Game Theory and Math Economics: A TCS Introduction

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Game Theory and Math EconomicsA TCS
Introduction

• Christos H. Papadimitriou
• UC Berkeley
• www.cs.berkeley.edu/christos

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Sources

• Osborne and Rubinstein A Course in Game Theory,
  MIT, 1994
• Mas-Colell, Whinston, and Greene
• Microeconomic Theory, Oxford, 1995
• these proceedings survey
• http//www.cs.berkeley.edu/christos/games/cs294.h
  tml and /focs01.ppt

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• Goal of TCS (1950-2000)
• Develop a mathematical understanding of the
  capabilities and limitations of the von Neumann
  computer and its software the dominant and most
  novel computational artifacts of that time
• (Mathematical tools combinatorics, logic)
• What should Theorys goals be today?

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The Internet

• built, operated and used by a multitude of
  diverse economic interests
• theoretical understanding urgently needed
• tools mathematical economics and game theory

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Game Theory
strategies
strategies
3,-2
payoffs
(NB also, many players)
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matching pennies
prisoners dilemma
e.g.
1,-1 -1,1
-1,1 1,-1
3,3 0,4
4,0 1,1
auction
chicken
1 n
1 . . n
0,0 0,1
1,0 -1,-1
0, v y
u x, 0
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concepts of rationality

• undominated strategy
• (problem too weak)
• (weakly) dominating srategy (alias duh?)
• (problem too strong, rarely exists)
• Nash equilibrium (or double best response)
• (problem may not exist)
• randomized Nash equilibrium
• Theorem Nash 1952 Always exists.

. . .
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• if a digraph with all in-degrees ?1 has a source,
• then it must have a sink
• ? Sperners Lemma
• ? Brouwers fixpoint Theorem
• (? Kakutanis Theorem ? market equilibrium)
• Nashs Theorem
• min-max theorem for zero-sum games
• linear programming duality

?
? P
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Sperners Lemma Any legal coloring of the
triangulated simplex has a trichromatic
triangle Proof
!
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Sperner ? Brouwer

• Brouwers Theorem Any continuous function from
  the simplex to itself has a fixpoint.
• Sketch Triangulate the simplex
• Color vertices according to which direction they
  are mapped
• Sperners Lemma means that there is a triangle
  that has no clear direction
• Sequence of finer and finer triangulations,
  convergent subsequence of the centers of Sperner
  triangles, QED

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Brouwer ? Nash

• For any pair of mixed strategies x,y
  (distributions over the strategies) define
• ?(x,y) (x, y), where x maximizes
• payoff1(x,y) - x x2,
• and similarly for y. Any Brouwer fixpoint is
  now a Nash equilibrium

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Nash ? von Neumann

• If game is zero-sum, then double best response is
  the same as max-min pair

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The critique of mixed Nash

• Is it really rational to randomize?
• (cf bluffing in poker, IRS audits)
• If (x,y) is a Nash equilibrium, then any y with
  the same support is as good as y.
• Convergence/learning results mixed
• There may be too many Nash equilibria

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is it in P?
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The price of anarchy
cost of worst Nash equilibrium
Koutsoupias and P, 1998
socially optimum cost
routing in networks
2 Roughgarden and Tardos, 2000
Also Spirakis and Mavronikolas 01, Roughgarden
01, Koutsoupias and Spirakis 01
The price of the Internet architecture?
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• More problems Nash equilibria may be
  politically incorrect Prisoners dilemma
• Repeated prisoners dilemma?
• Herb Simon (1969) Bounded Rationality
• the implicit assumption that reasoning and
• computation are infinitely cheap
• is often at the root of negative results in
  Economics
• Idea Repeated prisoners dilemma played by
  memory-limited players (e.g., automata)?

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d
d
c
d
c
d
c
d
c
c, d
c
punish once
tit-for-tat
d
d
c
d
c
d
c,d
c
c
c
d
punish forever
switch on d
Theorem These are the only undominated 2-state
strategies
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how about f(n)-g(n) equilibria?

• Theorem PY94
• (d - d)n
• complicated N.e.
• with payoffs 3 - ?
• tit-for-tat

g(n)
2n
n
f(n)
2n
n
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mechanism design(or inverse game theory)

• agents have utilities but these utilities are
  known only to them
• game designer prefers certain outcomes depending
  on players utilities
• designed game (mechanism) has designers goals as
  dominating strategies

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mechanism design (math)

• n players, set K of outcomes, for each player i a
  possible set Ui of utilities of the form u K ?
  R
• designer preferences P U1 ? ? Un ? 2K
• mechanism strategy spaces Si, plus a mapping G
  S1 ? ? Sn ? K

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Theorem (The Revelation Principle) If there is
a mechanism, then there is one in which all
agents truthfully reveal their secret
utilities. Proof common-sense simulation
Theorem (Gibbard-Satterthwaite) If the sets of
possible utilities are too rich, then only
dictatorial Ps have mechanisms. Proof Arrows
Impossibility Theorem
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• but if we allow mechanisms that use Nash
  equilibria instead of dominance, then almost
  anything is implementable
• but these mechanisms are extremely complex and
  artificial
• (TCS critique would be welcome here)

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• but if outcomes in K include payments (K K0 ?
  Rn ) and utilities are quasilinear (utility of
  core outcome plus payment) and designer prefers
  to optimize the sum of core utilities, then the
  Vickrey-Clarke-Groves mechanism works

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e.g., Vickrey auction

• sealed-highest-bid auction encourages gaming and
  speculation
• Vickrey auction Highest bidder wins,
• pays second-highest bid
• Theorem Vickrey auction is a truthful
  mechanism.
• Theorem It maximizes social benefit and
  auctioneer expected revenue.

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e.g., shortest path auction
3
6
5
4
t
s
6
10
3
11
pay e its declared cost c(e), plus a bonus equal
to dist(s,t)c(e) ?- dist(s,t)
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• Theorem Resulting mechanism is truthful and
  maximizes social benefit
• Theorem Suri Hershberger 01 Payments can
  be computed by one shortest path computation.

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e.g., 2-processor scheduling Nisan and Ronen
1998

• two players/processors, n tasks, each with a
  different execution time on each processor
• each execution time is known only to the
  appropriate processor
• designer wants to minimize makespan
• ( maximum completion time)
• each processor wants to minimize its own
  completion time

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• Idea Allocate each task to the most efficient
  processor (i.e., minimize total work). Pay each
  processor for each task allocated to it an amount
  equal to the time required for it at the other
  processor
• Fact Truthful and 2-approximate

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Theorem (Nisan-Ronen) No mechanism can achieve
ratio better than 2

• Sketch By revelation, such a mechanism would be
  truthful.
• wlog, Processor 1 chooses between proposals of
  the form (partition, payment), where the payment
  depends only on the partition and Processor 2s
  declarations

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Theorem (Nisan-Ronen, continued)

• Suppose all task lengths are 1, and Processor 1
  chooses a partition and a payment
• If we change the 1-lengths in the partition to ?
  and all others to 1 ?, it is not hard to see
  that the proposals will remain the same, and
  Processor 1 will choose the same one
• But this is 2-suboptimal, QED
• Also k processors, randomized 7/4 algorithm.

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e.g., pricing multicasts Feigenbaum, P.,
Shenker, STOC2000
52
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costs

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21
40
70
11, 10, 9, 9
14, 8
9, 5, 5, 3
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23, 17, 14, 9
17, 10
utilities of agents in the node
(u the intrinsic value of the information to
agent i, known only to agent i)
i
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• We wish to design a protocol that will result
• in the computation of
• x ( 0 or 1, will i get it?)
• v (how much will i pay? (0 if x 0) )
• protocol must obey a set of desiderata

i
i
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• 0 ? v ? u,
• lim x 1
• strategy proofness (w u ? x ? v )
• w (u u u ) ? w (u u'u )
• welfare maximization
• ? ui xi cT max
• marginal cost mechanism

i
i
i
u ??
i
def
i
i
i
i
i
i
i
1
n
1
i
n

• budget balance
• ? v c ( T x)
• Shapley mechanism

i
i
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But
In the context of the Internet, there is another
desideratum Tractability the protocol should
require few (constant? logarithmic?) messages
per link. This new requirement changes
drastically the space of available solutions.
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• 0 ? v ? u
• lim x 1
• strategy proofness (w u ? x ? v )
• w (u u u ) ? w (u u'u )
• welfare maximization
• ? w max
• marginal cost mechanism

i
i
i
u ??
i
def
i
i
i
i
i
i
i
1
n
1
i
n

• budget balance
• ? v c ( T x)
• Shapley mechanism

i
i
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Bottom-up phase
W ? u ? W ? c, if gt 0 0
otherwise
i
j
c
W
1
W
3
W
2
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Top-down phase

A
c
D min A, W
D
D
v max 0, u ? D
i
i
Theorem The marginal cost mechanism is
tractable.
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Theorem The Shapley value mechanism is
intractable. Model Nodes are linear decision
trees, and they exchange messages that are linear
combinations of the us and cs
c
1
It reduces to checking whether Au gt Bc by two
sites, one of which knows u and the other c,
where A, B are nonsingular
c
2
agents drop out one-by-one
c
n
u lt u lt lt u
1
2
n
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Algorithmic Mechanism Design

• central problem
• few results outside social welfare maximization
  framework (n.b.Archer and Tardos 01)
• VCG mechanism often breaks the bank
• approximation rarely a remedy (n.b.Nisan and
  Ronen 99, Jain and Vazirani 01)
• wide open, radical departure needed

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algorithmic aspects of auctions

• Optimal auction design Ronen 01
• Combinatorial auctions Nisan 00
• Auctions for digital goods
• On-line auctions
• Communication complexity of combinatorial
  auctions Nisan 01

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coalitional games
Game with players in n v (S) the maximum
total payoff of all players in S,
under worst case play by n
S How to split v (n) fairly?
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first idea the core
A vector (x1, x2,, xn) with ?i x i v(n) is
in the core if for all S we have xS ? v(S)
Problem It is often empty
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second idea the Shapley value
xi E?(vj ?(j) ? ?(i) - vj ?(j) lt ?(i))
e.g., power in the UN Security Council splitting
the cost of a trip
Theorem Shapley The Shapley value is the only
allocation that satisfies Shapleys axioms.
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third idea bargaining setfourth idea
nucleolus ...seventeenth idea the von
Neumann-Morgenstern solution
Deng and P. 1990 complexity-theoretic critique
of solution concepts
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some thoughts on privacy

• also an economic problem
• surrendering private information is either good
  or bad for you
• personal information is intellectual property
  controlled by others, often bearing negative
  royalty
• selling mailing lists vs. selling aggregate
  information false dilemma
• Proposal evaluate the individuals contribution
  when using personal data for decision-making

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e.g., marketing survey Kleinberg, Raghavan, P
2001
likes

• companys utility is proportional to the
  majority
• customers utility is 1 if in the majority
• how should all participants be compensated?

customers
possible versions of product
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the internet game
3, 2
capacity of the internal network to carry
traffic (edges have ? capacity)
1, 1
2, 0
1, 4
5, 9
3, 1
intensity of traffic to/from this
node, distributed to other nodes
proportionately to their intensity
3, 6
2, 2
7, 4
3, 1
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vS value of total flow that can be
handled by the subgraph induced by S

• Compute the Shapley flow
• Find a flow in the core
• Under what circumstances is the core
• nonempty? Contains all maximal flows?

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Game Theory and Math Economics

• Deep and elegant
• Different
• Exquisite interaction with TCS
• Relevant to the Internet
• Wide open algorithmic aspects
• Mathematical tools of choice
• for the new TCS