Game Theory, Internet and the WebA new Science?

Paul G. Spirakis (google: Paul Spirakis)

Research Academic Computer Technology Institute

&

University of Liverpool

(with help from C. H. Papadimitriou, Berkeley)

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• 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 TheoryGame = 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 Theorystrategies

strategies3,-2

payoffs

Similarly for many players

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1,-1 -1,1

-1,1 1,-1

3,3 0,4

4,0 1,1

This for that

Prisoner’s dilemma

e.g.

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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»

sources sink

t

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Sperner’s Lemma: Any legal coloring of a triangulated polytope contains a trichromatic triangle.

Proof:

!

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Sperner BrowerBrower’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 NashFor 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

uw

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 decidethen 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)

(max over all NE p).

[Koutsoupias, Papadimitriou, 1999] Coordination Ratio

[Mavronicolas, Spirakis, 2001]

[Roughgarden, Tardos, 2001]

1OPT

SC(p)maxR

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But also

The Price of Stability (PoS)

(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.

OPT

pSCT

)(min

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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, “free- riders”, 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.