1. 1 Robust Newsvendor Competition Houyuan Jiang (Cambridge Judge Business School) Serguei Netessine (The Wharton School of Business) Sergei Savin (Columbia Graduate School of Business)

2. 2 Outline  Newsvendor problem and robust optimization  Maximin/worst-case criterion  Absolute regret criterion  Relative regret criterion  Computational comparisons

3. 3 Newsvendor competition (Netessine-Rudi, 03) p i : Unit price c i : Unit cost D i : Demand random variable F i : CDF for D i Q i : Production quantity o ij: : Overflow rate from newsvendor j to newsvendor i

4. 4 What demand information is available  Moments: Mean, standard deviation, etc  Shape: Support, symmetry, uni-modality, etc We assume that we only know demand supports

5. 5 Robust approaches  Maximin/worst-case: Scarf (58), Gallego-Moon (93), Bental-Nemirovsky (99), Bertsimas- Sim (04), Aghassi-Bertsimas (06)  Absolute regret: Savage (51), Kasugi-Kasegai (61), Variraktarakis (00), Perakis-Roels (05), Perakis-Roels (06), Perakis-Roels (07), Yue et al (07)  Relative regret/competitive ratio: Ball-Queyranne (05), Lan et al (06), Zhu et al (06)  Maximum entropy: Jaynes (57), Eren-Maglaras (06)

6. 6 Maximin criterion  Monopoly: Optimal quantity order for newsvendor i is the lower bound of their demand support, A i  Competition: Optimal quantity order for newsvendor i is still the lower bound of their demand support, A i

7. 7 Absolute regret: Definition

8. 8 Absolute regret: Results 1. There exists a Nash equilibrium and the response function is 2. There exists a unique Nash equilibrium if 3. There exists one newsvendor such that its quantity order does not exceed the upper bound of its demand support 4. There exists a unique Nash equilibrium for a symmetric game:

9. 9 Absolute regret: Results (N = 2) There exists a unique closed-form Nash equilibrium  Which is

10. 10 Relative regret: Definition

11. 11 Regret regret: Results 1. There exists a Nash equilibrium and the response function is 2. There exists a unique Nash equilibrium if 3. There exists one newsvendor such that its quantity order does not exceed the upper bound of its demand support 4. There exists a unique Nash equilibrium for a symmetric game:

12. 12 Relative regret: Results (N = 2) There exists a closed-form Nash equilibrium  Which is unique  Which is unique for some cases

13. 13 Symmetric game: Support sensitivity Basic data: N = 2 [A,B]=[20,70] p=20 c=10 o=0.5

14. 14 Symmetric game: Price sensitivity

15. 15 Symmetric game: Overflow sensitivity

16. 16 Asymmetric game: Support sensitivity

17. 17 Symmetric game: Price sensitivity

18. 18 Symmetric game: Overflow sensitivity

19. 19 Ex-ante and ex-post versions Ex-ante: Evaluate performance measures before demand realization is known Ex-post: Evaluate performance measures after demand realization is known

20. 20 Conclusions and future research  Ex-ante and ex-post versions are equivalent when only demand supports are available  We obtained closed-form response functions and some closed-form Nash equilibrium  Absolute regret criterion is most robust  Robust games with additional information  Robust games with asymmetric information

21. 21 Thank you

22. 22 Maximin criterion: Definition Ex-ante Ex-post

23. 23 Absolute regret: Definition Ex-ante Ex-post

24. 24 Relative regret: Definition Ex-ante Ex-post