Minimizing%20Efficiency%20Loss%20in%20Mechanism%20and%20Protocol%20Design

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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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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
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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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 vi for winning
• Terminology
• surplus of S v(S) - C(S) where v(S) Si vi

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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/Krishnam
  urthy/Sami/Shenker 03

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Shapley Mechanism for Multicast

• collects bids (bi for each i)
• initialize S all players
• share each edge equally
  among its users
• if bi ? pi for all i, done.
• else drop a player i with
  bi lt pi and iterate

e3
e2
e1
Price c(e1) c(e2)/3 c(e3)/4
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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 ?(i,S) for every
  i in S (suggested
  prices)
• Defn ? is ß-budget-balanced (ß-BB)
• if prices charged within ß of C(S)
• Moulin mechanism simulate ascending auction
  using ? to compute prices at each iteration.

e3
e2
e1
Price c(e1) c(e2)/3 c(e3 )/4
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Moulin Mechanisms Good News

• Fact Moulin 99 if cost-sharing method ? is
  monotone (price for each player only increases),
  then the Moulin mechanism is truthful.
• utility vi- pi 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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Moulin Mechanisms Bad News

• Claim Moulin mechanisms (e.g., the Shapley
  mechanism) need not maximize surplus.

k players with valuations 1,1/2, 1/3, , 1/k
e1 1 e
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Moulin Mechanisms Bad News

• Claim Moulin mechanisms (e.g., the Shapley
  mechanism) need not maximize surplus.
• opt surplus ? (ln k) - 1, Shapley surplus 0

k players with valuations 1,1/2, 1/3, , 1/k
e1 1 e
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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.

k players with valuations 1,1/2, 1/3, , 1/k
e1 1 e
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Measuring Surplus Loss

• Goal minimize worst-case surplus loss.
• surplus of S v(S) - C(S)
• Defn social cost of S p(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

e1 1 e
1,1/2, 1/3, , 1/k
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Measuring Surplus Loss

• Goal minimize worst-case surplus loss.
• surplus of S v(S) - C(S)
• Defn social cost of S p(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 a-approximate if it is an
  a-approximation algorithm w.r.t. the social cost
  objective (in the usual sense).

e1 1 e
1,1/2, 1/3, , 1/k
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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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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,
  ChawlaRS WINE 06, RS IPCO
  07

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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,
  ChawlaRS WINE 06, RS IPCO
  07
• Ex multicast Shapley is optimal Moulin
  mechanism
• approximation factor of social cost Hk
• extends to all submodular cost functions

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More Examples

• Examples
• uncapacitated facility location the Pal-Tardos
  03 mechanism optimal Moulin mechanism
• optimal approximation T(log k)
• Steiner tree the Jain-Vazirani 01 mechanism
  optimal Moulin mechanism
• optimal approximation factor of social cost
  T(log2 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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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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A Natural Lower Bound

• consider a cost-sharing method ? for C
  corresponding Moulin mechanism
  M
• order the players of U 1,2,...,k
• let xi ?(i,1,2,...,i)
• set vi xi - e
• M outputs Ø, social cost ? Si xi OPT is C(U)
• ? Si ?(i,1,2,...,i)/C(U) lower bounds
  approximation factor

e1 1 e
1,1/2, 1/3, , 1/k
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A Natural Lower Bound

• consider a cost-sharing method ? for C
  corresponding Moulin mechanism
  M
• order the players of U 1,2,...,k
• let xi ?(i,1,2,...,i)
• set vi xi - e
• M outputs Ø, social cost ? Si xi OPT is C(U)
• ? Si ?(i,1,2,...,i)/C(U) lower bounds
  approximation factor
• Defn the summability a of ? for C is the largest
  lower bound arising in this way.

e1 1 e
1,1/2, 1/3, , 1/k
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A Key Theorem

• Summary a Moulin mechanism based on an
  a-summable cost-sharing method is no better than
  a-approximate.

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A Key Theorem

• Summary a Moulin mechanism based on an
  a-summable cost-sharing method is no better than
  a-approximate.
• Theorem Roughgarden/Sundararajan STOC 06 a
  Moulin mechanism based on an a-summable, ß-BB
  cost-sharing method is (aß)-approximate.
• Point for every O(1)-BB method ?, the parameter
  a completely characterizes the approximation
  factor of the corresponding mechanism.

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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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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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Shapley Network Design Games

• Given G (V,E), fixed costs ce
• k players vertex pairs (si,ti)
• each picks an si-ti 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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Inefficiency under Shapley

• Recall with Shapley cost sharing,
• POA k, even in undirected graphs
• POS Hk in directed graphs
• (unknown in undirected graphs)

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Inefficiency under Shapley

• Recall with Shapley cost sharing,
• POA k, even in undirected graphs
• POS Hk in directed graphs
• (unknown in undirected graphs)
• Question 1 can we do better?
• Question 2 subject to what?

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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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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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Directed Graphs

• Negative result worst-case POA k for every
  cost-sharing method, even non-uniform.

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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 ? Hk OR
• (2) there is a network game with no Nash eq.

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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 ? Hk 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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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(log2 k) for
  pairs (follows from IW 91, AA96)

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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(log2 k) for
  pairs (follows from IW 91, AA96)
• Theorem For every unform cost-sharing method,
  worst-case POA O(log k). even single-sink
• follows from complete characterization of uniform
  cost-sharing methods that always admit PNE

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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 O(log k).
• extremal graph construction
• lower bounds for "oblivious network design"

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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)