1 Introduction

Negotiation is a key form of interaction among agents that can be used for resolving conflicts and reaching agreements. Formal argumentation is a process based on the construction and comparison of arguments considering the conflicts that may emerge among them. Such conflict are called attacks. The idea is to determine set(s) of non-conflicting arguments (called extensions), which are considered acceptable or justified. The function in charge of calculating the extensions is called semantics [13].

Some works on negotiation argue that argumentation – using explanatory arguments – allows that an agent acquire additional information about his opponents, which can be used for attacking his opponent’s proposals or justifying his own proposals (e.g., [3, 12, 26, 29]). Besides explanatory arguments, there exist other kinds of arguments that can be used in negotiation dialogues and act as persuasive elements aiming to force or convince an opponent to accept readily a given proposal, these are called rhetorical arguments (e.g., in [20,21,22]). According to Ramchurn et al. [27], a negotiation involving these kinds of arguments is called persuasive negotiation.

We can describe the rhetorical arguments as follows: (i) threats, which try to persuade an opponent agent by using the argument that something negative will happen to him if he does not accept to do the requirement sent by the proponent; (ii) rewards, which try to persuade an opponent by using the argument that something positive will happen to him if he accepts to do the requirement sent by the proponent; and (iii) appeals, which try to persuade an opponent in the same form than rewards, but this positive event will depend on the opponent; hence, appeals can be seen as self-rewards [4].

Rhetorical arguments have been studied in terms of speech acts (e.g., [27]) and in other articles, a logical formalization has been given (e.g., [5, 6]). It was also studied how to evaluate their strength values (e.g., [6, 21, 22]). However, to the best of our knowledge, no study about a protocol involving these arguments has been proposed.

In order to better understand the problem, imagine a scenario where two agents, Maria (M) and Carlos (C), are discussing about household chores. Maria is trying to persuade Carlos to do the cleaning of their apartment. The following dialogue shows how agreement is reached:

(1) M: Carlos, could you please do the cleaning?

(2) C: No, I can not, I have to work.

(3) M: If you do the cleaning, I could help you with your reports and you can finish your work early.

(4) C: You can not help me.

(5) M: Why?

(6) C: Because these reports are about a topic you do not know.

(7) M: Well, if you do not do the cleaning, I will not go to your mother’s house on Saturday.

(8) C: If you will not go to my mother’s house, I will not talk to her about the work for your brother.

(9) M: That is not longer necessary, my brother got a job yesterday.

(10) C: OK. You win!

In this example, Maria succeeds in persuading her husband Carlos to do the cleaning of their apartment. On the first attempt to persuade Carlos, Maria uses a reward (line 3), which is not accepted, resulting in an attack to her reward (line 4). In this settings, an attack is a contradictory statement. Since her reward was not successful, she uses a more powerful argument, i.e. a threat(line 7), and then he also answers with another threat (line 8), which we can call a counter-threat. She answers attacking Carlos’ counter-threat (line 9). Notice that this attack is not a counter-threat, which indicates that there is more than one way to attack a rhetorical argument. Finally, Carlos accepts to do the cleaning.

Besides rhetorical arguments and their corresponding attacks, we can notice that an explanation is required during the conversation (line 5), this means that agents can make questions to each other and can use explanatory arguments to justify their opinions. Figure 1 shows the outline of the dialogue in terms of rhetorical arguments, attacks, and other illocutions.

Fig. 1.
figure 1

Outline of the dialogue between Maria and Carlos, which ends successfully for Maria.

Full size image

From this scenario, we can observe that during a persuasive negotiation dialogue agents can exchange rhetorical arguments, attacks to rhetorical arguments, questions, explanatory arguments, and attacks to explanatory arguments.

Most of the research about protocols in literature is focused on negotiation (e,g., [1, 11, 16, 28]), some others on persuasion (e.g., [10, 17]), and some others on argumentation-based dialogues (e.g., [8, 9, 14, 23, 25, 30, 31]). Although some of these protocols take into account explanatory arguments and attacks among them, these are not embedded in a persuasive negotiation dialogue (according to the definition of [27]) and do not interact with other kinds of arguments.

Thus, the research questions that are addressed in this paper are:

  1. 1.

    How can rhetorical arguments, explanatory arguments, attacks, and other illocutions be combined in a coherent dialogue?

  2. 2.

    How can argumentation techniques be used in this kind of dialogue?

In order to address the first question, we propose a protocol that has a dialogue game form where utterances are viewed as moves in a game, which is guided according to a defined set of rules. According McBurney and Parsons [18], formal dialogue games allow sufficient flexibility of expression while avoid state-space explosion. Regarding the second question, an important part of a dialogue is the outcome. In this case, the outcome has to determine the final status of the dialogue (that is, it ends with an agreement or not), the winner (if there is an agreement), and the set of commitments the winner agent has to fulfil after the dialogue, for example, Maria will go to the house of the mother-in-law since Carlos will do the cleaning. We will use argumentation semantics to determine the outcome of the dialogue.

Next section presents the main concepts about argumentation and argumentation semantics. Section 3 concerns with the type of arguments and attacks. Section 4 presents the proposed protocol, that is, the rules that govern the interactions among the agents, the argumentation framework that determines the outcome of the dialogue and the main properties of the approach. In Sect. 5, we illustrate our new protocol by applying it to the example given in Introduction. Finally, Sect. 6 is devoted to conclusions and future work.

2 Background

In this section, we present the concepts of argumentation framework and the acceptability semantics for linear dialogues.

An argumentation framework consists of a set of arguments and a attack relation between them. The following definitions were extracted from [2] and [13].

Definition 1

(Argumentation framework) An argumentation framework is an ordered pair , where \(\mathtt {ARG}\) is a finite set of arguments and a binary relation on \(\mathtt {ARG}\) (i.e., ). We call an attack relation and (A, B) means that argument A attacks argument B.

Before presenting the acceptability notion, it is important to study linear discussions. A linear discussion is a sequence of arguments such that each argument attacks the argument preceding it in the sequence. This sequence will determine which arguments can be considered acceptable and which cannot.

Definition 2

(Linear discussions) Let be an argumentation framework and \(A \in \mathtt {ARG}\). A linear discussion for A in \(\mathtt {ARG}\) is a sequence \(s = \langle A_1, ... , A_n \rangle \) of elements of \(\mathtt {ARG}\) (where n is a positive integer) such that \(A_1 = A\) and \(\forall i \in \{2, 3, .. . , n\}\) .

Next we present the semantics, which determines what arguments are considered acceptable or justified. First, it is necessary the notion of supporters. Given three arguments \(A_1\), \(A_2\), and \(A_3\). If and , then \(A_3\) supports \(A_1\). The acceptability notion is directly related to argument A because it represents the central point of the discussion. Thus, given a sequence s, we can say that \(\forall A_i \in s\), \(A_i\) supports A if i is odd and it attacks A if i is even. Let \(\mathtt {sup}(A)\) return all the supporters of \(A_1\) and \(\mathtt {att}(A)\) return all its attackers.

Definition 3

(Semantics) Let be an argumentation framework, \(A \in \mathtt {ARG}\), s a sequence for A, and n the length of s. It holds that (i) if n is odd, then \(A \cup \mathtt {sup}(A)\) are acceptable, or (ii) if n is even, then \( \mathtt {att}(A)\) are acceptable.

Let be a semantics function that returns the set of acceptable arguments.

3 Building Blocks

In this section, we present the topic language used to represent the content of illocutions exchanged by agents. We also present the definitions of the kinds of arguments that can be exchanged and study the possible attacks to each of them.

According to Van Veenen and Prakken [30], formal dialogue games have a topic language , expressed in a certain logic, and a communication language with a protocol , which specifies the allowed moves at each point in a dialogue. We can say that a persuasive negotiation dialogue happens between a proponent agent P and an opponent agent O about a topic . In this work, the topic language is specified using the classical logical language. Symbols \(\wedge \),\(\vee \) and \(\lnot \) denote the logical connectives conjunction, disjunction, and negation, respectively. Besides, \(\vdash \) stands for the classical inference and \(\equiv \) logical equivalence. From we can distinguish the three following sets of formulas:

  • contains the goals the agent pursues;

  • contains what the agent believes the goals of the other agent are (that is, his opponent’s goals); and

  • is the knowledge base of the agent, which gathers the information the agent has about the environment.

Goals and opponent’s goals are represented with positive literalsFootnote 1 from . Besides, and are finite sets such that .

Now, let us present the definitions of arguments, explanatory arguments, rhetorical arguments, and attacks involved during a persuasive negotiation dialogue. In what follows, for a given argument, the function \(\mathtt {SUPP}\) returns all the beliefs of (called support) used to build the argument and \(\mathtt {CONC}\) returns its conclusion. The first one is a basic definition where any element of can be an argument whereas explanatory and rhetorical arguments have a deductive form. Indeed, a fact or a goal is entailed from the support.

Definition 4

(Argument [19]) Let be a knowledge base. An argument A is \(\varphi \) if with \(\mathtt {SUPP}(A)\) = \(\{\varphi \}\), \(\mathtt {CONC}(A) = \varphi \).

Explanations are the most common category of arguments. They represent the reasons to believe in a fact.

Definition 5

(Explanatory argument [7]) Let be the knowledge base of an agent. An explanatory argument is a tuple such that (i) , (ii) , and (iii) is consistent and minimal.Footnote 2 With: and \(\mathtt {CONC}(A)=h\).

Unlike explanatory arguments, rhetorical arguments are also made up by goals and opponent’s goals.

Definition 6

(Rhetorical arguments [7]) Let be the knowledge base of an agent, be his goals base, and be his opponent’s goals base. A threat, reward, or appeal is a triple such that (i) , (ii) , (iii)  , and (iv) is consistent and minimal. Besides:

  • In the case of threats, it holds that

  • In the case of rewards and appeals, it holds that .

With: , \(\mathtt {CONC}(A)=g\), and \(\mathtt {OPGOAL}(A)=go\) returns the goal that is being threatened, rewarded, or appealed.

It is also necessary to define the attacks each kind of argument may receive. An attack relation between two arguments A and B denotes the fact that these arguments cannot be accepted simultaneously since they contradict each other. In the case of explanatory arguments, two kinds of attacks can be determined, undercuts and rebuttals. An undercutting argument is an argument whose conclusion contradicts some of the elements of the support of another argument, and a rebutting argument is an argument whose conclusion is the negation of the conclusion of another argument. Formally:

Definition 7

(Attacks to explanatory arguments [7]) Let and be two explanatory arguments:

  • undercuts iff such that \(h' \equiv \lnot h''\).

  • rebuts iff \(h \equiv \lnot h'\).

Regarding attacks to rhetorical arguments, we have distinguished three types of attacks. The first one occurs when a threat is attacked by another threat (we can call it a counter-threat). In this case, there is no a logical contradiction but we can notice that the goal threatened by an agent is used by the opponent to construct another threat as can be observed in lines (7) and (8) of the example given in Introduction.

Definition 8

(Counter-threat) Let be the set of goals of an proponent agent P, be a threat of P, and be a threat of opponent agent O. We say that \(th_O\) counter-threatens \(th_P\) when \(g'=go\) and .

The second type of attack occurs when the opponent answers disesteeming his threatened/rewarded/appealed goal and denies his interest in achieving it. For example, line (9) of the example given in Introduction. In line (8), Carlos threatens Maria with not talking with his mother about a job for Maria’s brother and in line (9), she says that her brother already got a job. This attack has the form of an explanatory argument.

Definition 9

(Disesteemation) Let be a rhetorical argument. An argument disesteems A when \(g'=\lnot go\).

To the best of our knowledge, the two previous types of attacks were not studied before. On the other hand, a rhetorical argument can also be undercut.

4 The Proposal

In this section, we present the language and the rules for the dialogue game. Besides, we present the argumentation framework that represents the dialogue and determines its outcome.

4.1 The Proposed Protocol

The game is mainly based on the following ideas. Each move in the dialogue, except the initial one, replies to the previous move of the other agent (we refer to the previous move as its target). In [24], the author proposes the idea of attack and surrender as a categorization of the possible replies to previous moves during the dialogue. A reply is a surrender when it is not against the previous move; otherwise, it is an attack. In summary, a reply can either be an attack or a surrender.

Table 1 presents the persuasion communication language , which takes into account the attacks and the surrenders. In this table, A, B, C, D, E and \(C'\) are arguments. Let us recall that an attack relation between two arguments denotes the fact that these arguments cannot be accepted simultaneously since they contradict each other. Following this idea, we can say that rejecting a request is an attack because it is a contradiction. We can also say that a rhetorical argument attacks a rejection because it defends a different position. Thus, we will consider an attack those defined in previous section – which are more related to logical inconsistency – and also those that support a different position during the dialogue. It is reasonable to think that all the illocutions used by the proponent P aim to support his position, which is contrary to the position of the opponent O. In the attack column, besides the possible attacks, the conditions that relate the attacked with its attacker are stated. In order to standardize the content of speech acts, all of them are arguments; whether they are explanatory, rhetorical, or the basic ones according to Definition 4.

The idea is that the proponent agent uses rhetorical arguments to try to convince his opponent. Thus, in this first version of the protocol only the proponent can use rhetorical arguments, that is threats, rewards, and appeals. Counter-threats act as attacks, in this sense, these can be used by the opponent. After a rhetorical argument, the opponent can accept the proposal, send an attack, or withdraw from the dialogue. The last case, may happen when he has no attack to send but does not want to accept the proposal or when he has an explanation for a questioning. Note that attacks can also be attacked and can be questioned in an element of their support. Only counter-threats cannot be questioned because it is based on goals and not in beliefs.

Table 1. Speech acts and possible replies in
Full size table

The third component of a dialogue game is the protocol, which specifies the allowed moves at each point in a dialogue. Thus, let us define first of all what a move is.

Definition 10

(Move) A move is a tuple \(m= \langle id, sd, tg, sp\rangle \) where:

  • \(id \in \mathbb {N}\) is the identifier of the move;

  • \(sd \in \{P, O\}\) is the sender of the message, i.e. the agent that makes the move;

  • \(tg \in \mathbb {N}\) is target of the move, i.e. a previous move to which it is directed. The target of a move is the identifier of some earlier move in the dialogue;

  • is an speech act.

Let be a set of moves. As for notation, we use id(m), sd(m), tg(m), and sp(m) to refer to each of the components of a given move m. For the sake of simplicity, when we want to refer to the i-th move in a sequence, we use \(m_i\). Besides, we use \(\mathtt {ARGUM}(sp(m))\) to refer to the argument associated to a given speech act.

A dialogue can be seen as a set of moves, which fulfil some conditions. Let us now present the formal definition of dialogue.

Definition 11

(Dialogue) A dialogue D between two agents P and O is a finite sequence \(\langle m_1, ..., m_n \rangle \), such that:

  • \(m_1= \langle 1, P, -, request(A) \rangle \). It means that the first utterance is sent by the proponent agent and has to be a request;

  • The content of \(m_k\) is request(A) iff \(k=1\). It means that a request can only be sent in the first move;

  • \(tg(m_1)=0\). It means that the first utterance has no target;

  • \(\forall k > 1\), it holds that \(tg(m_k)=j\), for \(j=k-1\). It means that the target of a move is always the previous move.

Let stand for the set of all dialogues.

In the illustrative example, we can see that the proponent and the opponent agents take the turn to speak one after another. These moves are controlled by a function that determines which of the agents will make the next move. Take into account that such move must agree with the possible replies defined in Table 1. Thus, a turn-taking function is a mapping , such that given a dialogue \(D=\langle m_1, ..., m_i \rangle \), it holds that (i) \(T(\emptyset ) = P\), (ii) \(T(D)=P\) if i is even, and (iii) \(T(D)=O\) if i is odd. We can notice that our definition of turn-taking forces a strict interleaving between agents P and O.

Next, we define our protocol in terms of legal moves the agents can perform. In Table 1, we can notice that the answer for a speech act why(A) is an explanation for it; however, there is a need for a stop condition \(\mathtt {COND}\) in order to avoid infinite questioning. This condition can be a maximum number of rounds.

Definition 12

(Legal-move function) A legal-move function is a mapping such that, given , for all , the following rules must be satisfied:

  • \(R_1:\) ;

  • \(R_2:\) sp(m) is a legal speech act after D (considering Table 1);

  • \(R_3:\) If \(\exists m_i | tg(m_i)=m_k\) – for \(1 < i \le n\), \(k<i\) – then \(\not \exists m_j | tg(m_j)=m_k\), for \(i \ne j\);

  • \(R_4:\) If \(sp(m)=threat(A)\), \(sp(m)=reward(A)\), or \(sp(m)=appeal(A)\), then \(sd(m)=P\);

  • \(R_5:\) If \(sp(m)=why(A)\) AND \(\mathtt {COND}==true\), then the sender agent has to change his move and use another speech act.

Rule 1 says that the sender of a move has to obey the turn-taking function. Rule 2 has to do with the valid answers to speech acts. Rule 3 means that there is no move with the same target in a dialogue. Rule 4 ensures that only the proponent can send a rhetorical argument. Finally, rule 5 concerns with avoiding infinite questions.

Besides the rules related to legal moves, it is important to define some rules about the beginning and the end of the dialogue.

  • \(R_6:\) If \(id(m)=1\), then \(sd(m)=P\);

  • \(R_7:\) If \(id(m)=1\), then \(sp(m)=request(A)\);

  • \(R_8:\) If \(sp(m) =accept(\varphi )\), then D ends with an agreement;

  • \(R_9:\) If \(sp(m) =concede(\varphi )\), then D ends with an agreement

  • \(R_{10}\) If , then D ends without an agreement;

Rule 6 says that the proponent agent always begins the dialogue and rule 7 asserts that the first movement is a request. Rules 8, 9, and 10 have to do with the termination of a dialogue.

A dialogue system also has effect or commitment rules, which specify the effects of moves on the participants’ commitments. A commitment store gathers the statements each agent have made and the challenges they have issued. Commitment rules define how these commitment stores have to be updated and whether particular illocutions can be uttered at a particular time.

Let and be the commitment stores of the proponent and the opponent agent, respectively. The set of commitment rules is the followings:

  1. 1.

    \(CR_1:\) If \(sp(m_i) = why(A)\) and \(sd(m_i)=P\) then

  2. 2.

    \(CR_2:\) If \(sp(m_i) = reward(A)\) and \(sd(m_i)=P\) then

  3. 3.

    \(CR_3:\) If \(sp(m_i) = threat(A)\) and \(sd(m_i)=P\) then

  4. 4.

    \(CR_4:\) If \(sp(m_i) = explanation(A)\) and \(sd(m_i)=P\) then

  5. 5.

    \(CR_5:\) If \(sp(m_i) = accept(\varphi )\) and \(sd(m_i)=P\) then

It also holds that: (i) \(CR_2\) also holds appeal(A) (ii) \(CR_2\) also holds counter-threat(B); (iii) \(CR_4\) also holds for undercut(B), rebuttal(B), and disesteemate(B); (iv) \(CR_5\) also holds for \(concede(\varphi )\); and (v) these rules hold for . Finally, it holds that both and are consistent.

4.2 Argumentation Framework and Dialogue Outcome

In this subsection, we present the argumentation framework for a persuasive negotiation dialogue and how to determine the outcome of the dialogue based on the semantics defined in Sect. 2.

Definition 13

(Dialogue Argumentation Framework) An AF for a negotiation persuasive dialogue is a tuple such that:

  • is a dialogue constructed under the rules of protocol and the commitment rules ;

  • and are the commitments sets of the proponent and the opponent, respectively;

  • \(\mathtt {ARG}=\{\mathtt {ARGUM}(sp(m_i)) \mid m_i \in D\), for \(1 \le i < n \}\);

  • ;

Recall that the speech acts that end a dialogue are not associated with an argument, which is reflected in the set of arguments. We can notice that \(\mathtt {ARG}\) and form a linear discussion with a sequence \(s = \langle A_1, ..., A_{n-1} \rangle \) where n is the number of movements of D, and \(A_1=\mathtt {ARGUM}(sp(m_1))\). This means that we can apply the semantics given in Definition 14 in order to define the acceptable arguments and based on these arguments, we can define the outcome of the dialogue, that is, the winner of the dialogue. If the proponent wins the dialogue (that is, he persuades his opponent), then the opponent has to perform the required action and the proposed threat, reward, or appeal has to be fulfilled. Note that the proponent can send more than one rhetorical argument, in this case, the last rhetorical argument sent during the dialogue is the one that has to be fulfilled. On the other hand, when the proponent loses the dialogue, the opponent does not have to perform the required action and no offer has to be fulfilled. Before define the outcome of the dialogue, we have to make a modification on the semantics due to the condition of the last movement.

Definition 14

(DAF Semantics) Let be a dialogue argumentation framework, \(A \in \mathtt {ARG}\), s a sequence for A, and \(n=|\mathtt {ARG}|+1\):

  • If n is odd and \(sp(m_n)=accept(\varphi )\) (or \(sp(m_n)=concede(\varphi )\)), then \(\mathtt {att}(A)\) are acceptable.

  • If n is even and \(sp(m_n)=accept(\varphi )\) (or \(sp(m_n)=concede(\varphi )\)), then \(A \cup \mathtt {sup}(A)\) are acceptable.

  • If \(sp(m_n)=withdraw\), then there are no acceptable arguments.

Let be a semantics function that returns the set of acceptable arguments.

We can now define the outcome of the dialogue.

Definition 15

(Dialogue Outcome) Let be a dialogue argumentation framework, \(A \in \mathtt {ARG}\) is an argument that represents a required action, and P and O the proponent and the opponent agent, respectively:

  • If then P wins the dialogue, and O has to perform \(\mathtt {CONC}(A)\).

  • If then P loses the dialogue.

4.3 Properties of the Proposal

In this section, we will study some properties of our proposal. The aim is to evaluate its legality in the sense of fulfillment of the rules and the soundness and completeness of the argumentation process.

The first proposition concerns with the legality of the moves exchanged during the dialogue.

Proposition 1

Given \(D=\langle m_1, ..., m_n \rangle \), and \(D'=\langle m_1, ..., n_m \rangle \), where \(1 \le m \le n\) and considering that is compatible with and fulfils all of the previously established rules. We can say that if \(\forall m\), \(D'\) is a legal dialogue, then D is also a legal dialogue.

Next propositions concerns with the soundness and completeness of the argumentation process.

Proposition 2

  be a dialogue argumentation framework:

  • If \(\forall A \in \mathtt {ARG}\), if , then \(\forall A'\) such that \(A' =\mathtt {ARGUS}(sp(m'))\), \(A =\mathtt {ARGUS}(sp(m))\), and \(sd(m)=sd(m')\), .

  • such that \(A' =\mathtt {ARGUS}(sp(m'))\) and \(A =\mathtt {ARGUS}(sp(m))\), \(sd(m)=sd(m')\).

The first item say that if an argument sent by one the agents is acceptable, then all the arguments sent by the same agent have to be acceptable as well. The second item says that all acceptable arguments were sent by the same agent.

5 Applying the Proposal to the Illustrative Example

In this section, we evaluate if the example given in Introduction fulfills the rules of the protocol. Besides, we determine the outcome of the dialogue.

We use P to refer to Maria because her role in the dialogue is to be the proponent and O to refer to Carlos because his role in the dialogue is to be the opponent. Next, we have the set of moves:

\(m_1\)::

\(\langle 1, P, 0, request(\langle \{cleaning\}, cleaning \rangle )\rangle \)

\(m_2\)::

\(\langle 2, O, 1, reject(\langle \{cleaning\}, cleaning \rangle )\rangle \)

\(m_3\)::

\(\langle 3, P, 2, reward(\langle \{cleaning \rightarrow can\_help, can\_help \rightarrow finish\_work\}, cleaning, finish\_work \rangle )\rangle \)

\(m_4\)::

\(\langle 4, O, 3, undercut(\langle \{\lnot can\_help\}, \lnot can\_help\rangle )\rangle \)

\(m_5\)::

\(\langle 5, P, 4, why (\langle \{\lnot can\_help\}, \lnot can\_help)\rangle \)

\(m_6\)::

\(\langle 6, O, 5, explanation(\langle \{work, work \rightarrow \lnot cleaning\}, \lnot cleaning\rangle )\rangle \)

\(m_7\)::

\(\langle 7, P, 6, threat (\langle \{\lnot cleaning \rightarrow \lnot going\_mother\_house\}, cleaning, going\_mother\_house \rangle )\rangle \)

\(m_{8}\)::

\(\langle 8, O, 7, counte\)-\(threat(\langle \{\lnot going\_mother\_house \rightarrow \lnot talking\_about\_brother\_work\}, going\_mother\_house, talking\_about\_brother\_work \rangle )\rangle \)

\(m_9\)::

\(\langle 9, P, 8, disesteemate(\langle \{brother\_has\_work \rightarrow \lnot talking\_about\_brother\_work \}, \lnot talking\_about\_brother\_work \rangle )\rangle \)

\(m_{10}\)::

\(\langle 10, O, 9, concede(\lnot talking\_about\_brother\_work)\rangle \)

We have a dialogue \(D=\{m_1, m_2, m_3, m_4, m_5, m_6, m_7, m_8, m_9, m_{10}\}\) where the request is the first move and the target of every move is the previous move. Regarding the rules of the legal-move function, we can say that all the moves of D follow these rules.

Let us now present the commitments sets:

  • .

We can notice that both commitment sets are consistent. Note also that includes the requested action as a positive literal (cleaning) whereas includes the requested action as a negative literal (\(\lnot cleaning\)).

Now, let us define the dialogue argumentation framework: where \(\mathtt {ARG}=\{A_1, A_2, A_3, A_4, A_5, A_6, A_7, A_8, A_9\}\) such that each argument is associated to the number of the move in the dialogue D; ; and D and the commitment sets were presented above.

The result of applying the semantics is: . We can now determine the outcome of the dialogue. We can notice that , this means that P (Maria) wins the dialogue and O (Carlos) has to do the cleaning of the apartment (cleaning).

6 Conclusions and Future Work

In this paper, we have presented a protocol for a persuasive negotiation dialogue. In the resulting dialogue game, agents can exchange rhetorical and explanatory arguments, can utter attacks for such arguments, can question an element of explanatory arguments, and also can use negotiation speech acts to request, reject, and finish the dialogue. The use of additional kinds of illocutions enriches the dialogue and allows the agents to not only try to persuade the other party but to defend their positions. The proposed protocol is also flexible since it allows for different alternative replies, which were categorized as attacks and surrenders.

For future research, we propose two possible directions: (i) the first one is to improve the protocol itself, for example, allowing that the target of an utterance to be any of the earlier moves and (ii) the second one is extending the protocol for more than two agents, which also has consequences on the possible attacks for arguments.