Decentralized Dynamics for Finite Opinion Games

Diodato Ferraioli, LAMSADEPaul Goldberg, University of LiverpoolCarmine Ventre, Teesside University

Opinion Formation in SAGT12 social network*

* All characters appearing in this talk are fictitious. Any resemblance to real persons, living or dead, is purely coincidental.

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Should carbonara have cream?

Y N

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Y

Y

Y NYN

(Aside note: The right answer is NO!)

PREVIOUS WORK

Repeated averaging: De Groot’s model

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…1

.50 1

.3

0

.45

.46

.23

.36

Econ question: Under what conditions repeated averaging leads to consensus?

Friedkin and Johnsen’s variation of De Groot’s model [Bindel, Kleinberg & Oren, FOCS 2011 ]

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1

.5

.3

0

.45

.46

.23

.5

0 1

Note: It is (0.23+0.3+0.46+1)/4 ≈ 0.5 (≠ 0.36)

Cost of disagreement [BKO11]

• “Selfish world viewpoint”: Consensus not reached because people will not compromise when this diminishes their utility

• To quantify the cost of absence of consensus they study the PoA of this game, where players have a continuum of actions available (i.e., numbers in [0,1])

0 1

bi

xj

xi

OUR CONTRIBUTION

Finite opinion games

0 1

0 1

Our assumption: bi in [0,1], xi in {0,1}

Convergence rate of best-response dynamics

• Potential game with a polynomial potential function

• Convergence of best-response dynamics to pure Nash equilibria is polynomial: at each step the potential decreases by a constant

0 1.5.25 .75

xj

xi

xi ≠ xj

Noisy best-responses

• Utilities hard to determine exactly in real life!– … or otherwise, elections would be less uncertain

• Introducing noise

no noise: selection of strategy which maximizes the utility

player’s strategy set

noise: probability distribution over strategies

player’s strategy set

Logit dynamics [Blume, GEB93], [Auletta, Ferraioli, Pasquale, (Penna) & Persiano , 2010-ongoing]

• At each time step, from profile x1. Select a player uniformly at random, call him i2. Update his strategy to si with probability

proportional to

• β is the “rationality level” (inverse of the noise)– β = 0: strategy selected u.a.r. (no rationality)– β ∞: best response selected (full rationality)– β > 0: strategies promising higher utility have higher

chance of being used

Convergence of logit dynamics

• Nash equilibria are not the right solution concept for Logit dynamics

• Logit dynamics defines an ergodic Markov chain– unique stationary distribution exists

• Better than (P)NE!

– this distribution is the fixed point of the dynamics (logit equilibrium)

• How fast do we converge to the logit equilibrium as a function of β?– The answer requires to bound the mixing time of the

Markov chain defined by logit dynamics

Results

Given an ordering o of the vertices of a graph G, cut(o) is defined as:

Cutwidth of G is the minimum cut(o) overall the possible orderings o

1 2

3 4

2

2 3 cut(o)=3

CW(G) = 2 (ordering 3,4,1,2)

• Upper bound for every β: (1+β) poly(n) eβΘ(CW(G))

• Upper bound for “small” β: O(n log n) • Lower bound for every β: (n eβ(CW(G)+f(beliefs)))/|R|

– Technicalities: • certain subset of profiles R, whose size is important to understand how

close the bounds are• f function of players’ beliefs, annulled for dubious players (bi=1/2, for

all i)

– “Tightness” for dubious players:• big β (|R| becomes insignificant)• Special social network graphs G for which we can relate |R| and CW(G)

– complete bipartite graphs– cliques

Upper bound for “small” β: some details

• Hypothesis:– Social network graph G connected– More than 2 players– β ≤ 1/max degree of G

• Proof technique: – Coupling of probability distributions

• Result determines a border value for β, for which logit dynamics “looks like” a random walk on an hypercube

Upper bound for every β: intuition

• Stationary distribution will visit both 0 and 1 • The chain will need to get from 0 to 1

– the harder (ie, more time needed) the higher the potential will get in this path (especially for β “big”)

• No matter the order in which players will switch from 0 to 1, at some point in this path we will have CW(G) “discording” edges in G

• The potential change for a “discording” edge is constant• Convergence takes time proportional to eβΘ(CW(G))

profiles

φ

0 1

Lower bound: intuition

(0,1, …,0)

(0,0, …,1)

(0, …,1,1)

(1,0, …,1)

(1, …,1,0)

… ……

… …(0,0, …,0)

(1,1, …,0)

(0,1, …,1)

(1,1, …,1)

(1,0, …,0)

T= profiles with potential at most CW(G)+f(b)

Bottleneck ratio of this set of profiles (measuring how hard it is for the chain to leave it) is

at most |R| e-β(CW(G)+f(b))

R = border of T

Mixing time of the chain at least the inverse of the b.r.

Lower bound for specific social networks

• For complete bipartite graphs and cliques, we express the cutwidth as a function of number of players

• We bound the size of R• We can then relate |R| and CW(G) and obtain

a lower bound which shows that the factor eβCW(G) in the upper bound is necessary

Conclusions & open problems

• We consider a class of finite games motivated by sociology, psychology and economics

• We prove convergence rate bounds for best-response dynamics and logit dynamics

• Open questions:– Close the gap on the mixing time for all β/network

topologies– Consider weighted graphs?– More than two strategies?– Metastable distributions?

• [Auletta, Ferraioli, Pasquale & Persiano, SODA12]