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Algorithmic Game Theory and Internet Computing

This text explores new market models for resource allocation in algorithmic game theory and internet computing, covering topics such as Fisher's model, Kelly's resource allocation model, TCP congestion control, and more.

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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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