Dynamic Pricing

1
Dynamic Pricing

• Peter R. Wurman
• North Carolina State University

2
E-commerce Big Picture
TCP/IP Databases HTTP HTML Encryption Ano
nymity
Infrastructure
3
E-commerce Big Picture
Web mining Data mining XML Recommendations
Make Contact
TCP/IP Databases HTTP HTML Encryption Ano
nymity
Infrastructure
4
E-commerce Big Picture
Web mining Data mining XML Recommendations
Auctions Agents
Contracts Payments
Make Contact
Negotiate
Exchange
TCP/IP Databases HTTP HTML Encryption Ano
nymity
Infrastructure
5
Why Auctions?

• Auctions enable dynamic pricing
• Let the demand (and supply) determine the market
  value

6
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

7
Discussion

• How did Sony pick 300?

8
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
9
What is the Market Value?

• Lets check ebay...

10
How Much Could Sony Have Made?

• Assume
• Purchasers are linearly distributed between x
  and 300.

x
370
Millions of units
.98
11
How Much Could Sony Have Made?

• Lost revenue 250000 (x - 300)

x
300
Millions of units
.5
12
Questions

• What was Sonys 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?

13
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

14
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

15
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

16
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

17
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

18
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.

19
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

20
Other types of Auctions

• Continuous Double Auction (CDA)
• Multiple buyers and sellers
• Clears continuously
• Call Market
• Multiple buyers and sellers
• Clears periodically

21
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 ?

22
Taxonomy
23
Core Auction Activities

• Auctions
• Receive bids
• Supply intermediate information (optional)
• Clear

24
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

25
Bidding Rules

• Individual rules govern
• Who can bid?
• Semantics of bid
• Beat-the-quote rules
• Beat-your-bid rules

26
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 bidders ids associated with their bids?
• How often is the information generated?
• Does everyone see the same information?

27
Clearing Rules

• Individual rules govern
• Who gets to trade?
• At what price?
• What events trigger a clear?
• When does the auction close?

28
Example Parametrizations
29
Purpose of an Auction

• Auctions
• Mediate communication

Auction
Agent
Agent
Agent
Agent
30
Purpose of an Auction

• Auctions
• Mediate communication
• Facilitate the multilateral exchange of resources

Auction
Agent

Agent
C
A
Agent
Agent
B
31
Analysis

• How do we predict the outcome?
• Characterize the decision problem each bidder
  faces
• Determine a rational strategy

32
Part 2 Decision Theory
33
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

34
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 humans
  preference structure

35
Example Who Wants to Be a Millionaire?
36
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?

37
Millionaire Scenario
?
38
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?

39
Properties of Preferences

• Orderability
• For any two states, either A B, B A, or AB
• 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

40
More Properties

• Substitutability
• If AB, then p, A (1-p) C p, B (1-p)
  Cfor 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

41
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

42
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

43
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?

44
Expected (Monetary) Value

• EMV(St. P) Sumi p(H on turn i) (2i)
  Sumi (1/ 2i) 2i 8
• If U ? EMV, then you should be willing to pay an
  infinite amount of money to play the game.

45
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

46
Human Utility for Money
47
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)

48
Quasilinear Utility

• Often assume that agents are risk neutral within
  the scope of the problem
• U(x,m) v(x) mwhere m is money

This allows us to use money as the scale of
utility
49
Quasilinear Utility

• For any allocation x,there is some amount of
  money, m,s.t. i is indifferent between x and m
  regardless of is endowment of money.
• We call vi(x) agent is willingness-to-pay for
  allocation x

50
Formal Model

• I the set of agents, i 1n
• G the set of resources, g 1m
• xig an amount of good g allocated to I
• xi
• eig agent is endowment of g
• ei

51
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

52
Agent Surplus

• An agents surplus is the change in its utility
  between two states
• Agent is surplus from the allocation xi
• si U(xi) - U(ei)

53
Solution

• Given a a particular allocation problem, what is
  a good distribution of the resources?

54
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(fi) Ui(fi) and for all other
  agents h Uh(fh) Uh(fh)

55
Pareto Efficient Solutions
U2
f 1
f 2
f 3
f 4
U1
56
Pareto Efficient Solutions
U2
f 1
f 2 Pareto dominates f 3
f 2
f 3
f 4
U1
57
Pareto Efficient Solutions
U2
f 1
The Pareto frontier
f 2
f 3
f 4
U1
58
Pareto Efficiency and Quasilinearity

• When agents have quasilinear preferences the
  Pareto solution satisfies max (Si
  v(fi)) subject to Si xig Si eig

59
Pareto Efficient Solutions
v2
f 1
ParetoSolutions
f 2
f 3
f 4
v1
60
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

61
Example Continued

• Definitions
• U1(e) v1(A) m1 3 m1
• U2(e) m2
• U1(f) m1
• U2(f) v2(A) m2 5 m2

62
Example Continued

• Conditions
• U1(f) U1(e) m1 3 m1m1 - m1 3 Dm1
• U2(f) U2(e) 5 m2 m2 5 - (m2
  -m2) -Dm2
• Constraint 3 Dm1 - Dm2 5

63
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

64
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 planners objective

65
How Do We Find f?

• In a distributed system
• We cant impose an allocation
• The valuations are private information
• We want a mechanism that encourages agents acting
  in their own self-interest to find f

66
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

67
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

68
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

69
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

70
Diagram
Price
4
3
2
1
Sellers
Buyers
Quantity
71
Interpretation of Bid Points
Price
4
Buyers will pay theirbid or less
3
Sellers will accept their bid or more
2
1
Quantity
72
One Set of Exchanges
Price
4
a3 sells 1 unit to a4 for 3 p1 4
3
2
a1 sells 1 unit to a2 for 1 p2 2
1
Quantity
73
Analysis

• Market is cleared
• Not a uniform price p1 ? p2
• Not efficient
• a3s bid is interpreted as its reserve value
• After the exchanges a3 would want to buy a2s
  unit
• There is a trade remaining
• The agents with the highest values (a3, a4) do
  not have the items in the end

74
Another Exchange Set
Price
4
a1 sells 1 unit to a4 for 1 p 4
3
2
1
Quantity
75
Analysis

• Market is cleared
• Uniform price
• Efficient
• a4 gets one, a3 keeps one
• Budget balanced
• But,
• If p
• If p 3, then a3 will want to sell one

76
Aggregate Demand
Price
4
3
2
1
Quantity
77
Equilibrium Prices
Price
4
Range of prices that balance supply and demand
3
2
1
Quantity
78
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. rangeThe
  (M1)st bid is the bottom of eq. range
• Any price in between is an eq. price

79
Mth and (M1)st Example
Price
L 4 M 2
4
Mth bid
3
(M1)st bid
2
1
Quantity
80
Another Example
Price
L 4 M 2
4
Mth bid
3
(M1)st bid
2
1
Quantity
81
Yet Another Example
Price
4
Mth bid
3
(M1)st bid
2
1
Quantity
82
Which Trades?
Price
4
Mth bid
3
(M1)st bid
2
1
Quantity
83
Which Trades?
Price
4
Mth bid
3
(M1)st bid
2
1
Quantity
84
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)

85
What Exchange Price?

• Let pM the value of the Mth bid
• Let pM1 the value of the (M1)st bid
• For any k, 0 k 1, p pM1 k(pM - pM1)is
  an equilibrium price

86
Properties

• Clears the market
• Budget balanced
• Uniform price
• Locally efficient
• Equilibrium prices

87
Price Quotes

• If the auction is not sealed bid, what price
  information should we reveal?

88
Inspiration CDA
Price
4
ask
3
bid
2
1
Quantity
89
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

90
Generalizing the Bid-Ask Quote

• Mapping
• Ask Mth price
• Bid (M1)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

91
Buyer Example
Price
4
4
5
ask
3
3
bid
2
2
1
1
Quantity
92
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 cant tell if you
  are winning
• Symmetric result for the seller

93
Multi-unit Bids

• When bids are divisible, the Mth, (M1)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

94
Multi-unit Bid Example
Price
3
3
2
2
1
1
1
Quantity
95
Some Take Home Messages

• Uniform prices are good
• Buyers and sellers are symmetrical

96
Part 3 Analysis
97
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

98
Proof

• Consider 2 agents
• Random valuations, v1, v2
• Place bids b1, b2
• Agent 1s utility for bidding b1
• U(b1) Pr(b1 b2)v1 - b2

99
Proof Case 1

• If v1 - b2 0, then agent 1 wants to maximize
  Pr(b1 b2)
• Does so by setting b1 v1

v1
b2
100
Proof Case 2

• If v1 - b2 Pr(b1 b2)
• Does so by setting b1 v1
• Thus, setting b1 v1 (truth-telling) is a
  dominant strategy

b2
v1
101
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
  doesnt win when it shouldnt

102
Extensions to the Mth (M1)st Price Auctions

• For single-unit buyers, it is a dominant strategy
  to bid truthfully in an (M1)st-price sealed-bid
  auction
• For single-unit sellers, it is a dominant
  strategy to bid truthfully in an Mth-price
  sealed-bid auction

103
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 (M1)st-price
  auction

104
Multi-unit Bid Example
Price
3
3
Mth, (M1)st
2
2
1
1
1
Quantity
105
Multi-unit Bid Example
Price
3
3
Mth
2
2
(M1)st
1
2
1
1
Quantity
106
Why Not Buyers and Sellers at the Same Time?

• One buyer, one seller
• Sealed bids b s
• Valuations, vs, vb, drawn from overlapping
  distributions

vb
vs
107
Desirable Properties

• Efficient
• If vs
• Truthful
• Dominant strategies to bid vs, vb
• Individually rational
• No agent will be made worse off
• Budget balanced
• Amount seller receives amount buyer pays

108
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
• Similarly, for the sellerto bid truthfully, ps
  b

vb
vs
109
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

110
McAfees Dual Price Auction

• We can get truthful behavior for both the buyers
  and sellers, and budget balance by sacrificing
  efficiency
• Let p pM1 k(pM - pM1)
• All exchanges occur except the lowest buyer at or
  above pM, and the highest seller at or below pM1

111
Dual Price Diagram
Price
4
Mth bid
3
(M1)st bid
2
Discard this trade
1
Quantity
112
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

113
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 buyers (or sellers)
  offer
• Common in multiunit auctions on the Internet

114
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

115
Chronological Match
Price
Earlier Bid Prices 1 ? 4 at 1 3 ? 2 at 3
Later Bid Prices 1 ? 4 at 4 3 ? 2 at 2
4
2
3
1
Quantity
116
Agents Care How Matches are Formed
Price
Earlier Bid Prices 3 ? 4 at 2 1 ? 2 at 1
Later Bid Prices 3 ? 4 at 4 1 ? 2 at 3
4
2
3
1
Quantity
117
Pay Buyers/Sellers Bid
Price
Sellers Bid Prices 1 ? 4 at 1 3 ? 2 at 2
Buyers Bid Prices 1 ? 4 at 4 3 ? 2 at 3
4
2
3
1
Quantity
118
Comparison

• 1 ? 4 3 ? 4 3 ? 2 1 ? 2
• Mth 3/3 3/3
• (M1)st 2/2 2/2
• Dual (.5) 2.5/-- N/A
• Earliest 1/3 2/1
• Latest 4/2 4/3
• Buyers 4/3 4/3
• Sellers 1/2 2/1

Uniform Prices
Discriminatory Prices
119
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

120
4-Heap Diagram
Bin
Sout
Bout
Sin
121
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)

122
Insert New Bid (1)
Bin
Sout
Bout
Sin
123
Insert New Bid (2)
Bin
Sout
Put
Bout
Sin
124
Insert New Bid (3)
Bin
Sout
Bout
Sin
Violates thecondition that units in Bin
units in Sin
125
Insert New Bid (4)
Bin
Sout
Get
Bout
Sin
126
Insert New Bid (5)
Bin
Sout
Put
Bout
Sin
127
Insert New Bid (6)
Bin
Sout
Bout
Sin
128
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

129
Indivisible bids

• Agent Bida1 buy exactly 2 units at
  3/eacha2 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?

130
Knapsack Problem

• How do we best fill a three unit knapsack

1
6
4
6
1
131
Another Example

• How do we best fill this four unit knapsack?
• Knapsack is a classic NP-complete problem

8
15
8
8
8
132
Indivisible Bids and Prices

• There is no price, p, such that a1 and a3 want to
  buy, and a2 doesnt
• For equilibrium, we need prices that are
  nonlinear functions of quantity

1
6
4
6
1
133
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