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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)

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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

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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

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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 ]

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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]

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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

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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

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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)

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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

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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

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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

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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

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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

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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

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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]

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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

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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]

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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

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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

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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

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

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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.

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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)

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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

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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

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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

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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

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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)

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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]

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

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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.

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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?

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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] )

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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

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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"

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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)

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