Consider the scenario in Figure 7(a): Alice (A) makes the proposal ϕ and Bob (B)
evaluates it, where B's reasoning is based upon two tests:

1. The relation between his CAF and ϕ. B's CAF may be in agreement (ϕ ↔ *flex* kB)
or not with ϕ, and B recognizes it by testing the condition listed above.

2. His stubbornness condition, i.e., if his CAF is stubB (flex kB ↔ *stub*B) or not.

Whenever B is stubborn, he performs the same counterproposal, otherwise he may relax his CAF by the (W) rule of Table 1 (flex kB → *flex* k+1 B) or change his theory by the (C) rule of Table 1 (¬(stubi ↔ flex k i
) and ¬(flex k i → *flex* k+1 i) and ¬(*flex* k+1 i →
flex k i
)).
At the end of his evaluation, B replies to A with a counterproposal ψ. When A
evaluates ψ she has to consider the relation between her CAF and ψ, her stubbornness condition (stubA ↔ *flex* kA) and B's evaluation. The evaluation of the opponent agent helps agents in choosing the new proposal. The choice of the action, weakening or changing theory, and of the next proposal depends on the agent's attitude: a *collaborative agent* chooses the proposal that improves the negotiation relation with the opponent, whereas a *competitive agent* chooses the proposal that changes the least the relation with the opponent. For instance, if B says to A that when A proposes ϕ they are in essence disagreement, and B makes the proposal ψ, A will propose ϕ1 or ϕ2, both inferred from ϕ by applying (W) or (C). When A is collaborative, she will propose ϕ1 because she knows that they will be in agreement. Conversely, A will propose ϕ2, if A is competitive, because she knows that they will remain in essence disagreement.

Suppose B says to A that when A proposes ϕ they are in relative disagreement
(ψ → ϕ) and B makes the proposal ψ, then A knows that they are in agreement when she proposes ψ.

To support the interaction sketched above, we define the system MND to consist of the standard introduction and elimination rules for the connectives of Li and L,
and of two sets of rules: one set for the second proposing agent (Table 2) and another set for the following proposing agents (Table 3). For the sake of space, we omit the assumption of non call-away conditions in negotiation rules and explain only some of the rules by example.

Assume that A begins a MN by proposing *flex* 0A to B. B evaluates τB,A(*flex* 0A)
with respect to his initial angle *flex* 0B and suppose B thinks that τB,A(*flex* 0A) is too strict, i.e., τB,A(flex 0A) → *flex* 0B. Thus, B cannot accept τB,A(*flex* 0A) and re-initiates the MN by the rule (I) and proposes flex 0B by B : *flex* 0B. Otherwise, suppose B thinks that τB,A(*flex* 0A) is entailed by his initial angle *flex* 0B and that τB,A(*flex* 0A) is not too general, i.e., it is not entailed by *stub*B. In this case, B knows that A cannot accept flex 0B because it is too strict with respect to her viewpoint (explained in the beginning of Section 3), thus if B accepts τB,A(*flex* 0A) by (Ag) because it satisfies the precondition (flex 0B → τB,A(*flex* 0A)), and says B : agree(A : *flex* 0A). This is the reason why there is no rule (RD) in Table 2 for relative disagreement relation. Consider the case in which B thinks that the proposal of A, *flex* 0A, is consistent to his initial angle *flex* 0B by (Co).

B says to A that they are in the compatibility relation by B : comp(A : *flex* 0A) and makes a new proposal B : *flex* 1B such that flex 0B → *flex* 1B (rule (W)). Now A thinks that τA,B(*flex* 1B) is an acceptable angle of her initial viewpoint, i.e. flex 1A ↔ τA,B(*flex* 1B).

Thus A agrees with B and says A : agree(B : *flex* 1B) by (*Co-Ag*). It may be the case that agents make proposals that become inconsistent with the received one. This inconsistency is tested by the opponent agent, not by the bidding one, because in MND