lecture13

lecture13 - Yinyu Ye, MS&E, Stanford MS&E211...

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Unformatted text preview: Yinyu Ye, MS&E, Stanford MS&E211 Lecture Note #13 1 OnlineOptimizationandIncentiveCompatibility YinyuYe DepartmentofManagementScienceandEngineering StanfordUniversity Stanford,CA94305,U.S.A. http://www.stanford.edu/˜yyye Yinyu Ye, MS&E, Stanford MS&E211 Lecture Note #13 2 Issues • Inrealapplications,data/informationisrevealed sequentially ,andonehasto makedecisionssequentiallybasedonwhatisknown–cannotwaitforsolving the offline problem. • Alloftheparticipants/palyersfarebestwhenthey truthfully revealprivate informationaskedforbythemechanism.Thisiscalled incentivecompatible . Yinyu Ye, MS&E, Stanford MS&E211 Lecture Note #13 3 CombinatorialAuction:offlinelinearprogram However,theproblemcanberewrittenas max π T x- y s.t. A T x- e · y ≤ , x ≤ q , x ≥ , where e isthevectorofallones.Thisisalinearprogram. π T x :the revenue amountcanbecollected. y :theworst-case cost (amountneedtopaytothewinnersiftheworst-casestate isrealized). Yinyu Ye, MS&E, Stanford MS&E211 Lecture Note #13 4 CombinatorialAuction:offlineutilityformulation max π T x- y + u ( s ) s.t. A T x- e · y + s = , x ≤ q , x ≥ . where u ( · ) isa concaveandincreasing vaulefunctionforthesurplusprofit s = e · y- A T x ineachstate i . Forexample, u ( s )= ∑ i u i ( s i ) where s i isanadditionalprofitifstate i is realized.Sincethisprofitis uncertain ,onecanvalueitusingaconcaveand increasingutilityfunctionfor riskaversion . Yinyu Ye, MS&E, Stanford MS&E211 Lecture Note #13 5 CombinatorialAuction:offlinebarrierformulation max π T x- y + ∑ i θ i log( s i ) s.t. A T x- e · y + s = , x ≤ q , ( x , s ) ≥ . where θ i represents initialseed moneyforstate i . Foranygiven θ i > forall i ,thepricevectorisnow unique . Yinyu Ye, MS&E, Stanford MS&E211 Lecture Note #13 6 CombinatorialAuction:onlineutilityformulation Giventheprevious ( t- 1) decisions ¯ x 1 ,..., ¯ x t- 1 ,andinput { π j , a j ,q j } t- 1 j =1 untiltime t ,the t th decisionistochoose x t suchthat, max π t x t- y + u ( s )+ ∑ t- 1 j =1 π j ¯ x j s.t. a t x t- e · y + s + ∑ t- 1 j =1 a j ¯ x j = , x t ≤ q t , x t ≥ . ( π t , a t ,q t ) :the newlyarrived biddingdata. y :theworst-casecost. ∑ t- 1 j =1 a j ¯ x j : outstandingshares ineachstatebeforethenewarrival. Yinyu Ye, MS&E, Stanford MS&E211 Lecture Note #13 7 CombinatorialAuction:simplifiedonlineformulation max π t x t- y + u ( s ) s.t. a t x t- e · y + s =- b t- 1 , x t ≤ q t , x t ≥ , where b t- 1 = ∑ t- 1 j =1 a j ¯ x j –outstandingsharesineachstate.Or max π t x t- y + u ( e · y- a t x t- b t- 1 ) s.t. x t ≤ q t , x t ≥ . Thisisa two-variable problem....
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This note was uploaded on 05/06/2010 for the course MSE 211 at Stanford.

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lecture13 - Yinyu Ye, MS&E, Stanford MS&E211...

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