On the Local and Global Price of Anarchy of Graphical Games

On the Local and Global Price of Anarchy of Graphical Games

Oren Ben-ZwiOren Ben-Zwi Amir RonenRonen

Oren Ben-ZwiOren Ben-Zwi Amir RonenRonen

Amir RonenAmir Ronen

Graphical Games

The ModelThe Model Each agent is identified with one vertexEach agent is identified with one vertex The utility of each agent depends solely The utility of each agent depends solely

on the actions of its neighborhoodon the actions of its neighborhood

GG give rise to many GG give rise to many structuralstructural properties of properties of gamesgames

Any game can be represented Any game can be represented

by a graphical gameby a graphical game

Price of Anarchy

WelfareWelfare We focus on positive utility gamesWe focus on positive utility games U(x) = U(x) = i ui(x) is called the welfare

Price of anarchyPrice of anarchyPoA(G) = min x Nash U(x)

max x U(x)

So, 0 PoA(G) 1

11 if i and all its neighbors play the sameif i and all its neighbors play the sameuui = =

otherwiseotherwise

Consensus Game

00 00

11

1111

11

00 00

Question what is the relation between the local and global price of anarchy?

Related Work

Structural properties (GT)Structural properties (GT)[GGJVY] symmetric graphical games under various assumptions [GGJVY] symmetric graphical games under various assumptions Existence results Existence results (pure Nash, symmetric Bayesian…)(pure Nash, symmetric Bayesian…) Connections between the degree of a player and its utilityConnections between the degree of a player and its utility

[…] Several, more specific games[…] Several, more specific games

Results are qualitativeResults are qualitative

Kearns et el.Kearns et el. Graphical generalizations of various models (Normal

form, Walrasian, Evolutionary) Structural phenomena in these models

Computational results Computational results PositivePositive [RP, P, …] [RP, P, …] HardnessHardness [GP, DGP, CD, …] [GP, DGP, CD, …]

Local global phenomenaLocal global phenomena [LPRS, P, …] Connections between local and global [LPRS, P, …] Connections between local and global

averagesaverages Property testing?Property testing?

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)

Covers

((,,)-cover )-cover A cover S = (S1, …, Sl) such that:1. i, LPoA(Si) 2. S is of width at most 3. The collection of interiors

S(-) = (S1(-), …, Sl

(-)) is also a coverS(-). S is of width 3

Thm If there exists an (,)-cover then PoA(G) /

Consider a set Si: Si can always obtain Uopt(Si

(-))! Thus, UxNash(Si) •Uopt(Si

(-))

Consider the sum i U(Si):

Each player is counted at most times

Q.E.D

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 Open: what about r > 1r > 1??

Reflections

Philosophical Philosophical If a decentralized system is composed of smaller, well behaved units, with small overlap betweenthem, then the whole system behaves well(e.g. departments within an organization)

ComputationalComputational It is much easier to analyze the smaller

units than the overall gameOperations on games Operations on games The theorem may give rise The theorem may give rise

to to

operations like compositions, replacements, etc.operations like compositions, replacements, etc.

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.

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

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

Averaging

Wastefulness Wastefulness in the basic theorem is the minimum LPoA(Si) is the maximum width

AveragingAveraging by the optimal welfares by the equilibrium utilities

(possibly less constructive)

Thm Thm If there exists an (*,*)-cover then PoA(G) */*

A Different Local Parameter

The The Nash ExpansionNash Expansion of a set of a set SS is is if for every set of if for every set of

actions of actions of N(S)N(S),, in equilibrium:in equilibrium:

ThmThm If S(-) is a disjoint cover such that The Nash expansion of every SSi is at least The average LPoA is *

Then PoA(G) *

We got rid of We got rid of When the game is relatively balanced,

becomes a becomes a combinatorialcombinatorial parameter parameter

Future Research

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

Structure Structure PoA of graphical games Other properties?