Game Theory, Internet and the Web
A new Science?
Paul G. Spirakis
(google: Paul Spirakis)
Research Academic Computer Technology Institute
&
University of Liverpool
(with help from C. H. Papadimitriou, Berkeley)
• Main Goal of Computer Science
(1950-2000):
To investigate the capabilities and limits of the
Computing Model of von Neumann – Turing
(and its software)
(Math Tools: Logic, Combinatorics, Automata )
• What is the goal of Computer Science for
the 21st century?
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The Internet and the Web
• Built, operated and used by a variety of
entities with diverse interests.
• Not yet understood deeply
“The Web is a huge arena of competition and
cooperation between many logical entities
with selfish interests” (C.H. Papadimitriou)
New Tool: Math Foundations of Economics,
Game Theory
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Game Theory
Game = Any interaction among rational and logical
entities each of which may have different motives
and goals.
Game Γ = (Ν, {Si}, {ui})
N = Set of “players”
Si = Set of pure strategics of player i
ui: XSi  R
= The utility function of
player i
(Expected Utility Theorem of Von Neumann &
Morgenstern)
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• A game is a system of rational and logical entities
in interaction
• Selfish entities: Each of them has a possibly
different utility function (and wants to maximize
it)
“People are expected utility maximizers”
• Such systems are very different from the “usual”
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Game Theory
strategies
strategies
3,-2
payoffs
Similarly for many players
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e.g.
This for that
1,-1 -1,1
-1,1 1,-1
Prisoner’s dilemma
3,3
0,4
4,0
1,1
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Rational Behaviour
• Dominant Strategies (but they do not always
exist)
• Nash Equilibria (mutual best response)
Each player will not benefit if she deviates
unilaterally
• Mixed Strategies allowed (i.e. prob.
distributions on the pure strategies of each
player).
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John Forbes Nash, Jr.
(A beautiful mind)
Theorem [Nash, 1952]
Every finite game has at least one
Nash Equilibrium
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The beauty of Mathematics
Discrete Math (Graphs)
Sperner Lemma (Combinatorics)
Fixpoint Theorem of Brower (Analyis)
Kakutani’s Theorem  Market Equilibria
Nash’s Theorem
zero sum games
duality, linear programming
?
P
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Discrete Mathematics
«Any directed graph with indegrees and
outdegrees at most 1, if it has a source
then it has a sink»
source
t
s
sink
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Sperner’s Lemma: Any legal coloring of a
triangulated polytope contains a trichromatic
triangle.
Proof:
!
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Sperner  Brower
Brower’s Thm:: Any continuous function from a
polytope to itself has a fix point.
Proof
Triangulate the polytope. Color the vertices
according to the direction indicated by the
function.
Sperner  There exist a triangle with “no exit”
Now make the triangulation dense
The subsequence of the centers of the Sperner
triangles converges
QED
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Brower  Nash
For each pair of mixed strategies x, y let:
(x,y) = (x’, y’), where x΄ maximizes
off1(x’,y) - |x – x’|2,
(off1 = expected payoff of player 1)
Similarly for y’.
Now any Brower fixpoint is a Nash
Equilibrium
QED
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Nash  von Neumann
If the game is zero – sum (constant sum) them
the mutual best responses are the same as a
max-min pair (and due to duality, the
solution of a Linear Program).
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The notion of Equilibrium is basic in
many Sciences
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Some Questions
• How logical is the probabilistic play?
(poker bluffs, tax evasion)
• Can we “learn” (or compute) an
equilibrium;
• What is the best (worst) Equilibrium;
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How do we prove existence?
• Induction (leads to an algorithm to find the solution)
• Pigeonhole principle
• Fixpoints f(x)=x
- convergent sequences
- equilibria
- Logic, λ-calculus, Tarski



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Given an existence proof for a solution, can we
find (compute) the solution?
• Not obvious
• A needle in a haystack
Inefficient Proofs of Existence
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COMPUTATIONAL COMPLEXITY
CLASSES
• A class = all problems whose worst-case time
complexity is “of the same order”
• Reductions A  B
If we have an algorithm for B then we can solve
A with the “same” effort
• Complete Problems in a class
A problem Π is complete in the class A iff
any problem Π΄ in A can be reduced to it.
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Two famous classes
Class P:
All problems algorithmically solvable in polynomial
time (on the length of the input)
Class NP
All problems solvable by a Non-deterministic
algorithm in polynomial time
(Guess and Verify)
Is P equal to NP?
(A millennium problem in CS/Math)
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The class PPAD
(for problems that have always a solution)
• Imagine a huge (exponential size) directed graph
• Simple structure: A union of directed paths
and cycles
u
w
u΄
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The class PPAD
(View vertices, σ, as possible solutions, sinks,
sources are solutions)
• An initial (artificial) source σ0
• Given σi we can efficiently compute σi+1
• Given σ, σ’ we can efficiently find if σ is the parent
of σ’
Complete Problem: Find another source / sink
(Another End of Line AEOL)
But the graph is very large!
(e.g. 2n vertices for n bits to name each σ)
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Theorem: Finding a Nash Equilibrium
(even for two player games) is Complete
in PPAD
(Papadimitriou,
Papadimitriou/Goldberg/Daskalakis,
Chen and Deng)
Till now we know only of exponential algorithms
to compute NE
(Lemke, Howson)
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• Same holds for Market Equilibria
• Problem becomes easy for special types of games
(e.g. congestion)
• What happens in Web’s games?
• What are the games and equilibria in the Web?
In its Social Nets?
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Approximate Equilibria
• ε-Nash: Each player stays at equilibrium decision,
even if she may gain at most “epsilon” by unilaterally
deviating
“We don’t change our mate for a slightly better”
• Can we compute ε-Nash equilibria efficiently?
BEST Poly-time result: ε = 0.34
[Tsaknakis, Spirakis, 07]
Sub exponential methods (Lipton, Markakis, Mehta, 03)
(Tsaknakis, Spirakis, 10)
• Still open to go below “1/3”
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The battlefield
• The “system”
• The Web
• The terrain
• Society
SOCIAL COST (Function of Social happiness)
SC : XCi  R
The function measures the social cost, given the
choices (strategies) yi of each player i.
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Examples of Social Cost
• Energy spent
• Max delay in streets
• Political cost for the country / EU given
the decisions of the leaders.
Altrouist: A player whose utility
function “agrees” with the social
cost function
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If God would order everybody how to decide
then we would get an Optimal Social Cost,
OPT
• But, actually, the “system” reaches
an equilibrium P
• How far is SC(p) from OPT?
(Usually OPT is not even an equilibrium!)
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The Price of Anarchy (PoA)
SC(p)
1
R  max
OPT
(max over all NE p).
[Koutsoupias, Papadimitriou, 1999]
Coordination Ratio
[Mavronicolas, Spirakis, 2001]
[Roughgarden, Tardos, 2001]
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But also
The Price of Stability (PoS)
SC ( p)
T  min
OPT
(min over all NE p)
[Schulz, Stier Moses, 2003]
[Anshelevich et al, 2004]
• Lots of results for PoA, PoS for congestion
games, network creation games etc.
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How to Control Anarchy
• Mechanisms design
• A set of rules and options put by game’s
designers. Does not affect the free will of
players. But appeals to their selfishess
(e.g. payments, punishments, ads). Aims
in “moving the game” to “good equilibria”
(desirable by the designer)
• New challenges in algorithms!
• Auctions
• Lies and truthfullness
• Stackelberg’s games (Leader plays first)
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Dynamics
• How can a Selfish System (e.g. the markets,
Society, the Web) approach an Equilibrium?
• Dynamics
Players interact, learn and do selfish choices,
and the “state” of the System changes with time
• Many, repeated, concurrent games all the time.
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The world is not perfect
• Players may be illogical and not so rational
• Players may have limited information
about the game (s), or limited knowledge.
• Errors are human / also for Computers
(“Trembling Hand”)
• Other factors (enemies of the System, “freeriders”, strange behaviour, …)
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But, fortunately:
• Players can learn, adapt, evolve
• Biology and “Self-regulation”
[Self-stabilization) [Dijkstra] [S. Dolev, E.
Schiller]
• Equilibria in animal, plants (microbes)
communities in antagonism or cooperation
• John Maynard Smith (1974)
(Evolutionary Games).
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Yet another Science
• Mathematical Ecology
(Alfred Lotka, Vito Volterra, 1920)
(dynamics of moskitos, also of hunter-prey
fish in Adriatic Sea ).
• Ancestor of Evolutionary Game Theory
• Evolutionary Methods in Economics
[Robert Axelrod, 1984]
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Relevant Math.
• Nonlinear dynamical systems
• Differential Equations
• Attractors, oscillations, Equilibria
• Chaotic Behaviour!
(and, again, fixpoints!)
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Dynamics of Selfish Systems
• Norms (Contracts, Social Rules)
• “Internal” causes for change:
- players’ selfish behaviour
- learning, adaptation
• Externalities
• “Final” result (equilibrium, stability, but
also complex behaviour, chaos)
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A New Science
• Deep and elegant
• Different
• Strong interaction with Foundations of CS
• Emerges everywhere (Research, Education,
funds)
(also new Industry: e-commerce, ads, Social
Nets …)
• A new light in Complexity
• Isaac Asimov’s “psychohistory”?
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MANY THANKS
FOR LISTENING TO ME.
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