where Ag is the set of the agents with | Ag |= n, ϕ ∈Si∈Ag Li and α is the minimum number of agreeing agents to consider positive the outcome of the MN.

## Theorem 4 Mnd Is Adequate To Represent The Mn Of N Agents.

Proof We consider n agents of which m agents have consistent viewpoints where n ≥
m ≥ α, namely such that there exists a possible common angle. Their stubbornness sets and their flexible sets of formulas are pairwise in one of the EGG/YOLK configurations except number 1. Suppose now that the MND system infers a Σiformula ξ. Then, ξ is a common angle for at least α agents. Conversely, suppose that MND does not infer any Σiformula. Then, some of the m agents are in call-away situation with the received proposal. Suppose now that there are more than n-α agents having pairwise inconsistent viewpoints (configuration 1). The relation established is absolute disagreement. The result is that no formula can be inferred through the system, which is consistent by Theorem 3. Hence, overall, the system is adequate. 

## 3.3 Mn Process Development

In this section, we show how the MN process develops in terms of the changing of the relations between the EGG/YOLKs of the agents. We model the multiparty MN among n agents as an English Auction Game in which the auctioneer negotiates simultaneously with n − 1 agents so that it can be considered as n − 1 bilateral MNs. For this reason we show here how the relation of the EGG/YOLKs changes during a MN process only for bilateral MN. As said above, the stubborn knowledge of the agents never changes during the negotiation but only the flexible part may differ from one step to the next one of a MN. The evolution of the relations between the EGG/YOLKs is different when the stubborn knowledge of the agents are inconsistent, or in a generalization relation, or just consistent or equivalent. We show below the MN development in all the cases.

Agents in MN make offers *flex* such that for each agent i:

## Flex I = Stubi ∧ Φ

where *stub*iis the stubbornness knowledge formula and ϕ is the flexible part of *flex* i
.

Whenever an agent receive the opponent proposal, she does not know which is its flexible part and which the stubbornness one. She only knows which is the relation of the received proposal with respect to her own stubbornness and flexible knowledge.

Only the supervisor system knows the stubbornness knowledge of all the agents. As said above, the stubbornness knowledge is unquestionable and it never changes during the negotiation. Therefore, the MN process can be represented as a path of a graph in which nodes are the EGG/YOLK configurations and edges are the result of the usage of a bidding rule (Table 2 and Table 3).

Suppose that the agents have inconsistent stubbornness knowledge; whatever the deductive rules they use, they remain related as in configuration number 1 and the knowledge is described by a logical formula involving the stubbornness formulas of the agents (see Table 7).

In the following subsections, we describe all the MN situations with respect to the relations of the stubbornness knowledge of the agents.