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Best Response Mechanisms

Noam Nisan, Michael Schapira, Gregory Valiant, and Aviv Zohar. Best Response Mechanisms. Motivation. Equilibrium is the basic object of study in game theory. Question : How is an equilibrium reached?

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Best Response Mechanisms

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  1. Noam Nisan, Michael Schapira, Gregory Valiant, and Aviv Zohar Best Response Mechanisms

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

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

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

  5. 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.

  6. The Setting

  7. The repeated best-response strategy: When a player’s turn arrives, it announces the best response to the latest announcements of others.

  8. Tie Breaking Rules

  9. Never Best Response (NBR) Strategies

  10. 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.

  11. Clear Outcomes

  12. 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. 3 3 2 1 4 2 1

  13. 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. S T 3 3 2 1 4 2 1

  14. 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 7 3.5 9 3.5 2 2 Ce =9

  15. Adjusting rate to fit bottleneck capacity: equivalent to best reply(with certain tie breaking rules) 1 3 1 2 1 3 3 4 2 2 4 2

  16. Results for Congestion Control Thm: The Congestion Control Game with routers that follow Fair-Queueing is NBR-Solvable with a clear outcome.

  17. Ce

  18. 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. Ce

  19. 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 TCP’s actual behavior in this setting can be seen as probing for the best-response.

  20. 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! 1 2 3 4 d

  21. 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.

  22. Thank You!

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