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On the Local and Global Price of Anarchy of Graphical Games Oren Ben-Zwi Ronen Oren Ben-ZwiAmir Ronen Amir Ronen

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Graphical Games The Model Each agent is identified with one vertex Each agent is identified with one vertex The utility of each agent depends solely on the actions of its neighborhood The utility of each agent depends solely on the actions of its neighborhood GG give rise to many structural properties of games Any game can be represented by a graphical game

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Price of Anarchy Welfare We focus on positive utility games We focus on positive utility games U(x) = U(x) = i u i (x) is called the welfare Price of anarchy PoA(G) = min x Nash U(x) max x U(x) So, 0 PoA(G) 1

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1 if i and all its neighbors play the same 1 if i and all its neighbors play the same u = u i = otherwise otherwise Consensus Game 0 0 1 1 1 100 Question what is the relation between the local and global price of anarchy?

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Related Work Structural properties (GT) [GGJVY] symmetric graphical games under various assumptions Existence results (pure Nash, symmetric Bayesian…) Existence results (pure Nash, symmetric Bayesian…) Connections between the degree of a player and its utility Connections between the degree of a player and its utility […] Several, more specific games Results are qualitative Kearns et el. Graphical generalizations of various models (Normal form, Walrasian, Evolutionary) Structural phenomena in these models

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Computational results Positive [RP, P, …] Positive [RP, P, …] Hardness [GP, DGP, CD, …] Hardness [GP, DGP, CD, …] Local global phenomena [LPRS, P, …] Connections between local and global averages [LPRS, P, …] Connections between local and global averages Property testing? Property testing?

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Locality in Graphical Games Consider a subset S of agents: Every choice of actions of N(S) induces a sub-game on S! A global NE induces a local equilibrium on S We say that LPoA(S) if for every choice of actions of N(S), the PoA of the induced game is at Least (i.e. S responds well to its environment)

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Covers (, )-cover A cover S = (S 1, …, S l ) such that: 1. 1. i, LPoA(S i ) 2. 2. S is of width at most 3. 3. The collection of interiors S (-) = (S 1 (-), …, S l (-) ) is also a cover S (-). S is of width 3 Thm If there exists an (, )-cover then PoA(G) /

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Consider a set S i : S i can always obtain U opt (S i (-) )! Thus, U x Nash (S i ) U opt (S i (-) ) Consider the sum i U(S i ): Each player is counted at most times Q.E.D

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Corollary Consider a game with a maximal degree d. Suppose that every ball S of radius r = 1 satisfies LPoA(S), then GPoA(G) / (d + 1) Open: what about r > 1?

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Reflections Philosophical Philosophical If a decentralized system is composed of smaller, well behaved units, with small overlap between them, then the whole system behaves well (e.g. departments within an organization) Computational Computational It is much easier to analyze the smaller units than the overall game Operations on games The theorem may give rise to operations like compositions, replacements, etc.

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The game The game is played e.g. on a d-regular torus The utilities are given by: A Biased Consensus Game Each agent prefers to play 1 unless all its neighbors play 0. I.e, there are only two pure Nash equilibria.

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Claim The local PoA of any ball of radius 1 is ~1/2 If at least one neighbor plays 1, everybody plays 1 in equilibrium and this is optimal If all neighbors play 0, everybody plays like the center. Thus, the optimal utility is 2. The bad equilibrium yields 1 + 1/d

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Tightness The theorem gives PoA(G) 1/[2(d+1)] The actual PoA is 1/d It is possible to make the gap arbitrarily small Thus, in general the bound is tight

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Averaging Wastefulness Wastefulness in the basic theorem is the minimum LPoA(S i ) is the maximum width Averaging by the optimal welfares by the equilibrium utilities (possibly less constructive) Thm Thm If there exists an ( *, * )- cover then PoA(G) * / *

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A Different Local Parameter The Nash Expansion of a set S is if for every set of actions of N(S), in equilibrium: Thm Thm If S (-) is a disjoint cover such that S The Nash expansion of every S i is at least The average LPoA is * Then PoA(G) * We got rid of We got rid of becomes a combinatorial parameter When the game is relatively balanced, becomes a combinatorial parameter

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Future Research Local Global Algorithms for finding good covers Dynamic behavior of locally good games Other properties (e.g. PoS) Property testing like phenomena? Structure PoA of graphical games Other properties?

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