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

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

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

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

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

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-Vazirani Nodes 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 2 10 5 4 1 4 2 7 5 1 t=0

16. 2 10 5 4 1 4 2 7 5 1 t=1

17. 2 10 5 4 1 4 2 7 5 1 t=3

18. 2 10 5 4 1 4 2 7 5 1 t=4

19. 2 10 5 4 1 4 2 7 5 1 t=5

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

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

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

23. 2 4 3 1 4 2 7 4 1 “10” 10

24. 2 4 3 1 10 2 7 4 1 “10” 10

25. 2 4 10 2 7 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 1-є є є є 10 1-є 1+є

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

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: 1. SP, IR, NPT, CS, PC 2. If, where, it finds a tree with profit at least where is decreasing in (the ratio increases as increases). 3. If for every tree T,, it demonstrates that no non- trivial positive surplus tree exists. 4. 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 1 1 1 4 5 4 6 44 7

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

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

36. Node Competition No node has only one user.

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

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

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

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: 1. Run a cancellable auction at each node, and fix that auction’s revenue as the node’s utility. 2. 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?