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Multistage bilateral bargaining model with incomplete information—A fuzzy approach
ARTICLE IN PRESS Int. J. Production Economics 117 (2009) 235–243
Contents lists available at ScienceDirect
Int. J. Production Economics journal homepage: www.elsevier.com/locate/ijpe
Multistage bilateral bargaining model with incomplete information—A fuzzy approach Francesco Costantino , Giulio Di Gravio Department of Mechanical and Aeronautical Engineering, University of Rome ‘‘La Sapienza’’, Via Eudossiana 18, 00184 Rome, Italy
a r t i c l e in fo
abstract
Article history: Received 9 September 2006 Accepted 7 September 2008 Available online 18 November 2008
The study proposes the implementation of an intermediation model in supply chains, integrating game theory and fuzzy logic, to represent the characteristic aspects of a bilateral bargaining with incomplete information where supplier–customer relationships are indirectly managed by a third party agent. The choice of combining these theories comes out from the necessity of smoothing the peculiar elements of the two analysis tools that, in describing real situations, present many potentialities of reciprocal adaptation. The scope is to combine a formal structure that could figure out the interrelations among actors involved in a strategic decisional context, with a mathematical elaboration of natural imprecision, uncertainty and incompleteness of data and information. The model derives from the theoretic foundation of Spulber [1999. Market Microstructure. Cambridge University Press, Cambridge, UK] and Rubinstein [1982. Perfect equilibrium in a bargaining model. Econometrica 50(1), 97–109] that, compared to the classical framework of asymmetric information and bid-spread problem by Harsanyi [1967. Games with incomplete information played by ‘‘Bayesian’’ players. I. The basic model. Management Science 14, 159–182], describe the process through the definition of new parameters such as bargaining power and breakdown probability. The contribution to the research is enriched by fuzzyfication process of data, considering Qi et al. [2005. Design retrieval technology of fuzzy customer requirements. In: World Congress on Mass Customization and Personalization] experiences, to build a framework that could transform inputs from the transaction, agents and market in an output that could regulate the possible concessions and the opportunity of accepting or refusing an offer. & 2008 Elsevier B.V. All rights reserved.
Keywords: Supply chain Intermediation Negotiation Fuzzy logic
1. Introduction Intermediation theory in supply chain management studies and describes the actions of an intermediary to allow the conclusion of a transaction among different contractors (Harsanyi, 1967; Rubinstein, 1982; Spulber, 1999; Qi et al., 2005). It has a specific role in supplier– customer relationships that the market requires when
Corresponding author.
E-mail address: [email protected] (F. Costantino). 0925-5273/$ - see front matter & 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.ijpe.2008.09.013
incomes that can be generated are higher than a direct negotiation (Simon, 1979). Generally, two different categories of processes can be individuated: transactional intermediation and informational intermediation. The first one groups all the relations that take place along the supply chain from the suppliers to the producers, distributors, wholesalers, retailers and customers: each agent that interacts with two sides plays the role of an intermediary, increasing the efficiency of the relation. For example, a wholesaler between a group of producers and different retailers can grant reduced delivery times and
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a risk pooling effect, aggregating demand and accumulating adequate stock. Its main features are:
higher reactivity to customer requirements; economies of scale and demand aggregation; reduction of uncertainty and definition of prices; reduction of partner research costs.
Differently, the second one tends to smooth the effect of information asymmetry that holds up transactions and causes adverse selection problems, like pre-contractual opportunism due to distorted and private information (Dutta, 1999). Its role is to regulate the transaction among parties, increasing the efficiency thanks to the emersion of hidden information, realized by synthesis and coordination mechanism. It is proved by Cachon and Lariviere (1999) and Cachon and Fisher (2000) that perfect information or extended flows of information are not sufficient to guarantee higher profits for the whole supply chain and its stages if this is not regulated. Including a third party agent, with a profit for intermediation, has the role to avoid short-term, selfish or utilitarian behaviours of privileged actors against a higher and shared revenue for the whole network. Even considering specific contract solution, as revenue sharing that could represent an attractive solution of profit growth and allocation, it does not have a real coordination effect, in particular where demand depends on costly retail effort (Cachon and Lariviere, 2001, 2002). A conventional approach to these aspects of stability and profit allocation problems in supply chain looks at game theory to find out the best solution in cooperative environments (Thun, 2003, 2005) or to identify potential unprofitable equilibriums (Estelle et al., 2005). Furthermore, Cachon and Harker (2002) explained that in some simple settings the cooperative solution is a Nash equilibrium and competition does not necessarily lead to supply chain inefficiency but, in other settings, competition leads to costs that are substantially higher than optimal, so requires the intervention of an intermediary. In any network where information sources are multiple, from the environment that influences the negotiation to the tendency of the players to deceive behaviours and criticalities, the application of analytic models is only a partial help to organizations in interpreting future scenarios. The integration of game theory and fuzzy logic gives the research an original contribution to define a bargaining model that could integrate the formal expressions described with mathematical elements that could manage uncertainty and vagueness. This approach was first formalized by Aubin (1979) with the introduction of fuzzy coalitions in cooperative games, defining the fuzzy core as the set of allocations that cannot be blocked by any coalition with an arbitrary rate of participation. Specific contributions were then given by Borges et al. (1997), Burns and Roszkowska (2002) and Garagic and Cruz (2003) with, respectively, the fuzzyfication of the prisoner’s dilemma, the investigation of the impact on Nash equilibrium and the incorporation of heuristic knowledge into conventional frameworks.
Wasfy and Hosni (1998) introduced fuzzy logic approach in a two-party negotiation model to handle imprecision in negotiations, considering the presence of an intermediary (umpire) but with no payoff for its role. 2. Intermediation models To investigate the evolution of the process, it is considered a first simple structure of a single supplier, a single customer and an intermediary. The first two have an interest in setting up a contract of sale for particular goods, whose negotiation defines the repartition of the surplus generated. The supplier has an opportunity cost s, while the customer is willing to pay v. Depending on the information state, public or private, we can identify two different situations:
perfect information, when s and v are known to the agents without asymmetry;
imperfect information, when the agents have only imprecise information, preferring to keep private the real entity of sA[s1, s2] and vA[v1, v2]. The intermediary can propose an alternative to direct negotiation, offering w to the supplier and asking p to the customer, if the spread (pw) is non-negative. 2.1. Bilateral bargaining with perfect information Supposing that the transaction generates costs T and a neat positive profit p ¼ v s T40, direct negotiation ends with a result of a quote b A[0,1] of p to the customer (bp) and the remaining to the supplier ((1b)p). The intermediary needs to define p and w to get them renounce to direct transaction: ( v p ¼ bp (1) w s ¼ ð1 bÞp Anyway, it has to reduce transaction costs, improving the efficiency (e.g. QpT), otherwise its income, equal to spread, is negative as p w ¼ ðv bpÞ ðð1 bÞp sÞ ¼ T
(2)
A bilateral bargaining process can be represented as a game with sequential choices, where the players alternate in proposing an offer to allocate profits: at any round, this can be accepted, terminating the process, or rejected with a counteroffer. The game repeats until an agreement is reached, unless one of the players prefers to exit as the transaction is no more convenient to continue (Fig. 1), represented by a certain breakdown probability. This feature describes the instability of negotiation that can be influenced by different factors as a bad reputation, in terms of constancy and reliability of one of the players, or the anticipation of a more profitable business. Outside the bargaining process, both sides have the opportunity to start a new transaction with other partners, where they know exactly their possible income (exit values). A customer would not ever accept an offer worse than its exit value, that is to say the expected minimum wage or
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Start
customer or supplier (equal probability) place an offer to divide surplus
the other agent...
refuses
accepts
exits
Stop Fig. 1. Bargaining process.
the best sure alternative, and it is not disposed to go below that level. For this reason, the value can represent a measure of the contractual power, in comparison to the competitors. An analytical description of the game (Ertogral and Wu, 2001) demonstrates the existence of a unique perfect equilibrium in sub-games that constitutes Nash equilibrium at every iteration. Denoting p as transaction surplus, Uc and Us the exit values of the customer and the supplier and (1r) the breakdown probability, they found the equilibrium Y c ¼ ðp U s Þ
Y s ¼ ðp U c Þ
r2 2ð2 rÞ
r2 2ð2 rÞ
ðp U c U s Þ
ðp U c U s Þ
for customer
for supplier
(3)
(4)
where Yc (Ys) is the customer (supplier) value of equilibrium that the other party accepts when offered. It is clear that negotiation power of a player, and the result that can be obtained from the game, depends on
its alternative of profit outside the transaction; its ability of influence breakdown probability.
of a direct negotiation. Thanks to the analysis effectuated, it is known that the customer expects to gain not less than (pYs) and, symmetrically, the supplier expects to gain not less than (pYc). To economically justify its intervention, the intermediary has to play with transaction costs not to pass a limit M, through a price offer (p,w): ( v pXðp Y s Þ (5) w sXðp Y c Þ where, substituting Eqs. (3) and (4) we can find the spread: p wpðv sÞ U c U s
r2 ð2 rÞ
ðp U c U s Þ
(6)
So, the higher the generable spread is, in particular its difference with the limit M, the more probable is the existence on the market of a regulator agent. 2.3. Intermediation with imperfect information In case of imperfect information a customer, an intermediary and a supplier develop a negotiation process in two different stages (Fig. 2), where the initial offer defines the highest desired profit for the parties:
2.2. Intermediation with perfect information
first, the intermediary starts a direct transaction with To induce the agents to accept the intervention of an intermediary, it has to propose advantageous conditions as to generate, at limit, an income not lower than the one
the supplier to define the purchasing price w* of the goods in relation to an opportunity cost sA[s1,s2] not revealed. The interaction is represented by a dynamic
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Start
intermediary offers
asks for counteroffer
supplier...
accepts
exits
definition of pricew*
Stop intermediary offers
asks for counteroffer
customer...
accepts
exits
definition of price p* Stop definition of spread
Fig. 2. Three agents negotiation model.
game with sequential choices, where the two parties alternate to propose offers and reach an agreement; then, the intermediary sets up a second transaction with the customer. The two parties negotiate a price p* that the client has to pay to get the products, in relation to its capability vA[v1, v2]. The sequence of offers tends to find a trade-off between the lower possible price for the customer and the maximum spread for the intermediary.
In the first stage, the two players negotiate to define the price the intermediary has to pay to purchase the product. At the beginning of the process, the intermediary inquires about the price the customer will be able to pay so to act coherently in both ways. We can assume that the
customer shows its lower price v1 due to its opportunistic behaviour. If it is higher than the presumable exit value s1 for the supplier, there are chances to reach an agreement that satisfies all the parties in the indicated interval of prices. The introduction of the intermediary generates a set of possible offers to the supplier [w1,w2], where w1 ¼ s1 and w2 ¼ v1T, and a set of possible prices to the customer [p1,p2], where p1 ¼ w*+T and p2 ¼ v2. According to the intermediary perceptions and depending on sA[s1,s2], any wA[w1,w2] represents an offer the supplier can accept, refuse or ask for a counteroffer, with different probabilities of occurrence. At the same time, depending on vA[v1,v2], any pA[p1,p2] represents a price the customer can accept, refuse or ask for a counteroffer.
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3. Fuzzy approach to intermediation with imperfect information Fuzzy version of the games defines that, at every round, the player determines its choice through an activity of fuzzyfication, inference and defuzzyfication of variables whose variation dominion is divided into classes to represent their uncertain nature. The output of the evaluation process is expressed in terms of ‘‘concession’’, that means how much the intermediary increase the offer to the supplier (or decrease the price to the customer). When t ¼ 0, at the beginning of the bargaining process, the intermediary places an offer wA[w1,w2] balancing its own best value w1 through the perception of the supplier’s expected profit. During the game, if the supplier asks for a counteroffer, it will be evaluated increasing this offer with a concession as a result of the inference engine of a fuzzy module. If the first stage ends with an agreement w*, in the second stage the intermediary starts a new bargaining process with the customer, defining a price pA[p1,p2] balancing its own best value p2 through the perception of the customer’s expected margin, reducing the price by a concession if the customer asks for a counteroffer. The decisional elements that govern the framework and contribute to define the correct proposal of offer or price are listed referring to a buyer–seller relationship:
unreliability: this parameter takes into account the perception of the buyer on the stability and respect of contractual clauses, also basing on past experiences. It defines the risk of incorrect behaviour of the seller where a low value of reliability reduces its contractual power; market penetration at time t: it is a measure of the possible alternatives that can present in relation with a general tendency of the market. At every stage of the bargaining it takes into account the plurality, visibility and behaviour of other potential sellers and buyers. Factors to consider are: J number of possible alternatives: if the seller has a dominant position in the market (for example, due to an exclusive licence of use or a patent), the buyer will tend to increase its offer, while a high number of alternatives decreases the concession; J number of existing relationships: along with the number of possible alternatives, it is to consider also the number of active connections with these actors, as an already existing network means more market availability and reduces the concession; J market information level: the knowledge of the structure of buyers’ and sellers’ behaviours, price strategies and inclinations, considering also similar markets, opens the possibility to act with more accuracy in the concession process; J product information level: where the buyer can easily determine the real product characteristics in relation to what is offered on the market, the perception of concession changes according to the quality dealt. J
offer at time t1 (price at time t1): the offer proposed by the intermediary to the supplier (or the price proposed to the customer) at t1 is the starting point to define the concession for each stage t of the game. The construction of the fuzzy set, that indicates how this value is perceived by the intermediary to influence the final concession, can be structured taking into account the following factors: J brand: if the brand of the product or service negotiated has a great impact, it tends to improve the position of the seller and increase its profit; J reputation: similar to brand, the reputation of the seller has a value on the market that can reflect on the buyer’s perception; J quality: characteristics of product or service negotiated have a direct impact on concession as the buyer is willing to offer more for high performances; J service level: at the same time, performances of the seller in terms of procurement time, perfect order and fill rate increase buyer’s offer; contractual power at time t: the equilibrium among parties depends on how much each agent can look like abandoning the stage or how much is the risk related to this event. It can be described by the following factors: J relative dimensions: if the buyer is notably greater than the seller in terms of relative size of organization, the smaller agent will suffer more for a failure of the process so it is pushed to a higher concession; J relative volumes: higher are the volumes negotiated, more the seller is risking to lose in terms of profit and tends to accept lower prices;
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In the first stage of the game the intermediary has to evaluate these parameters considering its position on the negotiation as a buyer. In the second stage, as it plays the role of the seller with a different agent, it is necessary to reinterpret and re-evaluate their impact. The concession output variable (c1A[0,w2w1] on offer and c2A[0,p2p1] on price) of the fuzzy inference engine represents a perception of the intermediary on the development of the negotiation where factors to consider are:
resistance to concession: according to the intermediary
attitude on times and risks of the bargain process, a concession policy in terms of price or offer sequences is settled; expected margin: the intermediary expectations about the results of the negotiations influence the perception on the concession value, considering also the possible economies on the transaction costs; tangibles/intangibles ratio: when further intangible benefits add to tangibles in concluding the process, the intermediary is pushed to a higher concession.
3.1. Fuzzy inference engine The construction of the fuzzy inference engine starts from the definition of the dominion of the input variables
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that represents the decisional elements, as formulated by Wasfy and Hosni (1998). First, consider a division of the dominion of a general variable x in fuzzy classes with a set of qualitative description. Each linguistic characteristic can be related to a fuzzy set with a specific membership function that defines its limits on three different states {low, medium, high} to describe the parameter. Denoting fk {k ¼ 1,y,n} the factors that have an influence on x, for each of them a triangular fuzzy quantifier can be associated, defined on a normalized interval (Fig. 3), and weighted {zr;r ¼ 1,y,m} to represent a degree of importance. A weighted sums S can be calculated by common operational rules on fuzzy and non-fuzzy numbers, whose result is still a triangular fuzzy number: S¼
n 1X z f n k¼1 rk k
(7)
It is to observe that, as every factor has an inverse influence on the decisional elements, the definition of the fuzzy standard quantifier has to associate low linguistic attributes to high values of the variable x. Denoting [x1,x2] the range of variation for x, inside this interval the definition of the fuzzy set medium can be effectuated averaging the influence of S: medium ¼ ðx2 x1 ÞS
(8)
After having determined the medium set, low and high sets have to be placed with a symmetric disposition. The definition of the fuzzy dominion has to grant that in each point of the interval:
no more than two sets compete; the total degree of the relative two membership function is equal to 100%. For the first stage of the process, the intermediary takes information from the market about the supplier’s opportunity cost sA[s1,s2] to define its offer wA[w1,w2]: on this second interval it is possible to build the fuzzy sets of the three decisional elements through their relative subfactors. The same process can be also applied on price pA[p1,p2] considering customer’s capability vA[v1,v2]. To effectuate the composition of the variables, any agent has to define the type and level of influence that inputs have on the output value, as to define a set of linguistic rules that can build up the framework. These
medium-high high medium low
very high 1
0
0.2
0.4
0.6
0.8
very low
1
Fig. 3. Fuzzy quantifier.
Table 1 Input–output fuzzy engine. IF
THEN
Contractual power at t
Market penetration at t
Offer at t1
Concession
Low Low Low Low Low Low Low Low Low Medium Medium Medium Medium Medium Medium Medium Medium Medium High High High High High High High High High
Low Low Low Medium Medium Medium High High High Low Low Low Medium Medium Medium High High High Low Low Low Medium Medium Medium High High High
Low Medium High Low Medium High Low Medium High Low Medium High Low Medium High Low Medium High Low Medium High Low Medium High Low Medium High
High High High High High Medium High Medium Medium High High Medium High Medium Medium Medium Medium Low High Medium Medium Medium Medium Low Medium Low Low
give a qualitative description of the actions to take place according to the environmental conditions and can be modified at any round according to the agent’s perceptions. From the intermediary point of view we can define an inverse influence of the three input variables on the output, as a high offer (price) at t1 tends to reduce the concession, as well as a high market penetration and contractual power, generating 33 inference rules shown in Table 1.
4. Case study Considering an international intermediary interested in buying high quality Italian wine to resell it to US distribution, a case study that shows the system capability is shown. The first stage starts from the definition of the range of possible offers to the supplier [w1,w2], assuming wA[h40.00; h60.00] reflecting on the expected wages of a high class wine and opportunities of penetration in the opening market. The definition of the fuzzy sets for the variables can be executed in accordance to the process described. For the variable offer at t1, it is to consider that the intermediary needs a high level of quality of the product and a company’s service level, to grant direct flow internationalization, brand and reputation does not affect
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very much this transaction as it can give good results also with small companies or newcomers. These considerations are settled to define weights of the quantifiers. Starting a negotiation with an affirmed company who grants good performances of its products and operations, it is possible to define the values of the quantifiers as shown in Table 2 and represented in Fig. 4.
These evaluations configure medium set of the offer at t1 according to Eqs. (7) and (8), while low and high ones derive from the symmetrical construction (Table 3 and Fig. 5). With a similar process it is possible to define the other input variables, contractual power and market penetration, and the output variable concession (Tables 4 and 5, Fig. 6).
Table 2 Values of offer at t1 factors.
Table 3 Results for offer at t1 fuzzy set definition.
Factor
Quantifier
Weight
Element
Results
Brand Reputation Quality Service level
High Medium High Medium high
0.15 0.15 0.4 0.3
Smedium Medium Low High
(0.12; 0.32; 0.52) (h42.40; h46.40; h50.40) (h40.00; h42.40; h46.40) (h46.40; h50.40; h60.00)
Fig. 4. Quantifier values for offer at t1 fuzzy set definition.
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Foe example, entering the model with a first offer by the intermediary of w0 ¼ h49, the supplier could accept, exit or ask for a counteroffer. In this third case, the inference engine proposes a value for concession of h2.30. This is the result of the centre of maximum combination of the rules in Table 1 with different activation weights, depending on the state of the fuzzy sets of the input variables. If a counteroffer is asked, the input value of w00 (price 0 w +concession ¼ w00 ¼ h51.30) generates a new concession of h1.15, and so on until the bargain ends. A sequence with four loops is reported in Table 6. If we assume for instance that supplier accept the first counteroffer (w* ¼ h51.30), we can enter the second stage of the game with the negotiation between the reseller and
Fig. 5. Offer at t1 fuzzy set.
Table 4 Quantifier values of contractual power at t, market penetration at t and concession factors. Variable
Quantifier
Weight
Contractual power Relative dimensions Relative volumes Unreliability
Medium Low Medium high
0.4 0.4 0.2
Market penetration Alternatives Existing relationships Market information level Product information level
High Medium high Very high Medium high
0.25 0.25 0.25 0.25
Concession Resistance to concession Expected margin Tangibles/intangibles ratio
Low Medium Very low
0.2 0.2 0.6
Table 5 Results for contractual power at t, market penetration at t and concession definition. Element
Results
Contractual power Medium Low High
(h48.80; h52.80; h56.80) (h40.00; h48.80; h52.80) (h52.80; h56.80; h60.00)
Market penetration Medium Low High
(h42.00; h45.00; h49.00) (h40.00; h42.00; h45.00) (h45.00; h49.00; h60.00)
Concession Medium Low High
(h0.80; h2.40; h6.40) (h0.00; h0.80; h2.40) (h2.40; h6.40; h20.00)
Fig. 6. Contractual power at t, market penetration at t and concession.
ARTICLE IN PRESS F. Costantino, G. Di Gravio / Int. J. Production Economics 117 (2009) 235–243
Table 6 Game steps in stage 1. Steps
Offer w (h)
Concession value (h)
1 2 3 4
49.00 51.30 52.45 53.02
2.30 1.15 0.57 0.40
US retailer. The price interval starts from p1 ¼ w*+T and the decisional elements in input have to be evaluated with reference to this new negotiation. In this set of solutions, the intermediary starts from a price, then applies discount equal to concession provided by fuzzy engine, until the second bargain ends. 5. Conclusions The model is able to analyse a possible strategic interaction context in a supply chain environment, integrating a fuzzy description of data in a formal framework of a bargaining game. The uncertainty of decisional parameters at any round and stage is evident: most of the actions derives from a qualitative analysis of the situation and, in any case, experience, intuition, ability of relieving any tone in other party behaviour play a fundamental role in getting a positive result of the negotiation. The dynamic structure of the system, the temporal variability and partial information of parameters need a framework to create an architecture for supplier– intermediary–customer interaction and evaluate the impact and define a sequence of offer-price proposition (known in literature as signaling game and screening game, Cachon and Netessine, 2003). The model represents a tool for analysis that can relate qualitative competencies with both quantitative and estimated information in a logic based on simple linguistic rules and components. It can be extended and refined evaluating the possibility of introducing further inputs, modifying the game structure, membership functions and relative dominions, eventually adding new levels, stages and players. The generalization of the methodology, easy to apply, allows a natural implementation in all the strategic contexts where it is critical to identify and correctly manage private information on prices, times and performance levels. In particular, the same dynamics can be replicated in any problem of scarce resource allocation to identify deceiving strategies in particular environment to define balanced contractual requirements. It is anyway evident how the fuzzyfication activity is the critic element to get the proper input for the system. The lack of a standard procedure and the necessary use of skilful users to evaluate classes and membership functions ask for a simplification and automation of the
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process. The possibility of giving to an expert system (knowledge-based or neural network-based) the activity of elaboration and formalization could increase notably the applicability of the framework, saving the end-user from errors in the decisional process, reducing risks of wrong interpretations and relieving from responsibilities. The inexperience, uncertainty, vagueness and ability of counterparties to give disguised indications could so be modified and interpreted with uniform criterions, through a system that gives an integrated and balanced vision of the different variables and parameters in any market environment. References Aubin, J.P., 1979. Mathematical Methods of Games and Economic Theory. North-Holland, Amsterdam. Borges, P.S.S., Pacheco, R.P.S., Barcia, R.M., Khator, S.K., 1997. A fuzzy approach to the prisoner’s dilemma. Biosystems 41 (2), 127–137. Burns, T.R., Roszkowska, E., 2002. Fuzzy games and equilibria: the perspective of the general theory of games on Nash and normative equilibria. In: Pal, S.K., Pollowski, L., Skowron, A. (Eds.), Rough-Nero Computing: Techniques for Computing with Words. Springer, Berlin. Cachon, G., Fisher, M., 2000. Supply chain inventory management and the value of shared information. Management Science 46 (8), 1032–1048. Cachon, G., Harker, P., 2002. Competition and outsourcing with scale economies. Management Science 48 (10), 1314–1333. Cachon, G., Lariviere, M., 1999. Capacity choice and allocation: strategic behavior and supply chain performance. Management Science 45 (8), 1091–1108. Cachon, G., Lariviere, M., 2001. Contracting to assure supply: how to share demand forecasts in a supply chain. Management Science 47 (5), 629–646. Cachon, G., Lariviere, M., 2002. Supply chain coordination with revenue sharing contracts. Management Science 51 (11), 30–44. Cachon, G., Netessine, S., 2003. Game theory in supply chain analysis. In: Simchi-Levi, D., Wu, S.D., Zuo-Jun, M.S. (Eds.), Handbook of Quantitative Supply Chain Analysis—Modeling in the e-business Era. Kluwer Academic Publishers, Dordrecht, (2004). Dutta, P.K., 1999. Strategies and Games: Theory and Practice. The MIT Press, Cambridge, MA. Ertogral, K., Wu, S.D., 2001. A bargaining game for supply chain contracting. Technical Report, Lehigh University, Department of Industrial and Systems Engineering, Bethlehem, PA. Estelle, J., Vorobeychik, Y., Wellman, M.P., Singh, S., Kiekintueld, C., Soni, V., 2005. Strategic interactions in a supply chain game. Computational Intelligence 21 (1), 1–26. Qi, F., Wang, Y., Tan, J., 2005. Design retrieval technology of fuzzy customer requirements. In: World Congress on Mass Customization and Personalization. Garagic, D., Cruz, J.B., 2003. An approach to fuzzy non-cooperative Nash games. Journal of Optimization Theory and Applications 118 (3), 475–491. Harsanyi, J.C., 1967. Games with incomplete information played by ‘‘Bayesian’’ players. I. The basic model. Management Science 14, 159–182. Rubinstein, A., 1982. Perfect equilibrium in a bargaining model. Econometrica 50 (1), 97–109. Simon, H.A., 1979. Rational decision making in business organizations. American Economic Review 69 (4), 493–513. Spulber, D.F., 1999. Market Microstructure. Cambridge University Press, Cambridge, UK. Thun, J.H., 2003. Analysis of cooperation in supply chain using game theory. In: EurOMA/POMS Joint International Conference. Thun, J.H., 2005. Comparing generic supply chain structures. In: EurOMA International Conference on Operations and Global Competitiveness. Wasfy, A.M., Hosni, A.Y., 1998. Two-party negotiation modeling: an integrated fuzzy logic approach. Group Decision and Negotiation 7 (6), 491–518.
Contents lists available at ScienceDirect
Int. J. Production Economics journal homepage: www.elsevier.com/locate/ijpe
Multistage bilateral bargaining model with incomplete information—A fuzzy approach Francesco Costantino , Giulio Di Gravio Department of Mechanical and Aeronautical Engineering, University of Rome ‘‘La Sapienza’’, Via Eudossiana 18, 00184 Rome, Italy
a r t i c l e in fo
abstract
Article history: Received 9 September 2006 Accepted 7 September 2008 Available online 18 November 2008
The study proposes the implementation of an intermediation model in supply chains, integrating game theory and fuzzy logic, to represent the characteristic aspects of a bilateral bargaining with incomplete information where supplier–customer relationships are indirectly managed by a third party agent. The choice of combining these theories comes out from the necessity of smoothing the peculiar elements of the two analysis tools that, in describing real situations, present many potentialities of reciprocal adaptation. The scope is to combine a formal structure that could figure out the interrelations among actors involved in a strategic decisional context, with a mathematical elaboration of natural imprecision, uncertainty and incompleteness of data and information. The model derives from the theoretic foundation of Spulber [1999. Market Microstructure. Cambridge University Press, Cambridge, UK] and Rubinstein [1982. Perfect equilibrium in a bargaining model. Econometrica 50(1), 97–109] that, compared to the classical framework of asymmetric information and bid-spread problem by Harsanyi [1967. Games with incomplete information played by ‘‘Bayesian’’ players. I. The basic model. Management Science 14, 159–182], describe the process through the definition of new parameters such as bargaining power and breakdown probability. The contribution to the research is enriched by fuzzyfication process of data, considering Qi et al. [2005. Design retrieval technology of fuzzy customer requirements. In: World Congress on Mass Customization and Personalization] experiences, to build a framework that could transform inputs from the transaction, agents and market in an output that could regulate the possible concessions and the opportunity of accepting or refusing an offer. & 2008 Elsevier B.V. All rights reserved.
Keywords: Supply chain Intermediation Negotiation Fuzzy logic
1. Introduction Intermediation theory in supply chain management studies and describes the actions of an intermediary to allow the conclusion of a transaction among different contractors (Harsanyi, 1967; Rubinstein, 1982; Spulber, 1999; Qi et al., 2005). It has a specific role in supplier– customer relationships that the market requires when
Corresponding author.
E-mail address: [email protected] (F. Costantino). 0925-5273/$ - see front matter & 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.ijpe.2008.09.013
incomes that can be generated are higher than a direct negotiation (Simon, 1979). Generally, two different categories of processes can be individuated: transactional intermediation and informational intermediation. The first one groups all the relations that take place along the supply chain from the suppliers to the producers, distributors, wholesalers, retailers and customers: each agent that interacts with two sides plays the role of an intermediary, increasing the efficiency of the relation. For example, a wholesaler between a group of producers and different retailers can grant reduced delivery times and
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a risk pooling effect, aggregating demand and accumulating adequate stock. Its main features are:
higher reactivity to customer requirements; economies of scale and demand aggregation; reduction of uncertainty and definition of prices; reduction of partner research costs.
Differently, the second one tends to smooth the effect of information asymmetry that holds up transactions and causes adverse selection problems, like pre-contractual opportunism due to distorted and private information (Dutta, 1999). Its role is to regulate the transaction among parties, increasing the efficiency thanks to the emersion of hidden information, realized by synthesis and coordination mechanism. It is proved by Cachon and Lariviere (1999) and Cachon and Fisher (2000) that perfect information or extended flows of information are not sufficient to guarantee higher profits for the whole supply chain and its stages if this is not regulated. Including a third party agent, with a profit for intermediation, has the role to avoid short-term, selfish or utilitarian behaviours of privileged actors against a higher and shared revenue for the whole network. Even considering specific contract solution, as revenue sharing that could represent an attractive solution of profit growth and allocation, it does not have a real coordination effect, in particular where demand depends on costly retail effort (Cachon and Lariviere, 2001, 2002). A conventional approach to these aspects of stability and profit allocation problems in supply chain looks at game theory to find out the best solution in cooperative environments (Thun, 2003, 2005) or to identify potential unprofitable equilibriums (Estelle et al., 2005). Furthermore, Cachon and Harker (2002) explained that in some simple settings the cooperative solution is a Nash equilibrium and competition does not necessarily lead to supply chain inefficiency but, in other settings, competition leads to costs that are substantially higher than optimal, so requires the intervention of an intermediary. In any network where information sources are multiple, from the environment that influences the negotiation to the tendency of the players to deceive behaviours and criticalities, the application of analytic models is only a partial help to organizations in interpreting future scenarios. The integration of game theory and fuzzy logic gives the research an original contribution to define a bargaining model that could integrate the formal expressions described with mathematical elements that could manage uncertainty and vagueness. This approach was first formalized by Aubin (1979) with the introduction of fuzzy coalitions in cooperative games, defining the fuzzy core as the set of allocations that cannot be blocked by any coalition with an arbitrary rate of participation. Specific contributions were then given by Borges et al. (1997), Burns and Roszkowska (2002) and Garagic and Cruz (2003) with, respectively, the fuzzyfication of the prisoner’s dilemma, the investigation of the impact on Nash equilibrium and the incorporation of heuristic knowledge into conventional frameworks.
Wasfy and Hosni (1998) introduced fuzzy logic approach in a two-party negotiation model to handle imprecision in negotiations, considering the presence of an intermediary (umpire) but with no payoff for its role. 2. Intermediation models To investigate the evolution of the process, it is considered a first simple structure of a single supplier, a single customer and an intermediary. The first two have an interest in setting up a contract of sale for particular goods, whose negotiation defines the repartition of the surplus generated. The supplier has an opportunity cost s, while the customer is willing to pay v. Depending on the information state, public or private, we can identify two different situations:
perfect information, when s and v are known to the agents without asymmetry;
imperfect information, when the agents have only imprecise information, preferring to keep private the real entity of sA[s1, s2] and vA[v1, v2]. The intermediary can propose an alternative to direct negotiation, offering w to the supplier and asking p to the customer, if the spread (pw) is non-negative. 2.1. Bilateral bargaining with perfect information Supposing that the transaction generates costs T and a neat positive profit p ¼ v s T40, direct negotiation ends with a result of a quote b A[0,1] of p to the customer (bp) and the remaining to the supplier ((1b)p). The intermediary needs to define p and w to get them renounce to direct transaction: ( v p ¼ bp (1) w s ¼ ð1 bÞp Anyway, it has to reduce transaction costs, improving the efficiency (e.g. QpT), otherwise its income, equal to spread, is negative as p w ¼ ðv bpÞ ðð1 bÞp sÞ ¼ T
(2)
A bilateral bargaining process can be represented as a game with sequential choices, where the players alternate in proposing an offer to allocate profits: at any round, this can be accepted, terminating the process, or rejected with a counteroffer. The game repeats until an agreement is reached, unless one of the players prefers to exit as the transaction is no more convenient to continue (Fig. 1), represented by a certain breakdown probability. This feature describes the instability of negotiation that can be influenced by different factors as a bad reputation, in terms of constancy and reliability of one of the players, or the anticipation of a more profitable business. Outside the bargaining process, both sides have the opportunity to start a new transaction with other partners, where they know exactly their possible income (exit values). A customer would not ever accept an offer worse than its exit value, that is to say the expected minimum wage or
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Start
customer or supplier (equal probability) place an offer to divide surplus
the other agent...
refuses
accepts
exits
Stop Fig. 1. Bargaining process.
the best sure alternative, and it is not disposed to go below that level. For this reason, the value can represent a measure of the contractual power, in comparison to the competitors. An analytical description of the game (Ertogral and Wu, 2001) demonstrates the existence of a unique perfect equilibrium in sub-games that constitutes Nash equilibrium at every iteration. Denoting p as transaction surplus, Uc and Us the exit values of the customer and the supplier and (1r) the breakdown probability, they found the equilibrium Y c ¼ ðp U s Þ
Y s ¼ ðp U c Þ
r2 2ð2 rÞ
r2 2ð2 rÞ
ðp U c U s Þ
ðp U c U s Þ
for customer
for supplier
(3)
(4)
where Yc (Ys) is the customer (supplier) value of equilibrium that the other party accepts when offered. It is clear that negotiation power of a player, and the result that can be obtained from the game, depends on
its alternative of profit outside the transaction; its ability of influence breakdown probability.
of a direct negotiation. Thanks to the analysis effectuated, it is known that the customer expects to gain not less than (pYs) and, symmetrically, the supplier expects to gain not less than (pYc). To economically justify its intervention, the intermediary has to play with transaction costs not to pass a limit M, through a price offer (p,w): ( v pXðp Y s Þ (5) w sXðp Y c Þ where, substituting Eqs. (3) and (4) we can find the spread: p wpðv sÞ U c U s
r2 ð2 rÞ
ðp U c U s Þ
(6)
So, the higher the generable spread is, in particular its difference with the limit M, the more probable is the existence on the market of a regulator agent. 2.3. Intermediation with imperfect information In case of imperfect information a customer, an intermediary and a supplier develop a negotiation process in two different stages (Fig. 2), where the initial offer defines the highest desired profit for the parties:
2.2. Intermediation with perfect information
first, the intermediary starts a direct transaction with To induce the agents to accept the intervention of an intermediary, it has to propose advantageous conditions as to generate, at limit, an income not lower than the one
the supplier to define the purchasing price w* of the goods in relation to an opportunity cost sA[s1,s2] not revealed. The interaction is represented by a dynamic
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Start
intermediary offers
asks for counteroffer
supplier...
accepts
exits
definition of pricew*
Stop intermediary offers
asks for counteroffer
customer...
accepts
exits
definition of price p* Stop definition of spread
Fig. 2. Three agents negotiation model.
game with sequential choices, where the two parties alternate to propose offers and reach an agreement; then, the intermediary sets up a second transaction with the customer. The two parties negotiate a price p* that the client has to pay to get the products, in relation to its capability vA[v1, v2]. The sequence of offers tends to find a trade-off between the lower possible price for the customer and the maximum spread for the intermediary.
In the first stage, the two players negotiate to define the price the intermediary has to pay to purchase the product. At the beginning of the process, the intermediary inquires about the price the customer will be able to pay so to act coherently in both ways. We can assume that the
customer shows its lower price v1 due to its opportunistic behaviour. If it is higher than the presumable exit value s1 for the supplier, there are chances to reach an agreement that satisfies all the parties in the indicated interval of prices. The introduction of the intermediary generates a set of possible offers to the supplier [w1,w2], where w1 ¼ s1 and w2 ¼ v1T, and a set of possible prices to the customer [p1,p2], where p1 ¼ w*+T and p2 ¼ v2. According to the intermediary perceptions and depending on sA[s1,s2], any wA[w1,w2] represents an offer the supplier can accept, refuse or ask for a counteroffer, with different probabilities of occurrence. At the same time, depending on vA[v1,v2], any pA[p1,p2] represents a price the customer can accept, refuse or ask for a counteroffer.
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3. Fuzzy approach to intermediation with imperfect information Fuzzy version of the games defines that, at every round, the player determines its choice through an activity of fuzzyfication, inference and defuzzyfication of variables whose variation dominion is divided into classes to represent their uncertain nature. The output of the evaluation process is expressed in terms of ‘‘concession’’, that means how much the intermediary increase the offer to the supplier (or decrease the price to the customer). When t ¼ 0, at the beginning of the bargaining process, the intermediary places an offer wA[w1,w2] balancing its own best value w1 through the perception of the supplier’s expected profit. During the game, if the supplier asks for a counteroffer, it will be evaluated increasing this offer with a concession as a result of the inference engine of a fuzzy module. If the first stage ends with an agreement w*, in the second stage the intermediary starts a new bargaining process with the customer, defining a price pA[p1,p2] balancing its own best value p2 through the perception of the customer’s expected margin, reducing the price by a concession if the customer asks for a counteroffer. The decisional elements that govern the framework and contribute to define the correct proposal of offer or price are listed referring to a buyer–seller relationship:
unreliability: this parameter takes into account the perception of the buyer on the stability and respect of contractual clauses, also basing on past experiences. It defines the risk of incorrect behaviour of the seller where a low value of reliability reduces its contractual power; market penetration at time t: it is a measure of the possible alternatives that can present in relation with a general tendency of the market. At every stage of the bargaining it takes into account the plurality, visibility and behaviour of other potential sellers and buyers. Factors to consider are: J number of possible alternatives: if the seller has a dominant position in the market (for example, due to an exclusive licence of use or a patent), the buyer will tend to increase its offer, while a high number of alternatives decreases the concession; J number of existing relationships: along with the number of possible alternatives, it is to consider also the number of active connections with these actors, as an already existing network means more market availability and reduces the concession; J market information level: the knowledge of the structure of buyers’ and sellers’ behaviours, price strategies and inclinations, considering also similar markets, opens the possibility to act with more accuracy in the concession process; J product information level: where the buyer can easily determine the real product characteristics in relation to what is offered on the market, the perception of concession changes according to the quality dealt. J
offer at time t1 (price at time t1): the offer proposed by the intermediary to the supplier (or the price proposed to the customer) at t1 is the starting point to define the concession for each stage t of the game. The construction of the fuzzy set, that indicates how this value is perceived by the intermediary to influence the final concession, can be structured taking into account the following factors: J brand: if the brand of the product or service negotiated has a great impact, it tends to improve the position of the seller and increase its profit; J reputation: similar to brand, the reputation of the seller has a value on the market that can reflect on the buyer’s perception; J quality: characteristics of product or service negotiated have a direct impact on concession as the buyer is willing to offer more for high performances; J service level: at the same time, performances of the seller in terms of procurement time, perfect order and fill rate increase buyer’s offer; contractual power at time t: the equilibrium among parties depends on how much each agent can look like abandoning the stage or how much is the risk related to this event. It can be described by the following factors: J relative dimensions: if the buyer is notably greater than the seller in terms of relative size of organization, the smaller agent will suffer more for a failure of the process so it is pushed to a higher concession; J relative volumes: higher are the volumes negotiated, more the seller is risking to lose in terms of profit and tends to accept lower prices;
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In the first stage of the game the intermediary has to evaluate these parameters considering its position on the negotiation as a buyer. In the second stage, as it plays the role of the seller with a different agent, it is necessary to reinterpret and re-evaluate their impact. The concession output variable (c1A[0,w2w1] on offer and c2A[0,p2p1] on price) of the fuzzy inference engine represents a perception of the intermediary on the development of the negotiation where factors to consider are:
resistance to concession: according to the intermediary
attitude on times and risks of the bargain process, a concession policy in terms of price or offer sequences is settled; expected margin: the intermediary expectations about the results of the negotiations influence the perception on the concession value, considering also the possible economies on the transaction costs; tangibles/intangibles ratio: when further intangible benefits add to tangibles in concluding the process, the intermediary is pushed to a higher concession.
3.1. Fuzzy inference engine The construction of the fuzzy inference engine starts from the definition of the dominion of the input variables
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that represents the decisional elements, as formulated by Wasfy and Hosni (1998). First, consider a division of the dominion of a general variable x in fuzzy classes with a set of qualitative description. Each linguistic characteristic can be related to a fuzzy set with a specific membership function that defines its limits on three different states {low, medium, high} to describe the parameter. Denoting fk {k ¼ 1,y,n} the factors that have an influence on x, for each of them a triangular fuzzy quantifier can be associated, defined on a normalized interval (Fig. 3), and weighted {zr;r ¼ 1,y,m} to represent a degree of importance. A weighted sums S can be calculated by common operational rules on fuzzy and non-fuzzy numbers, whose result is still a triangular fuzzy number: S¼
n 1X z f n k¼1 rk k
(7)
It is to observe that, as every factor has an inverse influence on the decisional elements, the definition of the fuzzy standard quantifier has to associate low linguistic attributes to high values of the variable x. Denoting [x1,x2] the range of variation for x, inside this interval the definition of the fuzzy set medium can be effectuated averaging the influence of S: medium ¼ ðx2 x1 ÞS
(8)
After having determined the medium set, low and high sets have to be placed with a symmetric disposition. The definition of the fuzzy dominion has to grant that in each point of the interval:
no more than two sets compete; the total degree of the relative two membership function is equal to 100%. For the first stage of the process, the intermediary takes information from the market about the supplier’s opportunity cost sA[s1,s2] to define its offer wA[w1,w2]: on this second interval it is possible to build the fuzzy sets of the three decisional elements through their relative subfactors. The same process can be also applied on price pA[p1,p2] considering customer’s capability vA[v1,v2]. To effectuate the composition of the variables, any agent has to define the type and level of influence that inputs have on the output value, as to define a set of linguistic rules that can build up the framework. These
medium-high high medium low
very high 1
0
0.2
0.4
0.6
0.8
very low
1
Fig. 3. Fuzzy quantifier.
Table 1 Input–output fuzzy engine. IF
THEN
Contractual power at t
Market penetration at t
Offer at t1
Concession
Low Low Low Low Low Low Low Low Low Medium Medium Medium Medium Medium Medium Medium Medium Medium High High High High High High High High High
Low Low Low Medium Medium Medium High High High Low Low Low Medium Medium Medium High High High Low Low Low Medium Medium Medium High High High
Low Medium High Low Medium High Low Medium High Low Medium High Low Medium High Low Medium High Low Medium High Low Medium High Low Medium High
High High High High High Medium High Medium Medium High High Medium High Medium Medium Medium Medium Low High Medium Medium Medium Medium Low Medium Low Low
give a qualitative description of the actions to take place according to the environmental conditions and can be modified at any round according to the agent’s perceptions. From the intermediary point of view we can define an inverse influence of the three input variables on the output, as a high offer (price) at t1 tends to reduce the concession, as well as a high market penetration and contractual power, generating 33 inference rules shown in Table 1.
4. Case study Considering an international intermediary interested in buying high quality Italian wine to resell it to US distribution, a case study that shows the system capability is shown. The first stage starts from the definition of the range of possible offers to the supplier [w1,w2], assuming wA[h40.00; h60.00] reflecting on the expected wages of a high class wine and opportunities of penetration in the opening market. The definition of the fuzzy sets for the variables can be executed in accordance to the process described. For the variable offer at t1, it is to consider that the intermediary needs a high level of quality of the product and a company’s service level, to grant direct flow internationalization, brand and reputation does not affect
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very much this transaction as it can give good results also with small companies or newcomers. These considerations are settled to define weights of the quantifiers. Starting a negotiation with an affirmed company who grants good performances of its products and operations, it is possible to define the values of the quantifiers as shown in Table 2 and represented in Fig. 4.
These evaluations configure medium set of the offer at t1 according to Eqs. (7) and (8), while low and high ones derive from the symmetrical construction (Table 3 and Fig. 5). With a similar process it is possible to define the other input variables, contractual power and market penetration, and the output variable concession (Tables 4 and 5, Fig. 6).
Table 2 Values of offer at t1 factors.
Table 3 Results for offer at t1 fuzzy set definition.
Factor
Quantifier
Weight
Element
Results
Brand Reputation Quality Service level
High Medium High Medium high
0.15 0.15 0.4 0.3
Smedium Medium Low High
(0.12; 0.32; 0.52) (h42.40; h46.40; h50.40) (h40.00; h42.40; h46.40) (h46.40; h50.40; h60.00)
Fig. 4. Quantifier values for offer at t1 fuzzy set definition.
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Foe example, entering the model with a first offer by the intermediary of w0 ¼ h49, the supplier could accept, exit or ask for a counteroffer. In this third case, the inference engine proposes a value for concession of h2.30. This is the result of the centre of maximum combination of the rules in Table 1 with different activation weights, depending on the state of the fuzzy sets of the input variables. If a counteroffer is asked, the input value of w00 (price 0 w +concession ¼ w00 ¼ h51.30) generates a new concession of h1.15, and so on until the bargain ends. A sequence with four loops is reported in Table 6. If we assume for instance that supplier accept the first counteroffer (w* ¼ h51.30), we can enter the second stage of the game with the negotiation between the reseller and
Fig. 5. Offer at t1 fuzzy set.
Table 4 Quantifier values of contractual power at t, market penetration at t and concession factors. Variable
Quantifier
Weight
Contractual power Relative dimensions Relative volumes Unreliability
Medium Low Medium high
0.4 0.4 0.2
Market penetration Alternatives Existing relationships Market information level Product information level
High Medium high Very high Medium high
0.25 0.25 0.25 0.25
Concession Resistance to concession Expected margin Tangibles/intangibles ratio
Low Medium Very low
0.2 0.2 0.6
Table 5 Results for contractual power at t, market penetration at t and concession definition. Element
Results
Contractual power Medium Low High
(h48.80; h52.80; h56.80) (h40.00; h48.80; h52.80) (h52.80; h56.80; h60.00)
Market penetration Medium Low High
(h42.00; h45.00; h49.00) (h40.00; h42.00; h45.00) (h45.00; h49.00; h60.00)
Concession Medium Low High
(h0.80; h2.40; h6.40) (h0.00; h0.80; h2.40) (h2.40; h6.40; h20.00)
Fig. 6. Contractual power at t, market penetration at t and concession.
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Table 6 Game steps in stage 1. Steps
Offer w (h)
Concession value (h)
1 2 3 4
49.00 51.30 52.45 53.02
2.30 1.15 0.57 0.40
US retailer. The price interval starts from p1 ¼ w*+T and the decisional elements in input have to be evaluated with reference to this new negotiation. In this set of solutions, the intermediary starts from a price, then applies discount equal to concession provided by fuzzy engine, until the second bargain ends. 5. Conclusions The model is able to analyse a possible strategic interaction context in a supply chain environment, integrating a fuzzy description of data in a formal framework of a bargaining game. The uncertainty of decisional parameters at any round and stage is evident: most of the actions derives from a qualitative analysis of the situation and, in any case, experience, intuition, ability of relieving any tone in other party behaviour play a fundamental role in getting a positive result of the negotiation. The dynamic structure of the system, the temporal variability and partial information of parameters need a framework to create an architecture for supplier– intermediary–customer interaction and evaluate the impact and define a sequence of offer-price proposition (known in literature as signaling game and screening game, Cachon and Netessine, 2003). The model represents a tool for analysis that can relate qualitative competencies with both quantitative and estimated information in a logic based on simple linguistic rules and components. It can be extended and refined evaluating the possibility of introducing further inputs, modifying the game structure, membership functions and relative dominions, eventually adding new levels, stages and players. The generalization of the methodology, easy to apply, allows a natural implementation in all the strategic contexts where it is critical to identify and correctly manage private information on prices, times and performance levels. In particular, the same dynamics can be replicated in any problem of scarce resource allocation to identify deceiving strategies in particular environment to define balanced contractual requirements. It is anyway evident how the fuzzyfication activity is the critic element to get the proper input for the system. The lack of a standard procedure and the necessary use of skilful users to evaluate classes and membership functions ask for a simplification and automation of the
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process. The possibility of giving to an expert system (knowledge-based or neural network-based) the activity of elaboration and formalization could increase notably the applicability of the framework, saving the end-user from errors in the decisional process, reducing risks of wrong interpretations and relieving from responsibilities. The inexperience, uncertainty, vagueness and ability of counterparties to give disguised indications could so be modified and interpreted with uniform criterions, through a system that gives an integrated and balanced vision of the different variables and parameters in any market environment. References Aubin, J.P., 1979. Mathematical Methods of Games and Economic Theory. North-Holland, Amsterdam. Borges, P.S.S., Pacheco, R.P.S., Barcia, R.M., Khator, S.K., 1997. A fuzzy approach to the prisoner’s dilemma. Biosystems 41 (2), 127–137. Burns, T.R., Roszkowska, E., 2002. Fuzzy games and equilibria: the perspective of the general theory of games on Nash and normative equilibria. In: Pal, S.K., Pollowski, L., Skowron, A. (Eds.), Rough-Nero Computing: Techniques for Computing with Words. Springer, Berlin. Cachon, G., Fisher, M., 2000. Supply chain inventory management and the value of shared information. Management Science 46 (8), 1032–1048. Cachon, G., Harker, P., 2002. Competition and outsourcing with scale economies. Management Science 48 (10), 1314–1333. Cachon, G., Lariviere, M., 1999. Capacity choice and allocation: strategic behavior and supply chain performance. Management Science 45 (8), 1091–1108. Cachon, G., Lariviere, M., 2001. Contracting to assure supply: how to share demand forecasts in a supply chain. Management Science 47 (5), 629–646. Cachon, G., Lariviere, M., 2002. Supply chain coordination with revenue sharing contracts. Management Science 51 (11), 30–44. Cachon, G., Netessine, S., 2003. Game theory in supply chain analysis. In: Simchi-Levi, D., Wu, S.D., Zuo-Jun, M.S. (Eds.), Handbook of Quantitative Supply Chain Analysis—Modeling in the e-business Era. Kluwer Academic Publishers, Dordrecht, (2004). Dutta, P.K., 1999. Strategies and Games: Theory and Practice. The MIT Press, Cambridge, MA. Ertogral, K., Wu, S.D., 2001. A bargaining game for supply chain contracting. Technical Report, Lehigh University, Department of Industrial and Systems Engineering, Bethlehem, PA. Estelle, J., Vorobeychik, Y., Wellman, M.P., Singh, S., Kiekintueld, C., Soni, V., 2005. Strategic interactions in a supply chain game. Computational Intelligence 21 (1), 1–26. Qi, F., Wang, Y., Tan, J., 2005. Design retrieval technology of fuzzy customer requirements. In: World Congress on Mass Customization and Personalization. Garagic, D., Cruz, J.B., 2003. An approach to fuzzy non-cooperative Nash games. Journal of Optimization Theory and Applications 118 (3), 475–491. Harsanyi, J.C., 1967. Games with incomplete information played by ‘‘Bayesian’’ players. I. The basic model. Management Science 14, 159–182. Rubinstein, A., 1982. Perfect equilibrium in a bargaining model. Econometrica 50 (1), 97–109. Simon, H.A., 1979. Rational decision making in business organizations. American Economic Review 69 (4), 493–513. Spulber, D.F., 1999. Market Microstructure. Cambridge University Press, Cambridge, UK. Thun, J.H., 2003. Analysis of cooperation in supply chain using game theory. In: EurOMA/POMS Joint International Conference. Thun, J.H., 2005. Comparing generic supply chain structures. In: EurOMA International Conference on Operations and Global Competitiveness. Wasfy, A.M., Hosni, A.Y., 1998. Two-party negotiation modeling: an integrated fuzzy logic approach. Group Decision and Negotiation 7 (6), 491–518.