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Foundations of Economics and Web Science

This work explores the intersection of economics and web science, discussing theoretical concepts and practical applications in algorithmic game theory, Nash equilibria, and economic behavior on the web. It references key works and discussions from the PCI 2008 conference in Samos, Greece, reflecting on the goals of theory in the present day. Topics include rationality, competitive behavior, the price of anarchy, and game theory strategies. The text delves into mathematical tools, including combinatorics and logic, that underpin the understanding of computational artifacts.

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Foundations of Economics and Web Science

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  1. 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

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

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

  4. 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

  5. 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. PCI 2008, Samos, August 28-30, 2008

  6. Game Theory strategies strategies 3,-2 payoffs (NB: also, many players) PCI 2008, Samos, August 28-30, 2008

  7. matching pennies prisoner’s dilemma e.g. auction chicken 0, v – y u – x, 0 PCI 2008, Samos, August 28-30, 2008

  8. 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. . . . PCI 2008, Samos, August 28-30, 2008

  9. (PPAD) ? 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  P PCI 2008, Samos, August 28-30, 2008

  10. Sperner’s Lemma: Any “legal” coloring of the triangulated simplex has a trichromatic triangle Proof: ! PCI 2008, Samos, August 28-30, 2008

  11. Sperner  Brouwer Brouwer’s 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” 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 PCI 2008, Samos, August 28-30, 2008

  12. 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 PCI 2008, Samos, August 28-30, 2008

  13. Nash  von Neumann If game is zero-sum, then double best response is the same as max-min pair PCI 2008, Samos, August 28-30, 2008

  14. 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 PCI 2008, Samos, August 28-30, 2008

  15. is it in P? PCI 2008, Samos, August 28-30, 2008

  16. 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… PCI 2008, Samos, August 28-30, 2008

  17. The Price of Anarchy cost of worst Nash equilibrium [Koutsoupias and P, 1998] “socially optimum” cost 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? PCI 2008, Samos, August 28-30, 2008

  18. 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. PCI 2008, Samos, August 28-30, 2008

  19. 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)? PCI 2008, Samos, August 28-30, 2008

  20. 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 PCI 2008, Samos, August 28-30, 2008

  21. 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 PCI 2008, Samos, August 28-30, 2008

  22. 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) PCI 2008, Samos, August 28-30, 2008

  23. Mechanism design (also implementation Theory) Aims to study how privately known preferences of many people can be aggregated towards a “Social choice” PCI 2008, Samos, August 28-30, 2008

  24. 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] PCI 2008, Samos, August 28-30, 2008

  25. 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 PCI 2008, Samos, August 28-30, 2008

  26. 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…) PCI 2008, Samos, August 28-30, 2008

  27. 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 PCI 2008, Samos, August 28-30, 2008

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

  29. Mechanisms with verification Declaration phase : agents “talk” and decide (on e.g. an allocation) Execution phase: agents actually execute the agreed output Payments need only to be given after the execution. (verify declarations, punish for lying) Unfortunately:Exponential Time allocation algorithms till now! PCI 2008, Samos, August 28-30, 2008

  30. 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] PCI 2008, Samos, August 28-30, 2008

  31. 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 individual’s contribution when using personal data for decision-making PCI 2008, Samos, August 28-30, 2008

  32. 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” PCI 2008, Samos, August 28-30, 2008

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