United for change: deliberative coalition formation to change the status quo

Social choice theory has provided us with a wealth of tools to design voting methods for democratic decision-making (see, e.g., Zwicker, 2016). However, besides voting, another important dimension of collective decision-making involves determining what should be put to vote in the first place. This aspect of democratic choice is referred to as the ‘right of initiative’ in a parliamentary context, and has received considerably less attention in the computational social choice literature.

Indeed, in practice, agents engaging in group decisions do not just vote on an externally determined set of choices: rather, they make proposals, deliberate over them, and join coalitions to push their proposals through. Understanding deliberative processes is essential in order to provide a more comprehensive picture of democratic decision-making. This is important in the context of established democratic institutions, such as parliaments, but is especially critical for digital democracy applications (Brill 2018) that provide support for democratic decision-making in more informal, less structured communities. Notable examples of this kind are the LiquidFeedback^{Footnote 1} (Behrens et al. 2014) and the Polis^{Footnote 2} platforms.

In this paper, we aim to provide a parsimonious model of deliberative processes in self-governed (online) systems. We abstract away from the communication mechanisms by means of which deliberation may be concretely implemented, and focus instead on the coalitional effects of deliberation, that is, on how coalitions in support of various proposals may be formed or broken in the face of new suggestions. In our model, agents and alternatives are located in a metric space, and there is one distinguished alternative, which we refer to as the status quo. We assume that the number of alternatives is large (possibly infinite), so that the agents cannot be expected to rank the alternatives or even list all alternatives they prefer to the status quo. Rather, during the deliberative process, some agents formulate new proposals, and then each agent can decide whether she prefers a given proposal p to the status quo, i.e., whether p is closer to her location than the status quo alternative; if so, she may join a coalition of agents supporting p. This process is democratic in that each participant who is capable of formulating a new proposal is welcome to do so; furthermore, participants who do not have the time or sophistication to work out a proposal can still take part in the deliberation by choosing which coalition to join.

These coalitions are dynamic and change over time: an agent may move to another coalition, or two coalitions may merge, possibly leaving some members behind. We assume that agents are (1) consensus-seeking, i.e., they aim to jointly identify a proposal that has large support, and (2) myopic, i.e., they make their moves without trying to predict their impact on other agents’ behavior in the subsequent steps. As a consequence, in our model agents always aim to increase the size of the coalition that they are part of, as long as this coalition supports a proposal they prefer to the status quo. In particular, they may move from a smaller coalition whose proposal is close to their ideal point to a larger coalition whose proposal is further away from their ideal point (as long as they still prefer the latter coalition’s proposal to the status quo). We consider a deliberation to be successful if it identifies a most popular alternative to the status quo. Thus, our results should be interpreted as an attempt to map out what stands in the way of successful deliberation even when assuming that the agents’ primary motivation is to reach a consensus.

To flesh out the broad outline of the deliberative coalition formation process described above, one needs to specify rules that govern the dynamics of coalition formation. We explore several such rules, ranging from single-agent moves to more complex transitions where coalitions merge behind a new proposal, possibly leaving some dissenting members behind. Our aim is to understand whether the agents can succeed at identifying credible alternatives to the status quo if they conduct deliberation in a certain way.

We concentrate on how the properties of the underlying abstract space of proposals and the coalition formation operators available to the agents affect the success of the deliberative coalition formation process. We show that, as the complexity of the proposal space increases, more sophisticated forms of coalition formation are required in order to assure success. Intuitively, this seems to suggest that complex deliberative spaces require more sophisticated coalition formation abilities on the side of the agents.

1.1 Paper contribution

We study five ways in which agents can form coalitions:

1. 1.

   by deviation, when a single agent from one coalition moves to another, weakly larger, coalition;

2. 2.

   by following, when a coalition joins another coalition in supporting the second coalition’s proposal;

3. 3.

   by merge, when two coalitions join forces behind a new proposal;

4. 4.

   by compromise, when agents belonging to two coalitions form a new larger coalition, possibly leaving dissenting agents behind;

5. 5.

   by multi-party compromise, when agents belonging to possibly more than two coalitions form a new larger coalition, perhaps leaving dissenting agents behind.

We refer to these types of coalition formation operations as transitions. Even though these transitions are all inspired by intuitive features of coalition formation in deliberation, we emphasise that we are not making empirical claims about the frequency or popularity of these transitions in real-world deliberations. Rather, they should be treated as abstract constructs that enable us to describe a hierarchy of coalition formation modes of increasing sophistication; this, in turn, puts us in a position to make claims to the effect that certain proposal spaces require more elaborate coalition formation procedures.

We show that, for each class of transitions, the deliberation process is guaranteed to terminate; in fact, for single-agent, follow, and merge transitions the number of steps until convergence is at most polynomial in the number of agents. Furthermore, follow transitions are sufficient for deliberation to succeed if the set of possible proposals is a subset of the 1-dimensional Euclidean space, but this is no longer the case in two or more dimensions; in contrast, compromise transitions are enough in sufficiently rich subsets of \({\mathbb R}^d\) for each \(d\ge 1\). The ‘richness’ condition, however, is essential: we provide an example where the space of proposals is a finite subset of \({\mathbb R}^3\), but compromise transitions are not capable of identifying a most popular proposal.

While our primary focus is on deliberation in Euclidean spaces, we also provide results for two classes of non-Euclidean spaces. The first one is the class of weighted trees, with the metric given by the standard path length. We show that, in these spaces, merge transitions suffice for successful deliberation. The second class of spaces arises naturally in combinatorial domains as well as in multiple referenda settings. These spaces, which we refer to as d-hypercubes, are defined by the set of all binary opinions on d issues with the discrete metric known as the Hamming, or Manhattan, distance. We show that deliberative success in d-hypercubes requires multi-party compromises involving at least d coalitions, and possibly more. In particular, we prove that d-compromises guarantee success if \(d\le 3\), but when \(d = 4\) we may need 5-compromises. Table 1 provides an overview of our findings, with pointers to the relevant results in the paper.

We view our work as an important step towards modeling a form of pre-vote deliberation usually not studied within the social choice literature: how voters can identify proposals with large support. We believe that our formal model provides the basis for a framework within which systems designed to support democratic deliberation could be modeled and analyzed. In particular, it captures a highly desirable common feature of such systems, including, e.g., LiquidFeedback and Polis: enabling participants to come up with novel proposals and to submit them for the approval (or disapproval) of other participants. Only proposals commanding large-enough support would then be considered as options to be voted upon in a formal voting phase. Hence, participants are nudged towards identifying positions that are mutually beneficial (see also the work of Speroni di Fenizio and Velikanov 2016) and can therefore elicit the approval of larger groups. Our theory aims to make the first steps towards providing analytical foundations for such systems.

1.2 Structure of the paper

After discussing related work (Sect. 2), we describe our formal model in Sect. 3. Then, we concentrate on Euclidean deliberation spaces and study the power of different deliberation operations: specifically, in Sect. 4, we consider single-agent transitions; in Sect. 5 we consider follow transitions; in Sect. 6 we consider merge transitions (also in the context of weighted trees); in Sect. 7 we consider compromise transitions. Finally, in Sect. 8, we study deliberation in hypercubes and multi-party compromise transitions. We conclude in Sect. 9.

2.1 Deliberation

Group deliberation has been an object of research in several disciplines, from economics to political theory and artificial intelligence. A wealth of different approaches to deliberation can be identified, reflecting the complexity of the concept. List (2011) has focused on axiomatic approaches to deliberation, viewed as an opinion transformation function. Experimental work—relying on either lab experiments (List et al. 2013) or simulations (Roy and Rafiee Rad 2020)—has then tried to assess the effects of deliberations on individual opinions (e.g., whether deliberation facilitates the set of opinions becoming single-peaked). Deliberation has also been approached from a game-theoretic perspective (Landa and Meirowitz 2009). With the tools of game theory, deliberation has been studied from a number of angles: as a process involving the exchange of arguments for or against positions (Chung and Duggan 2020; Patty 2008); as a process of persuasion (Glazer and Rubinstein 2001, 2004, 2006); as the pre-processing of inputs for voting mechanisms (Austen-Smith and Feddersen 2005; Perote-Peña and Piggins 2015; Karanikolas et al. 2019); as a mechanism enabling preference discovery (Hafer and Landa 2007). Deliberation has also been studied as a distributed protocol for large-scale decision-making involving only sequential local interaction in small groups, with a focus on algorithmic complexity (Goel and Lee 2016; Fain et al. 2017).

Our work sets itself somewhat apart from the above literature by focusing exclusively on the consensus-seeking aspect of deliberation. Deliberation is studied here as a distributed process whereby widely supported proposals can be identified in a decentralized way. In doing so, we abstract away from: the concrete interaction mechanisms (e.g., argumentation) that agents may use to communicate and assess proposals; strategic issues that may arise during communication; as well as how deliberation might interact with specific voting rules. Instead, we concentrate on the results of such interactions, as manifested by changes in the structure of coalitions supporting different proposals. This focus also sets our work apart from influential opinion dynamics models (e.g., Groot 1974) where deliberation is driven by social influence rather than by coalition formation via, for instance, compromise.

Importantly, we abstract away from strategic issues that agents may have to confront when deciding to join or leave coalitions; this differentiates our work from the related literature on dynamic coalition formation (Dieckmann and Schwalbe 2002; Chalkiadakis and Boutilier 2008). In particular, Chalkiadakis and Boutilier (2008) concentrate on uncertainties that the agents may experience; and Dieckmann and Schwalbe (2002) focus on agents that receive payoffs, which depend on the coalitions they are in.

2.2 Spatial voting

From a technical point of view, our contribution is closely related to work on spatial voting that stems from the Hotelling model (Hotelling 1929). Many social choice settings are naturally embedded in a metric space, and there is a large literature that considers preference aggregation and coalition formation in such scenarios (Coombs 1964; Merrill and Grofman 1999; de Vries 1999). In this context we mention, in particular, the work of Shahaf et al. (2019), which considers a framework for voting and proposing, instantiated to several metric spaces. It includes an agent population where each agent is associated with an ideal point. This framework models a broad range of social choice settings, and is intended to be used for deliberative decision-making. Shahaf et al. (2019) evaluate several general aggregation methods for their framework based on their computational and normative properties. Our work extends the framework of Shahaf et al. (2019) by accommodating the processes of coalition formation.

2.3 Coalition formation

Since we model deliberation as the evolution of coalition structures in response to deviations by groups of agents, our framework is closely related to that of hedonic games (Aziz and Savani 2016). Hedonic games offer a rich framework for studying stability properties of coalition structures where agents have preferences over coalitions that they can be part of; here, stability is interpreted as the absence of profitable deviations, by individual agents or groups of agents. In a similar spirit, we are also concerned with stability of coalition structures that arise in a chain of deviations. Indeed, some of the transitions types that we investigate are inspired by stability concepts developed in the hedonic games literature (e.g., single-agent transitions and \(\ell \)-compromise transitions are closely related to the deviations underpinning the concepts of, respectively, Nash stability and core stability in hedonic games). However, a key difference is that in our framework each coalition is associated with a specific proposal, and the choice of proposal is an important factor in determining which agents would like to join/abandon this coalition. In this regard, our setting is closer to that of group activity selection (Darmann et al. 2018), where agents form groups to engage in activities. However, to the best of our knowledge, there is no existing work on group activity selection with metric preferences, and some of the transition types we consider (such as merge, follow and subsume transitions) have not received much attention in the context of group activity selection.

We view deliberation as a process in which agents aim to find an alternative preferred over the status quo by as many agents as possible. Thus, we assume a (possibly infinite) domain X of alternatives, or proposals, which includes the status quo, or reality, \(r \in X\). We also assume a set \(V = \{v_1, \ldots , v_n\}\) of n agents. For each proposal \(x\in X\), an agent v is able to articulate whether she (strictly) prefers x over the status quo r (denoted as \(x >_v r\)); when \(x >_v r\) we say that v approves x. For each \(v\in V\), let \(X^v=\{x\in X: x>_v r\}\); the set \(X^v\) is the approval set of v. Given a subset of agents \(C \subseteq V\) and a proposal \(p\in X\), let furthermore \(C^p:= \{v \in C: p >_v r\}\). Then, the agents in \(C^p\) are the approvers of p in C.

Throughout this paper, in order to represent agents’ views on a possibly infinite set of proposals, we focus on the setting where X and V are contained in a metric space \((M, \rho )\), i.e., (1) \(X, V\subseteq M\), (2) the mapping \(\rho : M\times M\rightarrow {\mathbb R}^+\cup \{0\}\) satisfies (i) \(\rho (x, y)=0\) if and only if \(x=y\), (ii) \(\rho (x, y)=\rho (y, x)\), and (iii) \(\rho (x, y)+\rho (y, z)\le \rho (x, z)\) for all \(x, y, z\in M\), and (3) for every \(x\in X\) and every \(v\in V\) we have \(x>_v r\) if and only if \(\rho (v, x)<\rho (v, r)\). E.g., if \(M = {\mathbb R}^2\) and the metric is the usual Euclidean metric in \({\mathbb R}^2\), then the approval set of v consists of all points in X that are located inside the circle with center v and radius \(\rho (v, r)\) (see Fig. 1), whereas the set of supporters of a proposal p in C consists of all agents \(v\in C\) such that \(\rho (v, p) < \rho (v, r)\). Geometrically, consider the perpendicular bisector to [p, r], i.e., the line \(\ell \) that passes through the midpoint of the segment [p, r] and is orthogonal to it. Then, \(C^p\) consists of all agents \(v\in C\) such that v and p lie on the same side of \(\ell \).

Thus, an instance of our problem can be succinctly encoded by a 4-tuple \((X, V, r, \rho )\); we will refer to such tuples as deliberation spaces.

Remark 1

Note that we do not require that \(X=M\). By allowing X to be a proper subset of M we can capture the case where the space of proposals is, e.g., a finite subset of \({\mathbb R}^d\) for some \(d\ge 1\). Moreover, in our model it need not be the case that \(V\subseteq X\), i.e., we do not assume that for each agent there exists a ‘perfect’ proposal. Furthermore, while for each agent v the quantities \(\rho (v, x)\) are well-defined for each \(x\in X\), we do not expect the agents to compare different proposals based on distance; rather, the distance only determines which proposals are viewed as acceptable (i.e., preferred to the status quo).

Agents proceed by forming coalitions around proposals. Thus, at each point in the deliberation, agents can be partitioned into coalitions, so that each coalition C is identified with a proposal \(p_C\) and all agents in C support \(p_C\). Agents may then move from one coalition to another as well as select a proposal from X that is not associated with any of the existing coalitions and form a new coalition around it. We consider a variety of permissible moves, ranging from single-agent transitions (when an agent abandons her current coalition and joins a new one), to complex transitions that may involve agents from multiple coalitions and a new proposal. In each case, we assume that agent v is unwilling to join a coalition if this coalition advocates a proposal p such that v (weakly) prefers the status quo r to p.

We now introduce the two main objects to be studied in our work: deliberative coalitions and deliberative coalition structures.

Definition 1

(Deliberative Coalition) A deliberative coalition is a pair \(\textbf{d}=(C, p)\), where \(C\subseteq V\) is a set of agents, \(p\in X\) is a proposal, and either (i) \(p=r\) and \(x\not >_v r\) for all \(v\in C, x\in X{{\setminus }}\{r\}\), or (ii) \(p\ne r\) and \(p >_v r\) for all \(v\in C\). We refer to p as the supported proposal of \(\textbf{d}\). The set of all deliberative coalitions is denoted by \(\mathcal {D}\).

When convenient, we identify a coalition \(\textbf{d}= (C,p)\) with its set of agents C and write \(\textbf{d}^q:= C^q\) for \(q\in X\), \(|\textbf{d}|:= |C|\).

Remark 2

While we allow coalitions that support the status quo, we require that such coalitions consist of agents who weakly prefer the status quo to all other proposals in X. We discuss a relaxation of this constraint in Sect. 9.

A partition of the agents into deliberative coalitions is called a deliberative coalition structure.

Definition 2

(Deliberative Coalition Structure) A deliberative coalition structure (coalition structure for short) is a set \(\textbf{D}= \{\textbf{d}_1, \ldots , \textbf{d}_m\}\), \(m \ge 1\), such that:

• \(\textbf{d}_i = (C_i,p_i)\in \mathcal {D}\) for each \(i \in [m]\);

• \(\bigcup _{i\in [m]}C_i = V\), \(C_i \cap C_j = \emptyset \) for all \(i,j \in [m]\) with \(i\ne j\).

The set of all deliberative coalition structures over V and X is denoted by \(\mathfrak {D}\).

Note that a deliberative coalition structure may contain several coalitions supporting the same proposal; also, for technical reasons we allow empty deliberative coalitions, i.e., coalitions (C, p) with \(C=\emptyset \).

Example 1

Consider a set of agents \(V = \{v_1, v_2, v_3\}\) and a space of proposals \(X = \{a, b, c, d, r\}\), where r is the status quo. Suppose that \(X^{v_1} = \{a, b\}\), \(X^{v_2} = \{b, c\}\), and \(X^{v_3} = \{b, c, d\}\). Then for \(C = \{v_1, v_2\}\) we have \(C^a = \{v_1\}\) and \(C^b = \{v_1, v_2\}\). Furthermore, let \(C_1=\{v_1, v_2\}\), \(C_2=\{v_3\}\), and let \(\textbf{d}_1 = (C_1, b)\), \(\textbf{d}_2=(C_2, c)\). Then \(\textbf{D}= \{\textbf{d}_1, \textbf{d}_2\}\) is a deliberative coalition structure; see Fig. 1 for an illustration.

A deliberation space with \(X = \{a, b, c, d, r\}\). The circle with center \(v_i\), \(i \in [3]\), contains all proposals approved by \(v_i\)

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3.1 Deliberative transition systems

As suggested above, we model deliberation as a process whereby deliberative coalition structures change as their constituent coalitions evolve. We provide general definitions for deliberative processes, modeled as transition systems (Definitions 3–6). Subsequently, we explore several specific kinds of transitions. Generally, a transition system is characterized by a set of states S, a subset \(S_0 \subseteq S\) of initial states, and a set of transitions T, where each transition \(t \in T\) is represented by a pair of states \((s, s') \in S\times S\); we write \(t=(s, s')\) and \(s \xrightarrow {t} s'\) interchangeably. We use \(s \xrightarrow {T} s'\) to denote that \(s \xrightarrow {t} s'\) for some \(t \in T\). A run of a transition system is a (finite or infinite) sequence \(s_0 \xrightarrow {T} s_1 \xrightarrow {T} s_2 \cdots \); such a run is initialized if \(s_0 \in S_0\). The last state of a finite run is called its terminal state.

Definition 3

(Deliberative Transition System) A deliberative transition system over a set of proposals X and a set of agents V is a transition system that has \(\mathfrak {D}\) as its set of states, a subset of states \(\mathfrak {D}_0 \subseteq \mathfrak {D}\) as its set of initial states, and a set of transitions \(\mathcal {S}\).

Deliberative transition systems describe all possible sequences of deliberative coalition structures that can be obtained by executing a given set of transitions, starting from a given set of initial deliberative coalition structures. As such, they can be used to analyze the dynamics of the deliberation process. We define deliberations as maximal runs of such systems.

Definition 4

(Deliberation) A deliberation is a maximal run of a deliberative transition system, that is, a run that does not occur as a prefix of any other run (i.e., cannot be extended).

A successful deliberation is one that identifies some of the most popular proposals in \(X{\setminus }\{r\}\). In particular, if there is a majority-approved proposal, a successful deliberation process allows the agent population to identify some such proposal. Observe, however, that it may be the case that no alternative is approved by a majority of voters.

Definition 5

(Most-Approved Alternatives) For a set of agents V and a set of proposals X, the set of most-approved alternatives is \(M^* = {\text {argmax}}_{x \in X{{\setminus }}\{r\}} |V^x|\) and the maximum approval is \(m^* = {\text {max}}_{x \in X{\setminus }\{r\}} |V^x|\).

Note that, as long as all approval sets are non-empty, \(M^* \ne \emptyset \) and \(|V^x| = m^*\) for every \(x \in M^*\). We are now ready to define success in deliberation.

Definition 6

A coalition \(\textbf{d}= (C,p)\) is successful if \(p\in M^*\) and \(|C| = m^*\). A deliberative coalition structure is successful if it contains a successful coalition, and unsuccessful otherwise. A deliberation is successful if it is finite and its terminal coalition structure is successful.

Intuitively, a coalition is successful if it consists of all approvers of a most-approved proposal. Successful deliberations are maximal finite runs that lead to a successful coalition structure, i.e., to a coalition structure that contains some successful coalition.

Example 2

Consider a toy example of a deliberative transition system that consists of (1) a single initial coalition structure \(\textbf{D}_0 = \{\textbf{d}_1, \textbf{d}_2, \textbf{d}_3\}\), with \(\textbf{d}_1 = (C_1, a)\) where \(C_1 = \{v_1, v_2\}\), \(\textbf{d}_2 = (C_2, b)\) where \(C_2 = \{v_3, v_4\}\), and \(\textbf{d}_3 = (C_3, c)\) where \(C_3 = \{v_5, v_6\}\); and (2) the set of transitions \(\mathcal S\) only containing a transition t from \(\textbf{D}_0\) to \(\textbf{D}_1 = \{\textbf{d}_4, \textbf{d}_5\}\), with \(\textbf{d}_4 = (C_4, e)\) where \(C_4 = \{v_1, v_2, v_3, v_4, v_5\}\), \(\textbf{d}_5 = (C_5, f)\) where \(C_5 = \{v_6\}\). Then, assuming that there is no proposal approved by all agents, we have that \(\textbf{D}_0 \xrightarrow {t} \textbf{D}_1\) is a successful deliberation.

Remark 3

Since successful deliberations identify proposals that receive the maximum number of approvals, a successful deliberative coalition structure identifies a proposal that would be a (possibly tied) winner of an election if agents were to submit approval ballots over the entire domain X (or, more precisely, \(X{\setminus }\{r\}\)— our model implicitly assumes that no agent approves r). However, there are two crucial differences between our deliberation model and conducting an approval-based election over X:

• First, our model of deliberation is decentralized in the sense that, at every step, agents do not need to report their approval/disapproval of each proposal, as they would have to do under a centralized voting mechanism.

• Second, our model is parsimonious in the sense that, at any time during the deliberation, each agent supports just a single proposal in her (possibly infinite) approval set. Thus, the agents do not need to be aware of the entire (possibly infinite) space of proposals X, whereas under a voting mechanism they would have to form an opinion on each proposal in X.

Our aim is to formulate conditions guaranteeing that such a deliberative process succeeds, i.e., identifies proposals that could arise as outcomes of an approval election on the entire space X.

Importantly, groups of agents may differ with respect to the types of agreements they are able to negotiate. To understand what can be accomplished by groups with particular sets of negotiation skills, we consider specific types of transitions; each can be thought of as a deliberation operator that might be available to the agents. For each transition type, we aim to determine if a deliberation that only uses such transitions is guaranteed to terminate, and, if so, whether the final coalition structure is guaranteed to be successful. We show that the answer to this question depends on the underlying metric space: simple transition rules guarantee success in simple metric spaces, but may fail in richer spaces.

The simplest kind of transition we consider is a deviation by a single agent. As we assume that agents aim to form a successful coalition and are not necessarily able to distinguish among approved proposals, it is natural to focus on transitions where an agent moves from a smaller group to a larger group; of course, this move is only possible if the agent approves the proposal supported by the larger group.

Definition 7

(Single-Agent Transition) A pair \((\textbf{D}, \textbf{D}')\) of coalition structures forms a single-agent transition if there exist coalitions \(\textbf{d}_1, \textbf{d}_2\in \textbf{D}\) and \(\textbf{d}'_1, \textbf{d}'_2\in \textbf{D}'\) such that \(|\textbf{d}_2|\ge |\textbf{d}_1|\), \(\textbf{D}{{\setminus }}\{\textbf{d}_1, \textbf{d}_2\}=\textbf{D}'{{\setminus }}\{\textbf{d}'_1, \textbf{d}'_2\}\), and there exists an agent \(v\in \textbf{d}_1\) such that \(\textbf{d}'_1=\textbf{d}_1{{\setminus }}\{v\}\), and \(\textbf{d}'_2=\textbf{d}_2\cup \{v\}\).^{Footnote 3} We refer to v as the deviating agent.

Since \(\textbf{D}'\) is a deliberative coalition structure, agent v must approve the proposal supported by \(\textbf{d}_2\). As a consequence, no agent can deviate from a coalition that supports r to a coalition that supports some \(p\in X{\setminus }\{r\}\) or vice versa.

Next, we show that a sequence of single-agent transitions necessarily terminates after polynomially many steps.^{Footnote 4}

Proposition 1

A deliberation that consists of single-agent transitions can have at most \(n^2\) transitions.

Proof

Given a coalition structure \(\textbf{D}=\{\textbf{d}_1, \ldots , \textbf{d}_m\}\) such that \(\textbf{d}_i=(C_i, p_i)\) for each \(i\in [m]\), let

We will refer to \(\lambda (\textbf{D})\) as the potential of \(\textbf{D}\). Consider a single-agent transition where an agent moves from a coalition of size x to a coalition of size y; note that \(1\le x\le y\). This move changes the potential by \((y+1)^2+(x-1)^2-y^2-x^2=2 + 2(y-x)\ge 2\).

We claim that for every deliberative coalition structure \(\textbf{D}\) over n agents we have \(\lambda (\textbf{D})\le n^2\). Indeed, for the coalition structure \(\textbf{D}_0\) where all agents are in one coalition we have \(\lambda (\textbf{D}_0)=n^2\). On the other hand, if a coalition structure contains two non-empty coalitions \(\textbf{d}_1\), \(\textbf{d}_2\) with \(|\textbf{d}_1|\le |\textbf{d}_2|\), then the calculation above shows that we can increase the potential by moving one agent from \(\textbf{d}_1\) to \(\textbf{d}_2\). Further, every coalition structure can be transformed into \(\textbf{D}_0\) by a sequence of such moves, and hence \(\lambda (\textbf{D})\le \lambda (\textbf{D}_0)\le n^2\) for each \(\textbf{D}\in \mathfrak {D}\).

As every single-agent transition increases the potential by at least 2, and the potential takes values in \(\{1, \dots , n^2\}\), the result follows. \(\square \)

However, a deliberation consisting of single-agent transitions is not necessarily successful, even for very simple metric spaces. The next example shows that such a deliberation may fail to identify a majority-approved proposal even if the associated metric space is the 1D Euclidean space.

The metric space in Example 3

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Example 3

Suppose that \(X, V\subseteq {\mathbb R}\), and \(X=\{r, a, b, c\}\) with \(r=0\), \(a=1\), \(b=5\), \(c=-1\). There are three agents \(v_1, v_2, v_3\) located at a, four agents \(v_4, v_5, v_6, v_7\) located at b, and three agents \(v_8, v_9, v_{10}\) located at c (see Fig. 2). Observe that a majority of the agents prefer a to r. Consider the deliberative coalition structure \(\{\textbf{d}_1, \textbf{d}_2, \textbf{d}_3\}\) with \(\textbf{d}_1=(\{v_1, v_2, v_3\}, a)\), \(\textbf{d}_2=(\{v_4, v_5, v_6, v_7\}, b)\), \(\textbf{d}_3=(\{v_8, v_9, v_{10}\}, c)\). There are no single-agent transitions from this coalition structure: in particular, the agents in \(\textbf{d}_2\) do not want to deviate to \(\textbf{d}_1\) because \(|\textbf{d}_1|<|\textbf{d}_2|\), and the agents in \(\textbf{d}_1\) do not want to deviate to \(\textbf{d}_2\), because they do not approve b. Note that this argument still applies to any proposal space \(X'\) with \(X\subseteq X'\); e.g., we can take \(X'=\mathbb R\).

Thus, to ensure success, we need to consider more powerful transitions.

Instead of considering moves by a single agent, we will now focus on moves by entire coalitions; specifically, we consider transitions where all members of a coalition join another coalition in supporting that coalition’s current proposal.

Definition 8

(Follow transition) A pair of coalition structures \((\textbf{D}, \textbf{D}')\) forms a follow transition if there exist non-empty coalitions \(\textbf{d}_1, \textbf{d}_2\in \textbf{D}\) and \(\textbf{d}'_2\in \textbf{D}'\) such that \(\textbf{d}_1=(C_1, p_1)\), \(\textbf{d}_2=(C_2, p_2)\), \(\textbf{D}{\setminus }\{\textbf{d}_1, \textbf{d}_2\}=\textbf{D}'{\setminus }\{\textbf{d}'_2\}\), and \(\textbf{d}'_2 = (C_1\cup C_2, p_2)\).

Note that each follow transition reduces the number of coalitions by one, so a deliberation that consists of follow transitions only cannot take more than \(n-1\) steps. Also, \(|\textbf{d}'_2|^2 = (|\textbf{d}_1|+|\textbf{d}_2|)^2>|\textbf{d}_1|^2+|\textbf{d}_2|^2\), i.e., follow transitions increase the potential function \(\lambda (\cdot )\) given by Equation (1). This implies the following bound.

Proposition 2

A deliberation that consists of single-agent transitions and follow transitions can have at most \(n^2\) transitions.

However, in contrast to single-agent transitions, follow transitions are sufficient for successful deliberation in any subset of the 1D Euclidean space.

Theorem 1

Consider a deliberation space \((X, V, r, \rho )\), where \(X, V\subseteq {\mathbb R}\) and \(\rho (x, y)=|x-y|\). Then every deliberation that consists of follow transitions only, or of a combination of follow transitions and single-agent transitions, is successful.

Proof

Assume for convenience that \(r=0\). Consider a deliberation that consists of single-agent transitions and follow transitions. By Proposition 2 we know that it is finite; let \(\textbf{D}\) be its terminal state.

Suppose that \(\textbf{D}\) contains two deliberative coalitions \((C_1, p_1)\) and \((C_2, p_2)\) with \(p_1, p_2\in {\mathbb R}^+\); assume without loss of generality that \(p_1\le p_2\). Note that \(C_1, C_2\subseteq {\mathbb R}^+\): all agents in \({\mathbb R}^-\cup \{0\}\) prefer r to \(p_1, p_2\). Furthermore, every agent in \(C_2\) approves \(p_1\): indeed, if \(v\in C_2\) does not approve \(p_1\) then \(|v-r|\le |v-p_1|\), i.e., \(v\le |v-p_1|\). Since \(v, p_1>0\), this would imply \(2v\le p_1\le p_2\), in which case v would not approve \(p_2\) either. Hence there is a follow transition in which \(C_2\) joins \(C_1\), a contradiction with \(\textbf{D}\) being a terminal coalition structure.

Thus, \(\textbf{D}\) contains at most one coalition, say, \((C^+, q^+)\), that supports a proposal in \({\mathbb R}^+\); by the same argument, it also contains at most one coalition, say, \((C^-, q^-)\), that supports a proposal in \({\mathbb R}^-\), and at most one coalition, say, \((C^0, r)\), that supports r.

Let p be some proposal in \(M^*\), and assume without loss of generality that \(p>0\). Since agents in \({\mathbb R}^-\) prefer r to p, we have \(C^-\cap V^{p}=\emptyset \); also, by definition, all agents in \(C^0\) weakly prefer r to p. Hence \(V^p\subseteq C^+\) and therefore \(|C^+|=m^*\). \(\square \)

If we modify the definition of follow transitions to require \(|\textbf{d}_2|\ge |\textbf{d}_1|\) (i.e., that the joint proposal of the new coalition is the proposal originally supported by the larger of the two coalitions, a seemingly sensible condition), then the proof of Theorem 1 no longer goes through. In fact, Example 3 illustrates that in this case the transition system may be unable to reach a successful state.^{Footnote 5}

Unfortunately, Theorem 1 does not extend beyond one dimension. The following examples show that in the Euclidean plane a deliberation that only uses single-agent transitions and follow transitions is not necessarily successful.

Example 4

Consider a space of proposals \(\{a, b, p, r\}\) embedded into \({\mathbb R}^2\), where r is located at (0, 0), p is located at (0, 3), a is located at \((-3, 3)\), and b is located at (3, 3). There are four agents \(v_1, v_2, v_3, v_4\) located at \((-3, 3)\), \((-3, 4)\), (3, 3), and (3, 4), respectively (see Fig. 3). Note that all agents prefer p to r. Consider a deliberative coalition structure \(\textbf{D}= \{\textbf{d}_1, \textbf{d}_2\}\), where \(\textbf{d}_1=(\{v_1, v_2\}, a)\), \(\textbf{d}_2=(\{v_3, v_4\}, b)\). Then no agent in \(\textbf{d}_1\) approves b, and no agent in \(\textbf{d}_2\) approves a, so there are no follow transitions and no single-agent transitions from \(\textbf{D}\).

The metric space in Example 4

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Example 5

Consider the example in Fig. 4, where all proposals other than r as well as all agents are located on a circle with center r; the metric is the usual Euclidean distance in \({\mathbb R}^2\). The alternatives a, b, and c are equidistant from each other, and p is located halfway between a and b (so, in particular, \(d(a, p) = d(p, b) = d(p, r)\)). The deliberative coalition structure is \(\textbf{D}= (\textbf{d}_1, \textbf{d}_2, \textbf{d}_3)\), where \(\textbf{d}_1=(\{v_1, v_2, v_3\}, a)\), \(\textbf{d}_2=(\{v_4, v_5, v_6\}, b)\), and \(\textbf{d}_3=(\{v_7\}, c)\); one can verify that each agent approves the proposal supported by her deliberative coalition. Observe that more than half of the agents (namely, \(v_2, v_3, v_4, v_5\)) prefer p to r. However, there are no follow transitions or single-agent transitions from \(\textbf{D}\): for every pair of distinct coalitions \(\textbf{d}_i, \textbf{d}_j\) the agents in \(\textbf{d}_i\) do not approve the proposal supported by \(\textbf{d}_j\).

The metric space in Example 5

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So far, we have focused on transitions that did not introduce new proposals. Examples 4 and 5, however, show that new proposals may be necessary to reach success: indeed, none of the proposals supported by existing coalitions in these examples was approved by a majority of agents. Thus, next we explore transitions that identify new proposals.

As a first step, it is natural to relax the constraint in the definition of the follow transitions that requires the new coalition to adopt the proposal of one of the two component coalitions, and, instead, allow the agents to identify a new proposal that is universally acceptable.

We do not specify how the compromise proposal p is identified. We imagine that a new proposal could be put forward by one of the agents in \(\textbf{d}_1, \textbf{d}_2\) or by an external mediator whose goal is to help the agents reach a consensus.

Definition 9

(Merge Transition) A pair of coalition structures \((\textbf{D}, \textbf{D}')\) forms a merge transition if there exist non-empty coalitions \(\textbf{d}_1, \textbf{d}_2\in \textbf{D}\) and \(\textbf{d}'_2\in \textbf{D}'\) such that \(\textbf{d}_1=(C_1, p_1)\), \(\textbf{d}_2=(C_2, p_2)\), \(\textbf{D}{\setminus }\{\textbf{d}_1, \textbf{d}_2\}=\textbf{D}'{\setminus }\{\textbf{d}'_2\}\), and \(\textbf{d}'_2 = (C_1\cup C_2, p)\) for some proposal p.

One can verify that in Example 4 the agents have a merge transition to the deliberative coalition \((\{v_1, v_2, v_3, v_4\}, p)\). Thus, merge transitions are strictly more powerful than follow transitions. Moreover, by considering the potential function \(\lambda (\cdot )\) defined in Eq. (1), one can check that a deliberation that consists of single-agent transitions and merge transitions can have at most \(n^2\) steps. This implies the following bound.

Proposition 3

A deliberation that consists of single-agent transitions and merge transitions can have at most \(n^2\) transitions.

We will now describe a class of metric spaces where merge transitions guarantee convergence, but follow transitions do not.

6.1 Trees

Consider a weighted tree T with a set of vertices U, a set of edges E, and a weight function \(\omega : E\rightarrow {\mathbb R}^+\). This tree defines a metric space \((M_T, \rho _T)\), where \(M_T=U\), and for a pair of vertices \(x, y\in U\) the distance \(\rho _T(x, y)\) is the length of the path between x and y in T, where the length of an edge e is given by \(\omega (e)\). Consequently, a tree T defines a family of deliberation spaces \((X, V, r, \rho )\), where \(X\subseteq M_T\), \(V\subseteq M_T\), \(r\in X\) and \(\rho =\rho _T\). It turns out that for any such deliberation space merge transitions guarantee successful termination.

Theorem 2

Consider a deliberation space \((X, V, r, \rho )\) that corresponds to a tree \(T=(U, E, \omega )\) with \(r\in U\). Then every deliberation that consists of merge transitions only, or of a combination of merge transitions and single-agent transitions, is successful.

Proof

It will be convenient to think of T as a rooted tree with root r and, for each \(x\in U\), denote by \(T_x\) the subtree of T that is rooted in x.

Assume for convenience that \(X=M_T\), i.e., every vertex of T is a feasible proposal (later we will show that this assumption is not necessary). Let p be some proposal in \(M^*\), and let q be the child of r such that the path from p to r passes through q (it is possible that \(p=q\)). We claim that \(q\in M^*\). Indeed, an agent approves q if and only if her ideal point is located in \(T_q\). On the other hand, if an agent’s ideal point is not located in \(T_q\), then the path from her ideal point to p passes through the root r and hence she does not approve p. Hence, the set of agents who approve p forms a subset of the set of agents who approve q, establishing our claim.

This argument also shows that there are exactly \(m^*\) agents whose ideal point is located in \(T_q\). Each such agent can only be in a deliberative coalition that supports a proposal in \(T_q\): indeed, if an agent’s ideal point is in \(T_q\), then she does not approve proposals outside of \(T_q\).

Now, consider a terminal coalition structure \(\textbf{D}\) and suppose that there does not exist a deliberative coalition of size \(m^*\) supporting a proposal in \(T_q\). Then, there are multiple deliberative coalitions that support proposals in \(T_q\) and consist of agents whose ideal points are in \(T_q\). Let \((C, p')\) and \((C, p'')\) be two such coalitions. But then there is a merge transition in which coalition \((C'\cup C'', q)\) forms, a contradiction with \(\textbf{D}\) being a terminal coalition structure.

If \(X\ne M_T\), then we can use a similar argument; the only modification is that we have to define q by considering the ancestors of p (other than r) that belong to X, and, among these, pick one that is closest to r. \(\square \)

Note that, for trees, we do need the full power of merge transitions: follow or single-agent transitions are insufficient for successful deliberation. This holds even if all edges have the same weight, as demonstrated by the following example.

Example 6

Consider a tree T with vertex set \(U = \{r, u_1, u_2, u_3\}\) and edge set \(E = \{\{r, u_1\}, \{u_1, u_2\}, \{u_1, u_3\}\}\); the weight of each edge is 1. Let \(X=U\), and suppose that there are two agents with ideal points at \(u_2\) and \(u_3\), respectively. Consider a coalition structure where each of these agents forms a singleton coalition around her ideal point. Note that \(\rho (u_2, r)=\rho (u_3, r)=\rho (u_2, u_3)\) and hence the agents do not approve each other’s positions. Therefore, there are no follow transitions or single-agent transitions from this coalition structure, even though \(u_1\) is approved by both agents and therefore there is a merge transition where both agents form a coalition around \(u_1\).

In contrast, the next example shows that, for Euclidean spaces, merge transitions are insufficient for successful deliberation, even when combined with single-agent transitions.

Example 7

Consider a modification of Example 4, where we add an agent \(v_5\) at \((-4, 0)\) to \(\textbf{d}_1\) and an agent \(v_6\) at (4, 0) to \(\textbf{d}_2\); let \(\textbf{D}'\) be the resulting coalition structure. Note that \(v_5\) approves a, but does not approve p, thereby preventing a merge transition where agents in \(\textbf{d}_1\) and \(\textbf{d}_2\) form a coalition around p.

The same conclusion can be drawn from Example 5. Indeed, in that example each coalition contains an agent who does not approve of any proposals except for the one supported by her coalition (\(v_1\) does not approve p, b and c, \(v_6\) does not approve p, a and c, and \(v_7\) does not approve a, b and p). Hence, no merge transition is possible.

Examples 5 and 7 illustrate that, to reach a successful outcome, coalitions may need to leave some of their members behind when joining forces. The following definition formalizes this idea.

Definition 10

(Compromise Transition) A pair of coalition structures \((\textbf{D}, \textbf{D}')\) forms a compromise transition if there exist coalitions \(\textbf{d}_1, \textbf{d}_2\in \textbf{D}\), \(\textbf{d}'_1, \textbf{d}'_2, \textbf{d}'\in \textbf{D}'\) and a proposal p such that \(\textbf{D}{{\setminus }}\{\textbf{d}_1, \textbf{d}_2\}=\textbf{D}'{{\setminus }}\{\textbf{d}'_1, \textbf{d}'_2, \textbf{d}'\}\), \(\textbf{d}_1=(C_1, p_1)\), \(\textbf{d}_2=(C_2, p_2)\), \(\textbf{d}'=(C_1^p\cup C_2^p, p)\), \(\textbf{d}'_1=(C_1{{\setminus }} C_1^p, p_1)\), \(\textbf{d}_2=(C_2{{\setminus }} C_2^p, p_2)\), and \(|C_1^p\cup C_2^p|>|C_1|, |C_2|\).

Intuitively, under a compromise transition some of the agents in \(\textbf{d}_1\) and \(\textbf{d}_2\) identify a suitable proposal p, and then those of them who approve p move to form a coalition that supports p, leaving the rest of their old friends behind; a necessary condition for the transition is that the new coalition should be larger than both \(\textbf{d}_1\) and \(\textbf{d}_2\).

Example 8

In Example 5, there is a compromise transition from \(\textbf{D}\): agents \(v_2\), \(v_3\), \(v_4\) and \(v_5\) form a coalition around p, with \(v_1\) and \(v_6\) (as well as \(v_7\)) remaining in singleton coalitions. Similarly, in Example 7 there is a compromise transition from \(\textbf{D}'\) to the coalition structure \(\{\textbf{d}'_1, \textbf{d}'_2, \textbf{d}'\}\), where \(\textbf{d}'_1 = (\{v_5\}, a)\), \(\textbf{d}'_2 = (\{v_6\}, b)\), and \(\textbf{d}'=(\{v_1, v_2, v_3, v_4\}, p)\). In either case the new coalition has size 4, so the resulting coalition structure is successful.

Importantly, we assume that all agents in \(\textbf{d}_1\) and \(\textbf{d}_2\) who approve p join the compromise coalition; indeed, this is what we expect to happen if the agents myopically optimize the size of their coalition.

An important feature of compromise transitions is that they ensure termination.

Proposition 4

A deliberation that consists of compromise transitions can have at most \(n^n\) transitions.

Proof

Consider a coalition structure \(\textbf{D}=\{\textbf{d}_1, \ldots , \textbf{d}_m\}\) such that \(\textbf{d}_i=(C_i, p_i)\) for each \(i\in [m]\). Assume without loss of generality that \(|C_1|\ge \ldots \ge |C_m|\) and there exists an \(\ell \in [m]\) such that \(|C_i|>0\) for \(i\in [\ell ]\), \(|C_i|=0\) for \(i=\ell +1, \ldots , m\). Let \(\gamma (\textbf{D})=(|C_1|, \ldots , |C_\ell |)\). Note that \(\gamma (\textbf{D})\) is a non-increasing sequence of positive integers. Given two non-increasing sequences \((a_1, \dots , a_s)\), \((b_1, \dots , b_t)\) of positive integers we write

if either (a) there exists a \(j\le \min \{s, t\}\) such that \(a_i=b_i\) for all \(i<j\) and \(a_j<b_j\), or (b) \(s<t\), and \(a_i=b_i\) for all \(i\in [s]\). Note that \(<_{\textrm{lex}}\) is a total order on the space of non-increasing sequences of positive integers. Now, observe that if \((\textbf{D}, \textbf{D}')\) is a compromise transition, then for the respective coalitions \(\textbf{d}_1, \textbf{d}_2, \textbf{d}'\) we have \(|\textbf{d}'|>\max \{|\textbf{d}_1|, |\textbf{d}_2|\}\), and hence

Now, consider a sequence of deliberative coalition structures \(\textbf{D}_1, \textbf{D}_2, \ldots \) such that for each \(i\ge 1\) the pair \((\textbf{D}_i, \textbf{D}_{i+1})\) is a compromise transition. Since \(<_{\textrm{lex}}\) is transitive, it follows from Eq. 3 that \(\gamma (\textbf{D}_i)\ne \gamma (\textbf{D}_j)\) for each \(i<j\). As each \(\gamma (\textbf{D}_i)\) is a sequence of at most n numbers, with each number taking values between 1 and n, our claim follows. \(\square \)

In contrast to the case of single-agent transitions and follow/merge transitions, we are unable to show that a deliberation consisting of compromise transitions always terminates after polynomially many steps.^{Footnote 6} We note that a compromise transition does not necessarily increase the potential function \(\lambda (\cdot )\) given by Eq. (1): e.g., in Example 7 the transition where \(v_1, v_2, v_3, v_4\) form a coalition around p does not change the value of \(\lambda \)

We say that a deliberation space \((X, V, r, \rho )\) is a Euclidean deliberation space if \(X={\mathbb R}^d\), \(V\subseteq {\mathbb R}^d\) for some \(d\ge 1\), and \(\rho \) is the standard Euclidean metric on \({\mathbb R}^d\). The main result of this section is that, in every Euclidean deliberation space, every maximal run of compromise transitions is successful. To prove it, we need two auxiliary lemmas. In what follows, for a coalition structure \(\textbf{D}\), \(|\textbf{D}|\) denotes the number of non-empty coalitions in \(\textbf{D}\) that do not support r.

Lemma 1

In every deliberation space, a deliberation that consists of compromise transitions and has a coalition structure \(\textbf{D}\) with \(|\textbf{D}|=2\) as its terminal state is successful.

Proof

Consider a coalition structure \(\textbf{D}\) with \(|\textbf{D}|=2\), and suppose that \(\textbf{D}\) is not successful. Let \(\textbf{d}_1\), \(\textbf{d}_2\) be the two non-empty coalitions in \(\textbf{D}\) that do not support r; we have \(|\textbf{d}_1|<m^*\), \(|\textbf{d}_2|<m^*\). For each \(p\in M^*\) we have \(V^p=\textbf{d}_1^p\cup \textbf{d}_2^p\) and hence \(|\textbf{d}_1^p\cup \textbf{d}_2^p|=m^* >|\textbf{d}_1|, |\textbf{d}_2|\). Thus, there exists a compromise transition from \(\textbf{D}\) in which agents in \(\textbf{d}_1^p\cup \textbf{d}_2^p\) form a coalition around p. \(\square \)

Lemma 2

In every Euclidean deliberation space, a deliberation that consists of compromise transitions and has a coalition structure \(\textbf{D}\) with \(|\textbf{D}|\ge 3\) as its terminal state is successful.

Proof

For the case \(d=1\) our claim follows from the proof of Theorem 1. We will now provide a proof for \(d=2\); it generalizes straightforwardly to \(d>2\).

Fix a coalition structure \(\textbf{D}\) that contains at least three non-successful coalitions none of which support r; we will show that \(\textbf{D}\) is not terminal. Let \(\textbf{d}_a=(C_a, a)\) be a maximum-size coalition in \(\textbf{D}\) that does not support r. Let \(\ell \) be the perpendicular bisector to the a–r segment, i.e., \(\ell \) passes through the middle of the a–r segment and is orthogonal to it. Note that \(\ell \) separates \({\mathbb R}^2\) into two open half-planes, so that r lies in one of these half-planes, while all points in \(C_a\) lie in the other half-plane (see Fig. 5). Let \(\ell '\) be the line that passes through r and is parallel to \(\ell \). For a positive \(\alpha \), let \(\ell '_1\) be the line obtained by rotating \(\ell '\) about r clockwise by \(\alpha \), and let \(\ell '_2\) be the line obtained by rotating \(\ell '\) counterclockwise by \(\alpha \). The line \(\ell '_1\) (respectively, \(\ell '_2\)) partitions \({\mathbb R}^2\) into open half-planes \(H_1\) and \(H'_1\) (respectively, \(H_2\) and \(H'_2\)). We can choose \(\alpha \) to be small enough so that \(C_a\subset H_1, C_a\subset H_2\) and so that no agent lies on \(\ell '_1\) or on \(\ell '_2\).

Now, if there exists a coalition \(\textbf{d}_b=(C_b, b)\in \textbf{D}\), \(\textbf{d}_b\ne \textbf{d}_a\), \(b\ne r\), such that \(v\in H_1\) or \(v\in H_2\) for some \(v\in C_b\), then r is not in the convex hull of \(C_a\) and v, and hence there is a line that separates \(C_a\cup \{v\}\) from r; by projecting r onto this line, we obtain a proposal \(r'\) that is approved by v and all agents in \(C_a\). Thus, there is a compromise transition in which a non-empty subset of agents in \(C^{r'}_b\) joins \(C_a\) to form a deliberative coalition around \(r'\).

Otherwise, \(\textbf{d}_x\subseteq H'_1\cap H'_2\) for all \(\textbf{d}_x\in \textbf{D}{{\setminus }}\{\textbf{d}_a\}\). Consider two distinct coalitions \(\textbf{d}_b, \textbf{d}_c\in \textbf{D}\) with \(\textbf{d}_b=(C_b, b)\), \(\textbf{d}_c=(C_c, c)\) and \(b, c\ne r\). As \(H'_1\cap H'_2\) is bounded by two rays that start from r, and the angle between these rays is \(2\pi -2\alpha <2\pi \), there is a line \(\ell ^*\) that divides \({\mathbb R}^2\) into two open half-planes so that r is in one half-plane and the set \(C_b\cup \mathcal {C}_c\) is in the other half-plane; thus, all agents in \(C_b\cup C_c\) approve the proposal p obtained by projecting \(r''\) onto \(\ell ^*\), and hence there is a merge transition. \(\square \)

Proof of Lemma 2

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Combining Lemmas 1 and 2, we obtain the main result of this section.

Theorem 3

In every Euclidean deliberation space, every maximal run of compromise transitions is a successful deliberation.

Proof

Consider a maximal run of compromise transitions, and let \(\textbf{D}\) be its terminal state. Suppose for the sake of contradiction that \(\textbf{D}\) is not successful. Note that this implies that \(|\textbf{D}|>1\). If \(|\textbf{D}|=2\) there is a transition from \(\textbf{D}\) by Lemma 1 and if \(|\textbf{D}|\ge 3\), then there is a transition from \(\textbf{D}\) by Lemma 2. \(\square \)

For the proof of Lemma 2 to go through, the underlying metric space should be sufficiently rich: to obtain the proposal approved by the new coalition, we project the status quo r on a certain line. The argument goes through if we replace \({\mathbb R}^d\) with \({\mathbb Q}^d\); however, it does not extend to the case where X is an arbitrary finite subset of \({\mathbb R}^d\). Intuitively, for deliberation to converge, at least some agents should be able to spell out nuanced compromise proposals.

Observe that the compromise transitions in Lemma 2 have a special form: when two coalitions join forces, at least one of them is fully behind the new proposal. This motivates the following definition.

Definition 11

(Subsume Transition) A pair of coalition structures \((\textbf{D}, \textbf{D}')\) forms a subsume transition if there exist coalitions \(\textbf{d}_1, \textbf{d}_2\in \textbf{D}\), \(\textbf{d}'_1, \textbf{d}'\in \textbf{D}'\) and a proposal p such that \(\textbf{D}{{\setminus }}\{\textbf{d}_1, \textbf{d}_2\}=\textbf{D}'{{\setminus }}\{\textbf{d}'_1, \textbf{d}'\}\), \(\textbf{d}_1=(C_1, p_1)\), \(\textbf{d}_2=(C_2, p_2)\), \(C_2^p=C_2\), \(C_1^p\ne \emptyset \), \(\textbf{d}'=(C_1^p\cup C_2, p)\), \(\textbf{d}'_1=(C_1{{\setminus }} C_1^p, p_1)\), and \(|C_1^p\cup C_2|>|C_1|\).

By construction, every subsume transition is a compromise transition. Since every merge transition is also a subsume transition (with \(\textbf{d}'_1\) being empty), it follows that Lemma 2 holds for deliberations that consist of subsume transitions.

While subsume transitions by themselves are not sufficient for successful deliberation in \({\mathbb R}^d\) (as the transition in Lemma 1 is not necessarily a subsume transition), they are nearly sufficient: the proof of Theorem 3 shows that we need at most one general compromise transition. Since every subsume transition increases the potential defined by Eq. (1) by

we obtain the following corollary.

Corollary 1

For every Euclidean deliberation space there exists a successful deliberation that consists of compromise transitions and has length at most \(n^2+1\).

Proof

Suppose that agents perform subsume transitions until no such transitions are available; as every subsume transition increases the potential \(\lambda (\cdot )\) (see Eq. 1), this process ends after at most \(n^2\) steps. By Lemma 2, if the resulting coalition structure \(\textbf{D}\) is not successful, then \(|\textbf{D}|=2\), in which case, by Lemma 1, there is a compromise transition from \(\textbf{D}\) to a successful coalition structure. \(\square \)

Given our positive results for Euclidean deliberation spaces, it is natural to ask whether compromise transitions are sufficient for convergence in other metric spaces. The following example, however, illustrates that this is not the case even if X is a finite subset of \({\mathbb R}^d\).

The deliberation space in Example 9

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Example 9

Figure 6 depicts a deliberation space that is embedded in \({\mathbb R}^3\), so that \(\rho \) is the usual Euclidean metric in \({\mathbb R}^3\), and \(X=\{r, a, b, c, d, p, e\}\), \(V=\{v_1, \ldots , v_9\}\) with

Then \(\textbf{D}=\{\textbf{d}_1, \textbf{d}_2, \textbf{d}_3, \textbf{d}_4, \textbf{d}_5\}\), where \(\textbf{d}_1=(\{v_1, v_5\}, a)\), \(\textbf{d}_2=(\{v_2, v_6\}, b)\), \(\textbf{d}_3=(\{v_3, v_7\}, c)\), \(\textbf{d}_4=(\{v_4, v_8\}, d)\), \(\textbf{d}_5=(\{v_9\}, e)\) is a deliberative coalition structure, and there are no compromise transitions from \(\textbf{D}\). However, \(\textbf{D}\) is not successful, as agents \(v_5, \dots , v_9\) all approve p.

Thus, for general metric spaces one needs an even richer class of transitions to identify credible alternatives to the status quo.

We closed the previous section by showing that ‘sparse’ subspaces of Euclidean spaces may be challenging even for compromise transitions (Example 9). In this section we explore other ‘sparse’ spaces, but move away altogether from the Euclidean framework. Specifically, we focus on spaces that arise naturally in decision-making on combinatorial domains (Lang and Xia 2016): d-dimensional hypercubes. These consist of the set \(\{0, 1\}^d\) of binary vectors of length d (d being a positive integer), endowed with the discrete metric h, defined as

This metric is known as the Hamming, or Manhattan, distance. Intuitively, each element of \(\{0, 1\}^d\) denotes one possible position on a set of d binary issues. For any positive integer d, we refer to the space \((\{0, 1\}^d, h)\) as the d-dimensional hypercube or simply d-hypercube. In what follows, for readability, we will often write vectors in \(\{0, 1\}^d\) as strings of 1 s and 0 s of length d: e.g., we will write 0110 instead of (0, 1, 1, 0). Also, without loss of generality we will assume throughout that the status quo r is the all-0 vector.

8.1 Deliberation on d-Hypercubes

We start with a simple observation:

Proposition 5

In d-hypercubes with \(d \le 2\), every deliberation consisting of compromise transitions is successful.

Proof

For \(d = 1\) the claim holds trivially. Indeed, a terminal coalition structure cannot contain two non-empty coalitions that both support the same proposal \(a\ne r\) (as they could then merge and form a bigger coalition around a). Hence, since the deliberation space only contains one point other than \(r=0\), all agents that support 1 must be in the same coalition.

For \(d = 2\), assume towards a contradiction that there exists a terminal deliberative coalition structure \(\textbf{D}\) that is not successful. Let p be a most approved alternative, \(p\ne 00\), and suppose that p is approved by \(m^*\) agents. Note that we can choose \(p\ne 11\), as all agents who approve 11 also approve both 10 and 01; assume without loss of generality that \(p=10\).

Since \(\textbf{D}\) cannot contain two non-empty coalitions that support the same proposal \(a\ne r\), it contains at most three non-empty coalitions that do not support 00. Suppose first that \(\textbf{D}\) contains a deliberative coalition (C, 11) with \(C\ne \emptyset \). Note that all agents in C have 11 as their ideal point, as agents whose ideal point is 01 or 10 do not approve 11. Since \(|C|<m^*\), there must exist a coalition \((C', q')\) with \(C'\ne \emptyset \) and \(q'=01\) or \(q'=10\). But then there is a follow transition in which C joins \(C'\), so \(\textbf{D}\) is be terminal.

Hence, \(\textbf{D}\) contains two non-empty deliberative coalitions: \((C_{10}, 10)\) and \((C_{01}, 01)\). Moreover, \(C_{01}\) contains some agents whose ideal point is 11 (as otherwise \(C_{10}\) would contain all agents who approve \(p=10\), a contradiction with \(\textbf{D}\) not being successful). But then there is a compromise transition where all agents in \(C_{01}\) whose ideal point is 11 move to \((C_{10}, 10)\), to form a coalition of size \(m^*>|C_{01}|\), a contradiction. \(\square \)

Observe that the proof of Proposition 5 holds for subsume transitions: indeed, the proof shows that if a coalition structure \(\textbf{D}\) is not successful, then there is a subsume transition from it. However, we will now show that other types of transitions considered in our work do not necessarily result in successful deliberation on 2-hypercubes.

Remark 4

Deliberations that consist of single-agent, follow, or merge transitions may fail to be successful on 2-dimensional hypercubes. Indeed, suppose that \(r=00\), \(v_1=v_2=10\), \(v_3=01\), and \(v_4=v_5=11\). Note that 10 is approved by agents \(v_1\), \(v_2\), \(v_4\), and \(v_5\), 01 is approved by agents \(v_3\), \(v_4\), and \(v_5\), and 11 is approved by agents \(v_4\) and \(v_5\), so the most approved proposal is 10. Consider the deliberative coalition structure \((\textbf{d}_1, \textbf{d}_2)\), where \(\textbf{d}_1\) consists of \(v_1\) and \(v_2\), and supports 10, whereas \(\textbf{d}_2\) consists of \(v_3\), \(v_4\), and \(v_5\), and supports 01. There are no single-agent, follow or merge transitions from this coalition structure, even though 01 is not the most approved project.

In higher dimensions, however, compromise transitions no longer guarantee success.

Example 10

(Compromise failure on 3-hypercube) Consider the 3-hypercube \(\{ 0,1 \}^3\) and let N consist of the following agents: \(v_1 = 100\), \(v_2 = 010\), \(v_3 = 110\), \(v_4 = 001\), \(v_5 = 101\) (see Fig. 7). Let \(\textbf{D}\) be the deliberative coalition structure that contains the following three coalitions: \((\{ v_1 \}, 100)\), \((\{ v_2, v_3 \}, 010)\), and \((\{ v_4, v_5 \}, 001)\). This coalition structure is terminal. It is not successful, however, as \(v_1\), \(v_3\) and \(v_5\) approve 100.

Consequently, single-agent, follow, merge and subsume transitions also fail to guarantee success in the above setting. However, a careful inspection of Example 10 suggests that a successful outcome would be within reach if compromises that involve three coalitions were allowed. This motivates us to study a natural multi-party generalization of compromise transitions.

The deliberative space in Example 10

Full size image

8.2 Multi-party compromises

Definition 12

(\(\ell \)-Compromise Transition) Given an integer \(\ell \ge 2\), we say that a pair of coalition structures \((\textbf{D}, \textbf{D}')\) forms an \(\ell \)-compromise transition if there exist coalitions \(\textbf{d}_1, \ldots , \textbf{d}_\ell \in \textbf{D}\), \(\textbf{d}'_1, \ldots , \textbf{d}'_\ell , \textbf{d}' \in \textbf{D}'\) and a proposal p such that, for all \(1 \le i \le \ell \) it holds that \(\textbf{D}{{\setminus }}\{\textbf{d}_1, \ldots , \textbf{d}_\ell \} = \textbf{D}' {{\setminus }} \{ \textbf{d}'_1, \ldots , \textbf{d}'_\ell , \textbf{d}' \}\), \(\textbf{d}_i = (C_i, p_i)\), \(\textbf{d}' = (\bigcup _{1 \le i \le \ell } C_i^p, p)\), \(\textbf{d}'_i = (C_i{{\setminus }} C_i^p, p_i)\), and \(|\bigcup _{1 \le i \le \ell } C_i^p|>|C_i|\).

Accordingly, a compromise transition (Definition 10) is then a 2-compromise transition. Note that our definition of an \(\ell \)-compromise does not require that \(C_i^p\ne \emptyset \) for all \(i\in [\ell ]\), i.e., some coalitions \(C_i\) may actually contribute an empty set \(C_i^p\) of agents to the newly formed coalition (though for \(\ell = 2\) this cannot happen). Hence, for every \(\ell , \ell '\) with \(2\le \ell <\ell '\le |\textbf{D}|\), it holds that an \(\ell \)-compromise transition \((\textbf{D}, \textbf{D}')\) is also an \(\ell '\)-compromise transition, as any coalition C not involved in the \(\ell \)-compromise trivially contributes \(\emptyset \) to the newly formed coalition.

Recall now the total order \(<_{\textrm{lex}}\) defined in Eq. (2). Observe that if \((\textbf{D}, \textbf{D}')\) is an \(\ell \)-compromise transition with \(\ell \ge 2\) then \(\gamma (\textbf{D})<_{\textrm{lex}}\gamma (\textbf{D}')\), because the newly formed coalition is strictly larger than all coalitions that have contributed to it. Consequently, all coalition structures that arise in a deliberation that consists of \(\ell \)-compromise transitions are pairwise distinct, and hence the length of such deliberation is bounded by \(n^n\).

Corollary 2

For each \(\ell \ge 2\), a deliberation that consists of \(\ell \)-compromise transitions can have at most \(n^n\) transitions.

At the extreme, if the agents can engage in n-compromises, then a successful coalition structure can be reached in one step, by having all agents who approve a proposal \(p\in M^*\) form a coalition around p. However, we are interested in outcomes that can be achieved when each interaction only involves a few coalitions. Thus, given a deliberation space, we are interested in finding the smallest value of \(\ell \) such that \(\ell \)-compromise transitions can reach a successful outcome. Formally, let \(\ell ^*(d)\) be the smallest value of \(\ell \) such that in a d-hypercube a deliberation that consists of \(\ell \)-compromise transitions is necessarily successful.

We will now establish some lower and upper bounds on \(\ell ^*(d)\), which hold for all \(d\ge 3\). The proofs are rather technical, so they are relegated to Appendix A. We note that, after the conference version of our paper has been published, Elkind et al. (2022) strengthened the lower bound on \(\ell ^*(d)\) (Proposition 6) to \(2^{\varTheta (d)}\).

Proposition 6

For each \(d\ge 3\) we have \(\ell ^*(d)\ge d\).

It is instructive to contrast Proposition 6 with the earlier Theorem 3: While d-Euclidean spaces are ‘rich’ enough for 2-compromises to guarantee success in any dimension, the d-hypercubes are ‘sparse’, so we need to facilitate interactions among at least d coalitions to guarantee success.

For the upper bound, an easy observation is that \(\ell ^*(d)\le 2^d\): this is because in a terminal coalition structure there cannot be two distinct coalitions supporting the same proposal. We will now show how to strengthen this bound by (almost) a factor of two.

Proposition 7

For each \(d\ge 2\) we have \(\ell ^*(d) \le 2^{d-1}+\frac{d+1}{2}\).

In what follows, we compute the exact value of \(\ell ^*(d)\) for \(d=3\) and \(d=4\). For \(d=3\), the lower bound of Proposition 6 turns out to be tight; however, for \(d=4\) this is no longer the case.

Proposition 8

We have \(\ell ^*(3)=3\) and \(\ell ^*(4)=5\).

While the lower bound shown by Elkind et al. (2022) and the upper bound of Proposition 7 are quite close, it remains an open problem to fully close the gap between them.

We proposed a formal model of deliberation for agent populations forming coalitions around proposals in order to change the status quo. We identified several natural modes of coalition formation, capturing the many coalitional effects that deliberative processes could support. We studied sufficient conditions for them to succeed in identifying maximally approved proposals. We intend our model as a foundation for the study of mechanisms and systems allowing communities to self-govern.

To the best of our knowledge, ours is the first model of deliberation focusing on iterative processes of coalition formation. The model lends itself to several avenues for future research.

As we have mentioned earlier in the paper, several technical challenges left open by our work, such as upper and lower bounds on the length of a sequence of compromise transitions and the values of \(\ell ^*(d)\), have been addressed by the follow-up work of Elkind et al. (2022). However, there are many conceptual questions that have not been answered yet.

In particular, one can consider natural extensions of the model, such as other transition types or other types of metric spaces. Specifically, for each type of deliberation (as captured by allowable transitions), it is important to understand exactly how ‘rich’ a space should be in order to support deliberation success. One can also ask whether our positive results are preserved if we impose additional conditions on the structure of the deliberative process, e.g., require new proposals to be ‘close’ to the original proposals. We may also revisit our approach to modeling coalitions that support the status quo: we now assume that each agent v with \(X^v\ne \emptyset \) is capable of identifying some proposal in \(X^v\), and this assumption may be too strong for many deliberation scenarios. It is perhaps more realistic to assume that some agents start out by supporting the status quo, but then they learn about a new proposal p that they prefer to the status quo by observing a coalition that supports p, and subsequently move to join this coalition. A formal model that can capture this form of preference discovery is a challenge for future work.

Further afield, it would be interesting to consider a stochastic variant of our model, in which each transition from a state is assigned a certain probability. A Markovian analysis of such systems might shed further light on the behavior of deliberative processes.

Crucially, in our current model agents are cooperative and non-strategic: they share a motivation to form larger coalitions and they truthfully reveal whether they approve a given proposal. Yet, agents may prefer to form a smaller group around proposals they strongly prefer to being in a larger coalition that supports a proposal they are lukewarm about. Furthermore, agents may be strategic about which proposals to support. This calls for exploring game-theoretic extensions of our framework.

Yet another ambitious direction for future work is to design a practical tool for deliberation and self-governance of an online community that incorporates the insights from our analysis. Such a tool could take the form of an AI bot over existing on-line deliberation platforms such as the LiquidFeedback and Polis platforms mentioned earlier in the paper. The bot would suggest proposals to agents in order to engender compromise, hopefully fostering successful deliberations.

In what follows, given a proposal \(p = (i_1, \dots , i_d)\in \{0, 1\}^d\) and a coalition structure \(\textbf{D}\), we denote the deliberative coalition in \(\textbf{D}\) that supports p by \(\textbf{d}_p=(C_p, p)\); also, if an agent’s ideal point is p, we will say that this agent is of type p. We will refer to the quantity \(i_1+\dots +i_d\) as the weight of proposal p.

Proof

(Proposition 6) Fix some \(d\ge 3\). We prove the claim by constructing a coalition structure in the d-hypercube that is terminal for \((d-1)\)-compromise transitions, but not successful. The construction generalizes Example 10.

For each \(i\in [d]\), let \(e_i\) be the point in the hypercube that has 1 in the i-th coordinate and 0 in all other coordinates. Also, for each \(i\in [d]{\setminus }\{1\}\), let \(f_i\) be the point in the hypercube that has 1 in coordinates 1 and i, and 0 in all other coordinates.

We create one agent of type \(e_1\), and place her into a singleton coalition that supports \(e_1\). For each \(i=2, \ldots , d\) we create \(d-2\) agents of type \(e_i\) and one agent of type \(f_i\), and place them into a coalition that supports \(e_i\); this coalition has \(d-1\) members. Let \(\textbf{D}\) be the resulting coalition structure.

Note that agents of type \(e_i\), \(i\in [d]\), do not approve any proposals other than \(e_i\). An agent of type \(f_i\) for some \(i\in [d]{{\setminus }}\{1\}\) approves \(f_i\), \(e_1\) and \(e_i\). Hence, \(e_1\) is the most-approved proposal, as it is approved by d agents. However, there are no \(\ell \)-compromise transitions from \(\textbf{D}\) for \(\ell <d\): unless each coalition in \(\textbf{D}\) contributes an agent to a coalition around \(e_1\), the resulting coalition would have at most \(d-1\) members, and hence it would not be attractive to agents of type \(f_i\) with \(i>1\). \(\square \)

The proofs of Propositions 7 and 8 make use of the two lemmas stated and proved below.

Lemma 3

In every deliberation space, every deliberation that consists of \(\ell \)-compromise transitions and terminates in a coalition structure \(\textbf{D}\) with \(|\textbf{D}| = \ell \), is successful.

Proof

Consider a coalition \(\textbf{D}\) with \(|\textbf{D}|= \ell \) that is not successful. Suppose first that \(\textbf{D}\) does not contain a coalition that supports r. Let \(\textbf{d}_1\), ..., \(\textbf{d}_\ell \) be the \(\ell \) coalitions in \(\textbf{D}\). We have \(|\textbf{d}_i|<m^*\) for \(i\in [\ell ]\). For each \(p \in M^*\) we have \(V^p = \bigcup _{i\in [\ell ]} \textbf{d}_i^p\) and hence \(|\bigcup _{i\in [\ell ]} \textbf{d}_i^p|=m^* >|\textbf{d}_i|\). Thus, by Definition 12, there exists an \(\ell \)-compromise transition from \(\textbf{D}\) in which agents in \(\bigcup _{i\in [\ell ]} \textbf{d}_i^p\) form a coalition around p. If \(\textbf{D}\) contains a coalition that supports r, the same argument applies with \(\ell -1\) coalitions. \(\square \)

Lemma 4

Let \(\textbf{D}\) be a terminal coalition structure in the d-hypercube with respect to \(\ell \)-compromises for some \(d\ge 2\), \(\ell \ge 2\). Then for each point p in the d-hypercube it holds that \(\textbf{D}\) contains at most one deliberative coalition that contains agents of type p.

Proof

Suppose \(\textbf{D}\) contains two deliberative coalitions \(\textbf{d}_1\), \(\textbf{d}_2\) that contain agents of type p. We can assume without loss of generality that \(|\textbf{d}_1|\ge |\textbf{d}_2|\). But then the agents of type p approve the proposal supported by \(\textbf{d}_1\), so the agents of type p in \(\textbf{d}_2\) have an incentive to move to \(\textbf{d}_1\), a contradiction with \(\textbf{D}\) being terminal. \(\square \)

Proof

(Proposition 7) The claim is clearly true for \(d=2\), so we can assume \(d\ge 3\). Consider a deliberative coalition structure \(\textbf{D}\) that does not admit any \(\ell \)-transitions for \(\ell =2^{d-1}+\frac{d+1}{2}\). We say that a coalition \(\textbf{d}=(C, p)\) in \(\textbf{D}\) is pure if all agents in C have the same type. We claim that \(\textbf{D}\) can contain at most \(d+1\) pure coalitions.

Indeed, suppose that \(\textbf{D}\) contains \(d+2\) pure coalitions. By the pigeonhole principle, there are two pure coalitions \(\textbf{d}=(C, p)\) and \(\textbf{d}'=(C', p')\) in \(\textbf{D}\) such that all agents in \(\textbf{d}\) are of type \(i_1i_2\dots i_d\), all agents in \(\textbf{d}'\) are of type \(j_1j_2\dots j_d\), and there exists an \(s\in [d]\) such that \(i_s=j_s=1\). Let \(e_s\) be a point in the hypercube that has 1 in position s and 0 in all other positions. Then there is a transition in which all agents in C, all agents in \(C'\) and all agents currently supporting s form a coalition that supports s; this is a contradiction, as this transition only involves 3 coalitions, and \(2^{d-1}+\frac{d+1}{2}>3\) for \(d\ge 3\).

Thus, all but \(d+1\) coalitions in \(\textbf{D}\) are not pure, and hence contain agents of at least two different types. Further, by Lemma 4, for each \(p\in \{0, 1\}^d\) it cannot be the case that two coalitions in \(\textbf{D}\) contain agents of type p. Hence, if \(\textbf{D}\) contains t pure coalitions, it contains at most \((2^{d}-t)/2\) non-pure coalitions, i.e., at most \(2^{d-1}+t/2\) coalitions altogether; as \(t\le d+1\), our bound on \(\ell ^*(d)\) now follows from Lemma 3. \(\square \)

Proof

(Proposition 8) We partition the proof into four claims.

Claim

\(\ell ^*(3)\ge 3\).

Proof

This follows immediately from Proposition 6. \(\square \)

Claim

\(\ell ^*(3)\le 3\).

Proof

Let \(\textbf{D}\) be a terminal coalition structure, and suppose that \(\textbf{D}\) is not successful.

Let \(\mathcal {C}=\{C_{110}, C_{101}, C_{011}, C_{111}\}\). Note that all agents in \(C_{110}\) are of type 110 or 111, and similarly for \(C_{101}\) and \(C_{011}\). On the other hand, \(C_{111}\) may contain agents of type 111, 110, 101 or 011.

Suppose first that at least two of the coalitions in \(\mathcal C\) are not empty. Then these two coalitions have a merge transition (which is also a 3-compromise transition) in which they form a coalition around 111, a contradiction with \(\textbf{D}\) being a terminal coalition structure. On the other hand, if all coalitions in \(\mathcal C\) are empty, then \(|\textbf{D}|=3\), and we obtain a contradiction by Lemma 3.

Thus, we can focus on the case where exactly one coalition in \(\mathcal C\)—say, C—is non-empty. Lemma 3 then implies that each of the coalitions \(C_{001}\), \(C_{010}\), and \(C_{100}\) is non-empty. As argued above, if there is an agent in C of type (i, j, k) then \(i+j+k\ge 2\). Now, if C contains no agent of type 110, then all agents in C approve 001 and hence there is a follow transition where C joins \(C_{001}\), a contradiction with \(\textbf{D}\) being a terminal coalition. Similarly, if C contains no agent of type 101, then C can join \(C_{010}\), and if C contains no agent of type 011, then C can join \(C_{100}\). We conclude that C contains agents of types 110, 101, and 011; hence, it has to be the case that C supports 111, i.e., \(C=C_{111}\).

Now, suppose that \(|C_{111}|\le |C_{ijk}|\) for some i, j, k with \(i+j+k=1\); assume without loss of generality that \(|C_{111}|\le |C_{001}|\). We have already argued that \(C_{111}\) contains some agents of type 011; note that these agents approve 001. Hence there is a subsume transition in which all agents in \(C_{111}\) who approve 001 move to 001, a contradiction with \(\textbf{D}\) being a terminal coalition structure.

The remaining possibility is that \(|C_{111}|> \max \{|C_{001}|, |C_{010}|, |C_{100}|\}\). In this case, if for some i, j, k with \(i+j+k=1\) the coalition \(C_{ijk}\) contained some agents who approved 111, then there would be a subsume transition in which all these agents would move to 111. Thus \(C_{111}\) already contains all agents who approve 111. As we assume that \(\textbf{D}\) is not successful, we have \(|C_{111}|<m^*\) and hence \(111\not \in M^*\). It follows that 110, 101, and 011 are not in \(M^*\) either, as every agent who approves one of these proposals also approves 111. Moreover, as \(C_{111}\) contains all agents that approve 111, each coalition \(C_{ijk}\) with \(i+j+k=1\) consists exclusively of agents of type (i, j, k). Thus, if, say, 100 is in \(M^*\), then all agents who approve 100 are either in \(C_{100}\) or in \(C_{111}\), and hence there is a follow transition in which all these agents join forces around 100, forming a coalition of size \(m^*\), a contradiction with \(\textbf{D}\) being terminal. \(\square \)

Claim

\(\ell ^*(4)\ge 5\).

Proof

Consider a coalition structure \(\textbf{D}= \{\textbf{d}_{1000}, \textbf{d}_{0100}, \textbf{d}_{0010}, \textbf{d}_{0001}, \textbf{d}_{1110}\}\), where

• \(C_{1000}\) contains 21 agents of type 1000 and 10 agents of type 1001;

• \(C_{0100}\) contains 21 agents of type 0100 and 10 agents of type 0101;

• \(C_{0010}\) contains 21 agents of type 0010 and 10 agents of type 0011;

• \(C_{0001}\) contains one agent of type 0001; and

• \(C_{1110}\) contains 30 agents of type 1101, 30 agents of type 1011, 30 agents of type 0111, 10 agents of type 1100, 10 agents of type 1010, and 10 agents of type 0110.

One can verify that each agent approves the proposal supported by her deliberative coalition.

There are 121 agents who approve 0001: this includes the one agent in \(C_{0001}\), 10 agents from each of the coalitions \(C_{1000}\), \(C_{0100}\) and \(C_{0001}\), and 90 agents from \(C_{1110}\). For these agents to gather at 0001, all five coalitions need to be involved. In particular, it cannot be the case that only the agents of types 1001, 0101 and 0011 move to 0001, as this would result in a coalition of size 31, which is equal to the size of their current coalitions.

It remains to argue that \(\textbf{D}\) is stable with respect to transitions that involve at most four coalitions. The argument in the previous paragraph already shows that every such transition should involve agents in \(C_{1110}\).

For each \(s=1, 2, 3, 4\) we will argue that there is no transition in which agents move to a point of weight s.

• \(s=4\): It is immediate that there is no transition to the unique point of weight 4, i.e., 1111, as all agents who approve 1111 are currently in \(C_{1110}\).

• \(s=3\): No agent not currently in \(C_{1110}\) approves 1110, so there is no transition in which some agents move to 1110. Now, consider a point of weight 3 that differs from 1110; for concreteness, take 0111 (the remaining two cases can be analyzed in the same way). There are 100 agents in \(C_{1110}\) that approve 0111, as well as 10 agents in \(C_{0100}\) and 10 agents in \(C_{0011}\). Thus, the coalition at 0111 would contain 120 agents, which is equal to the size of \(C_{1110}\). Therefore, agents in \(C_{1110}\) would not benefit from this transition.

• \(s=2\): For every proposal of weight 2, there are at most 60 agents in \(C_{1110}\) who approve this proposal and at most 10 other agents who approve it; as \(|C_{1110}|>70\), there is no transition to a point of weight 2 that is attractive to agents in \(C_{1110}\).

• \(s=1\): There are 80 agents in \(C_{1110}\) who approve 1000; these agents could move to 1000, but this would result in a coalition of size 111, which is smaller than their current coalition. Hence, a transition in which agents who approve 1000 move to 1000 is not possible; for a similar reason, there is no transition where agents who approve 0100 or 0010 move to the respective points.

\(\square \)

Claim

\(\ell ^*(4)\le 5\).

Proof

Let \(\textbf{D}\) be a terminal coalition structure with respect to 5-compromise transitions, and suppose for the sake of contradiction that \(\textbf{D}\) is not successful. By Lemma 3, \(\textbf{D}\) contains at least six coalitions.

Consider an agent who approves 1111. Then her ideal point has weight 3 or 4. Consequently, this agent approves all proposals of weight 3 or 4: indeed, her distance to 0000 is at least 3, while her distance to any proposal of weight at least 3 is at most 2. It follows that if \(\textbf{D}\) contains a coalition \(\textbf{d}_{1111}= (C_{1111}, 1111)\) then it does not contain any coalitions that support a proposal of weight 3: indeed, the ideal point of each agent in \(C_{1111}\) has weight 3 or 4, so if there is another coalition \(C'\) that supports a proposal p of weight 3, then there is a follow transition in which \(C_{1111}\) and \(C'\) merge around p.

Furthermore, \(\textbf{D}\) can contain at most two coalitions that support proposals of weight at least 3. Indeed, assume for the sake of contradiction that there are three coalitions in \(\textbf{D}\) that support proposals of weight at least 3. As argued in the previous paragraph, it cannot be the case that one of them supports 1111, so we can assume without loss of generality that \(\textbf{D}\) contains coalitions \(\textbf{d}_{1110}= (C_{1110}, 1110)\), \(\textbf{d}_{1101}= (C_{1101}, 1101)\), and \(\textbf{d}_{1011}= (C_{1011}, 1011)\), with \(|C_{1110}|\ge |C_{1101}|\ge |C_{1011}|\). Then \(C_{1011}\) can only contain agents of type 0011: all other agents who approve 1011 also approve either 1110 or 1101, so they would have an incentive to move to the weakly larger coalitions that support these proposals. Similarly, \(C_{1101}\) can only contain agents of type 1001 or 0101, as all other agents who approve 1101 also approve 1110, so they can move to the weakly larger coalition \(C_{1110}\). But then all agents in \(C_{1101}\) and \(C_{1011}\) approve 0001, so there is a 3-compromise in which all agents in \(C_{1101}\), \(C_{1011}\) and \(C_{0001}\) merge around 0001, a contradiction.

Thus, we have argued that \(\textbf{D}\) contains at most two coalitions that support proposals of weight 3 or higher. Now, consider coalitions in \(\textbf{D}\) that support proposals of weight at most 2.

Suppose \(\textbf{D}\) contains a coalition \(\textbf{d}_{1100}= (C_{1100}, 1100)\). Note that all agents in \(C_{1100}\) approve 1000 and 0100. Thus, if \(\textbf{D}\) contains a coalition that supports 1000 or 0100, then there would be a follow transition in which all agents in \(C_{1100}\) adopt the proposal of this coalition, a contradiction with \(\textbf{D}\) being a terminal coalition structure. Similarly, if \(\textbf{D}\) contains a coalition \(\textbf{d}_{1010}\) or \(\textbf{d}_{1001}\), there would be a merge transition in which \(C_{1100}\) merges with this coalition around 1000, and if \(\textbf{D}\) contains a coalition \(\textbf{d}_{0110}\) or \(\textbf{d}_{0101}\), there would be a merge transition in which \(C_{1100}\) merges with this coalition around 0100.

We conclude that if \(\textbf{D}\) contains \(\textbf{d}_{1100}\) then it contains at most two other coalitions that support proposals of weight at most 2: namely, it may contain \(\textbf{d}_{0011}\) (in which case it contains no coalitions that support a proposal of weight 1), or it may contain one or both of the coalitions \(\textbf{d}_{0010}\) and \(\textbf{d}_{0001}\). By symmetry, it follows that if \(\textbf{D}\) contains a coalition that supports a proposal of weight 2, then it contains at most three coalitions that support proposals of weight at most 2. As we have argued that \(\textbf{D}\) contains at most two proposals of weight at least 3, we obtain a contradiction with \(|\textbf{D}|\ge 6\). Hence, we may assume that \(\textbf{D}\) contains no coalitions that support a proposal of weight exactly 2.

Thus, if \(|\textbf{D}|\ge 6\), it must be the case that \(\textbf{D}\) contains four coalitions supporting proposals of weight 1, as well as two coalitions supporting proposals of weight at least 3. We have argued that among these two coalitions, one (a weakly smaller one) only contains agents of two types of weight 2 each; moreover, our analysis shows that all agents in this coalition approve some proposal p of weight 1. But then there is a follow transition in which this coalition joins the coalition that currently supports p, a contradiction again. Thus, we can conclude that \(\textbf{D}\) is successful.

Proposition 8 now follows by combining Claims 1–4. \(\square \)