join a proto-coalition or not as specified above is optimal. (As a matter of fact the joining strategy of a party <mml:math><mml:mrow><mml:mpadded width="+1.7pt"><mml:mi>i</mml:mi></mml:mpadded><mml:mo>∈</mml:mo><mml:mi>T</mml:mi></mml:mrow></mml:math>i∈T is optimal for any belief party <mml:math><mml:mi>i</mml:mi></mml:math>i might hold.)

Acknowledgments

We thank Scott de Marchi, Xiaochen Fan, Tim Feddersen, Arturas Rozenas, and Kenneth Shepsle for helpful comments and suggestions. All errors are ours.

Footnotes

  • ?1To whom correspondence may be addressed. Email: tiberiu.dragu{at}nyu.edu or michael.laver{at}nyu.edu.

References

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