Abstract
We study a repeated newsvendor game with transshipments. In every period n, retailers face a stochastic demand for an identical product and independently place their inventory orders before demand realization. After observing the actual demand, each retailer decides how much of her leftover inventory or unsatisfied demand she wants to share with the other retailers. Residual inventories are then transshipped in order to meet residual demands, and dual allocations are used to distribute residual profit. Unsold inventories are salvaged at the end of the period. While in a single-shot game retailers in an equilibrium withhold their residuals, we show that it is a subgame-perfect Nash equilibrium for the retailers to share all of the residuals when the discount factor is large enough and the game is repeated infinitely many times. We also study asymptotic behavior of the retailers’ order quantities and discount factors when n is large. Finally, we provide conditions under which a system-optimal solution can be achieved in a game with n retailers, and develop a contract for achieving a system-optimal outcome when these conditions are not satisfied. This chapter is based on Huang and Sošić (European Journal of Operational Research 204(2):274–284, 2010).
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Notes
- 1.
For some exceptions, see Yang and Schrage (2009), which show that the inventory levels can increase after centralization when demand follows right-skewed distributions, or when the newsvendor ratio is low.
- 2.
If D has a finite support with upper bound \(\bar{D}\) , then \(M =\bar{ D}\).
- 3.
Note that in our repeated-game setting we were able to achieve \(\vec{{X}}^{\mathrm{d}}\) as a SPNE, by utilizing the fact that J i d(X i D) ≥ J i 1(X i 1). Unfortunately, J i d(X i C) can be greater or smaller than J i 1(X i 1), hence a first-best ordering quantity cannot, in general, be obtained as a SPNE.
- 4.
The amount of transfer payments d i ≤ 0 (realized when a player benefits from a defection) removes from a retailer all possible gains from that defection. Δ it > 0 (which leads to d i > 0) implies that a retailer observes a loss as a result of someone’s defection (and is, therefore, compensated from payments of those who benefit); this retailer receives a fraction of total transfer payments proportional to her loss as compared to the total losses observed by the system.
- 5.
In the whole contract lifetime, the discretionary transfer payment happens at most n − 1 times, as the number of inventory-sharing retailers is reduced from n to 1.
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Huang, X., Sošić, G. (2012). Repeated Newsvendor Game with Transshipments. In: Choi, TM. (eds) Handbook of Newsvendor Problems. International Series in Operations Research & Management Science, vol 176. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-3600-3_4
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