Mechanism Design with Allocative, Informational and Learning Constraints May / 46

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1 Mechanism Design with Allocative, Informational and Learning Constraints May 2017 Mechanism Design with Allocative, Informational and Learning Constraints May / 46

2 Joint work with Ph.D. student Abhinav Sinha (Graduating in September)

3 Allocation problems in networks A number of interesting problems in networks involve optimization of a social utility function (sum of agents utilities) under resource constraints, privacy constraints, and strategic behavior by agents Examples include: - power production, distribution, consumption on the smart grid, - Bandwidth allocation to cellular service providers - unicast, multi-rate multicast service on the Internet - advertisement on social networks - economies with public or local public goods (e.g., investment on clean air or cyber-security) Mechanism Design with Allocative, Informational and Learning Constraints May / 46

4 Spectrum allocation Total Bandwidth B x 1 x 2 x 3... x N max x R N + s.t. v i (x i ) i N x i B i N private consumption goods: utilities v i ( ) depend only on their own allocation Mechanism Design with Allocative, Informational and Learning Constraints May / 46

5 Unicast service on the Internet Tx Rx x 1 T1 R1 x 1 x 2 x 3 T2 T3 x 1 x 2 x1 x 2 x 3 x 3 x 4 R2 x 2 R3 x 3 x 4 T4 R4 x 4 s.t. max x R N + v i (x i ) i N i N l x i c l l L Mechanism Design with Allocative, Informational and Learning Constraints May / 46

6 Multi-rate multicast service on the Internet Tx Rx x 11 T1 R1 x 11 R2 x 22 x 22 x 23 x 34 x 35 T2 T3 x 34 x 35 x 11 x 22 x 11 x 23 x 22 R3 x 23 R4 x 34 R5 x 35 s.t. max x R N + k K l k K i G k v ki (x ki ) max {x ki } c l l L i G l k Mechanism Design with Allocative, Informational and Learning Constraints May / 46

7 Power allocation in wireless networks Each agent is a (transmitter, receiver) pair. x x 2 x 3 x N 1 = {1, 2, 3} N 2 = {3, 4} max x v i ({x i } i Nk(i) ) i N s.t. x R N + The vector of transmission powers x = (x 1,..., x N ) is a public (or local public) good Mechanism Design with Allocative, Informational and Learning Constraints May / 46

8 NUM and Salient Features Maximize sum of utilities subject to network constraints max v i (x) s.t. x X. x i N Linear constraints - common knowledge. v i ( ) - Private information and known only to agent i Designer can impose taxes and quasi-linear utilities u i = v i (x) t i. Mechanism Design with Allocative, Informational and Learning Constraints May / 46

9 Informal Framework of Mechanism Design Designer wishes to allocate optimally. Agents are strategic and... Possess private information relevant to optimal allocation. Designer wishes to cover this Informational gap Design Message Space and Contract. Agents then announce a selected message and receive outcomes based on the contract. Mechanism Design with Allocative, Informational and Learning Constraints May / 46

10 Hurwicz-Reiter Model Envinronment CP Outcome V = V i X Mechanism Design with Allocative, Informational and Learning Constraints May / 46

11 Hurwicz-Reiter Model Envinronment CP Outcome V = V i X Contract : h M = M i Messages Mechanism Design with Allocative, Informational and Learning Constraints May / 46

12 Hurwicz-Reiter Model Envinronment V = V i v CP Outcome X = X X x i (NE) h M = M i m Messages Design M & h : M X to replicate CP mapping at NE. Mechanism Design with Allocative, Informational and Learning Constraints May / 46

13 Motivating Question The motivating question, in this work, is whether it is possible to design Nash implementation mechanisms that posses certain properties... Mechanism Design with Allocative, Informational and LearningMay Constraints / 46

14 Motivating Design Principles (a) Reduction in communication overhead for NUM problems Models with large type spaces - Private info. v i : R A R. (b) Systematic Design to deal with Allocative constraints Integrate different network constraints under one design umbrella. Special focus: Can off-equilibrium feasibility be accommodated? Mechanism Design with Allocative, Informational and LearningMay Constraints / 46

15 Motivating Design Principles (contd) (c) Informational constraints on messaging What if not all agents can talk to a common designer? What if they don t want to? (privacy) 2 m 2 m 4 4 m 2 m 3 m3 m4 m 1 m (Not to be confused with Unicast or Multicast links or dependence of utility on local allocation only.) Mechanism Design with Allocative, Informational and LearningMay Constraints / 46

16 Motivating Design Principles (contd) (d) Learning of Nash equilibrium Solution concept of NE and convergence of a learning dynamic go together for models with stable environments Design for two Off-equilibrium properties - Off-equilibrium Feasibility - Convergence of a large class of Learning algorithms. (e) Designing beyond sum of utilities - fairness What about mechanism design for other objectives?, e.g., max x X i N ( log (v i (x)) or max x X min i N ( v i (x)) ). Mechanism Design with Allocative, Informational and LearningMay Constraints / 46

17 Overview (a) Reduced communication overhead (b) Systematic Design Part 1: Multirate/Multicast (c) Informational constraints on messaging (d) Learning of NE Part 2: Distributed M.D. and Learning (e) Designing beyond SoU Fairness (not in this talk) Mechanism Design with Allocative, Informational and LearningMay Constraints / 46

18 Overview (a) Reduced communication overhead (b) Systematic Design Part 1: Multirate/Multicast (c) Informational constraints on messaging (d) Learning of NE Part 2: Distributed M.D. and Learning (e) Designing beyond SoU Fairness (not in this talk) Mechanism Design with Allocative, Informational and LearningMay Constraints / 46

19 Overview (a) Reduced communication overhead (b) Systematic Design Part 1: Multirate/Multicast (c) Informational constraints on messaging (d) Learning of NE Part 2: Distributed M.D. and Learning (e) Designing beyond SoU Fairness (not in this talk) Mechanism Design with Allocative, Informational and LearningMay Constraints / 46

20 Overview (a) Reduced communication overhead (b) Systematic Design Part 1: Multirate/Multicast (c) Informational constraints on messaging (d) Learning of NE Part 2: Distributed M.D. and Learning (e) Designing beyond SoU Fairness (not in this talk) Mechanism Design with Allocative, Informational and LearningMay Constraints / 46

21 Systematic Design for Allocative constraints: Multi-rate/multicast transmission Overview 1 Systematic Design for Allocative constraints: Multi-rate/multicast transmission 2 Distributed Mechanisms with Learning Guarantees Mechanism Design with Allocative, Informational and LearningMay Constraints / 46

22 Systematic Design for Allocative constraints: Multi-rate/multicast transmission Relevant Literature Earliest works by [Groves, Ledyard 1977] for Lindahl allocation and [Hurwicz 1979] for Walrasian allocation. Also [Tian 1989, Chen 2002] for Lindahl allocation. Recent works motivated by [Kelly, Maulloo, Tan 1998] - Competitive equilibria, only for price-takers. Two classifications for recent works: based on constraint set, based on full/partial implementation. Single-link Unicast [Yang, Hajek 2006b], [Maheshwaran, Başar 2004] - full implementation. Mechanism Design with Allocative, Informational and LearningMay Constraints / 46

23 Systematic Design for Allocative constraints: Multi-rate/multicast transmission Relevant Literature (contd) More General Constraint set [Yang, Hajek 2007], [Johari, Tsitsiklis 2009] - arbitrary convex constraint sets but with partial implementation. Off-equilibrium Feasibility. [Jain, Walrand 2010] - Partial Implementation. [Kakhbod, Teneketzis 2012a,b] - Full implementation for general unicast and multicast. Mechanism Design with Allocative, Informational and LearningMay Constraints / 46

24 Systematic Design for Allocative constraints: Multi-rate/multicast transmission The Multirate/Multicast Model Tx T1 max{x 11, x 12} T2 A B max{x 23, x 24} max{x 11, x 12 } + max{x 23, x 24 } max{x 11, x 12 } + x x 23 C 12 x 24 x 11 x 23 Rx R1 x 11 R2 x 12 R3 x 23 R4 x 24 Agent - a specific transmitter-receiver pair. Multicast Group - Group of agents requesting the same content. (e.g. same TV program). Multi-rate - Agents within the same multicast group can request different QoS (e.g. high/standard definition video). Watching on mobile phone or on HDTV. Mechanism Design with Allocative, Informational and LearningMay Constraints / 46

25 Systematic Design for Allocative constraints: Multi-rate/multicast transmission Notation Tx T1 max{x 11, x 12} T2 A B max{x 23, x 24} max{x 11, x 12 } + max{x 23, x 24 } max{x 11, x 12 } + x x 23 C 12 x 24 x 11 x 23 Rx R1 x 11 R2 x 12 R3 x 23 R4 x 24 Set of multicast groups K = {1,..., K}. Agents indexed group-wise (k, i) K G k (called agent ki). Each agent uses a fixed route L ki L. Set of groups active on link l L denoted by K l K. Agents from group k that are active on link l are identified by G l k G k. Mechanism Design with Allocative, Informational and LearningMay Constraints / 46

26 Systematic Design for Allocative constraints: Multi-rate/multicast transmission Centralized Problem Tx T1 T2 max{x 11, x 12 } A max{x 23, x 24 } max{x 11, x 12 } + max{x 23, x 24 } max{x 11, x 12 } + x x 23 C 12 B x 24 x 11 x 23 Rx R1 x 11 R2 x 12 R3 x 23 R4 x 24 Agents T1-R1 T1-R2 T2-R3 T2-R4 max x R N + s.t. k K k K l i G k v ki (x ki ) (CP MM ) max {x ki } c l l L i G l k Mechanism Design with Allocative, Informational and LearningMay Constraints / 46

27 Systematic Design for Allocative constraints: Multi-rate/multicast transmission Public and Private goods Two levels of interaction in the Multi-rate Multicast problem 1 Contest between groups through the highest rate at each link. 2 No contest within a group, since only maximum gets charged. Due to the max{ }, there s a possible case for free-riding. Mechanism Design with Allocative, Informational and LearningMay Constraints / 46

28 Systematic Design for Allocative constraints: Multi-rate/multicast transmission Systematic Mechanism Design max x,s ki N v ki (x ki ) s.t. x ki 0 ki N (C 1 ) and k K l s l k cl l L (C 2 ) and x ki s l k i G l k, k, l (C 3) s l k { } - proxy for max xki. i G l k Mechanism Design with Allocative, Informational and LearningMay Constraints / 46

29 Systematic Design for Allocative constraints: Multi-rate/multicast transmission KKT - Necessary and Sufficient Assuming utilities are strictly concave (and monotonic) and differentiable There exist Primal variables (x, s ) and Dual variables (λ l ) l L and (µ l ki )ki N,l Lki such that: 1. Primal Feasibility (x, s ) satisfy multicast constraints. 2. Dual Feasibility λ, µ 0. Mechanism Design with Allocative, Informational and LearningMay Constraints / 46

30 Systematic Design for Allocative constraints: Multi-rate/multicast transmission KKT - Necessary and Sufficient Assuming utilities are strictly concave (and monotonic) and differentiable 3. Complimentary Slackness λ l ( k K l s l k c l ) = 0 l L µ l ki ( x ki sl k ) = 0 ki N, l L ki 4. Stationarity v ki (x ki ) = λ l = µ l ki l L ki µ l ki ki N if x ki > 0 k K l, l L i G l k Mechanism Design with Allocative, Informational and LearningMay Constraints / 46

31 Systematic Design for Allocative constraints: Multi-rate/multicast transmission Mechanism - Message Space ( M ki = R + R 2L ki + where m ki = y ki, ( ) p l ki, ql ki )l L ki. y ki agent ki s demand for allocation. p l ki individual price for ki. (proxy for µl ki ) Best not to have agents pay at prices quoted by themself. So q l ki used in place of pl ke for the next agent on link l. p l kj q l kj p l ki q l ki p l ke q l ke Mechanism Design with Allocative, Informational and LearningMay Constraints / 46

32 Systematic Design for Allocative constraints: Multi-rate/multicast transmission Mechanism: Proportional Allocation A new allocation scheme... that guarantees off-equilibrium feasibility! Demand y is translated into allocation x using a scaling factor, i.e., x ki (m) = r y ki with r = min l L { r l (m) }. x 2 y x y x y x x 1 Mechanism Design with Allocative, Informational and LearningMay Constraints / 46

33 Systematic Design for Allocative constraints: Multi-rate/multicast transmission Mechanism: Taxes Tax for any agent ki is the sum of taxes t l ki over route L ki. Mechanism Design with Allocative, Informational and LearningMay Constraints / 46

34 Systematic Design for Allocative constraints: Multi-rate/multicast transmission Mechanism: Taxes Tax for any agent ki is the sum of taxes t l ki over route L ki. Consider agents kj ki ke from G l k. Mechanism Design with Allocative, Informational and LearningMay Constraints / 46

35 Systematic Design for Allocative constraints: Multi-rate/multicast transmission Mechanism: Taxes Consider agents kj ki ke from G l k. t l ki = x kiq l kj + (ql ki pl ke )2 + q l ( kj p l ki q l ) ( ) kj s l k x ki + (w l k wl k )2 + w l ( k w l k w l ) ( k c l ) s l k k K l Mechanism Design with Allocative, Informational and LearningMay Constraints / 46

36 Systematic Design for Allocative constraints: Multi-rate/multicast transmission Mechanism: Taxes Consider agents kj ki ke from G l k. x ki q l kj Payment for x ki Mechanism Design with Allocative, Informational and LearningMay Constraints / 46

37 Systematic Design for Allocative constraints: Multi-rate/multicast transmission Mechanism: Taxes Consider agents kj ki ke from G l k. (q l ki pl ke )2 Equal Prices (individual) p l kj q l kj p l ki q l ki p l ke q l ke Mechanism Design with Allocative, Informational and LearningMay Constraints / 46

38 Systematic Design for Allocative constraints: Multi-rate/multicast transmission Mechanism: Taxes Consider agents kj ki ke from G l k. q l ( kj p l ki q l ) ( ) kj s l k x ki Complimentary Slackness (individual) Mechanism Design with Allocative, Informational and LearningMay Constraints / 46

39 Systematic Design for Allocative constraints: Multi-rate/multicast transmission Mechanism: Taxes Consider agents kj ki ke from G l k. Define w l k = i G l k (w l k wl k )2 Equal Prices (group) p l ki and w l k = 1 K l 1 k K l \{k} w l k. Mechanism Design with Allocative, Informational and LearningMay Constraints / 46

40 Systematic Design for Allocative constraints: Multi-rate/multicast transmission Mechanism: Taxes Consider agents kj ki ke from G l k. w l ( k w l k w l ) ( k c l ) s l k k K l Complimentary Slackness (group) Mechanism Design with Allocative, Informational and LearningMay Constraints / 46

41 Systematic Design for Allocative constraints: Multi-rate/multicast transmission Summary of Results Theorem (Full Implementation+IR+WBB/SBB) At any Nash equilibrium m of the induced game, The allocation x(m ) is the unique solution to (CP MM ). Individual Rationality is satisfied for all agents. Weak Budget Balance ki N t ki(m ) 0. Strong Budget Balance ki N t ki(m ) = 0, with an augmented message space. Mechanism Design with Allocative, Informational and LearningMay Constraints / 46

42 Systematic Design for Allocative constraints: Multi-rate/multicast transmission Summary of Results Theorem (Full Implementation+IR+WBB/SBB) At any Nash equilibrium m of the induced game, The allocation x(m ) is the unique solution to (CP MM ). Individual Rationality is satisfied for all agents. Weak Budget Balance ki N t ki(m ) 0. Strong Budget Balance ki N t ki(m ) = 0, with an augmented message space. Basic idea behind proof of Implementation F.O.C. for NE gives KKT as necessary conditions. Existence using S.O.C. and invertibility. Mechanism Design with Allocative, Informational and LearningMay Constraints / 46

43 Distributed Mechanisms with Learning Guarantees Overview 1 Systematic Design for Allocative constraints: Multi-rate/multicast transmission 2 Distributed Mechanisms with Learning Guarantees Mechanism Design with Allocative, Informational and LearningMay Constraints / 46

44 Distributed Mechanisms with Learning Guarantees Mechanism Design, Learning and Networks Mechanism Design / Strategic behavior Learning / Transient behavior Networks / Local messages Mechanism Design with Allocative, Informational and LearningMay Constraints / 46

45 Distributed Mechanisms with Learning Guarantees Learning and Modeling trade-offs Fully Compliant Agents Sophistication & Guarantees Design Choices Fully Strategic Agents Good enough guarantee and large enough design space Mechanism Design with Allocative, Informational and LearningMay Constraints / 46

46 Distributed Mechanisms with Learning Guarantees Mechanism Design, Learning and Networks Well-known algorithms for finding roots and/or optimization Stochastic Approximation: [Robbins, Monro 1951] Gradient Descent: [Nesterov 1983] Simulated Annealing: [Khachaturyan et al 1979, Kirkpatrick et al 1983] Online algorithms: prediction with advice [Littlestone, Warmuth 1994, Vovk 1992]; exp3 [Auer et al 2002] Learning / Transient Behavior Mechanism Design with Allocative, Informational and LearningMay Constraints / 46

47 Distributed Mechanisms with Learning Guarantees Mechanism Design, Learning and Networks Decentralized system Message passing is done locally Realistic and scalable architecture Several well-known graph problems such as Consensus on a Network [Fischer, Lynch, Paterson 1985] Byzantine Generals [Lamport, Shostak, Pease 1982] Network / Local Messages Mechanism Design with Allocative, Informational and LearningMay Constraints / 46

48 Distributed Mechanisms with Learning Guarantees Learning + Networks State of the art design has accommodated at most two out of three requirements Learning on a network with non-strategic agents: distributed optimization - Distributed Stochastic Gradient descent: [Tsitsiklis, Bertsekas 1986] - Consensus and Optimization: [Nedic, Ozdaglar, Parrilo 2008] - ADMM: [Boyd, Parikh, Chu, Peleato, Eckstein 2011]. Learning / Transient Behavior Network / Local Messages Mechanism Design with Allocative, Informational and LearningMay Constraints / 46

49 Distributed Mechanisms with Learning Guarantees Mechanism Design + Learning Learning Mechanism Design Mechanism Design with Learning guarantees in Broadcast environments (i.e., fully connected networks). Supermodularity has played a major role in providing learning guarantees in games. [Milgrom, Roberts 1990; Chen 2002] Mechanism Design with Allocative, Informational and LearningMay Constraints / 46

50 Distributed Mechanisms with Learning Guarantees Mechanism Design + Learning Theorem (Milgrom and Roberts 1990) For a supermodular game i.e., compact action space and an increasing best-response, any learning strategy within the adaptive dynamics (AD) class converges to a point between two most extreme NE. Limitations Requirement of a compact action space, and A region for the convergent point if multiple equilibria exist. Experimental results have shown that supermodularity does not result in fast convergence Mechanism Design with Allocative, Informational and LearningMay Constraints / 46

51 Distributed Mechanisms with Learning Guarantees Goal - Design simultaneously for all three Mechanism Design / Strategic behavior Learning / Transient behavior Networks / Local messages Mechanism Design with Allocative, Informational and LearningMay Constraints / 46

52 Distributed Mechanisms with Learning Guarantees Walrasian and Lindahl allocation Two centralized problems - private and public goods Private good max v i (x i ) x R N i N s.t. x i = 0. i N (Walrasian) Public good max v i (x). x R i N (Lindahl) Mechanism Design with Allocative, Informational and LearningMay Constraints / 46

53 Distributed Mechanisms with Learning Guarantees Walrasian and Lindahl allocation Two centralized problems - private and public goods Private good max v i (x i ) x R N i N s.t. x i = 0. i N (Walrasian) Public good max v i (x). x R i N (Lindahl) Environment assumption: Utilities v i : R R are assumed to be strictly concave with continuous second derivatives such that ( v i ( ) η, 1 ). η Mechanism Design with Allocative, Informational and LearningMay Constraints / 46

54 Distributed Mechanisms with Learning Guarantees Summarizing our goals We aim to design, as before, reduced message space Nash implementation mechanisms such that... A. Allocation and Tax function depend only on neighborhood messages x i, t i : M R, with x i (m i, m N(i) ) and t i (m i, m N(i) ) B. There is guaranteed convergence, to the Nash equilibrium, when agents choose their learning strategies within a class L of learning strategies. Mechanism Design with Allocative, Informational and LearningMay Constraints / 46

55 Distributed Mechanisms with Learning Guarantees Walrasian Mechanism - Proxy via neighbor Message: m i = (y i, q i ) R N+1, - Demand y i R, - Proxy q i = (q 1 i,..., qn i ) RN. Proxy q i are included to collect demand y j from non-neighbor agents. q k j y k q k i i j k Mechanism Design with Allocative, Informational and LearningMay Constraints / 46

56 Distributed Mechanisms with Learning Guarantees Mechanism - Allocation and Tax ( Allocation: x i = y i 1 y k ). N 1 k i Mechanism Design with Allocative, Informational and LearningMay Constraints / 46

57 Distributed Mechanisms with Learning Guarantees Mechanism - Allocation and Tax ( ) Allocation: x i = y i 1 q k n(i,k). N 1 k i n(i, k) neighbor of i closest to k, Mechanism Design with Allocative, Informational and LearningMay Constraints / 46

58 Distributed Mechanisms with Learning Guarantees Mechanism - Allocation and Tax ( ) Allocation: x i = y i 1 q k n(i,k). N 1 k i Tax: t i = p i x i price p i = ( y i + k i y k ) Mechanism Design with Allocative, Informational and LearningMay Constraints / 46

59 Distributed Mechanisms with Learning Guarantees Mechanism - Allocation and Tax ( ) Allocation: x i = y i 1 q k n(i,k). N 1 k i Tax: t i = p i x i price p i = ( q i n(i,i) + k i y k ). Mechanism Design with Allocative, Informational and LearningMay Constraints / 46

60 Distributed Mechanisms with Learning Guarantees Mechanism - Allocation and Tax ( ) Allocation: x i = y i 1 q k n(i,k). N 1 k i Tax: t i = p i x i price p i = ( q i n(i,i) + k i q k n(i,k) ). Mechanism Design with Allocative, Informational and LearningMay Constraints / 46

61 Distributed Mechanisms with Learning Guarantees Mechanism - Allocation and Tax ( ) Allocation: x i = y i 1 q k n(i,k). N 1 k i Tax: t i = p i price p i = ( x i + k N(i), k=i q i n(i,i) + k i q k n(i,k) ( ) q k 2 i y k + explicit duplication ). k/ N(i) k i ( q k i qk n(i,k) duplication via message passing ) 2, Mechanism Design with Allocative, Informational and LearningMay Constraints / 46

62 Distributed Mechanisms with Learning Guarantees Mechanism - Allocation and Tax ( Allocation: x i = y i 1 q k n(i,k) + q k ) n(i,k). N 1 ξ ξ d(i,k) 1 k N(i) k/ N(i) k i Tax: t i = p i price x i + d(i, k) shortest distance to k k N(i), k=i ( p i = 1 q i n(i,i) + δ ξ k N(i) ( ) q k 2 i ξy k + explicit duplication q k n(i,k) ξ + k/ N(i) k i k/ N(i) k i q k n(i,k) ξ d(i,k) 1 ( q k i ξqk n(i,k) duplication via message passing ). ) 2, Mechanism Design with Allocative, Informational and LearningMay Constraints / 46

63 Distributed Mechanisms with Learning Guarantees Main Result Theorem 1 The induced game has unique NE and the corresponding allocation is x. 2 Best-Response for the induced game is contractive and hence every learning dynamics in the ABR class, converges to the NE. 3 Budget Balance: the total tax paid at NE is zero, t t N = 0. Same results for Lindahl allocation. Mechanism Design with Allocative, Informational and LearningMay Constraints / 46

64 Distributed Mechanisms with Learning Guarantees Sketch of the proof Efficient Nash equilibrium arguments same as before. Contraction verify BR < 1. Row-sum matrix norm. Explicitly write down best-response (y i, q i ) = BR i (y i, q i ). Quadratic tax and Linear allocation make this easy. Show there exist parameters ξ (0, 1) and δ (0, ) for any given η (1, ). Mechanism Design with Allocative, Informational and LearningMay Constraints / 46

65 Distributed Mechanisms with Learning Guarantees Contraction and ABR Theorem (Healy and Mathevet (2012)) If a game is contractive i.e., BR < 1, then all ABR dynamics converge to the unique Nash equilibrium. Mechanism Design with Allocative, Informational and LearningMay Constraints / 46

66 Distributed Mechanisms with Learning Guarantees Contraction and ABR Theorem (Healy and Mathevet (2012)) If a game is contractive i.e., BR < 1, then all ABR dynamics converge to the unique Nash equilibrium. Cournot Best-Response Dynamics. Best-Response to empirical distrib. from past k periods. Best-Response to any convex combination of past k periods. Fictitious Play (under concave utilities). Mechanism Design with Allocative, Informational and LearningMay Constraints / 46

67 Distributed Mechanisms with Learning Guarantees Adaptive Best-Response Dynamics (ABR) Asymptotically, the support of randomized actions must not be further than the best-response to the worst observed action in the finite past. m 2 m m 1 Mechanism Design with Allocative, Informational and LearningMay Constraints / 46

68 Distributed Mechanisms with Learning Guarantees Adaptive Best-Response Dynamics (ABR) Asymptotically, the support of randomized actions must not be further than the best-response to the worst observed action in the finite past. m 2 C m m 1 Mechanism Design with Allocative, Informational and LearningMay Constraints / 46

69 Distributed Mechanisms with Learning Guarantees Adaptive Best-Response Dynamics (ABR) Asymptotically, the support of randomized actions must not be further than the best-response to the worst observed action in the finite past. m 2 C BR(C) m m 1 Mechanism Design with Allocative, Informational and LearningMay Constraints / 46

70 Distributed Mechanisms with Learning Guarantees Adaptive Best-Response Dynamics (ABR) Asymptotically, the support of randomized actions must not be further than the best-response to the worst observed action in the finite past. m 2 C BR(C) m C m 1 Message at t should be inside C Mechanism Design with Allocative, Informational and LearningMay Constraints / 46

71 Distributed Mechanisms with Learning Guarantees Numerical example N = 31, η = 25. Agents utility function as quadratic v i (x) = θ i x 2 i + σ ix i. Two graphs: (1) full binary tree and (2) Erdős-Reńyi random graph where any two edges are connected with probability p = 0.3 Two types of learning dynamics: (a) action taken is the best-response to an exponentially weighed average of past actions; (b) best-respond to the arithmetic mean of past 10 rounds Distance to optimum allocation Binary Tree and Exponential Weight Binary Tree and 10-period average Erdos-Renyi and Exponential Weight Erdos-Renyi and 10-period average Iteration number 10 4 Mechanism Design with Allocative, Informational and LearningMay Constraints / 46

72 Distributed Mechanisms with Learning Guarantees Conclusions and Future Research Directions We try to systematize the design of mechanisms for NUM problems with small message spaces, allocative constraints and off-equilibrium feasibility, informational constraints and provide learning guarantees. Can we further reduce average message space from (N + 1) to ( deg + 1 )? r R(i) y r = s L j i k y i + s L + s R s R = Impossibility of Learning for small and continuous mechanisms. What about dynamic environments? (dynamic mechanism design/dynamic games/perfect Bayesian Equilibria...) l L(i) y l Mechanism Design with Allocative, Informational and LearningMay Constraints / 46

73 Distributed Mechanisms with Learning Guarantees Thank you. Mechanism Design with Allocative, Informational and LearningMay Constraints / 46