Constantinos Costis Daskalakis MIT Yang Cai McGill Matt Weinberg Princeton Algorithm Algorithm Design desired Output given Input Algorithm Agents Reports Agents Payoffs ID: 590012
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Slide1
Reducing Mechanism Design to Optimization
Constantinos (Costis) Daskalakis (MIT)
Yang Cai
(McGill)
Matt Weinberg (Princeton)Slide2
Algorithm
Algorithm Design
(desired)
Output
(given)
InputSlide3
Algorithm
Agents’
Reports
Agents’
Payoffs
Mechanism Design
(desired)
Output
(given)
Input
MechanismSlide4
E.g. Computing MaxInput: x
1, x2,…, xnGoal:
compute max(x1
,…,xn)
Algorithm: TrivialBut what if inputs are strategic?suppose input
i
has value
x
i
for being
selected&algorithm
doesn’t know
x
ifacing trivial algorithm, every input reports
+∞A better Algorithm
[Vickrey’61 ]:collect reported inputs:
b
1,…, bn (can’t enforce bi=x
i a priori)select i* = arg max bi
charge winner
i
* the 2nd highest report: arg maxj≠i* bjClaim: It is in every i’s best interest to report bixi (regardless of the reports of the other inputs). Vickrey auction is the new max.Slide5
Agents’
Reports
Agents’
Payoffs
Mechanism vs Algorithm Design
(desired)
Output
(given)
Input
MechanismSlide6
[Nisan-Ronen’99]:How much more difficult are optimization problems on “strategic” input compared to “honest” input?
Algorithmic Mechanism DesignInformation:what information does the mechanism have about the inputs?what information do the inputs have about each other?
does the mechanism also have some private information whose release may
influence the inputs’ behavior (e.g. quality of a good in an auction)?
Complexity:computational, communication, …
centralized:
complexity to run the mechanism
vs distributed
: complexity for each input to optimize own behaviorSlide7
Combinatorial Auctions
Setting: bidders e.g. att,
verizon, t-mobile
and several regional providers.set
of non-identical items.
e.g. licenses for broadcasting at a certain frequency in a given region.
each bidder
has valuation function
Constraints:
Each item can be given to one bidder
An allocation is a
-dimensional vector
of disjoint sets, where
is the bundle of items allocated to bidder
.
There are
different allocations
Goal:
Choose allocation maximizing
[VCG ’73]:
Optimal auction:
Ask bidders to report their valuation functions
Choose
maximizing
Charge “Clarke payments”
e
nsures 1 is truthful
Generalizes
Vickrey’s
auctionSlide8
VCG vs Communication vs ComputationObvious Questions:How are valuation functions communicated in Step 1
A valuation
requires
numbers to be specifiedEven when Step 1 can be carried out with reasonable communication, Step 2 might be computationally intractablee.g. maybe bidders are single-minded, desiring the whole neighborhood of a node in a network, at value
Finding welfare maximizing allocation is as hard as finding a maximum independent set (NP-complete)
[VCG ’73]:
Optimal auction:
Ask bidders to report their valuation functions
Choose
maximizing
Charge “Clarke payments”
Cheat Sheet:
= #bidders
=
#items
= bidder ’s valuation Slide9
The Menu
Computing on Strategic Inputs
Black-Box Reductions for Bayesian Mechanism Design
Truthfulness vs Computation vs Communication
Statement of Main Result & Proof Vignette
Beyond Bayes?Slide10
The Menu
Computing on Strategic Inputs
Black-Box Reductions for Bayesian Mechanism Design
Truthfulness vs Computation vs Communication
Statement of Main Result & Proof Vignette
Beyond Bayes?Slide11
Reducing CommunicationSolution 1: Assume bidder valuations are not arbitraryHow can a bidder maintain
numbers in his head anyway?Chances are there is some succinct description of the valuationsOnly target succinct valuations, such as single-minded, additive, unit-demand, which can be communicated efficientlyNow only roadblock is computational (optimizing welfare given these valuations)Solution 2: Resort to indirect mechanismsE.g. design mechanisms that interact w/ bidders in several rounds, asking questions such as value queries: “what is your value for bundle
?”demand queries: “Given item prices
what bundle
optimizes
?”
Cheat Sheet:
= #bidders
=
#items
= bidder
’s
valuation
Slide12
Communication vs TruthfulnessDef:
is submodular iff
[Vondrak’08]:
Consider a combinatorial auction with submodular bidders. With value query access to
true
bidder valuations can achieve
-fraction of optimal welfare in polynomial (in both
and
) #queries/time.
[Dughmi-Vondrak’11]:
If a truthful mechanism makes
value queries
to bidders and guarantees
-fraction of optimal welfare, then it must make exponentially many queries.
In other words, clear separation in the communication complexity of mechanisms vs algorithms under value queries.
truthfulness, communication and approximation are at odds w/ each other
Cheat Sheet:
= #bidders
= #items = bidder ’s valuation
submodular
Some hope:
[…,
Dobzinski’16]
:
-
appx using poly time, poly
demand
queries.
“Given item prices
what
bundle optimizes
?”
Slide13
Truthfulness vs ComputationLast slide: lower bounds on communication.What if valuations can be described succinctly?
[Dobzinski-Vondrak’12]: Even if each bidder’s valuation can be succinctly described (w/ communication):No deterministic poly-time truthful mechanism can get better than
-fraction of optimal welfare (unless NP=RP)No
randomized poly-time truthful mechanism can get better than
-fraction of optimal welfare (unless NP
P/poly
)
[Papadimitriou,
Schapira
, Singer’08;
Buchfuhrer
et al’10;
Dobzinski
, Vondrak’12; Daniely,
Schapira, Shahaf’15]:
APX
APX
IC
problems that can be -approximated in poly-time, when input is known
problems that can be
-approximated in poly-time via truthful algorithms
Cheat Sheet: = #bidders = #items
= bidder
’s
valuation
submodularSlide14
Bayes to the RescueSame setting, except for every bidder
a distribution over valuations is known.To ease discussion and leave communication issues aside, suppose:
is set of possible allocationseach valuation
(normalized, explicit)
each
has finite support
New goal:
Design mechanism that optimizes expected welfare.
“expected” w.r.t. the
’s, the randomness in bidders’ strategies (if any), and the randomness employed by the mechanism (if any)
[
Hartline-Lucier’10,
Bei
,
Huang’11, Hartline-Malekian-Kleinberg’11]:
Suppose auctioneer has (black-box access to) algorithm which, given valuations
,
outputs allocation achieving -fraction of optimal welfare.Then,
, can construct poly-time truthful mechanism whose expected welfare is (runtime is polynomial in
)
So…
Bayes-APX Bayes-APXIC ! Cheat Sheet: = #bidders = #items
= bidder
’s
valuation
Slide15
The Menu
Computing on Strategic Inputs
Black-Box Reductions for Bayesian Mechanism Design
Truthfulness vs Computation vs Communication
Statement of Main Result & Proof Vignette
Beyond Bayes?Slide16
[Nisan-Ronen’99]:How much more difficult are optimization problems on “strategic” input compared to “honest” input?
Mechanism Design via ReductionsBlack-box reduction from mechanism- to algorithm-design for all optimization problemsThe Dream:
Agent 1
Input
Agent n
Input
Output
Algorithm
that works on strategic input
…
Want:
Algorithm that works on honest input
Have:
Known Input
Output Slide17
[Nisan-Ronen’99]:
How much more difficult are optimization problems on “strategic” input compared to “honest” input?
Mechanism Design via ReductionsBlack-box reduction from mechanism- to algorithm-design for all optimization problems
The Dream:
Agent 1
Input
Agent n
Input
Output
Algorithm that works on strategic input
Algorithm that works on honest input
…
Want:
Have:
Known Input
Output Slide18
[Nisan-Ronen’99]:
How much more difficult are optimization problems on “strategic” input compared to “honest” input?
Mechanism Design via ReductionsBlack-box reduction from mechanism- to algorithm-design for all optimization problems
The Dream:
Agent 1
Input
Agent n
Input
Output
Algorithm that works on strategic input
Algorithm that works on honest input
…
Want:
Have:
Known Input
Output 1
Chosen Input 1
Output k
Output
Chosen Input k
…Slide19
Why Black-Box Reductions?
Mechanism Design
Algorithm
Design
More is known about algorithms than mechanisms
Hope: unsolved problems might reduce to already-solved problems
Allows larger community to tackle important problems
Without necessarily learning game theory
Provides deeper understanding of Mechanism Design
What makes incentives so difficult to deal with?
[
Bei
,
Huang’11; Hartline-Malekian-Kleinberg’11]:
Dream true for
Bayesian
welfare
maximization.
[Cai-Daskalakis-Weinberg’12-’14]:
Mechanism to algorithm design reduction exists for any Bayesian optimization problem! (with right qualifications
)Slide20
Black-Box Reductions – Beyond Welfare[Cai-Daskalakis-Weinberg’12-’14]:
Mechanism to algorithm design reduction exists for any Bayesian optimization problem! (w/ right qualifications) [Important Corollaries]: 1. Exist poly-time revenue-optimal multi-item auctions with additive bidders (
Provides a generalization of
[Myerson’81]
revenue-optimal single-item auctions
c
an also accommodate budget constraints
[Bhalgat-Gollapudi-Munagala’13,Daskalakis-Devanur-Weinberg’15]
, envy-freeness, non-standard objectives, e.g.
Makespan
, fairness
[Daskalakis-Weinberg’15]
2. w/ submodular bidders revenue optimization is highly
inapproximable
for truthful mechanisms
3. [Cai-Devanur-Weinberg’16]
deduce structural understanding of approximately optimal auctions, unifying and strengthening [
Chawla-Hartline-Kleinberg’07,Chawla et al’10,Hart-Nisan’12,Babaioff et al ’14, Yao’15]
N.B. Parallel work of [Alaei-Fu-Haghpanah-Hartline-Malekian’12] obtains results for special case of “service-constraint-environments”
Slide21
The Menu
Computing on Strategic Inputs
Black-Box Reductions for Bayesian Mechanism Design
Truthfulness vs Computation vs Communication
Statement of Main Result & Proof Vignette
Beyond Bayes?Slide22
The Menu
Computing on Strategic Inputs
Black-Box Reductions for Bayesian Mechanism Design
Truthfulness vs Computation vs Communication
Statement of Main Result
& Proof Vignette
Beyond Bayes?Slide23
Mechanism Design
Given: 1. Objective O : X n ×
S
, where
X is the input set, and S
is the solution set
2. access to strategic agents
1
…
n
s.t.
agent
i
:
knows the ith
input
xi has keen interest xi(s)
in our choice of s S Bayesian assumption:
x
i
Fi (Fi known) Goal: Find truthful mechanism optimizing objective O (in expectation over × Fi ), among all possible mechanisms A General Reduction
Algorithm Design
Given:
1. same
X
,
S, O
2.
known
inputs
x
1
,…,
x
n
X 3. known
inputs y1,…,yn X ± Goal: Find s S to optimize O(x1,...,xn,s) + i
yi
(
s
)
Impossible [CIL’12]
= reduction
e.g.
Makespan
(
n
strategic machines,
m
jobs)
Input
set:
X
=
(processing times for m jobs)
Solution
set:
S
= {
s
: [
m
]
[
n
]} (schedules)
Objective:
O
(
x
1
,…
x
n
;
s
) =
max
i
j
s
ij
x
ij
Agent
i
:
knows
x
i
= (
x
i1
,…,
x
im
)
has value
-
j
s
ij
x
ij
for schedule
s
Goal:
Minimize
O
in expectation, truthfully
~
F
i
Makespan
with costs
Input
:
x
1
,…,
x
n
y
1
,…,
y
n
Minimization Objective:
max
i
j
s
ij
x
ij
-
j
j
s
ij
y
ij
Slide24
Mechanism Design
Given: 1. Objective O : X n ×
S
, where X
is the input set, and S is the solution set
2. access to strategic agents
1
…
n
s.t.
agent
i
:
knows the ith
input x
i
has keen interest xi(s) in our choice of s
S Bayesian assumption: xi
F
i
(Fi known) Goal: Find truthful mechanism optimizing objective O (in expectation over × Fi ), among all possible mechanisms A General ReductionAlgorithm Design
Given:
1. same
X
,
S, O
2.
known
inputs
x
1
,…,
x
n
X 3. known inputs y
1,…,yn
X ± Goal: Find s S to optimize O(x1,...,xn,s) + i yi (s
)
[Cai-Daskalakis-Weinberg’13]:
Polynomial-time, black-box reduction from mechanism design for arbitrary objective
O
to algorithm design for
same objective
O
plus linear cost function.
i.e. if RHS tractable, then LHS tractable
also approximation preserving:
i.e.
α
-approximation to RHS
α
-approximation to LHS
techniques
:
probability and convex programming: approximation-sensitive versions of the equivalence of optimization and separation
[Grötschel-Lovász-Schrijver’80, Karp-Papadimitriou’80
];
constructive versions of Border’s theorem
[…,Border’91,’07, CKM’11, CDW’12,
Alaei
et al’12]
and multi-item extensions thereof
[CDW’12]
Impossible [CIL’12]
= reductionSlide25
Proof Vignette
Mechanism design as an optimization problemSimple Setting: item, bidders, bidder ’s value
Goal:
Find revenue-optimal auction
Combinatorial optimization approach: describe mechanism through its
Allocation function:
Price function:
Interpretation:
Mechanism will ask bidders to report their values
probability item goes to bidder
when bidders report
price charged to bidder
when bidders report
Feasibility constraints:
Truthfulness constraints:
Revenue:
Slide26
Proof
Vignette (cont.)Feasibility constraints:
Truthfulness constraints:
Revenue:
Issues: Too many variables/constraints
If
types per agent,
variables
Interim
a
llocation rule:
Represents expected prob
. of allocation to bidder
, conditioning on reporting
, in expectation over other bidders’ reports assuming that they report truthfully
Interim
price rule:
Feasibility
constraints:
ex-post allocation rule consistent w/
Truthfulness:
Revenue:
Slide27
example: 2 bidders w/ values uniform in V
1={A, B, C} and V2={D, E, F}Question: Is
shown below feasible?
A vs D/E/F
A wins.
B/C vs D
D wins.
A
B
⅓
⅓
C
⅓
D
E
⅓
⅓
F
⅓
B vs F
B wins.
C vs E
E wins.
B vs E
B needs to win
w.p
. ½, E needs to win
w.p
. ⅔
Single-Item Interim Feasibility
so infeasible !
✔
✔
Slide28
[
Border ’91, Border ’07, Che-Kim-Mierendorff ’11]: A (single-item) interim allocation rule
is feasible
iff
BUT
still too many constraints
if
types per agents,
constraints
[Cai-Daskalakis-Weinberg’12]:
constraints suffice.
[
Alaei
et al’12]:
Polynomial-time
algorithm for
checking feasibility.
Feasibility constraints:
"Interim allocation feasible"
Truthfulness:
Revenue:
Now a poly-size LP!
Single-Item Interim Feasibility (cont.)
p
robability item goes to some bidder
whose value
probability there appears some bidder
whose value
Slide29
Multi-Item Interim FeasibilitySo far: single-item interim feasibilityHow about multi-item settings?
pertinent question: what information to include in interim rule in general settings?Approach 1:Maintain for each bidder , for each possible valuation , for each item
, the expected probability
that item is allocated to bidder
, when he reports
to the mechanism, in expectation w.r.t. the other bidders’ valuations
Question: Can you express bidder’s expected utility w.r.t. these variables?
Yes, if bidder additive/unit demand
e.g. additive valuation:
No, in general
e.g.
,
Allocation 1: allocate both shoes w/
prob
½
Allocation 2: allocate a random shoe
Same interim probabilities, but different expected values
Slide30
Multi-Item Interim Feasibility (cont.)Approach 2:Maintain for each bidder
, for each possible valuation , for each set of items , the expected probability
that subset
is allocated to bidder
, when he reports to the mechanism, in expectation w.r.t. the other bidders’ valuations
Works for general valuations, but computationally expensive
Our approach:
let’s not be literal w/ “allocation rule”
Write LP w.r.t. “interim swap valuation function”
Maintain for each bidder
, for each possible
pair of valuations
,
the expected
value
of bidder
when his real valuation is
but he reports
to the mechanism, in expectation w.r.t. the other bidders’ valuations
[CDW’13]:
Can always write poly-size LP in terms of interim swap valuation functions.
Slide31
Approach (really sketchy)
Algorithm Design Given: 1. same X , S, O
2. known inputs x1
,…,xn
X
3.
known
inputs
y
1
,…,
y
n
X
±
Goal: Find
s S to optimize O(
x1,...,xn,s) +
i
yi (s)Write Polynomial Size LP using interim swap valuationissue: - feasibility hard to check Given an interim swap valuation function, is there a feasible mechanism that induces it?
need separation oracle
= reduction
Mechanism Design
Given:
1. Objective
O
:
X
n
×
S
, where
X is the input set, and S is the solution set 2. access to strategic agents 1…n s.t. agent i : knows the ith input
xi has keen interest x
i
(
s
)
in our choice of
s
S
Bayesian assumption:
x
i
F
i
(
F
i
known)
Goal:
Find truthful mechanism optimizing objective
O
(in expectation over
×
F
i
), among all possible mechanisms
s
uffices to get separation oracleSlide32
The Menu
Computing on Strategic Inputs
Black-Box Reductions for Bayesian Mechanism Design
Truthfulness vs Computation vs Communication
Statement of Main Result & Proof Vignette
Beyond Bayes?Slide33
The Menu
Computing on Strategic Inputs
Black-Box Reductions for Bayesian Mechanism Design
Truthfulness vs Computation vs Communication
Statement of Main Result & Proof Vignette
Beyond Bayes?Slide34
XOS
Non-Bayesian Welfare Optimization
additive
Sub-modular
unit-demand
Sub-additive
Strong Communication/Computational LB for value queries
-appx under demand queries
VCG gets OPT in poly-time, poly-communication, dominant strategies
[w/ Syrgkanis’16]:
no-regret learning cannot be efficiently implemented in
SiSPAs
[w/ Syrgkanis’16]: N
o-envy learning
can be efficiently implemented; gives 0.5 approximation
No-regret against best bid vector in hindsight
No-regret against best bundle in hindsight
, for some vectors
[Bik’99,CKS’08, BR’11, HKMN’11,FKL’12,ST’13, FFGL’13]:
no-regret learning
results in 0.25 OPT in simultaneous second price auctions (
SiSPAs
)
Sell items separately through second price auctionsSlide35
No-Envy Learning Outcomes
World view for XOS bidders in SiSPAs
Nash Eq.
Correlated Eq.
Coarse correlated Eq.
No-Regret Learning Outcomes
welfare
Still
welfare
but no-regret learning intractable
and no-envy learning tractableSlide36
SummaryThis Talk: Optimization on strategic inputStandard mechanism design theory may not be implementable once computational/communication considerations are introducedThe fields of communication and computational complexity provide tools for mechanism design w/ these considerations
Bayesian MD gets around computational intractability barriersw/ Cai and Weinberg, we obtain a generic, poly-time reduction from mechanism to algorithm design“optimizing on strategic inputs is no harder than adding linear cost to your objective”Important Corollary: revenue optimal multi-item auctions; generalizes [Myerson’81]Interaction of worst-case and Bayesian guarantees not well-understoodE.g. dominant strategy truthful mechanisms in Bayesian settings with optimal expected revenue?Non-Bayesian Setting: Syrgkanis and I broke the submodular bidder barrier using no-envy learning. Beyond XOS bidders? Other applications?Slide37
A little Econ Aesthetic ?Structure vs Computation?Closed-form Solutions:
universal claims about optimal solutionAlgorithms: computation of optimal solution on instance-by-instance basisOpen Problem: Understand output of our algorithm for revenue optimization in multi-item settingsIn single-bidder settings:
[Daskalakis-Deckelbaum-Tzamos’13, Koutsoupias-Giannakopoulos’14, Haghpanah-Hartline’14,Babaioff et al’14] [
Daskalakis-Deckelbaum-Tzamos’16]: Use of optimal transport theory (generalizations of Monge-Kantorovich duality) to characterize the structure of optimal single-buyer multi-item mechanisms.
See my survey: “Auctions Defying Intuition?”, in SIGECOM July’15upcoming Econometrica paper: “Strong Duality for a Multi-Good Monopoly”Multi-bidder settings?
see
[Yao’15, Cai-Devanur-Weinberg’16]
for a characterization of approximately optimal mechanisms
Exact?
Thanks!Slide38
Approach (really sketchy)
Algorithm Design Given: 1. same X , S, O 2. known
inputs x1,…,x
n
X 3. known inputs y
1
,…,
y
n
X
±
Goal: Find
s
S to optimize
O(x1,...,xn,s
) + i yi (
s
)
Write Configuration LP- incentives, feasibility linearissue: - exponential size (distribution over S for every possible report by the strategic inputs)Compress to Polynomial Size (non-configuration) LP via projectionissue: - feasibility hard to check
need separation oracle
= reduction
Mechanism Design
Given:
1. Objective
O
:
X
n
×
S
, where
X is the input set, and S is the solution set 2. access to strategic agents 1…n s.t. agent i : knows the ith input xi
has keen interest xi(
s
)
in our choice of
s
S
Bayesian assumption:
x
i
F
i
(
F
i
known)
Goal:
Find truthful mechanism optimizing objective
O
(in expectation over
×
F
i
), among all possible mechanisms
Slide39
Threshold price to win set
No-Envy vs No-Regret Learning
Fix mechanism
Suppose
bidders engage in a repeated execution of mechanism
bidder
’s action in round
in
SiSPAs
, this is a vector of bids on each item
i
n more complex mechanisms, more complex
Def:
An algorithm that chooses bid
after observing
for all
is
“no-regret”
iff
for any (adaptively chosen)
Def:
An algorithm that chooses bid
after observing
for all
is
“no-regret”
iff
for any (adaptively chosen)
Provenance:
Walrasian
equilibrium
in general, incomparable
(for
SiSPAs
, XOS bidders)
Slide40
No-envy Learning in SiSPAsSetting:
items sold to bidders each item sold via a second price auctionbidders bid simultaneously on each item2nd price auctions run in parallelassumption: each bidder has an XOS valuation (
submodular)access to valuations: each bidder is capable of making demand queries to his valuation function
“under prices
what’s my favorite bundle?”
[Daskalakis-Syrgkanis’16
]:
Consider some bidder
with an XOS valuation, participating in a sequence of
SiSPAs
. After each round
the bidder sees the other bidders’ bids
(can relax) before choosing his bid
in round
There is a polynomial-time learning algorithm guaranteeing average utility:
(Each step requires a demand query)
When all bidders use above algorithm, average bidder utilities converge to at least half
of optimal welfare
.
Simultaneous Second-Price Auction (
SiSPA
)
Average competing bid on item j
Enabled by
PoA
type analysis
Enabled by new online learning algorithms