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

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

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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

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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

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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)

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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

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7 Absolute regret: Definition

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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:

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

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10 Relative regret: Definition

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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:

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

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

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14 Symmetric game: Price sensitivity

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15 Symmetric game: Overflow sensitivity

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16 Asymmetric game: Support sensitivity

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17 Symmetric game: Price sensitivity

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18 Symmetric game: Overflow sensitivity

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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

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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

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21 Thank you

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22 Maximin criterion: Definition Ex-ante Ex-post

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23 Absolute regret: Definition Ex-ante Ex-post

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24 Relative regret: Definition Ex-ante Ex-post

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