Multicast networks profit maximization and strategyproofness
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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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Multicast networks profit maximization and strategyproofness

Multicast NetworksProfit Maximization and Strategyproofness

David Kitchin, Amitabh Sinha

Shuchi Chawla, Uday Rajan, Ramamoorthi Ravi

ALADDIN

Carnegie Mellon University


The multicast network problem

The Multicast Network Problem

root node


The multicast network problem1

u

i

The Multicast Network Problem

6

18

10

other nodes, with

utilities

30

12

20


The multicast network problem2

c

e

The Multicast Network Problem

30

4

5

3

15

edges, with

costs

6

16

18

10

19

14

8


The multicast network problem3

The Multicast Network Problem

6

Build a multicast tree T which maximizes:

4

18

10

3

15

(net worth)

10

30

20

8


The multicast network game

c

u

e

i

The Multicast Network Game

?

Edges and nodes

are agents.

?

?

?

?

?

?

We don’t know

‘s or ‘s

?

?

?

?

?

?

?

?

?

?

?


The multicast network game1

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”


Mechanism design

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 profit

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


Strategyproofness

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.


Other conditions

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.


A note on pc hardness

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

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 vazirani nodes as agents

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.


Jain vazirani

Jain-Vazirani

10

2

1

t=0

5

4

1

2

5

4

7


Jain vazirani1

Jain-Vazirani

10

2

1

t=1

5

4

1

2

5

4

7


Jain vazirani2

Jain-Vazirani

10

2

1

t=3

5

4

1

2

5

4

7


Jain vazirani3

Jain-Vazirani

10

2

1

t=4

5

4

1

2

5

4

7


Jain vazirani4

Jain-Vazirani

10

2

1

t=5

5

4

1

2

5

4

7


Properties of j v

Properties of J-V

  • Satisfies all of our earlier conditions: SP, NPT, IR, CS, PC.

  • Budget-balanced, not profit maximizing.


Vickrey spanning tree edges as agents

Vickrey Spanning TreeEdges as agents

VST: Descending auction for edges as agents

Charges edges their “second price” to ensure

strategyproofness.


Vickrey spanning tree

Vickrey Spanning Tree

10

2

1

“15”

4

4

1

2

4

3

7


Vickrey spanning tree1

Vickrey Spanning Tree

10

2

1

“10”

4

4

1

2

4

3

7


Vickrey spanning tree2

Vickrey Spanning Tree

10

2

1

“10”

4

10

1

2

4

3

7


Vickrey spanning tree3

Vickrey Spanning Tree

4

2

10

7

2

4


Vst is strategyproof

VST is strategyproof

  • Edges in T have no incentive to bid higher

  • Edges outside T have no incentive to bid lower


Vst j v

VST + J-V

We have SP for edges and for nodes…

why not just combine the two?


Vst j v1

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


Vst j v2

VST + J-V

VST + J-V gives this tree:

1

є

10

1

1

1

є

є


Vst j v3

VST + J-V

But we could have gotten this (better) tree:

10

1+є

Need to be able to evaluate mechanisms!


Guarantees

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

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).


Guarantee

ß-guarantee

1

8

7

4

4

4

1

1

4

6

5

1


Competition

Competition

To obtain reasonable bounds, we need competition.

  • Edges – Competition across cuts

  • Nodes – Multiple users at each node


Edge competition

Є-Edge Competition

y

x

x < y < x(1 + є)


Node competition

Node Competition

No node has only one user.


Edge agents m1

Edge-agents (M1)

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

+5

4

-8

4

u

+7

Differences between GW and J-V


Edge agents m11

Edge-agents (M1)

2. Build a VST on the node set

4

2

7

2


Edge agents m12

Edge-agents (M1)

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

+3

-5

+1

+1

+7

+6

-10

+2


Edge agents m13

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.


Edge agents m14

Edge-agents (M1)

Edge-agents is a profit

guaranteeing mechanism, on any

є-edge competitive graph.


All agents m2

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.


Cancellable auctions

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

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 m21

All-agents (M2)

All-agents is a profit

guaranteeing mechanism, on any

є-edge competitive and node competitive

graph.


No competition

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

Conclusions

  • Need approximations to ensure computability

  • Need competition to ensure profitability

  • Solution is possible, but bounds are impractical.


Multicast networks profit maximization and strategyproofness

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