Dynamics, robustness and fragility Private trust Public trust of - PowerPoint PPT Presentation

Dynamics of trust Dusko Pavlovic Introduction Dynamics, robustness and fragility Private trust Public trust of trust Conclusions Dusko Pavlovic Kestrel Institute and Oxford University FAST Malaga, October 2008

Dynamics of trust Outline Dusko Pavlovic Introduction Private trust Public trust Introduction Conclusions Private trust process Public trust process Conclusions

Dynamics of trust Outline Dusko Pavlovic Introduction Motivation Introduction Problem Private trust Motivation Public trust Conclusions Problem Private trust process Public trust process Conclusions

Dynamics of trust Adverse selection Dusko Pavlovic Introduction Motivation Problem Private trust Public trust Conclusions T RUST E-certified uncertified honest 94.6% 97.5% malicious 5.4% 2.5 % Table: Trustworthyness of T RUST E [Edelman 2007]

Dynamics of trust Adverse selection Dusko Pavlovic Introduction Motivation Problem Private trust Google Public trust Conclusions sponsored organic top 4.44% 2.73% top 3 5.33% 2.93 % top 10 5.89% 2.74 % top 50 5.93% 3.04 % Table: Malicious search engine placements [Edelman 2007]

Dynamics of trust Adverse selection Dusko Pavlovic Introduction Motivation Problem Private trust Yahoo! Public trust Conclusions sponsored organic top 6.35% 0.00% top 3 5.72% 0.35 % top 10 5.14% 1.47 % top 50 5.40% 1.55 % Table: Malicious search engine placements [Edelman 2007]

Dynamics of trust Adverse selection Dusko Pavlovic Introduction Motivation Problem Private trust Ask Public trust Conclusions sponsored organic top 7.99% 3.23% top 3 7.99% 3.24 % top 10 8.31% 2.94 % top 50 8.20% 3.12 % Table: Malicious search engine placements [Edelman 2007]

Dynamics of trust Adverse selection Dusko Pavlovic Introduction Motivation Problem Private trust Public trust Conclusions "Pillars of the society" Social hubs are are often corrupt.

Dynamics of trust Questions Dusko Pavlovic Introduction Motivation Problem Private trust Public trust Conclusions ◮ Why does adverse selection happen? ◮ Can it be eliminated? Limited? ◮ Can we hedge against it?

Dynamics of trust Outline Dusko Pavlovic Introduction Introduction Private trust Trust dynamics Trust distribution Interpretation Private trust process Public trust Conclusions Trust dynamics Trust distribution Interpretation Public trust process Conclusions

� � � � � Dynamics of trust Trust (rating) vectors Dusko Pavlovic Introduction Private trust trustors trustees Trust dynamics Trust distribution Interpretation � � Public trust � � � � � � � � � � � Conclusions � 4 � � � � � � � � � � � � 11 � � � • � � � � � � � � � � � 6 � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � 1 � � � � � � � � • � � � � � � � � � � � � � � � � � � � 2 � � � � � � � � � � � � � � � � τ 1 4 11 6 0 τ 2 0 1 0 2

� � � Dynamics of trust Private trust dynamics Dusko Pavlovic Introduction Private trust trustors trustees Trust dynamics Trust distribution Interpretation � � Public trust � � � � � � � � � � � Conclusions � 4 � � � � � � � � � � � � 11 � � � • � � � � � � � � 6 � � � � � � � � � � � � � � � � � � � � � � � � � � � τ ( t ) 4 11 6 0

� � � � � � � � � � � � � � � � � � � � � Dynamics of trust Private trust dynamics Dusko Pavlovic Introduction Private trust trustors trustees Trust dynamics Trust distribution Interpretation � Public trust Conclusions X • � i � � � � X ( t + 1 ) = i = C ( t ) τ i ( t ) Prob 1 − α (where C ( t ) = i ∈ J τ i ( t ) ) �

� � � � � � � � � � � � � � � � � � � � � � Dynamics of trust Private trust dynamics Dusko Pavlovic Introduction Private trust trustors trustees Trust dynamics Trust distribution Interpretation � Public trust Conclusions • � � X � new � � X ( t + 1 ) = new = Prob α

Dynamics of trust Private trust dynamics Dusko Pavlovic Introduction Trust updating process Private trust Trust dynamics Trust distribution Interpretation Public trust Conclusions  τ i ( t ) if i � X ( t + 1 )      0 if i = X , not satisfactory    τ i ( t + 1 ) =   1 if i = X , satisfactory, new       1 + τ i ( t ) if i = X , satisfactory, not new  

Dynamics of trust Trust distribution Dusko Pavlovic Introduction Task Private trust Trust dynamics Trust distribution Interpretation Estimate Public trust Conclusions w ℓ ( t ) = # { i ∈ J | τ i ( t ) = ℓ }

Dynamics of trust Trust distribution Dusko Pavlovic Introduction Private trust Trust dynamics Trust distribution Interpretation Public trust Conclusions � � w 1 ( t + 1 ) − w 1 ( t ) = J · Prob X ( t + 1 ) = i | i new · γ ⊥ � � − w 1 ( t ) · Prob X ( t + 1 ) = i | τ i ( t ) = 1 = J αγ ⊥ − w 1 ( t ) C ( t )

Dynamics of trust Trust distribution Dusko Pavlovic Introduction Private trust Trust dynamics Trust distribution Interpretation Public trust Conclusions � � w ℓ ( t + 1 ) − w ℓ ( t ) = w ℓ − 1 ( t ) · Prob X ( t + 1 ) = i | τ i ( t ) = ℓ − 1 · γ ℓ − 1 � � − w ℓ ( t ) · Prob X ( t + 1 ) = i | τ i ( t ) = ℓ = w ℓ − 1 ( t ) C ( t )( ℓ − 1 ) γ ℓ − 1 − w ℓ ( t ) C ( t ) ℓ

Dynamics of trust Trust distribution Dusko Pavlovic Introduction Private trust The system Trust dynamics Trust distribution Interpretation ∆ t w 1 ( t ) = J αγ ⊥ − C ( t ) w 1 ( t ) Public trust Conclusions ∆ t w ℓ ( t ) = w ℓ − 1 ( t ) C ( t )( ℓ − 1 ) γ ℓ − 1 − w ℓ ( t ) C ( t ) ℓ

Dynamics of trust Trust distribution Dusko Pavlovic Introduction Private trust . . . divided by J becomes Trust dynamics Trust distribution Interpretation ∆ t v 1 ( t ) = αγ ⊥ − C ( t ) v 1 ( t ) Public trust Conclusions ∆ t v ℓ ( t ) = v ℓ − 1 ( t ) C ( t )( ℓ − 1 ) γ ℓ − 1 − v ℓ ( t ) C ( t ) ℓ where v ℓ ( t ) = w ℓ ( t ) = Prob ( i ∈ J | τ i ( t ) = ℓ ) J form a stochastic process v : N −→ D R

Dynamics of trust Trust distribution Dusko Pavlovic Introduction Private trust . . . and since v : N −→ D R is a martingale, Trust dynamics Trust distribution it extends to v : R −→ D R and the system becomes Interpretation Public trust dv 1 αγ ⊥ − c Conclusions = t v 1 dt dv ℓ γ ℓ − 1 c ( ℓ − 1 ) v ℓ − 1 − c ℓ v ℓ = dt t where C ( t ) ≈ c 1 − α t , for c = 1 + αγ ⊥ (see Appendix)

Dynamics of trust Trust distribution Dusko Pavlovic The steady state of v : R −→ D R will be in the form Introduction v ℓ ( t ) = t · υ ℓ , where Private trust Trust dynamics Trust distribution = αγ ⊥ − c υ 1 υ 1 Interpretation = γ ℓ − 1 c ( ℓ − 1 ) υ ℓ − 1 − c ℓυ ℓ Public trust υ ℓ Conclusions

Dynamics of trust Trust distribution Dusko Pavlovic The steady state of v : R −→ D R will be in the form Introduction v ℓ ( t ) = t · υ ℓ , where Private trust Trust dynamics αγ ⊥ Trust distribution υ 1 = Interpretation c + 1 Public trust ( ℓ − 1 ) γ ℓ − 1 c Conclusions = υ ℓ υ ℓ − 1 ℓ c + 1

Dynamics of trust Trust distribution Dusko Pavlovic . . . which expands into Introduction γ 1 c αγ ⊥ Private trust = υ 2 c + 1 · 2 c + 1 Trust dynamics Trust distribution 2 c + 1 · 2 γ 2 c γ 1 c αγ ⊥ Interpretation υ 3 = c + 1 · Public trust 3 c + 1 Conclusions . . .  n − 1  ( n − 1 )!  �  c n − 1 ·  =   υ n αγ ⊥ γ ℓ     � n   k = 1 ( kc + 1 )    ℓ = 1 αγ ⊥ G n − 1 ( n − 1 )! = · c � n � k + 1 � k = 1 c � 1 + 1 � Γ( n )Γ αγ ⊥ G n − 1 c = · c � n + 1 + 1 � Γ c αγ ⊥ G n − 1 � n , 1 + 1 � = · B c c

Dynamics of trust Trust distribution Dusko Pavlovic The solution Introduction αγ ⊥ = υ 1 Private trust c + 1 Trust dynamics Trust distribution αγ ⊥ G n − 1 � n , 1 + 1 � Interpretation = B υ n Public trust c c Conclusions αγ ⊥ G n →∞ n − ( 1 + 1 c ) −→ c where ∞ � G = γ ℓ > 0 follows from ℓ = 1 1 e s ℓ ≤ γ ℓ ≤ 1 for some ∞ � s ℓ < ∞ ℓ = 1

Dynamics of trust Trust distribution Dusko Pavlovic Introduction Private trust Trust dynamics Theorem Trust distribution Interpretation The described process of trust building leads, in the long Public trust Conclusions run, to the power law distribution of the number of trustees with the trust rating n αγ ⊥ GJ n − ( 1 + 1 c ) w n ≈ c