Power and Stability in Network Connectivity Games

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Power and Stability in Network Connectivity Games

• Yoram Bachrach
• Jeffrey S. Rosenschein
• Ely Porat

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Outline

• Network Reliability
• Motivation
• Reliability hotspots
• Distributing profits
• Game Theoretic Solution Concepts
• Power indices
• Stability the Core
• Power, stability and network reliability
• Vertex Connectivity Games (CGs)
• The Banzhaf power index in CGs
• P-Completeness
• Polynomial algorithm for trees
• The core in CGs
• Related work
• Conclusions and future directions

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Network Reliability

• Servers connected to each other via network links
• Within a time unit, links have a certain
  probability of failure
• Typically networks have some redundancy
• Classical network reliability works
• Each link has a survival probability, of
  remaining in the graph
• Compute the probability of obtaining the goal in
  the surviving graph
• Various network goals
• Source target connectivity (STC-P), full
  connectivity (FC-P), full connectivity given a
  backbone infrastructure, connectivity of specific
  servers, allowing a certain bandwidth between the
  source and target,
• Complexity
• Valiant STC-P is P-hard
• Provan and Ball FC-P is P hard

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Network Reliability Hotspots

• Servers and links may fail
• Maintenance may allow improving the survival
  probability of a link
• Or even eliminate failures altogether
• Maintenance resources are limited
• Which link / server should be allocated these
  resources first?
• Remove single points of failure
• What about more complex failure types (pairs,
  triplets, k-subsets)
• Maximize probability of obtaining goal
• Power indices allow finding such reliability
  hotspots

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Networks and Self-Interested Companies

• Servers and links are owned by self interested
  companies (agents)
• Companies care about their own profit
• Promise a certain reward for achieving the goal
• Some companies may form a coalition to achieve
  the goal
• The companies must decide how the rewards should
  be distributed among them
• The Core would constitute a stable allocation

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TU Coalitional Games

• Agents
• Coalition
• Characteristic function
• Simple coalitional games
• Coalitions either win or lose
• Increasing games gt
• The more agents you have, the more money you make
• Super-additive
• It is always worthwhile for coalitions to merge
• The Grand Coalition would form

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Payoffs

• Define how the total utility is distributed
• A payoff vector such that
• Individual rationality
• Otherwise, an agent can do better alone
• The payoff of a coalition C is
• A coalition C is blocking if p(C) lt v(C)

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The Core (Stability)

• The set of all payment vectors that are not
  blocked by any coalition
• For any coalition C, p(C) v(C)
• No coalition has an incentive to split off from
  the grand coalition
• Proposed by Gillies (1953) and von Neumann
  Morgenstein (1947)

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Simple Games, Cores and Veto Agents

• The core is a difficult requirement for simple
  games
• Give something to a non veto agent
• Some coalition C wins without him, and thus is
  blocking
• The core is empty if there are no veto agents
• Give all the reward to the veto agents
• Suppose some coalition C blocks
• C must contain all the veto agents, so it gets
  all the reward
• C gets all the reward, which is exactly what it
  can do by itself
• Thus C cannot block
• The core of a simple game is the set of all
  payoff vectors that give everything to the veto
  agents an nothing to non-veto agents

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The Banzhaf Index Power

• Measures real power in weighted voting systems
• Suitable to any simple coalitional game
• Counts the number of coalition when an agent is
  pivotal out of all wining coalitions containing
  that agent
• Fulfils the dummy axiom and symmetry axiom
• Can be normalized to fulfill the efficiency axiom
  as well
• Doesnt fulfill the additivity axiom

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Example Network (1)
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Example Network (2)
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Vertex Connectivity Games

• Games played on a graph GltV,Egt representing a
  communication network
• Focus on server failures
• Rather than network link failures
• Servers are vertices, network links are edges
• Different types of servers
• Standard vertices the network servers
• Backbone vertices servers that are guaranteed
  to never fail
• Primary vertices important servers required to
  be connected
• Simple characteristic function
• Agents are the standard vertices
• A coalition wins if its servers as well as the
  backbone allow connectivity between any two
  primary servers

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Example CG
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Example CG
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Example CG (Winning)
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Example CG
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Example CG
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Example CG (Winning)
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Example CG
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Example CG (Losing)
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Power and Network Reliability

• Suppose a failure for every unmaintained server
  is equally probable
• We have limited maintenance resources
• Which servers should be maintained first?
• Powerful servers are more critical
• Server failure is more likely to cause a failure
  in maintaining the required connectivity
• Assuming failure probability is 0.5, maintained
  severs never fail
• Suppose we can only maintain one server
• Each subset of servers is equally likely to
  remain
• Probability of connectivity give that we maintain
  a server is its Banzhaf power index

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CG Banzhaf

• Computing the Banzhaf index of a server in a
  vertex connectivity game is P-complete
• Reduction from SET-COVER
• Count the number of different set covers
• Request connectivity of b and the tis
• No backbone vertices
• Query regarding as Banzhaf index
• Vertex a is critical for all set covers, and only
  them

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CG Banzhaf

• Polynomial algorithm for the Banzhaf index in
  trees
• A standard vertex in a simple path between two
  primaries is veto agent
• Coalitions containing all standard vertices
  between two primaries connects them
• Polynomial to test if a vertex is on a path
  between two primaries
• If not it is a dummy and has a power index of 0
• If so, it is critical in all winning coalitions
  that contain it, and has a Banzhaf index of 1

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Stability in CGs the Core

• Given a certain reward for maintaining
  connectivity in a CG domain, how should we divide
  it among the companies owning the servers?
• The Core constitutes a stable allocation
• A distribution not in the core would break down
  the grand coalition
• Companies would split away, and form smaller
  coalitions
• InternetA, InternetB, InternetC
• Can we compute the core in a CG?

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CG Core

• CGs are simple games
• The core can be represented as a list of veto
  agents
• CGs are increasing coalitional games
• Adding servers only helps you connect more
  primary vertices
• An agent is veto if and only if the coalition of
  all the agents except that aget, , is losing
• Veto agents can be found in polynomial time
• Remove tested vertex and all the connected edges
• Test connectivity between all primary servers
• The core of CGs can be computed in polynomial
  time

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Conclusion Future Directions

• Suggested a game theoretic model for network
  reliability, based on server failures
• Computing the Banzhaf index in CGs is P-complete
• Polynomial algorithm for Banzhaf in tree CGs
• Polynomial algorithm for computing the core of
  CGs
• Possible future work
• Other power indices
• Approximation for general CGs
• Power indices in other network domains