Best Response Mechanisms

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Best Response Mechanisms

• Noam Nisan, Michael Schapira, Gregory Valiant,
  and Aviv Zohar

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Motivation

• Equilibrium is the basic object of study in game
  theory.
• Question How is an equilibrium reached?
• In a truly satisfactory answer each players rule
  of behavior is simple and locally rational
• repeated best-response
• repeated better-response
• regret-minimization

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Motivation

• Repeated best-response is often employed in
  practice
• e.g., Internet routing
• We ask When is such locally-rational behavior
  really rational?

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• Repeated best-response is not always best.
• the game is solvable through elimination of
  dominated strategies.

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Overview of Results

• We identify a small class of games for which
• Repeated best-response converges (quickly) from
  any initial point.
• It is a rational choice in the long run (an
  equilibrium).
• While small, this class covers several important
  examples
• Internet Routing, Cost Sharing, Stable Roommates,
  Congestion Control.

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The Setting

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• The repeated best-response strategy
• When a players turn arrives, it announces the
  best response to the latest announcements of
  others.

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Tie Breaking Rules

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Never Best Response (NBR) Strategies

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NBR-Solvability

• Def A game G is NBR-solvable (under some
  tie-breaking rule) if there exists a sequence of
  eliminations of NBR strategies from the game that
  leaves each player with only a single strategy.
• There must be such a sequence for every type
  configuration of the players.

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Clear Outcomes

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Example Congestion Control

• A crude model of TCP congestion control.
  Godfrey, Schapira, Zohar, Shenker SIGMETRICS
  2010
• A protocol responsible for scaling back
  transmission rate in cases of congestion.
• The network is represented by a graph with
  capacities on the edges.

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• Each player is a pair of source target nodes,
  connected by a simple path, and has some maximal
  rate of transmission.
• Actions of players selecting transmission
  rate(up to limit).
• Utility amount of flow that reaches
  destination.

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• Flow is handled as if routers use Fair Queuing
• Capacity on each link is equally divided between
  players that use the link.
• Unused capacity by some player is divided equally
  among others

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• Adjusting rate to fit bottleneck capacity
  equivalent to best reply(with certain tie
  breaking rules)

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Results for Congestion Control

• Thm The Congestion Control Game with routers
  that follow Fair-Queueing is NBR-Solvable with a
  clear outcome.

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• Eliminate all transmission rates below ?e for
  them.
• If they all transmit at least ?e, none will
  manage to get more through. Eliminate all rates
  above ?e.
• Repeat with the residual graph and remaining
  players.

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Results for Congestion Control

• Thm The Congestion Control Game with routers
  that follow Fair-Queueing is NBR-Solvable with a
  clear outcome.
• Corollaries
• Best-response is incentive compatible
• Converges fast regardless of topology
• TCPs actual behavior in this setting can be seen
  as probing for the best-response.

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Other Games

• Matching
• Uncorrelated markets, interns and hospitals
• Cost-sharing games
• BGP interdomain routing in the internet.
• See the paper for more details and references!

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• Open Questions
• Explore other dynamics (e.g., regret
  minimization) and other equilibria (e.g., mixed
  Nash, correlated).
• Find an exact characterization of games where
  repeated best-response is rational.

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Thank You!