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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

root node

u

i

6

18

10

other nodes, with

utilities

30

12

20

c

e

30

4

5

3

15

edges, with

costs

6

16

18

10

19

14

8

6

Build a multicast tree T which maximizes:

4

18

10

3

15

(net worth)

10

30

20

8

c

u

e

i

?

Edges and nodes

are agents.

?

?

?

?

?

?

We don’t know

‘s or ‘s

?

?

?

?

?

?

?

?

?

?

?

“6”

“35”

“5”

“6”

“17”

…so the agents

give us bids

“8”

“4”

“16”

“8”

“18”

“20”

“12”

“19”

“22”

“17”

“10”

“16”

“18”

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

Mechanism and agents have different goals:

- We want to maximize (profit)
- They want to maximize (or )
Mechanism must also satisfy some conditions

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.

- 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.

- 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

- 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

J-V: A timed, ‘moat-growing’ algorithm for nodes as agents

Distributes costs to users based on

how their moats grow.

10

2

1

t=0

5

4

1

2

5

4

7

10

2

1

t=1

5

4

1

2

5

4

7

10

2

1

t=3

5

4

1

2

5

4

7

10

2

1

t=4

5

4

1

2

5

4

7

10

2

1

t=5

5

4

1

2

5

4

7

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

VST: Descending auction for edges as agents

Charges edges their “second price” to ensure

strategyproofness.

10

2

1

“15”

4

4

1

2

4

3

7

10

2

1

“10”

4

4

1

2

4

3

7

10

2

1

“10”

4

10

1

2

4

3

7

4

2

10

7

2

4

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

We have SP for edges and for nodes…

why not just combine the two?

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

VST + J-V gives this tree:

1

є

10

1

1

1

є

є

But we could have gotten this (better) tree:

10

1+є

Need to be able to evaluate mechanisms!

- 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

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).

1

8

7

4

4

4

1

1

4

6

5

1

To obtain reasonable bounds, we need competition.

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

y

x

x < y < x(1 + є)

No node has only one user.

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

+5

4

-8

4

u

+7

Differences between GW and J-V

2. Build a VST on the node set

4

2

7

2

3. Prune out any unprofitable subtrees, and return T.

+3

-5

+1

+1

+7

+6

-10

+2

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.

Edge-agents is a profit

guaranteeing mechanism, on any

є-edge competitive graph.

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.

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.

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 is a profit

guaranteeing mechanism, on any

є-edge competitive and node competitive

graph.

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

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

Questions?