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

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

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

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 vi for winning

Terminology: surplus of S = v(S) - C(S) [where v(S) = Σi vi]

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

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 < pi and iterate

Price =

c(e1) + c(e2)/3 + c(e3)/4

e2

e1

e3

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(e1) + c(e2)/3 + c(e3 )/4

e2

e1

e3

8

Moulin Mechanisms: Good NewsFact: [Moulin 99] if cost-sharing method χ 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)

9

Moulin Mechanisms: Bad NewsClaim: Moulin mechanisms (e.g., the Shapley

mechanism) need not maximize surplus.

e1 = 1 + k players with valuations:1,1/2, 1/3, … , 1/k

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Moulin Mechanisms: Bad NewsClaim: Moulin mechanisms (e.g., the Shapley

mechanism) need not maximize surplus.

opt surplus (ln k) - 1, Shapley surplus = 0

e1 = 1 + k players with valuations:1,1/2, 1/3, … , 1/k

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Moulin Mechanisms: Bad NewsClaim: 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.

e1 = 1 + k players with valuations:1,1/2, 1/3, … , 1/k

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

e1 = 1 +

1,1/2, 1/3, … , 1/k

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).

e1 = 1 +

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],

[Chawla+R+S WINE 06], [R+S 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],

[Chawla+R+S WINE 06], [R+S 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 = Θ(log k)

Steiner tree: the [Jain-Vazirani 01] mechanism = optimal Moulin mechanism optimal approximation factor of social cost = Θ(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 χ for C + corresponding Moulin mechanism M

order the players of U = {1,2,...,k} let xi = χ(i,{1,2,...,i}) set vi = xi - M outputs Ø, social cost Σi xi ; OPT is ≤ C(U)

Σi χ(i,{1,2,...,i})/C(U) lower bounds approximation factor

e1 = 1 +

1,1/2, 1/3, … , 1/k

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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 xi = χ(i,{1,2,...,i}) set vi = xi - M outputs Ø, social cost Σi xi ; OPT is ≤ C(U)

Σi 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.

e1 = 1 +

1,1/2, 1/3, … , 1/k

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A Key Theorem

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

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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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

25

Shapley Network Design GamesGiven: 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)

t

s

1+ k

1 1k12 13

= =

t

0 0 0 0

1+ . . .0

1k-1

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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?

t

s

1+ k

1 1k12 13

= =

t

0 0 0 0

1+ . . .0

1k-1

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

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.

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 Hk OR

(2) there is a network game with no Nash eq.

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 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: UniformTheorem: 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])

34

Undirected Graphs: UniformTheorem: 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 = Ω(log k). [even single-sink]

follows from complete characterization of uniform cost-sharing methods that always admit PNE

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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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)