A Bayesian negotiation model for quality and price in a multi-consumer context
Introduction
The problem of reliability/quality demonstration of products or a system has gained large interest in recent years, due to the increasing industrial demands (see [2]). At the same time, reliability demonstration of a system can be viewed as a Bayesian decision problem in which the acceptance of the system with respect to its reliability is analyzed (see [25]). Thus, the principles of Bayesian decision theory can be applied (see [3]). Rufo et al. [31] presented a Bayesian decision making approach for the sale of a product batch in the context of life testing. Specifically, a Bayesian sequential negotiation model for which a manufacturer offers a product batch to a consumer is considered. The decision of accepting or rejecting the product batch is based on the reliability of the product, which is represented by its lifetime.
Negotiation can be defined as a process of joint decision making (see [39], [35]). That is, a process by which two or more parties try to reach compromises and to come to an agreement. Negotiations occur in all domains of life (see, e.g., [37]). However, the basic structure of negotiations applied to different contexts is basically the same (see [23]). Thus, any negotiation process has the following common characteristics (see [26]): (i) there are two or more parties involved in the entire procedure; (ii) the payoffs for each party depend either on the consequences of the joint decisions or on alternatives external to the negotiations; (iii) the parties can reciprocally and directly exchange information (this communication can be honest or not); (iv) the parties can be creative in the decisions they make in order to arrive at a joint decision. Current reviews on negotiation analysis are provided by Tsay and Bazerman [38] and Agarwal [1].
Zheng et al. [41] pointed out that the existing literature about mathematical study of negotiation can be divided into two classes: mediated and non-mediated negotiation. In the first case, the parties interact with each other through a mediator. In non-mediated negotiation, the parties communicate with one another directly. By considering the literature for this last type of negotiation, different suppositions can be made. They are related to the number of parties, i.e., bilateral or multilateral negotiation (see, e.g., [22]) and the number of issues to be negotiated, i.e., single issue or multi-issue negotiation (see, for instance, [8]). Most literature focuses on negotiation model by considering two parties and one issue (see, e.g., [40]) or two parties and several issues (see [9], [6], [12], [42], among others). However, nowadays there exists literature dealing with several parties and issues. Moreover, it applies negotiation to different fields (see, for instance, [21], [28], [27]). Other applications that consider negotiation teams are provided by Sánchez-Anguix et al. [33], [34].
A difficulty when dealing with both single issue and multi-issue negotiation is to characterize the preferences of a party through the utility function. Therefore, the parties will make decisions based on the corresponding utility function. In addition, another matter to take into account is to assume whether parties that negotiate have complete or incomplete information about their opponents׳ preferences (utility functions). Observe that, in the last case, it is harder to come to an optimal agreement. Lai et al. [16] presented a more detailed review.
In a wide variety of industrial and commercial applications, the product reliability has an influence on the parties׳ preferences. Much of the literature on reliability and survival analysis deals with the topics of life testing and the analysis of failure data. Furthermore, in most cases involving industrial settings, life-testing is performed with the aim of making decisions (see, e.g., [36]). Lindley and Singpurwalla [18], [19] proposed a Bayesian framework which involved a single issue and two adversarial decision makers: the manufacturer and the consumer. The scenario in which the consumer demands a batch of items from the manufacturer for testing is considered. The consumer could either accept or reject the batch provided by the manufacturer. This decision is based on a single issue that is the product reliability. If the consumer chooses to reject the batch, then the two main decisions brought up are the following: (i) whether the manufacturer should offer a sample to the consumer for testing, and (ii) how large it should be. Lindley and Singpurwalla [18] discuss the problem above in the context of acceptance sampling for quality control by using Bernoulli, Poisson and Normal distributions, whereas Lindley and Singpurwalla [19] consider exponential lifetimes in a reliability context.
This paper presents a Bayesian sequential negotiation model among multiple parties (a manufacturer and several consumers) on two issues: price and product reliability/quality. Thus, the procedure presented in Rufo et al. [31] is extended and modified by considering different consumers and by introducing the product price as well as its quality as decision criteria. The mediator׳s presence is not required. Thus, the manufacturer directly interacts with the consumers. In addition, it is considered that each consumer does not have knowledge about the preferences and judgements of the remaining consumers. The decision problem is solved under the manufacturer viewpoint. Firstly, it is assumed that the manufacturer offers the consumers a product batch by considering an initial price. Then, the consumers can choose between two alternatives that consist of accepting or rejecting the product batch. This decision is made based on the consumers׳ preferences and beliefs. If the second alternative is selected, then the manufacturer considers the possibility of convincing at least one of the consumers that the manufacturer׳s perspective is more correct. In order to do it, she/he has two options: to modify the initial price and/or to show that the product reliability/quality is high, providing the consumers with a sample.
A Bayesian approach is implemented to obtain both the price and the optimal sample size that the manufacturer will offer the consumer. The model is computationally complex since it involves the computation of some distributions together with some expected utilities. Thus, a direct implementation is possible when analytical expression can be obtained. In other cases, efficient Monte Carlo methods will be used. Finally, it is shown through two applications that the proposed approach can be used in several decision making settings.
The outline of the paper is as follows. In Section 2, the general model is described together with a suitable negotiation protocol. Moreover, a straightforward approach is proposed in order to obtain both an appropriate price and an optimal sample size. Section 3 applies the developed methodology to several distributions. Firstly, a set of distributions coming from the exponential family is taken. Hence, a unified framework is provided. That is applied to the Weibull case. For the second application, Bernoulli sampling is proposed. Finally, conclusions are presented in Section 4.
Section snippets
A general negotiation framework
Suppose that several parties (one manufacturer and different consumers) are negotiating the sale of a product. The main criteria for this sale are based on the price and the reliability/quality of the product. Both manufacturer and consumers are involved in a negotiation process in which they are expected to come to an agreement. In addition, it is assumed that the sale is made so that only one consumer is buying all products. On the other hand, the manufacturer is the only party who has
Applications
In this section, two applications are considered in order to illustrate the Bayesian negotiation model previously presented. Firstly, a short description is carried out for a class of exponential families. Next, a practical application for a distribution of this family is performed, specifically, for the Weibull distribution. Finally, an example for the binomial distribution is considered.
Conclusion
The main goal of this paper is to develop a general Bayesian model of decision making by considering sequential negotiation in a context of industrial and commercial market. It is a natural extension of the approach in Rufo et al. [31], where several consumers are considered. In addition, the price to be offered and the product quality are additional decision criteria. Specifically, the aim is the sale of a product batch based on two criteria: price and product reliability/quality. This last
Acknowledgments
This research has been supported by Ministerio de Economía y Competitividad, Spain (Projects MTM2011-28983-C03-02 and MTM2014-56949-C3-3-R), Gobierno de Extremadura, Spain (Project GR15106) and European Union (European Regional Development Funds).
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