# Multi-Issue Negotiation Protocol with Pre-Domain Narrowing
## 5 Experiments
### Experimental Settings  
To evaluate the proposed pre-narrowing protocol, we performed a negotiation simulation experiment using the 31 types of domains we created. Table 6 compares four patterns of negotiations, including both regular negotiations without a narrowing phase (baseline) and negotiations with the narrowing phase. In the main negotiation phase of (A), (B), and (C), the deadline for negotiation was set to 200 rounds. For (D) and (E), the deadline was set to 100 rounds each for the pre-negotiation and main negotiation, yielding a total of 200 rounds. In the narrowing phase, each agent follows the strategy described in Section 4.4. The main negotiation phase adopts a concession strategy that proposes the offer randomly selected from the ones \begin{table}
\begin{tabular}{l c c} \hline \hline  & **Narrowing** & **Narrowing** \\  & **of Issues** & **of Options** \\ \hline (A) Regular negotiation without narrowing (baseline) & No & No \\ (B) Narrowing of issues using the simultaneous submission & Simultaneous & No \\ (C) Narrowing of issues using the simultaneous submission & Yes & \\ + Narrowing of options & Pre- & No \\ (D) Narrowing of issues by pre-negotiation & negotiation & Yes \\ + Narrowing of options & & \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Outline of comparative experiment. \begin{table}
\begin{tabular}{l c c c c c c c c c} \hline \hline Domain name & 6-1 & 6-2 & 6-3 & 6-4 & 6-5 & 6-6 & 6-7 & 6-8 & 6-9 \\ \hline Parameter \(\alpha\) & 5 & 5 & 3 & 3 & 3 & 3 & 2 & 2 \\ Parameter \(\beta\) & 2 & 2 & 3 & 3 & 4 & 4 & 4 & 5 & 5 \\ Domain size & 46,656 & 46,656 & 46,656 & 46,656 & 46,656 & 46,656 & 46,656 & 46,656 & 46,656 \\ Bid distribution & 0.139379 & 0.148945 & 0.198747 & 0.365260 & 0.308778 & 0.264349 & 0.503257 & 0.575785 & 0.418446 \\ Degree of conflict & 0.129969 & 0.165049 & 0.379772 & 0.103607 & 0.326277 & 0.413255 & 0.100535 & 0.284029 & 0.436136 \\ \hline Domain name & 7-1 & 7-2 & 7-3 & 7-4 & 7-5 & 7-6 & 7-7 & 7-8 \\ \hline Parameter \(\alpha\) & 5 & 5 & 3 & 3 & 3 & 3 & 2 & 2 \\ Parameter \(\beta\) & 2 & 2 & 3 & 4 & 4 & 4 & 5 & 5 \\ Domain size & 279,936 & 279,936 & 279,936 & 279,936 & 279,936 & 279,936 & 279,936 & 279,936 \\ Bid distribution & 0.158757 & 0.158722 & 0.363537 & 0.330697 & 0.283796 & 0.490113 & 0.521724 & 0.429182 \\ Degree of conflict & 0.115045 & 0.173343 & 0.093432 & 0.300188 & 0.377834 & 0.121725 & 0.258319 & 0.426091 \\ \hline Domain name & 8-1 & 8-2 & 8-3 & 8-4 & 8-5 & 8-6 & 8-7 & 8-8 \\ \hline Parameter \(\alpha\) & 5 & 5 & 3 & 3 & 3 & 3 & 2 & 2 \\ Parameter \(\beta\) & 2 & 2 & 3 & 4 & 4 & 4 & 5 & 5 \\ Domain size & 1,679,616 & 1,679,616 & 1,679,616 & 1,679,616 & 1,679,616 & 1,679,616 & 1,679,616 & 1,679,616 \\ Bid distribution & 0.177688 & 0.173045 & 0.384115 & 0.297828 & 0.310945 & 0.461515 & 0.510020 & 0.427693 \\ Degree of conflict & 0.122420 & 0.161021 & 0.130693 & 0.336388 & 0.365515 & 0.109451 & 0.274070 & 0.425334 \\ \hline Domain name & 9-1 & 9-2 & 9-3 & 9-4 & 9-5 & 9-6 & \\ \hline Parameter \(\alpha\) & 5 & 5 & 3 & 3 & 2 & 2 & \\ Parameter \(\beta\) & 2 & 2 & 4 & 3 & 5 & 5 & \\ Domain size & 10,077,696 & 10,077,696 & 10,077,696 & 10,077,696 & 10,077,696 & \\ Bid distribution & 0.173481 & 0.204348 & 0.382773 & 0.409996 & 0.547979 & 0.439740 & \\ Degree of conflict & 0.087268 & 0.108184 & 0.286353 & 0.131805 & 0.314927 & 0.420430 & \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Evaluation domain characteristics. whose utility is more than the value of a time-dependent concession function at that time. The time-dependent concession function for agents is given by  
\[U_{target}(t)=U_{max}-(U_{max}-U_{min})\times t^{1/\epsilon}\ \ (e\in\{0.2,1.0,5.0\}), \tag{7}\]  
where \(U_{max}\) denotes the maximum obtainable utility. In regular negotiations, \(U_{max}=1\), whereas it is possible that \(U_{max}<1\) when the negotiation domain space is narrowed in the narrowing phase. \(U_{min}\) is the parameter representing the lowest utility an agent is willing to concede to.