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Multicast Networks Profit Maximization and Strategyproofness

Multicast Networks Profit Maximization and Strategyproofness. David Kitchin, Amitabh Sinha Shuchi Chawla, Uday Rajan, Ramamoorthi Ravi ALADDIN Carnegie Mellon University. The Multicast Network Problem. root node. u. i. The Multicast Network Problem. 6. 18. 10. other nodes, with

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Multicast Networks Profit Maximization and Strategyproofness

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  1. Multicast NetworksProfit Maximization and Strategyproofness David Kitchin, Amitabh Sinha Shuchi Chawla, Uday Rajan, Ramamoorthi Ravi ALADDIN Carnegie Mellon University

  2. The Multicast Network Problem root node

  3. u i The Multicast Network Problem 6 18 10 other nodes, with utilities 30 12 20

  4. c e The Multicast Network Problem 30 4 5 3 15 edges, with costs 6 16 18 10 19 14 8

  5. The Multicast Network Problem 6 Build a multicast tree T which maximizes: 4 18 10 3 15 (net worth) 10 30 20 8

  6. c u e i The Multicast Network Game ? Edges and nodes are agents. ? ? ? ? ? ? We don’t know ‘s or ‘s ? ? ? ? ? ? ? ? ? ? ?

  7. The Multicast Network Game “6” “35” “5” “6” “17” …so the agents give us bids “8” “4” “16” “8” “18” “20” “12” “19” “22” “17” “10” “16” “18”

  8. Mechanism Design We write an algorithm which: • Decides T based on bids b. • Gives (or takes) payments p for all agents in T. This is a mechanism

  9. For Fun and Profit Mechanism and agents have different goals: • We want to maximize (profit) • They want to maximize (or ) Mechanism must also satisfy some conditions

  10. Strategyproofness The most important condition is strategyproofness: A mechanism is strategy-proof (SP) if for all clients, is a dominant strategy irrespective of the bids of other agents and for all edges, is a dominant strategy. i.e., nobody lies.

  11. Other conditions • No Positive Transfers (NPT) • All , and all (we don’t subsidize agents) • Individual Rationality (IR) • All , and all (no agent takes a loss) • Consumer Sovereignty (CS) • If a node bids high enough, it must be included in T. • Polynomial Computability (PC) • All computation must be done in polynomial time.

  12. A note on PC (hardness) • PCST (Prize Collecting Steiner Tree), a related graph problem, is NP-hard • PCST has a 2-approximation • Net Worth, the actual underlying graph problem, is NP-hard • Also NP-hard to separate around zero • Also NP-hard to approximate to any constant

  13. Previous research • Solved: • Nodes are agents, edges are fixed (Jain-Vazirani) • Edges are agents, nodes are non-valued (VST) • Unsolved: • Edges are agents, nodes are fixed • Both are agents

  14. Jain-VaziraniNodes as agents J-V: A timed, ‘moat-growing’ algorithm for nodes as agents Distributes costs to users based on how their moats grow.

  15. Jain-Vazirani 10 2 1 t=0 5 4 1 2 5 4 7

  16. Jain-Vazirani 10 2 1 t=1 5 4 1 2 5 4 7

  17. Jain-Vazirani 10 2 1 t=3 5 4 1 2 5 4 7

  18. Jain-Vazirani 10 2 1 t=4 5 4 1 2 5 4 7

  19. Jain-Vazirani 10 2 1 t=5 5 4 1 2 5 4 7

  20. Properties of J-V • Satisfies all of our earlier conditions: SP, NPT, IR, CS, PC. • Budget-balanced, not profit maximizing.

  21. Vickrey Spanning TreeEdges as agents VST: Descending auction for edges as agents Charges edges their “second price” to ensure strategyproofness.

  22. Vickrey Spanning Tree 10 2 1 “15” 4 4 1 2 4 3 7

  23. Vickrey Spanning Tree 10 2 1 “10” 4 4 1 2 4 3 7

  24. Vickrey Spanning Tree 10 2 1 “10” 4 10 1 2 4 3 7

  25. Vickrey Spanning Tree 4 2 10 7 2 4

  26. VST is strategyproof • Edges in T have no incentive to bid higher • Edges outside T have no incentive to bid lower

  27. VST + J-V We have SP for edges and for nodes… why not just combine the two?

  28. VST + J-V We have SP for edges and for nodes… why not just combine the two? 1+є 1-є є 10 1 1-є 1-є 1-є є є

  29. VST + J-V VST + J-V gives this tree: 1 є 10 1 1 1 є є

  30. VST + J-V But we could have gotten this (better) tree: 10 1+є Need to be able to evaluate mechanisms!

  31. Guarantees • Can’t approximate Net Worth to any constant… • …how do we compare mechanisms? • We make guarantees • If there is a very profitable tree, guarantee some fraction of its profit. • If all possible trees are too unprofitable, prove that there is no good solution. • Tighter bounds == better mechanism

  32. Profit Guaranteeing Mechanisms An -profit guaranteeing mechanism, where and satisfies the following criteria: • SP, IR, NPT, CS, PC • If , where , it finds a tree with profit at least where is decreasing in (the ratio increases as increases). • If for every tree T, , it demonstrates that no non-trivial positive surplus tree exists. • If neither 2 nor 3 is true, it simply returns a solution with non-negative profit (possibly the empty solution).

  33. ß-guarantee 1 8 7 4 4 4 1 1 4 6 5 1

  34. Competition To obtain reasonable bounds, we need competition. • Edges – Competition across cuts • Nodes – Multiple users at each node

  35. Є-Edge Competition y x x < y < x(1 + є)

  36. Node Competition No node has only one user.

  37. Edge-agents (M1) 1. Run Goemans-Williamsen (GW) to decide node set +5 4 -8 4 u +7 Differences between GW and J-V

  38. Edge-agents (M1) 2. Build a VST on the node set 4 2 7 2

  39. Edge-agents (M1) 3. Prune out any unprofitable subtrees, and return T. +3 -5 +1 +1 +7 +6 -10 +2

  40. Edge-agents (M1) 4. If user set was empty, rerun GW with 2u. If this still returns an empty tree, we state that all possible trees are unprofitable.

  41. Edge-agents (M1) Edge-agents is a profit guaranteeing mechanism, on any є-edge competitive graph.

  42. All-agents (M2) All-agents is surprisingly simple: • Run a cancellable auction at each node, and fix that auction’s revenue as the node’s utility. • Run Edge-agents using those fixed utilities.

  43. Cancellable auctions But what’s a cancellable auction? An auction is cancellable if the auctioneer has the option of cancelling the auction if some condition is not met, and this does not affect the strategy of the participants. Want to cancel auctions at every node that doesn’t end up in T.

  44. SCS auction Sampling Cost Sharing (SCS) Auction • Satisfies our conditions (NPT, etc.) • Guarantees at least ¼ of maximum revenue we could raise with any SP mechanism. • Requires at least two buyers (node competition)

  45. All-agents (M2) All-agents is a profit guaranteeing mechanism, on any є-edge competitive and node competitive graph.

  46. No Competition What if nodes aren’t competitive? • We can no longer give an guarantee • Build a VST first and then run J-V to allocate costs to nodes. • The mechanism is (0,4)-guaranteeing

  47. Conclusions • Need approximations to ensure computability • Need competition to ensure profitability • Solution is possible, but bounds are impractical.

  48. Questions?

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