Foundations of Economics and Web Science Paul G. Spirakis CTI Patras Joint work with Spyros Kontogiannis (Univ. Ioannina) inspired also by a talk of Christos

Foundations of Economics and Web Science

Paul G. Spirakis CTI Patras

Joint work with Spyros Kontogiannis (Univ. Ioannina)

inspired also by a talk of Christos Papadimitriou (2002)

12th Pan-Hellenic Conference on Informatics (PCI 2008), Samos, Greece, August 28-30, 2008

PCI 2008, Samos, August 28-30, 2008 2

• 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 Theory’s goals be today?

3PCI 2008, Samos, August 28-30, 2008

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The Web

• built, operated and used by a multitude of diverse economic interests

• theoretical understanding urgently needed

• tools: mathematical economics and game theory

PCI 2008, Samos, August 28-30, 2008

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Web Science[J. Hendler, N. Shadbolt, W. Hall, T. Berners-Lee, and D. Weitzner, CACM 2008]

Web Science is an interdisciplinary approach to understanding the Web

BUT: The Web has become a large-scale scene of social interaction.

SO: Competitive behavior of entities in the Web calls for Algorithmic Game Theory, whose role is to study the (algorithmic aspects of) the effect of selfishness on the performance of the whole system.


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Game Theorystrategies

strategies3,-2

payoffs

(NB: also, many players)


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1,-1 -1,1

-1,1 1,-1

0,0 0,1

1,0 -1,-1

3,3 0,4

4,0 1,1

matching pennies prisoner’s dilemma

chicken

auction

1 … n

1

.

.

nu – x, 0

0, v – y

e.g.


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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 Sperner’s Lemma

Brouwer’s fixpoint Theorem( Kakutani’s Theorem market equilibrium)

Nash’s Theorem min-max theorem for zero-sum games

linear programming duality

(PPAD)

?

P


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Sperner’s Lemma: Any “legal” coloring of the triangulated simplex has a trichromatic triangle

Proof:

!


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Sperner Brouwer

Brouwer’s Theorem: Any continuous function from the simplex to itself has a fixpoint.

Sketch: Triangulate the simplexColor vertices according to “which direction they are

mapped”Sperner’s 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 – x’|2,

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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Nash Approximations• [Lipton, Markakis, Mehta (2003)] Subexponential

(but not polynomial) Approximation Scheme.• [Tsaknakis, Spirakis (2007)] polynomial time

computable 0.3393-approximate Nash eq.• [Kontogiannis, Spirakis (2007)] polynomial time

computable 0.658-approximate Nash eq. (for stronger notion of approximation).

• Many other polynomial time constructions of NE for special cases…


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The Price of Anarchy

cost of worst Nash equilibrium“socially optimum” cost

[Koutsoupias and P, 1998]

routing in networks= 4/3 [Roughgarden and Tardos, 2000]

Also: [Spirakis and Mavronikolas 01,Roughgarden 01, Koutsoupias, Mavronikolas and Spirakis 01]

The price of the Internet architecture?


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Atomic Congestion Games[Fotakis, Kontogiannis,Spirakis, 2004 / 2005]

• No convergence of BR dynamics to PNE, for non-linear delays.

• Price of Anarchy in linear-delay single commodity nets): Θ(logm / loglogm)

• Only linear delays assure bounded PoA

[Fotakis, Kontogiannis,Spirakis, 2006] Coalitions in parallel link nets:

• Existence of PNE

• Threshold behavior (wrt max coalition size) on the quality of the game.


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• More problems: Nash equilibria may be “politically incorrect:” Prisoner’s dilemma

• Repeated prisoner’s 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 prisoner’s dilemma played by memory-limited players (e.g., automata)?


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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 designer’s 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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-Algorithmic Problems in a Distributed Setting

-participants (agents) cannot be assumed to follow the algorithm but rather their own self – interests (They are capable of manipulating the algorithm)

-Algorithm designer: should make sure in advance that the agents’ interests are best served by behaving correctly

-Solution: algorithm plus payments to participants (Mechanism)


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Mechanism design (also implementation Theory)

Aims to study how privately known preferences of many people can be aggregated towards a “Social choice”


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Stackelberg Routing (An “abuse” of the notion)

- Noncooperative networks - Notion of Network Manager - Manager controls part of the flow, is aware - of the noncooperative behavior of the agents and performs her routing aiming at improving the overall system performance.

[Korilis, Lazar, Orda, ’95][Roughgarden, ’01]


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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 P’s have mechanisms.

Proof: Arrow’s 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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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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Mechanisms with verification Declaration phase : agents “talk” and decide (on e.g. an allocation) Execution phase: agents actually execute the agreed outputPayments need only to be given after the execution.

(verify declarations, punish for lying)

Unfortunately:Exponential Time allocation algorithms till now!


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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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• 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 individual’s contribution

when using personal data for decision-making

Some Thoughts on Privacy


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Game Theory and Math Economics:

• Deep and elegant• Different• Exquisite interaction with TCS• Relevant to the Internet and other selfish networks • Wide open algorithmic and complexity aspects• Mathematical tools of choice

for the “new TCS”


Documents

Foundations of Economics and Web Science Paul G. Spirakis CTI Patras Joint work with Spyros Kontogiannis (Univ. Ioannina) inspired also by a talk of Christos