Transcript 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),

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)
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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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) = Σi 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/Krishnamurthy/Sami/Shenker 03]
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Shapley Mechanism for Multicast

collects bids (bi for each i)

initialize S = all players

e3
e2
share each edge equally
among its users



if bi  pi for all i, done.
else drop a player i with
bi < pi and iterate
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
e3
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)
e2
e1
Price =
c(e1) + c(e2)/3 + c(e3 )/4
Moulin mechanism: simulate ascending auction
using χ to compute prices at each iteration.
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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 = 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.
e1 = 1 + e
k players with valuations:
1,1/2, 1/3, … , 1/k
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Moulin Mechanisms: Bad News
Claim: Moulin mechanisms (e.g., the Shapley
mechanism) need not maximize surplus.
e1 = 1 + e

k players with valuations:
1,1/2, 1/3, … , 1/k
opt surplus  (ln k) - 1, Shapley surplus = 0
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Moulin Mechanisms: Bad News
Claim: Moulin mechanisms (e.g., the Shapley
mechanism) need not maximize surplus.
e1 = 1 + e

k players with valuations:
1,1/2, 1/3, … , 1/k
opt surplus  (ln k) - 1, Shapley surplus = 0
Negative result [GL,R,FKSS]: no truthful mechanism
gets non-trivial approximation of BB + surplus.
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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: π(S) = C(S) + v(U\S)


U = set of all players
note: social cost = -surplus + v(U)
e1 = 1 + e
1,1/2, 1/3, … , 1/k
Bad example: opt social cost  1, Shapley social cost  ln 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: π(S) = C(S) + v(U\S)


U = set of all players
note: social cost = -surplus + v(U)
e1 = 1 + e
1,1/2, 1/3, … , 1/k
Bad example: opt social cost  1, Shapley social cost  ln k
Defn: a mechanism is α-approximate if it is an αapproximation algorithm w.r.t. the social cost
objective (in the usual sense).
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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
e1 = 1 + e
order the players of U = {1,2,...,k}
let xi = χ(i,{1,2,...,i})
1,1/2, 1/3, … , 1/k
set vi = xi - e
M outputs Ø, social cost  Σi xi ; OPT is ≤ C(U)
 Σi χ(i,{1,2,...,i})/C(U) lower bounds approximation factor
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A Natural Lower Bound





consider a cost-sharing method χ for C
+ corresponding Moulin mechanism M
e1 = 1 + e
order the players of U = {1,2,...,k}
let xi = χ(i,{1,2,...,i})
1,1/2, 1/3, … , 1/k
set vi = xi - e
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.
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A Key Theorem
Summary: a Moulin mechanism based on an αsummable cost-sharing method is no better than
α-approximate.
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A Key Theorem
Summary: a Moulin mechanism based on an αsummable cost-sharing method is no better than
α-approximate.
Theorem [Roughgarden/Sundararajan STOC 06]: a
Moulin mechanism based on an α-summable, ßBB 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
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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
t
Recall: with Shapley cost sharing,
 POA = k, even in undirected graphs
 POS = Hk in directed graphs

1+e
k
s
(unknown in undirected graphs)
t
1
1+e
0
12
13
=
= . . .1
0 0 0
0
1 k-
1k
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Inefficiency under Shapley
t
Recall: with Shapley cost sharing,
 POA = k, even in undirected graphs
 POS = Hk in directed graphs

1+e
s
(unknown in undirected graphs)
t
1
1+e
Question #1: can we do better?
k
0
12
13
=
= . . .1
0 0 0
0
1 k-
1k
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 costsharing 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 costsharing 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 = Ω(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, worstcase 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)
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