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In this paper, we extend and improve the formal, executable framework for automated multi-issue negotiation between two autonomous competitive software agents proposed by Cadoli. This model is based on the view of negotiation spaces (or “areas”), representing the admissible values of the goods involved in the process as convex regions. However, in order to speed up the negotiation process and guarantee convergence, there was the restriction of potential agreements to vertices included in the intersection of the two areas. We present and assess experimentally an extension to Cadoli's approach where, for both participating agents, interaction is no longer vertex based, or at least not necessarily so. This eliminates the asymmetry among parties and the limitation to polyhedral negotiation areas. The extension can be usefully integrated to Cadoli's framework, thus obtaining an enhanced algorithm that can be effective in many practical cases. We present and discuss a number of experiments, aimed at assessing how parameters influence the performance of the algorithm and how they relate to each other. We discuss the usefulness of the approach in relevant application fields, such as, for instance, supply chain management in the fashion industry, which is a field of growing importance in economy and e-commerce.

Negotiation is a decision-making process in which multiple parties jointly make decisions to cope with conflicting interests. In particular, automated negotiation can be considered as a kind of interaction (i.e., a dialogue) in which some agents, with a desire to cooperate but with conflicting interests, work together with the aim of reaching a common goal [

Negotiation can also be defined as “a distributed search in a space of potential agreements” [

The particular mechanism of negotiation considered in this work is

The research reported in this paper presents the extension (preliminary introduced in [

The method developed in [

In this paper, we present and assess experimentally an extension to Cadoli’s approach (that we call ERMP for “Extended RMP”) where interaction is no longer vertex-based, or at least not necessarily so. That is, we allow both agents to potentially make offers that are an internal point of its negotiation space and then try to approach the opponent’s counterproposal “step by step.” The extension presented here overcomes some problems of the original one, such as the asymmetry among the parties (only one agent is allowed to use RMP otherwise problematic situations occur), the “flat” nature of RMP, where no agents’ preferences and utilities are considered (this aspect is also considered in [

ERMP has a potential practical applicability in many areas related to e-commerce or supply chain management (SCM), where there is a growing interest in autonomous interacting software agents and their potential application to support actual negotiation and contract making. For instance, negotiation within strict time-to-market constraints is a crucial component of the supply chain in the fashion industry, a field which has a growing importance in economy in general and in e-commerce in particular. Agent-based or agent-supported negotiation can help overworked human managers, where intelligent agents can be equipped with their own strategies and objectives. Decision makers being represented by agents can help to make better decisions in a shorter time, as argued, for example, in [

This paper is structured as follows. Section

In this section we first present the approach by Cadoli [

Negotiation involves two or more parties that exchange proposals until either an agreement is found, that is, the last proposal is acceptable by all the parties involved, or there is an evidence of the fact that no agreement is possible. Without loss of generality it is going to be assumed that negotiation involves only two parties, that later on will be called “agents.”

Negotiation involves variables (also called negotiation issues). A proposal (or “offer”) consists in communicating to the other party a possible assignment of values to the involved variables (each one is called a “variable assignment”).

Negotiation is restricted without loss of generality to involve only two variables. The approach however could be easily generalized.

The negotiation space (also called negotiation region or feasibility region) associated with each party coincides with the set of variable assignments that are considered to be acceptable, that is, where the value assigned to each variable is within the range that the party considers to be acceptable. Variables are considered to denote real numbers.

As only two variables are involved, negotiation spaces are restricted to be regions in the Cartesian plane.

A possible proposal is, in principle, any point of the negotiation space.

Negotiation spaces are restricted to be convex regions; that is, all points (pairs of real numbers) included within the boundaries of each individual region belong to the region itself and thus are equally acceptable as potential agreements. Therefore, each negotiation space can be described by a set of constraints which describe the region perimeter by describing the acceptable range of values for each variable.

Negotiation spaces are considered to be polyhedral. Thus, negotiation spaces admit a finite number of vertices.

Possible proposals are asymmetrical for the two agents. One of them is supposed to have the objective of minimizing the number of iterations, and in this perspective it offers only the vertices of negotiation spaces. The other one can instead offer any point of the negotiation space. The reasons of this asymmetry will be discussed in the following.

Of course, there are other assumptions that determine the negotiation scenario. For instance, it is assumed (again without loss of generality) that the two parties have agreed in advance on the issues which are involved. Also, parties associate a meaning to the variables, that in their view may represent prices, time, number of items, and so forth.

A negotiation starts when one of the two agents makes a proposal. The other one will respond with a counterproposal, and the process will go on in subsequent

As offers are restricted for one of the two agents to be vertices of a polyhedral region, in the RMP approach the process will necessarily end whenever this agent has no more vertices to offer. This means that each negotiation process always converges to an end in a finite number of steps. However, the number of vertices can be, in the worst case, exponential in the number of variables. Thus, the process has an exponential worst-case complexity in the number of negotiation steps. Therefore, the approach defines a negotiation protocol aimed at obtaining on average large savings in terms of number of proposed vertices. The underlying assumptions about the participating agents are the following.

Agents are perfect rational reasoners.

Agents communicate only by exchanging proposals. There is no other form of shared knowledge. When a proposal has been issued, it becomes common knowledge for all involved parties.

Agents are partially cooperative, in the sense that they are aware of the negotiation protocol that they apply faithfully, that is:

they do not make offers that they are not really willing to accept;

they do not cheat; that is, they do not make proposals that are not implied by the protocol at that step.

Agents are able to reason by means of projections.

Let us now discuss the nature of

So, the projection of

Line segments’ projections.

In the example, there are considered the four points

the projection of the segment

the projection of

Suppose, for instance, referring to Figure

To exemplify it, assume the following exchange: agent

From proposals

Also,

Excluding portions of the interaction area leads to excluding many potential offers and thus reducing the number of steps. In fact, agent

Notice that this example shows that the asymmetry in the nature of the offers from the two parties is an essential part of the approach. In fact, agent

If both parties are aimed at minimizing the number of steps and thus use the same protocol and offer only vertices, possible agreements might be excluded. The agreements might fail to be reached also for other reasons. This is one of the reasons leading to the extension of this approach. In summary, the interactions among negotiating agents involve the following factors.

Notice that, in this setting, finding an agreement corresponds to solving a

Let us now consider a more complex (and complete) example. Consider agents

An example of negotiation.

In this example, the negotiation area (indicated as

Assume that the negotiation process starts with a proposal from the agent

Agent

Since agent

Also this new proposal is not accepted from the agent

After two interactions, the agents have exchanged four proposals, namely,

The two proposals

Since agent

Agent

Agent

Finally, this last offer belongs to the negotiation area of the agent Seller, and therefore this proposal will be accepted. In this case, the negotiation process terminated successfully.

Notice that the generalization of the approach to

There are however other ways to cope with a larger number of variables while staying within the two-issue case discussed before. One of the two issues might be the most relevant variable (e.g., price or time of delivery), and the other issue might represent a combination of the other variables, possibly according to some utility function (the reader may refer, e.g., to [

Some proposed extensions to the basic RMP approach are motivated by the following observations.

Limiting the possible proposals to vertices is efficient but has some limits. One of them is that only one agent can adopt this strategy, that in fact works only if the other agent is instead capable of offering any point of the feasibility region. Actually, RMP makes the implicit assumption that the parties have agreed in advance of the respective strategies. If both agents instead offer only vertices, in some cases this induces problematic trade-offs: in particular, whenever the intersection area is not empty but includes no vertices. For example, in Figure

The

As mentioned in [

In [

A first problematic interaction.

A second problematic interaction.

Mancini [

The aim of the present work is that of coping with some of the problematic aspects and extending the basic approach under an important respect. In particular, proposals are assumed to be not only vertices but also internal points of the convex regions. However, since an infinite number of possible offers exists those points cannot be randomly selected. Rather, a protocol that carefully chooses the offers and keeps the advantages of reasoning by means of projections is proposed. Then, the introduced algorithm is more flexible: in fact, both agents are allowed to follow the same protocol, thus relaxing the main limitation of RMP. However, as we will demonstrate by means of extensive experiments, it is still reasonably efficient. We also intend to cope with the fact that the interest of agents in a contract usually declines gradually with distance from their ideal contract. Thus, we will introduce a form of “local search” starting from preferred vertices.

In the new algorithm, the next offer selection is based on recent proposals by the same agent, which will be increased (or decreased) by a

This section is dedicated to the introduction of a formal and executable extended approach to automated multi-issue negotiation between two partially cooperative agents. The proposed extension to the original Cadoli’s approach tries to respect its

Firstly, the set of agents involved in the process is defined as

The extension to [

In the extended algorithm the number of offers in the worst case (no agreement possible) is infinite. Nevertheless, the large number of experiments performed leads to the confidence about the fact that the proposed strategy speeds up the negotiation process whenever an agreement exists.

The approach proposed is based upon relaxing the limitation of offers to be vertices. Thus, it can be applied to any kind of convex negotiation areas, without restriction to polyhedral ones. But, considering that each negotiation area contains a huge (infinite) number of possible proposals, in which way and under which criteria are those points to be selected?

Here an example is proposed in order to explain the crucial point of the algorithm, that is, the choice-of-proposals phase. Assume that there are two agents (

To do so, agent

it computes the vector

it models and resolves the system (set) of equations composed of one of the inner products (

Finally, by merging (

In this way, each agent tries to approach the counterpart’s offer by proposing a point that is more likely to be accepted and by adapting the individual profile to the one that can be assumed for the opponent. To do this, the agent has to add new constraints to its knowledge base. If no such point is found, that is, in the case where the entire semicircumference is not included in the negotiation area, then the next proposal will be a new random point of the feasibility region.

After that, each agent exploits the reasoning on projection in the same way as in [

Otherwise, subsequent proposals will be selected in the same way as the second one (where the center of new circumference will be the agent’s last proposal) by adding, however, one further condition: the new proposals must not be included in the areas of the projections made so far. The projection areas may be described in terms of a set of linear equations. In this way, the agent will find the new variable values to offer by working out a new, or extended, DCSP.

The trade-off strategy in multi-variable regions.

An advantage of the proposed algorithm is that the agent does not have to store all past proposals. Rather, the only information that the agent needs in order to construct the new projections consists of the two most recent proposals made by each party. If an agreement has not been found yet, the agent continues with the next proposal and so on.

Since there is a huge (infinite, in principle) number of points included in an agent feasibility region, convergence of the extended algorithm is not guaranteed. To overcome this problem, two solutions are prima facie available.

A first simple empirical solution can be to introduce an upper bound to the number of allowed interactions. The upper bound is strictly needed in case there is no possible agreement: here in fact, subsequent attempts would otherwise go on indefinitely.

A second possible solution relies on an assumption about the size of the margin

ERMP can be usefully integrated with the original RMP approach. Assume in fact that an agent has private preferences and objectives. This may often result in the fact that some vertices of the feasibility region are preferred to the others and that the part of the region nearby these vertices is preferred to the rest, in the sense that the agent would prefer to reach an agreement there.

An integration between the two approaches might allow the agent to still try to minimize the number of iterations, while also trying to fulfill its objective of reaching a “preferred” agreement. A possible integration is, for example, the following:

the agent might initially offer the preferred vertices;

if no agreement has been found, it might start ERMP not from a random point, but from a preferred vertex;

If step 2 has been repeated for all preferred vertices with no agreement, the remaining (nonpreferred) vertex will be offered;

if there are no more vertices to offer, the agent may optionally (as a last chance) start ERMP from a random point.

In this case, it looks reasonable to stop EMRT after a predefined number of steps. This integration captures the important aspect (emphasized, e.g., in [

This integration can be considered to be a “Local Search variant” of RMP. In fact, local search techniques are a family of general-purpose techniques for the solution of optimization problems [

This section describes our pilot implementation of ERMP by mapping it into the DALI language for multiagent systems (MAS) modeling and implementation. DALI [

To support this claim, a snapshot of the code is presented, namely, the part that exploits the reactive capability of DALI agents. In order to improve elaboration tolerance of the implementation, the private knowledge of each agent participating in a negotiation (such as the constraints and the size of the

The connective

If the received proposal is included in the negotiation area of the agent (i.e.,

If instead the received proposal is not included in the negotiation area, then it is not accepted. Abstracting away from the details of the code, with the following rule the agent computes a counterproposal according to the above-illustrated algorithm, sends this counterproposal to the opponent, and updates its constraint knowledge base:

More precisely,

We have performed a number of experiments in order to analyze and validate the behavior of ERMP. In the experiments, that have involved a great number of instances, it is tried to be established which parameters influence the performance of the algorithm and in which way. It is also intended to establish different parameters that are related to each other.

Experiments have been performed on a real-world case study involving variables with a clear intuitive meaning. The case study is in the context of the fruit market. In this experiment, two agents,

Assume that agent

At this experiment 500 instances of such case are introduced, and the results obtained are as follows:

the average number of interactions necessary to conclude the process is

the minimum number of interactions observed during the experiments is 1, with a frequency of 56% (i.e., 56 negotiation processes out of 100 ended in just 1 step with an agreement);

the maximum number of interactions observed is 15, with a frequency of 3%.

We have then tried to understand which parameters of the process can influence, and if so in which way, the number of interactions needed to conclude the process. The significant parameters are at least (i) the size of the intersection area, (ii) the distance between the first proposals of the two agents, (iii) the size of the negotiation area, and (iv) the choice of the circumference radius.

In all experiments, there are considered 500 instances per each subcase

First, proposals are considered to be random points of the negotiation areas, and they are forced to be at the maximum possible distance. Then, the size of the intersection area is varied in order to make a point on the algorithms behavior.

By comparing the average, the maximum, and the minimum numbers of interaction where first proposals are random points of the negotiation areas or instead where first proposals are considered to be more distant, it may be noticed that the distance between the first proposals actually affects the algorithm performance.

In the former case, where the first proposals are random points, the average number of iterations roughly decreases with the increase of the size of the intersection area, as the random factor prevents a more decided correlation. However, the statistics table shows that most instances succeed within a low-medium number of iterations.

In the latter case, where there is the maximum possible distance between the first proposals, the average number of iterations clearly decreases with the increase of the size of the intersection area. However, most instances succeed within a medium-high number of step, except when the intersection area becomes really wide.

The average number of steps increases with the size of the negotiation areas, which however is what one may reasonably expect. So, as a rule of thumb we propose to cope with large areas by increasing the

The last experiment mentioned previously is useful for a comparison between the extended algorithm ERMP and the original RMP. It should be noticed that the extended algorithm is able to cope with the first problematic situation discussed in Section

With respect to the basic approach, the performance of the extended algorithm is worse in case of negotiation spaces with a limited number of vertices, but works better, in average, in the opposite case (high number of vertices). This feature addresses one of the main challenges in developing effective negotiation protocols, that is, scalability; often in fact a protocol (including RMP) can produce excessively high failure rates, when there are many possible offers, due to computational intractability.

The interesting survey about automated negotiation by Beam and Segev [

The first negotiation support systems (NSSs) [

While NSS was aimed at supporting the users during their negotiation processes, in the last years the first fully automated systems were proposed: in these systems, negotiation involves intelligent agents, that is, entities capable of a certain autonomy. Sycara and Zeng [

After Kasbah, a variety of architectures are proposed, among them are [

Game-theoretic techniques are based on strategies which have been extensively studied in game theory. The heuristics are mainly based on empirical testing and evaluation. In argumentation-based approaches, the negotiating parties can exchange any kind of feedback rather than just proposals and counterproposals. Among them we mention [

We may notice that ERMP can be adopted either as a stand-alone strategy of negotiation or as a constraint-based technique to be used in the context of more general architectures like, for instance, the one in [

Faratin et al. in [

A recent research work that proposes intelligent agents for negotiation in the fashion business field is that of [

This section briefly discusses our RM framework in the light of the specific needs of the fashion and textile (FT) supply chain management. As discussed, for example, in [

As argued in [

The specific issues of this supply chain scenario are both promising and potentially problematic, we believe, for all agent-based approaches to management. First of all, the online nature of the problem, with its almost-real-time reaction requirements, is against analytic models which, by being high-level, pay a price (if implemented) in terms of raw performance. On the other hand, the changeable nature of the supply chain and the fact that bidders may enter or leave a call for tender at any time call exactly for computational models where the compositionality property, that is, the formalization is fully parametric with respect to the bids, is guaranteed.

One feature that is rather specific to the fashion and textile supply chain is the fact that negotiation is in general not needed in all circumstances and on all issues. For example, the buyer (e.g., a fashion retailer) may especially predesign the terms, products specification, fabric, production quantity, and so forth and ask for the quotation from a panel of selected sellers (i.e., the manufacturers). Usually, the one who quotes the lowest price upfront will be the final sellers.

However, following the analysis of Bruce and Daly [

the retailer requires a specific technology to produce an innovative fashionable product, and the number of this kind of sellers is limited in the market;

the availability of raw material (e.g., cotton) and price fluctuate significantly.

As emphasized in [

Market requirements on fast fashion and short lead time call instead for efficient computational models such as the ones discussed in this paper. In these models, an agents’ motivation is usually to maximize its own utility in terms of utility function or preferences structure while staying within almost-real-time time bounds. We believe that, at present, efficient computational models (such as ERMP and the one of [

In perspective, however, an integration with analytic methods might better contribute to a full (or almost full) automation of a whole supply chain framework. A promising direction is that proposed by Sycara and Dai [

The ERMP model proposed in this paper is based upon a heuristic algorithm that considers not only the vertices as possible offers but also internal points of the feasibility regions. We have shown that the extended approach can be usefully combined with the original one, thus obtaining a local search variant that is able to cope with, for example, preferences, while still aiming at minimizing the number of steps.

By comparing the proposed algorithm with the Cadoli’s original RMP [

The additional complexity, according to the experiments, appears to be a reasonable price to pay for the extra features and for the possibility of other extensions. In fact, the approach can be further extended, for example, by adding new protocols and objective and utility functions. We in fact envisage an integration with [

Applications of such an efficient and effective computational model of negotiation can be found in many fields where there is a large number of negotiation rounds, almost impossible to be optimally coped with by humans. These fields include contexts where, as it is the case with the fashion supply chain negotiation scenario, the time-to-market aspect is crucial.

This work started as a cooperation with Marco Cadoli. Unfortunately, Marco left us in November 2006 after a long illness. This paper is dedicated to him.