a study on Robust Mechanisms for Social Coordination

Transcription

1 a study on Robust Mechanisms for Social Coordination Philip N. Brown and Jason R. Marden University of California, Santa Barbara CDC Workshop: Distributed Autonomy and Human-Machine Networks December 14, 2015

2 Multiagent coordination Central Goal Derive efficient system-wide behavior through the design of admissible control algorithms Smart Grid Sensor coverage Traffic network socio-technical systems engineering systems social systems

3 Multiagent coordination Central Goal Derive efficient system-wide behavior through the design of admissible control algorithms Traffic network efficient: desirable allocation for any network demands admissible: local/aggregate incentive mechanism challenge: uncontrollable entities, unknown sensitivity social systems

4 Multiagent coordination Central Goal Derive efficient system-wide behavior through the design of admissible control algorithms Are there robust mechanisms for coordinating social behavior? Traffic network (guarantees irrespective of populations demands/sensitivities) social systems

5 Case study: Pigou s network Motivation: - Uninfluenced systems ofter exhibit poor system behavior c(x) =x unit flow of traffic S D c(x) =1 optimal outcome vs. self-interested outcome

6 Case study: Pigou s network Motivation: - Uninfluenced systems ofter exhibit poor system behavior c(x) =x 1/2 unit flow of traffic S 1/2 D c(x) =1 optimal outcome vs. self-interested outcome 3/4

7 Case study: Pigou s network Motivation: - Uninfluenced systems ofter exhibit poor system behavior c(x) =x unit flow of traffic S 1 c(x) =1 D optimal outcome vs. self-interested outcome 3/4 1

8 Case study: Pigou s network Motivation: - Uninfluenced systems ofter exhibit poor system behavior c(x) =x unit flow of traffic S 1 c(x) =1 D optimal outcome vs. self-interested outcome self-interested outcome 3/4 33% worse than optimal outcome 1

9 Case study: Braess paradox Motivation: - - Uninfluenced systems ofter exhibit poor system behavior Natural influencing mechanisms need not lead to intuitive outcomes

10 Case study: Braess paradox Motivation: - - Uninfluenced systems ofter exhibit poor system behavior Natural influencing mechanisms need not lead to intuitive outcomes c(x) =x c(x) =1 c(x) =x c(x) =1 c(x) =0 c(x) =1 c(x) =x c(x) =1 c(x) =x original network vs. original network + extra edge

11 Case study: Braess paradox Motivation: - - Uninfluenced systems ofter exhibit poor system behavior Natural influencing mechanisms need not lead to intuitive outcomes c(x) =x c(x) =1 1/2 c(x) =1 1/2 c(x) =x c(x) =x c(x) =1 c(x) =0 c(x) =1 c(x) =x original network vs. original network + extra edge 1.5

12 Case study: Braess paradox Motivation: - - Uninfluenced systems ofter exhibit poor system behavior Natural influencing mechanisms need not lead to intuitive outcomes c(x) =x c(x) =1 c(x) =1 1/2 1/2 c(x) =x c(x) =x c(x) =1 1 c(x) =0 c(x) =1 c(x) =x original network vs. original network + extra edge 1.5 2

13 Case study: Braess paradox Motivation: - - Uninfluenced systems ofter exhibit poor system behavior Natural influencing mechanisms need not lead to intuitive outcomes c(x) =x c(x) =1 c(x) =1 1/2 1/2 c(x) =x c(x) =x c(x) =1 1 c(x) =0 c(x) =1 c(x) =x original network vs. original network + extra edge 1.5 additional resources resulted in 33% worse system performance 2

14 Taxation mechanism Can taxes be employed to help coordinate behavior? Structure of Mechanism path based vs. edge based? anonymous vs. discriatory? fixed vs. behavior dependent? Informational Demand network demands? network structure? population sensitivity?

15 Taxation mechanism Can taxes be employed to help coordinate behavior? Structure of Mechanism path based vs. edge based? anonymous vs. discriatory? fixed vs. behavior dependent? Informational Demand network demands? network structure? population sensitivity? None. None. None.

16 Recall: Pigou s network Can taxes be employed to help coordinate behavior? (discourage users from using top edge) c(x) =x S D Nash flow 1 c(x) =1 Optimal flow 1/2 0 33% degradation in performance 1/2 1 3/4

17 Recall: Pigou s network Can taxes be employed to help coordinate behavior? (discourage users from using top edge) c(x) =x t(x) =x S D c(x) =1

18 Recall: Pigou s network Can taxes be employed to help coordinate behavior? (discourage users from using top edge) c(x) =2x c(x) =x t(x) =x S D c(x) =1

19 Recall: Pigou s network Can taxes be employed to help coordinate behavior? (discourage users from using top edge) c(x) =2x c(x) =x t(x) =x S D c(x) =1 Optimal flow 1/2 1/2 3/4

20 Recall: Pigou s network Can taxes be employed to help coordinate behavior? (discourage users from using top edge) c(x) =2x c(x) =x t(x) =x S D Nash flow 1/2 c(x) =1 Optimal flow 1/2 1/2 1/2 3/4 3/4

21 Recall: Pigou s network Can taxes be employed to help coordinate behavior? (discourage users from using top edge) c(x) =2x c(x) =x t(x) =x S D Nash flow 1/2 c(x) =1 Optimal flow 1/2 1/2 3/4 tolls are an effective remedy for efficiency loss 1/2 3/4

22 Recall: Pigou s network Can taxes be employed to help coordinate behavior? (discourage users from using top edge) c(x) =2x c(x) =x t(x) =x S D c(x) =1 Structure of Mechanism path based vs. edge based? anonymous vs. discriatory? fixed vs. behavior dependent? Informational Demand network demands? network structure? population sensitivity? None. None. None.

23 Recall: Pigou s network Can taxes be employed to help coordinate behavior? (discourage users from using top edge) c(x) =2x c(x) =x t(x) =x S D c(x) =1 Structure of Mechanism path based vs. edge based? anonymous vs. discriatory? fixed vs. behavior dependent? Informational Demand network demands? network structure? population sensitivity? None. None. None.

24 Recall: Pigou s network Can taxes be employed to help coordinate behavior? (discourage users from using top edge) c(x) =2x c(x) =x t(x) =x S D c(x) =1 Structure of Mechanism path based vs. edge based? anonymous vs. discriatory? fixed vs. behavior dependent? Informational Demand network demands? network structure? population sensitivity? None. None. None.

25 Recall: Pigou s network Can taxes be employed to help coordinate behavior? (discourage users from using top edge) c(x) =2x c(x) =x t(x) =x S D c(x) =1 Structure of Mechanism path based vs. edge based? anonymous vs. discriatory? fixed vs. behavior dependent? Informational Demand network demands? network structure? population sensitivity? None. None. None.

26 Recall: Pigou s network Can taxes be employed to help coordinate behavior? (discourage users from using top edge) c(x) =2x c(x) =x t(x) =x S D c(x) =1 Structure of Mechanism path based vs. edge based? anonymous vs. discriatory? fixed vs. behavior dependent? Informational Demand network demands? network structure? population sensitivity? None. None. None.

27 Recall: Pigou s network Can taxes be employed to help coordinate behavior? (discourage users from using top edge) c(x) =2x c(x) =x t(x) =x S D c(x) =1 Structure of Mechanism path based vs. edge based? anonymous vs. discriatory? fixed vs. behavior dependent? Informational Demand network demands? network structure? population sensitivity? None. None. None.

28 Recall: Pigou s network Can taxes be employed to help coordinate behavior? (discourage users from using top edge) c(x) =2x c(x) =x t(x) =x S D c(x) =1 Structure of Mechanism path based vs. edge based? anonymous vs. discriatory? fixed vs. behavior dependent? Informational Demand network demands? network structure? population sensitivity? None. None. None.

29 Recall: Pigou s network Can taxes be employed to help coordinate behavior? Structure of Mechanism path based vs. edge based? anonymous vs. discriatory? fixed vs. behavior dependent? Informational Demand network demands? network structure? population sensitivity? None. None. None.

30 Model Model: Users sensitivities between congestion and tolls unknown Cost (agent i) = QoS + i Incentives (congestion) (tolls) (unknown/heterogenous sensitivity) :[0, 1]! [, max ] Fleischer et al., Tolls for Heterogeneous Selfish Users in Multicommodity Networks and Generalized Congestion Games, R. Cole et al., Pricing Network Edges for Heterogeneous Selfish Users, 2003.

31 Model Model: Users sensitivities between congestion and tolls unknown Cost (agent i) = QoS + i Incentives (congestion) (tolls) (unknown/heterogenous sensitivity) :[0, 1]! [, max ] Nash flow: A flow such that no agent has a unilateral incentive to deviate. Fleischer et al., Tolls for Heterogeneous Selfish Users in Multicommodity Networks and Generalized Congestion Games, R. Cole et al., Pricing Network Edges for Heterogeneous Selfish Users, 2003.

32 Model Model: Users sensitivities between congestion and tolls unknown Cost (agent i) = QoS + i Incentives (congestion) (tolls) (unknown/heterogenous sensitivity) :[0, 1]! [, max ] Nash flow: A flow such that no agent has a unilateral incentive to deviate. structure of mechanism informational demands efficiency guarantees? Fleischer et al., Tolls for Heterogeneous Selfish Users in Multicommodity Networks and Generalized Congestion Games, R. Cole et al., Pricing Network Edges for Heterogeneous Selfish Users, 2003.

33 Performance taxation mechanism information demands fixed flow dependent network structure network demands users sensitivities best efficiency guarantees?

34 Example taxation mechanism information demands fixed flow dependent network structure network demands users sensitivities best efficiency guarantees (example) previous example c(x) =2x c(x) =x t(x) =x S D c(x) =1

35 Pigovian taxes taxation mechanism information demands fixed flow dependent network structure network demands users sensitivities best efficiency guarantees (Pigovian tax) (homogenous) Pigovian Tax c(x) =2x c(x) =x t(x) =x t e (f e )=f e c 0 e(f e ) c(x) =1 M. Beckman, C. McGuire, and C. B. Winsten, Studies in the Economics of Transportation, Sandholm, Evolutionary Implementation and Congestion Pricing, 2002.

36 Fixed taxes taxation mechanism information demands fixed flow dependent network structure network demands users sensitivities best efficiency guarantees (Pigovian tax) (Fixed taxes) (homogenous) Fixed Tax t e (f e )=c e not robust to mischaracterizations L. Fleischer, K. Jain, and M. Mahdian, Tolls for Heterogeneous Selfish Users in Multicommodity Networks and Generalized Congestion Games, Karakostas, G. and Kolliopoulos, Edge pricing of multicommodity networks for heterogeneous selfish users, R. Cole, Y. Dodis, and T. Roughgarden, Pricing Network Edges for Heterogeneous Selfish Users, 2003.

37 Our contributions taxation mechanism information demands fixed flow dependent network structure network demands users sensitivities best efficiency guarantees (Pigovian tax) (Fixed taxes) (Brown & JRM) (homogenous) Brown and JRM, Optimal Mechanisms for Robust Coordination in Congestion Games, 2015.

38 Our contributions taxation mechanism information demands fixed flow dependent network structure network demands users sensitivities best efficiency guarantees (Pigovian tax) (Fixed taxes) (Brown & JRM) (homogenous) (Pigovian tax) c e (f e )+t e (f e ) c e (f e )+f e c 0 e(f e ) efficiency guarantees Brown and JRM, Optimal Mechanisms for Robust Coordination in Congestion Games, 2015.

39 Our contributions taxation mechanism information demands fixed flow dependent network structure network demands users sensitivities best efficiency guarantees (Pigovian tax) (Fixed taxes) (Brown & JRM) (homogenous) (Pigovian tax) c e (f e )+t e (f e ) c e (f e )+f e c 0 e(f e ) efficiency guarantees (our setting) c e (f e )+ x t e (f e ) x 2, max Brown and JRM, Optimal Mechanisms for Robust Coordination in Congestion Games, 2015.

40 Our contributions taxation mechanism information demands fixed flow dependent network structure network demands users sensitivities best efficiency guarantees (Pigovian tax) (Fixed taxes) (Brown & JRM) (homogenous) (Pigovian tax) c e (f e )+t e (f e ) c e (f e )+f e c 0 e(f e ) efficiency guarantees (our setting) c e (f e )+ x t e (f e ) x 2, max Brown and JRM, Optimal Mechanisms for Robust Coordination in Congestion Games, 2015.

41 Our contributions taxation mechanism information demands fixed flow dependent network structure network demands users sensitivities best efficiency guarantees (Pigovian tax) (Fixed taxes) (Brown & JRM) (homogenous) (Pigovian tax) c e (f e )+t e (f e ) c e (f e )+f e c 0 e(f e ) efficiency guarantees (our setting) c e (f e )+ x t e (f e ) x 2, max t e (f e )= lim k!1 k c e(f e )+f e c 0 e(f e ) Brown and JRM, Optimal Mechanisms for Robust Coordination in Congestion Games, 2015.

42 Our contributions taxation mechanism information demands fixed flow dependent network structure network demands users sensitivities best efficiency guarantees (Pigovian tax) (Fixed taxes) (Brown & JRM) (homogenous) (Pigovian tax) c e (f e )+t e (f e ) c e (f e )+f e c 0 e(f e ) efficiency guarantees (our setting) c e (f e )+ x t e (f e ) x 2, max t e (f e )= lim k!1 k c e(f e )+f e c 0 e(f e ) (requires large tolls) Brown and JRM, Optimal Mechanisms for Robust Coordination in Congestion Games, 2015.

43 Our contributions taxation mechanism information demands fixed flow dependent network structure network demands users sensitivities best efficiency guarantees (Pigovian tax) (Fixed taxes) (Brown & JRM) (homogenous) What if we have constraints on the maximum toll? Brown and JRM, Optimal Mechanisms for Robust Coordination in Congestion Games, 2015.

44 Our contributions Setting: Parallel networks, affine latency functions c e (f e )=a e f e + b e Theorem: Let B be the maximum toll. Tolls that optimize worst-case efficiency are t e (f e )=k 1 (, max,b)f e + k 2 (, max,b) 95% efficiency guarantees 90% 85% knowledge of population 80% 1 max max max max 0 75% increasing tolls Brown and JRM, Optimal Mechanisms for Robust Coordination in Congestion Games, 2015.

45 Discriatory tolls Previous setting: Anonymous tolls (every driver sees same price) Question: Could we exploit discriatory pricing to improve efficiency guarantees? Brown and JRM, A Study on Price-Discriation for Robust Social Coordination, 2015.

46 Discriatory tolls Previous setting: Anonymous tolls (every driver sees same price) Question: Could we exploit discriatory pricing to improve efficiency guarantees? Cost (agent i) = QoS + i Incentives (congestion) (tolls) (unknown/heterogenous sensitivity) :[0, 1]! [, max ] Brown and JRM, A Study on Price-Discriation for Robust Social Coordination, 2015.

47 Discriatory tolls Previous setting: Anonymous tolls (every driver sees same price) Question: Could we exploit discriatory pricing to improve efficiency guarantees? Cost (agent i) = QoS + i Incentives (congestion) (tolls) (unknown/heterogenous sensitivity) :[0, 1]! [, max ] Approach: Discriate according to price sensitivity max Brown and JRM, A Study on Price-Discriation for Robust Social Coordination, 2015.

48 Discriatory tolls Previous setting: Anonymous tolls (every driver sees same price) Question: Could we exploit discriatory pricing to improve efficiency guarantees? Cost (agent i) = QoS + i Incentives (congestion) (tolls) (unknown/heterogenous sensitivity) :[0, 1]! [, max ] Approach: Discriate according to price sensitivity P 1 P 2 P 3 max t 1 e( ) t 2 e( ) t 3 e( ) Brown and JRM, A Study on Price-Discriation for Robust Social Coordination, 2015.

49 Discriatory tolls Previous setting: Anonymous tolls (every driver sees same price) Questions: How should you discriate? What are the optimal taxes within each bin? What are the optimal bin boundaries? How finely should we discriate? (errors?) Adaptive tolling mechanisms? Comparison between discriatory and non-discriatory mechanisms? Approach: Discriate according to price sensitivity P 1 P 2 P 3 max t 1 e( ) t 2 e( ) t 3 e( ) Brown and JRM, A Study on Price-Discriation for Robust Social Coordination, 2015.

50 A lemma Existing design: PoA(t,, max ) t e ( ) max Brown and JRM, A Study on Price-Discriation for Robust Social Coordination, 2015.

51 A lemma Existing design: PoA(t,, max ) t e ( ) max New specifications: PoA( t,, max ) t e ( ) max Can we exploit the original design/analysis for new domain? Brown and JRM, A Study on Price-Discriation for Robust Social Coordination, 2015.

52 A lemma Existing design: PoA(t,, max ) scale of uncertainty driving factor of PoA t e ( ) max max New specifications: PoA( t,, max ) t e ( ) max Can we exploit the original design/analysis for new domain? Brown and JRM, A Study on Price-Discriation for Robust Social Coordination, 2015.

53 A lemma Existing design: PoA(t,, max ) scale of uncertainty driving factor of PoA t e ( ) max max New specifications: P 1 PoA( t,, max ) P 2 P 3 max max 2 max Brown and JRM, A Study on Price-Discriation for Robust Social Coordination, 2015.

54 A lemma Existing design: PoA(t,, max ) scale of uncertainty driving factor of PoA t e ( ) max max New specifications: PoA( t,, max ) P 1 P 2 P 3 max t 1 e( ) = t e ( ) scale tax by normalizing left boundary Brown and JRM, A Study on Price-Discriation for Robust Social Coordination, 2015.

55 A lemma Existing design: PoA(t,, max ) scale of uncertainty driving factor of PoA t e ( ) max max New specifications: PoA( t,, max ) P 1 P 2 P 3 max t 2 e( ) = max max t e ( ) Brown and JRM, A Study on Price-Discriation for Robust Social Coordination, 2015.

56 A lemma Existing design: PoA(t,, max ) scale of uncertainty driving factor of PoA t e ( ) max max New specifications: PoA( t,, max ) P 1 P 2 P 3 t 3 e( ) = max max 2 t e ( ) Brown and JRM, A Study on Price-Discriation for Robust Social Coordination, 2015.

57 A lemma Existing design: PoA(t,, max ) scale of uncertainty driving factor of PoA t e ( ) max max New specifications: PoA( t,, max ) P 1 P 2 P 3 max t 1 e( ) t 2 e( ) t 3 e( ) Lemma: PoA( t,, max ) apple PoA(t,, max ) (irrespective of network characteristics) Brown and JRM, A Study on Price-Discriation for Robust Social Coordination, 2015.

58 Discriatory taxes What is the best discriatory taxation mechanism to use? max Brown and JRM, A Study on Price-Discriation for Robust Social Coordination, 2015.

59 Discriatory taxes What is the best discriatory taxation mechanism to use? max scale of uncertainty efficiency of resulting equilibria Brown and JRM, A Study on Price-Discriation for Robust Social Coordination, 2015.

60 Discriatory taxes What is the best discriatory taxation mechanism to use? max scale of uncertainty efficiency of resulting equilibria Approach: Discriate to imize maximum scale of uncertainty Brown and JRM, A Study on Price-Discriation for Robust Social Coordination, 2015.

61 Discriatory taxes What is the best discriatory taxation mechanism to use? P 1 P 2 P m max max 1/m max 2/m max (m 1)/m all bins have the same scale of uncertainty Brown and JRM, A Study on Price-Discriation for Robust Social Coordination, 2015.

62 Discriatory taxes What is the best discriatory taxation mechanism to use? P 1 P 2 P m t 1 t 2 t e( ) e( ) m e ( ) max Brown and JRM, A Study on Price-Discriation for Robust Social Coordination, 2015.

63 Discriatory taxes What is the best discriatory taxation mechanism to use? P 1 P 2 P m t 1 t 2 t e( ) e( ) m e ( ) max Recall Previous Theorem Let B be the maximum toll. Tolls that optimize worst-case efficiency are t e (f e )=k 1 (, max,b)f e + k 2 (, max,b) lower boundary (parallel affine networks) upper boundary Brown and JRM, A Study on Price-Discriation for Robust Social Coordination, 2015.

64 Discriatory taxes What is the best discriatory taxation mechanism to use? P 1 P 2 P m t 1 t 2 t e( ) e( ) m e ( ) max Recall Previous Theorem Let B be the maximum toll. Tolls that optimize worst-case efficiency are t e (f e )=k 1 (, max,b)f e + k 2 (, max,b) lower boundary (parallel affine networks) upper boundary Theorem: PoA(m, B,, max ) decreasing in m goes to 1 if B sufficiently large Brown and JRM, A Study on Price-Discriation for Robust Social Coordination, 2015.

65 Discriatory taxes What is the best discriatory taxation mechanism to use? P 1 P 2 P m t 1 t 2 t e( ) e( ) m e ( ) max 95% efficiency guarantees 90% 85% 80% 75% m =1 m =2 m =3 max increasing maximum toll max =fixed Brown and JRM, A Study on Price-Discriation for Robust Social Coordination, 2015.

66 Discriatory taxes What is the best discriatory taxation mechanism to use? P 1 P 2 P m t 1 t 2 t e( ) e( ) m e ( ) GOAL improve efficiency resulting equilibria max non-discriatory tolls (requires large tolls) discriatory tolls (requires smaller tolls) discriation can compensate for magnitude of tolls Brown and JRM, A Study on Price-Discriation for Robust Social Coordination, 2015.

67 Discriatory taxes What is the best discriatory taxation mechanism to use? P 1 P 2 P m t 1 t 2 t e( ) e( ) m e ( ) GOAL improve efficiency resulting equilibria max non-discriatory tolls (requires large tolls) discriatory tolls (requires smaller tolls) discriation can compensate for magnitude of tolls Open Question What is the optimal discriatory toll? Brown and JRM, A Study on Price-Discriation for Robust Social Coordination, 2015.

68 Multiagent coordination Central Goal Derive efficient system-wide behavior through the design of admissible control algorithms Smart Grid Sensor coverage Traffic network socio-technical systems engineering systems social systems

69 Multiagent coordination Central Goal Derive efficient system-wide behavior through the design of admissible control algorithms challenges Traffic network Is there a general theory for robust social coordination? What is the role of discriation in multiagent coordination? What about learning and influencing at the same time? social systems

70 Thank you Game theoretic methods for distributed control: JRM & Shamma, ``Game Theory and Distributed Control," Handbook of Game Theory, Volume IV, Trade-offs in multiagent coordination: Brown & JRM, Optimal Mechanisms for Robust Coordination in Congestion Games, Brown & JRM, A Study on Price-Discriation for Robust Social Coordination, Brown & JRM, The Robustness of Marginal-Cost Taxes in Affine Congestion Games, JRM, The Role of Information in Multiagent Coordination, 2015.