3.1 Threats

The use of threats is a well-known strategy in negotiation (e.g., Sycara Reference Sycara1990; Sierra et al. Reference Sierra, Jennings, Noriega and Parsons1998; Ramchurn et al. Reference Ramchurn, Jennings and Sierra2003); however, unlike rewards and appeals, threats have a negative nature.

Based on the three threats given in the example of the Introduction, we can say that a threat is mainly made up of two goals:

1. • An opponent’s goal: It is the goal of the opponent that is being threatened by the proponent. It is a goal that the opponent wants to achieve or maintain. For example, ‘maintaining a good reputation’, ‘gaining customers fidelity’, and ‘avoiding legal problems’.

2. • An outsourced goal of the proponent: It is the goal of the proponent that needs the opponent involvement to be achieved. For example, ‘getting that $\mathtt{COMPANY}$ refunds my ticket’s money’.

Following, we present the formal definition of a threat. This is based on the definition given in Amgoud and Prade (Reference Amgoud and Prade2004), with some modifications that consider the mental states of the negotiating BBGP-based agent and the rule-based approach.

According to this definition, a threat is constructed under the hypothesis that the proponent’s goal will not be achieved, which has a negative effect not only on the proponent but also on the opponent. The negative effect on the proponent is obviously that he does not achieve a goal and the negative effect on the opponent is that he will not achieve one of his goals either. Thus, both agents need each other to achieve their goals.

3.2 Rewards and appeals

Rewards and appeals are also used during a negotiation dialogue as positive persuasive elements (e.g., Sierra et al. Reference Sierra, Jennings, Noriega and Parsons1997; Shi et al. Reference Shi, Tao and Lu2006; Florea & Kalisz Reference Florea and Kalisz2007; Ramchurn et al. Reference Ramchurn, Sierra, Godo and Jennings2007). Both rewards and appeals result in a clear benefit for the opponent agent.

We can construct rewards and appeals in the same way as we construct threats. This means that rewards and appeals are also based on an opponent’s goal and on an outsourced goal of the proponent.

Below, we present the formal definition of rewards and appeals. This is also based on the definition given in Amgoud and Prade (Reference Amgoud and Prade2004), with the necessary modifications that consider the mental states of the negotiating BBGP-based agent.

4.1 Pre-conditions of credibility and preferability

According to Guerini and Castelfranchi (Reference Guerini and Castelfranchi2006), a rhetorical argument has to meet some pre-conditions in order the proponent can reach a negotiation favorable to him. Consequently, the chosen rhetorical argument has to belong to the set of rhetorical arguments that meet such pre-conditions. These pre-conditions are related to the credibility of the proponent agent and to the preferability of the opponent’s goal regarding the requested action.

4.1.2 Preferability

The second pre-condition a proponent agent has to meet in order to attain a favorable negotiation is the preferability (Guerini & Castelfranchi Reference Guerini and Castelfranchi2006). This pre-condition is based on the relation between the opponent’s goal and the action he is required to perform. According to Guerini and Castelfranchi (Reference Guerini and Castelfranchi2006), the opponent’s goal must be more valuable for him (the opponent) than performing the required action.

We first present the criteria that will be evaluated in order to estimate how valuable a goal is for the opponent. Below, we analyze each criteria and indicate how the value of the opponent’s goal will be estimated.

1. 1. Importance of the opponent’s goal: It is related to how meaningful the goal is for the opponent. The value of the importance of a given goal go is a real number represented by: $\mathtt{IMP}(go) \in [0,1]$ where zero means that the goal is not important at all, and one is the maximum importance of the goal. The more important a goal is for the opponent, the more threatenable, rewardable, or appealable this goal is.

2. 2. Effectiveness of the opponent’s goal: It is related to the degree to which an opponent’s goal is successful for persuasion and it is based on the status of the goal in the intention formation process. Let us recall that we are working with BBGP-based agents; therefore, the goals base of the opponent is divided into five sub-sets: active goals, pursuable goals, chosen goals, executive goals, and cancelled goals. A goal is close of being achieved when its status is chosen or executive and it is far of being achieved when its status is active or pursuable. Thus, depending on its status, a goal can be considered more or less threatenable, rewardable, or appealable. Let us analyze each case:

   1. • Threatenable goal: Recall that threats have a negative nature. In terms of the status of a goal, it means that a threat may make a goal go back to a previous status. In this work, we assume that every threatened goal will become cancelled. Therefore, a goal is considered more threatenable when its status is executive and less threatenable when its status is active. This is because an agent has more to lose when an executive goal is threaten than when an active goal is threaten. Regarding a cancelled goal, it is not threatenable at all.

   2. • Rewardable and appealable goal: In this case, both rewards and appeals have a positive nature. In terms of the status of a goal, it means that a reward/appeal may make a goal go forward to an advanced status. In this work, we assume that every rewarded/appealed goal will become executive. Therefore, a goal is considered more rewardable/appealable when its status is cancelled and less rewardable/appealable when its status is chosen. This is because an agent has more to win when a cancelled goal is rewarded/appealed than when a chosen goal is rewarded/appealed. Executive goals cannot be rewarded/appealed because the proponent has nothing to offer that makes them go forward. Therefore, executive goals are not rewardable/appealable at all.

The value of the effectiveness of a goal go depends on the argument that is built from it. We denote by $\mathtt{arg}(go) \in \{th, rw, ap\}$ the type of argument that can be built where th means that the type of argument is a threat, rw means that the type of argument is a reward, and ap means that the type of argument is an appeal. The effectiveness of an opponent’s goal go is represented by $\mathtt{eff}(go) \in \{0,0.25,0.5,0.75,1\}$ such that zero means that go is not effective at all and one means that go is completely effective. The effectiveness of an opponent’s goal is evaluated as follows:

\begin{equation*}\mathtt{EFF}(go)= \left\{ \begin{array}{ll} & \ \textrm{if\ } \mathtt{arg}(go)=th\ \textrm{and\ } go \in \mathcal{G}\mathcal{O}_{canc}, \textrm{or} \\[5pt] 0 & \ \textrm{if\ } \mathtt{arg}(go)=rw\ \textrm{and\ } go \in \mathcal{G}\mathcal{O}_{e}, \textrm{or} \\[5pt] & \ \textrm{if\ } \mathtt{arg}(go)=ap\ \textrm{and\ } go \in \mathcal{G}\mathcal{O}_e \\[2pt] \hline & \ \textrm{if\ } \mathtt{arg}(go)=th\ \textrm{and\ } go \in \mathcal{G}\mathcal{O}_a, \textrm{or} \\[5pt] 0.25 & \ \textrm{if\ } \mathtt{arg}(go)=rw\ \textrm{and\ } go \in \mathcal{G}\mathcal{O}_c, \textrm{or} \\[5pt] & \ \textrm{if\ } \mathtt{arg}(go)=ap\ \textrm{and\ } go \in \mathcal{G}\mathcal{O}_c \\[2pt] \hline & \ \textrm{if\ } \mathtt{arg}(go)=th\ \textrm{and\ } go \in \mathcal{G}\mathcal{O}_p, \textrm{or} \\[5pt] 0.5 & \ \textrm{if\ } \mathtt{arg}(go)=rw\ \textrm{and\ } go \in \mathcal{G}\mathcal{O}_p, \textrm{or} \\[5pt] & \ \textrm{if\ } \mathtt{arg}(go)=ap\ \textrm{and\ } go \in \mathcal{G}\mathcal{O}_p \\[2pt] \hline & \ \textrm{if\ } \mathtt{arg}(go)=th\ \textrm{and\ } go \in \mathcal{G}\mathcal{O}_c, \textrm{or} \\[5pt] 0.75 & \ \textrm{if\ } \mathtt{arg}(go)=rw\ \textrm{and\ } go \in \mathcal{G}\mathcal{O}_a, \textrm{or} \\[5pt] & \ \textrm{if\ } \mathtt{arg}(go)=ap\ \textrm{and\ } go \in \mathcal{G}\mathcal{O}_a \\[2pt] \hline & \ \textrm{if\ } \mathtt{arg}(go)=th\ \textrm{and\ } go \in \mathcal{G}\mathcal{O}_e, \textrm{or} \\[5pt] 1 & \ \textrm{if\ } \mathtt{arg}(go)=rw\ \textrm{and\ } go \in \mathcal{G}\mathcal{O}_{canc}, \textrm{or} \\[5pt] & \ \textrm{if\ } \mathtt{arg}(go)=ap\ \textrm{and\ } go \in \mathcal{G}\mathcal{O}_{canc} \\ \end{array} \right. \end{equation*}

Based on the importance and the effectiveness of a opponent’s goal, it can be estimated how valuable this goal is. Thus, the worth of an opponent’s goal is represented by $\mathtt{WORTH}: \mathcal{G} \mathcal{O} \rightarrow [0,1]$ and it is estimated as follows.

Definition 6 (Worth of the opponent’s goal). Let go be an opponent’s goal, $\mathtt{IMP}(go)$ its importance, and $\mathtt{EFF}(go)$ its effectiveness. The equation for calculating the worth of go is

(1) \begin{equation} \mathtt{WORTH}(go)= \frac{\mathtt{IMP}(go) + \mathtt{EFF}(go)}{2}\end{equation}

We use the average value in order to obtain the final value of the worth of an opponent’s goal because we consider that both criteria are equally significant to make the calculation and they do not overlap each other, since each of them characterizes a different aspect of the goal. We also want to keep the values of the worth of the goal in the same interval than the values of the both criteria, namely importance and effectiveness.

So far, we have analyzed the criteria to estimate how valuable an opponent’s goal is. In order to evaluate the pre-condition preferability, the proponent should also know the value the opponent gives to the required action in order to compare both values. If the value of the opponent’s goal is greater than the value of the required action then, the argument that uses that goal is considered preferable. Let us explain it with human examples. During an assault, the thief threatens the victim with the following sentence: ‘If do not give me your bag, I hurt you’. In this situation, it is rational to think that the physical well-being is above all. Hence, the value of the goal of the victim (the opponent) is greater than the value of the required action. In another scenario, we have a boss who is trying to convince one of his employees to work on Saturdays with the following reward: ‘If you work every Saturday, then I give you a chocolate’. In this situation, it is reasonable to believe that the value of the opponent’s goal is not greater than the value of the required action.

Guerini and Castelfranchi (Reference Guerini and Castelfranchi2006) claim that a threat, reward, or appeal can be considered convincing when it is credible and preferable. However, depending on the scenario, the proponent agent may or not have information about the real value of an action for his opponent. Thus, we can divide the scenarios into: (i) fully informed scenarios, in which the proponent knows both the value of the actions for his opponent and the value of the opponent’s goals; and (ii) partially informed scenarios, in which the proponent only know the value of the opponent’s goals. Therefore, the preferability of a given goal can only be evaluated in fully informed scenarios. An example of this kind of scenario may be a robots scenario, where the agents share the same actions and values for that actions. In partially informed scenarios, the preferability cannot be evaluated; however, the proponent agent has the value of the opponent’s goal. While it is true that the proponent will not know if an argument is convincing, he can base on the value of its strength to decide which argument to choose and send to his opponent.

For fully informed scenarios, the preferability of a given opponent’s goal is determined in the following definition.

Definition 7 (Preferability of an opponent’s goal). Let $go \in \mathcal{G}\mathcal{O}$ be an opponent’s goal and $ac \in \mathcal{A}\mathcal{O}$ an opponent’s action. Goal go is preferable if $\mathtt{WORTH}(go) > \mathtt{VALUE}(ac)$ ; otherwise, it is not preferable.

Notice that in the rescue robots scenario, the base of actions may be the same for all of them. Therefore, we may have that $\mathcal{A} = \mathcal{A}\mathcal{O}$ .

4.2 Steps of the model

In the previous sub-section, we have studied the pre-conditions for a rhetorical argument to be considered convincing. In other words, if a rhetorical argument meets the pre-conditions previously presented, then the proponent agent believes that he is able to convince his opponent to perform the requested action. Assuming that the agent has more than one convincing rhetorical argument, he still needs a way to compare such arguments. Therefore, a strength value for each argument is still necessary. Thus, in this sub-section we will study how these pre-conditions can be combined to obtain the strength of each argument. We propose a strength calculation model based on the previously studied pre-conditions and that can be applied when the proponent has only one possible opponent or when the proponent can choose an opponent to send an argument. The output of this proposal is a set of rhetorical arguments with their respective strength values.

The first step of the calculation model is related to the proponent’s credibility. Let us recall that an opponent agent has a credibility threshold that indicates the lowest value of the reputation of the proponent agent so the arguments of him may be considered credible. Let us suppose that a proponent agent P—with reputation value $\mathtt{REP}(P)=0.6$ —has two opponents $O_1$ and $O_2$ and let $\mathtt{THRES}(O_1)=0.5$ , $\mathtt{THRES}(O_2)=0.8$ be the thresholds of agents $O_1$ and $O_2$ , respectively. Since $\mathtt{REP}(P)\geqslant\mathtt{THRES}(O_1)$ , the proponent P can continue in the process of evaluation of the strength of the arguments addressed to $O_1$ , but it does not occur for agent $O_2$ because $\mathtt{REP}(P)\ngeq \mathtt{THRES}(O_2)$ .

When the proponent is considered credible by his opponent(s), the next step of the model is related to the preferability notion. It is important to highlight that only when the proponent believes that he is credible he can continue to the next step. Thus, regarding preferability two possibilities can be distinguished:

1. 1. Fully informed scenarios: We can differentiate two sets of arguments. One set includes the arguments that are constructed using a preferable opponent’s goal and the other set includes the arguments that are constructed using non-preferable goals. All the arguments of the first set are considered convincing. This means that any of these arguments can make the opponent performs the required action. The strength value of these arguments is considered an absolute value. Thus, in this kind of scenarios the agent is sure that will convince his opponent if the first set has at least one argument.

2. 2. Partially informed scenarios: In this kind of scenario, we only have one set of arguments whose strength values are considered relative values because the agent is not sure about the convincing power of his arguments.

We can notice that the preferability concept has a direct impact on the assurance of the agent that his arguments are convincing or not. However, the preferability itself is not a value that represents the strength of an argument. Since the preferability evaluation is based on the worth of the opponent’s goal, we will use this value in order to rank the arguments addressed to a given opponent. Thus, we will call this value the basic strength of an argument.

Definition 8 (Basic strength). Let $A=\langle T, g, go \rangle$ be a rhetorical argument; the basic strength of A is obtained by applying the same formula used for calculating the worth of go.

(2) \begin{equation} \mathtt{ST\_BASIC}(A)= \mathtt{WORTH}(go)= \frac{\mathtt{IMP}(go) + \mathtt{EFF}(go)}{2}\end{equation}

A direct consequence of the above definition is that the value of the basic strength of a rhetorical argument is a real number between 0 and 1. Formally:

The basic strength is useful for ranking the arguments; however, for a more accurate value of the strength of the arguments, we can also take into account the credibility value. We will call this value the combined strength of an argument.

Before presenting the formula for calculating the combined strength of an argument, let us analyze the following situation. P is a proponent agent and O his opponent, let $\mathtt{REP}(P)=0.6$ be the reputation of agent P, and $\mathtt{THRES}_1(O)=0.5$ and $\mathtt{THRES}_2(O)=0.2$ be two possible thresholds of O. We can notice that $\mathtt{THRES}_1$ reflects a less credulous attitude than $\mathtt{THRES}_2$ ; thus, although P is credible in both cases, the ‘accurate’ value of P’s credibility is different for each case since the difference between $\mathtt{REP}(P)$ and $\mathtt{THRES}_1$ is less than the difference between $\mathtt{REP}(P)$ and $\mathtt{THRES}_2$ . Therefore, the credibility value of P should have an impact on the calculation of the strength of the arguments because the higher the difference between the threshold value and the reputation value is, the higher the credibility of the proponent is.

We use next equation to calculate the ‘accurate’ value of the credibility of P with respect to an opponent O, whose threshold is $\mathtt{THRES}(O)$ .

(3) \begin{equation}\mathtt{ACCUR\_CRED}(P, O)=\mathtt{REP}(P)-\mathtt{THRES}(O)\end{equation}

This value is used to obtain the combined strength of the arguments. Thus, the combined strength of an argument depends on the basic strength of the argument and the ‘accurate’ value of the proponent’s credibility.

Definition 9 (Combined strength). Let $A=\langle T, g, go \rangle$ be a rhetorical argument and $O \in \mathcal{O} pp$ be an opponent whose threatened/rewarded/appealed goal is go. The combined strength of A is obtained by applying:

(4) \begin{equation} \mathtt{ST\_COMB}(A)=\mathtt{ST\_BASIC}(A) \times \mathtt{ACCUR\_CRED}(P,O)\end{equation}

We can say that the value of the combined strength of an argument is a real number that is between zero and the product of the basic strength times the proponent reputation value. Thus, the combined strength has its maximum value when the opponent’s threshold is zero and the basic strength of the argument is maximal. Formally:

Figure 1 depicts a workflow of our approach for the strength evaluation. In summary, first of all, the proponent has to evaluate his credibility with respect to his opponent, if he is not credible enough he stops the process; otherwise, he continues. Depending on the type of scenario, the preferability is evaluated or not. Besides, the agent may choose to take into account the accuracy, in such case the combined strength is calculated; otherwise, he only calculates the basic strength.

Figure 1 Workflow of the proposed strength calculation model

5.1 Consumer Complaint Website scenario

In this scenario, we work with threats and rewards. Recall that this scenario is an example of partially informed scenario; therefore, the value of the arguments strength is considered relative.

THREATS

Next, we present the mental state of agent $\mathtt{CONSUMER}$ and the logical formalization of each threat. Hereafter, we omit some elements from the mental state because they are not necessary.

$\mathtt{CONSUMER}=\langle \mathcal{T}, \mathcal{G}, \mathcal{O} pp, \mathcal{G}\mathcal{O}, \mathcal{S}_{\mathcal{O} pp},\mathcal{S}_{\mathcal{G}\mathcal{O}}, \mathcal{A}, \mathcal{A}\mathcal{O}, \mathtt{REP}\rangle$ where:

$\mathcal{T}=\{\mathcal{F}, \mathcal{S}, \mathcal{D}\}$ such that

$\mathcal{S}=\{\neg$ refund(money) $\rightarrow \neg buy\_again(ticket),$

$\neg$ refund(money) $\rightarrow destroy(rep\_social\_net)$ ,

$\neg$ refund(money) $\rightarrow take(legal\_actions)$ ,

$\neg buy\_again(ticket) \rightarrow \neg gain($ costu_fidel),

$destroy(rep\_social\_net) \rightarrow \neg have(good\_rep)$ ,

$take(legal\_actions) \rightarrow \neg avoid(legal\_probs)\}$

$\mathcal{G}=\{g\}$ such that $g=get(\mathtt{COMPANY},$ ‘refund(money)’)

$\mathcal{O} pp=\{\mathtt{COMPANY}\}$

$\mathcal{G}\mathcal{O}_a=\{go_3\}, \mathcal{G}\mathcal{O}_c=\{go_1\}, \mathcal{G}\mathcal{O}_e=\{go_2\}$ such that $go_1=gain(costu\_\,fidel),$

$go_2=have(good\_rep)$ , and $go_3=avoid(legal\_probs)$

$\mathcal{S}_{\mathcal{O} pp}=\{(\mathtt{COMPANY}, 0.75, \{go_1,go_2,go_3\})\}$ such that $\mathtt{THRES}(\mathtt{COMPANY})=0.75$ , and

$\{go_1, go_2,go_3\}\in TH_{\mathcal{G}\mathcal{O}}$

$\mathcal{S}_{\mathcal{G}\mathcal{O}}=\{(go_1,0.85),(go_2,0.85),(go_3,0.7)\}$ , and $\mathtt{REP}=0.8$ .

From this mental state, the following threats can be generated:

$th_1=\langle T_1, g, go_1 \rangle$ where:

$T_1 \cup \neg \mathtt{SECOND}(g)= \{(\neg$ refund(money) $, \emptyset),$

$ (\neg buy\_again(ticket), \neg$ refund(money) $\rightarrow \neg buy\_again(ticket)),$

$(\neg gain(costu\_\,fidel),\neg buy\_again(ticket) \rightarrow \neg gain(costu\_\,fidel))\}$

$th_2=\langle T_2, g, go_2 \rangle$ where:

$T_2 \cup \neg \mathtt{SECOND}(g)= \{(\neg$ refund(money) $, \emptyset),$

$ \qquad(destroy(rep\_social\_net), \neg$ refund(money) $\rightarrow destroy(rep\_social\_net)),$

$\qquad(\neg have(good\_rep),destroy(rep\_social\_net) \rightarrow \neg have(good\_rep))\}$

$th_3=\langle T_3, g, go_3 \rangle$ where:

$T_3 \cup \neg \mathtt{SECOND}(g)= \{(\neg$ refund(money) $, \emptyset),$

$ \qquad(take(legal\_actions),\neg$ refund(money) $ \rightarrow take(legal\_actions)),$

$\qquad(\neg avoid(legal\_probs),take(legal\_actions) \rightarrow \neg avoid(legal\_probs))\}$

According to the calculation model, firstly the credibility of $\mathtt{CONSUMER}$ has to be evaluated. Since $\mathtt{REP(CONSUMER)}>\mathtt{THRES(COMPANY)}$ (i.e., 0.8 $>$ 0.75), we can proceed to calculate the basic strength values of the threats generated by $\mathtt{CONSUMER}$ . Since there is only one possible opponent, then it is not necessary to make the calculation of the combined strength values. Table 1 shows the basic and combined values of the strength of the threats generated by agent $\mathtt{CONSUMER}$ ; the combined strength is calculated considering that $\mathtt{ACCUR\_CRED}(\mathtt{CONSUMER, COMPANY})=0.8-0.75=0.05$ .

Table 1. Strength values of the threats of agent $\mathtt{CONSUMER}$ in the software agents scenario

Thus, we have that threat $th_2$ —whose opponent’s goal is $have(good\_rep)$ —is the strongest one and threat $th_3$ —whose opponent’s goal is $avoid(legal\_probs)$ —is the least strong threat.

REWARDS

Let us recall the rewards that agent $\mathtt{COMPANY}$ can construct to try to convince agent $\mathtt{CONSUMER}$ to accept only the 20% refund. If $\mathtt{CONSUMER}$ decides to send his strongest threat (i.e., threat $th_2$ ), $\mathtt{COMPANY}$ can construct a counter-threat to such threat. Thus, $\mathtt{COMPANY}$ would have both rewards and threats to support his position. In natural language, the rewards and threat that $\mathtt{COMPANY}$ can generate are

1. • $rw_1: $ If you agree with the 20% of refund, we will give you 10 000 miles.

2. • $rw_2: $ If you agree with the 20% of refund, we will sell you an executive ticket for the price of an economic one for any national destination.

3. • $rw_3: $ If you agree with the 20% of refund, we will give you our service of assistance for elderly for free for any destination.

4. • $th_4:$ You should not destroy my reputation, otherwise I will denounce you for defamation and I will ask for a payment of civil reparations amounting to $1000 in favor of me.

Next, we present the mental state of agent $\mathtt{COMPANY}$ and the logical formalization of the three rewards and the threat.

$\mathtt{COMPANY}=\langle \mathcal{T}, \mathcal{G}, \mathcal{O} pp, \mathcal{G}\mathcal{O}, \mathcal{S}_{\mathcal{O} pp},\mathcal{S}_{\mathcal{G}\mathcal{O}}, \mathcal{A}, \mathcal{A}\mathcal{O}, \mathtt{REP}\rangle$ where:

$\quad\mathcal{T}=\{\mathcal{F}, \mathcal{S}, \mathcal{D}\}$ such that

$\qquad\mathcal{S}=\{accept(refund\_20) \rightarrow gain(miles),$

$\qquad accept(refund\_20) \rightarrow get\_discount(exec\_ticket)$ ,

$\qquad accept(refund\_20) \rightarrow free(elderly\_assist)$ ,

$\qquad\neg drop(destroy\_rep) \rightarrow make(denounce\_difam)$ ,

$\qquad make(denounce\_difam) \rightarrow pay\_repar(1000)$ ,

$\qquad pay\_repar(1000) \rightarrow \neg avoid(extra\_budget)\}$

$\mathcal{G}=\{g_1, g_2\}$ such that $g_1=get(\mathtt{CONSUMER},$ ‘ $accept(refund\_20)$ ’) and

$g_2=get(\mathtt{CONSUMER},$ ‘ $drop(destroy\_rep)$ ’)

$\mathcal{O} pp=\{\mathtt{CONSUMER}\}$

$\mathcal{G}\mathcal{O}_a=\{go_6\}, \mathcal{G}\mathcal{O}_c=\{go_4,go_7\}, \mathcal{G}\mathcal{O}_{canc}=\{go_5\}$ such that $go_4=gain(miles),$

$go_5=get\_discount(exec\_ticket)$ , $go_6=free(elderly\_assist)$ , and $go_7=avoid(extra\_budget)$

$\mathcal{S}_{\mathcal{O} pp}=\{(\mathtt{CONSUMER}, 0.7, \{go_4,go_5,go_6, go_7\})\}$ such that $\mathtt{THRES}(\mathtt{CONSUMER})=0.7$ ,

$\{go_4, go_5,go_6\}\in RW_{\mathcal{G}\mathcal{O}}$ , and $\{go_7\}\in TH_{\mathcal{G}\mathcal{O}}$

$\mathcal{S}_{\mathcal{G}\mathcal{O}}=\{(go_4,0.8),(go_5,0.7),(go_6,0.4),(go_7,0.9)\}$ , and $\mathtt{REP}=0.9$

From this mental state, the following rewards and threat can be generated:

$rw_1=\langle T_1, g_1, go_4 \rangle$ where:

$T_1 \cup \mathtt{SECOND}(g_1)= \{(accept(refund\_20), \emptyset), (gain(miles), accept(refund\_20) \rightarrow$

$ \quad gain(miles))\}$

$rw_2=\langle T_2, g_1, go_5 \rangle$ where:

$T_2 \cup \mathtt{SECOND}(g_1)= \{(accept(refund\_20), \emptyset),$

$ (get\_discount(exec\_ticket), accept(refund\_20) \rightarrow get\_discount(exec\_ticket))\}$

$rw_3=\langle T_3, g_1, go_6 \rangle$ where:

$T_3 \cup \mathtt{SECOND}(g_1)= \{(accept(refund\_20), \emptyset), (free(elderly\_assist), accept(refund\_20) \rightarrow$

$\quad free(elderly\_assist))\}$

$th_4=\langle T_4, g_2, go_7 \rangle$ where:

$T_4 \cup \neg \mathtt{SECOND}(g_2)= \{(\neg drop(destroy\_rep), \emptyset),$

$(make(denounce\_difam), \neg drop(destroy\_rep) \rightarrow make(denounce\_difam))$

$(pay\_repar(1000),make(denounce\_difam) \rightarrow pay\_repar(1000))$

$(\neg avoid(extra\_budget),pay\_repar(1000) \rightarrow \neg avoid(extra\_budget))\}$

First of all, the credibility of $\mathtt{COMPANY}$ has to be evaluated. Since $\mathtt{REP(COMPANY)}>\mathtt{THRES(CONSUMER)}$ (i.e., $ 0.9 > 0.7$ ), we can proceed to calculate the basic strength values of the arguments generated by $\mathtt{COMPANY}$ . Like in the previous case, since there is only one possible opponent, then it is not necessary to make the calculation of the combined strength values. Table 2 shows the basic and combined values of the strength of the rewards and the threat generated by agent $\mathtt{COMPANY}$ ; the combined strength is calculated considering that $\mathtt{ACCUR\_CRED}(\mathtt{COMPANY,CONSUMER})=0.9-0.7=0.2$ .

Thus, we have that reward $rw_2$ —whose opponent’s goal is $get\_discount(exec\_ticket)$ —is the strongest rhetorical argument and reward $rw_1$ —whose opponent’s goal is gain(miles)—is the least strong argument. However, notice that the strength value of the unique threat is very close to the strength value of reward $rw_2$ . This means that depending on the strategy of the agent, he can choose to send a threat or a reward.

Table 2. Strength values of the rewards and the threat of agent $\mathtt{COMPANY}$ in the software agents scenario

5.2 Rescue robots scenario

This is a scenario of a natural disaster, where a set of robot agents have a set of tasks such as: (i) looking through rubble to find survivors, (ii) wandering the area in search of people needing help, (iii) helping disabled people do tasks, and (iv) bringing supplies for survivors. When they find a person who is seriously injured, the robots take him/her to the hospital, otherwise he/she is sent to a shelter. The robots can communicate with each other in order to ask for/send information or to ask for help.

The disaster area is divided into numbered zones, which are named by using ordered pairs. In the disaster area, there is also a robot workshop, where they can supply of power to keep working and be fixed, in case of a damage or failure.

Each agent is in charge of a certain zone and must achieve its own goals with respect to that zone, which are closely related to its tasks. However, robots can help each other in certain situations, for example, to remove heavy debris. It is under these conditions where a persuasive negotiation dialogue may arise, because robots have to decide whether to continue with their tasks and accomplish their own goals or stop to help another robot.

In this scenario, we work with appeals. Notice that this scenario is a sample of fully informed scenario; therefore, the concept of preferability is employed and the arguments strength is considered absolute.

Next, we present three appeals that a robot agent $\mathtt{TOM}$ can generate in order to try to convince another robot agent $\mathtt{BOB}$ to help him with a heavy debris.

1. • $ap_1: $ If you help me, you can win utility points.

2. • $ap_2: $ If you help me, you can recharge your battery since the workshop is next to this zone.

3. • $ap_3: $ If you help me, you can fix your sensor since the workshop is next to this zone.

Next, we present the mental state of agent $\mathtt{TOM}$ and the logical formalization of the three appeals. $\mathtt{TOM}=\langle \mathcal{T}, \mathcal{G}, \mathcal{O} pp, \mathcal{G}\mathcal{O}, \mathcal{S}_{\mathcal{O} pp},\mathcal{S}_{\mathcal{G}\mathcal{O}}, \mathcal{A}, \mathcal{A}\mathcal{O}, \mathtt{REP}\rangle$ where:

$\mathcal{T}=\{\mathcal{F}, \mathcal{S}, \mathcal{D}\}$ such that

$\mathcal{S}=\{help\_with(debris) \rightarrow gain(util\_points),$

$\quad help\_with(debris) \rightarrow go(workshop)$ ,

$\quad go(workshop) \rightarrow recharge(battery)$ ,

$\quad go(workshop) \rightarrow fix(sensor)$ ,

$\mathcal{G}=\{g_3\}$ such that $g_3=get(\mathtt{BOB},$ ‘ $help\_with(debris)$ ’)

$\mathcal{O} pp=\{\mathtt{BOB}\}$

$\mathcal{G}\mathcal{O}_a=\{go_9\}, \mathcal{G}\mathcal{O}_p=\{go_8\},\mathcal{G}\mathcal{O}_c=\{go_{10}\}$ such that

$go_8=gain(util\_points),$ $go_9=recharge(battery)$ , and $go_{10}=fix(sensor)$

$\mathcal{S}_{\mathcal{O} pp}=\{(\mathtt{BOB}, 0.7, \{go_8,go_9,go_{10}\})\}$ such that $\mathtt{THRES}(\mathtt{BOB})=0.7$ , and $\{go_8, go_9, go_{10}\}\in AP_{\mathcal{G}\mathcal{O}}$

$\mathcal{S}_{\mathcal{G}\mathcal{O}}=\{(go_8,0.7),(go_9,0.9),(go_{10},0.75)\}$

$\mathcal{A}=\mathcal{A}\mathcal{O}=\{help\_with(debris)\}$

$\mathcal{A}_{val}=\{(help\_with(debris),0.55)\}$

$\mathtt{REP}=0.8$

From this mental state, the following appeals can be generated:

$ap_1=\langle T_1, g_3, go_8 \rangle$ where:

$T_1 \cup \mathtt{SECOND}(g_3)= \{(help\_with(debris), \emptyset), (gain(util\_points), help\_with(debris) \rightarrow$

$gain(util\_points))\}$

$ap_2=\langle T_2, g_3, go_9 \rangle$ where:

$T_2 \cup \mathtt{SECOND}(g_3)= \{(help\_with(debris), \emptyset),$

$ (go(workshop), help\_with(debris) \rightarrow go(workshop))$

$ (recharge(battery), go(workshop) \rightarrow recharge(battery))\}$

$ap_3=\langle T_3, g_3, go_{10} \rangle$ where:

$T_3 \cup \mathtt{SECOND}(g_3)= \{(help\_with(debris), \emptyset),$

$ (go(workshop), help\_with(debris) \rightarrow go(workshop))$

$ (fix(sensor), go(workshop) \rightarrow fix(sensor))\}$

Like in previous scenarios, the credibility of $\mathtt{TOM}$ has to be evaluated. We have that $\mathtt{TOM}$ is considered credible by $\mathtt{BOB}$ based on $\mathtt{REP(TOM)}>\mathtt{THRES(BOB)}$ (i.e., $0.8 > 0.7$ ). Thus, we can proceed to calculate the basic strength values of the appeals generated by $\mathtt{TOM}$ . Table 3 shows the basic and combined values of the strength of the appeals generated by agent $\mathtt{TOM}$ ; the combined strength is calculated considering that $\mathtt{ACCUR\_CRED}(\mathtt{TOM,BOB})=0.8-0.7=0.1$ .

Table 3. Strength values of the appeals of agent $\mathtt{TOM}$ in the rescue robots scenario

Recall that $\mathtt{ST\_BASIC}(ap)=\mathtt{WORTH}(go)$ such that go is the opponent’s goal that makes up the appeal ap. Thus, we can compare the values of the opponent’s goals and the value of the required action. The result is the following:

$\mathtt{WORTH}(go_8)>\mathtt{VALUE}(help\_with(debris))$ ( $0.6>0.55$ )

$\mathtt{WORTH}(go_9)>\mathtt{VALUE}(help\_with(debris))$ ( $0.825>0.55$ )

$\mathtt{WORTH}(go_{10})\ngtr\mathtt{VALUE}(help\_with(debris))$ ( $0.5<0.55$ )

Therefore, we have that appeals $ap_1$ and $ap_2$ are convincing ones, whereas appeal $ap_3$ is not convincing. This means that the set of arguments that agent $\mathtt{TOM}$ can use during the negotiation dialogue has been reduced to two.

5.3 Patients medication scenario

This is a scenario of a smart Medication Coach Intelligent Agent (MCIA) that supports patients in handling their medicine (Ingeson et al. Reference Ingeson, Blusi and Nieves2018; Blusi & Nieves Reference Blusi and Nieves2019). The MCIA manages different types of information such as the medication plan of the patients, medication restrictions, and the patient’s preferences. Besides it perceives input data about the environment and the user activities through an ARFootnote ⁵ -headset (see Figure 2 for an illustration where a patient interacts with the MCIA through Microsoft HoloLens). The goal of the MCIA is to make sure that patients take their medicines at the times they are specified. With this aim, MCIA sends reminders to the patients; however, they can intentionally dismiss such reminders. It is in this point that rhetorical arguments can be used to try to convince patients to take their medicine. For example, assuming that a patient—let us call him John—needs to take his pills for osteoporosis; however, he intentionally dismisses the reminders. So agent $\mathtt{MCIA}$ has to try to convince him to follow the treatment, he may use one of the following rhetorical arguments:

1. • $ap_{md}$ : If you take your medicine, I will talk with your son to come to visit you more frequently and you can talk more with him.

2. • $th_{md}$ : You should take your medicine, otherwise I will tell your son not to bring your preferred cake.

3. • $rw_{md}$ : If you take your medicine, I will talk with the administrator to buy you the rocking chair you want so much.

Figure 2 A MCIA used in a homelike environment. Extracted from Blusi and Nieves (Reference Blusi and Nieves2019)

Next, we present the mental state of agent $\mathtt{MCIA}$ and the logical formalization of the rhetorical arguments.

$\mathtt{MCIA}=\langle \mathcal{T}, \mathcal{G}, \mathcal{O} pp, \mathcal{G}\mathcal{O}, \mathcal{S}_{\mathcal{O} pp},\mathcal{S}_{\mathcal{G}\mathcal{O}}, \mathcal{A}, \mathcal{A}\mathcal{O}, \mathtt{REP}\rangle$ where:

$\mathcal{T}=\{\mathcal{F}, \mathcal{S}, \mathcal{D}\}$ such that

$\mathcal{S}=\{take(calcium) \rightarrow visit\_of(son, weekly),$

$visit\_of(son, weekly) \rightarrow talk\_more\_with(son)$ ,

$\neg take(calcium) \rightarrow \neg recieve(pre\,f\_cake)$ ,

$take(calcium) \rightarrow have(rocking\_chair)$ ,

$\mathcal{G}=\{g_4\}$ such that $g_4=get(John,$ ‘take(calcium)’)

$\mathcal{O} pp=\{Jonh\}$

$\mathcal{G}\mathcal{O}_a=\{go_{11}\}, \mathcal{G}\mathcal{O}_c=\{go_{12},go_{13}\}$ such that

$go_{11}=have(rocking\_chair),$ $go_{12}=receive(pref\_cake)$ , and $go_{13}=talk\_more\_with(son)$

$\mathcal{S}_{\mathcal{O} pp}=\{(John, 0.85, \{go_{11},go_{12},go_{13}\})\}$ such that $\mathtt{THRES}(John)=0.85$ , $\{go_{11}\}\in RW_{\mathcal{G}\mathcal{O}}$ , $\{go_{12}\}\in TH_{\mathcal{G}\mathcal{O}}$ , and $\{go_{13}\}\in AP_{\mathcal{G}\mathcal{O}}$

$\mathcal{S}_{\mathcal{G}\mathcal{O}}=\{(go_{11},0.85),(go_{12},0.7),(go_{13},0.9)\}$

$\mathtt{REP}=0.9$

From this mental state, the following rhetorical arguments can be generated:

$ap_{md}=\langle T_1, g_4, go_{13} \rangle$ where:

$T_1 \cup \mathtt{SECOND}(g_4)= \{(take(calcium), \emptyset), $

$(visit\_of(son,weekly), take(calcium) \rightarrow visit\_of(son,weekly))\}$

$(talk\_more\_with(son), visit\_of(son, weekly) \rightarrow talk\_more\_with(son))$

$th_{md}=\langle T_2, g_4, go_{12} \rangle$ where:

$T_2 \cup \mathtt{SECOND}(g_4)= \{(\neg take(calcium), \emptyset),$

$ (\neg recieve(pre\,f\_cake), \neg take(calcium) \rightarrow \neg recieve(pre\,f\_cake))$

$rw_{md}=\langle T_3, g_4, go_{11} \rangle$ where:

$T_3 \cup \mathtt{SECOND}(g_4)= \{(take(calcium), \emptyset),$

$ (have(rocking\_chair), take(calcium) \rightarrow have(rocking\_chair))$

First of all, the credibility of $\mathtt{MCIA}$ has to be evaluated. We have that $\mathtt{MCIA}$ is considered credible by John based on $\mathtt{REP(MCIA)}>\mathtt{THRES(John)}$ (i.e., $0.9 > 0.85$ ). Thus, we can proceed to calculate the basic strength values of the rhetorical arguments generated by $\mathtt{MCIA}$ . Table 4 shows the basic and combined values of the strength; the combined strength is calculated considering that $\mathtt{ACCUR\_CRED}(\mathtt{MCIA},John)=0.9-0.85=0.05$ .

Table 4. Strength values of the rhetorical arguments of agent $\mathtt{MCIA}$ in the patients medication scenario

Thus, we have that reward $rw_{md}$ —whose opponent’s goal is $go_{11}=have(rocking\_chair)$ —is the strongest rhetorical argument. Even when goal $go_{13}= talk\_more\_with(son)$ is more important, the fact that it is close to be achieved determines that offering the reward can be more persuasive. We can think in the following manner, $go_{13}$ is chosen, so it is likely that John and her son have already agreed to talk more and the only thing missing is to set the dates; on the other hand, $go_{11}$ is only active, this means that it is still missing to evaluate the goal and to allocate resources (e.g., money) for buying the chair. Therefore, John will gain more by having a rocking chair bought, which can impact on the persuasive power of agent $\mathtt{MCIA}$ .

6.1 Experiment 1

In this experiment, we consider that there are some BBGP-based agents that are credible and others that are not. Besides, we use the basic strength calculation. This leads to three possible situations:

1. 1. Both the proponent and the opponent agents are credible. In this case, a negotiation dialogue begins.

2. 2. The proponent agent is credible, whereas the opponent agent is not credible. In this case, any argument used by the opponent will be evaluated by the proponent due to the opponent low credibility. This means that the proponent does not believe that any of his goals can be threatened/rewarded/appealed. On the other hand, the goals the arguments used by the proponent can impact on the goals of the opponent. Thus, we settled that the opponent has to accept to do the required action.

3. 3. The proponent agent is not credible, whereas the opponent agent is credible. In this case, the negotiation does not even begin, because the proponent will never convince the opponent.

Figures 3 and 4 show the behavior of the variables number of exchanged arguments and number of reached agreements. Recall that for each experiment, we run 1000 negotiation encounters; however, BBGP-based agents only engage in a negotiation when either both are credible or the proponent is credible. We run experiments taking into account different reputation values for the agents and we have noticed that the less the reputation value is the less the number of negotiation encounters is. This is quite rational because low reputation values mean that it is more difficult that agents engage in a negotiation. For the results presented in this experiment, we used a reputation value of 0.8 for both agents and the thresholds are generated randomly in the interval [0, 1] before each negotiation encounter.

Figure 3 Experiment 1: comparison of the variable number of exchanged arguments

Figure 4 Percentage of negotiations that end in an agreement vs. percentage of negotiations that do not end in an agreement. (a) For BBGP-based agents. (b) For IMP-based agents

The fact that BBGP-based agents do not engage in all the negotiations impacts on the experimental variables. Thus, the number of exchanged arguments is indeed less in negotiation between BBGP-based agents (Figure 3). We could believe that it may impact negatively on the variable number of reached agreements because IMP-based agents participate in all the possible negotiations, that is, 1000 negotiation encounters, while BBGP-based agents only participate in some negotiation encounters. However, the results show that even in that conditions, BBGP-based agents reach more agreements than IMP-based agents. Figure 4 shows the percentages of reached agreements, non-reached agreements, and the percentage of the negotiations the BBGP-agents do not engage in. These values reflect the average behavior of the agents. Notice that, although BBGP-based agents do not engage in all the negotiation, they reach more agreements than IMP-based agents.

6.2 Experiment 2

In this experiment, we focus on comparing the performance of the BBGP-based agents considering that they negotiate either in fully informed scenarios or in partially informed scenarios. Let us recall that in fully informed scenarios the proponent agent employs convincing arguments to try to convince his opponent, whereas in partially informed scenarios the agent does not know which arguments are convincing and which are not.

The experimental variables that are taken into account in this experiment are the number of arguments used by the proponent and the number of arguments exchanged during the negotiation. In this experiment, the first variable is especially important given that in fully informed scenarios the proponent knows the value of the required action while in partially informed scenarios the proponent does not know this information. This fact impacts directly on the number of arguments used by the proponent because when he knows this value his persuasive strategy only includes those rhetorical arguments that fulfil the preferability condition, that is, the value of these rhetorical arguments is greater than the value of the action. Thus, agents in fully informed scenarios employ a modified conservative strategy, in which their first rhetorical argument to be sent is the least valued preferable argument.

Figures 5 and 6 show the results of this experiment. In Figure 5, we can observe the behavior of the variable number of proponent arguments. The difference of amount of arguments used by the proponent is very notorious. In average, BBGP-agents in fully informed scenarios use 70 282 arguments, whereas in partially informed scenarios, they use 205 179 arguments during the negotiation encounters. This means that in partially informed scenarios the amount of used arguments is almost three more times than the amount of arguments used in fully informed scenarios. The number of arguments used by the proponent has also an impact on the total number of exchanged arguments (see Figure 6). For this variable, on average, we have that the number of exchanged arguments in fully informed scenarios is 140 030, whereas in partially informed scenarios it is 274 926, which is almost double of arguments.

Figure 5 Experiment 2: comparison of the variable number of arguments sent by the proponent

Figure 6 Experiment 2: comparison of the variable number of exchanged arguments

We defined fully informed scenarios under the premise that the value of the actions is the same for all the participant agents. We are conscientious that in a scenario of robot agents, this premise may be more easily satisfied than in scenarios involving humans. However, with this experiment we wanted to show that the preferability condition has a big impact on the number of arguments used during the negotiation encounters. We have considered that in partially informed scenarios, the proponent agents do not know the value of the actions for their opponents; nevertheless, we could consider that a proponent agent may employ the value he gives to his actions as a reference point to select their arguments. All this aiming at distinguishing convincing arguments and non-convincing arguments.

6.3 Experiment 3

In this experiment, we again evaluate the performance of BBGP-based agents. However, in this experiment, we compare the performance of the basic strength to the performance of the combined strength. Let us recall that for calculating the basic strength, we only take into account the opponent’s goal, whereas for calculating the combined strength, besides the opponents goal, we take into account the accurate credibility of the agent. Recall that the reputation is an evidence of the proponent’s past behavior of an agent with respect to his opponents. We assume that this value is already estimated and it is not private information; thus, the reputation value of an agent is visible for any other agent. On the other hand, the ‘accurate’ value of the credibility of an agent P with respect to an opponent O—whose threshold is $\mathtt{THRES}(O)$ —is given by $\mathtt{ACCUR\_CRED}(P,O) = \mathtt{REP}(P) - \mathtt{THRES}(O)$ . Thus, we want to know how the value of the accurate credibility impacts on the studied variables, that is, the number of arguments sent by the proponent and the number of exchanged arguments during the dialogue.

The settings in this experiment are a little different from the settings in previous experiments. This is because we want to focus on the impact of the accurate credibility. Thus, the settings of this experiment are each agent generates 10 arguments, the reputation of each agents is 1, the threshold of the proponent agent is always 0.8, and the threshold of the opponent goes down from 0.8 to 0.55. Decreasing the opponent’s threshold makes him more credulous and persuadable. We have run six scenarios:

1. • In the first scenario, both agents have the same reputation and threshold values; therefore, the value of the accurate credibility is the same for both agents (i.e., $1-0.8=0.2$ ).

2. • In the second scenario, the threshold of the proponent agent is 0.8 and the threshold of the opponent is 0.75; hence, the value of the accurate credibility is different. Thus, the value of the accurate credibility that the proponent uses for calculating the combined strength of the arguments is 0.25, whereas the value of the accurate credibility that the opponent uses for calculating the combined strength of the arguments is 0.2. This means that there is a difference of 0.05 between both values of the accurate credibility and this also means that in the second scenario the opponent is more persuadable than in the first scenario.

3. • In the remaining scenarios, the threshold of the proponent is 0.8 and the threshold of the proponent goes down in 0.05 in each scenario. Thus, in the third scenario, the threshold of the opponent is 0.7, in the fourth scenario it is 0.65, in the fifth scenario it is 0.6, and in the sixth scenario it is 0.55. This means that the difference between the value of the accurate credibility increases. Thus, in the third scenario, the difference is 0.1, in the fourth scenario it is 0.15, in the fifth scenario it is 0.2, and in the last scenario it is 0.25.

Table 5 shows a resume of these six scenarios, which includes the values of the reputation, threshold, and accurate credibility of agents proponent and opponent.

Table 5. Experiment 3: values of the reputation, threshold, and accurate credibility of agents proponent and opponent

Table 6. Percentage of negotiations that end favorably for the proponent P vs. percentage of negotiations that end favorably for the opponent O

We have run 1000 negotiation encounters for each scenario and besides evaluating the variables of efficiency; we also compare the number of times that the proponent succeeds in persuading the opponent.

Figures 7 and 8 illustrate the results of this experiment. Figure 7 shows the behavior of the variable number of arguments sent by the proponent. When the calculation is done by applying the combined strength equation, we can notice that the greater the difference between the values of the accurate credibility, the fewer the number of arguments the proponent sends. On the other hand, when the calculation is done by applying the basic strength equation, the number of arguments is very similar. Figure 8 shows the behavior of the variable number of exchanged arguments. This result reaffirms that the performance of the combined strength calculation improves as the difference of the values of the accurate credibility increases.

Figure 7 Experiment 3: comparison of the variable number of arguments sent by the proponent

Figure 8 Experiment 3: comparison of the variable number of exchanged arguments

In Table 6, we compare the number of times that the proponent succeeds in persuading the opponent. Each line corresponds to one of the scenarios defined in Table 5. In all the scenarios, when the calculation is based on the basic strength, the percentages of succeed of proponent and opponent are balanced whereas when the negotiation is on the combine strength, the percentage of succeed of the proponent increases as the difference of the accurate credibility values increases. The difference in success percentage is even more notorious in the last scenarios. Indeed, in the fourth scenario the percentage of succeed of the proponent is almost 100% and in the last scenario it is 100%. We can say that if the difference between the values of the accurate credibility is equal to or greater than 0.2, the proponent always succeeds.

We can conclude that while it is true that our approach has a better performance, it is also true that it is necessary to model further knowledge about the opponent. This need of further modeling may be seen as a weakness of the model; however, we can notice that in all the evaluated variables, the model is always more efficient and effective than the other approach. Indeed, when more criteria are employed both the efficiency and the effectiveness increase. Let us recall Experiment 1, IMP-based agents engage in all negotiation encounters (i.e., 1000 encounters) whereas BBGP-based agents only engage in a negotiation encounter when the proponent agent is credible enough (i.e., around 800 encounters). This would mean that IMP-based agents may always achieve more agreements than BBGP-based agents; nevertheless, the results show that the higher the number of generated arguments is, the more agreements the BBGP-agents achieve. The efficiency of the model is even more notorious when the scenario is a fully informed scenario. In this type of scenario, the number of negotiation cycles and the number of exchanged arguments are less than in partial informed scenarios. Thus, when we take into account the criterion credibility and the type of scenario, the efficiency of the model increases even more.