Dynamics of Consensus in Finite Opinion Games

1. Decentralized Dynamics for Finite Opinion Games DiodatoFerraioli, LAMSADE Paul Goldberg, University of Liverpool Carmine Ventre, Teesside University

2. 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.

3. Should carbonara have cream? Y Y N … … … N Y N Y Y (Aside note: The right answer is NO!)

4. Previous work

5. 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?

6. 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)

7. 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

8. Our contribution

9. Finite opinion games 0 1 1 0 Our assumption: bi in [0,1], xi in {0,1}

10. 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

11. 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

12. 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

13. 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

14. 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

15. 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

16. 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

17. 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.

18. 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

19. 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]