Constant Price of Anarchy in Network Creation Games via Public Service Advertising Presented by Sepehr Assadi Based on a paper by Erik D. Demaine and Morteza Zadimoghaddam [Demaine et. al’] Sharif University Of Technology 1
Dec 22, 2015
Sharif University Of Technology 1
Constant Price of Anarchy in Network Creation Games via Public Service Advertising
Presented bySepehr Assadi
Based on a paper byErik D. Demaine and Morteza Zadimoghaddam
[Demaine et. al’]
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Intro. To Game Theoretic Concepts
• Game: – A Set of selfish agents– A strategy profile for each agent• Showing possible actions
– A cost function for each agent• Showing cost of each agent in the game
• Dynamic:– Each agents try to minimize her cost
selfishly!
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Game Theoretic Concepts (cont.)
• Stable State:– A state which no one wants to change!– Has different mathematical notions
• Nash equilibrium ( pure / approximate / strong …)
• Pair wise stability• …
• Social cost:– A cost defined for the game in a state
• Usually the sum of all the agents cost.
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Nash equilibrium
• Definition:– State of the game where no agent can
change its strategy and gain lower cost• Somehow everyone are satisfied of their
current state!
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Price of Stability & Anarchy
• Optimum social cost:– The minimum social cost in a state
designed by an authority• Maybe not a stable state!
• Price of Stability:– The ratio of minimum social cost in a
stable state over optimum social cost
• Price of Anarchy:– The ratio of maximum social cost in a
stable state over optimum social cost
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Network Creation Games
• Network Creation Games:– Agents:
• nodes of a graph
– Strategy profile: • creating edges to other nodes and accepting/rejecting the edge connected to them.
– Cost function: • cost of establishing network + cost of using
network
– Social Cost:• Usually sum/max of the cost of the agents
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(n,k)-uniform bounded budget connection game
• Definition:– n agent creating the nodes of graph– Each agent strategy is subset of at most k other
nodes• Each node create edges to other and accepts/rejects the edge connected to it
– Agent cost:• Sum of its distances to all other nodes in the underlying
graph
– Social cost:• Sum of the cost of all agents in a state
– Stable State• Nash equilibrium
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Known Results
• Price of Stability:
• Price of Anarchy:
• What is the meaning?– Good Nash equilibrium vs Bad Nash
equilibrium!
• Natural question:– How can we shift the decision of the agents
towards the better Nash equilibrium?
Adver
tise!
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Public Service Advertising (PSA)
• Introduced in SODA 2009 [Balcan et.al’]• Previously studied on:– Fair cost sharing, selfish routing, scheduling
games
• Idea:– Use an advertising campaign – Introduce better strategies to agents
• Hopefully they will accept!
– Agents may share only a fraction of their budgets in the campaign • Even a small fraction can goes a long way!
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Public Service Adverting (cont.)
• Formal Definition:– Each agent is advertised a strategy
• Strategies might differs with each other!
– Each agent accept the advertised strategy with probability of α independently.• These nodes are called Receptive Nodes.
– Each receptive node is willing to share β fraction of its budget. • Therefore in this game there are βk edge for each
receptive node.
– The final state again should be Stable state.• In this game Nash equilibrium.
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Motivation
• Social Networks!• Political Campaigns!• Peer-to-peer Networks!
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Result of this work
• Result:–With proper method of advertising the
price of anarchy would be O(1/α) – Even without knowing α and β
beforehand• Not mentioned in this lecture
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Overview of method
1. Partition nodes to different set – Each set is advertised uniquely
2. Create a core graph using receptive nodes– Examine the cost in this graph
3. Wait for the others to reach a stable state– Examine the cost of this newly created
graph
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Method of Advertising
• Let:– K” = αβ /c log(n) • for sufficiently large c, i.e. c ≥ 5.
• Partitioning:– Partition nodes to l ≤ logk (n) sets S1, S2,
…,Sl
• Such that: – |S1| = βk / 2
– |Si+1| / |Si| = k”
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Method of Advertising (Cont.)
• Advertised strategy:– S1:
• Connect βk / 2 -1 edges to other nodes in S1.
– Si for i > 1:• Choose randomly c log(n)/2α from Si-1
– c log(n)/2α ≤ βk / 2
– Si i ≥ 1:• Accepts βk / 2 edges from Si+1
• Notes:– If a node is non-receptive node we assume that it deletes
the edge!– Even if a node is receptive it might be overwhelmed!
• Gets more than βk edge
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Lemma: Hierarchical Tree
• Lemma: – Above strategy creates a hierarchical
tree shaped subgraph– Diameter of the subgraph is 2logk” (n)
– Covers all the receptive nodes with high probability• High probability: 1 – 1/nc for some c ≥ 1
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Lemma: Hierarchical Tree (Example)
• Sketch:
• Diameter!– Shown in green!
S1
S3
S2
Sl
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Lemma: Hierarchical Tree (Proof)
• Proof:– Every receptive node is connected to a
receptive node in higher level– For each node in Si : • expected number of receptive nodes it
connects is c log(n)/2α * α = c log(n) / 2• Chernouff bound: At least log (n) of them are
receptive!
–What if they delete this edge? • Happens if it is overwhelmed!
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Lemma: Hierarchical Tree (Proof)
• Proof:– Probability of a receptive node being
overwhelmed:– Expected number of edges from higher
level:• α|Si| (c log(n)/2α ) / |Si-1|
• |Si| / |Si-1| = k” = αβk / (c log(n)/2α )
• = αβk / 2.
–Markov inequality: • P(Overwhelmed) = P(edge > βk ) < α/2 < 1/2.
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Lemma: Hierarchical Tree (Proof)
• Proof:– Probability of a receptive node being
overwhelmed:– Expected number of edges from higher
level:• α|Si| (c log(n)/2α ) / |Si-1|
• |Si| / |Si-1| = k” = αβk / (c log(n)/2α )
• = αβk / 2.
–Markov inequality: • P(Overwhelmed) = P(edge > βk ) < α/2 < 1/2.
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Lemma: Hierarchical Tree (Proof)
• Proof:– Probability of a receptive node being
overwhelmed:– Expected number of edges from higher
level:• α|Si| (c log(n)/2α ) / |Si-1|
• |Si| / |Si-1| = k” = αβk / (c log(n)/2α )
• = αβk / 2.
–Markov inequality: • P(Overwhelmed) = P(edge > βk ) < α/2 < 1/2.
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Lemma: Hierarchical Tree (Proof)
• Proof:– Probability of a receptive node being
overwhelmed:– Expected number of edges from higher
level:• α|Si| (c log(n)/2α ) / |Si-1|
• |Si| / |Si-1| = k” = αβk / (c log(n)/2α )
• = αβk / 2.
–Markov inequality: • P(Overwhelmed) = P(edge > βk ) < α/2 < 1/2.
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Lemma: Hierarchical Tree (Proof)
• Proof:– Probability of a receptive node being
overwhelmed:– Expected number of edges from higher
level:• α|Si| (c log(n)/2α ) / |Si-1|
• |Si| / |Si-1| = k” = αβk / (c log(n)/2α )
• = αβk / 2.
–Markov inequality: • P(Overwhelmed) = P(edge > βk ) < α/2 < 1/2.
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Lemma: Hierarchical Tree (Proof)
• Proof:– Probability of a node not connected to any
receptive and not overwhelmed node• X: number of receptive and not overwhelmed
nodes• xi: i-th receptive node it connects is not
overwhelmed
• X = x1 + x2 + x3 + … + x log(n) & E[X] >= log(n)/2
• P(X <= 1) ? • Chernouff bound ?
– They are not independent!
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Lemma: Hierarchical Tree (Proof)
• Proof:– What should we do?
• xi are negatively correlated
• Negative correlation: – E[XY] – E[X]E[Y] < 0– If one happens, the probability of the other became lower
• Generalized Chernouff bound: holds for negative correlated xi (and some more families too)
– Thus, every receptive node is connected to at least one another receptive node in lower level• And it is not overwhelmed!
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Lemma: Hierarchical Tree (Proof)
• Figure:• Requesting initial edges!
A node in Si
Set Si-1
(c log(n) / 2α ) random node!
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Lemma: Hierarchical Tree (Proof)
• Figure:• Expected of receptive nodes
A node in Si
Set Si-1
(c log(n) / 2α ) * α = c log(n) / 2
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Lemma: Hierarchical Tree (Proof)
• Figure:• High probable number of receptive nodes
A node in Si
Set Si-1
Chernouff Bound:log(n)
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Lemma: Hierarchical Tree (Proof)
• Figure:• Some are overwhelmed
A node in Si
Set Si-1
Probability of overwhelming 1/2
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Lemma: Hierarchical Tree (Proof)
• Figure:• One remains at least!
A node in Si
Set Si-1
Negative CorrelationChernouff BoundOne node!
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Lemma: Hierarchical Tree (Proof)
• Proof:–We have:• Each receptive node is connected to at least
one on the top level• All the nodes in first level are connected to
each other• Therefore, all receptive nodes are in the
subgraph with diameter 2*l (l is number of Si)
• End of proof ☐
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Handling Non-receptive nodes
• Now, what about others?• Lemma:– In any stable graph maximum distance of
other nodes from any receptive node v, is O(l + logk(n)/α)
– Diameter of stable graph now is at most O(logk”(n)/α)
• Proof: – Proof is completely combinatorial. (not of our
interest here)
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Final Result
• Theorem:– Price of anarchy is at most
O(logk”(n)/α logk(n)) = O(logk”(n)/α)
• Proof:
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Final Result
• Theorem:– Price of anarchy is at most
O(logk”(n)/α logk(n)) = O(logk”(n)/α)
• Proof:– Previous lemma,
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Final Result
• Theorem:– Price of anarchy is at most
O(logk”(n)/α logk(n)) = O(logk”(n)/α)
• Proof:– Previous lemma,– Average distant in the optimal graph is
Ω(logk(n)) [Laotaris et.al’]
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Final Result
• Theorem:– Price of anarchy is at most
O(logk”(n)/α logk(n)) = O(logk”(n)/α)
• Proof:– Previous lemma,– Average distant in the optimal graph is
Ω(logk(n)) [Laotaris et.al’]
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Final Result (Cont.)
• Corollary:– If k is Ω(log1+ε(n)), the price of anarchy is
O(1/αε)
• Finally:– Public service advertising leads to
O(1/α) Price of Anarchy!
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References
• [Demaine et. al’]:– “Constant Price of Anarchy in Network Creation Games via Public Service
Advertising”, Erik D. Demaine and Morteza Zadimoghaddam . Algorithms
and Models for the Web-Graph.
• [Balcan et.al’]:– “Improved equilibria via public service advertising”, Balcan, Blum,
Mansour. In Proceedings of the 20th Annual ACM-SIAM Symposium on Discrete Algorithms.
• [Laotaris et.al’]:– “Bounded budget connection (BBC) games or how to make friends and
influence people, on a budget”, Laoutaris, Poplawski, Rajamaran, Sundaram, Teng. In Proceedings of the 27th ACM Symposium on Principles of Distributed Computing.