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					Dynamic Pricing


          Peter R. Wurman
    North Carolina State University
E-commerce Big Picture




 Infrastructure
        TCP/IP        Databases
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        Anonymity
E-commerce Big Picture


 Make Contact
    Web mining
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    Recommendations


 Infrastructure
        TCP/IP        Databases
        HTTP & HTML   Encryption
        Anonymity
E-commerce Big Picture


 Make Contact         Negotiate            Exchange
    Web mining
                        Auctions             Contracts
    Data mining
                        Agents               Payments
    XML
    Recommendations


 Infrastructure
        TCP/IP                Databases
        HTTP & HTML           Encryption
        Anonymity
Why Auctions?
   Auctions enable dynamic pricing
    –   Let the demand (and supply) determine the
        market value
Current Example
 Sony introduced the Playstation2 in the US
  yesterday
 Lines began the day before
 Expect to sell 3 million by March
 Only expect to have half the units they need
  for the holidays
Discussion
   How did Sony pick $300?
How much did Sony Make?
 Assume 500,000 units the first day
 Cost = $300/unit
 Revenue = $300 * 500,000 = $150,000,000




    $300
                              Millions of units
                 .5
What is the Market Value?
   Let’s check ebay...
How Much Could Sony Have
Made?
   Assume
    –    Purchasers are linearly distributed between $x
         and $300.
    $x


    $370
                                          Millions of units
                         .98
How Much Could Sony Have
Made?
   Lost revenue = 250000 * (x - 300)



    $x


    $300
                                  Millions of units
                    .5
Questions
 What was Sony’s allocation policy?
 For what price should Sony have sold the
  Playstation2?
 Where did all of that other money go?
 How could Sony have gotten more of it?
More Motivation
 $2.7 trillion in Internet commerce by 2003
  (Forrester)
 B2B exchanges developing in nearly every
  industry
    –   Automotive, textiles, steel, farm equipment,
        chemicals, used laboratory equipment, etc.
   Many of these have or plan auction
    capabilities
What is an Auction?
 Auctions are mediated negotiation
  mechanisms in which one negotiable
  parameter is price
 Note:
    –   Mediated implies messages are sent to
        mediator, not directly between participants
    –   Mediator follows a strict policy for determining
        outcome based on messages
    –   Single seller auctions are a special case
Classic English
   An auctioneer stands up in front of the room
    –   Outcry: bidders call out prices
    –   Silent: auctioneer calls prices and bidders signal
        silently
 Highest bidder gets the object
 Pays e more than the next highest bidder
Classic Dutch
 Price clock starts at too high a price
 Price descends in real time
 First bidder to signal gets the goods at the
  price on the clock
Sealed Bid Auction
 Everyone puts their bid in an envelope and
  submits it to the auctioneer
 At a designated time, the auctioneer opens
  the envelopes and determines the highest
  bidder
 Winning bidder pays its bid
 Often used in reverse for procurement
Vickrey Auction
 Aka. Second-Price Sealed Bid Auction
 Everyone puts their bid in an envelope and
  submits it to the auctioneer
 At a designated time, the auctioneer opens
  the envelopes and determines the highest
  bidder
 Winner pays the price of the second highest
  bidder.
Ten-Dollar Auction
 Object for sale: a $10 bill
 Rules
    –   Highest bidder gets it
    –   Highest bidder and the second highest bidder
        pay their bids
    –   New bids must beat old bids by 50¢.
    –   Bidding starts at $1
Other types of Auctions
   Continuous Double Auction (CDA)
    –   Multiple buyers and sellers
    –   Clears continuously
   Call Market
    –   Multiple buyers and sellers
    –   Clears periodically
Other types of Auctions
   Reverse Auction
    –   Single buyer
    –   Lowest seller gets to sell the object
    –   Used in many procurement situations
   Multi-item Auctions
    –   Single seller
    –   Multiple units for sale
    –   N highest bidders get objects and pay ?
Taxonomy
Core Auction Activities
   Auctions:
    –   Receive bids
    –   Supply intermediate information (optional)
    –   Clear



                           Information Revealed



                                       Clearing Policy


                       Bidding Rules
Core Auction Activities (2)
   Receive bids
    –   Enforce any bidding rules
   Release intermediate information (optional)
    –   Produce quotes
    –   List of winning bidders
   Clear
    –   Determine who trades with who and at what
        price
Bidding Rules
   Individual rules govern
    –   Who can bid?
    –   Semantics of bid
    –   Beat-the-quote rules
    –   Beat-your-bid rules
Information Revelation Rules
   Individual rules govern:
    –   Are price quotes generated and if so, in what
        format?
    –   Are the winners identified?
    –   Are the current winning bids identified?
    –   Are the bidder’s id’s associated with their bids?
    –   How often is the information generated?
    –   Does everyone see the same information?
Clearing Rules
   Individual rules govern
    –   Who gets to trade?
    –   At what price?
    –   What events trigger a clear?
    –   When does the auction close?
Example Parametrizations
                          English Outcry     Vickrey         CDA
Bidding Rules
   Participation            Many:1          Many:1        Many:many
   Bid semantics           Single-unit     Single-unit    Single-price
   Beat-the-quote            Buyer            NA              NA
Information Re velation
   Price quotes               Ask            None          Bid-Ask
   Quote schedule            Activity        None          Activity
   Order book                 Open           Closed        Closed
Clearing Policy
   Clearing schedule        Inactivity      Fixed time     Activity
   Closing Conditions       Inactivity      Fixed time      Never
   Matching function        First-price    Second-price    By time
Purpose of an Auction
   Auctions:
    –   Mediate communication



                        Auction

           Agent

                                    Agent
                Agent       Agent
Purpose of an Auction
   Auctions:
    –   Mediate communication
    –   Facilitate the multilateral exchange of resources

                           Auction

            Agent              $

               C                             Agent
                                         A
                   Agent
                           B   Agent
Analysis
   How do we predict the outcome?
    –   Characterize the decision problem each bidder
        faces
    –   Determine a “rational” strategy
Part 2: Decision Theory
What is a Rational Decision?
   We assume that agents have preferences
    over states of the world
    –   A>B   A is strictly preferred to B
    –   A~B   agent is indifferent between A & B
    –   A≥B   A is weakly preferred to B
Lotteries
   A lottery is a combination of a probability
    and an outcome
    –   L = [p, A; 1 – p, B]
    –   L = [1, A]
    –   L = [p, A; q, B, 1 – p – q, C]
   Lotteries can be used to asses a human’s
    preference structure
Example: Who Wants to Be a
Millionaire?
Millionaire Scenario
 You have just achieved $500,000
 You have have no idea on the last question
 If you guess
    –   [3/4, $100,000; 1/4, $1,000,000]
   If you quit
    –   [1, $500,000]
   What do you do?
Millionaire Scenario
Maximizing the Expected Payoff
   Maximize expected monetary value (EMV):
    –   EMV(guess) = pcorrect * U(guess correct) +
        pwrong * U(guess wrong)
    –             =1/4(1,000,000) + 3/4 (100,000)
    –             = 325,000
    –   EMV(quit) = 500,000


   What if you had narrowed the choice to two
    alternatives?
Properties of Preferences
   Orderability
    –   For any two states, either A > B, B > A, or A~B
   Transitivity
    –   If A > B and B > C, then A > C
   Continuity
    –   If A > B > C, then there is some p, s.t.
        [p, A; (1-p) C] ~ B
More Properties
   Substitutability
    –   If A~B, then [p, A; (1-p) C] ~ [p, B; (1-p) C]
        for any value of p
   Monotonicity
    –   If A > B and p ≥ q then [p, A; (1-p) B] ≥ [q, A;
        (1-q) B]
   Decomposibility
    –   Compound lotteries can be reduced to simpler
        ones using laws of probability
Utility Functions
   If the agents preferences satisfy the
    properties, there is a real valued function U
    such that
    –   U(A) > U(B) implies that A > B
    –   U(A) = U(B) implies that A ~ B
Human Utility for Money
   The evidence suggests that humans do not
    have a linear utility for money
    –   We have regret
    –   Our utility seems to depend upon our existing
        wealth
          The first million has more of an effect on our
          lifestyle than the 100th million
St. Petersburg Paradox
 Bernoulli, 1738
 Game:
    –   A fair coin is tossed
    –   If tails, you double the pot & flip again
    –   If heads, the game ends and you keep pot


   How much would you pay for a chance to
    play this game?
Expected (Monetary) Value
   EMV(St. P) = Sumi p(H on turn i) * (2i)
              = Sumi (1/ 2i) 2i
             =∞

   If U  EMV, then you should be willing to
    pay an infinite amount of money to play the
    game.
Expected Utility
   If utility for money has the form:
    –   U(m) = log2 m
   Then
    –   EU(St. P) = Sum (1/ 2i) log2 2i
                     =2
   So a player with this utility function should
    be willing to pay 2 utiles (= $4) to play
Human Utility for Money
              U+




  $-                      $+



              U-
Risk
   Let
    –   S = [1, x]
    –   L = [p, y; 1 – p, z]
    –   Where x =EMV(L) = py + (1 – p)z
 Risk averse: U(S) > U(L)
 Risk neutral: U(S) = U(L)
 Risk seeking: U(S) < U(L)
Quasilinear Utility
   Often assume that agents are risk neutral
    within the scope of the problem
    –   U(x,m) = v(x) + m
        where m is money
                       +
                    U
                                 This allows us
                                 to use money
        $-                  $+   as the scale of
                                 utility
                    U-
Quasilinear Utility
   For any allocation x,
    there is some amount of money, m’,
    s.t. i is indifferent between x and m’
    regardless of i’s endowment of money.

   We call vi(x) agent i’s willingness-to-pay
    for allocation x
Formal Model
 I = the set of agents, i = 1…n
 G = the set of resources, g = 1…m
 xig = an amount of good g allocated to I
 xi = <xi1,…xiM>
 eig = agent i’s endowment of g
 ei = <ei1,…eiM>
Implicit Assumptions
   We have assumed:
    –   Agents know their valuations
    –   Valuations are independent
    –   Valuations are private


   Other choices include
    –   Valuations are correlated
    –   Valuations depend on externalities
Agent Surplus
 An agent’s surplus is the change in its
  utility between two states
 Agent i’s surplus from the allocation xi:
    –   si = U(xi) - U(ei)
Solution
   Given a a particular allocation problem,
    what is a good distribution of the resources?
Pareto Efficient Solutions
   An allocation, f, is Pareto efficient (optimal)
    if
    –   No agent can be made better off without
        making some agent worse off
    –   i.e. there is no solution f’ and agent i for which
          Ui(f’i) > Ui(fi)
          and for all other agents h
           Uh(f’h) ≥ Uh(fh)
Pareto Efficient Solutions

     U2
                f1


                     f2

           f3
                          f4



                               U1
Pareto Efficient Solutions

     U2
                f1

                               f 2 Pareto
                     f2        dominates f 3

           f3
                          f4



                                U1
Pareto Efficient Solutions

     U2
                f1

                          The Pareto frontier
                     f2

           f3
                              f4



                                   U1
Pareto Efficiency and
Quasilinearity
   When agents have quasilinear preferences
    the Pareto solution satisfies:
        max (Si v     (fi))
        subject to Si xig = Si eig
Pareto Efficient Solutions

     v2
                f1
                           Pareto
                          Solutions
                     f2

           f3
                             f4



                                  v1
A Simple Example
   Two agents, one unit of resource A
    –   e1A = 1, e2A = 0
    –   v1(A) = $3, v2(A) = $5


   Claim:
    –   Pareto Efficient solution gives A to agent 2
Example Continued
   Definitions:
    –   U1(e) = v1(A) + m1 = 3 + m1
    –   U2(e) = m2
    –   U1(f*) = m*1
    –   U2(f*) = v2(A) + m*2 = 5 + m*2
Example Continued
   Conditions:
    –   U1(f*) ≥ U1(e)
        m*1 ≥ 3 + m1
        m*1 - m1 ≥ 3          Dm1

    –   U2(f*) ≥ U2(e)
        5 + m*2 ≥ m2
        5 ≥ - (m*2 -m2)       -Dm2

   Constraint: 3 ≤ Dm1 = - Dm2 ≤ 5
Example Conclusion
   As long as agent 2 compensates agent 1 $x,
    $3 ≤ x ≤ $5, both agents are better of
    exchanging A.
    –   s1 = x - 3
    –   s2 = 5 - x
No Further Gains
 Pareto efficiency and quasilinearity implies
  that no more surplus can be created
 which implies that there are no further gains
  from trade to be made

   Finding f* is the social planner’s objective
How Do We Find f*?
   In a distributed system:
    –   We can’t impose an allocation
    –   The valuations are private information
   We want a mechanism that encourages
    agents acting in their own self-interest to
    find f*
Core Auction Activities
Revisited
   Receive bids
    –   Enforce any bidding rules
   Release intermediate information (optional)
    –   Produce quotes
    –   List of winning bidders
   Clear
    –   Determine who trades with who and at what
        price
Clearing Policies
 Input: the set of bids
 Output: a set of exchanges:
    –   a1 gives x units to a2 for $p


   The policy for determining the exchanges
    from the bids is called the matching policy
Properties of Matching Policies
 Buyers pay no more than their bid, sellers
  receive no less
 Exchanges are budget balanced
 The market is cleared
 The exchanges represent a locally efficient
  allocation
 All exchanges occur at a uniform
  equilibrium price
An Example
   Agent   Bid
    a1      sell 1 unit at $1
    a2      buy 1 unit at $2
    a3      sell 1 unit at $3
    a4      buy 1 unit at $4
Diagram
                  Price
              4


          3


              2


          1


                          Quantity
 Interpretation of Bid Points
                              Price
                          4

                                      Buyers will pay their
                      3
                                      bid or less
Sellers will accept
their bid or more         2


                      1


                                      Quantity
One Set of Exchanges
                   Price
               4
                     a3 sells 1 unit to a4
           3         for $3 ≤ p1 ≤ $4

               2
                     a1 sells 1 unit to a2
           1         for $1 ≤ p2 ≤ $2

                           Quantity
Analysis
 Market is cleared
 Not a uniform price p1 ≠ p2
 Not efficient:
    –   a3’s bid is interpreted as its reserve value
    –   After the exchanges a3 would want to buy a2’s
        unit
    –   There is a trade remaining
    –   The agents with the highest values (a3, a4) do
        not have the items in the end
Another Exchange Set
                   Price
               4


           3
                     a1 sells 1 unit to a4
               2
                     for $1 ≤ p ≤ $4

           1


                           Quantity
Analysis
 Market is cleared
 Uniform price
 Efficient
    –   a4 gets one, a3 keeps one
 Budget balanced
 But,
    –   If p < $2, then a2 will want to buy one
    –   If p > $3, then a3 will want to sell one
Aggregate Demand
                  Price
              4


          3


              2


          1


                          Quantity
Equilibrium Prices
                     Price
                4


            3                 Range of prices
                              that balance supply
                2             and demand

            1


                             Quantity
General Procedure to Find
Equilibrium Prices
 Let there be L single-unit bids
 Let M be the number of sell offers
 Method:
    –   Sort all bids by price
    –   Count down M bids
 The Mth bid is the top of the eq. range
  The (M+1)st bid is the bottom of eq. range
 Any price in between is an eq. price
Mth and (M+1)st Example
                    Price
                4
                                       L=4
                                       M=2

  Mth bid   3


                2           (M+1)st bid

            1


                            Quantity
Another Example
                    Price
            4
                                       L=4
                                       M=2

  Mth bid   3


                2           (M+1)st bid

                1


                            Quantity
Yet Another Example
                      Price
 L=4
                  4
 M=2

                  3           Mth bid

(M+1)st bid   2


              1


                              Quantity
Which Trades?
                      Price
 L=4
                  4
 M=2

                  3           Mth bid

(M+1)st bid   2


              1


                              Quantity
Which Trades?
                      Price
 L=4
                  4
 M=2

                  3           Mth bid

(M+1)st bid   2


              1


                              Quantity
Which Trades?
   If all exchanges occur at price p, agents do
    not care who they trade with
    –   Let x = {0, 1}
    –   Agents utility: ui(x) = vi(x) - p(x - e)
What Exchange Price?
 Let pM = the $ value of the Mth bid
 Let pM+1 = the $ value of the (M+1)st bid


   For any k, 0 ≤ k ≤ 1,
        p = pM+1 + k(pM - pM+1)
    is an equilibrium price
Properties
 Clears the market
 Budget balanced
 Uniform price
 Locally efficient
 Equilibrium prices
Price Quotes
   If the auction is not sealed bid, what price
    information should we reveal?
Inspiration: CDA
                   Price
           4


     ask   3


               2           bid

               1


                           Quantity
Bid-Ask “Spread”
 The ask quote is what a buyer needs to offer
  to form an exchange
 The bid quote is what a seller needs to offer
  to form an exchange

   In a CDA, it represents the spread between
    the buyers and the sellers
Generalizing the Bid-Ask Quote
   Mapping:
    –   Ask = Mth price
    –   Bid = (M+1)st price


   The interpretation works even if the
    standing bids overlap
    –   Beating the ask price either matches an
        unmatched seller, or displaces a matched buyer
Buyer Example
                    Price
          4                     4
                                5
          3   ask               3


bid   2                     2


      1                     1


                Quantity
More About Quotes
 Suppose you have a buy bid, b.
 You are winning if:
    –   b > pask
    –   b = pask, and pask > pbid
   But, if b = pask = pbid ,
    you can’t tell if you are winning

   Symmetric result for the seller
Multi-unit Bids
 When bids are divisible, the Mth, (M+1)st
  pricing still works
 M = the number of units for sale
 Essentially, treat each unit as a separate bid
      Agent       Bid
      a1          sell 3 units at $1
      a2          buy 2 unit at $2
      a3          buy 2 unit at $3
Multi-unit Bid Example
                    Price
 L=7
 M=3

                3 3


                2   2


        1 1 1


                            Quantity
Some Take Home Messages
 Uniform prices are good
 Buyers and sellers are symmetrical
Part 3: Analysis
The Vickrey Auction
 Single seller (M = 1)
 N buyers
 Highest bidder pays the second highest
  (M +1)st price
 Sealed bids


   Property: dominant strategy for buyers to
    bid true valuation
Proof
   Consider 2 agents
    –   Random valuations, v1, v2
    –   Place bids b1, b2
   Agent 1’s utility for bidding b1
    –   U(b1) = Pr(b1 > b2)[v1 - b2]
Proof: Case 1
   If [v1 - b2] > 0, then agent 1 wants to
    maximize Pr(b1 > b2)
    –   Does so by setting b1 = v1


                                                   v1


                                              b2
Proof: Case 2
   If [v1 - b2] < 0, then agent 1 wants to
    minimize Pr(b1 > b2)
    –   Does so by setting b1 = v1
                                              b2

   Thus, setting b1 = v1                          v1
    (truth-telling)
    is a dominant strategy
Intuition
   The amount that an agent pays is not a
    function of their bid

   The only thing an agent controls is the
    probability that it wins when it should, and
    doesn’t win when it shouldn’t
Extensions to the Mth & (M+1)st
Price Auctions
   For single-unit buyers, it is a dominant
    strategy to bid truthfully in an
    (M+1)st-price sealed-bid auction

   For single-unit sellers, it is a dominant
    strategy to bid truthfully in an Mth-price
    sealed-bid auction
Why Not Multiunit Bidders?
   A bidder whose bid is setting the price, may
    benefit by lowering its bid on some of its
    units

   Example: a multiunit buyer in an (M+1)st-
    price auction
Multi-unit Bid Example
                    Price
 L=7
 M=3

                3 3


                2   2           Mth, (M+1)st

        1 1 1


                            Quantity
Multi-unit Bid Example
                    Price
 L=7
 M=3

                3 3


                2   2           Mth

        1 1 1       2           (M+1)st

                            Quantity
Why Not Buyers and Sellers at
the Same Time?
 One buyer, one seller
 Sealed bids b & s
 Valuations, vs, vb, drawn
  from overlapping
                                   vb
  distributions
                              vs
Desirable Properties
   Efficient
    –   If vs < vb, the agents trade
   Truthful
    –   Dominant strategies to bid vs, vb
   Individually rational
    –   No agent will be made worse off
   Budget balanced
    –   Amount seller receives = amount buyer pays
No Perfect Mechanism
                            –   Meyerson & Satterthwaite, 1983

 Ub(b) = Pr(b > s)[ vb - pb]
 For the buyer to bid truthfully,
  from the previous result,
  pb = s
                                                             vb
   Similarly, for the seller          vs
    to bid truthfully,
     ps = b
Not Budget Balanced
   For both agents to bid truthfully, we
    –   give the seller b
    –   take from the buyer s
   But b > s, so the mechanism runs a deficit
McAfee’s Dual Price Auction
   We can get truthful behavior for both the
    buyers and sellers, and budget balance by
    sacrificing efficiency

 Let p = pM+1 + k(pM - pM+1)
 All exchanges occur except the lowest
  buyer at or above pM, and the highest seller
  at or below pM+1
Dual Price Diagram
                      Price
                  4


                  3            Mth bid

(M+1)st bid   2

                              Discard this trade
              1


                               Quantity
Dual Price Properties
 Everyone bids truthfully
 Individually rational
 Budget balanced
 Not efficient, but only sacrifices the lowest
  valued exchange
    –   This can be arbitrarily bad
    –   i.e. if there was only one trade available
Other matching functions
   Chronological
    –   Exchange occurs at the price of the earlier/later
        bid
    –   Model used by the stock market in conjunction
        with immediate clears
   Pay buyers, pay sellers bid
    –   All exchanges occur at the buyer’s (or seller’s)
        offer
    –   Common in multiunit auctions on the Internet
The Example
   Agent   time   Bid
    a1      1      sell 1 unit at $1
    a2      2      buy 1 unit at $3
    a3      3      sell 1 unit at $2
    a4      4      buy 1 unit at $4
Chronological Match
                              Price
Earlier Bid Prices:       4           Later Bid Prices:
1  4 at $1
                                      1  4 at $4
                          2
3  2 at $3
                                      3  2 at $2
                      3


                      1


                                      Quantity
Agents Care How Matches are
Formed
                              Price
Earlier Bid Prices:       4           Later Bid Prices:
3  4 at $2
                                      3  4 at $4
                          2
1  2 at $1
                                      1  2 at $3
                      3


                      1


                                      Quantity
Pay Buyers/Sellers Bid
                              Price
Sellers Bid Prices:       4           Buyers Bid Prices:
1  4 at $1
                                      1  4 at $4
                          2
3  2 at $2
                                      3  2 at $3
                      3


                      1


                                      Quantity
Comparison
          14       34
          32       12
Mth       $3/$3     $3/$3   Uniform
(M+1)st $2/$2       $2/$2   Prices
Dual (.5) $2.5/--   N/A

Earliest   $1/$3    $2/$1
Latest     $4/$2    $4/$3   Discriminatory
                            Prices
Buyers     $4/$3    $4/$3
Sellers    $1/$2    $2/$1
4-Heap Algorithm
 Straightforward algorithm for managing all
  of the previous types of auctions
 Keep all bids in four heaps
    –   Bin    current winning buy bids
    –   Bout   current non-winning buy bids
    –   Sin    current winning sell bids
    –   Sout   current non-winning sell bids
4-Heap Diagram

                 Bin
   Sout


                 Bout
   Sin
Properties of Heaps
 Bout, Sin ordered so highest price is top
 Bin, Sout ordered so lowest price is “top”
 Constraints
    –   # units in Bin = # units in Sin
    –   top(Bout) ≤ top(Bin)
    –   top(Sin) ≤ top(Sout)
    –   top(Sin) ≤ top(Bin)
    –   top(Bout) < top(Sout)
Insert New Bid (1)

                     Bin
   Sout


                     Bout
   Sin
Insert New Bid (2)

                     Bin
   Sout                Put



                     Bout
   Sin
Insert New Bid (3)

                                  Bin
        Sout


                                  Bout
        Sin
Violates the
condition that
# units in Bin = # units in Sin
Insert New Bid (4)

                     Bin
   Sout
         Get


                     Bout
   Sin
Insert New Bid (5)

                     Bin
   Sout

          Put
                     Bout
   Sin
Insert New Bid (6)

                     Bin
   Sout


                     Bout
   Sin
Complexity Analysis
 Insert new bids in O(log L)
 Remove a bid in O(log L)
 Quote in constant time
 Clear in O(size of Bin)


   Can be used for all of the above matching
    functions
Indivisible bids
   Agent   Bid
    a1      buy exactly 2 units at $3/each
    a2      buy exactly 2 units at $2/each
    a3      buy exactly 1 unit at $1/each

 Three units for sale
 Who do we give them to?
Knapsack Problem

   How do we best fill a three unit knapsack


                                      $1

           $6   $4                    $6
                      $1
Another Example

   How do we best fill this four unit knapsack?


                                      $8

          $15
                $8    $8              $8


   Knapsack is a classic NP-complete problem
Indivisible Bids and Prices

                                       $1

          $6    $4                     $6
                      $1


 There is no price, p, such that a1 and a3
  want to buy, and a2 doesn’t
 For equilibrium, we need prices that are
  nonlinear functions of quantity
Summary
 Can get dominant strategies for one side or
  the other (w/ single units)
 Cannot get desirable properties for both
  sides
 Several methods to set prices
 4-Heap algorithm handles many efficiently
 Optimal allocations with indivisible bids is
  an NP-complete problem

				
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