Multicast Networks Profit Maximization and Strategyproofness

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Multicast NetworksProfit Maximization and
Strategyproofness

• David Kitchin, Amitabh Sinha
• Shuchi Chawla, Uday Rajan, Ramamoorthi Ravi
• ALADDIN
• Carnegie Mellon University

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The Multicast Network Problem
root node
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The Multicast Network Problem
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other nodes, with utilities
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The Multicast Network Problem
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edges, with costs
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The Multicast Network Problem
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Build a multicast tree T which maximizes
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(net worth)
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The Multicast Network Game
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Edges and nodes are agents.
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We dont know s or s
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The Multicast Network Game
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so the agents give us bids
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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

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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

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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.

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Other conditions

• No Positive Transfers (NPT)
• All , and all (we dont
  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.

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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

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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

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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.
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Jain-Vazirani
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Jain-Vazirani
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Jain-Vazirani
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Jain-Vazirani
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Jain-Vazirani
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Properties of J-V

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

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Vickrey Spanning TreeEdges as agents
VST Descending auction for edges as agents
Charges edges their second price to
ensure strategyproofness.
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Vickrey Spanning Tree
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Vickrey Spanning Tree
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Vickrey Spanning Tree
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Vickrey Spanning Tree
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VST is strategyproof

• Edges in T have no incentive to bid higher
• Edges outside T have no incentive to bid lower

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VST J-V
We have SP for edges and for nodes why not just
combine the two?
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VST J-V
We have SP for edges and for nodes why not just
combine the two?
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1-?
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VST J-V
VST J-V gives this tree
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VST J-V
But we could have gotten this (better) tree
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Need to be able to evaluate mechanisms!
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Guarantees

• Cant 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

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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).

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ß-guarantee
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Competition

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

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?-Edge Competition
y
x
x lt y lt x(1 ?)
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Node Competition
No node has only one user.
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Edge-agents (M1)
1. Run Goemans-Williamsen (GW) to decide node set
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Differences between GW and J-V
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Edge-agents (M1)
2. Build a VST on the node set
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Edge-agents (M1)
3. Prune out any unprofitable subtrees, and
return T.
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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.
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Edge-agents (M1)

• Edge-agents is a profit
• guaranteeing mechanism, on any
• ?-edge competitive graph.

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All-agents (M2)

• All-agents is surprisingly simple
• Run a cancellable auction at each node, and fix
  that auctions revenue as the nodes utility.
• Run Edge-agents using those fixed utilities.

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Cancellable auctions

• But whats 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
  doesnt end up in T.

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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)

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All-agents (M2)

• All-agents is a profit
• guaranteeing mechanism, on any
• ?-edge competitive and node competitive
• graph.

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No Competition

• What if nodes arent 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

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Conclusions

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

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Questions?