# Multi-Issue Negotiation Protocol with Pre-Domain Narrowing
## 5 Experiments
### Numerical Experimental Results and Discussion
The reason for this is that if the partial consensus decided in advance while narrowing the issues has low utility for the agents, it is difficult to search for a bid that could be agreed upon under the given conditions. Moreover, when option-narrowing was performed in addition to issue-narrowing, the agreement rate was higher than that of the baseline. By narrowing the options, thereby narrowing the number of bids with low utility for each agent, it became easier to reach an agreement. In addition, when the agents were hard-headed, both having \(U_{min}\) values of 0.9, the agreement rate was less than 1/20 in the baseline but was significantly improved by the introduction of the negotiation domain space-narrowing phase. This is because the negotiation domain space-narrowing phase can determine the parts that make it difficult to reach a consensus, making it easier to search for a bid that leads to a consensus. The results of individual utility (Figure 6) indicate the same trend as the agreement rate results. Introducing the negotiation domain space-narrowing phase makes it more likely that a consensus is reached and increases the individual utility. In other words, this phase gives agents motivation to agree to adopt the negotiation domain space narrowing. Figure 7 shows the change in negotiation time compared with the baseline. By introducing the negotiation domain space-narrowing phase, for all methods, negotiation time became shorter than that of the baseline. By narrowing a large negotiation domain space in advance and then performing the main negotiation, the space searched by each agent during the negotiation became smaller, reducing negotiation time. In particular, when issues were narrowed using the pre-negotiation method and options were narrowed afterward, the negotiation time was reduced to under 1/5 on average. Rather than negotiating in a large negotiation domain space as it is, by deciding in advance the issues that could be narrowed, we could shorten the negotiation time taken to reach a consensus. The results for the distance between the agreed-upon bid and Pareto optimal solution (Figure 8) differed depending on the method used to narrow the issues. In a friendly setting where the \(U_{min}\) for each agent is small, narrowing the issues using the pre-negotiation method yielded better results. Meanwhile, in a setting where agents were more residue in their conflict, the simultaneous submission method yielded better results. When narrowing issues using the pre-negotiation method, only the issues that both agents agree to delete are targeted during the narrowing down process, whereas issues are narrowed down if at least one agent agrees using the simultaneous submission method. Therefore, agents who are \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Agent’s \(U_{min}\)/** & & & & & & \\
**Opponent Agent’s \(U_{min}\)** & **0.9** & **0.8** & **0.7** & **0.6** & **0.5** \\ \hline
0.9 & 0.002998 & 0.015326 & 0.015326 & 0.012549 & 0.010929 \\
0.8 & 0.015326 & 0.028840 & 0.024309 & 0.023478 & 0.023913 \\
0.7 & 0.015326 & 0.024309 & 0.026052 & 0.029582 & 0.033781 \\
0.6 & 0.012549 & 0.023478 & 0.029582 & 0.037623 & 0.042478 \\
0.5 & 0.010929 & 0.023913 & 0.033781 & 0.042478 & 0.045074 \\ \hline \hline \end{tabular}
\end{table} 
Table 10: Results from negotiation without narrowing: distance to Pareto optimal solution. friendly toward one another can allocate issues in such a manner that utility is high for both parties, and the final consensus tends to be close to the Pareto optimal solution. However, when an antagonistic agent participated in the negotiation, narrowing issues using the pre-negotiation was difficult, and we could not narrow the bids far from the Pareto optimal solution. In this case, the result was that it was easier to reach a consensus closer to the Pareto optimal solution when narrowing issues using the simultaneous submission method.