# Noam Nisan, Michael Schapira, Gregory Valiant, and Aviv Zohar.

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

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

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

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

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.

The Setting 

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

Tie Breaking Rules 

Never Best Response (NBR) Strategies  0-202 3-15-10 3-3 121

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.

Clear Outcomes 

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

 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

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

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

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

 CeCe

 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

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.

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

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.