Presentation is loading. Please wait.

Presentation is loading. Please wait.

Minimizing Efficiency Loss in Mechanism and Protocol Design Tim Roughgarden (Stanford) includes joint work with: Shuchi Chawla (Wisconsin), Ho-Lin Chen.

Similar presentations


Presentation on theme: "Minimizing Efficiency Loss in Mechanism and Protocol Design Tim Roughgarden (Stanford) includes joint work with: Shuchi Chawla (Wisconsin), Ho-Lin Chen."— Presentation transcript:

1 Minimizing Efficiency Loss in Mechanism and Protocol Design Tim Roughgarden (Stanford) includes joint work with: Shuchi Chawla (Wisconsin), Ho-Lin Chen (Stanford), Aranyak Mehta (IBM Almaden), Mukund Sundararajan (Stanford), Gregory Valiant (UC Berkeley)

2 2 Reasons for Efficiency Loss Non-cooperative equilibria: no control of underlying game, players' actions Auction design: players have private "valuations" for goods can use VCG mechanism to maximize efficiency but suboptimality inevitable if goal includes: poly-time + hard allocation (combinatorial auctions) different (e.g. maxmin) objective [Nisan/Ronen 99] revenue constraints

3 3 Quantifying Efficiency Loss Early applications: price of anarchy [Kousoupias/Papadimitriou 99], etc. approximation mechanisms both poly-time combinatorial auctions and maxmin objectives This talk: mechanism/protocol design to minimize worst-case efficiency loss. mechanism design s.t. revenue constraint protocol design to minimize price of anarchy full information but implementation constraints

4 4 Cost-Sharing Problems general case: set U of players, cost function C defined on U (incurred by mechanism) special case: fixed-tree-multicast rooted tree T with fixed edge costs c; C(S) = cost of subtree spanning S [Feigenbaum/Papadimitriou/Shenker 00] player i has valuation v i for winning Terminology: surplus of S = v(S) - C(S) [where v(S) = Σ i v i ]

5 5 Cost-Sharing Mechanisms cost-sharing mechanism: collect bids, pick winning set S, determines prices for winners Natural goals: truthful + "individually rational" economically efficient (maximizes surplus) "budget-balance" (revenue covers cost incurred) VCG fails miserably here fact: 3 goals mutually incompatible [Green/Laffont, Roberts 70s], [Feigenbaum/Krishnamurthy/Sami/Shenker 03]

6 6 Shapley Mechanism for Multicast collects bids (b i for each i) initialize S = all players share each edge equally among its users if b i p i for all i, done. else drop a player i with b i < p i and iterate Price = c(e 1 ) + c(e 2 )/3 + c(e 3 )/4 e2e2 e1e1 e3e3

7 7 Moulin Mechanisms [Moulin 99] Given: cost fn C(S) on subsets S of U Cost-Sharing Method: for every set S, defines a cost share χ(i,S) for every i in S (suggested prices) Defn: χ is ß-budget-balanced (ß-BB) if prices charged within ß of C(S) Moulin mechanism: simulate ascending auction using χ to compute prices at each iteration. Price = c(e 1 ) + c(e 2 )/3 + c(e 3 )/4 e2e2 e1e1 e3e3

8 8 Moulin Mechanisms: Good News Fact: [Moulin 99] if cost-sharing method χ is monotone (price for each player only increases), then the Moulin mechanism is truthful. utility = v i - p i if i wins, 0 otherwise reason: same as a classical ascending auction Also: groupstrategyproof (form of collusion-resistance) prices charged cover cost incurred (up to ß factor)

9 9 Moulin Mechanisms: Bad News Claim: Moulin mechanisms (e.g., the Shapley mechanism) need not maximize surplus. e 1 = 1 + k players with valuations: 1,1/2, 1/3, …, 1/k

10 10 Moulin Mechanisms: Bad News Claim: Moulin mechanisms (e.g., the Shapley mechanism) need not maximize surplus. opt surplus (ln k) - 1, Shapley surplus = 0 e 1 = 1 + k players with valuations: 1,1/2, 1/3, …, 1/k

11 11 Moulin Mechanisms: Bad News Claim: Moulin mechanisms (e.g., the Shapley mechanism) need not maximize surplus. opt surplus (ln k) - 1, Shapley surplus = 0 Negative result [GL,R,FKSS] : no truthful mechanism gets non-trivial approximation of BB + surplus. e 1 = 1 + k players with valuations: 1,1/2, 1/3, …, 1/k

12 12 Measuring Surplus Loss Goal: minimize worst-case surplus loss. surplus of S: v(S) - C(S) Defn: social cost of S: π(S) = C(S) + v(U\S) U = set of all players note: social cost = -surplus + v(U) Bad example: opt social cost 1, Shapley social cost ln k e 1 = 1 + 1,1/2, 1/3, …, 1/k

13 13 Measuring Surplus Loss Goal: minimize worst-case surplus loss. surplus of S: v(S) - C(S) Defn: social cost of S: π(S) = C(S) + v(U\S) U = set of all players note: social cost = -surplus + v(U) Bad example: opt social cost 1, Shapley social cost ln k Defn: a mechanism is α-approximate if it is an α- approximation algorithm w.r.t. the social cost objective (in the usual sense). e 1 = 1 + 1,1/2, 1/3, …, 1/k

14 14 Goal + Main Result High-level goal: subject to reasonable BB, design mechanism with smallest approximation factor. note: requires both upper + lower bound results precisely quantifies inevitable surplus loss

15 15 Goal + Main Result High-level goal: subject to reasonable BB, design mechanism with smallest approximation factor. note: requires both upper + lower bound results precisely quantifies inevitable surplus loss Main result: complete soln for Moulin mechanisms. [Roughgarden/Sundararajan STOC 06], [Chawla+R+S WINE 06], [R+S IPCO 07]

16 16 Goal + Main Result High-level goal: subject to reasonable BB, design mechanism with smallest approximation factor. note: requires both upper + lower bound results precisely quantifies inevitable surplus loss Main result: complete soln for Moulin mechanisms. [Roughgarden/Sundararajan STOC 06], [Chawla+R+S WINE 06], [R+S IPCO 07] Ex: multicast: Shapley is optimal Moulin mechanism approximation factor of social cost = H k extends to all submodular cost functions

17 17 More Examples Examples: uncapacitated facility location: the [Pal-Tardos 03] mechanism = optimal Moulin mechanism optimal approximation = Θ(log k) Steiner tree: the [Jain-Vazirani 01] mechanism = optimal Moulin mechanism optimal approximation factor of social cost = Θ(log 2 k) also extends to Steiner forest mechanism of [Konemann/Leonardi/Schaefer SODA 05] and rent-or buy mechanism of [Gupta/Srinivasan/Tardos 03]

18 18 Proof Techniques Part I: (problem-independent) identify parameter of a monotone cost-sharing method that controls approximation factor of Moulin mechanism [upper and lower bounds] reduces property of mechanism to property of method Part II: (problem-dependent) prove upper bound on parameter for favorite mechanisms, lower bound for all mechanisms has flavor of analysis of online algorithms

19 19 A Natural Lower Bound consider a cost-sharing method χ for C + corresponding Moulin mechanism M order the players of U = {1,2,...,k} let x i = χ(i,{1,2,...,i}) set v i = x i - M outputs Ø, social cost Σ i x i ; OPT is C(U) Σ i χ(i,{1,2,...,i})/C(U) lower bounds approximation factor e 1 = 1 + 1,1/2, 1/3, …, 1/k

20 20 A Natural Lower Bound consider a cost-sharing method χ for C + corresponding Moulin mechanism M order the players of U = {1,2,...,k} let x i = χ(i,{1,2,...,i}) set v i = x i - M outputs Ø, social cost Σ i x i ; OPT is C(U) Σ i χ(i,{1,2,...,i})/C(U) lower bounds approximation factor Defn: the summability α of χ for C is the largest lower bound arising in this way. e 1 = 1 + 1,1/2, 1/3, …, 1/k

21 21 A Key Theorem Summary: a Moulin mechanism based on an α- summable cost-sharing method is no better than α-approximate.

22 22 A Key Theorem Summary: a Moulin mechanism based on an α- summable cost-sharing method is no better than α-approximate. Theorem [Roughgarden/Sundararajan STOC 06] : a Moulin mechanism based on an α-summable, ß- BB cost-sharing method is (α+ß)-approximate. Point: for every O(1)-BB method χ, the parameter α completely characterizes the approximation factor of the corresponding mechanism.

23 23 Beyond Moulin Mechanisms Question: why obsessed with Moulin mechanisms? only general technique to achieve truthful + BB strong lower bounds for approximation for some problems [Immorlica/Mahdian/Mirrokni SODA 05] non-trivial to design (e.g., for UFL)

24 24 Beyond Moulin Mechanisms Question: why obsessed with Moulin mechanisms? only general technique to achieve truthful + BB strong lower bounds for approximation for some problems [Immorlica/Mahdian/Mirrokni SODA 05] non-trivial to design (e.g., for UFL) Acyclic Mechanisms [Mehta/Roughgarden/Sundararajan EC 07] : generalizes Moulin mechanisms. idea: order offers within iteration of ascending auction most "off-the-shelf" primal-dual algorithms work as is exponentially better BB + efficiency for e.g. Set Cover

25 25 Shapley Network Design Games Given: G = (V,E), fixed costs c e k players = vertex pairs (s i,t i ) each picks an s i -t i path Shapley cost sharing: cost of each edge of formed network split equally among users [Anshelevich et al FOCS 04] full-information noncooperative game

26 26 Inefficiency under Shapley Recall: with Shapley cost sharing, POA = k, even in undirected graphs POS = H k in directed graphs (unknown in undirected graphs) t s 1+ k 1 1 k == t k- 1

27 27 Inefficiency under Shapley Recall: with Shapley cost sharing, POA = k, even in undirected graphs POS = H k in directed graphs (unknown in undirected graphs) Question #1: can we do better? Question #2: subject to what? t s 1+ k 1 1 k == t k- 1

28 28 In Defense of Shapley Essential properties: (non-negotiable) "budget-balanced" (total cost shares = cost) "separable" (cost shares defined edge-by-edge) pure-strategy Nash equilibria exist Bonus good properties: (negotiable) "uniform" (same definition for all networks) "fair" (characterizes Shapley)

29 29 Key Question The Problem: design edge cost-sharing methods to minimize worst-case POA and/or POS. directed vs. undirected uniform vs. non-uniform single-sink vs. terminal pairs [Chen/Roughgarden/Valiant 07] Related work: coordination mechanisms [Christodoulou/Koutsoupias/Nanavati ICALP 04], [Immorlica/Li/Mirrokni/Schulz 05], [Azar et al 07] resource allocation [Johari/Tsitsiklis 07]

30 30 Directed Graphs Negative result: worst-case POA = k for every cost-sharing method, even non-uniform.

31 31 Directed Graphs Negative result: worst-case POA = k for every cost-sharing method, even non-uniform. Theorem: Shapley is the optimal uniform cost- sharing method! For every method, either: (1) there is a network game s.t. POS H k OR (2) there is a network game with no Nash eq.

32 32 Directed Graphs Negative result: worst-case POA = k for every cost-sharing method, even non-uniform. Theorem: Shapley is the optimal uniform cost- sharing method! For every method, either: (1) there is a network game s.t. POS H k OR (2) there is a network game with no Nash eq. Shapley can be justified on efficiency grounds, not just usual fairness/simplicity reasons open: what's up with non-uniform methods?

33 33 Undirected Graphs: Uniform Theorem: in undirected graphs, can reduce the worst-case POA to polylogarithmic! simple uniform priority-based scheme POA = O(log k) in with single sink, O(log 2 k) for pairs (follows from [IW 91], [AA96] )

34 34 Undirected Graphs: Uniform Theorem: in undirected graphs, can reduce the worst-case POA to polylogarithmic! simple uniform priority-based scheme POA = O(log k) in with single sink, O(log 2 k) for pairs (follows from [IW 91], [AA96] ) Theorem: For every unform cost-sharing method, worst-case POA = Ω(log k). [even single-sink] follows from complete characterization of uniform cost-sharing methods that always admit PNE

35 35 Undirected: Non-Uniform Theorem: Can reduce POA to 2 in single-sink networks via non-uniform method. idea: use Prim MST to define priority scheme easy: matching lower bound Theorem: For every non-uniform method, worst- case POA is general networks is Ω(log k). extremal graph construction lower bounds for "oblivious network design"

36 36 Open Questions Cost-Sharing Mechanism Design: lower bounds for non-Moulin mechanisms more applications of acyclic mechanisms profit-maximization Optimal Protocol Design: non-uniform methods in directed graphs lower bounds for scheduling mechanisms new applications (selfish routing, fair queuing)


Download ppt "Minimizing Efficiency Loss in Mechanism and Protocol Design Tim Roughgarden (Stanford) includes joint work with: Shuchi Chawla (Wisconsin), Ho-Lin Chen."

Similar presentations


Ads by Google