Featured Story March-April 2014
Quantity Discounts in a Two-Node Supply Chain with Complete and Asymmetric Information

by Mr. Dimitris Zissis, Doctoral Candidate

We consider a two node supply chain, with one manufacturer that produces a single product using a lot-for-lot policy, and one retailer that orders items to fulfill forecasted market demand. Both nodes (players) are rational, base their decisions on either known or anticipated cost data, and have set-up costs (production-related for the manufacturer and order-related for the retailer). The material flow between the two players is performed by a specialized enterprise, e.g., a third party logistics -3PL- provider. The manufacturer does not own a warehouse facility, nor can it accommodate inventory at other premises; thus, completed lots are directly forwarded to the retailer. The latter can either handle inventory at his own warehouse or use the storage facility of the 3PL, who can store inventory at a premium (i.e., per unit cost higher than that of the retailer's own facilities). So, the retailer has private information, which is often encountered in practice (in general, for some or all the players to have private information about the supply chain in which they participate).

In a typical Bayesian approach, the business relationship between the manufacturer and the retailer can be modeled via a game with asymmetric information; the asymmetry reflects the two levels of warehousing cost that the retailer knows (own warehouse or 3PL storage facilities - private information) when opting for it, while the manufacturer assumes a probability function for the two levels of the retailer's warehousing cost (with probability p for the low case and 1-p for the case of high holding cost). Obviously, the manufacturer prefers higher orders from the retailer, since this would reduce his set-up costs; on the other hand, the retailer must consider both ordering and storage costs, and determine preferable quantity levels according to the values of all cost parameters.

One lever that the manufacturer can employ to force the retailer to increase orders levels is quantity discounts that reduce the per-unit buying price. This is exactly the issue we investigate and attempt to derive analytical results for the underlying optimization problem. We decided to use quantity discounts (in contrast to returns policies, back-up agreements or other forms of rebates and incentives) in order to promote coordination of the two nodes without resorting to complex contracts or bi-directional information flow and negotiations, since such discounts require only the initial statement of the prices and the discount levels, and then the players either opt to enter or reject the business relationship.

Opportunities for mutual benefits cannot be found, unless the retailer shares his private information. To proceed in sharing private information, the retailer must be provided with appropriate incentives. Towards this end, the Revelation Principle (RP), which is a tool of Game Theory, offers significant insights. The RP was first proposed by Gibbard (1973) and Myerson (1979, 1982). Myerson, Hurwicz and Maskin, were extended the RP's attributes and were awarded the 2007 Nobel Prize in economics.

In our work, first we prove the typical EOQ result for the case of complete information according to a novel theoretical perspective - that of game theory (Gibbons, 1992). In this case, it is optimal for the manufacturer to provide to the retailer a specific quantity discount. Subsequently, we devise exact expressions for the retailer's orders and the manufacturer's discounts given the retailer's private information. We prove that it is feasible to achieve perfect coordination under specific values of the model's parameters. This result is particularly interesting since it goes against several intuitive statements of previous researchers (Corbett and Tang 1999, and Ha, 2001) that claim non-existence of perfect coordination under information asymmetry. These analytical results are evaluated using practical data. In addition, the role of the 3PL company in the business interaction between the two players is examined, since the service costs it offers and the inventory holding prices it provides to the retailer may affect the outcome of the business interaction among the players. Finally, sensitivity analysis and an industrial example offer evidence of the robustness of the results and of the applicability of the approach to real-life business ventures.


Corbett, C. and C. Tang. 1999. Designing supply contracts: Contract type and information asymmetry. In S. R. G. Tayur and M. Magazine (Eds.), Quantitative Models for Supply Chain Management. Kluwer Academic Publishers.

Gillard, A. 1979. "Manipulation of Voting Schemes: A General Result." Econometrica 41: 587-601.

Gibbons, R. 1992. A Primer in Game Theory, Prentice Hall.

Ha, A.  2001. Supplier-buyer contracting: Asymmetric cost information and cutoff level policy for buyer participation. Naval Research Logistics 48 (1), 41-64.

Prize Committee of the Royal Swedish Academy of Sciences, 2007. "Mechanism Design Theory".

Myerson, R. 1979. Incentive-Compatibility and the Bargaining Problem, Econometrica, 47: 61-73.

Myerson, R. 1982. Optimal Coordination Mechanisms in Generalized Principal - Agent Problems, Journal of Mathematical Economics, 10: 67-81.

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