

![0_image_0.png](0_image_0.png)

j : ϕ ¬(stubi ∧ τi,j (ϕ)) i : absDis(j : ϕ) ∩ i : flex 1 i (AD) j : ϕ ¬(flex 0 i ∧ τi,j (ϕ)) ∧ (stubi ∨ τi,j (ϕ) i : essDis(j : ϕ) ∩ i : flex 1 i (ED) j : ϕ ¬(flex 0 i → τi,j (ϕ)) ∧ (τi,j (ϕ) → flex 0 i ) i : flex 0 i (I) j : ϕ (flex 0 i → τi,j (ϕ)) i : agree(j : ϕ) ∩ i : τi,j (ϕ) (Ag) j : ϕ (flex 0 i ∨ τi,j (ϕ)) ∧ ¬(flex 0 i → τi,j (ϕ)) ∧ ¬(τi,j (ϕ) → flex 0 i ) i : comp(j : ϕ) ∩ i : flex 1 i (Co)

![0_image_1.png](0_image_1.png)

Table 2 Rules for the second proposing agent.

## 3.2 Mn Rules

In this section, we provide the deductive rules for the MN process. We distinguish between pairwise MN and one-to-many MN because in the latter case an agent behaves differently when she is the auctioneer. Moreover, in a 1 − n MN the supervisor system ends the negotiation in a positive way when all or an acceptable part of the agents share a common angle: in the former case the negotiation is totally positive and in the latter it is partially positive.

## 3.2.1 Mn Rules: 1-1 Mn

We give the transition rules the agents use to negotiate depending on the mutual negotiation position they test and on their flexibility; these rules are coupled with those in Table 1. No rules are needed for the first proposing agent because she only makes a proposal; conversely the second and the following proposing agents make proposals and assert the evaluation of the received offer.

There are different rules for the second proposing agent and the following ones.

The second proposing agent has to check if the first proposal is too general or too restrictive and thus if the negotiation has to end immediately in the former case (callaway relation), or to re-initiate with a new proposal in the latter case ((I) rule).

Therefore, a following proposing agent receives proposals that are not restrictions of her initial one and she has only to test if a received proposal is not too general.