Multi-Issue Negotiation Protocol with Pre-Domain Narrowing

Abstract

Consensus building among agents is crucial in multi-agent systems because each agent acts independently according to its utility function, and conflict among agents can occur. Therefore, automated negotiation is an essential technology for efficiently resolving conflicts and forming consensuses while also maintaining agents’ privacy. As the domain to be negotiated is large, the computational cost of reaching a consensus increases and the agreement rate decreases. Some negotiation protocols have been proposed wherein a mediator collects the utility information of each agent and creates multiple alternatives of agreements to handle large-scale multi-issue negotiations. However, in such protocols, a limitation is placed on agents’ privacy because all agents have to disclose their private information by following the mediator and pre-decided negotiation rules. In this study, we propose a negotiation protocol with a predomain-narrowing phase to enable efficient negotiations in large-scale domains which can maintain the privacy of information that agents should not disclose to their opponents or the mediator. The proposed protocol divides the negotiation process into a predomain-narrowing phase and the main negotiation phase. In the proposed protocol, the parts subject to negotiation are first narrowed in upon, and then the main negotiation is performed. We also propose two narrowing methods: issue- and option-narrowing. Further, we propose naive agent strategies considering the predomain-narrowing phase. We perform comparative simulation experiments between the baseline negotiation protocol without a domain-narrowing phase and the proposed negotiation protocol with the predomain-narrowing phase. The experimental results show that the proposed protocol achieves higher agreement rates in less negotiation time than the baseline.

1. Introduction

Consensus building among agents has attracted increased attention in recent studies on multi-agent systems. Because autonomous agents act independently according to the goals and preferences of their user (owner), conflicts among agents can occur. Therefore, there is a need for technology that efficiently resolves conflict situations and makes agreements among agents while also maintaining their independence and privacy. To achieve this, automated negotiation is attracting attention [1,2,3]. Agents are also employed in negotiations when a human negotiator is starting a crucial task. They can assist humans in finding and making an agreement during negotiations in real life. They can apply to real-world scenarios, such as e-markets [4,5], cooperative behavior among robots [6], and flexible supply chain networks [7]. Therefore, the development of automated negotiating agents requires further study considering their potential in applications.

A basic automated negotiation problem is the multi-issue bargaining problem, wherein some agents negotiate, attempting to make an agreement. The negotiation domain is shared among agents comprising multiple issues, with each issue having multiple options. Each agent has its preference and aims to maximize its utility in making an agreement. In such a problem, where agents’ preferences differ, if an agent discloses its preferences, exploitation may occur, whereby other agents’ acquisition utilities are unilaterally enhanced. Because of this, agents prefer not to disclose their preferences during negotiation.

In a multi-issue negotiation problem, the problem size, which is the same as the domain size, is expressed by ${\displaystyle \sum _{i=1}^n{k}_i}$, where n is the number of issues and $k_i$ is the number of options for the $i\mathrm{th}$ issue. Therefore, as the number of issues and number of options for each issue increase, the utility space the agent must search through during negotiation widens. In other words, the larger the domain, the more time the processes require, thereby decreasing the negotiation efficiency. Some negotiation protocols have been proposed wherein a mediator collects the utility information of each agent and creates multiple alternatives of agreements to handle large-scale multi-issue negotiation [8,9,10,11]. However, in such protocols, a limitation is placed on agents’ privacy because all agents have to disclose their private information by following the mediator and pre-decided negotiation rules.

In addition, few existing works attempt to define multistage negotiation protocols for narrowing the domain through the negotiations among agents without the mediator. To realize efficient negotiations, there is a need to narrow down the domain space before the actual negotiations are performed and narrow down the negotiation domain space by an efficient negotiation protocol. Moreover, the protocol would be more likely to be adopted if the narrowing phase increases the utility and social welfare each agent could obtain.

In this study, we propose a novel automated negotiation protocol for efficient negotiation in large-scale domain problems. The protocol is designed to decrease the amount of information that agents must disclose to other agents or mediators during a negotiation. In the proposed protocol, the negotiation session is divided into two phases: the “negotiation domain space-narrowing phase” and “main negotiation phase”. The negotiation domain space-narrowing phase is further divided into the narrowing of issues and the narrowing of options for each issue. We propose two methods for narrowing issues. The first is for each agent to inform the mediator of the issues it does not prioritize. The second is to narrow the number of issues through negotiation among agents. Regarding the narrowing of options, each agent submits to the mediator a list of options that it deems unnecessary for negotiation, and the mediator narrows the options based on the lists. We perform comparative simulation experiments between the baseline negotiation protocol without a domain-narrowing phase and the proposed negotiation protocol with a predomain-narrowing phase. We evaluate three measures representing characteristics and hence prepare a well-balanced domain set in the experiments.

The main contributions of this study are as follows.

• We propose a negotiation protocol with a predomain-narrowing phase that can perform efficient negotiations in large-scale domains and keep the private information that agents should not disclose to their opponents and mediators.
• We perform simulation experiments with the proposed protocol, evaluating with multiple measures. When performing negotiations by narrowing the negotiation domain space in advance, the utility obtained by the agents and the agreement rate increase. Further, in the negotiations based on the proposed protocol, the negotiation time can be reduced compared with the baseline.

The remainder of this article is organized as follows. Section 2 explains the problem setup and basic negotiation protocols for a bilateral multi-issue negotiation problem. Section 3 describes relevant existing negotiation protocols. In Section 4, we propose a protocol for negotiation domain space narrowing to facilitate the handling of negotiations in large-scale domains. Section 5 describes the experimental settings for evaluating the proposed protocol as well as the results and considerations. Finally, Section 6 summarizes this study and presents possible future works.

2. Bilateral Multi-Issue Closed Bargaining Problem

This section shows the essential preliminaries related to automated negotiation. It shows the important background of this study to explain the proposed protocol.

2.1. Domain of Negotiation

In bilateral multi-issue closed bargaining problems, two agents, who perform negotiations within a common domain, are considered. The negotiation domain comprises n number of issues $I_1,I_2,\dots ,I_n$ and $k_i$ number of options for each issue $I_i$. The acceptable solutions for negotiations or the alternatives of agreement (bids) are represented by the vector $\overrightarrow{b}=[v_{c_1}^1,v_{c_2}^2,\dots ,v_{c_n}^n]$ comprising each of the options from each issue.

2.2. Utility Function for Agents

Each negotiating agent is assigned its utility function. The utility function with respect to the alternatives of agreement (bid) $\overrightarrow{b}=[v_{c_1}^1,v_{c_2}^2,\dots ,v_{c_n}^n]$ is expressed as follows:

$$U(\overrightarrow{b})=\sum _{i=1}^n\left(w_i\times eval(v_{c_i}^i)\right)$$

(1)

where $w_i$ denotes issue $I_i$’s weight, and $eval(v_{c_i}^i)$ is the evaluation value for each option $v_{c_i}^i$. Issue weight $w_i$ satisfies ${\sum }_{i=1}^n{w}_i=1$ and $w_i\ge 0$, and the evaluation value satisfies $0\le eval(v_{c_i}^i)\le 1$. The objective function of each agent in this study is to maximize its utility obtained by the bid for which a consensus is reached.

Depending on the negotiation scenario, sometimes discount factors or reservation values are assigned. Discount factor $\delta (0\le \delta \le 1)$ is a coefficient that decreases the utility for each bid as the negotiation time progresses. Discount utility $U_{\delta }(\overrightarrow{b},t)$ for a consensus bid at time t is expressed as follows:

$$U_{\delta }(\overrightarrow{b},t)=U(\overrightarrow{b})\times {\delta }^t$$

(2)

The deadline for negotiation is determined by either the actual time or number of proposals made by each agent. For time t, the deadline is normalized to the range of $t\in [0,1]$. The reservation value is the minimum utility that can be obtained when negotiations are ended without a consensus being reached. Discounting is also applied to the reservation value via the discount coefficient, similar to the regular utility.

2.3. Alternating Offers Protocol

For the negotiations covered in this study, we use the alternating offers protocol [12]. This protocol is widely used in bilateral negotiations, wherein two agents take turns choosing actions. Agents may choose one of the following actions when it is their turn: Accept, Offer, or EndNegotiation.

• Accept: accept the (most recent) proposal made by the other agent. This action, however, cannot be selected as the first action at the beginning of a negotiation.
• Offer: reject the other agent’s proposal and propose a new bid.
• EndNegotiation: abandon the negotiation without reaching a consensus.

The deadline for the negotiation is determined by either the execution time or the number of rounds (the number of times agents send proposals to each other). Until the deadline is reached, agents take turns selecting actions. When an agent’s offer is accepted by the other agent, a consensus is built according to that offer (bid). When this occurs, each agent obtains a utility calculated by evaluating the bid using its utility function. Meanwhile, consensus building is considered to have failed if the deadline is reached without a consensus being reached or if one agent chooses EndNegotiation. In such cases, the utility obtained by each agent is either 0 or the reservation value.

2.4. Bargaining Solutions

Some solutions for bilateral bargaining problems include the Pareto optimal solution and Kalai-Smorodinsky solution.

• Pareto optimal solution: this is a solution in which there is no means of improving an agent’s utility without degrading at least one other agent’s utility. The line connecting the Pareto optimal solutions, or the set of all Pareto optimal solutions, is called a Pareto front.
• Kalai-Smorodinski solution: this is the point where the line segment connecting the reference point $[0,0]$ and the ideal point $[1,1]$ of the utility space intersects the Pareto front [13]. The Kalai-Smorodinsky solution may not be acceptable in the problem set up of this study where the bids are discrete.

3. Related Works

In large-scale multi-issue negotiation, the negotiation domain size increases exponentially as the number of issues and options increases. To solve this problem, some negotiation protocols have been studied. Some existing negotiation protocols for multiple interdependent issue negotiations introduce the effect of heuristic methods that aim to find suboptimal agreements. We briefly cover this section’s most relevant related works and provide a comparison table in

Table 1. Comparison matrix for selected relevant related studies.

	Our Work	Faratin 1998 [14] etc.	Hattori 2007 [8]	Hara 2013 [9]	Fujita 2014 [10]
Bi/Multi-lateral	Bilateral	Bilateral	Multilateral	Multilateral	Multilateral
Utility Function	Weighted-sum	Linear	Constraint	Constraint	Constraint
Mediator in main
negotiation phase	No	Yes	Yes	Yes	Yes
Privacy	Yes	No	No	No	No
Primary Approach of
improving efficiency	Pre-domain narrowing	Meta heuristics	Multi narrowing	Genetic algorithm	Issue clustering

.

Faratin et al. [14] proposed methods based on the linear combinations of simple functions, called tactics, that manipulate the utility of contracts. These methods were divided into tradeoffs and issue manipulation mechanisms [15]. They later presented a tradeoff strategy, where multiple negotiation decision variables were traded off against one another to develop a heuristic computational model for this strategy and showed that it could increase social welfare for the considered system [16]. Their algorithm used the notion of fuzzy similarity to approximate the preference structures of other agents and used hill-climbing to explore the space of possible tradeoffs for the most acceptable. Klein et al. [17] proposed negotiation protocols focusing on nonlinear utility functions and described a simulated annealing-based approach that achieved near-optimal social welfare for negotiations.

Hattori et al. [8] proposed a protocol that repeatedly narrows the negotiation domain space. Each agent performs a rough search of the negotiation domain space, identifying the range wherein its utility is expected to be high and submits this to the mediator. The mediator presents a range within the negotiation domain space that is an intersection of the ranges submitted by each agent. By repeating these operations multiple times, the scope of negotiation is gradually narrowed, until a single consensus is ultimately decided upon. When using this protocol, there is a possibility that no intersection exists when agents submit extremely narrow ranges, and that narrowing fails. Moreover, if agents submit ranges that are too extensive, it is then difficult to efficiently narrow the domain.

Hara et al. [9] proposed a meditation protocol that employs a genetic algorithm for negotiations based on utility that changes over time. In this protocol, the mediator prepares multiple alternatives of agreement and presents them to the agents. Each agent reorders these alternatives in descending order of their utilities and submits them to the mediator. Based on the submitted orders, the mediator uses a genetic algorithm to search for Pareto-dominated bids. The Pareto optimal solution is obtained by repeating this operation.

Fujita et al. [10] proposed a method of searching for the final offer by simulated annealing, where the negotiation domain space is first partitioned using clustering. In a utility space where a dependent relationship exists among issues, an agent submits a group of issues with a strong dependent relationship to the mediator. The mediator obtains the union of the issue groups submitted by each agent and then searches for a consensus for each group of issues. In the final search for an alternative of agreement, the mediator creates a bid and presents it to the agent. The agent replies regarding whether it agrees with the proposed consensus. Based on this reply, simulated annealing is used to search for a better consensus.

Maestre et al. [18,19] improved an auction-based protocol that was first proposed by Ito et al. [20], using weighted constraints and addressing highly-rugged utility spaces. However, achieving high scalability in negotiations with multiple interdependent issues remained a problem. Maestre et al. [21] also proposed a recursive nonmediated bargaining mechanism involving two agents who simultaneously exchange proposals defined as negotiation domain spaces. Further, Lang et al. [22] presented two configurable multi-issue negotiation protocols inspired by heuristic optimization algorithms for centralized problems, the so-called metaheuristics. In particular, both proposed protocols could efficiently reach a consensus, even for complex, nonlinear negotiation domain spaces. However, this centralized situation is sometimes unrealistic for real-world negotiations, so we consider a decentralized approach for multi-issue negotiation problems. Kakimoto et al. [11] proposed a protocol where the negotiation domain space is recursively partitioned. Each agent submits a graph expressing the interdependence of issues in the negotiation domain space to the mediator. The mediator clusters issues based on the submitted information, creating a dendrogram. The negotiation domain space is partitioned recursively based on this dendrogram, a partial consensus is sought, and then a consensus is decided upon.

In many negotiation protocols, including the aforementioned, the search for an alternative of agreement depends on the mediator. In such protocols, an agent must submit the required information to the mediator as shown in Table 1. It is often the case that the mediator also creates the bids. Using such protocols, on the one hand, an alternative of agreement can be efficiently sought in complex negotiation problems. On the other hand, the privacy of each agent becomes limited. Autonomous agents may not prefer to disclose information about their utility. Moreover, if agents cannot create a bid among themselves, whether a consensus that is better for all agents can be reached will depend on the mediator. Thus, there is a need for a negotiation protocol where offers can be efficiently sought while maintaining the agents’ privacy.

As automated negotiation has attracted attention, the International Automated Negotiating Agents Competition (ANAC), a competition where agents are created under common protocols and rules, has been held more than ten times. The agents negotiate and compete for better results in the negotiations. This competition serves as a place where various negotiation strategies are proposed worldwide. The agent strategies submitted to the competition are shared within the community and contribute to the development and evaluation of new negotiation strategies [23,24]. In this competition, some automated negotiation platforms have been developed. For example, Genius [25] provided a common platform for automated negotiating agents, standardizing domains (the field of negotiation), and negotiation protocols. In addition, NegMAS [7] developed as a general-purpose negotiation platform, which allows users to simulate parallel or complex negotiations that reflect actual negotiation environments. Ebrahimnezhad and Fujita proposed NegoSim, a negotiation simulator that introduces a negotiation framework called EUBOA. Different APIs allow users complete control over developing and modifying automated negotiation agents, protocols, preferences, and the ability to create or modify different GUIs [26]. Recently, agent negotiation strategies considering reinforcement learning have attracted attention because of their ability to adapt to various scenarios and opponents [27,28,29].

Recently, some negotiation protocols and negotiation frameworks were proposed focusing on the application of automated negotiation (e.g., concurrent bilateral negotiation, electric vehicle (EV), supply chain finance (SCF), and supply chain management (SCM)). Baarslag et al. [30] presented Bargaining Chips: a framework for one-to-many concurrent composite negotiations, where multiple deals can be reached and combined. It aims to evaluate general asynchronous negotiation and coordination strategies in a setting that generalizes over some existing negotiation approaches. Mohammad [31] considered a set of concurrent negotiations with a utility function defined only for the complete set of agreements and no locally defined ordering of outcomes in any negotiation. The paper presents an algorithm that allows agents to maximize their expected global utility by orchestrating its behavior in all negotiation threads. Khan et al. [32] proposed a multi-issue negotiation protocol between active consumers and a management platform to establish load coordination in a highly congested network for EVs. The multi-issue negotiation protocol simultaneously considers the consumption interval, the price, and the size of the energy packages. Fiedler [33] proposed a multi-agent system to automate and facilitate selecting the best financing options in SCF. An automated negotiation process in the form of an auction realizes it by modeling the information asymmetry between external and internal supply chain actors and additional external effects for internal supply chain actors while taking over financing. Usha et al. [34] presented a lifecycle model of a negotiation agent, identifying the components of automated negotiation and their interactions by surveying methods used in the automated negotiation literature.

4. Negotiation Protocol with Pre-Domain Narrowing

We propose a negotiation pre-narrowing protocol to enable efficient negotiation in a large negotiation domain space. In this protocol, we introduce a negotiation domain space-narrowing phase, where issues and options are narrowed before the main negotiation. By narrowing the negotiation domain space before the main negotiation, the computational cost of searching for offers in a large negotiation domain space is reduced. In addition, it is possible to prevent inconvenient agreements during the main negotiation by eliminating options from which each agent would obtain little utility in advance. We assume that the issues are independent to work effectively in multi-issue negotiation because each agent needs to calculate the narrowing issues and options with high computational costs considering the dependencies among issues.

Figure 1 shows the outline of the proposed protocol. A negotiation domain space-narrowing phase is newly set up before the main negotiation phase, where regular negotiations are performed. In the negotiation domain space-narrowing phase, issues are narrowed down in the first step, and then options within each issue are narrowed. In the first step, it is decided which issues will not be a subject in the main negotiation, and agreements on these issues are concluded in advance. In the next step, for each issue, options to be excluded as major negotiation subjects are decided. We propose two methods for narrowing issues: the “simultaneous submission method” and “pre-negotiation method”.

4.1. Narrowing Issues Using the Simultaneous Submission Method

In the simultaneous submission method, both agents simultaneously submit issues that do not need to be subject to negotiation, thereby narrowing the number of issues to be negotiated.

Table 2. An example of issue-narrowing using the simultaneous submission method (“×” represents an unnecessary issue).

Issue	Agent		Determination Method
	A	B	for Agreeing on an Option
Destination			Main negotiation
Travel expense	×		Agent B chooses
Travel method			Main negotiation
Accommodation	×	×	Select randomly
Meals		×	Agent A chooses

shows an example of narrowing issues using the simultaneous submission method. Each agent submits to the mediator a list of issues in the negotiation domain space, which, from their perspective, do not need to be subject to negotiation. Among the submitted issues, for issues submitted by only one agent, the option that the other agent prefers is decided upon. For issues that both agents have submitted as unnecessary, the mediator randomly decides from the options, and an agreement is determined. Issues that neither party submits become a subject for the main negotiation.

This process is finished after one round of submissions, and messages exchanged for narrowing the issues do not go back and forth multiple times. Further, the number of issues in the negotiation domain space is very small compared with the domain size. Therefore, it is possible to narrow down issues with a small computational cost. However, because an agreement for issues that both agents submit as unnecessary is decided randomly, there is a possibility that win–win alternatives of agreement that would have been convenient for both agents are lost. In addition, because both agents make their submissions simultaneously, they are unable to surmise each other’s intentions or make deals such as gaining the right to decide on one issue by allowing the other agent to decide on another issue.

4.2. Narrowing Issues Using the Pre-Negotiation

We also propose a method for determining how to narrow the issues before the main negotiation using pre-negotiation. During the pre-negotiation, negotiation is performed for each issue in the negotiation domain space, in a domain where the options are “subject to the main negotiation”, “Agent A chooses”, and “Agent B chooses”. This pre-negotiation domain is shown in

Table 3. An example of Pre-negotiation.

Issue	Option
Destination	Main negotiation
	Agent A chooses
	Agent B chooses
Travel expense	Main negotiation
	Agent A chooses
	Agent B chooses
⋮	⋮

. In the pre-negotiation method, negotiations are performed in the domain using the alternating offers protocol, as usual, and the alternatives of agreements are decided upon.

Based on the consensus reached during pre-negotiations, the issues that will not be the main negotiation subjects are decided. For example, consider the case where the following agreement was obtained in the pre-negotiation: [Travel destination: Main negotiation, Travel expense: Agent B, Travel method: Main negotiation, Accommodation: Agent A, Meals: Agent A]. In such a case, Agent A chooses the issues “accommodation” and “meals”, whereas Agent B chooses the issue “travel expense”. The remaining issues, “travel destination” and “travel method” are the main negotiation subjects. Notably, if no agreement is reached during the pre-negotiation, the narrowing of issues does not occur, and all issues will be negotiated during the main negotiation phase.

The domain size of the pre-negotiation method is ${(a+1)}^n$, where a is the number of agents participating in the negotiation, and n is the number of issues. If the number of options $k_i$ for each issue in the original domain is greater than 3, the pre-negotiation domain would be smaller than the original domain, making pre-negotiation easier. Compared with the simultaneous submission method, where the mediator makes a random decision for issues submitted as unnecessary by both agents, in this method, the selection is always made by either agent. This allows agents to strategize during the narrowing process; for example, an agent could forgo the right to decide on some issues in exchange for the right to decide on an issue it considers relevant. Thus, the pre-negotiation method makes it easier for both agents to reach a win-win agreement.

4.3. Narrowing of Options

Each agent submits to the mediator a list of all options in an issue that were not narrowed during the issue-narrowing phase to be excluded from the negotiation. Then, the mediator narrows from the negotiation domain space all options common to the lists submitted by all participating agents and discloses them to the agents.

Table 4. An example of narrowing options (× represents elimination).

Issue	Option	Agent		Option Eliminated
		A	B	Option
Travel destination	Hokkaido	×	×	×
	Tokyo	×	×	×
	Kyoto
Travel method	Flight	×
	Train		×
	Ferry	×	×	×

shows an example of narrowing options. Options commonly submitted by both agents are eliminated, and the negotiation domain space is narrowed.

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.

4.4. Naive Agent Strategies for the Negotiation Domain Space Narrowing

We design naive agent strategies considering the negotiation domain space for the proposed pre-narrowing protocol. In the main negotiation phase of our 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 we 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, we 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”.

4.4.1. Strategy for Narrowing Issues Using the Simultaneous Submission Method

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, we 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 $I_i$, we 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, we 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:

$${\displaystyle U_{expected}=\sum _{i=1}^m\left(w_{o_i}\times \underset{j}{min}eval(v_j^{o_i})\right)+\sum _{i=m+1}^n\left(w_{o_i}\times \frac{1}{k_{o_i}}\sum _{j=1}^{k_{o_i}}eval(v_j^{o_i})\right)}$$

(3)

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.

4.4.2. Strategy for Narrowing Issues Using the Pre-Negotiation Method

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.

$$\begin{array}{cc}\hfill {\displaystyle U_{pre}(\overrightarrow{b})=}& \sum _{i\in (Issuechosenbyoneself)}\left(w_i\times \underset{j}{max}eval(v_j^i)\right)\hfill \\ \hfill & +\sum _{i\in (Issuechosenbyopponent)}\left(w_i\times \underset{j}{min}eval(v_j^i)\right)\hfill \\ \hfill & +\sum _{i\in (Issuetobedeterminedinthemainnegotiation)}\left(w_i\times \frac{1}{k_i}\sum _{j=1}^{k_i}eval(v_j^i)\right)\hfill \end{array}$$

(4)

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.

$$U_{target}(t)=1-(1-U_{min})\times t$$

(5)

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.

4.4.3. Strategy for Narrowing Options

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 study, we 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:

$${\displaystyle eval(v_j^i)<\frac{{\displaystyle U_{min}-\sum _{k\in (Issuessubjectedtonarrowing)}\left(w_k\times eval(v_{selected}^k)\right)}}{{\displaystyle \sum _{k\in (Issuesnotsubjectedtonarrowing)}{w}_k}}}$$

(6)

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.

5. Experiments

5.1. 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 ${\displaystyle \prod _{i=1}^n{k}_i}$, where n is the number of issues and $k_i$ is the number of options for the $i\mathrm{th}$ 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. Evaluation domain characteristics.

Domain name	6-1	6-2	6-3	6-4	6-5	6-6	6-7	6-8	6-9
Parameter $\alpha$	5	5	3	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
Domain name	7-1	7-2	7-3	7-4	7-5	7-6	7-7	7-8
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
Domain name	8-1	8-2	8-3	8-4	8-5	8-6	8-7	8-8
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	0.160121	0.130693	0.336388	0.365515	0.109451	0.274070	0.425334
Domain name	9-1	9-2	9-3	9-4	9-5	9-6
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	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

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. It is necessary to narrow the negotiation domain space for such negotiations using the proposed pre-narrowing protocol to improve the efficiency of the search for alternatives of agreement.

5.2. 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. Outline of comparative experiment.

	Narrowing of Issues	Narrowing of Options
(A) Regular negotiation without narrowing (baseline)	No	No
(B) Narrowing of issues using the simultaneous submission	Simultaneous submission	No
(C) Narrowing of issues using the simultaneous submission + Narrowing of options	Yes
(D) Narrowing of issues by pre-negotiation	Pre-negotiation	No
(E) Narrowing of issues by pre-negotiation + Narrowing of options	Pre-negotiation	Yes

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 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

$$U_{target}(t)=U_{max}-(U_{max}-U_{min})\times t^{1/e}(e\in \{0.2,1.0,5.0\}),$$

(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. In this experiment, agents use a common $U_{min}$, consistently from the narrowing phase to the main negotiation phase. Thus, the $U_{min}$ used by an agent to narrow issues, narrow options, and perform the main negotiation is the same. The value for $U_{min}$ shall be $U_{min}\in \{0.9,0.8,0.7,0.6,0.5\}$. Regarding the $U_{min}$ of the two negotiation agents, we experiment with all possible combinations.

Based on the above, negotiations using the five negotiation protocols shown in Table 6 are performed 31 (domains) × 25 ($U_{min}$ combinations) × 9 (combinations of e) × 100 (repetitions) = 697,500 times each. Then, we compare the domain-narrowing results, negotiation results, etc.

In our experiment, an evaluation based on domain-narrowing results from the narrowing phase and an evaluation based on the results of the main negotiation phase were performed. The evaluation measures for the domain-narrowing results are as follows.

• Ratio of narrowed issues: the ratio of narrowed issues to the total issues. The number of issues in the original domain is 1.
• Ratio of narrowed options: the ratio of options narrowed to the total options.
• Domain size reduction from narrowing: the ratio of domain size (bid quantity) reduction that occurs due to the narrowing phase to the original domain size. The original domain size is 1.

The evaluation measures for the results of the main negotiation are as follows.

• Agreement rate: the ratio of instances where a final consensus is reached in the main negotiation.
• Obtained individual utility: the utility obtained by an agent as a result of the main negotiation.
• Negotiation time: the time taken to reach a consensus in a negotiation. In the case of narrowing issues using the pre-negotiation, the sum of the pre-negotiation time and main negotiation time is used.
• Distance to Pareto optimal solution: the distance between the agreed upon bid and the nearest Pareto optimal solution. The Pareto front before narrowing down occurs is used as the reference.

5.3. 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 ${\displaystyle \prod _{i=1}^n{k}_i}$ 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.

Table 7. Results from negotiation without narrowing: agreement rate.

Agent’s ${\mathit{U}}_{\mathit{min}}$/
Opponent Agent’s ${\mathit{U}}_{\mathit{min}}$	0.9	0.8	0.7	0.6	0.5
0.9	0.046165	0.346165	0.490717	0.568548	0.722975
0.8	0.346165	0.461290	0.541183	0.661344	0.816720
0.7	0.490717	0.541183	0.609821	0.733118	0.889391
0.6	0.568548	0.661344	0.733118	0.850287	0.949534
0.5	0.722975	0.816720	0.889391	0.949534	0.984946

,

Table 8. Results from negotiation without narrowing: individual utility.

Agent’s ${\mathit{U}}_{\mathit{min}}$/
Opponent Agent’s ${\mathit{U}}_{\mathit{min}}$	0.9	0.8	0.7	0.6	0.5
0.9	0.042800	0.325729	0.467834	0.544317	0.693393
0.8	0.297192	0.409201	0.490975	0.601909	0.743827
0.7	0.398693	0.455376	0.523709	0.631905	0.767807
0.6	0.440490	0.522624	0.589657	0.687014	0.777985
0.5	0.518589	0.598473	0.663449	0.722877	0.770972

,

Table 9. Results from negotiation without narrowing: negotiation time (s).

Agent’s ${\mathit{U}}_{\mathit{min}}$/
Opponent Agent’s ${\mathit{U}}_{\mathit{min}}$	0.9	0.8	0.7	0.6	0.5
0.9	4.50856	3.88385	3.43793	3.10461	2.80801
0.8	3.88385	3.43074	3.06399	2.75062	2.48414
0.7	3.43793	3.06399	2.75156	2.46658	2.22320
0.6	3.10461	2.75062	2.46658	2.21432	2.00482
0.5	2.80801	2.48414	2.22320	2.00482	1.81897

and

Table 10. Results from negotiation without narrowing: distance to Pareto optimal solution.

Agent’s ${\mathit{U}}_{\mathit{min}}$/
Opponent Agent’s ${\mathit{U}}_{\mathit{min}}$	0.9	0.8	0.7	0.6	0.5
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

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 Figure 5, Figure 6, Figure 7 and Figure 8. The ratios are represented by the numbers in the figures when the results of regular negotiation without narrowing are set to 1.

Table 7, Table 8 and Table 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. 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 resolute 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 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.

Further, the main contributions of our proposed negotiation protocol where we introduced the domain narrowing phase are clarified the following points. Figure 9 shows the comparative results of line graphs among approaches. The bar means the average of all scores.

• Agreement rate and individual utility from negotiations increased compared with a regular negotiation without the narrowing phase.
• By narrowing the subject domain before performing the main negotiation, negotiation time can be reduced.
• Narrowing unnecessary parts of the domain in advance makes it easier to reach a consensus that is close to the Pareto optimal solution.
• Improvements in agreement rate, negotiation time, and closeness to the Pareto optimal solution depend on the combination of agent negotiation strategies.

5.4. Discussion for Privacy Issue of Our Proposed Negotiation Protocol

Our proposed negotiation protocol only uses the mediator in the pre-negotiation phase. The negotiation protocols the mediator is used in the main negotiation phase have the limitation of the agents’ privacy because all agents have to disclose their private information by following the mediator and pre-decided negotiation rules. Conversely, the only information an agent must submit to the mediator is “whether an option may be deleted”. Therefore, private information (e.g., utility information) can be disclosed to the mediator. In addition, 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.

Suppose we consider the other existing negotiation protocol with the mediator in the main negotiation phase to solve the multilateral and complex domain cases. In that case, our proposed approach can escape to reveal the private information related to each option, issue, and bid. The reason for this is that negotiation the domain-narrowing phase, the domain size to be negotiated in the main negotiation phase was reduced from 1/100 to 1/1000 compared with the approach without the pre-negotiation phase. From this point of view, our proposed method contributes to solving the privacy issue.

6. Conclusions

6.1. Summary

In this study, for multi-issue negotiation problems in a large negotiation domain space, we proposed a protocol for narrowing the negotiation domain space in advance for efficient negotiation while ensuring that agents do not have to disclose too much information to their opponents or the mediator. As the number of issues and the number of options for each issue increase, the negotiation domain space grows exponentially in multi-issue negotiation problems. To realize efficient negotiation in such a large negotiation domain space, we introduced a negotiation domain space-narrowing phase before the main negotiation and proposed a protocol for narrowing the issues and options subject to negotiation. The negotiation domain space-narrowing phase consisted of narrowing the issues and narrowing the options. To narrow down the issues, we designed two methods: a simultaneous submission method and a pre-negotiation method. We also proposed naive agent strategies considering the negotiation domain space-narrowing phase.

To evaluate the proposed protocol, we performed a negotiation simulation experiment, using multiple large negotiation domains and naive agent strategies. By narrowing the issues and their options via the negotiation domain space-narrowing phase, we confirmed that the size of the negotiation domain space subjected to the negotiation could be narrowed from $1/100$ to $1/1000$. When performing negotiations by narrowing the negotiation domain space in advance, in many cases, the utility obtained by agents increased, and the agreement rate increased. Further, in the negotiations based on the proposed protocol, we confirmed that negotiation time was reduced compared with the baseline. From the above results, the proposed protocol improved negotiation efficiency.

6.2. Future Works

6.2.1. More Diverse Negotiation Environments

In the proposed protocol, the narrowing of the negotiation domain space depends on the agent’s narrowing strategy. Thus, if an agent is reluctant toward narrowing down, a large negotiation domain space cannot be sufficiently narrowed. Moreover, even if the agents have a preference to narrow down, if the original negotiation domain is very large, there is a limit as to how much the space can be narrowed during the narrowing phase. Further, when narrowing issues using the pre-negotiation method, negotiations are performed in a pre-negotiation domain with (number of agents + 1) options for each issue. This means that, when negotiations are performed in a domain where the number of options for each issue is small, the negotiation may lose efficiency. The proposed protocol is designed to minimize the amount of information an agent must disclose; however, to better support the various negotiation environments, a flexible negotiation protocol that considers the tradeoff between efficiency and the extent to which agents must disclose their information should be designed.

6.2.2. Multilateral Negotiations (With Three or More Agents)

The proposed protocol is not limited to bilateral negotiations but can be extended to negotiations with three or more agents. However, in negotiations with three or more agents, the range that can be narrowed in the domain-narrowing phase would be small. A protocol design is required so that the negotiation domain space narrowing remains easy even when the number of agents increases.

6.2.3. Development of Agent Strategies

Our main focus was to develop a new negotiation protocol. Regarding the development of agent strategies compatible with this protocol, the proposed strategies are simple. In the future, it will be necessary to design more effective strategies that are compatible with our negotiation protocol. For instance, more strategic agents for the predomain-narrowing phase or more sophisticated concession strategies are crucial to analyze negotiation results in more realistic settings.