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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)
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
10
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
11
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
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
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
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
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(log2 k) also extends to Steiner forest mechanism of
[Konemann/Leonardi/Schaefer SODA 05] and rent-or buy mechanism of [Gupta/Srinivasan/Tardos 03]
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
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
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 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
21
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.
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
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
26
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
27
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]
30
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?
33
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"
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)