Decentralized Dynamics for Finite Opinion Games

1 / 19

# Decentralized Dynamics for Finite Opinion Games - PowerPoint PPT Presentation

Decentralized Dynamics for Finite Opinion Games. Diodato Ferraioli , LAMSADE Paul Goldberg, University of Liverpool Carmine Ventre , Teesside University. Opinion Formation in SAGT12 social network*. …. …. ….

I am the owner, or an agent authorized to act on behalf of the owner, of the copyrighted work described.

## PowerPoint Slideshow about 'Decentralized Dynamics for Finite Opinion Games' - halona

An Image/Link below is provided (as is) to download presentation

Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.

- - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - -
Presentation Transcript

### Decentralized Dynamics for Finite Opinion Games

Paul Goldberg, University of Liverpool

Carmine 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.

Should carbonara have cream?

Y

Y

N

N

Y

N

Y

Y

(Aside note: The right answer is NO!)

Repeated averaging: De Groot’s model

1

0

.46

.5

.3

.45

.36

1

0

.23

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 ]

0

1

.5

.46

.3

.45

1

.5

0

.23

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])

bi

1

0

xi

xj

Finite opinion games

0

1

1

0

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

xi

xj

xi ≠ xj

0

1

.25

.5

.75

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

noise: probability distribution over strategies

player’s strategy set

player’s strategy set

Logitdynamics [Blume, GEB93], [Auletta, Ferraioli, Pasquale, (Penna) & Persiano , 2010-ongoing]
• At each time step, from profile x
• Select a player uniformly at random, call him i
• Update his strategy to siwith 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
Upper bound for every β: (1+β) poly(n) eβΘ(CW(G))

Upper bound for “small” β: O(n log n)

Lower bound for everyβ: (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

Results

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

Cutwidthof G is the minimum cut(o) overall the possible orderings o

2

1

2

cut(o)=3

3

2

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

3

4

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 “small” β: some details
Upper bound for every β: intuition

φ

• 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 forβ “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))

1

profiles

0

Lower bound: intuition

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

(1,1, …,0)

(1,0, …,0)

(0,1, …,1)

(0,1, …,0)

(1,0, …,1)

(0,0, …,0)

(1,1, …,1)

(0,0, …,1)

(1, …,1,0)

(0, …,1,1)

R = border of T

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))

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

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

Lower bound for specific social networks
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]