# American Institute of Mathematical Sciences

April  2018, 14(2): 731-742. doi: 10.3934/jimo.2017072

## A stochastic newsvendor game with dynamic retail prices

 School of Physical and Mathematical Sciences Division of Mathematical Sciences, Nanyang Technological University, 21 Nanyang Link Singapore 637371

* Corresponding author: Nicolas Privault

Received  November 2016 Revised  March 2017 Published  September 2017

Fund Project: This research was supported by Singapore MOE Tier 1 Grant MOE2015-T1-2-130 RG122/15

We extend stochastic newsvendor games with information lag by including dynamic retail prices, and we characterize their equilibria. We show that the equilibrium wholesale price is a nonincreasing function of the demand, while the retailer's output increases with demand until we recover the usual equilibrium. In particular, it is then optimal for retailer and wholesaler to have demand at least equal to some threshold level, beyond which the retailer's output tends to an upper bound which is absent in fixed retail price models. When demand is given by a delayed Ornstein-Uhlenbeck process and price is an affine function of output, we numerically compute the equilibrium output and we show that the lagged case can be seen as a smoothing of the no lag case.

Citation: Ido Polak, Nicolas Privault. A stochastic newsvendor game with dynamic retail prices. Journal of Industrial & Management Optimization, 2018, 14 (2) : 731-742. doi: 10.3934/jimo.2017072
##### References:
 [1] F. Baghery and B. Øksendal, A maximum principle for stochastic control with partial information, Stochastic Analysis and Applications, 25 (2007), 705-717. doi: 10.1080/07362990701283128. [2] B. Øksendal, L. Sandal and J. Ubøe, Stochastic Stackelberg equilibria with applications to time-dependent newsvendor models, Journal of Economic Dynamics and Control, 37 (2013), 1284-1299. doi: 10.1016/j.jedc.2013.02.010. [3] B. Øksendal and A. Sulem, Forward-backward stochastic differential games and stochastic control under model uncertainty, Journal of Optimization Theory and Applications, 161 (2014), 22-55. doi: 10.1007/s10957-012-0166-7. [4] N. C. Petruzzi and M. Dada, Pricing and the newsvendor problem: A review with extensions, Operations Research, 47 (1999), 183-194. doi: 10.1287/opre.47.2.183. [5] I. Polak and N. Privault, Cournot games with limited demand: From multiple equilibria to stochastic equilibrium, preprint (2017), 25 pages. [6] Y. Qin, R. Wang, A. J. Vakharia, Y. Chen and M. H. Seref, The newsvendor problem: Review and directions for future research, European Journal of Operational Research, 213 (2011), 361-374. doi: 10.1016/j.ejor.2010.11.024.

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##### References:
 [1] F. Baghery and B. Øksendal, A maximum principle for stochastic control with partial information, Stochastic Analysis and Applications, 25 (2007), 705-717. doi: 10.1080/07362990701283128. [2] B. Øksendal, L. Sandal and J. Ubøe, Stochastic Stackelberg equilibria with applications to time-dependent newsvendor models, Journal of Economic Dynamics and Control, 37 (2013), 1284-1299. doi: 10.1016/j.jedc.2013.02.010. [3] B. Øksendal and A. Sulem, Forward-backward stochastic differential games and stochastic control under model uncertainty, Journal of Optimization Theory and Applications, 161 (2014), 22-55. doi: 10.1007/s10957-012-0166-7. [4] N. C. Petruzzi and M. Dada, Pricing and the newsvendor problem: A review with extensions, Operations Research, 47 (1999), 183-194. doi: 10.1287/opre.47.2.183. [5] I. Polak and N. Privault, Cournot games with limited demand: From multiple equilibria to stochastic equilibrium, preprint (2017), 25 pages. [6] Y. Qin, R. Wang, A. J. Vakharia, Y. Chen and M. H. Seref, The newsvendor problem: Review and directions for future research, European Journal of Operational Research, 213 (2011), 361-374. doi: 10.1016/j.ejor.2010.11.024.
Retailer output $d \longmapsto q^*_t(d,w) = \psi_d (w)$ in (38) with $w=0.5$
Wholesaler profit $w\longmapsto wq^*_t(d,w) = w \psi_d (w)$ with $d=0.2 < a/(4b)$
Wholesaler profit $w\longmapsto wq^*_t(d,w) = w \psi_d (w)$ with $d=0.35>a/(4b)$
Wholesaler vs retailer profits (46) and (47) as functions of $d_t$ with $\delta=0$
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