Multicast networks profit maximization and strategyproofness
Download
1 / 48

Multicast Networks Profit Maximization and Strategyproofness - PowerPoint PPT Presentation


  • 78 Views
  • Uploaded on

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

loader
I am the owner, or an agent authorized to act on behalf of the owner, of the copyrighted work described.
capcha
Download Presentation

PowerPoint Slideshow about 'Multicast Networks Profit Maximization and Strategyproofness' - carnig


An Image/Link below is provided (as is) to download presentation

Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.


- - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - -
Presentation Transcript
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 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.