# Multi-Issue Negotiation Protocol with Pre-Domain Narrowing
## 5 Experiments  
### Domain  
Agent performance and negotiation outcomes in automated negotiations are known to be influenced by the following three domain elements [35; 36]. **Domain size:**: the number of bids that exist in a domain. The larger the domain size, the larger the amount of computation required for negotiation. The domain size in a multi-issue negotiation problem is expressed by \(\prod_{i=1}^{n}k_{i}\), where \(n\) is the number of issues and \(k_{i}\) is the number of options for the \(i\)th issue. **Bid distribution:**: how bids are distributed in a domain. If the distribution is large, many bids are distanced from the Pareto front. Bid distribution is defined as the average distance from each bid in the domain to the nearest Pareto optimal bid. **Degree of conflict:**: this shows whether there is an alternative of agreement that allows two agents to both obtain high utilities (a win-win proposal). If the degree of conflict is high, the number of bids that can be agreed upon by mutual concession will decrease. The degree of conflict is defined as the distance between the Kalai-Smorodinsky solution and the ideal point \((1.0,1.0)\). In our simulation experiments, to evaluate the proposed protocol regardless of the domain's characteristics, we created a domain set in which the above elements have various values. A total of 31 types of domains were created and used as our evaluation domain set while ensuring that there would be different combinations of the three elements. The method for creating the evaluation domain set is as follows. First, for each issue \(I_{i}\), decide the number of issues \(n\) and the number of options \(k_{i}\). When creating the domains for this study, all \(k_{i}\)s are 6. For example, for a domain with 7 issues, each issue has 6 options, and the domain size is 6\({}^{7}\) = 279,936. The weights for each issue and the evaluation values for each option were randomly generated using a probability distribution, as in the method in PRIANAC [24]. The weights of each issue were generated by taking the Dirichlet distribution to be a probability density function. The parameter of the Dirichlet distribution is a vector with length \(n\). All elements were set to 5. For example, the probability density function that determines the weight of issues in a domain comprising seven issues is \(Dir(5,5,5,5,5,5,5)\). The evaluation value of each option is generated using the beta distribution as a probability density function. Beta distribution parameters \(\alpha\) and \(\beta\) are determined according to the characteristics of the utility function that is to be created. Further, our evaluation domain has no discount factor (\(\delta=1\)), and the reservation value is 0. Table 5 shows the parameters used to create our evaluation domain, and the values for the three elements of the generated domains. All profiles in each domain are named (_number of issues_)-(_profile number_). For each category of the number of issues, domains with various characteristics were generated. Domains 6-8, 7-7, 8-7, and 9-5 were domains with a particular large bid distribution. In these domains, many bids were far from the Pareto front. This measure represents an increased likelihood that an agreement would be detrimental to both agents. In our pre-narrowing protocol, it is expected that eliminating such bids before the main negotiation will allow for a better consensus to be reached. Meanwhile, domains 6-9, 7-8, 8-8, and 9-6 had a particularly high degree of conflict. Because there is no bid in the negotiation domain space for which the utility of both parties is high, it can be said that these are domains where reaching a consensus is difficult. In such domains, if there is insufficient time to search for an alternative of agreement, there is a high possibility that negotiations will break down and no agreement will be reached.