Multicast NetworksProfit Maximization and Strategyproofness

David Kitchin, Amitabh Sinha

Shuchi Chawla, Uday Rajan, Ramamoorthi Ravi

ALADDINCarnegie Mellon University

The Multicast Network Problem

root node

The Multicast Network Problem

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other nodes, with utilities u i

The Multicast Network Problem

edges, with costs ce

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The Multicast Network Problem

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Build a multicast tree T which maximizes:

T eT i cu

(net worth)

The Multicast Network Game

Edges and nodesare agents.

ce

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The Multicast Network Game

…so the agents give us bids

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“8”“17”

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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

For Fun and ProfitMechanism and agents have different

goals:

We want to maximize (profit) They want to maximize (or )

Mechanism must also satisfy some conditions

T eT i pp

ii pu ee cp

Strategyproofness

The most important condition is strategyproofness:

A mechanism is strategy-proof (SP) if for all clients, is adominant strategy irrespective of the bids of other agents and forall edges, is a dominant strategy.

i.e., nobody lies.

ii ub

ee cb

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.

0ip 0ep

0 ii pu 0 ee cp

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

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

Jain-VaziraniNodes as agents

J-V: A timed, ‘moat-growing’ algorithm for nodes as agentsDistributes costs to users based on

how their moats grow.

Jain-Vazirani

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5511t=0t=0

Jain-Vazirani

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5511t=1t=1

Jain-Vazirani

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5511t=3t=3

Jain-Vazirani

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5511t=4t=4

Jain-Vazirani

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5511t=5t=5

Properties of J-V Satisfies all of our earlier conditions: SP, NPT,

IR, CS, PC. Budget-balanced, not profit maximizing.

Vickrey Spanning TreeEdges as agents

VST: Descending auction for edges as agents

Charges edges their “second price” to ensurestrategyproofness.

Vickrey Spanning Tree

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4411““15”15”

Vickrey Spanning Tree

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““10”10”

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Vickrey Spanning Tree

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““10”10”

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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

VST + J-VWe have SP for edges and for nodes…why not just combine the two?

VST + J-VWe have SP for edges and for nodes…why not just combine the two?

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VST + J-VVST + J-V gives this tree:

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VST + J-VBut we could have gotten this (better) tree:

10 1+є

Need to be able to evaluate mechanisms!

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

Profit Guaranteeing Mechanisms

An -profit guaranteeing mechanism, where and satisfies the following criteria:

1. SP, IR, NPT, CS, PC2. 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).

),( ]1,0[1

RTf )1()( * Rk )( 0)( k

)( *Tf

)()( TrTc

ß-guarantee1

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Competition

To obtain reasonable bounds, we need competition.

Edges – Competition across cuts Nodes – Multiple users at each node

Є-Edge Competition

xx

x < y < x(1 + x < y < x(1 + є)є)

yy

Node Competition

41 u

92 u

83 u

No node has only one user.

Edge-agents (M1)

1. Run Goemans-Williamsen (GW) to decide node set

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Differences between GW and J-V

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 allpossible trees are unprofitable.

Edge-agents (M1)

Edge-agents is a profitguaranteeing mechanism, on any є-edge competitive graph.

)4,()1(2

1

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.

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.

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)

All-agents (M2)

All-agents is a profitguaranteeing mechanism, on any є-edge competitive and node

competitive graph.

)4,()1(8

1

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

Conclusions Need approximations to ensure

computability Need competition to ensure profitability Solution is possible, but bounds are

impractical.

Questions?