1. Algorithmic Game Theory and Internet Computing
Market Equilibrium and
Pricing of Goods
Algorithmic Game Theory and Internet Computing
Vijay V. Vazirani
Georgia Tech

2. Adam Smith The Wealth of Nations, 1776.
“It is not from the benevolence of the butcher, the brewer, or the baker, that we expect our dinner, but from their regard for their own interest.” Each participant in a competitive economy is “led by an invisible hand to promote an end which was no part of his intention.”

3. What is Economics? ‘‘Economics is the study of the use of
scarce resources which have alternative uses.’’
Lionel Robbins
(1898 – 1984)

4. How are scarce resources assigned to alternative uses?

6. How are scarce resources assigned to alternative uses?
Prices!

8. How are scarce resources assigned to alternative uses?
Prices
Parity between demand and supply

9. How are scarce resources assigned to alternative uses?
Prices
Parity between demand and supply equilibrium prices

10. Leon Walras, 1874
Pioneered general
equilibrium theory

11. Mathematical ratification!
General Equilibrium Theory Occupied center stage in Mathematical Economics for over a century
Mathematical ratification!

12. Central tenet
Markets should operate at equilibrium

13. Central tenet Markets should operate at equilibrium i.e., prices s.t.
Parity between supply and demand

14. Do markets even admit equilibrium prices?

15. Do markets even admit equilibrium prices?
Easy if only one good!

16. Supply-demand curves

17. Do markets even admit equilibrium prices?
What if there are multiple goods and multiple buyers with diverse desires and different buying power?

18. Irving Fisher, 1891 Defined a fundamental market model
Special case of Walras’
model

21. Concave utility function
(Of buyer i for good j)
amount of j
utility

22. total utility

23. For given prices, find optimal bundle of goods

24. Several buyers with different utility functions and moneys.

25. Several buyers with different utility functions and moneys
Several buyers with different utility functions and moneys. Equilibrium prices

26. Several buyers with different utility functions and moneys
Several buyers with different utility functions and moneys. Show equilibrium prices exist.

27. Arrow-Debreu Theorem, 1954
Celebrated theorem in Mathematical Economics
Established existence of market equilibrium under
very general conditions using a deep theorem from
topology - Kakutani fixed point theorem.

28. First Welfare Theorem Competitive equilibrium =>
Pareto optimal allocation of resources
Pareto optimal = impossible to make
an agent better off without making some
other agent worse off

29. Second Welfare Theorem
Every Pareto optimal allocation of resources
comes from a competitive equilibrium
(after redistribution of initial endowments).

30. Kenneth Arrow
Nobel Prize, 1972

31. Gerard Debreu
Nobel Prize, 1983

32. Agents: buyers/sellers
Arrow-Debreu Model
Agents: buyers/sellers

33. Initial endowment of goods
Agents
Goods

34. Prices
= $ = $ = $10
Agents
Goods

35. Incomes
Agents
$50
$60
Goods
Prices
=$ =$15 =$10
$40
$40

36. Maximize utility
Agents
$50
$60
Goods
Prices
=$ =$15 =$10
$40
$40

37. Find prices s.t. market clears
Agents
$50
$60
Goods
Prices
=$ =$15 =$10
$40
Maximize utility
$40

38. Arrow-Debreu Model
n agents, k goods

39. Arrow-Debreu Model n agents, k goods
Each agent has: initial endowment of goods,
& a utility function

40. Arrow-Debreu Model n agents, k goods
Each agent has: initial endowment of goods,
& a utility function
Find market clearing prices, i.e., prices s.t. if
Each agent sells all her goods
Buys optimal bundle using this money
No surplus or deficiency of any good

41. Utility function of agent i
Continuous, quasi-concave and
satisfying non-satiation.
Given prices and money m,
there is a unique utility maximizing bundle.

42. Proof of Arrow-Debreu Theorem
Uses Kakutani’s Fixed Point Theorem.
Deep theorem in topology

43. Proof Uses Kakutani’s Fixed Point Theorem.
Deep theorem in topology
Will illustrate main idea via Brouwer’s Fixed
Point Theorem (buggy proof!!)

44. Brouwer’s Fixed Point Theorem
Let be a non-empty, compact, convex set
Continuous function
Then

45. Brouwer’s Fixed Point Theorem

46. Brouwer’s Fixed Point Theorem

47. Observe: If p is market clearing
prices, then so is any scaling of p
Assume w.l.o.g. that sum of
prices of k goods is 1.
k-1 dimensional
unit simplex

48. Idea of proof Will define continuous function
If p is not market clearing, f(p) tries to
‘correct’ this.
Therefore fixed points of f must be
equilibrium prices.

49. When is p an equilibrium price?
s(j): total supply of good j.
B(i): unique optimal bundle which agent i wants to buy after selling her initial
endowment at prices p.
d(j): total demand of good j.

50. When is p an equilibrium price?
s(j): total supply of good j.
B(i): unique optimal bundle which agent i wants to buy after selling her initial
endowment at prices p.
d(j): total demand of good j.
For each good j: s(j) = d(j).

51. What if p is not an equilibrium price?
s(j) < d(j) => p(j)
s(j) > d(j) => p(j)
Also ensure

52. Let
s(j) < d(j) =>
s(j) > d(j) =>
N is s.t.

53. is a cts. fn.
=> is a cts. fn. of p
=> f is a cts. fn. of p

54. is a cts. fn.
=> is a cts. fn. of p
=> f is a cts. fn. of p
By Brouwer’s Theorem, equilibrium prices exist.

55. is a cts. fn.
=> is a cts. fn. of p
=> f is a cts. fn. of p
By Brouwer’s Theorem, equilibrium prices exist.
q.e.d.!

56. Bug??

57. Boundaries of

58. Boundaries of
B(i) is not defined at boundaries!!

59. Kakutani’s fixed point theorem
S: compact, convex set in
upper hemi-continuous

60. Fisher reduces to Arrow-Debreu
Fisher: n buyers, k goods
AD: n +1 agents
first n have money, utility for goods
last agent has all goods, utility for money only.

61. Pricing of Digital Goods
Music, movies, video games, …
cell phone apps., …, web search results,
… , even ideas!

62. Pricing of Digital Goods
Music, movies, video games, …
cell phone apps., …, web search results,
… , even ideas!
Once produced, supply is infinite!!

63. What is Economics? ‘‘Economics is the study of the use of
scarce resources which have alternative uses.’’
Lionel Robbins
(1898 – 1984)

64. Jain & V., 2010:
Market model for digital goods,
with notion of equilibrium.
Proof of existence of equilibrium.

65. Idiosyncrasies of Digital Realm
Staggering number of goods available with
great ease, e.g., iTunes has 11 million songs!
Once produced, infinite supply.
Want 2 songs => want 2 different songs,
not 2 copies of same song.
Agents’ rating of songs varies widely.

66. Game-Theoretic Assumptions
Full rationality, infinite computing power:
not meaningful!

67. Game-Theoretic Assumptions
Full rationality, infinite computing power:
not meaningful!
e.g., song A for $1.23, song B for $1.56, …

68. Game-Theoretic Assumptions
Full rationality, infinite computing power:
not meaningful!
e.g., song A for $1.23, song B for $1.56, …
Cannot price songs individually!

69. Market Model Uniform pricing of all goods in a category.
Assume g categories of digital goods.
Each agent has a total order over all songs
in a category.

70. Arrow-Debreu-Based Market Model
Assume 1 conventional good: bread.
Each agent has a utility function over
g digital categories and bread.

71. Optimal bundle for i, given prices p
First, compute i’s optimal bundle, i.e.,
#songs from each digital category
and no. of units of bread.
Next, from each digital category,
i picks her most favorite songs.

72. Agents are also producers
Feasible production of each agent
is a convex, compact set in
Agent’s earning:
no. of units of bread produced
no. of copies of each song sold
Agent spends earnings on optimal bundle.

73. Equilibrium (p, x, y) s.t. Each agent, i, gets optimal bundle &
“best” songs in each category.
Each agent, k, maximizes earnings,
given p, x, y(-k)
Market clears, i.e., all bread sold &
at least 1 copy of each song sold.

74. Theorem (Jain & V. , 2010): Equilibrium exists
Theorem (Jain & V., 2010): Equilibrium exists. (Using Kakutani’s fixed-point theorem)

75. Arrow-Debreu Theorem, 1954 Highly non-constructive!
Celebrated theorem in Mathematical Economics
Established existence of market equilibrium under
very general conditions using a theorem from
topology - Kakutani fixed point theorem.
Highly non-constructive!

76. Leon Walras Tatonnement process:
Price adjustment process to arrive at equilibrium
Deficient goods: raise prices
Excess goods: lower prices

77. Leon Walras Tatonnement process:
Price adjustment process to arrive at equilibrium
Deficient goods: raise prices
Excess goods: lower prices
Does it converge to equilibrium?

78. GETTING TO ECONOMIC EQUILIBRIUM: A PROBLEM AND ITS HISTORY
For the third International Workshop on Internet and Network Economics
Kenneth J. Arrow

79. OUTLINE BEFORE THE FORMULATION OF GENERAL EQUILIBRIUM THEORY
PARTIAL EQUILIBRIUM
WALRAS, PARETO, AND HICKS
SOCIALISM AND DECENTRALIZATION
SAMUELSON AND SUCCESSORS
THE END OF THE PROGRAM

80. Part VI: THE END OF THE PROGRAM
Scarf’s example
Saari-Simon Theorem: For any dynamic system depending on first-order information (z) only, there is a set of excess demand functions for which stability fails. (In fact, theorem is stronger).
Uzawa: Existence of general equilibrium is equivalent to fixed-point theorem
Assumptions on individual demand functions do not constrain aggregate demand function (Sonnenschein, Debreu, Mantel)

81. Several buyers with different utility functions and moneys
Several buyers with different utility functions and moneys. Find equilibrium prices!!

82. The new face of computing

83. Today’s reality New markets defined by Internet companies, e.g.,
Microsoft
Google
eBay
Yahoo!
Amazon
Massive computing power available.
Need an inherently-algorithmic theory of
markets and market equilibria.

84. Standard sufficient conditions
on utility functions (in Arrow-Debreu Theorem):
Continuous, quasiconcave,
satisfying non-satiation.

85. Complexity-theoretic question
For “reasonable” utility fns.,
can market equilibrium be computed in P?
If not, what is its complexity?

86. Several buyers with different utility functions and moneys
Several buyers with different utility functions and moneys. Find equilibrium prices.

88. “Stock prices have reached what looks like a permanently high plateau”

89. “Stock prices have reached what looks like a permanently high plateau”
Irving Fisher, October 1929

92. Linear Fisher Market Assume: Buyer i’s total utility,
mi : money of buyer i.
One unit of each good j.

93. Eisenberg-Gale Program, 1959

94. Eisenberg-Gale Program, 1959
prices pj

95. Why remarkable? Equilibrium simultaneously optimizes for all agents.
How is this done via a single objective function?

96. Rational convex program
Always has a rational solution,
using polynomially many bits,
if all parameters are rational.
Eisenberg-Gale program is rational.

97. KKT Conditions Generalization of complementary slackness
conditions to convex programs.
Help prove optimal solution to EG program:
Gives market equilibrium
Is rational

98. Lagrange relaxation technique
Take constraints into objective with a penalty
Yields dual LP.