When narrowing options, the options that agents want to decide whether to delete are at most ${\displaystyle \sum _{i=1}^n{k}_i}$ , where n is the number of issues, and $k_i$ is the number of options for each issue in the original domain. This computational cost is less than the domain size. Therefore, the computational cost for narrowing the options is very small because the lists are submitted to the mediator only once. In addition, the only information an agent must submit to the mediator is “whether an option may be deleted”. The only information the other agent can know is regarding “the options that both agents agreed to delete” and “the options that an opposing agent agreed to delete, but that this agent did not agree to”. In other words, the other agent cannot know more information unless it increases the options it agrees to delete. Therefore, each agent can efficiently narrow the options in the negotiation domain space while maintaining privacy.

Researchers design naive agent strategies considering the negotiation domain space for the proposed pre-narrowing protocol. In the main negotiation phase of the protocol, existing agent strategies for multi-issue negotiations can be applied to the pre-domain narrowing phase. Moreover, the predomain-narrowing phase is a part that researchers newly introduced, so the utility functions cannot be used directly as they are. In addition, an agent strategy that is compatible with the predomain-narrowing narrowing phase is required. This naive agent should be based on time-dependent concessions and narrowing within the range of the value the agent can make a maximum concession. It should not consider the opponent’s modeling and future predictions. Therefore, researchers design a simple and effective strategy for each of the following: “narrowing issues using the simultaneous submission method”, “narrowing issues using the pre-negotiation method”, and “narrowing options”.

When using the simultaneous submission method for narrowing issues, agents submit a list of issues that do not need to be negotiated from their perspective to the mediator. The offer related to a submitted issue is selected by the other agent or mediator. Therefore, the agreement on the submitted issues is probably disadvantageous to the submitter. As a naive strategy for the simultaneous submission method, researchers consider a method of submitting a list of issues that are unlikely to disadvantage the submitter. Here, the minimum utility $U_{min}$ that the agent can concede will be used as a parameter, and narrowing down is performed within the range where the acquired utility is expected to be $U_{min}$  or greater.

For a given issue $I_i$ , researchers assume that the probability of each option being selected follows a uniform distribution when deciding an alternative of agreement during the main negotiation; the expected utility is ${\displaystyle w_i\times \frac{1}{k_i}\sum _{j=1}^{k_i}eval(v_j^i)}$ . Moreover, if an agent submits an issue and the option that is most unfavorable to the agent is selected, the utility is ${\displaystyle w_i\times \underset{j}{min}eval(v_j^i)}$ . It can be considered that the smaller the difference between these values, the less likely it is that the agent will be disadvantaged. Therefore, researchers rearrange the issues as $I_{o_1},I_{o_2},\dots ,I_{o_n}$ in ascending order of the differences.

Next, from these issues, the number of issues m to be submitted is decided. If m issues are decided upon and submitted according to the order above, the utilities expected to be obtained are as follows:

The maximum m within the range where the expected utilities do not fall below $U_{min}$ is determined, and the list of issues $I_{o_1},I_{o_2},\dots ,I_{o_m}$  is submitted to the mediator.

When narrowing issues using the pre-negotiation method, each issue is negotiated in a domain where the options are “subject to the main negotiation”, “Agent A chooses”, and “Agent B chooses”. An agent needs to define its utility function for this pre-negotiation domain. In a naive strategy, the utility function for pre-negotiation is defined by the utility expected to be finally obtained. Whenever an agent has the right to choose an issue, it can always choose the option that maximizes its utility. In this case, the expected utility is the maximum. Meanwhile, when the other agent has the right to choose, the expected utility is set as the minimum value, under the assumption that the most unfavorable choice will be made. If an issue is to be a subject in the main negotiation, then assuming that the probability that each option is selected follows a uniform distribution, the average is taken to be the expected utility. Based on the above, Equation (4) is the utility function for pre-negotiation.

In pre-negotiation, the concession strategy is achieved using a concession function wherein the target utility decreases with time. Equation (5) shows the relationship between time t during the pre-negotiation and the target utility. where $U_{min}$ is a parameter representing the lowest utility to which an agent is willing to make a concession. At the start of pre-negotiations, the target utility is the maximum value of 1. The target value then decreases linearly so that it reaches $U_{min}$  at the end of the negotiation. At a given time t, an agent accepts an opponent’s proposal if the utility is $U_{target}(t)$ or greater; otherwise, it replies with a counterproposal that has a utility of $U_{target}(t)$  or greater.

When narrowing options, a list of all options in the issues that was not eliminated when narrowing the issues is submitted to the mediator. In the strategy for deciding a list in this research, researchers use the same parameter $U_{min}$ for narrowing issues. To ultimately obtain a utility of $U_{min}$ or greater in the main negotiation, it is advisable to eliminate all options whose evaluation values are $eval(v_j^i)<U_{min}$

. However, for options where decisions have already been made in the issue-narrowing phase, the decisions cannot be changed. Therefore, the options to be submitted are the set of options $v_j^i$ whose evaluation values satisfy the following:

If all of these options were to be excluded from the scope of the main negotiation, the utility gained when an arbitrary agreement is reached will be $U_{min}$  or greater.