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Algorithmic Game Theory and Internet Computing

Algorithmic Game Theory and Internet Computing. New Market Models Resource Allocation Markets. Vijay V. Vazirani. Fisher’s Model. n buyers, with specified money, m(i) for buyer i k goods (unit amount of each good) Linear utilities: is utility derived by i

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Algorithmic Game Theory and Internet Computing

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  1. Algorithmic Game Theoryand Internet Computing New Market Models Resource Allocation Markets Vijay V. Vazirani

  2. Fisher’s Model • n buyers, with specified money, m(i) for buyer i • k goods (unit amount of each good) • Linear utilities: is utility derived by i on obtaining one unit of j • Total utility of i,

  3. Fisher’s Model • n buyers, with specified money, m(i) • k goods (each unit amount, w.l.o.g.) • Linear utilities: is utility derived by i on obtaining one unit of j • Total utility of i, • Find prices s.t. market clears

  4. Eisenberg-Gale Program, 1959

  5. Via KKT Conditions can establish: • Optimal solution gives equilibrium allocations • Lagrange variables give prices of goods

  6. Eisenberg-Gale program helps establish: • Equilibrium exists (under mild conditions) • Equilibrium utilities and prices are unique

  7. Eisenberg-Gale program helps establish: • Equilibrium exists (under mild conditions) • Equilibrium utilities and prices are unique • Rational!!

  8. Kelly’s resource allocation model, 1997 Mathematical framework for understanding TCP congestion control

  9. Kelly’s model Given: network G = (V,E) (directed or undirected) capacities on edges source-sink pairs (agents) m(i): money/unit time agent i is willing to pay

  10. Kelly’s model Network determines: f(i): flow rate of agent i Assume utility u(i) = m(i) log f(i) Total utility is additive

  11. Convex Program for Kelly’s Model

  12. Kelly’s model Lagrange variables: p(e): price/unit flow

  13. Kelly’s model Optimum flow and edge prices are in equilibrium: 1). p(e)>0 only if e is saturated 2) flows go on cheapest paths 3) money of each agent is fully used Let rate(i) = cost of cheapest path for i m(i) = f(i) rate(i)

  14. Kelly’s model Optimum flow and edge prices are in equilibrium: 1). p(e)>0 only if e is saturated 2) flows go on cheapest paths 3) money of each agent is fully used Let rate(i) = cost of cheapest path for i f(i)’s andrate(i)’s are unique!

  15. TCP Congestion Control • f(i): source rate • prob. of packet loss (in TCP Reno) queueing delay (in TCP Vegas) p(e):

  16. TCP Congestion Control • f(i): source rate • prob. of packet loss (in TCP Reno) queueing delay (in TCP Vegas) Kelly: Equilibrium flows are proportionally fair: only way of increasing an agent’s flow by 5% is to decrease other agents’ flow by at least 5% p(e):

  17. TCP Congestion Control • f(i): source rate • prob. of packet loss (in TCP Reno) queueing delay (in TCP Vegas) • Low, Doyle, Paganini: continuous time algs. for computing equilibria (not poly time). p(e):

  18. TCP Congestion Control • f(i): source rate • prob. of packet loss (in TCP Reno) queueing delay (in TCP Vegas) • Low, Doyle, Paganini: continuous time algs. for computing equilibria (not poly time). • AIMD + RED converges to equilibrium primal-dual (source-link) alg. p(e):

  19. TCP Congestion Control • f(i): source rate • prob. of packet loss (in TCP Reno) queueing delay (in TCP Vegas) • Low, Doyle, Paganini: continuous time algs. for computing equilibria (not poly time). • FAST: for high speed networks with large bandwidth p(e):

  20. Combinatorial Algorithms • Devanur, Papadimitriou, Saberi & V., 2002: for Fisher’s linear utilities case • Kelly & V., 2002: Kelly’s model is a generalization of Fisher’s model. Find comb. poly time algs!

  21. Irrational for 2 sources & 3 sinks $1 $1 $1

  22. Irrational for 2 sources & 3 sinks Equilibrium prices

  23. 1 source & multiple sinks • 2 source-sink pairs

  24. $5 $5

  25. $30 $10 $40

  26. Jain & V., 2005: strongly poly alg • Primal-dual algorithm • Usual: linear programs & LP-duality • This: convex programs & KKT conditions • Ascending price auction • Buyers: sinks (fixed budgets, maximize flow) • Sellers: edges (maximize price)

  27. rate(i): cost of cheapest path

  28. Capacity of edge =

  29. min s-t cut

  30. nested cuts

  31. Find s-t max flow • Flow and prices will: • Saturate all red cuts • Use up sinks’ money • Send flow on cheapest paths

  32. $30 $10 $40

  33. Rational!!

  34. Max-flow min-cut theorem

  35. Other resource allocation markets • 2 source-sink pairs (directed/undirected) • Branchings rooted at sources (agents) • Spanning trees • Network coding

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