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Argumentation for Common Ground: Finding Zones of Possible Agreement between Individuals in Conflict
Elisa Cavatorta, Antonio Rago
Intelligence
Status: succeeded | Model: Gemma-4-26B-A4B | Prompt: intel-v1 | Confidence: 92%
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Summary
This paper introduces a novel approach using quantitative bipolar argumentation frameworks (QBAFs) to identify Zones of Possible Agreement (ZOPA) in conflicts, specifically applied to the Palestinian-Israeli conflict. The method models individual citizens' reasoning about peace agreement clauses as arguments with intrinsic strengths and support/attack relations. By merging QBAFs from conflicting parties, the framework identifies mutually acceptable agreements based on gradual semantics (DF-QuAD and QEM), demonstrating that common ground can be found even when parties hold opposing views on specific clauses.
Entities (7)
Relation Signals (5)
Quantitative Bipolar Argumentation Framework → appliedto → Palestinian-Israeli Conflict
confidence 95% · To evaluate our approach under conditions of real-world relevance, we focus on the Palestinian-Israeli conflict
Quantitative Bipolar Argumentation Framework → usedfor → Zone of Possible Agreement
confidence 95% · we introduce a quantitative bipolar argumentation framework tailored to represent each side's reasoning about peace agreements... merging these frameworks can enable negotiators to identify peace agreements that are mutually acceptable.
DF-QuAD → istypeof → Gradual Semantics
confidence 90% · we assess the suitability of two of the most popular gradual semantics, DF-QuAD (49) and QEM (44).
QEM → istypeof → Gradual Semantics
confidence 90% · we assess the suitability of two of the most popular gradual semantics, DF-QuAD (49) and QEM (44).
Large Language Model → usedfordataretrieval → Survey Data
confidence 85% · preliminary experiments using survey data from both existing work and retrieved by a large language model.
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Abstract
Abstract:How can common ground between societies in conflict be identified when citizens' acceptability of peace agreements is shaped by contested narratives? Such acceptability is mediated not only by the clauses that agreements include or exclude, but crucially by citizens' subjective reasoning concerning agreements' clauses. In this paper, we leverage computational argumentation to introduce a novel approach to identifying mutually acceptable agreements among individuals in conflict, i.e. a Zone of Possible Agreement (ZOPA). First, we introduce a quantitative bipolar argumentation framework tailored to represent each side's reasoning about peace agreements. We then show how merging these frameworks can enable negotiators to identify peace agreements that are mutually acceptable. To evaluate our approach under conditions of real-world relevance, we focus on the Palestinian-Israeli conflict, where long-standing policy, practitioner and public interest underscores the demand for methods capable of analysing polarised public reasoning. We show how our framework identifies a ZOPA through theoretical analysis and preliminary experiments using survey data from both existing work and retrieved by a large language model. The results illustrate how argumentation can empower negotiators and conflict-resolution teams in mapping feasible ZOPAs grounded in citizens' reasoning.
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- Source: https://arxiv.org/abs/2608.15634v1
- Canonical: https://arxiv.org/abs/2608.15634v1
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Argumentation for Common Ground: Finding Zones of Possible Agreement between Individuals in Conflict Elisa Cavatortaand Antonio Rago Abstract How can common ground between societies in conflict be identified when citizens’ acceptability of peace agreements is shaped by contested narratives? Such acceptability is mediated not only by the clauses that agreements include or exclude, but crucially by citizens’ subjective reasoning concerning agreements’ clauses. In this paper, we leverage computational argumentation to introduce a novel approach to identifying mutually acceptable agreements among individuals in conflict, i.e. a Zone of Possible Agreement (ZOPA). First, we introduce a quantitative bipolar argumentation framework tailored to represent each side’s reasoning about peace agreements. We then show how merging these frameworks can enable negotiators to identify peace agreements that are mutually acceptable. To evaluate our approach under conditions of real-world relevance, we focus on the Palestinian-Israeli conflict, where long-standing policy, practitioner and public interest underscores the demand for methods capable of analysing polarised public reasoning. We show how our framework identifies a ZOPA through theoretical analysis and preliminary experiments using survey data from both existing work and retrieved by a large language model. The results illustrate how argumentation can empower negotiators and conflict-resolution teams in mapping feasible ZOPAs grounded in citizens’ reasoning. 1 Introduction Sustainable peace agreements fundamentally depend on citizens’ willingness to comply with new institutions that depart from the status quo. When societies are not ready for compromise, agreements face backlash, rejection, or non-ratification (52). Public acceptability is mediated not only by the provisions agreements include (or exclude), but by narratives about their promised outcomes, subjective arguments about how provisions address grievances, and the risks they are perceived to entail. These subjective arguments are inherently difficult to systematise, yet it is precisely therein that the bottlenecks to compromise reside (37). Meanwhile, computational argumentation (20) is a field within AI which excels in representing knowledge and resolving conflicts therein. The formalisms offered by this rich area of research have been deployed in related settings to that we consider here, e.g. opinion modelling (16; 51) and judgmental forecasting (30; 24). However, to our knowledge, its technologies have not been deployed in real-world conflict resolution, unlike other areas of AI such as Markov decision processes and linear temporal logic (34), or large language models (LLMs) (53; 35). In this paper, we build on the contributions of 13, who examine the acceptability of prospective peace agreements in both Israeli and Palestinian societies and identify the Zone of Possible Agreement (ZOPA), the set of agreements in which two parties can find common ground. Using nationally representative samples and experimentally controlled clause values, the authors estimate causal effects of clause inclusion on agreement endorsement. We extend this work by focusing on the theoretical and empirical evaluation of the reasoning underpinning these population-level parameters, leveraging argumentation to analyse individuals’ reasoning about agreement clauses. To do so, we use quantitative bipolar argumentation frameworks (QBAFs) (2; 6), i.e. formal argumentation frameworks that represent arguments with an intrinsic strength and positive or negative relations between them. Gradual semantics, i.e. quantitative evaluation methods, may then be applied to evaluate an argument’s acceptability, which have been shown to be useful in settings from explainable AI (17) to online review aggregation (48). In doing so, we make theoretical and experimental contributions that justify the use of computational argumentation as a means for supporting tools to assist negotiators and conflict-resolution teams in mapping feasible ZOPAs grounded in citizens’ reasoning. The intended users of these tools are those undertaking analysis of negotiations such as professional mediators, peace negotiators and non-governmental organisations who assess the public viability of specific agreements’ clauses. We believe that by leveraging the reasoning behind citizens’ narratives, our method can help to develop tools for revealing ZOPAs which were previously inaccessible to the designers of acceptable peace agreements. After giving the necessary preliminaries (§2), we make the following contributions: • We introduce novel QBAFs tailored to represent conflicting individual’s reasoning about agreements and show theoretically that, if equipped with suitable gradual semantics, they intuitively represent individuals’ views (§3). • We demonstrate how a set of QBAFs representing individual citizens’ conflicting reasoning can be merged into a single QBAF to indicate ZOPAs, a subset of the agreements, proving formal guarantees thereon and further restricting the set of suitable gradual semantics (§4). • We perform preliminary experiments to evaluate our approach using survey data taken from (13), in addition to data from the reports of nationally-representative surveys retrieved using an LLM, illustrating the theoretical results and suitability for real-world deployment with negotiators (§5). We then consider the related work in the literature (§6), before concluding and looking ahead to future work (§7). 2 Preliminaries Application Context Peace agreements are contracts between conflicting parties that aim to resolve the underlying issues causing the conflict. In (13), a peace agreement, P∈P , where P is the set of all peace agreements, is a set of clauses, i.e. proposed changes in, or continuations of, the status quo (allowing for one-hot representations of multi-value variables). We let =C1,…,CnC=\C_1,…,C_n\ be a set of n possible clauses representing changes in the status quo, where any P∈P is such that P⊆P and P is the power set of C. For example, let us consider a situation in the Israeli-Palestinian context in which we have =C1,C2C=\C_1,C_2\. Here, C1C_1 may be the clause calling for a freeze on settlement building in the West Bank, and C2C_2 may be the clause requiring Palestinians to officially recognise Israel, both changes in the status quo. The absence of such clauses in our peace agreements represents continuations of the status quo, i.e. continuation of settlement building and no recognition of Israel, resp. Note that the absence of a clause CiC_i from a peace agreement PjP_j, i.e. Ci PjC_i P_j, means that the peace agreement contains the negation of the clause CiC_i, i.e. the continuation of the status quo. For example, if we have Pi,Pj∈P_i,P_j , where Pi=∅P_i= and Pj=P_j=C, PiP_i represents an agreement with maximal continuation of the status quo (i.e. no change from status quo) and PjP_j represents an agreement with maximal change from the status quo (i.e. all clauses representing changes in the status quo are contained in PjP_j). For each clause, a citizen uiu_i either endorses it or does not. This partitions the set of clauses such that =+i∪−iC=C_+^i _-^i with +i∩−i=∅C_+^i _-^i= , where +iC_+^i is the set of clauses uiu_i endorses and −iC_-^i the set they do not. We treat endorsement as binary, so a citizen is never undecided or indifferent about a clause. Quantitative Argumentation We use the notion of a QBAF ⟨,,,τ⟩ ,A,S,τ where: X is a finite set of arguments; ⊆×A\! \!X\!×\!X (⊆×S\! \!X\!×\!X) is a binary, directed relation of attack (support, resp.) between arguments, where A and S are disjoint; τ:→[0,1]τ\!:\!X\!→\![0,\!1] ascribes base scores to arguments, representing their intrinsic acceptabilities.11 1 Note that in (6), base scores are defined for more general preorders, but here, for simplicity and in line with the majority of existing work, we restrict to [0,1][0,1]. For any argument xi∈x_i\!∈\!X, we use (xi)=xj∈|(xj,xi)∈A(x_i)\!=\!\x_j\!∈\!X|(x_j,x_i)\!∈\!A\ to denote xix_i’s set of attackers, and (xi)=xj∈|(xj,xi)∈S(x_i)\!=\!\x_j\!∈\!X|(x_j,x_i)\!∈\!S\ to denote xix_i’s set of supporters. We deploy gradual semantics, denoted by σ, which, for a given QBAF =⟨,,,τ⟩Q= ,A,S,τ , assigns each argument xi∈x_i a strength σ(,xi)∈[0,1]σ(Q,x_i)∈[0,1] representing its acceptability. In the remainder of this section, we assume as given a generic QBAF =⟨,,,τ⟩Q= ,A,S,τ with a gradual semantics σ. Gradual semantics’ suitability for specific applications is examined using their theoretical properties. Some of these properties are defined parametrically based on a comparison measure between sets of arguments’ strengths. In this paper, we opt for one such measure (without loss of generality) which discounts arguments which zero strength, ≥σ _σ, as in (7). Formally, for X⊆X , we define a function returning a multiset by removing zero strength attackers z(X)=σ(,xi)|xi∈X,σ(,xi)≠0z(X)=\σ(Q,x_i)|x_i∈ X,σ(Q,x_i)≠ 0\. Then, we use a comparison measure such that for A,B⊆A,B , we denote: A=σBA=_σB iff z(A)=z(B)z(A)=z(B); A≥σBA _σB iff there exists an injective mapping f from z(B)z(B) to z(A)z(A) such that ∀xi∈z(B)∀ x_i∈ z(B), σ(,f(xi))≥σ(,xi)σ(Q,f(x_i))≥σ(Q,x_i); and A>σBA>_σB, iff A≥σBA _σB and B≱σAB _σA. Any σ satisfies balance (6) iff ∀xi∈∀ x_i : if (xi)=σ(xi)A(x_i)=_σS(x_i) then σ(,xi)=τ(xi)σ(Q,x_i)=τ(x_i); if (xi)>σ(xi)A(x_i)>_σS(x_i) then σ(,xi)≤τ(xi)σ(Q,x_i)≤τ(x_i); and if (xi)<σ(xi)A(x_i)<_σS(x_i) then σ(,xi)≥τ(xi)σ(Q,x_i)≥τ(x_i). Any σ satisfies monotonicity (6) iff ∀xi,xj∈∀ x_i,x_j : if τ(xi)=τ(xj)τ(x_i)=τ(x_j), (xi)=σ(xj)A(x_i)=_σA(x_j) and (xi)=σ(xj)S(x_i)=_σS(x_j), then σ(,xi)=σ(,xj)σ(Q,x_i)=σ(Q,x_j); and if τ(xi)≤τ(xj)τ(x_i)≤τ(x_j), (xi)≥σ(xj)A(x_i) _σA(x_j) and (xi)≤σ(xj)S(x_i) _σS(x_j), then σ(,xi)≤σ(,xj)σ(Q,x_i)≤σ(Q,x_j). Any σ satisfies strict monotonicity (6) iff σ satisfies monotonicity and ∀xi,xj∈∀ x_i,x_j such that τ(xi)≤τ(xj)τ(x_i)≤τ(x_j), (xi)≥σ(xj)A(x_i) _σA(x_j) and (xi)≤σ(xj)S(x_i) _σS(x_j), and at least one of these relations is strict, then σ(,xi)<σ(,xj)σ(Q,x_i)<σ(Q,x_j). Any σ satisfies duality (44) iff ∀xi,xj∈∀ x_i,x_j such that τ(xi)=1−τ(xj)τ(x_i)=1-τ(x_j), (xi)=(xj)A(x_i)=S(x_j) and (xi)=(xj)S(x_i)=A(x_j), σ(,xi)=1−σ(,xj)σ(Q,x_i)=1-σ(Q,x_j). In this paper, we assess the suitability of two of the most popular gradual semantics, DF-QuAD (49) and QEM (44). Both of these semantics satisfy monotonicity, balance and duality, but only QEM satisfies strict monotonicity (7; 42). 3 Representing Citizens’ Reasoning on Peace Agreements with QBAFs In this section, we define a framework for representing citizens’ reasoning about the acceptability of agreements, before undertaking theoretical analysis to identify which properties characterise desirable gradual semantics in this setting. Our framework is defined as follows. Figure 1: Two QBAFs i=⟨i,i,i,τi⟩Q^i= ^i,A^i,S^i,τ^i (left) and p=⟨p,p,p,τp⟩Q^p= ^p,A^p,S^p,τ^p (right) representing the reasoning of users uiu_i and upu_p, resp., on =P1,P2,P3,P4P=\P_1,P_2,P_3,P_4\, in which =C1,C2C=\C_1,C_2\ where C1=C_1= Settlement building stops and C2=C_2= Palestinians officially recognise Israel, +i=−j=C2C_+^i=C_-^j=\C_2\, −i=+j=C1C_-^i=C_+^j=\C_1\, i=p=p1,p2,p3,p4X_P^i=X_P^p=\p_1,p_2,p_3,p_4\, i=c1i,c2iX_C^i=\c_1^i,c_2^i\, p=c1p,c2pX_C^p=\c_1^p,c_2^p\, ℛi=r1i,r2i,r3iX_R^i=\r_1^i,r_2^i,r_3^i\ and ℛp=r1p,r2p,r3pX_R^p=\r_1^p,r_2^p,r_3^p\. Arguments are represented by nodes, attacks by red edges labelled “−-” and supports by green edges labelled “++”. The QEM gradual semantics is used to calculate argument strengths. Definition 1. Given a set of peace agreements P and a citizen ui∈u_i , a QBAF representing uiu_i’s reasoning on P is a QBAF =⟨,,,τ⟩Q= ,A,S,τ with gradual semantics σ such that: 1. =∪ℛX=X_P _C _R, where: • X_P is the set of agreement arguments where ||=|||X_P|=|P| and ∀Pj∈∀ P_j , ∃pj∈∃ p_j _P; • X_C is the set of clause arguments where ||=|||X_C|=|C|, ∀Cj∈∀ C_j , ∃cj∈∃ c_j _C and =+∪−X_C=X_C^+ _C^- such that: – |+|=|+||X_C^+|=|C_+| and ∀Cj∈+∀ C_j _+, ∃cj∈+;∃ c_j _C^+; – |−|=|−||X_C^-|=|C_-| and ∀Cj∈−∀ C_j _-, ∃cj∈−;∃ c_j _C^-; • ℛX_R is the set of reasoning arguments; 2. ⊆(×)∪(ℛ×(ℛ∪))A\! \!(X_C\!×\!X_P)\!∪\!(X_R\!×\!(X_R\!∪\!X_C)) and ⊆(×)∪(ℛ×(ℛ∪))S\! \!(X_C\!×\!X_P)\!∪\!(X_R\!×\!(X_R\!∪\!X_C)) where: • ∀cj∈+∀ c_j _C^+, ∀pk∈∀ p_k _P, (cj,pk)∈(c_j,p_k) iff Cj PkC_j P_k; • ∀cj∈+∀ c_j _C^+, ∀pk∈∀ p_k _P, (cj,pk)∈(c_j,p_k) iff Cj∈PkC_j∈ P_k; • ∀cj∈−∀ c_j _C^-, ∀pk∈∀ p_k _P, (cj,pk)∈(c_j,p_k) iff Cj∈PkC_j∈ P_k; • ∀cj∈−∀ c_j _C^-, ∀pk∈∀ p_k _P, (cj,pk)∈(c_j,p_k) iff Cj PkC_j P_k; 3. τ(pj)=0.5τ(p_j)=0.5 ∀pj∈∀ p_j _P. The intuition for each of the points above is as follows. (1) Our framework represents the agreements, the clauses and reasoning thereon as arguments. (2) The clause arguments represent uiu_i’s opinion of whether the corresponding clause should or should not happen, based on whether the clause is in +iC_+^i or −iC_-^i, resp. (3) Attacks and supports are such that an argument representing a clause with (without) endorsement from uiu_i supports (attacks, resp.) arguments representing agreements that contain the clause, and attacks (supports, resp.) arguments representing agreements that do not contain the clause. Meanwhile, (arguments representing22 2 We may informally refer to agreement/clause/reasoning arguments as agreements/clauses/reasoning, resp., where it is clear we are referring to the QBAF and not the entities being represented.) reasoning may attack or support clauses or other reasoning. Base scores of the agreements are fixed to 0.50.5, the midpoint of the range, representing neutral prior acceptance. In this paper, we assume that τ(cj)=0.5τ(c_j)=0.5 ∀cj∈∀ c_j _C, other choices are discussed in §7 as directions for future work. In line with other works (49; 16), we limit to acyclic QBAFs in this paper, i.e. not allowing for circular reasoning from citizens, though Definition 1 has no such restriction. While we note that |||P| is combinatorial in |||C|, we take the logical first step of evaluating of all possible agreements, leaving to future work the investigation of algorithms for improved scaling, and noting that strengths in the gradual semantics studied here can be computed in linear time for acyclic graphs (45). Next, we introduce a ranking over the peace agreements. Definition 2. Given a set of peace agreements P, a citizen ui∈u_i and a QBAF representing uiu_i’s reasoning on P, =⟨,,,τ⟩Q= ,A,S,τ with σ, an argumentative ranking by Q and σ is a total ordering over P, ⪯σ _σ^Q, such that ∀Pi,Pj∈∀ P_i,P_j , Pi≃σPjP_i _σ^QP_j iff σ(,pi)=σ(,pj)σ(Q,p_i)=σ(Q,p_j) and Pi≺σPjP_i _σ^QP_j iff σ(,pi)<σ(,pj)σ(Q,p_i)<σ(Q,p_j). Intuitively, argumentative rankings order the peace agreements based on their strengths, giving a ranking based on their acceptabilities within the QBAF. Figure 1 gives two examples of QBAFs representing the reasoning of a hypothetical Israeli (left) and a hypothetical Palestinian (right) citizen (superscript i and p, resp.). Note that here, +i=−p=C2C_+^i=C_-^p=\C_2\ and −i=+p=C1C_-^i=C_+^p=\C_1\, meaning uiu_i and upu_p disagree on both of the two clauses, resulting in opposite attack and support relations between the corresponding clause and agreement arguments. The argumentative rankings representing the two citizens’ perspectives are P2≺σiP1≺σiP4≺σiP3P_2 _σ^Q^iP_1 _σ^Q^iP_4 _σ^Q^iP_3 and P3≺σpP1≺σpP4≺σpP2P_3 _σ^Q^pP_1 _σ^Q^pP_4 _σ^Q^pP_2. Though at first glance, it seems that there is no common ground between the two citizens since they endorse completely different clauses, the argumentative ranking demonstrates that some compromise may be found between the two, i.e. agreement P4P_4 in this case. In the remainder of the paper, we demonstrate how a set of agreements which is mutually acceptable to both parties can be identified by applying gradual semantics in a principled manner, i.e. ensuring that they satisfy certain properties, and then merging the QBAFs. We will now assess the behaviour of gradual semantics, as defined by their theoretical properties. The notation in this section uses a generic QBAF =⟨,,,τ⟩Q= ,A,S,τ with gradual semantics σ representing the reasoning of a citizen ui∈u_i on agreements P. When comparing QBAFs for different citizens, we use superscripts: we refer to the Q for uiu_i as i=⟨i,i,i,τi⟩Q^i= ^i,A^i,S^i,τ^i , and to any clause or reasoning argument therein as cji∈ic_j^i _C^i and rki∈ℛir_k^i _R^i, resp. Agreement arguments are not assigned superscripts as the same set of agreements is present for all citizens. We first consider each of the properties mentioned in §2 in turn. Balance requires that if an argument’s attackers are stronger than its supporters, then the argument’s strength should be less than or equal to its base score, and vice versa. A violation of balance would create an inconsistency within our setting, e.g. in Figure 1 if p2p_2 were assigned a higher strength than its base score in iQ^i when it has stronger opposition than support. We thus believe balanced semantics are essential for intuitive interpretations of citizens’ reasoning. (Strict) Monotonicity requires that increasing the base score, removing/weakening the attackers or adding/strengthening the supporters of an argument can only increase (always increases, resp.) its strength, and vice versa. These properties thus guarantee an intuitive monotonic relationship between an argument’s strength and its attackers, supporters and base score. For example, in Figure 1, for pQ^p we would expect that increasing the strength of c2pc_2^p (i.e. increasing upu_p’s negative sentiment towards C2C_2, a clause P1P_1 does not contain) or decreasing the strength of c1pc_1^p (i.e. decreasing upu_p’s positive sentiment towards C1C_1, a clause P1P_1 does not contain) could only strengthen p1p_1 (which represents P1P_1). Whether the stronger condition, strict monotonicity, is required, i.e. p1p_1 is always strengthened under these changes, or whether the weaker condition is sufficient, is a question we address in §4. Duality requires that two arguments which are “mirror images” of one another, in terms of its base score, attackers and supporters, should have strengths which are also mirrored about the midpoint 0.50.5 of the [0,1][0,1] scale. For example, in Figure 1, from the Israeli citizen’s perspective (iQ^i), if we take a pair of arguments which have complementary base scores, attackers and supporters, e.g. p1p_1 and p4p_4, it must be the case that σ(i,p1)=1−σ(i,p4)σ(Q^i,p_1)=1-σ(Q^i,p_4), given that p1p_1 and p4p_4’s attackers, supporters and base scores are complements of one another. Likewise for p2p_2 and p3p_3, and for the same argument pairs from the Palestinian citizen’s perspective (pQ^p). We thus require duality because clause endorsement and non-endorsement are constructed as exact mirrors, and we have no principled reason to break that symmetry in the semantics. Next, we give some theoretical results that further justify monotonicity and duality. First, Corollary 1 shows that the intuitive base score condition holds by default for agreement arguments as their base scores are fixed at 0.50.5.33 3 All proofs are given in the supplementary material. Corollary 1. For any pj,pk∈p_j,p_k\!∈\!X_P, if (pj)=(pk)A(p_j)\!=\!S(p_k), (pj)=(pk)S(p_j)\!=\!A(p_k) and σ satisfies duality, then σ(,pj)=1−σ(,pk)σ(Q,p_j)=1-σ(Q,p_k). Our next result concerns the attackers and supporters of peace agreements, and thus the endorsement of their claims. Lemma 1. For any pj,pk∈p_j,p_k _P, if (pj)⊃(pk)S(p_j) (p_k) (and thus (pj)⊂(pk)A(p_j) (p_k)) and σ satisfies monotonicity, then σ(,pj)≥σ(,pk)σ(Q,p_j)≥σ(Q,p_k). This result shows that an agreement containing more endorsed clauses and fewer non-endorsed clauses will be more acceptable. For example, in Figure 1, in iQ^i we expect that σ(i,p1)≥σ(i,p2)σ(Q^i,p_1)≥σ(Q^i,p_2), while in pQ^p we expect that σ(p,p1)≤σ(p,p2)σ(Q^p,p_1)≤σ(Q^p,p_2). Since citizen’s clause endorsement is a fundamental basis of agreement acceptability, gradual semantics’ satisfaction of monotonicity seems crucial. The next implication concerns the agreement ranking. Proposition 1. For Pj,Pk∈P_j\!,\!P_k\!\!∈\!P, if Pj=+iP_\!j\!\!=\!C_+^i, Pk=−i\!P_k\!\!=\!C_-^i and σ satisfies monotonicity, then Pj⪰σPlP_j\!\! _σ^Q\!\!P_l ∀Pl∈∀\!P_l\!\!∈\!P and Pk⪯σPmP_k\!\! _σ^Q\!\!P_m ∀Pm∈∀\!P_m\!\!∈\!P. An agreement with total endorsement of its clauses will rank highest amongst all clauses, while one with zero endorsement will rank lowest. In Figure 1, this means that in iQ^i (in pQ^p) agreement P3P_3 is ranked highest (lowest, resp.) amongst the agreements given that it has a minimal (maximal, resp.) set of attackers and a maximal (minimal, resp.) set of supporters, which we believe is intuitive behaviour. In summary, we have established that balance, monotonicity (though not necessarily strict monotonicity) and duality are essential properties for gradual semantics in our framework, in that they enforce intuitive behaviour in the representation of individual citizens’ opinions on peace agreements. Both DF-QuAD and QEM satisfy these requirements, and so would be considered suitable gradual semantics at this point. 4 Merging QBAFs to Find ZOPAs This section describes how we merge QBAFs representing the opinions of citizens in conflict in order to turn the disagreement into a search for common ground. We first combine the reasoning of multiple citizens into a single merged QBAF, defining the ZOPA therein. We then undertake theoretical analysis to support the choice of gradual semantics. We merge citizens’ QBAFs as follows. Definition 3. Given a set of n QBAFs 1,…,n\Q^1,…,Q^n\, where i=⟨i,i,i,τi⟩Q^i= ^i,A^i,S^i,τ^i for i∈1,…,ni∈\1,…,n\, with a gradual semantics σ representing the reasoning of a corresponding set of citizens U=u1,…,un⊆U=\u_1,…,u_n\ on P, the merged QBAF representing U’s reasoning on P is a QBAF ∗=⟨∗,∗,∗,τ∗⟩Q^*= ^*,A^*,S^*,τ^* such that: • ∗=1∪…∪nX^*=X^1∪… ^n; • ∗=(1∪…∪n)∩(∗×∗)A^*=(A^1∪… ^n)∩(X^*×X^*); • ∗=(1∪…∪n)∩(∗×∗)S^*=(S^1∪… ^n)∩(X^*×X^*); • τ∗τ^* is such that for any x∈∩ix ^i, τ∗(x)=τi(x)τ^*(x)=τ^i(x). Figure 2: Merged QBAF representing the reasoning of ui,up\u_i,u_p\ on P from Figure 1, where the values in each argument represent its base score (in normal font) and its strength (in bold font, calculated with the QEM semantics). Intuitively, merging combines several citizens’ reasoning into a single graph: the agreement layer, the set of candidate agreements, is shared across citizens. In contrast, the clause arguments and reasoning arguments are individual-specific and disjoint, and the merging preserves the union of all arguments, along with their corresponding relations and base scores. For the remainder of this section, we assume as given a generic merged QBAF ∗=⟨∗,∗,∗,τ∗⟩Q^*= ^*,A^*,S^*,τ^* with gradual semantics σ representing the reasoning of U⊆U on P. With a slight abuse of notation, we allow argumentative rankings to be applied to merged QBAFs. Figure 2 illustrates a merged QBAF from the two QBAFs shown in Figure 1. Despite the impression of total disagreement with no common ground between uiu_i and upu_p when the QBAFs were viewed individually, the merged QBAF reveals agreements which are mutually acceptable to both citizens based on their own reasoning, i.e. if we take the argumentative ranking for this merged QBAF, P1≺σ∗P2≺σ∗P3≺σ∗P4P_1\!\! _σ^Q^*\!\!P_2\!\! _σ^Q^*\!\!P_3\!\! _σ^Q^*\!\!P_4, we see that P4P_4 is the most mutually acceptable agreement, while P1P_1 is the least mutually acceptable. Next, we introduce a ZOPA, i.e. a classification of which arguments might be considered acceptable by all citizens. Definition 4. The ZOPA between U in ∗Q^* with σ is (∗,σ)=Pk∈∣σ(∗,pk)>0.5Z(Q^*,σ)=\P_k σ(Q^*,p_k)>0.5\. The ZOPA is the set of candidate agreements whose strength in the merged QBAF exceeds the agreements’ fixed base score of 0.50.5, i.e. the neutral midpoint. Clearing this threshold means the reasoning from citizens on both sides is on balance supportive of the agreement. The ZOPA for the example in Figure 2 is (∗,σ)=P3,P4Z(Q^*\!\!,σ)=\P_3,P_4\. Agreement P4P_4 is the more acceptable of the two because, while its clauses received mixed endorsement from the citizens, its supporting reasoning was stronger: the attackers of this agreement are the two clauses on which the citizens compromised somewhat in their reasoning (c1ic_1^i for uiu_i, compromising with the attacker r3ir_3^i, and c2pc_2^p for upu_p, compromising with the attacker r3pr_3^p). This example demonstrates how reasoning, and the argumentative strength thereof, drives our identification of a ZOPA. The inclusion of P4P_4 in the ZOPA demonstrates the importance of compromises in narratives. Nevertheless, this effect raises the obvious question of the system’s susceptibility to strategic manipulation, e.g. if compromises are purposely hidden or strengths are exaggerated, but at this stage, we assume the access to truthful opinions. One limitation of our approach is that Definition 3 takes the disjoint union of clause and reasoning arguments across citizens, meaning they could potentially include duplicates. This could be addressed by merging similar arguments as in (26), which could allow for the extraction new relations between them with argument mining (11; 25) or the adjustment of base scores based on aggregating citizens’ endorsement (50). Also, our threshold-based ZOPA is one of several ZOPA notions our framework supports natively: it is the most parsimonious choice consistent with balance. The same merged QBAF accommodates threshold-based, top-k and Pareto improvement on the status-quo as direct variants. We leave an investigation of their formal guarantees to future work. We now theoretically analyse our merged QBAF, determining the properties needed in the selected gradual semantics to guarantee intuitive behaviour. Firstly, our decision to merge the arguments in a simple manner gives the following. Proposition 2. If σ satisfies balance, then σ(∗,cij)=σ(j,cij)σ(Q^*,c_i^j)=σ(Q^j,c_i^j) ∀cij∈∗∀ c_i^j _C^*. Intuitively, the strengths of clause and reasoning arguments will be preserved in the merged QBAF, giving provenance to the merged QBAF in that reasoning can be traced back to the citizen from whom it came. Figure 2 shows why this is the case, with arguments “upstream” of the clause and reasoning arguments remaining separate from the others due to the direction of the reasoning. To capture the dynamics of bilateral disagreements, for the remainder we restrict merged QBAFs to two citizens, i.e. U=ui,ujU\!\!=\!\!\u_i,u_j\ (which may represent two homogenous parties). Theorem 1. For any pk∈∗p_k\!∈\!X_P^* in ∗Q^* with σ, where σ satisfies balance and strict monotonicity: if ∗(pk)<σ∗(pk)A^*\!(p_k)\!<_σ\!S^*\!(p_k), then Pk∈(∗,σ) [rgb]0,0,0P_k\!∈\!Z(Q^*\!,σ); and if ∗(pk)≥σ∗(pk)A^*\!(p_k)\! _σ\!S^*\!(p_k), then Pk (∗,σ) [rgb]0,0,0P_k\! \!Z(Q^*\!,σ). An agreement is part of the ZOPA if its attackers in the merged graph are weaker than its supporters. Consequently, the ZOPA identifies the common ground in the form of agreements of which the collective reasoning is in support, rather than in opposition. For example, in Figure 2, the stronger supports from c2ic_2^i and c1pc_1^p, compared to the weaker attacks from c1ic_1^i and c2pc_2^p, mean P4P_4 is in the ZOPA (and ranked highest). We will now assess the argumentative ranking induced by ∗Q^* and σ in Definition 2 by considering two extreme cases. Theorem 2. If +i=−jC_+^i=C_-^j, −i=+jC_-^i=C_+^j, σ(i,cki)=σ(j,ckj)σ(Q^i,c_k^i)=σ(Q^j,c_k^j) ∀k∈1,…,||∀ k∈\1,…,|C|\ and σ satisfies balance, then Pl≃σ∗PmP_l _σ^Q^*P_m ∀Pl,Pm∈∀ P_l,P_m and (∗,σ)=∅Z(Q^*,σ)= . When two citizens disagree on every clause with reasoning of identical strength, the merged graph is symmetric: every agreement faces exactly as much support as it does attack and the ZOPA is empty. This shows the framework behaving appropriately in the worst-case: perfectly opposed views with no strength asymmetry results in no common ground. Theorem 3. If Pk=+i=+jP_k=C_+^i=C_+^j, Pl=−i=−jP_l=C_-^i=C_-^j, |z()|=|||z(X_C)|=|X_C| and σ satisfies balance and strict monotonicity, then Pk≻σ∗PmP_k _σ^Q^*\!\!~P_m ∀Pm∈∖Pk∀ P_m \P_k\ and Pl≺σ∗PnP_l _σ^Q^*\!\!~P_n ∀Pn∈∖Pl∀ P_n \P_l\. Further, Pk∈(∗,σ)P_k (Q^*,σ) and Pl (∗,σ)P_l (Q^*,σ). Meanwhile, when both sides of the conflict endorse the same clauses, intuitively, the agreement that includes precisely those clauses is ranked strictly highest and lies in the ZOPA, while the agreement which includes precisely none of those clauses is ranked strictly lowest and lies outside it. In this section, we have demonstrated how both parties’ QBAFs can be merged to reveal ZOPAs between citizens, proving intuitive behaviour can be guaranteed. These results, in addition to those from §3, show that gradual semantics which satisfy the properties of balance, (strict) monotonicity and duality are suitable for our application in real-world conflict resolution. Thus, the QEM semantics is suitable, while DF-QuAD is not given its violation of strict monotonicity. 5 Empirical Evaluation We now perform preliminary experiments to assess the suitability of our method for real-world deployment. We do so with survey data from (13) (§5.1) and retrieved data from LLMs (§5.2). In both experimental settings, we use the same set-up as 13, with Israeli respondents on one side and Palestinian respondents on the other, and eight binary clauses forming each agreement. The clauses (and the corresponding status quo variant) were: 1) settlement freeze (or continuation); 2) recognition of Israel as the nation state of the Jewish people (or lack thereof) ; 3) establishment of an independent Palestinian state with equitable land swaps (or current jurisdiction); 4) increased freedom of movement for all people (or current restrictions); 5) unrestricted rights to access to Holy sites (or current restrictions); 6) Jerusalem as joint capital (or separate and divided capital cities); 7) mutual amnesty for prisoners (or current practices of detention); and 8) proportionality on water rights (or current distribution). We use the QEM semantics given the findings from §3-§4. 5.1 Survey Data from Existing Work In our first experiment, we assess whether our merged QBAF is able to recover respondent preferences that were measured independently of it. To do so, we use the data of 13, who fielded a 64-agreement, rank-ordering task over the same eight binary clauses with balanced samples of Israelis (n=1152n=1152) and Palestinians (n=1152n=1152). We ask whether our merged QBAF, with the clause arguments populated using the analysis from (13), is able to produce an argumentative ranking which corresponds to a “ground truth” empirical ranking that we infer from the raw ranking data by measuring, for each agreement, the share of respondents (from both populations) who rank the agreement above the status quo agreement. To populate the clause arguments for the Israeli and Palestinian sides, we assign them strengths, assuming the reasoning upstream is implicit since 13 do not record reasoning. However, their empirical design identifies, for each clause, the proportion πja _j^a, for each party a∈i,pa∈\i,p\, who prefer an agreement containing clause j∈1,…,8j∈\1,…,8\ to the otherwise identical agreement without it. We use those 16 causal estimates to populate the clauses directly. Party a endorses the change variant of clause j when 13’s coefficient βja>0 _j^a>0. A clause argument cjac_j^a is given strength σ(cja,a)=|πja−(1−πja)|σ(c_j^a,Q^a)=| _j^a-(1- _j^a)| with πja=Λ(βja) _j^a= ( _j^a) and Λ the logistic function, so the strength is the margin by which the endorsed variant wins: if everyone in party a prefers an agreement with clause j’s compared to the agreement without clause j, the strength is 1; if everyone is indifferent, the strength is 0. Over the 64 agreements, the argumentative ranking and the empirical ranking correlate at Spearman ρ=0.448ρ=0.448 (p=0.000p=0.000) and Kendall τ=0.293τ=0.293 (p=0.000p=0.000). Turning the ranking into a set, a pooled majority puts 56 of the 64 deals above the status quo and the merged QBAF puts 63 above the strength of the status quo. This yields a precision of 0.8730.873 and recall 0.9820.982. However, because the target is 56/64 it is an easy target. While this assessment is by no means perfect, we believe it shows encouraging correlation. 5.2 Retrieved Data from LLMs Our second experiment examines whether our method gives intuitive results that would be useful for a peace negotiator. To assess whether it is feasible we built an LLM-driven pipeline that retrieves reasoning arguments from published survey reports, material a mediator typically possesses, demonstrating how our method could provide information without costly fieldwork. The output is a ranking over agreements, together with the clause-level strengths behind it, which is potentially crucial information for a negotiator. To elicit public reasoning that reflects contemporary Palestinian and Israeli views, we prompted an LLM (Opus 4.7 from Anthropic44 4 https://w.anthropic.com/news/claude-opus-4-7) to retrieve reasoning arguments about the eight clauses from the reports of nationally-representative opinion polls conducted by a curated list of well-reputed institutes. Every candidate argument was manually validated against the source document before it was admitted to the QBAFs.55 5 Technically, we ask for a literal substring check that the quoted statement appears verbatim in the document, an identity check on the source name and URL, and a context check that the statement carries a percentage figure and a token identifying the Israeli or Palestinian population. Prevalence is recorded as reported and never inverted: 20% support for annexation is recorded as 0.2, never as 0.8 opposition. Each retained argument is then encoded along two attributes. Its base score is the prevalence quoted for the relevant population, so a reported 69% becomes 0.690.69. Its stance is the polarity of its content towards the side’s endorsed clause: a reason favouring the clause enters as a supporter, a reason against it as an attacker. For example, the statement “69% of Palestinians indicated satisfaction with prisoner release” enters as a supporter of the corresponding clause with strength 0.690.69. While this approach is token-intensive and relies on human oversight, we believe that it makes good use of public reports and LLMs, grounding arguments in citable polling data, while providing a reasonable preliminary assessment of our approach before it is deployed in the real world. Figure 3: For the data retrieved by the LLM, a comparison, for each clause, of the difference in change endorser versus change opposer clause strength with number of agreements in the ZOPA which contain the clause. The approach retrieves 29 validated reasoning arguments, 14 for Israelis (2 in favour of changes, 12 opposing changes) and 15 for Palestinians (11 in favour of changes, 4 opposing changes), drawn from seven distinct opinion-poll report documents, published in 2024-2025. Figure 3 illustrates the results from this experiment. The chart illustrates the potential of our approach in that clauses where the strengths of the clause arguments endorsing a change outweighed those opposing the change (determined by the reasoning) resulted in that change being included in more ZOPA agreements, and vice versa for the opposite case. Particularly encouraging are the facts that: the mutually endorsed change (Clause 5) was in the most ZOPA agreements; the clause with the most positive combined reasoning was in the next most ZOPA agreements (Clause 7) and the clause with the most negative combined reasoning was in the fewest ZOPA agreements (Clause 2). We believe this provides encouraging evidence for the real-world suitability of our approach. 6 Related Work There is a vast body of work on gradual semantics, e.g. considering only relations of attack (9; 36) or support (1), or those which do not include a base score (3). Those for QBAFs are arguably more popular (23; 59; 56), which potentially align with human reasoning (41; 55). Various analyses of gradual semantics’ behaviour have been undertaken (39; 18; 40; 58; 33; 5), the findings from which may be useful in our setting, e.g. explanations of strengths for deeper analysis of reasoning. Gradual semantics’ handling of uncertainty and incomplete information has also led to a number of applications in real-world contexts, e.g. fraud detection (14), judgmental forecasting (30) and various forms of explainable AI (46; 43; 47; 48). To our knowledge, they have not yet been applied to real-world peace agreements. Argumentation has also been deployed successfully in negotiation (31; 4; 10; 21) and automated persuasion (27; 29; 12; 19; 32). None of these approaches use gradual argumentation, highlighting the potential of cross-fertilisations with our work. 7 Conclusions In this paper, we introduced a novel, tailored QBAF for representing citizens’ reasoning about peace agreements and showed how merging opposing parties’ QBAFs can reveal ZOPAs grounded in evidence-based reasoning. Our theoretical analysis demonstrates that gradual semantics satisfying balance, (strict) monotonicity and duality naturally produce intuitive rankings over agreements. The empirical evaluation on the Israeli-Palestinian conflict tests the framework against survey data from both existing work and retrieved from an LLM, showing reasonable correlation with the existing data and its suitability for real-world deployment. This work shows that argumentation has the potential to assist negotiators in identifying feasible common ground, even amid deeply polarised public discourse. Our study opens several avenues for future work. One is an empirical evaluation involving reasoning elicited from actual survey respondents or structured interviews processed via NLP. Scaling the approach to nationally representative samples would require developing efficient methods to merge thousands of individual QBAFs. Methodologically, future work could include developing principled protocols for base score elicitation, potentially informed by behavioural principles from behavioural economics. These advances would naturally lend themselves to empirical analyses that pinpoint which reasoning arguments are the strongest barriers to agreements and identify arguments that, if introduced or reframed, would shift both sides’ QBAFs towards mutual acceptability and ultimately support conflict resolution. We believe that our contributions highlight the potential of argumentation in general in this setting. For example, allowing for set-attacks and set-supports (8) may allow us to model conditional dependencies between clauses. 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Additional Definitions In the proofs, in order to formalise chains of reasoning from one argument to another via the attack and support relations, for any xi,xj∈x_i,x_j , we let a path from xix_i to xjx_j be defined as (x0,x1),(x_0,x_1), …,…, (xn−1,xn)(x_n-1,x_n) for some n>0n>0, where x0=xix_0=x_i, xn=xjx_n=x_j and, for any 1≤k≤n1≤ k≤ n, (xk−1,xk)∈∪(x_k-1,x_k) . We will use (,xi,xj) paths(Q,x_i,x_j) to denote the set of all paths between any xi,xj∈x_i,x_j , and we treat paths as sets of pairs. The DF-QuAD semantics (49) is a gradual semantics such that for any xi∈x_i , σ(,xi)=c(τ(xi),Σ(σ(,(xi))),Σ(σ(,(xi))))σ(Q,x_i)=c(τ(x_i), (σ(Q,A(x_i))), (σ(Q,S(x_i)))) where, for any S⊆S , σ(,S)=(σ(,x1),…,σ(,xk))σ(Q,S)=(σ(Q,x_1),…,σ(Q,x_k)) for (x1,…,xk)(x_1,…,x_k), an arbitrary permutation of S, and: Σ is such that Σ(())=0 (())=0, where ()() is an empty sequence, and, for v1,…,vn∈[0,1]v_1,…,v_n∈[0,1] (n≥1n≥ 1), if n=1n=1, then Σ((v1))=v1 ((v_1))=v_1; if n=2n=2, then Σ((v1,v2))=v1+v2−v1⋅v2 ((v_1,v_2))=v_1+v_2-v_1· v_2; and if n>2n>2, then Σ((,,,,,))=Σ(Σ((,,,,,)),vn) ((v_1,…,v_n))= ( ((v_1,…,v_n-1)),v_n); c is such that, for v0,v−,v+∈[0,1]v^0,v^-,v^+∈[0,1], if v−≥v+v^-≥ v^+, then c(v0,v−,v+)=v0−v0⋅|v+−v−|c(v^0,v^-,v^+)=v^0-v^0·|v^+-v^-| and if v−<v+v^-<v^+, then c(v0,v−,v+)=v0+(1−v0)⋅|v+−v−|c(v^0,v^-,v^+)=v^0+(1-v^0)·|v^+-v^-|. The QEM semantics66 6 We define a simplified gradual semantics here for the case of acyclic graphs. (44) is a gradual semantics such that for any xi∈x_i , σ(,xi)=τ(xi)+(1−τ(xi))⋅h(Exi)−τ(xi)⋅h(−Exi)σ(Q,x_i)=τ(x_i)+(1-τ(x_i))· h(E_x_i)-τ(x_i)· h(-E_x_i) where Exi=∑xj∈(xi)σ(,xj)−∑xk∈(xi)σ(,xk)E_x_i= _x_j (x_i)σ(Q,x_j)- _x_k (x_i)σ(Q,x_k) and for all v∈ℝv , h(v)=maxv,021+maxv,02h(v)= \v,0\^21+ \v,0\^2. Proofs Corollary 1. For any pj,pk∈p_j,p_k _P, if (pj)=(pk)A(p_j)=S(p_k), (pj)=(pk)S(p_j)=A(p_k) and σ satisfies duality, then σ(,pj)=1−σ(,pk)σ(Q,p_j)=1-σ(Q,p_k). Proof. By Definition 1, ∀pl∈∀ p_l _P, τ(pl)=0.5τ(p_l)=0.5. Then, the proof follows directly from the definition of duality. ∎ Lemma 1. For any pj,pk∈p_j,p_k _P, if (pj)⊃(pk)S(p_j) (p_k) (and thus (pj)⊂(pk)A(p_j) (p_k)) and σ satisfies monotonicity, then σ(,pj)≥σ(,pk)σ(Q,p_j)≥σ(Q,p_k). Proof. By Definition 1, pjp_j and pkp_k are such that (pj)∪(pj)=(pk)∪(pk)=A(p_j) (p_j)=A(p_k) (p_k)=X_C and τ(pj)=τ(pk)=0.5τ(p_j)=τ(p_k)=0.5. Then, by monotonicity, σ(,pj)≥σ(,pk)σ(Q,p_j)≥σ(Q,p_k). ∎ Proposition 1. For Pj,Pk∈P_j,P_k , if Pj=+iP_j=C_+^i, Pk=−iP_k=C_-^i and σ satisfies monotonicity, then Pj⪰σPlP_j _σ^QP_l ∀Pl∈∀ P_l and Pk⪯σPmP_k _σ^QP_m ∀Pm∈∀ P_m . Proof. By Definition 1, ∀pn∈∀ p_n _P, (pn)∪(pn)=A(p_n) (p_n)=X_C and τ(pn)=0.5τ(p_n)=0.5. By the same definition, for any Co∈+iC_o _+^i, co∈(pj)c_o (p_j) since Co∈PjC_o∈ P_j and co∈(pk)c_o (p_k) since Co PkC_o P_k, and conversely for any Cp∈−iC_p _-^i, cp∈(pj)c_p (p_j) since Cp PjC_p P_j and cp∈(pk)c_p (p_k) since Cp∈PkC_p∈ P_k. Thus, it must be the case that (pj)=(pk)=∅A(p_j)=S(p_k)= and (pj)=(pk)=S(p_j)=A(p_k)=X_C. Then, also by Definition 1, any pl∈∖pjp_l _P \p_j\ is such that (pl)⊂(pj)S(p_l) (p_j) and thus (pl)⊃(pk)A(p_l) (p_k). Similarly, any pm∈∖pkp_m _P \p_k\ is such that (pm)⊂(pk)A(p_m) (p_k) and thus (pm)⊃(pk)S(p_m) (p_k). Then, by Lemma 1, it must be the case that σ(,pj)≥σ(,pl)σ(Q,p_j)≥σ(Q,p_l) and σ(,pk)≤σ(,pm)σ(Q,p_k)≤σ(Q,p_m), and thus, by Definition 2, Pj⪰σPlP_j _σ^QP_l and Pk⪯σPmP_k _σ^QP_m. ∎ Proposition 2. If σ satisfies balance, then σ(∗,cij)=σ(j,cij)σ(Q^*,c_i^j)=σ(Q^j,c_i^j) ∀cij∈∗∀ c_i^j _C^*. Proof. By Definitions 1 and 3 it can be seen that ∀rkl∈ℛ∗∀ r_k^l _R^* such that (∗,rkl,cij)≠∅ paths(X^*,r_k^l,c_i^j)≠ , l=jl=j, τ∗(rkl)=τl(rkl)τ^*(r_k^l)=τ^l(r_k^l), ∗(rkl)=l(rkl)A^*(r_k^l)=A^l(r_k^l) and ∗(rkl)=l(rkl)S^*(r_k^l)=S^l(r_k^l). By balance, it must then be the case that σ(∗,rkl)=σ(j,rkl)σ(Q^*,r_k^l)=σ(Q^j,r_k^l) and, by the same logic, we can deduce that σ(∗,cij)=σ(j,cij)σ(Q^*,c_i^j)=σ(Q^j,c_i^j). ∎ Theorem 1 (Balance of ZOPAs). For any pk∈∗p_k _P^* in ∗Q^* with σ, where σ satisfies balance and strict monotonicity: if ∗(pk)<σ∗(pk)A^*(p_k)<_σS^*(p_k), then Pk∈(∗,σ)P_k (Q^*,σ); and if ∗(pk)≥σ∗(pk)A^*(p_k) _σS^*(p_k), then Pk (∗,σ)P_k (Q^*,σ). Proof. Let us first prove that if ∗(pk)<σ∗(pk)A^*(p_k)<_σS^*(p_k), then Pk∈(∗,σ)P_k (Q^*,σ). Let us compare with some pl∈∗p_l _P^* such that ∗(pl)=σ∗(pl)=σ∗(pk)A^*(p_l)=_σS^*(p_l)=_σA^*(p_k). Here, balance would require that σ(∗,pl)=τ∗(pl)=0.5σ(Q^*,p_l)=τ^*(p_l)=0.5. Then, for it to hold that ∗(pk)<σ∗(pk)A^*(p_k)<_σS^*(p_k), it must be the case that (xk)>σ(xl)S(x_k)>_σS(x_l) since ∗(pk)=∗(pl)A^*(p_k)=A^*(p_l). By Definition 1, τ∗(pk)=τ∗(pl)=0.5τ^*(p_k)=τ^*(p_l)=0.5 and so, by strict monotonicity, σ(,xk)>σ(∗,xl)=0.5σ(Q,x_k)>σ(Q^*,x_l)=0.5. Then, by Definition 4, Pk∈(∗,σ)P_k (Q^*,σ). Next, let us prove that if ∗(pk)≥σ∗(pk)A^*(p_k) _σS^*(p_k), then Pk (∗,σ)P_k (Q^*,σ). Straightforwardly, balance requires that σ(∗,xk)≤τ∗(xk)=0.5σ(Q^*,x_k)≤τ^*(x_k)=0.5. Then, by Definition 4, Pk (∗,σ)P_k (Q^*,σ). ∎ Theorem 2 (ZOPAs under Total Disagreement). If +i=−jC_+^i=C_-^j, −i=+jC_-^i=C_+^j, σ(i,cki)=σ(j,ckj)σ(Q^i,c_k^i)=σ(Q^j,c_k^j) ∀k∈1,…,||∀ k∈\1,…,|C|\ and σ satisfies balance, then Pl≃σ∗PmP_l _σ^Q^*P_m ∀Pl,Pm∈∀ P_l,P_m and (∗,σ)=∅Z(Q^*,σ)= . Proof. By Definition 1, ∀pn∈∗∀ p_n _P^*, ∗(pn)∪∗(pn)=∗A^*(p_n) ^*(p_n)=X_C^* and τ∗(pn)=0.5τ^*(p_n)=0.5. By the same definition, for any Co∈+i∩−jC_o _+^i _-^j, coi∈∗(p)c_o^i ^*(p_p) and coj∈∗(p)c_o^j ^*(p_p) for any Pp∈P_p such that Co∈PpC_o∈ P_p. Meanwhile, coi∈∗(pq)c_o^i ^*(p_q) and coj∈∗(pq)c_o^j ^*(p_q) for any Pq∈P_q such that Co PqC_o P_q. Conversely, for any Cr∈−i∩+jC_r _-^i _+^j, cri∈∗(ps)c_r^i ^*(p_s) and crj∈∗(ps)c_r^j ^*(p_s) for any Ps∈P_s such that Cr∈PsC_r∈ P_s. Meanwhile, cri∈∗(pt)c_r^i ^*(p_t) and crj∈∗(pt)c_r^j ^*(p_t) for any Pt∈P_t such that Cr PtC_r P_t. Then, since σ(∗,cki)=σ(∗,ckj)σ(Q^*,c_k^i)=σ(Q^*,c_k^j) ∀k∈1,…,||∀ k∈\1,…,|C|\, it must be the case that ∗(pl)=σ∗(pl)A^*(p_l)=_σS^*(p_l) and ∗(pm)=σ∗(pm)A^*(p_m)=_σS^*(p_m). Balance then requires that σ(∗,pl)=τ∗(pl)=0.5σ(Q^*,p_l)=τ^*(p_l)=0.5 and σ(∗,pm)=τ∗(pm)=0.5σ(Q^*,p_m)=τ^*(p_m)=0.5. Then, by Definition 2, Pl≃σ∗PmP_l _σ^Q^*P_m and, by Definition 4, (∗,σ)=∅Z(Q^*,σ)= . ∎ Theorem 3 (ZOPAs under Total Agreement). If Pk=+i=+jP_k=C_+^i=C_+^j, Pl=−i=−jP_l=C_-^i=C_-^j, |z()|=|||z(X_C)|=|X_C| and σ satisfies balance and strict monotonicity, then Pk≻σ∗PmP_k _σ^Q^*\!\!~P_m ∀Pm∈∖Pk∀ P_m \P_k\ and Pl≺σ∗PnP_l _σ^Q^*\!\!~P_n ∀Pn∈∖Pl∀ P_n \P_l\. Further, Pk∈(∗,σ)P_k (Q^*,σ) and Pl (∗,σ)P_l (Q^*,σ). Proof. By Definition 1, ∀po∈∗∀ p_o _P^*, ∗(po)∪∗(po)=∗A^*(p_o) ^*(p_o)=X_C^* and τ∗(po)=0.5τ^*(p_o)=0.5. By the same definition, for any Cp∈+i∩+jC_p _+^i _+^j, cpi,cpj∈∗(pk)c_p^i,c_p^j ^*(p_k) since Cp∈PkC_p∈ P_k and cpi,cpj∈∗(pl)c_p^i,c_p^j ^*(p_l) since Cp PlC_p P_l. Conversely, for any Cq∈−i∩−jC_q _-^i _-^j, cqi,cqj∈∗(pk)c_q^i,c_q^j ^*(p_k) since Cq PkC_q P_k and cqi,cqj∈∗(pl)c_q^i,c_q^j ^*(p_l) since Cq∈PlC_q∈ P_l. Thus, it must be the case that ∗(pk)=∗(pl)=∅A^*(p_k)=S^*(p_l)= and ∗(pk)=∗(pl)=∗S^*(p_k)=A^*(p_l)=X_C^*. Then, also by Definition 1, ∗(pm)⊂∗(pk)S^*(p_m) ^*(p_k) and thus ∗(pm)⊃∗(pk)A^*(p_m) ^*(p_k). Similarly, ∗(pn)⊂∗(pl)A^*(p_n) ^*(p_l) and thus ∗(pn)⊃∗(pl)S^*(p_n) ^*(p_l). Then, by similar logic to Lemma 1 but taking into account that all clauses have non-zero strength, i.e. |z(∗)|=|∗||z(X_C^*)|=|X_C^*|, it must be the case that ∗(pk)<σ∗(pm)A^*(p_k)<_σA^*(p_m), ∗(pk)>σ∗(pm)S^*(p_k)>_σS^*(p_m), ∗(pl)>σ∗(pn)A^*(p_l)>_σA^*(p_n) and ∗(pl)<σ∗(pn)S^*(p_l)<_σS^*(p_n). Given that τ∗(pk)=τ∗(pm)τ^*(p_k)=τ^*(p_m) and τ∗(pl)=τ∗(pn)τ^*(p_l)=τ^*(p_n), strict monotonicity requires that σ(∗,pk)>σ(∗,pm)σ(Q^*,p_k)>σ(Q^*,p_m) and σ(∗,pl)<σ(∗,pn)σ(Q^*,p_l)<σ(Q^*,p_n). Then, by Definition 2, Pk≻σ∗PmP_k _σ^Q^*P_m and Pl≺σ∗PnP_l _σ^Q^*P_n, resp. Further, given that ∗(pk)=∅<σ∗(pk)A^*(p_k)= <_σS^*(p_k) and ∗(pl)>σ∗(pl)=∅A^*(p_l)>_σS^*(p_l)= , by Theorem 1, Pk∈(∗,σ)P_k (Q^*,σ) and Pl (∗,σ)P_l (Q^*,σ), resp. ∎