# Argument-based Multi-Issue Negotiation
(Supplementary Material)

Thalya Fossey, Jean-Guy Mailly, Pavlos Moraitis

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Thalya Fossey, Jean-Guy Mailly, Pavlos Moraitis. Argument-based Multi-Issue Negotiation (Supplementary Material). 2025. hal-05064307

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<figure><img src="image_1.png" /></figure>
# Argument-based Multi-Issue Negotiation
## Supplementary material

Thalya Fossey<fnref n="1" />, Jean-Guy Mailly<fnref n="2" /> and Pavlos Moraitis<fnref n="1" />,<fnref n="3" />

## 1 Control Argumentation Frameworks

2 Control Argumentation Frameworks (CAFs) [Dimopoulos et al., 2018] generalize AFs in various ways. The first one is the

3 integration of several kinds of uncertainties in the framework: about the existence of an argument, about the existence of an attack, or about the direction of an attack. Then, a set of control arguments and attacks allows an agent to influence the acceptability of some target arguments, whatever the actual state of the uncertain parts of the framework.

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10 Definition 1 (Control Argumentation Framework [Dimopoulos et al., 2018]). A Control Argumentation Framework

11 (CAF) is a triple $\mathcal{CAF} = (\mathcal{F}, \mathcal{C}, \mathcal{U})$ where $\mathcal{F}$ is the fixed

12 part, $\mathcal{U}$ is the uncertain part and $\mathcal{C}$ is the control part of $\mathcal{CAF}$

13 with:

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* $\mathcal{F} = (A_F, \rightarrow)$ where $A_F$ is a set of arguments that we know they belong to the system and $\rightarrow \subseteq (A_F \cup A_U) \times (A_F \cup A_U)$ is an attack relation representing a set of attacks for which we are aware both of their existence and their direction.

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* $\mathcal{U} = (A_U, (\rightleftarrows \cup \dashrightarrow))$ where $A_U$ is a set of arguments for which we are not sure that they belong to the system, $\rightleftarrows \subseteq (((A_U \cup A_F) \times (A_U \cup A_F))\setminus \rightarrow)$ is an attack relation representing a set of attacks for which we are aware of their existence but not of their direction and $\dashrightarrow \subseteq (((A_U \cup A_F) \times (A_U \cup A_F))\setminus \rightarrow)$ is an attack relation representing a set of attacks for which we are not aware of their existence but we are aware of their direction, with $\rightleftarrows \cap \dashrightarrow = \emptyset$.

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* $\mathcal{C} = (A_C, \Rightarrow)$ where $A_C$ is a set of arguments that the agent can choose to use or not, and $\Rightarrow \subseteq \{(a_i, a_j) \mid a_i, a_j \in (A_F \cup A_C \cup A_U) \text{ and } a_i \in A_C \text{ or } a_j \in A_C\} \setminus (\rightarrow \cup \rightleftarrows \dashrightarrow))$ is an attack relation.

33 $A_F, A_U$ and $A_C$ are disjoint sets of arguments.

34 Reasoning with CAFs is based on completions, i.e. standard AFs that represent all the possible combinations for taking into account the uncertain arguments and attacks from the CAF uncertain part.

38 Definition 2 (Completion [Dimopoulos et al., 2018]). Given $\mathcal{CAF} = (\mathcal{F}, \mathcal{C}, \mathcal{U})$, a completion of $\mathcal{CAF}$ is an AF $\langle A, R \rangle$, s.t.

41 • $A = A_F \cup A_C \cup A_{comp}$ where $A_{comp} \subseteq A_U$;

• if $(a, b) \in R$, then $(a, b) \in \rightarrow \cup \rightleftarrows \dashrightarrow \cup \Rightarrow$;

• if $(a, b) \in \rightarrow$, then $(a, b) \in R$;

• if $(a, b) \in \rightleftarrows$ and $a, b \in A$, then $(a, b) \in R$ or $(b, a) \in R$;

• if $(a, b) \in \Rightarrow$ and $a, b \in A$, then $(a, b) \in R$.

The reasoning of an agent using a CAF is based on the notion of control configuration, which means selecting some of the control arguments in order to influence the acceptability of arguments. When it is possible to choose a control configuration such that a given set of arguments is accepted in all completions, we say that the CAF is controllable.

**Definition 3** (Controllability [Dimopoulos et al., 2018]). A control configuration of $\mathcal{CAF} = (\mathcal{F}, \mathcal{C}, \mathcal{U})$ is a subset $A_{conf} \subseteq A_C$. Given a set of arguments $T \subseteq A_F$ and a semantics $\sigma$, we say that $T$ is skeptically (resp. credulously) reached by the configuration $A_{conf}$ w.r.t. $\sigma$ if $T$ is included in every (resp. at least one) $\sigma$-extension of every completion of $\mathcal{CAF}' = (\mathcal{F}', \mathcal{C}', \mathcal{U}')$, with $\mathcal{C}' = \langle A_{conf}, \{(a_i, a_j) \in \Rightarrow | a_i, a_j \in (A_F \cup A_U \cup A_{conf})\} \rangle$. We say that $\mathcal{CAF}$ is skeptically (resp. credulously) controllable w.r.t. $T$ and $\sigma$.

In this work, we focus on credulous controllability. With a slight abuse of language, we say that a set of arguments $T$ is accepted in a CAF when this CAF is credulously controllable w.r.t. $T$ and the chosen semantics. Computational approaches based on Quantified Boolean Formulas (QBFs) [Dimopoulos et al., 2018] or SAT-based Counterexample Guided Abstract Refinement (CEGAR) [?] have been proposed to solve controllability issues. Other forms of controllability have been proposed in the literature [Mailly, 2020; Gaignier et al., 2021] and could be of interest for argument-based negotiation, but are kept for future work.

## 2 Argumentation Theories: From Single-Issue to Multi-Issue

We show how to adapt to our multi-issue setting the definition of the agents argumentation theories originally proposed by [Dimopoulos et al., 2021] for single-issue negotiation.

**Definition 4** (Agent's argumentation theory). An agent argumentation theory is defined as $T^{ag} = \langle A^{ag}, \rightarrow_{ag} \rangle$, with:

• $A^{ag} = A_p^{ag} \cup A_e^{ag}$,

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<footnote n="1">¹Université Paris Cité, LIPADE, F-75006 Paris, France</footnote>

<footnote n="2">²Université Toulouse Capitole, IRIT, Toulouse, France</footnote>

<footnote n="3">³Argument Theory, Paris, France</footnote>

<footnote>thalya.fossey, pavlos.moraitis}@u-paris.fr, jean-guy.mailly@irit.fr</footnote>
• $\rightarrow_{ag} = \rightarrow_{p}^{ag} \cup \rightarrow_{e}^{ag} \cup \rightarrow_{m}^{ag}$ where:

- $\rightarrow_{p}^{ag} = \bigcup_{j \in I(x)} \rightarrow_{p,j}^{ag}$ where $\rightarrow_{p,j}^{ag} \subseteq A_{p,j}^{ag} \times A_{p,j}^{ag}$ are attacks between practical arguments of the same issue $i_j(x)$ s.t. $(a, b) \in \rightarrow_{p,j}^{ag}$ if $a \in A_{p,v_i}^{ag}$ and $b \in A_{p,v_i}^{ag}$ with $i_k \neq i_l$,

- $\rightarrow_{e}^{ag} \subseteq A_{e}^{ag} \times A_{e}^{ag}$ are attacks between epistemic arguments,

- $\rightarrow_{m}^{ag} \subseteq A_{e}^{ag} \times A_{p}^{ag}$ are attacks from epistemic arguments to practical arguments.

In Definition 4, the constraints on $\rightarrow_{p,j}^{ag}$ guarantee that two arguments supporting different values of the same issue are attacking each other (for instance, one cannot accept together one argument supporting the color blue and one argument supporting the color red), but there is no such constraint for arguments supporting the same value of the same issue (there may be several arguments in favor of the same color). Notice also that an epistemic argument can attack practical arguments associated with different issues.

**Definition 5 (Agent representation of Opponent's Argumentation Theory).** In order to represent an agent's beliefs over its opponent theory, we introduce new sets of arguments:

• $A_{F_e}^{ag,op} \subseteq A_e^{op}$ (resp. $A_{F_p}^{ag,op} \subseteq A_p^{op}$) are epistemic (resp. practical) arguments s.t. the agent is certain about their existence in their opponent's personal theory,

• $A_{U_e}^{ag,op} \subseteq A_e^{op}$ (resp. $A_{U_p}^{ag,op} \subseteq A_p^{op}$) are epistemic (resp. practical) arguments s.t. the agent is not certain about their existence in their opponent's personal theory.

An agent's representation of its opponent is a $CAF^{ag,op}$ s.t. $CAF^{ag,op} = \langle F^{ag,op}, C^{ag,op}, U^{ag,op} \rangle$ where:

• $F^{ag,op} = \langle A_F^{ag,op}, \rightarrow \rangle$, with $A_F^{ag,op} = A_{F_e}^{ag,op} \cup A_{F_p}^{ag,op}$, and $\langle A_{F_e}^{ag,op}, \rightarrow_e^{ag,op} \rangle$ defining the epistemic arguments subpart s.t. $\rightarrow_e^{ag,op} \subseteq (A_{F_e}^{ag,op} \cup A_{U_e}^{ag,op}) \times (A_{F_e}^{ag,op} \cup A_{U_e}^{ag,op})$. The above also holds for the practical arguments subpart.

• $U^{ag,op} = \langle A_U^{ag,op}, \rightleftarrows_{ag,op} \cup \leftarrow_{ag,op} \rangle$ with $A_U^{ag,op} = A_{U_e}^{ag,op} \cup A_{U_p}^{ag,op}$, $\rightleftarrows_{ag,op} = \rightleftarrows_e \cup \leftarrow_p$, $\leftarrow_{ag,op} = \leftarrow_e \cup \leftarrow_p$, and $\langle A_{U_e}^{ag,op}, \rightleftarrows_e \cup \leftarrow_e \rangle$, with $\rightleftarrows_e \subseteq ((A_{U_e}^{ag,op} \cup A_{F_e}^{ag,op}) \times (A_{U_e}^{ag,op} \cup A_{F_e}^{ag,op}) \setminus \rightarrow_e^{ag,op})$, $\leftarrow_e \subseteq ((A_{U_e}^{ag,op} \cup A_{F_e}^{ag,op}) \times (A_{U_e}^{ag,op} \cup A_{F_e}^{ag,op}) \setminus \rightarrow_e^{ag,op})$ defining the epistemic arguments subpart. The same holds for the practical arguments subpart. $\rightleftarrows_e \cap \leftarrow \neq \emptyset$.

• $C^{ag,op} = \langle A_c^{ag}, \Rightarrow_{ag,op} \rangle$ where $A_c^{ag} \subseteq A_e^{ag}$ is the set of control arguments and $\Rightarrow_{ag,op} \subseteq \{(a_i, a_j) | a_i \in A_c^{ag}$ and $a_j \in A_c^{ag} \cup A_{F_e}^{ag,op} \cup A_{U_e}^{ag,op}\} \setminus (\rightarrow_e^{ag,op} \cup \rightleftarrows_e \cup \leftarrow_e)$.

## 3 Benchmark generation

The experimental evaluation is based on benchmark generator implemented in Python, that generates a multi-issue negotiation instances by computing weights and utilities profiles for

two agents, as well as the argumentation theories and CAFs associated to it.

The theories $T^\alpha = \langle A^\alpha, \rightarrow_\alpha \rangle$ and $T^\beta = \langle A^\beta, \rightarrow_\beta \rangle$ can share a common part, i.e there exist $N_{\alpha,\beta} = \langle A^{N_{\alpha,\beta}}, \rightarrow_{N_{\alpha,\beta}} \rangle$, such that $A^{N_{\alpha,\beta}} = A^\alpha \cap A^\beta$ and $\rightarrow_{\alpha,\beta} = \rightarrow_\alpha \cap \rightarrow_\beta$. Moreover, control arguments are not attacked.

The structure of the theories and the weights' and utilities' profile rely on a number of defined parameters. For our experiments we define a number of issues $n$ in the negotiation. Each issue $i$ is assigned a random number of values $m_i \in [min, max]$. The number of practical arguments is set to be the sum of the maximum number of values for each issue, i.e. $n \times max$, so each value is guaranteed to be supported by at least one practical argument. If there are more practical arguments than there are values (i.e. for at least one issue $i$, $m_i < max$) the remaining practical arguments are randomly assigned to one of the existing values. The number of epistemic and control arguments are also set depending on the number of issues in the negotiation. We define the *density* of the attacks from epistemic to practical arguments and between epistemic arguments. The *density* is a ratio of the number of attacks in the theory to the number of all the possible attacks (which is the square of the number of arguments). The extensions of those theories are computed to retrieve the practical arguments that are mostly acceptable for each issue. We determine the utility of the values based on the acceptability of the practical arguments: we assign a higher utility to the values whose supporting practical arguments are found in many extensions than to the values whose supporting practical arguments are found in fewer (or no) extensions. The weights of the issues are randomly generated. The CAFs theories are generated based on the argumentation theories, $CAF^{\alpha,\beta} = \langle F^{\alpha,\beta}, U^{\alpha,\beta}, C^{\alpha,\beta} \rangle$ is generated from $T^\beta$ (similarly $CAF^{\beta,\alpha} = \langle F^{\beta,\alpha}, U^{\beta,\alpha}, C^{\beta,\alpha} \rangle$ is generated from $T^\alpha$). We define the rate of uncertain arguments, uncertain attacks, and undirected attacks. Moreover, the arguments in the $CAF^{\alpha,\beta}$ satisfy $A_p^\beta \subseteq A_F^{\alpha,\beta}$ (and symmetrically for $CAF^{\beta,\alpha}$). As in [Dimopoulos et al., 2019; Dimopoulos et al., 2021], the attack relations satisfy the following conditions: a) $\rightarrow_{\alpha,\beta} \subseteq \rightarrow_\beta$, b) $\rightarrow_\beta \cap (A_F^{\alpha,\beta} \times A_F^{\alpha,\beta} \subseteq \rightarrow_{\alpha,\beta})$, c) $(\rightleftarrows_{\alpha,\beta} \cup \leftarrow_{\alpha,\beta}) \subseteq (\rightarrow_\beta \setminus \rightarrow_{\alpha,\beta})$, d) $\rightarrow_{\alpha,\beta} \cap \leftarrow_{\alpha,\beta} = \emptyset$. These conditions allow agents to have an accurate model of their opponent.

Arguments in $A_F^{\alpha,\beta} \cup A_U^{\alpha,\beta}$ have a probability of being attacked by a control argument that is either 0 if the arguments does not need to be controlled or a probability $pr$ if they should. This probability allows for more freedom in the creation of the theories, an agent does not automatically have all the control arguments it needs to make the practical arguments supporting the values it wants accepted in its opponent's CAF.

Our experiments include 100 multi-issue negotiations with 2 issues and 100 negotiations with 3 issues, with the number of values for each issue included between $min = 3$ and $max = 5$, where the negotiations 1 to 50 (resp 51 to 100) have the following parameters: for the personal argumentation theories, the density of epistemic arguments attacking practical arguments is 0.05 (resp 0.05), the density of epis-
temic arguments attacking one another is 0.1 (resp 0.1) and agents do not share any arguments (resp. share 5% of epis- temic and practical arguments). For the CAFs, the rate of uncertain arguments and uncertain attacks is 0.2 (resp 0.2), the rate of undirected attacks is 0.1 (resp 0.1), the rate of un- known arguments is 0 (resp. is 0.1) and the probability *pr* that the epistemic arguments are attacked by a control argument is 0.8 (resp 0.85).

## 4 Negotiation Example

Now we run an example of our negotiation protocol in order to provide an illustration of its main features.

**Agents profiles.** Consider a negotiation between partners deciding on a house for retirement. The negotiation object (i.e. the retirement house) have two issues: the location of the house (waterfront or city) and the size of the house (1 or 2 floors).

More formally the agents $\alpha$ and $\beta$ representing one partner each, negotiate about two issues $\mathcal{I}(x) = \{i_1, i_2\}$, where $i_1 = \text{location}$ and $i_2 = \text{size}$, with their respective domains $\mathcal{V}_1 = \{v_1^1, v_2^1\}$, where $v_1^1 = \text{city}$ and $v_2^1 = \text{waterfront}$ and $\mathcal{V}_2 = \{v_1^2, v_2^2\}$, where $v_1^2 = \text{onefloor}$ and $v_2^2 = \text{twofloors}$; both agents have a reservation value equal to 0.6.

The weights of the issues for $\alpha$ are $w_1^\alpha = 0.7$ and $w_2^\alpha = 0.3$, and the utilities of the values are $u_1^\alpha(v_1^1) = 0.6$, $u_1^\alpha(v_2^1) = 0.88$ (for the values of issue $i_1$); and $u_2^\alpha(v_1^2) = 0.79$, $u_2^\alpha(v_2^2) = 0.45$ for the values of issue $i_2$. There are four possible combinations of values, for which we compute the utility for agent $\alpha$:

* $U^\alpha(\{\langle i_1, v_1^1 \rangle, \langle i_2, v_2^1 \rangle\}) = 0.7 \times 0.6 + 0.3 \times 0.79 = 0.657$,

* $U^\alpha(\{\langle i_1, v_1^1 \rangle, \langle i_2, v_2^1 \rangle\}) = 0.7 \times 0.88 + 0.3 \times 0.79 = 0.853$,

* $U^\alpha(\{\langle i_1, v_1^1 \rangle, \langle i_2, v_2^2 \rangle\}) = 0.7 \times 0.6 + 0.3 \times 0.45 = 0.555$,

* $U^\alpha(\{\langle i_1, v_2^1 \rangle, \langle i_2, v_2^2 \rangle\}) = 0.7 \times 0.88 + 0.7 \times 0.45 = 0.751$.

Then, for agent $\beta$, the weights of the issues are $w_1^\beta = 0.6$ and $w_2^\beta = 0.4$, and the utilities are $u_1^\beta(v_1^1) = 0.67$, $u_1^\beta(v_1^2) = 0.84$, $u_2^\beta(v_1^2) = 0.7$ and $u_2^\beta(v_2^2) = 0.75$. The utilities of the combinations of values are as follows:

* $U^\beta(\{\langle i_1, v_1^1 \rangle, \langle i_2, v_2^1 \rangle\}) = 0.6 \times 0.67 + 0.4 \times 0.7 = 0.682$,

* $U^\beta(\{\langle i_1, v_2^1 \rangle, \langle i_2, v_2^1 \rangle\}) = 0.6 \times 0.84 + 0.4 \times 0.7 = 0.784$,

* $U^\beta(\{\langle i_1, v_1^1 \rangle, \langle i_2, v_2^2 \rangle\}) = 0.6 \times 0.67 + 0.4 \times 0.75 = 0.702$,

* $U^\beta(\{\langle i_1, v_2^1 \rangle, \langle i_2, v_2^2 \rangle\}) = 0.6 \times 0.84 + 0.4 \times 0.75 = 0.804$.

At the beginning of the negotiation $\alpha$'s (resp. $\beta$'s) utility target $U_{\text{target}}^\alpha = 0.853$ (resp. $U_{\text{target}}^\beta = 0.804$) is its highest possible utility.

Their argumentation theories ($T^\alpha$ and $T^\beta$ respectively) and their beliefs about their opponent ($CAF^{\alpha,\beta}$ and $CAF^{\beta,\alpha}$ respectively) are depicted by Figure 1, and instantiated below.

Arguments labelled with a value are the practical arguments supporting these values (for instance $p_1$ supports $v_1^1$), and other arguments are epistemic arguments.

* $p_1$: Prefer to live in the city. 247

* $p_2$: Prefer to live in the waterfront because there is more sunlight. 248

* $p_3$: Prefer one floor because it is more practical. 250

* $p_4$: Prefer two floors because it is bigger. 251

* $a$: The city is far from the sea. 252

* $e$: The city is close to the shops. 253

* $b$: Houses are too hot in the summer when there is too much sunlight. 254

* $c$: Two floors are better because with more windows there is more sunlight and the house is brighter. 256

* $d$: Two floors are not the best because stairs can be tiresome. 258

* $f$: Exercise is good for the health. 260

* $p_5$: Prefer to live in the city because the weather is more tempered. 261

* $p_6$: Prefer to live in the waterfront. 263

* $p_7$: Prefer one floor. 264

* $p_8$: Prefer two floors because it gives a better view of the sea. 265

* $g$: I think we can have a better insulated house in the waterfront than in the city. 267

* $h$: With the current legislation, houses that have more than one floor are not allowed in the waterfront so we will not find any. 269

<figure><img src="image_2.png" /><figcaption>Figure 1: The theories of agents $\alpha$ and $\beta$ before the negotiation</figcaption></figure>

For simplification, the weight's and utilities' modeling of the opponent will not be described in this example, but 272

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it is used when choosing the best combination of pairs
$\langle issues, values \rangle$ to send to the opponent. Furthermore, this
example aims at showing the contribution of argumentation
in multi-issue negotiations.

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In the rest of this example, we will focus on the abstract
first part of this example to show how our protocol works.

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**First negotiation round.** We assume that $\alpha$ starts the negotiation. Using the utility-based mechanism and comparing the utility to its current $U_{\text{target}}^{\alpha}(t)$, its best combination of pairs $\langle issues, values \rangle$ is $\omega = \{\langle i_1, v_2^1 \rangle, \langle i_2, v_1^2 \rangle\}$. It must determine whether there are jointly acceptable practical arguments $\Phi = \{\phi^1, \phi^2\}$ supporting these values. It is not the case, $p_2$ which supports $v_1^1$ is not in any stable extension of $T^\alpha$, so it is disregarded. $\alpha$ searches for its next best combination (by decreasing its utility target to the next step) which is $\omega = \{\langle i_1, v_2^1 \rangle, \langle i_2, v_2^2 \rangle\}$. Similarly as for the previous combination, $p_2$ which supports $v_2^2$ so the combination is disregarded. The next best option for $\alpha$ is $\omega = \{\langle i_1, v_1^1 \rangle, \langle i_2, v_1^2 \rangle\}$. It must, again, determine whether there are jointly acceptable practical arguments $\Phi = \{\phi^1, \phi^2\}$ supporting these values. It is the case, e.g., considering the stable extension $S = \{b, d, e, p_1, p_3\}$ (so, $\phi^1 = p_1$ supports $v_1^1$ and $\phi^2 = p_3$ supports $v_2^2$). Now $\alpha$ must search for practical arguments $\Theta = \{\theta^1, \theta^1\}$ in its CAF that would support these values, and make $\beta$ accept the offer built from $\omega$. The only possibility here is $\theta^1 = p_5$ and $\theta^2 = p_8$. However, the target $\Theta = \{p_5, p_8\}$ is not acceptable in all the completions of the CAFs (for instance, in the completions where the attack $(g, p_5)$ exists); and $\alpha$ does not have access to control arguments that could make $\Theta$ acceptable. So $U_{\text{target}}^{\alpha}(t)$ decreases at the next step, and $\alpha$ checks the next best combination of pairs $\langle issues, values \rangle$. There is no other such combination with supporting arguments that are acceptable in an extension. So $\alpha$'s turn is over, $U_{\text{target}}^{\alpha} = 0.6$ (i.e. the reservation value), and the token goes to $\beta$.

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**Second negotiation round.** Now $\beta$ acts as proponent, and identifies its best combination of pairs $\langle issues, values \rangle$ to be $\omega' = \{\langle i_1, v_2^1 \rangle, \langle i_2, v_2^2 \rangle\}$, with a utility equal to its $U_{\text{target}}^{\beta}(t)$.

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In $\beta$'s theory, this combination of values is supported by $\Phi' = \{p_6, p_8\}$ which is not included in any extension of $T^\beta$. So, $\beta$ searches for its next best combination (by decreasing its utility target to the next step). In this case, the next best combination of pairs $\langle issues, values \rangle$ which is $\omega' = \{\langle i_1, v_2^1 \rangle, \langle i_2, v_2^2 \rangle\}$. In $\beta$'s theory, this combination of values is supported by $\Phi' = \{p_6, p_7\}$ which is included in the single stable extension of $T^\beta$, $S' = \{g, h, p_6, p_7\}$. When using the CAF to identify practical arguments supporting these values, agent $\beta$ can use the arguments $\Theta' = \{p_2, p_3\}$. It first checks whether $\Theta'$ is acceptable in $CAF^{\beta,\alpha}$ without using control arguments; this is not the case because of completions where $b$ exists (and thus where $p_2$ is defeated by $b$). Then, searching for a control configuration that makes $\Theta'$ acceptable leads to selecting the configuration $\{g, h\}$ (indeed it can be checked that including these control arguments implies that $b$ and $c$ will always be rejected, so $p_2$ and $p_3$ will be credulously accepted since they defend themselves against other attackers than $b$ and $c$). Agent $\beta$ can thus make an offer $o(\beta, \alpha)$

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consisting of the combination of issues, values and supporting practical arguments ({$\langle i_1, v_2^1, p_2 \rangle$, $\langle i_2, v_1^2, p_3 \rangle$}), the extension containing the practical arguments ({$a, g, h, p_2, p_3$}) and the attacks defending the arguments in the extension ({$\langle a, p_1 \rangle$, $(a, e)$, $(g, b)$, $(h, c)$, $(p_3, p_4)$}).

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When receiving the offer, agent $\alpha$ updates its theory and its CAF to take into account the information received. Then, it checks that the utility of the offer is above $U_{\text{target}}^{\alpha}$ (0.853 > 0.6). In $T^\alpha$, the addition of $g$ and $h$ with the corresponding attacks make the practical arguments $\Theta' = \{p_2, p_3\}$ acceptable (for instance, in the stable extension $\{a, d, g, h, p_2, p_3\}$), so agent $\alpha$ accepts the offer.

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**Without argumentation.** In a protocol without argumentation, $\alpha$ would have sent the offer containing the combination $\omega = \{\langle i_1, v_2^1 \rangle, \langle i_2, v_1^2 \rangle\}$ (and lowers its utility target to the next offer it could accept, i.e. $U_{\text{target}}^{\alpha}(t) = 0.751$). However, the utility for $\beta$ is below its current utility target $U_{\text{target}}^{\beta}(t)$ (0.784 < 0.804). So $\beta$ can not accept the offer, and sends its best offer which contains the combination $\omega = \{\langle i_1, v_2^1 \rangle, \langle i_2, v_2^2 \rangle\}$. The utility for $\alpha$ for this offer is 0.751 which is equal to $U_{\text{target}}^{\alpha}(t)$ so $\alpha$ accepts the offer. When we take into account the arguments of both agents, we can see that the values in the combination $\omega$ of this offer are not acceptable in $T^\beta$. It means this agreement is not coherent with $\beta$'s beliefs and would not hold in a real life situation: the agents would agree on choosing a two-floors house on the waterfront, despite the current legislation preventing it.

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## References

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[Dimopoulos et al., 2018] Yannis Dimopoulos, Jean-Guy Mailly, and Pavlos Moraitis. Control argumentation frameworks. In *Proceedings of the Thirty-Second AAAI Conference on Artificial Intelligence*, (AAAI-18), New Orleans, Louisiana, USA, February 2-7, 2018, pages 4678–4685. AAAI Press, 2018.

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