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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 player’s 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. 2,12,10,00,0 3,03,01,11,1

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Overview of Results We identify a small class of games for which: 1. Repeated best-response converges (quickly) from any initial point. 2. 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 player’s 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 0-202 3-15-10 3-3 121

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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. 2 3 2 1 1 4 3

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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. 2 3 2 1 1 4 3 ST

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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 9 2 7 C e =9 3.5 2

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

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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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CeCe

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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. CeCe

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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 TCP’s 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! 1 2 3 4 d

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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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