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Algorithmic Game Theory & Internet Computing: Market Equilibria and Nash Bargaining Solutions

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102 diapositives · 1.1K Views · Jan 04, 2020

Explore the intersection of economics, scarce resources, market equilibrium, and computational algorithms in the context of game theory and internet computing, as presented by Vijay V. Vazirani from Georgia Tech. Delve into the depths of how scarce resources are allocated, equilibrium prices are determined, and the groundbreaking Arrow-Debreu Theorem of 1954. Discover the pivotal role of mathematical economics, including the Eisenberg-Gale convex program of 1959, in understanding market equilibria. Uncover the complexities of multiple goods, diverse buyer preferences, and the algorithmic underpinnings needed for today's dynamic markets driven by internet giants like Google, eBay, Yahoo!, and Amazon. Learn about the innovative Combinatorial Algorithm for Linear Case of Fisher's Model by Devanur, Papadimitriou, Saberi & V. (2002), which offers efficient solutions for capturing equilibrium allocations in Fisher's model. Witness the power of algorithmic game theory in deciphering market equilibria and Nash bargaining solutions, showcasing the dynamic balance between demand and supply in economic transactions.

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  1. Combinatorial Algorithms for Convex Programs (Capturing Market Equilibria and Nash Bargaining Solutions) Algorithmic Game Theoryand Internet Computing Vijay V. Vazirani Georgia Tech

  2. What is Economics? ‘‘Economics is the study of the use of scarce resources which have alternative uses.’’ Lionel Robbins (1898 – 1984)

  3. How are scarce resources assigned to alternative uses?

  4. How are scarce resources assigned to alternative uses? Prices!

  5. How are scarce resources assigned to alternative uses? Prices Parity between demand and supply

  6. How are scarce resources assigned to alternative uses? Prices Parity between demand and supplyequilibrium prices

  7. Do markets even admitequilibrium prices?

  8. General Equilibrium TheoryOccupied center stage in MathematicalEconomics for over a century Do markets even admitequilibrium prices?

  9. Arrow-Debreu Theorem, 1954 • Celebrated theorem in Mathematical Economics • Established existence of market equilibrium under very general conditions using a theorem from topology - Kakutani fixed point theorem.

  10. Do markets even admitequilibrium prices?

  11. Easy if only one good! Do markets even admitequilibrium prices?

  12. Supply-demand curves

  13. What if there are multiple goods and multiple buyers with diverse desires and different buying power? Do markets even admitequilibrium prices?

  14. Irving Fisher, 1891 • Defined a fundamental market model

  15. linear utilities

  16. For given prices,find optimal bundle of goods

  17. Several buyers with different utility functions and moneys.

  18. Several buyers with different utility functions and moneys.Find equilibrium prices.

  19. “Stock prices have reached what looks likea permanently high plateau”

  20. “Stock prices have reached what looks likea permanently high plateau” Irving Fisher, October 1929

  21. Arrow-Debreu Theorem, 1954 • Celebrated theorem in Mathematical Economics • Established existence of market equilibrium under very general conditions using a theorem from topology - Kakutani fixed point theorem. • Highly non-constructive!

  22. General Equilibrium Theory An almost entirely non-algorithmic theory!

  23. The new face of computing

  24. Today’s reality • New markets defined by Internet companies, e.g., • Google • eBay • Yahoo! • Amazon • Massive computing power available. • Need an inherenltly-algorithmic theory of markets and market equilibria.

  25. Combinatorial Algorithm for Linear Case of Fisher’s Model • Devanur, Papadimitriou, Saberi & V., 2002 Using the primal-dual paradigm

  26. Combinatorial algorithm • Conducts an efficient search over a discrete space. • E.g., for LP: simplex algorithm vs ellipsoid algorithm or interior point algorithms.

  27. Combinatorial algorithm • Conducts an efficient search over a discrete space. • E.g., for LP: simplex algorithm vs ellipsoid algorithm or interior point algorithms. • Yields deep insights into structure.

  28. No LP’s known for capturing equilibrium allocations for Fisher’s model

  29. No LP’s known for capturing equilibrium allocations for Fisher’s model • Eisenberg-Gale convex program, 1959

  30. No LP’s known for capturing equilibrium allocations for Fisher’s model • Eisenberg-Gale convex program, 1959 • Extended primal-dual paradigm to solving a nonlinear convex program

  31. Linear Fisher Market • B = n buyers, money mifor buyer i • G = g goods, w.l.o.g. unit amount of each good • : utility derived by i on obtaining one unit of j • Total utility of i, • Find market clearing prices.

  32. Eisenberg-Gale Program, 1959

  33. Eisenberg-Gale Program, 1959 prices pj

  34. Why remarkable? • Equilibrium simultaneously optimizes for all agents. • How is this done via a single objective function?

  35. Theorem • If all parameters are rational, Eisenberg-Gale convex program has a rational solution! • Polynomially many bits in size of instance

  36. Theorem • If all parameters are rational, Eisenberg-Gale convex program has a rational solution! • Polynomially many bits in size of instance • Combinatorial polynomial time algorithm for finding it.

  37. Theorem • If all parameters are rational, Eisenberg-Gale convex program has a rational solution! • Polynomially many bits in size of instance • Combinatorial polynomial time algorithm for finding it. Discrete space

  38. AllocationsPrices (Money) Idea of algorithm • primal variables: allocations • dual variables: prices of goods • iterations: execute primal & dual improvements

  39. How are scarce resources assigned to alternative uses? Prices Parity between demand and supply

  40. Yin & Yang

  41. Nash bargaining game, 1950 • Captures the main idea that both players gain if they agree on a solution. Else, they go back to status quo. • Complete information game.

  42. Example • Two players, 1and 2, have vacation homes: • 1: in the mountains • 2: on the beach • Consider all possible ways of sharing.