1. Game Theoretic Problems in Network Economics and Mechanism Design Solutions
Y. Narahari
Co-Researchers: Dinesh Garg, Rama Suri, Hastagiri, Sujit Gujar
September 2007
E-Commerce Lab
Computer Science and Automation,
Indian Institute of Science, Bangalore
E-Commerce Lab, CSA, IISc

2. E-Commerce Lab, CSA, IISc
OUTLINE
Examples of Game Theoretic Problems in Network Economics
Mechanism Design
Case Study: Sponsored Search Auctions
Future Work
E-Commerce Lab, CSA, IISc

3. Talk Based on
Y. Narahari, Dinesh Garg, Rama Suri, Hastagiri
Game Theoretic Problems in Network Economics and Mechanism Design Solutions
Research Monograph in the AI & KP Series
To Be Published by
Springer, London, 2008
E-Commerce Lab, CSA, IISc

4. Supply Chain Network Formation
Supply Chain Network Planner
Stage
Manager
E-Commerce Lab, CSA, IISc

5. E-Commerce Lab, CSA, IISc
Indirect Materials Procurement
Suppliers with
Volume Contracts
Purchase Reqs
Vendor
identified
IISc
PReqs
CSA
PROC.
MARKET
Catalogued
Suppliers without
Volume Contracts
RFQ
Reqs
PURCHASE
SYSTEM EE
Quotes
PHY
Auction
Non Catalogued
Suppliers
Optimized
Order(s)
recommendations
ADM
PO’s to Suppliers
E-Commerce Lab, CSA, IISc

6. Ticket Allocation in Software Maintenance
Customer .
Team of
Maintenance Engineers
Web
Interface
Product #1
Queue
Product
Lead #1 .
Based on Type of Application Or product, problems are distributed to various Queues . .
Product #100
Queue
Product
Lead #100
Level 1
Product Maintenance Processes
E-Commerce Lab, CSA, IISc

7. Ticket Allocation Game
effort, time
effort, time
effort, time
Project lead (Ticket Allocator) (rational and intelligent)
Maintenance Engineers (rational and intelligent)
E-Commerce Lab, CSA, IISc

8. Resource Allocation in Grid Computing
E-Commerce Lab, CSA, IISc

9. E-Commerce Lab, CSA, IISc
Incentive Compatible Broadcast in Ad hoc Wireless Networks ?
E-Commerce Lab, CSA, IISc

10. Internet Routing Tier 3 Tier 2 Tier 1
Tier 1: UU Net, Sprint, AT&T, Genuity
Tier 2: Regional/National ISPs
Tier 3: Residential/Company ISP
E-Commerce Lab, CSA, IISc

11. Web Service Composition E-Commerce Lab, CSA, IISc
Service Providers1, 2
Service Providers 2,3
Service Providers 3,4
There could be alternate service providers for
each web service
How do we select the best mix of web service providers
so as to execute the end-to-end business process at
minimum cost taking into account QOS requirements?
E-Commerce Lab, CSA, IISc

12. Web Services Composition Game E-Commerce Lab, CSA, IISc
A, B, AB 1
A, B, C 2
A, C, AC
Web Service Requestor (client) (rational and intelligent) 3
A, B, C, ABC 4
Web Service Providers (rational and intelligent)
E-Commerce Lab, CSA, IISc

13. Web Services Market Game E-Commerce Lab, CSA, IISc
QoS
SLA
Cost
Penalties
Web
Services
Market
Web Service Providers
Web Service Requestors
(rational and intelligent)
(rational and intelligent)
E-Commerce Lab, CSA, IISc

14. Sponsored Search Auction E-Commerce Lab, CSA, IISc

15. E-Commerce Lab, CSA, IISc
Sequence of Queries
User 1
User 2
User N
Google Q1 Q2 Q1 Q3 Q2 Q1 Q2 Q3
E-Commerce Lab, CSA, IISc

16. Sponsored Search Auction Game E-Commerce Lab, CSA, IISc
CPC
Advertisers
E-Commerce Lab, CSA, IISc

17. Some Important Observations
Players
are rational and intelligent
Conflict and cooperation
are both relevant issues
Some information is
common knowledge
Some information is
is private and distributed
(incomplete information)
Our Objective: Design a social choice function
With desirable properties, given that the
players are rational, intelligent, and strategic
E-Commerce Lab, CSA, IISc

18. E-Commerce Lab, CSA, IISc
Game Theory
Mathematical framework for rigorous study of conflict and cooperation among rational, intelligent agents
Market
Buying Agents (rational and intelligent)
Selling Agents (rational and intelligent)
E-Commerce Lab, CSA, IISc

19. E-Commerce Lab, CSA, IISc
Strategic form Games S1
U1 : S R Sn
Un : S R
N = {1,…,n}
Players
S1, … , Sn
Strategy Sets
S = S1 X … X Sn
Payoff functions
(Utility functions)
Players are rational : they always strive to maximize their individual payoffs
Players are intelligent : they can compute their best responsive strategies
Common knowledge
E-Commerce Lab, CSA, IISc

20. Example 1: Matching Pennies
(1,-1)
(-1,1)
Two players simultaneously put down a coin, heads up or tails up. Two-Player zero-sum game
S1 = S2 = {H,T}
E-Commerce Lab, CSA, IISc

21. Example 2: Prisoners’ Dilemma
E-Commerce Lab, CSA, IISc

22. Example 3: Hawk - Dove 2 1 0,0 20,5 5,20 10,10 H Hawk D Dove
Models the strategic conflict when two players are fighting over a
company/territory/property, etc.
E-Commerce Lab, CSA, IISc

23. Example 4: Indo-Pak Budget Game
India
Healthcare
Defence
10,10
-10, 20
20, -10
0,0
Models the strategic conflict when two players
have to choose their priorities
E-Commerce Lab, CSA, IISc

24. Example 5: Coordination
In the event of multiple equilibria, a certain equilibrium becomes a focal equilibrium based on certain environmental factors
College
MG Road
100,100
0,0
5,5
E-Commerce Lab, CSA, IISc

25. E-Commerce Lab, CSA, IISc
Nash Equilibrium
(s1*,s2*, … , sn*) is a Nash equilibrium if si* is a best response for player ‘i’ against the other players’ equilibrium strategies
Prisoner’s Dilemma
(C,C) is a Nash Equilibrium. In fact, it is
a strongly dominant strategy equilibrium
E-Commerce Lab, CSA, IISc

26. E-Commerce Lab, CSA, IISc
Nash’s Theorem
Every finite strategic form game has at least one mixed strategy Nash equilibrium
Mixed strategy of a player ‘i’ is a probability distribution on Si
is a mixed strategy Nash equilibrium if
is a best response against ,
E-Commerce Lab, CSA, IISc

27. E-Commerce Lab, CSA, IISc
John von Neumann
( )
Founder of Game theory with Oskar Morgenstern
E-Commerce Lab, CSA, IISc

28. E-Commerce Lab, CSA, IISc
John F Nash Jr. ( )
Landmark contributions to Game theory: notions of Nash Equilibrium and Nash Bargaining
Nobel Prize : 1994
E-Commerce Lab, CSA, IISc

29. E-Commerce Lab, CSA, IISc
John Harsanyi
( )
Defined and formalized Bayesian Games
Nobel Prize : 1994
E-Commerce Lab, CSA, IISc

30. E-Commerce Lab, CSA, IISc
Reinhard Selten
( )
Founding father of experimental economics and bounded rationality
Nobel Prize : 1994
E-Commerce Lab, CSA, IISc

31. E-Commerce Lab, CSA, IISc
Thomas Schelling
( )
Pioneered the study of bargaining and strategic behavior
Nobel Prize : 2005
E-Commerce Lab, CSA, IISc

32. E-Commerce Lab, CSA, IISc
Robert J. Aumann
( )
Pioneer of the notions of common knowledge, correlated equilibrium, and repeated games
Nobel Prize : 2005
E-Commerce Lab, CSA, IISc

33. E-Commerce Lab, CSA, IISc
Lloyd S. Shapley
( )
Originator of “Shapley Value” and Stochastic Games
E-Commerce Lab, CSA, IISc

34. E-Commerce Lab, CSA, IISc
William Vickrey
(1914 – 1996 )
Inventor of the celebrated Vickrey auction
Nobel Prize : 1996
E-Commerce Lab, CSA, IISc

35. E-Commerce Lab, CSA, IISc
Roger Myerson
( )
Fundamental contributions to game theory, auctions, mechanism design
E-Commerce Lab, CSA, IISc

36. E-Commerce Lab, CSA, IISc
MECHANISM DESIGN
E-Commerce Lab, CSA, IISc

37. E-Commerce Lab, CSA, IISc
Mechanism Design Problem
O<M<L
L<O<M
M<L<O
Yuvraj
Dravid
Laxman
O: Opener
M: Middle-order
L: Late-order
Greg
How to transform individual preferences into social decision?
How to elicit truthful individual preferences ?
E-Commerce Lab, CSA, IISc

38. The Mechanism Design Problem E-Commerce Lab, CSA, IISc
agents who need to make a collective choice from outcome set
Each agent privately observes a signal which determines preferences over the set
Signal is known as agent type.
The set of agent possible types is denoted by
The agents types, are drawn according to a probability distribution function
Each agent is rational, intelligent, and tries to maximize its utility function
are common knowledge among the agents
E-Commerce Lab, CSA, IISc

39. Two Fundamental Problems in Designing a Mechanism
Preference Aggregation Problem
For a given type profile of the agents, what outcome should be chosen ?
Information Revelation (Elicitation) Problem
How do we elicit the true type of each agent , which is his private information ?
E-Commerce Lab, CSA, IISc

40. Information Elicitation Problem E-Commerce Lab, CSA, IISc

41. Preference Aggregation Problem (SCF) E-Commerce Lab, CSA, IISc

42. E-Commerce Lab, CSA, IISc
Indirect Mechanism
E-Commerce Lab, CSA, IISc

43. Social Choice Function and MechanismSn θ1 θn
Outcome Set
Outcome Set
f(θ1, …,θn) Є X
g(s1(.), …,sn() Є X
(S1, …, Sn, g(.))
x = (y1(θ), …, yn(θ), t1(θ), …, tn(θ))
A mechanism induces a Bayesian game and is designed to implement a social choice function in an equilibrium of the game.
E-Commerce Lab, CSA, IISc

44. Equilibrium of Induced Bayesian Game E-Commerce Lab, CSA, IISc
Dominant Strategy Equilibrium (DSE)
A pure strategy profile is said to be dominant strategy equilibrium if
Bayesian Nash Equilibrium (BNE)
A pure strategy profile is said to be Bayesian Nash equilibrium
Observation
Dominant Strategy-equilibrium Bayesian Nash- equilibrium
E-Commerce Lab, CSA, IISc

45. E-Commerce Lab, CSA, IISc
Implementing an SCF
Dominant Strategy Implementation
We say that mechanism implements SCF in dominant strategy equilibrium if
Bayesian Nash Implementation
We say that mechanism implements SCF in Bayesian Nash equilibrium if
Observation
Dominant Strategy-implementation Bayesian Nash- implementation
Andreu Mas Colell, Michael D. Whinston, and Jerry R. Green, “Microeconomic Theory”, Oxford University Press, New York, 1995.
E-Commerce Lab, CSA, IISc

46. E-Commerce Lab, CSA, IISc
Properties of an SCF
Ex Post Efficiency
For no profile of agents’ type does there exist an
such that and for some
Dominant Strategy Incentive Compatibility (DSIC)
If the direct revelation mechanism has a dominant strategy equilibrium in which
Bayesian Incentive Compatibility (BIC)
If the direct revelation mechanism has a Bayesian Nash equilibrium in which
E-Commerce Lab, CSA, IISc

47. E-Commerce Lab, CSA, IISc
Outcome Set
Project Choice Allocation
I0, I1,…, In : Monetary Transfers
x = (k, I0, I1,…, In )
K = Set of all k
X = Set of all x
E-Commerce Lab, CSA, IISc

48. Social Choice Function
where,
E-Commerce Lab, CSA, IISc

49. E-Commerce Lab, CSA, IISc
Values and Payoffs
Quasi-linear Utilities
E-Commerce Lab, CSA, IISc

50. Quasi-Linear Environment E-Commerce Lab, CSA, IISc
Valuation function of agent 1
Policy Maker
project choice
Monetary transfer to agent 1
E-Commerce Lab, CSA, IISc

51. Properties of an SCF in Quasi-Linear Environment
Ex Post Efficiency
Dominant Strategy Incentive Compatibility (DSIC)
Bayesian Incentive Compatibility (BIC)
Allocative Efficiency (AE)
SCF is AE if for each , satisfies
Budget Balance (BB)
SCF is BB if for each , we have
Lemma 1
An SCF is ex post efficient in quasi-linear environment iff it is AE + BB
E-Commerce Lab, CSA, IISc

52. A Dominant Strategy Incentive Compatible Mechanism
Let f(.) = (k(.),I0(.), I1(.),…, In(.)) be allocatively efficient.
Let the payments be :
Groves Mechanism
E-Commerce Lab, CSA, IISc

53. VCG Mechanisms (Vickrey-Clarke-Groves)
Groves Mechanisms
Clarke Mechanisms
Generalized Vickrey Auction
Vickrey Auction
Allocatively efficient, individual rational, and dominant strategy incentive compatible with quasi-linear utilities.
E-Commerce Lab, CSA, IISc

54. A Bayesian Incentive Compatible Mechanism
Let f(.) = (k(.),I0(.), I1(.),…, In(.)) be allocatively efficient.
Let types of the agents be statistically independent of one another
dAGVA Mechanism
E-Commerce Lab, CSA, IISc

55. Basic Types of Procurement Auctions
Reverse Dutch Auction 1 n
100, 95, 90, 85,
80, 75, 70, 65,
60, stop.
Auctioneer or Buyer
Reverse English Auction
Sellers 1 n
0, 10, 20, 30,
40, 45, 50, 55,
58, 60, stop.
Buyer
Sellers
Reverse Second Price Auction (Reverse Vickrey Auction)
Reverse First Price Auction 1 80 75 1 2 75
Winner = 4 Price = 60 2 65
Winner = 4 Price = 60 70 3 3 60 4 60 4 50
Sellers
Sellers
E-Commerce Lab, CSA, IISc

56. E-Commerce Lab, CSA, IISc
WBB
SBB AE
EPE
dAGVA
BIC IR
GROVES
DSIC
MOULIN
E-Commerce Lab, CSA, IISc

57. Sponsored Search Auctions E-Commerce Lab, CSA, IISc

58. E-Commerce Lab, CSA, IISc
OUTLINE
Sponsored Search Auctions
SSA as a Mechanism Design Problem
Three Different Auction Mechanisms: GFP, GSP, VCG
A New Mechanism: OPT
Comparison of Different Mechanisms
Ongoing Work
E-Commerce Lab, CSA, IISc

59. Sponsored Search Auction Game E-Commerce Lab, CSA, IISc
CPC
Advertisers
E-Commerce Lab, CSA, IISc

60. Some Important Observations
Players
are rational and intelligent
Conflict and cooperation
are both relevant issues
Some information is
common knowledge
Some information is
is private and distributed
(incomplete information)
Our Objective: Design a social choice function
With desirable properties, given that the
players are rational, intelligent, and strategic
E-Commerce Lab, CSA, IISc

61. Sponsored Search Auction as a Mechanism Design Problem
(Allocation Rule, Payment Rule)
E-Commerce Lab, CSA, IISc

62. E-Commerce Lab, CSA, IIScX
E-Commerce Lab, CSA, IISc

63. E-Commerce Lab, CSA, IIScX
E-Commerce Lab, CSA, IISc

64. Bayesian Game Induced by the Auction Mechanism
Induces a Bayesian game
among advertisers
where
Set of advertisers
Valuation set of advertiser
Set of bids for advertiser
A pure strategy of advertiser
Prior distribution of advertiser valuations
Utility payoff of advertiser
E-Commerce Lab, CSA, IISc

65. Strategic Bidding Behavior of Advertisers E-Commerce Lab, CSA, IISc
If all the advertisers are rational and intelligent and this fact is common knowledge then each advertiser’s expected bidding behavior is given by
Dominant Strategy Equilibrium
Strategy profile is said to be Dominant Strategy equilibrium iff
Bayesian Nash Equilibrium
Strategy profile is said to be Bayesian Nash equilibrium iff
E-Commerce Lab, CSA, IISc

66. Advertisers’ Bidding Strategy E-Commerce Lab, CSA, IISc
VCG: Follow irrespective of what the others are doing
OPT: Follow if all rivals are also doing so
GSP: Never follow strategy Use the following
E-Commerce Lab, CSA, IISc

67. E-Commerce Lab, CSA, IISc
Properties of a Sponsored Search Auction Mechanism
E-Commerce Lab, CSA, IISc

68. E-Commerce Lab, CSA, IISc
Google’s Objectives
Short Term
Long Term Q1 Q2 Q3 Q1 Q2 Q3
Revenue Maximization
Click Fraud Resistance
Individual Rationality
Incentive Compatibility
E-Commerce Lab, CSA, IISc

69. Click Fraud Resistance E-Commerce Lab, CSA, IISc
Google’s Objectives
Revenue Maximization
Choose auction mechanism
such that despite
strategic bidding behavior of advertisers, expected revenue is maximum
Click Fraudulence
increase the spending of rival advertisers without increasing its own.
is click fraudulent if an advertiser finds a way to
Click Fraud Resistance
is click fraud resistant If it is not click fraudulent
E-Commerce Lab, CSA, IISc

70. Individual Rationality E-Commerce Lab, CSA, IISc
Google’s Objectives
Individual Rationality
Advertiser’s participation is voluntary
Will bid only if the participation constraint is satisfied
Why should bother about it ?
Advertisers may decide to quit !!
What can do about it ?
Choose an auction mechanism
which is IR
E-Commerce Lab, CSA, IISc

71. Incentive Compatibility E-Commerce Lab, CSA, IISc
Google’s Objectives
Incentive Compatibility
Difficulties faced by an Advertiser
In practice, the assumptions like rationality, intelligence, and common knowledge are hardly true
Need to invoke sophisticated but impractical software agents to compute the optimal
Why should bother about it ?
Low ROI switch to other search engines !!
E-Commerce Lab, CSA, IISc

72. Incentive Compatibility E-Commerce Lab, CSA, IISc
What can do about it ?
Choose an auction mechanism
which is IC
Dominant Strategy Incentive Compatibility
Incentive compatible if truth telling is a dominant strategy equilibrium
Auction mechanism
is said to be dominant strategy
Bayesian Incentive Compatibility
Auction mechanism
is said to be Bayesian incentive
compatible if truth telling is a Bayesian Nash equilibrium
E-Commerce Lab, CSA, IISc

73. Properties of Auction Mechanisms E-Commerce Lab, CSA, IISc
Bayesian IC
Dominant
Individual
Strategy
Rationality IC
VCG
GFP
OPT
GSP
E-Commerce Lab, CSA, IISc

74. Four Different Auction Mechanisms E-Commerce Lab, CSA, IISc
GFP
GSP
VCG
OPT
(Overture, 1997)
(Google, 2002)
Notation
Feasibility Condition:
E-Commerce Lab, CSA, IISc

75. Generalized First Price (GFP) E-Commerce Lab, CSA, IISc1 2 m
Allocation Rule
Allocated the slots in decreasing order of bids
Payment Rule
Every time a user clicks on the Ad, the advertiser’s account is automatically billed the amount of the advertiser’s bid
E-Commerce Lab, CSA, IISc

76. E-Commerce Lab, CSA, IISc
Example: GFP Q
Search Results
Sponsored Links 1 2
E-Commerce Lab, CSA, IISc

77. Generalized Second Price (GSP) E-Commerce Lab, CSA, IISc
Allocation Rules
Rule: 1 2 m
Allocate the slots in decreasing order of bids
Greedy Rule:
Allocate 1st slot to advertiser
Allocate 2nd slot to advertiser
Rule:
Allocate the slots in decreasing order of Ranking Score
Ranking Score =
E-Commerce Lab, CSA, IISc

78. Generalized Second Price (GSP) E-Commerce Lab, CSA, IISc
Greedy
Observation 1:
Greedy
Click probability is independent of the
identity of advertisers
Greedy
E-Commerce Lab, CSA, IISc

79. Generalized Second Price (GSP) E-Commerce Lab, CSA, IISc
Payment Rule
For every click, charge next highest bid + $0.01
The bottom most advertiser is charged highest disqualified bid +$0.01
charge 0 if no such bid
E-Commerce Lab, CSA, IISc

80. E-Commerce Lab, CSA, IISc
Example: GSP Q
Search Results
Sponsored Links 1 2
E-Commerce Lab, CSA, IISc

81. Vickrey-Clarke-Groves (VCG) E-Commerce Lab, CSA, IISc
Allocation Rule 1 2 m
In decreasing order of bids
Payment Rule
Case 1
Case 2
E-Commerce Lab, CSA, IISc

82. E-Commerce Lab, CSA, IISc
Example: VCG Q
Search Results
Sponsored Links 1 2
E-Commerce Lab, CSA, IISc

83. E-Commerce Lab, CSA, IISc
Optimal (OPT) 1 2 m
Allocation Rule
Where is the highest value among
(Assumption: is non decreasing: True for Uniform, Exponential)
Observation 4:
Advertisers are symmetric i.e.
Allocation Rule
OPT
E-Commerce Lab, CSA, IISc

84. E-Commerce Lab, CSA, IISc
Optimal (OPT)
Payment Rule
Assumptions:
Advertisers are symmetric, i.e.
Whenever an advertiser bids charge him for every query irrespective of whether his Ad is displayed or not
Where is the probability that advertiser will receive a click if he bids and rest of the advertisers bid their true values
E-Commerce Lab, CSA, IISc

85. E-Commerce Lab, CSA, IISc
Example: OPT Q
Search Results
Sponsored Links 1 2
E-Commerce Lab, CSA, IISc

86. E-Commerce Lab, CSA, IISc
Example: OPT
E-Commerce Lab, CSA, IISc

87. E-Commerce Lab, CSA, IISc
Example: OPT
E-Commerce Lab, CSA, IISc

88. Expected Revenue of Seller E-Commerce Lab, CSA, IISc
Case 1
Observation
E-Commerce Lab, CSA, IISc

89. Expected Revenue of Seller E-Commerce Lab, CSA, IISc
Case 1
E-Commerce Lab, CSA, IISc

90. E-Commerce Lab, CSA, IISc
Conclusions
Allocation
Payment
DSIC
BIC IR
CFR
GSP
Decreasing order of the bids
Next Highest bid
(PPC) X
VCG
Marginal Contribution
OPT
(PPP)
E-Commerce Lab, CSA, IISc

91. E-Commerce Lab, CSA, IISc
Ongoing Work
Deeper Mechanism Design
Repeated Games Model
Learning Bidding Strategies
Cooperative Bidding
E-Commerce Lab, CSA, IISc

92. E-Commerce Lab, CSA, IISc
Questions and Answers …
Thank You …
E-Commerce Lab, CSA, IISc

93. Vickrey Auction for Ticket Allocation
effort, time
effort, time
effort, time
Project lead Ticket Allocator
Maintenance Engineers
E-Commerce Lab, CSA, IISc

94. E-Commerce Lab, CSA, IISc
Incentive Compatible Broadcast Problem: Successful broadcast requires appropriate forwarding of the packets by individual selfish wireless nodes. Reimbursing the forwarding costs incurred by the nodes is a way to make them forward the packets. For this, we need to know the exact transit costs of the nodes. We can design an incentive compatible broadcast protocol by embedding appropriate incentive schemes into the broadcast protocol. We shall refer to the problem of designing such robust broadcast protocols as the incentive compatible broadcast (ICB) problem. ?
Line Network
Bi-connected ad hoc network
Source Rooted Broadcast Tree
E-Commerce Lab, CSA, IISc

95. Vickrey Auction for Ticket Allocation E-Commerce Lab, CSA, IISc
Maintenance Engineers
Bid 1 – Rs. 1000
Bid 2 – Rs. 1500
Bid 3 – Rs. 1200
Allocation
Engineer 1 is selected
as winner (lowest bid)
Payment
Engineer 1 is paid
(1200 – 1000)
= 1200
Vickrey Auction is Dominant Strategy
Incentive Compatible --
Truth revelation is a best response for each agent
Irrespective of what is reported by the other agents
E-Commerce Lab, CSA, IISc

96. Vickrey Auction as a Strategic Form Game
E-Commerce Lab, CSA, IISc

97. GVA for Web Services Composition E-Commerce Lab, CSA, IISc
A, B, AB 1
A, B, C 2
A, C, AC
Web Service Requestor (client) 3
A, B, C, ABC 4
Web Service Providers
E-Commerce Lab, CSA, IISc

98. GVA for Web Services CompositionAB AC
ABC 1 30 20 - 40 2 25 3 35 50 4 70
Optimal Allocation: 1AB; 4 C
Optimal Cost: = 60
Optimal Cost without 1 = 70
Optimal Cost without 4 = 65
Payment to provider 1 = – 60 = 50
Payment to provider 4 = = 25
E-Commerce Lab, CSA, IISc