# Multi-Issue Negotiation Protocol with Pre-Domain Narrowing
## 5 Experiments
### Numerical Experimental Results and Discussion  
Figure 2 shows the rate of issue-narrowing using the simultaneous submission and pre-negotiation methods. The average is taken for each combination of the parameter \(U_{min}\) for the two agents. Figure 3 shows the rate of option-narrowing when options are narrowed in addition to issue-narrowing. Figure 4 shows the rate of domain size reduction that occurs due to the narrowing phase. Figure 2 compares the results from narrowing issues using the simultaneous submission and pre-negotiation methods. When narrowing issues, the smaller the \(U_{min}\) of an agent, the greater the number of issues the agent will be willing to delete. Thus, as the \(U_{min}\) of each agent becomes smaller, the rate of issues narrowed increases. Using the simultaneous submission method, an issue is narrowed if at least one agent agrees to delete it. Because of this, the rate of deleted issues increases based on the smaller of the two agents' \(U_{min}\) values. Meanwhile, using the pre-negotiation method, the deletion of issues is decided based on  
the agreement of both agents. Therefore, the smaller the \(U_{min}\) of both agents, the more issues could be narrowed. When comparing the rate of issues narrowed, in many instances, the pre-negotiation method narrowed down more issues. This is because the rights to select the necessary issues are allocated to each agent, thereby increasing the expected utility for both agents. When options are narrowed after the issue-narrowing, Figure 3 shows the ratio of options narrowed to the total options in the original domain. For the option-narrowing, the larger the \(U_{min}\) of each agent or the smaller the utility of partial consensuses decided by issue-narrowing, the more options are eliminated. Therefore, many options were eliminated not only for parts with small \(U_{min}\) values, where issue-narrowing causes the deletion of several options, but also in cases where both agents' \(U_{min}\) values were large. Figure 4 shows the ratio of domain size reduction to the original domain due to the domain-narrowing phase. By narrowing issues (reducing \(n\)) and narrowing options (reducing \(k_{i}\)), the domain size decreases exponentially because the domain size is represented as \(\prod_{i=1}^{n}k_{i}\) in this problem. From the results of our experiment, we confirmed that owing to the negotiation domain space-narrowing phase, comprising issue- and option-narrowing, the domain size was reduced from 1/100 to 1/1000. Next, we evaluate the negotiation results. Tables 7-10 show the negotiation results for domains that were not narrowed down (A). Meanwhile, the negotiation results for when the negotiation domain space narrowing phase was used, (B)-(E), are shown in Figures 5-8. The ratios are represented by the numbers in the figures when the results of regular negotiation without narrowing are set to 1. Tables 7-9 show the results for agreement rate, individual utility, and negotiation time for regular negotiations without narrowing the domain. The smaller the value of \(U_{min}\), the easier it becomes for both agents to reach a consensus. Therefore, the agreement rate and individual utility increase, whereas the negotiation time shortens. Table 10 shows the results for the distances from the agreed-upon bid to the nearest Pareto optimal solution. If agents' \(U_{min}\) values are small, the consensus is reached earlier. It was often impossible to reach a consensus that was close to Pareto optimal solution, a solution that is good for both agents. Figure 5 shows the change in the agreement rate for negotiations where the negotiation domain space-narrowing phase was introduced in the form of the proposed protocol. When only issue-narrowing was performed using the simultaneous submission method, no significant change was observed in the agreement rate. Meanwhile, when issues were narrowed using the pre-negotiation method, the agreement rate decreased.