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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 - PowerPoint PPT Presentation
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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
1018
1230
20
6
other nodes, with utilities u i
The Multicast Network Problem
edges, with costs ce
4
14
1618
106
8
19
3
5
15
30
The Multicast Network Problem
1018
30
20
6
4
10
8
315
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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?We don’t know ‘s or ‘su i
?
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The Multicast Network Game
…so the agents give us bids
“5”
“17”
“18”“20”
“12”“8”
“18”
“19”
“4”
“6”
“16”
“35”
“8”“17”
“10”
“22”
“16”
“6”
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
22
1010
55
44
11
4422
77
5511t=0t=0
Jain-Vazirani
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1010
55
44
11
4422
77
5511t=1t=1
Jain-Vazirani
22
1010
55
44
11
4422
77
5511t=3t=3
Jain-Vazirani
22
1010
55
44
11
4422
77
5511t=4t=4
Jain-Vazirani
22
1010
55
44
11
4422
77
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
22
1010
44
33
11
4422
77
4411““15”15”
Vickrey Spanning Tree
22
44
33
11
4422
77
4411
““10”10”
1010
Vickrey Spanning Tree
22
44
33
11
101022
77
4411
““10”10”
1010
Vickrey Spanning Tree
22
44
1010
22
77
44
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?
1
1-є
єє
є
10
1-є
1-є
1-є
1+є
VST + J-VVST + J-V gives this tree:
1
єє
є
10
1
1
1
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
8
1
1
1
4
5
4
6
4 4
7
4
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
+5
-8
+7
4
4 u
Differences between GW and J-V
Edge-agents (M1)
2. Build a VST on the node set
22
22
77
44
Edge-agents (M1)
3. Prune out any unprofitable subtrees, and return T.
+3+3-5-5
+7+7
+1+1
+6+6
+2+2
+1+1
-10-10
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?