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Dynamic and Non-Uniform Pricing Strategies for Revenue

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					Dynamic and Non-Uniform Pricing
    Strategies for Revenue
         Maximization
         Tanmoy Chakraborty
              Zhiyi Huang
            Sanjeev Khanna
       University of Pennsylvania


                                    1
                 Introduction
                 distinct items.
• A seller with ￿￿
• ￿￿                 has                 .
     buyers, buyer ￿￿ private valuation ￿￿
                                         ￿￿


• Seller designs a mechanism to determine
  allocation of items and prices.
• Buyers’ objective: maximize quasi-linear utility,
  i.e. valuation - price.
• Seller’s objective:
   – maximize social welfare, i.e. å￿￿(￿￿
                                     ￿￿).
                                      ￿￿￿￿


   – maximize revenue.

                                                      2
                General Buyers
• Sandholm ’02 Finding an allocation maximizing
  NP-Hard to approximate limited 1/2-￿￿ setting
  social welfare/revenue inwithin ￿￿ factor if is
                                   supply

  unconstrained valuations)are allowed.
   – Consider a graph ￿￿ , ￿￿
                       =(￿￿

  – Each node maps to a buyer
  – Each edge maps to anonly if it gets all incident edges.
  – Each buyer will pay 1 item

  – Finding the profit-maximizing subset of buyers ,
    finding maximal independent set.

                                                              3
      Restricted Types of Buyers
• Additive= ∑ ￿￿
        )
      (￿￿ ￿￿ (￿￿
   – ￿￿       ∈￿￿ )
• XOS (￿￿-XOS)
      (￿￿ )     1(￿￿ 2(￿￿      (￿￿
   – ￿￿ = max {￿￿), ￿￿), … , ￿￿)}
                              ￿￿
       ’s are
   – ￿￿ additive functions
      ￿￿


• Sub-additive ￿￿ + ￿￿
      (￿￿ ) ≤    )
   – ￿￿⋃ ￿￿ (￿￿ (￿￿   )



• In this talk we will focus on sub-additive buyers.
                                                       4
       Maximizing Social Welfare
  – Lavi et al. Social Welfare with Limitedfor general
• Maximizing’05          truthful mechanism Supply

   – buyers. et al. ’06 ’07 ￿￿ ￿￿
     Dobzinski               (log ) truthful mechanism

   – for sub-additive buyers. algorithm for sub-additive
     Feige ’06 2-approximation

     buyers.




                                                           5
           Maximizing Revenue
• Maximizing Revenue with Unlimited Supply
                          (log ) truthful mechanism for
  – Guruswami et al. ’05 ￿￿ ￿￿

     Balcan et al. or ￿￿ approximation algorithm for
   – unit-demand’06 single-minded buyers.
                        (￿￿)
     single-minded buyers who want at most ￿￿ items.
                        (log ) truthful mechanism for
   – Balcan et al. ’08 ￿￿ ￿￿

     general buyers.




                                                        6
                      Item Pricing
                           for         .
• Seller chooses a price ￿￿ each item ￿￿
                          ￿￿


• Buyers come in sequence (adversarial/random/best
• order) ￿￿ a subset ￿￿the remaining items
  Buyer buys                of
                                        .
  maximizing the utility, i.e. ￿￿ - ∑￿￿￿￿
                                  )
                                (￿￿ ∈￿￿￿￿
                                ￿￿


• Objective: Find a pricing that maximizes revenue.
   – Dobzinski et al. ’06          for general buyers in limited

     supply setting.
   – Briest et al. ’09 pricing for lotteries.

• Our focus: maximizing revenue by item pricing for sub-
  additive buyers and limited supply.
                                                                   7
              Competitive Ratio
• Optimal Social Welfare all allocation ￿￿2, … , ￿￿
                  ￿￿ for
  – OPT = max ∑￿￿ ￿￿
                   (￿￿
                     )                   1, ￿￿     .
                                                  ￿￿
   – OPT upper bounds maximal revenue.

• Competitive Ratio has competitive ratio ￿￿
   – A pricing strategy                    if the
     expected revenue is ￿￿     ).
                          (OPT/￿￿

• Intrinsic Gap: Social Welfare vs. Revenue
                       (log ) ratio.
     get better than ￿￿ ￿￿
   – There exists a sub-additive buyer s.t. no pricing can

   – So we aim for poly-logarithmic competitive ratio.

                                                             8
   What’s Known in This Model?
• Not much until recently…
  – Balcan, Blum, and Mansour ’08 considered static
  – uniform pricing for sub-additive buyers. … , H/￿￿
    Pick a random price from {H, H/2, H/4,          2
                                                      }.
  – H is max￿￿ ￿￿):
              ,￿￿(￿￿
                ￿￿
    (a) no one will buy at a price above H;
                                          to
    (b) some item has value at least H/￿￿some buyer.
  – This strategy gets            competitive ratio.

  – They also proved            lower bound.

                                                           9
    Lower Bound for Static Uniform
              Pricing
• We strengthen the BBM lower bound to
  essentially match their upper bound.
  Thm 1. There exists an instance s.t. no static
  than          even competitive ratio better
  uniform pricing haswith complete information

  and identical buyers.
• The lower bound also holds for 2 distinct
  buyers with 3-XOS valuations.


                                                   10
  Can We Beat This Lower Bound?
• We need to go beyond static uniform pricings
  to beat this lower bound…
• Two natural extensions:
  – static ) dynamic: allow changing price upon the
    arrival of each buyer.
  – uniform ) non-uniform: allow using different
    prices for different items.



                                                      11
                 Our Results
Thm 2. There exists an ￿￿ 2 ￿￿
                        (log ) competitive

                                 (log /loglog2
dynamic pricing is pricing.than ￿￿ 2no dynamic
uniform uniform better Moreover, ￿￿
 ) even
￿￿ given complete information.

Thm 3. There exists an ￿￿ 3 ￿￿
                        (log ) competitive

static non-uniform pricing with complete
information and identical buyers.
– Also get poly-logarithmic ratio for constrained types
  of buyers without knowing the valuations.

                                                          12
       Dynamic Uniform Pricing
  The try: pick
• FirstAlgorithm: a threshold ￿￿
   – Randomly                  *
                                 from
  – Randomly pick a price from set, H/￿￿
                {H, H/2, H/4, …        2
                                         }.
                                         }

    on the arrival of each buyer.     *
                                     ￿￿


  Counter Example: The first ￿￿ buyers value all
                       (log2 ￿￿
  items at H, and the last ￿￿ )
                     ￿￿ buyers values each
                                      buyer
  item at H, w.h.p. one of the first ￿￿ buys all

  items at low price.
                                                   13
                Proof Sketch
  ￿￿items supported by price ￿￿ picking set
    of                           , then
  Lemma 1. Consider a single buyer, if there is aa
  price randomly from {H, H/2, H/4, … , ￿￿ gives
                                          /2}
  ￿￿(￿￿         ) expected revenue.
       |S|/log ￿￿
  Lemma 2. Consider the allocation ￿￿2, … , ￿￿
                                         1, ￿￿    ￿￿
                                              be
  that maximizes the social welfare, let ￿￿ the
              Buyerat price ￿￿entire
                     will buy if
             subset B wouldthethey￿￿presented
  set of items that i                 are
                              buy if ￿￿ is
                                             ￿￿￿￿

  at price H/2￿￿, then
                              available
               the only items￿￿
                   ,￿￿ |(H/2
                    |￿￿
                ∑￿￿ ￿￿￿￿ ) ≥ OPT

• The proofs rely on sub-additive valuations.
                                                   14
           Proof Sketch (Cont’d)
• Suppose the threshold is               , and let
             denote the subset of remaining items in
      when       arrives
   – All items in        are sold with a “high” price


   – The expected revenue from      is (Lemma 1)

   – So we have all ￿￿
   – Sum up for        (Lemma 2)
                    , ￿￿


                                                        15
   LB for Dynamic Uniform Pricing
                                   (log ) lower
• Warm up: Two examples that give ￿￿ ￿￿

            uniform pricing with any ￿￿
• bound for ￿￿ ￿￿= 1/2￿￿ single buyer.
    ￿￿ ∙2 , and (￿￿
  |￿￿ ￿￿
     |=             )        for       .
                                     ∈￿￿
                                       ￿￿
  1                             1
             OPT = log2 ￿￿                OPT = log ￿￿
             ALG = log ￿￿                 ALG = 1

       1/2                          1/2
   1                            1

              1/4                          1/4
       4                            4
              12        1/￿￿               12        1/￿￿
                         log
                        ￿￿ ￿￿                         log
                                                     ￿￿ ￿￿
  S1 S2       S3    …   Slog ￿￿ T1 T2       T3   …    Tlog ￿￿
                                                             16
  LB for Dynamic Uniform Pricing
• Combining two example:
  – Each buyer has its own copy of    and .
  – Also interested in   ,      , at much lower price.
  p    1                                      1
                                                              /loglog2 ￿￿
                                                    X = log2 ￿￿

           1/X                                    1/X


                  1/X2                                    1/X2

                                1/￿￿                                    1/￿￿

      Si 1 Si 2   Si 3   …   Silog ￿￿ ￿￿
                                    /loglog   T i1 T i2   T i3   …   Tilog ￿￿ ￿￿
                                                                            /loglog

                                                                                  17
                  Summary
• Static uniform pricing has strong lower
  bounds, even when buyers are very simple, or
  buyers have identical valuation –
• Changing the price (dynamic uniform pricing)
  improves the revenue significantly –
• Static non-uniform pricing gets poly-
  logarithmic ratio in some constrained setting.
  – Understanding the power of static non-uniform
    pricing in the general setting is still open.
                                                    18
Thank you!




             19

				
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