1 Introduction
Egalitarian decisionmaking requires not only equality in voting on alternatives, but also equality in deciding on which alternatives to vote upon. Social choice theory typically focuses on the former at the expense of the latter. The process of defining the alternatives to vote upon is often deliberative, even though it is often informal and/or not egalitarian, being driven by agents with special functions or authority. Agents participating in such collective decision mechanisms make proposals, deliberate over them, and join coalitions to push their proposals through.
Here we consider an agent population wishing to change the status quo. A natural approach would be to accept a new proposal to change the status quo as soon as such a proposal enjoys the support of an agent majority. But how can such an agent majority be established, and how can a suitable proposal be identified? We present a deliberative decision making process in which agents are equal not only as voters but also as proposers. Proposals as well as the status quo are elements of some, possibly infinite, abstract space. All voters are required to be able to articulate if they prefer a given proposal over the status quo; also, at least some—and possibly all—voters need to be able to articulate new proposals. Agents form coalitions around proposals that they prefer over the status quo (or the status quo itself). Specifically (see Section 2 for formal definitions), our setting consists of a space of proposals , one of which is the status quo , and a set of agents, each initially supporting a proposal (possibly the status quo).
Our focus is on coalitions of agents that may form based on communication and deliberation among the agents. Following such deliberation, agents can then join forces to create larger coalitions by merging two coalitions around a new compromise proposal, possibly leaving dissenting agents behind. The goal of this process is to select a majoritysupported proposal. Thus, the deliberation and coalition formation process terminates once some proposal reaches an agent majority. This proposal is then selected, resulting in the agent population success in changing the status quo.
Contribution
Many social choice settings are naturally embedded in a metric space. For instance, in the analysis of singlewinner elections it is common to assume that voters and alternatives are points on a line, and preferences are induced by Euclidean distances [4, 15]. Thus, we first consider a model in which the space of proposals is an amicable space: this is a special class of metric spaces, which includes Euclidean spaces.
We show that, for amicable spaces, a very simple deliberation process always results in a success: Deliberation terminates with a majority coalition, as long as a majoritysupported proposal exists. We then consider a more general space of proposals in which our deliberation process may fail, and study a more complex deliberative process under which the guarantee of success can be recovered. For completeness, we describe also a centralized solution, based on dynamic programming, that identifies a majoritysupported proposal if one exists, provided that all agents can articulate all the proposals they prefer over the status quo and that such sets of proposals have succinct representations.
We view our work as an important step towards modeling a form of prevote deliberation that is usually not studied within the social choice literature: how voters can identify majoritysupported proposals. Our goal is to provide the foundation for the design and development of practical systems that can support egalitarian deliberative decisionmaking, for instance by helping agents to identify mutually beneficial compromise positions (cf. [6]).
Related Work
Works on group deliberation have appeared in several disciplinary areas, from economics, to social choice and artificial intelligence. Deliberation via sequential interaction in small groups has been studied
[12, 8] as a means to implement largescale decisionmaking when alternatives form a median graph. Our work is not based on interaction in small groups, and focuses on how majority proposals can be identified in a decentralized way.In social choice theory and economics, papers have developed axiomatic [14], experimental [13], as well as gametheoretic approaches to deliberation. The latter have focused, for instance, on persuasion [9, 10, 11] or on the way in which deliberation interacts with specific voting rules in determining group decisions [1]. The interaction between deliberation and voting has also been recently investigated by karanikolas2019voting [karanikolas2019voting] using tools from argumentation theory. We, however, abstract from the concrete interaction mechanism by which agents discuss the proposals themselves, as well as on how deliberation might interact with specific voting rules, and concentrate on the results of such interactions, as they are manifested by changes in the structure of coalitions and the preferences of the agents.
The fact that we abstract from strategic issues that agents may be confronted with in deciding to join or leave coalitions differentiates our work also from related literature on dynamic coalition formation [7, 3].
Finally, unlike influential opinion dynamics models (e.g., [5]), in our model deliberation is driven by compromise rather than by influence or trust.
2 Coalition Formation by Compromise
We view deliberation as a process in which agents aim to find an alternative preferred by a majority of agents over the status quo. To achieve this, agents are willing to (i) join a coalition that supports a proposal they prefer over the status quo (ii) shift their support to another proposal in order to gain strength in numbers, i.e., form a larger coalition around a compromise proposal (iii) change their mind (i.e., change their preferences over the status quo) in the face of a new idea or a new argument.
We assume a domain, or space , of alternatives, or proposals, that includes the status quo (reality) , and a set of agents. We make a very weak assumption on the capabilities of all agents: For each proposal an agent is able to articulate whether she (strictly) prefers over the status quo (denoted as ). When we say that supports . For each , we let ; the set is the approval set of , i.e., the set of proposals that supports.
2.1 Coalitions and Coalition Structures
Definition 1 (Supporting Agents).
Given a subset of voters and a proposal , let ; the agents in are the supporters of in .
Definition 2 (Deliberative Coalition).
A deliberative coalition over and is a pair where and . We refer to as the supported proposal of . The set of all deliberative coalitions is denoted by .
That is, a deliberative coalition is a set of agents and a proposal that is approved by all agents within the coalition. When convenient, we identify a coalition with its set of agents and write , .
Definition 3 (Deliberative Coalition Structure).
A deliberative coalition structure (coalition structure for short) over and is a set , with , such that:

for each ;

, for all .
Given a coalition structure , we refer to as the set of proposals of and to as the partition of induced by . We say that is atomic if each coalition in is a singleton (i.e., if ). The set of all deliberative coalition structures over and is denoted by .
Example 1.
Consider a set of agents and a space of proposals . Suppose that , , and . Then for we have and . Further, let , , and let , . Then is a deliberative coalition structure.
2.2 Deliberative Transition Systems
We model deliberation as a process whereby deliberative coalition structures change as their constituent coalitions change. We provide general definitions for deliberative processes (Definitions 47), propose compromise transitions (Definition 8) and regret transitions (Definition 10), and explore their induced transition systems.
2.2.1 Successful Deliberations
A transition system is characterized by a set of states , a subset of which are initial states, and a set of transitions , where each transition is represented by a pair of states ; we write and interchangeably. We use to denote that for some . A run of a transition system is a (finite or infinite) sequence . Such a run is initialized if .
Definition 4 (Deliberative Transition System).
A deliberative transition system (over and ) is a transition system that has as its set of states, a subset of states as its set of initial states ( need not contain the atomic coalition structure), and a set of transitions . Runs are denoted by letter .
Definition 5 (Deliberation).
A deliberation is a maximal run, that is, a run that does not occur as a prefix of any other run (i.e., cannot be extended).
We are interested in successful deliberations, i,.e., deliberations that end in a majoritysupported proposal. A successful deliberation process allows the agent population to identify a proposal that can then result in a majoritysupported change to the status quo. More generally (and even if there are no majoritysupported proposals), we wish our deliberative systems to find mostsupported proposals, as captured by the following definitions.
Definition 6 (MostSupported Alternatives).
For a set of agents and a set of alternatives , the set of mostsupported alternatives is and the maximum support is . If and , then every is a consensus.
Note that and for every .
Definition 7.
A coalition is successful if and . A 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 supporters of a mostsupported proposal. Successful deliberations are maximal finite runs that lead to a successful coalition structure.
Example 2.
Consider a toy example of a deliberative transition system that consists of (1) a single initial coalition structure , with where , where , and where ; and (2) the set of transitions only containing a transition from to , with where , where . Then, assuming that there is no proposal approved by all agents, we have that is a successful deliberation.
2.2.2 Compromise Transitions
Next we consider a specific type of transition, named compromise transition, that would instantiate our first deliberative system. The intuition behind compromise transitions, defined below, is that agents are willing to join and form larger coalitions, even if at the expense of supporting proposals that are not their mostpreferred ones, as they are aware that only majoritysupported proposals will be selected to change the status quo. Note that these transitions are quite simple, in the sense that (1) we do not specify how the compromise proposal is identified and (2) we assume that agents from the original coalitions would always move to the new compromise coalition, given that they prefer over the status quo; we discuss more refined transitions later.
Definition 8 (Compromise Transition).
Let be a coalition structure on and . A triple is a valid compromise for if , and the compromise coalition with satisfies . Given a triple that is a valid compromise for , let , ; we refer to agents in and as noncompromising agents, and refer to the transition , where as a compromise transition. We also say that is the result of applying compromise to (in symbols: ), and call the proposal the compromise proposal. The set of valid compromises for is denoted by .
Intuitively, a compromise transition takes two coalitions and a compromise proposal , and forms a larger compromise coalition around , possibly leaving behind some members of and that do not support . Intuitively, a compromise transition covers two realworld scenarios relevant in deliberative contexts: One, in which coalitions and negotiate and tentatively agree on , following which they confirm that sufficiently many members of and support and hence would join the new coalition, in which case is formed. Another, in which members of one coalition (say, ), unilaterally defect to another coalition , since they all support and is larger than (or becomes lager than after the defection). In this scenario the proposal of prevails, , and hence has no defectors, .
Remark 1.
Agents need not reveal all their preferences upfront, but need only to articulate their preferences over the status quo when presented with an alternative. Agent preferences may even be cyclic. However, members of a coalition cannot defect to another coalition (possibly completing a cycle) just because they feel that its supported proposal has more advantages over the status quo than the proposal of their current coalition; such defection constitutes a compromise transition only if the resulting coalition is larger than and . Thus, as each agent initially supports the proposal of the coalition she is in, once an agent is in a coalition that supports a proposal different than the status quo, she will remain in a coalition with a supported alternative for which .
This is a rather realistic scenario, considering that deliberative domains, such as enacting new laws or creating a new budget, may be rather complex [16]: It is certainly a stretch to expect citizens to articulate, e.g., their preferred codeoflaw. On the other hand, informed citizens, when presented with a new law or a new budget, should be able to say whether they prefer them to the legal status quo/last year’s budget, especially if their merits and disadvantages have been publicly discussed. At the same time, legal or financial experts such as law makers, public officials, or members of the academia could very well propose their own alternative legislation or budget and argue publicly in their favor, as part of the public deliberative process.
2.3 Compromise Deliberations Terminate
Proposition 1.
A deliberation in a transition system containing only compromise transitions is finite.^{1}^{1}1Note that this holds even if preferences are cyclic.
Proof.
Given a coalition structure , let
be the vector of coalition sizes, ordered in nonincreasing order. Consider a compromise transition from
to associated with a valid compromise . Then is lexicographically strictly greater than , since the compromise coalition is larger than and . Further, the size of each coalition is at most . ∎2.3.1 Compromise Deliberations May Fail
Unfortunately, compromise transitions do not guarantee the success of deliberation, as they may fail to find a majoritysupported proposal.
Example 3.
Consider the example depicted in Figure 1. While is majoritysupported, no compromise transition exists from the current coalition structure.
3 Deliberative Success via Amicability
We will now consider a restricted class of proposal spaces and show that deliberation always terminates successfully in such spaces. In these spaces, whenever we have three or more coalitions that do not support the status quo, there is compromise transition that is either a (1) merge transition, where two coalitions merge together and choose a new proposal or a (2) defect transition, where an agent joins the maximumsize coalition and the enlarged coalition agrees on a new proposal.
Definition 9 (Amicable Space).
Let be a proposal space (that contains ). We say that is amicable if for every set of three disjoint voter coalitions each of which is not successful and is not supporting , there exist two coalitions and a proposal such that (1) and or (2) is a maximumsize coalition in and , .
To illustrate this idea, consider , where each agent is associated with a point so that , where is the Euclidean distance in ; we will refer to this space as the dimensional Euclidean preference space [16].
For , for each set of three coalitions in there are at least two coalitions whose associated proposals and are on the same side of . If is closer to than is, then all voters in both coalitions prefer to , i.e., they can perform a merge transition. Thus, the onedimensional Euclidean preference spaces is amicable. We will now argue that this holds for any dimension.
Theorem 1.
For every , the dimensional Euclidean preference space is amicable.
Proof.
We have already seen a proof for . We will now provide a proof for ; it generalizes straightforwardly to .
Consider a set that contains at least three disjoint voter coalitions, all of which are not successful and none of which supports ; let be a maximumsize coalition in . Let be the line that passes through the middle of the – segment and is orthogonal to it; this line separates into two open halfplanes so that lies in one of these halfplanes while all points in lie in the other halfplane (see Figure 2). Let be the line that passes through and is parallel to . For a positive , let be the line obtained by rotating about clockwise by , and let be the line obtained by rotating counterclockwise by . The line (resp. ) partitions into open halfplanes and (resp. and ). We can choose to be small enough that and so that no voter lies on or on .
Now, if there exists a coalition , , such that or for some , then is not in the convex hull of and , and hence there is a line that separates from ; by projecting on this line, we obtain a proposal that is approved by and all agents in .
Otherwise, we have for all . Consider two distinct coalitions . As is bounded by two rays that start from , and the angle between these rays is , there is a line that divides into two open halfplanes so that is in one halfplane while are in the other halfplane; thus, all agents in approve the proposal obtained by projecting onto . ∎
Deliberation in amicable spaces is always successful.
Theorem 2.
In amicable spaces, every maximal run of compromise transitions is a successful deliberation.
Proof.
Consider an unsuccessful configuration over a voter set and an amicable space . We will argue that there is compromise transition from . Let be the list of coalitions in that do not support , where for . Note that , since otherwise would be successful. Suppose that . Since is unsuccessful, we have for . Then there is a mostsupported proposal for which ; hence, is a valid compromise. On the other hand, if , then, since is amicable, there is a merge transition or a defect transition, as described in Definition 9. ∎
However, not all spaces are amicable; in fact, even not all metric spaces are amicable. (We leave a more refined characterization of amicable spaces for future research.)
Example 4.
Consider a metric space with elements , set of coalitions where and all agents in are located at , and the following distances (denoted with distance function ): , . Then there is no proposal in this metric space that is attractive to agents in more than one coalition, so amicability does not hold.
4 Deliberative Success via Regrets
A fundamental aspect of a successful deliberative process is that it may cause people to change their mind, for example when exposed to a new idea (new alternative) or a new argument in favor of a different alternative. Hence we add the possibility for agents to change their mind by introducing regret transitions.
Definition 10 (Regret Transition).
Given a coalition structure on , we say that a coalition structure is obtained from by a regret transition if , there is a coalition with such that , and , where and ; we write . We denote by the set of all regret transitions for ; the set of all possible regret transitions for coalition structures in is denoted by . If can be obtained from by means of a regret transition, we write .
Regret transitions model agents that change their mind regarding the proposal supported by their current coalition, and, as a result, defect in order to support a different proposal (including possibly the status quo). First we note that finitely many regret transitions do not affect convergence; the observation below follows by applying the argument of Proposition 1 to a suffix of the run that has no regrets.
Observation 1.
A run of a compromise and regret transitions with a finite number of regrets is finite.
We now explore how regrets can be used to ensure the success of deliberations even if the space is not amicable.
Definition 11 (Trajectory, Parsimonious Deliberation).
The trajectory of a run is the sequence of partitions of induced by the coalition structures occurring in the run. A parsimonious deliberation is a run which is maximal among the runs whose trajectories do not contain any repeating pairs of partitions of . A parsimonious deliberation is successful if its terminal coalition structure is successful.
A parsimonious deliberation is a sequence of transitions that cannot be further extended without performing a transition that would repeat the effects, in terms of the coalitions it generates, of an earlier transition; e.g., it rules out a regret transition where an entire coalition changes the proposal it supports. Parsimony captures the intuition that agents may not be willing to explore proposals that would lead to partitions that have already been explored. It builds into the notion of deliberation a memory of the previous coalition formation attempts. Parsimonious deliberations must be finite as is finite.
Lemma 1.
In a transition system with compromise and regret transitions, for every unsuccessful coalition structure there exists a successful parsimonious deliberation that starts from .
Proof.
Let be an unsuccessful coalition structure, and let be a successful coalition. By our assumption, . Consider a minimumsize set of coalitions , where for all , such that . As is successful, is the set of all supporters of (Definition 7).
First, for each coalition , , such that we perform a regret transition in which the voters in split off from and form a coalition that supports . After this step, each coalition either consists of voters who approve or contains no such voters, so we can form by a sequence of merge transitions.
Clearly, the resulting run does not contain repeating pairs of partitions of . If it is not a maximal run with this property, we can extend it to a longer run with no repeating pairs of partitions; this process will terminate since is finite. ∎
Theorem 3.
A parsimonious deliberation of a transition system containing compromise and regret transitions is successful.
Proof.
We argue towards a contradiction and assume that there exists a deliberation which is not successful. Let be the terminal coalition structure of such deliberation. By Definition 6, does not contain any successful coalitions; let be a successful coalition. By assumption, . There exists a minimumsize set , with , , such that . As is successful, is the set of all supporters of (Definition 7). By Lemma 1 there exists a deliberation from to a coalition structure containing . Since was assumed to be a deliberation, the first transition in should be such that there already exists a transition in with and as otherwise could be extended without inducing a trajectory with repeating pairs. The same argument applies for all subsequent transitions of . Hence the trajectory of is a suffix of the trajectory of , and is therefore successful. Contradiction. ∎
5 A Centralized Solution
We think of deliberation as a multiagent process that may replace the centralized identification of mostsupported proposals by analytic methods. If is finite, then it is possible to go over all elements , ask each agent whether they support , and identify the mostsupported alternatives, at a cost of .
If is infinite, then an analytic solution may still be possible in principle, provided that agents not only can articulate whether they prefer a given proposal to the status quo, but also can specify the set of all proposals that they prefer over the status quo (their approval sets). This can be done, of course, only if such sets can be represented efficiently, as is the case, e.g., for the dimensional Euclidean preference space; an interesting point that we do not consider here is when such efficient representation exist.
Here, we show that if voters can articulate their approvals sets, if these sets can be represented succinctly, and if their intersection and union can be computed efficiently, then the subset of with the support of the majority of (and, more generally, the mostsupported alternatives) can be computed efficiently, using dynamic programming. We note, however, that this possibility does not detract from the value of the deliberative process, since voters generally are not able to articulate the set of their supported proposals. Also, it is possible to go over all sets of agents and look for the largest set of agents who approval sets have a nonempty intersection; however, the running time of this procedure scales exponentially with , while the dynamic program we describe next runs in time polynomial in .
Definition 12.
For a set of agents , let (so is the intersection of the approval sets of ). For , , let , and define:
Intuitively, contains all the sets of proposals that have the support of at least agents with indices between and inclusive. Note that, by definition, .
Remark 2.
If we can compute the sets , then we can compute the set , , of mostsupported proposals. Doing so can help find proposals supported by an agent supermajority; this can be useful in various circumstances, including sybil resilience [17]. Note that, if , then there is no need to vote over the proposals: the status quo has majority support and is retained. However, if a single majority (or supermajority) supported proposal needs to be elected, tiebreaking, possible employing the status quo [18], should be applied.
The overall idea of our DP is to consider singletons as the base case; then, to find an alternative supported by many agents within some set of agents, we divide the set and find an alternative wellsupported in both parts.
Definition 13.
For every , and , define
Lemma 2.
For all with and all we have .
Lemma 3.
can be computed with a number of set operations that is polynomial in and .
Combining Lemmas 2 and 3 and Remark 2, we conclude that our dynamic program can identify the set of mostsupported proposals in polynomial time.
Observe that any deliberative coalition structure containing the DP solution is successful. Furthermore, any run of compromise transitions containing such a coalition structure must be a successful deliberation.
6 Outlook
We proposed a formal model of deliberation for agent populations forming coalitions around proposals supported over the status quo. We showed that, while deliberation might not be successful in general, it is always successful in amicable spaces or with additional regret transitions. We mention some future research directions.
Stochastic deliberation
It is natural to extend our nondeterministic model to a stochastic model in which certain compromise transitions are more likely to happen than others. A Markovian analysis of such systems might shed further light on further properties of deliberation.
Strategic issues
In our current model agents are not strategic. They truthfully reveal whether they support a given proposal or not. Yet revealed support for proposals may well be object of manipulation. Such a gametheoretic extension of the model is left to future work.
Artificial mediators
An ambitious direction for future work is to design a practical tool for deliberation support building on top of our model and analysis. Such a tool can take the form of an AI bot over existing online deliberation platforms (e.g., [6, 2]). The bot suggests proposals to agents in order to support compromise, hopefully leading to successful deliberations.
Acknowledgements
We thank Schloss Dagstuhl—Leibniz Center for Informatics. The paper develops ideas discussed by the authors and other participants of the Dagstuhl Seminar 19381 (ApplicationOriented Computational Social Choice), summer 2019. We thank the generous support of the Braginsky Center for the Interface between Science and the Humanities. Edith Elkind was supposed by an ERC Starting Grant ACCORD (GA 639945). Nimrod Talmon was supported by the Israel Science Foundation (ISF; Grant No. 630/19).
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