Transcript Revenue Optimization in MultiDimensional Settings Constantinos Daskalakis EECS, MIT Reference: http://arxiv.org/abs/1207.5518 ► ► Revenue-optimization, so far Jason’s talk:  Bayesian mechanism design: Auction design in presence of.

Revenue Optimization in MultiDimensional Settings
Constantinos Daskalakis
EECS, MIT
Reference: http://arxiv.org/abs/1207.5518
►
►
Revenue-optimization, so far
Jason’s talk:
 Bayesian mechanism design: Auction design in presence of stochastic
information about the bidders.
 Analysis tools: Bayes-Nash equilibrium.
 Single-dimensional setting: 1 type of item, many bidders, constraints on
who can be given an item simultaneously.
 [Myerson’81]: Revenue-optimal auction in all single-dimensional settings.
► In fact, a reduction from auction- design to algorithm- design:
Revenue optimization reduces to (virtual) welfare optimization.
Tim’s talk:
 Single-dimensional settings are not only optimally solvable, but also
amenable to simple approximations.
 “Exist constant-factor approximations which use little information about
the bidders.”
Beyond Single-Dimensional Settings
online marketplaces
- heterogeneous items
- complicated allocation constraints
sponsored search
online inventory allocation
spectrum auctions
Multi-dimensional Auction Setting
1
1
…
…
j
i
-
…
…
m
n
Bidders have values on items and bundles of items.
Bidder’s valuation (or type) encodes that information.
revenue/social
welfare/other
objective
Example Valuations
- Example 1: Additive bidder
- described by vector of values
- their valuation is:
(one value per item)
- Example 2: Single-minded combinatorial bidder
- described by the subset of items they desire S, and their value v
- their valuation is:
 For simplicity, throughout talk assume bidders are additive.
Multi-dimensional Auction Setting
1
…
…
j
i
-
…
…
m
revenue/social
welfare/other
objective
1
n
universe of possible
valuations for bidder i
Bidders have values on items and bundles of items.
Bidder’s valuation (or type)
encodes that information.
additive bidder:
Bidders’ types (t1,…,tm) come from known product distribution
.
Multi-dimensional Auction Setting
1
…
…
j
i
…
…
m
revenue/social
welfare/other
objective
1
n
universe of possible
valuations for bidder i
-
Bidders have values on items and bundles of items.
Bidder’s valuation (or type)
encodes that information.
additive bidder:
Bidders’ types (t1,…,tm) come from known product distribution
-
Auctioneer will decide some allocation A  [m] x [n], and charge prices.
There are (possibly combinatorial) constraints
on what
allocations are allowed.
.
Example 1: selling paintings
1
1
…
…
j
i
-
…
…
m
n
Items are paintings.
No painting should be given to more than one bidder
so:
Example 2: selling houses
1
1
…
…
j
i
-
…
…
m
n
Items are houses.
No house should be given to more than one bidder.
No bidder should receive more than one house.
so:
Example 3: selling doctor appointments
1
1
…
…
i
j
-
…
…
m
n
Items are slots with doctors in a hospital.
No slot should be given to more than one bidder.
No bidder should get more than one slot with same doctor, or overlapping
slots with different doctors.
Example 4: building bridges
1
…
i
…
m
-
Items are possible locations for building a bridge L = {l1, l2, …,ln}.
If a location is given to one bidder, it is given to all bidders (as every bidder
will use a bridge if it is built).
so:
Multi-dimensional Auction Setting
revenue/social
welfare/other
objective
1
1
…
…
j
i
…
…
m
n
universe of possible
valuations for bidder i
-
Bidders have values on items and bundles of items.
Bidder’s valuation (or type)
encodes that information.
Bidders’ types (t1,…,tm) come from known product distribution
-
Auctioneer will decide some allocation A  [m] x [n], and charge prices.
There are (possibly combinatorial) constraints
on what
allocations are allowed.
-
INPUT: m, n, T1,…,Tm ,
, and some access to
.
.
Auction in Action
1
1
expected welfare:
…
…
i
expected revenue:
-
chosen
j
by auction
- Commits
to an auction design,
specifying
allowed
over bidders’
types t1,(i)…,
tm, the bidder actions,
(ii)the
allocation
price rule;
randomness
in the
auction, and (iii)the
the bidders’
- in
Asks
bidders
play the auction;
strategies
Bayes
Nashtoequilibrium
- Implements the allocation and price
n
rule specified by the auction;
- Goal: Optimize revenue/welfare.
payment made by bidder
i to the auctioneer
Uses as input: the auction specification, her own type, and her beliefs about
over bidders’ types t1, …, tm, the
the types of the other bidders;
randomness in the auction, and the bidders’
Chooses how to play;
strategies
in Bayesbundle
Nash –equilibrium
Goal: optimize her own utility (= value
for allocated
price charged).
Each Bidder:
-
…
…
m
outcome in
Auctioneer:
background
welfare vs revenue optimization in multidimensional settings
Welfare-Optimization
►
[Vickrey-Clarke-Groves]: Mechanism design for welfare-optimization
is no harder than algorithm design for welfare-optimization.
►
the VCG auction (as a reduction):
 bidders are asked to report their types: t1, t2,…, tm ;
 the mechanism chooses the allocation
► obtained via
;
a call to a welfare optimization algorithm
 bidders are charged so that: reported types = true types.
► truthfulness-inducing payments
can be computed via calls to a welfare
optimization algorithm (e.g. Clarke pivot payments)
►
Corollary: the only bottleneck to tractable welfare-optimizing
mechanisms is whether there is a computationally efficient algorithm for
the underlying welfare optimization problem.
► Approximation preserving mechanism-to-algorithm reduction ?
 yes! [Hartline-Lucier’10, Hartline-Kleinberg-Malekian’11, Bei-Huang’11]
 Spoiler Alert: Nicole will present these results.
Revenue-Optimization
► [Myerson ’81]: In all single-dimensional (i.e. single-item / single-item
multi-unit) settings, mechanism design for revenue optimization reduces to
algorithm design for welfare optimization.
►
Myerson’s auction (as a reduction):
 bidders are asked to report their types
;
 reported types are transformed to virtual-types
 the virtual-welfare maximizing allocation is chosen;
► this is
;
a call to a welfare optimization algorithm
 and prices are charged to make sure bidders report truthfully.
► truthfulness-inducing payments
can be computed via calls to a welfare
optimization algorithm
►
►
Corollary: If the underlying welfare-maximization problem is
tractable, then so is the revenue-optimal auction.
Unanswered: Beyond single-item settings? Robustness to approximation;
►
Myerson’s proof is “mystical;” result comes mysteriously out of algebra…
Multi-Dimensional Auction Phenomena
optimal mechanism: sell item for 1/2
expected revenue: 1/4
optimal mechanism ?
idea 1: run n different auctions as above
…
…
additive
expected revenue: n/4
idea 2: offer the grand bundle of all the
items at price
expected revenue:
moral of the story: bundling helps
Multi-Dimensional Auction Phenomena (2)
[Thanassoulis’04]:
- optimal deterministic mechanism:
post price of $5.097 on each item
expected revenue:  $5.05
unit demand
- a better randomized mechanism
additionally sell for $5.057 the lottery (1/2,1/2)
expected revenue:  $5.06
moral of the story: randomization helps
[Briest-Chawla-Kleinberg-Weinberg’10]: gap may be arbitrarily high in general
[Chawla-Malec-Sivan’10]: for independent values, unit-demand
bidders, gap is at most a factor of 34.
Multi-dimensional Mechanisms
(known results prior to FOCS’12)
►
Large body of work in Economics:
multi- to single [McAfee-McMillan’88], [Wilson’93], [Armstrong’96], [Rochet-Chone’98],
dimensional reduction
[Armstrong’99],[Zheng’00], [Basov’01], [Kazumori’01],
[Thanassoulis’04],[Vincent-Manelli’06], [Vincent-Manelli’07], …
 Progress sporadic.
►
Recently (2007-today), algorithmic tools enabled progress.
 constant-factor approximations (limited settings)
► [Chawla-Hartline-Kleinberg ’07],
[Chawla et al’10], [Bhattacharya et al’10],
[Alaei’11], [Hart-Nisan ’12], [Kleinberg-Weinberg ’12], [Alaei et al’12]
 exact solutions (limited settings)
’11], [D-Weinberg ’12], [Cai-D-Weinberg ’12a],
[Alaei
al ’12],
multitoetsingle[Cai-Huang’13]
► [Cai-D
 “limited settings”:
► additive bidders and
bidder reductions
no allocation constraints
► unit-demand bidders and matroid constraints on who can be served
Setting
paintings
Previous
Results
OPT: [Cai-D-Weinberg STOC’12]
(matroid)
APPX: [A,BGGM,CHMS,KW]
houses
(matching)
doctor
appointments
(downwards-closed)
OPT: OPEN
APPX: OPEN
OPT: OPEN
APPX: OPEN
bridges
OPT: OPEN
Challenge
►
Revenue optimization in general multi-item settings.
 ideally: (i) unified solution for all settings
 (ii) Robustness of solutions to approximation, complexity.
A General Solution
►
A generic reduction:
 Mech. for Revenue Optimization  Alg. for Welfare Optimization
►
[Cai-D-Weinberg FOCS’12]: Suppose that:
 bidder types are independent;
 [the number of bidders m, items n, and the set-system
allocations are unrestricted.]
►
►
of feasible
then the revenue-optimal auction can be computed with
queries to a welfare-optimization algorithm A for .
The optimal auction has the following form:






bidders are asked to report their types;
reported types are transformed into virtual types via bidder-specific functions;
the virtual-welfare optimizing allocation in is chosen with a call to A;
prices are charged to enforce truth-telling in Nash equilibrium.
in Myerson’s theorem: virtual function = deterministic, closed-form
here, randomized, computed during execution of Ellipsoid.
A General Solution
►
A generic reduction:
 Mech. for Revenue Optimization  Alg. for Welfare Optimization
►
[Cai-D-Weinberg FOCS’12]: Suppose that:
 bidder types are independent;
 [the number of bidders m, items n, and the set-system
allocations are unrestricted.]
►
►
►
of feasible
then the revenue-optimal auction can be computed with
queries to a welfare-optimization algorithm A for .
The optimal auction is a virtual welfare maximizer.
Corollary: If the underlying welfare-maximization problem for
is
tractable (ignoring incentives), then so is the revenue-optimal auction.
Today’s menu
General Auction Setting
Background on Optimal MD
Challenge, General Result
Some Technical Ideas
Combinatorial Optimization Viewpoint
Specifying an auction
(explicit but expensive)
►
via the ex-post allocation & price rule:
: probability distribution over feasible sets from which
allocation is chosen when bidders’ types are
: price that bidder i pays under type profile
►
►
►
►
►
description size:
feasibility trivial to check (in fact hardwired into A)
Folklore: can write LP on A and P to find a revenue-optimal auction
why? expected revenue is linear in P, truthfulness constraints linear in A, P
trouble: exponentially many variables
 solving LP takes exponential time
 solution is bad: “laundry-list” auction
Specifying an auction
(lossy and still expensive)
►
via the ex-post allocation probabilities:
: marginal probability that item j is allocated to
bidder i when bidders’ types are
: price that bidder i pays when bidders’ types are
►
description size still prohibitive:
► and feasibility is hard to check now efficiently:
1. is there a distribution over feasible allocations with these marginals, in view of
the possibly combinatorial constraints on feasible allocations given by ?
2. bright exception: for simple allocation constraints, e.g. matching constraints,
checking feasibility is easy using the Birkhoff-von Neumann theorem.
►
can’t write LP anymore and, even if I could, its solution would be useless.
the reduced form of an auction
►
a.k.a. the interim allocation & price rule :
: marginal allocation probability of item j to bidder i when his
type is ti (over the randomness in the other bidders’ types, and the
randomness in mechanism)
: expected price paid by bidder i when his type is ti (over the
other bidders’ types, and the randomness in the mechanism)
►
description size:
;
► c.f. description complexity of ex-post allocation rule
;
► feasibility hard to check:
1. Can the per-bidder marginal probabilities be reconciled?
2. …in a way that also respects the allocation constraints given by ?
i.e. when can interim probabilities be converted to a feasible mechanism?
► wishful thinking: what if we could check feasibility efficiently?
Mechanism Design with the Reduced Form
Variables:
expected value of bidder i of
type
for being given
(additivity of bidders)
the reduced form
of sought auction
Constraints:
Bayesian Nash Equilibrium:
Objective:
- the expected revenue
- Need: (i) Separation oracle for feasible reduced forms
- (ii) Efficient map from feasible reduced form to mechanism
(optimal feasible reduced form is useless in itself)
feasibility of reduced forms
Feasibility of Interim Rules (example)
►
►
►
easy setting: single item, two bidders with types uniformly
distributed in T1={A, B} and T2={C, D} respectively
allocation constraints = item cannot be given to more than one bidder
Question: Are the following interim allocation probabilities feasible?
½
bidder 1
½
A
C ½
B
D ½
bidder 2
whenever types are A, C: A needs to get item
whenever types are A, D: A needs to get item
type A satiated
whenever types are B, C: C needs to get item
type C satiated
whenever types are B, D: B needs to get item with prob. 0.4 and
D needs to get item with prob. 0.8
Feasibility of Interim Rules
- Single-item reduced forms:
- [Border ’91, Border ’07, CKM ’11]: Necessary and sufficient conditions.
-
linear constraints OR separation oracle w/ runtime
- [Cai-D-Weinberg’12, Alaei et al ’12]: SO w/ runtime
.
.
- Any hope for multi-item reduced forms, w/ arbitrary allocation constraints
- [Cai-D-Weinberg FOCS’12 ]: Given black-box access to max-welfare
algorithm for
can decide feasibility of reduced-forms efficiently.
Moreover, given feasible reduced form can efficiently find a mechanism
with this reduced form.
- the combinatorial optimization perspective
- geometric view:
?
Feasibility of Interim Rules
Claim 1:
set of feasible
interim rules
Claim 2: Given max-welfare algorithm for allocation constraints
can find a separation oracle for
(and vice versa).
MD version of Grötschel-Lovász-Schrijver
equivalence of optimization and separation
Claim 3: Every vertex of the polytope is the reduced form of a virtual
welfare maximizing allocation rule.
Characterizing the Vertices
Characterizing the Vertices
virtual welfare maximizing
interim rule “in direction ”
expected virtual
welfare achieved by
allocation rule with
interim rule
interpretation: virtual value derived by bidder i
when given item j, if his type is A
Characterizing the Vertices
A virtual VCG allocation rule is defined by virtual functions
, where
, for all i.
virtual welfare maximizing
It takes
a type-vector
t1, t2,
interim
ruleas
“ininput
direction
”
…, tm
- transforms it into the virtual type-vector
- then optimizes welfare using virtual types instead of true ones
Q: OK understood what corner
does, but what mechanism has this ?
A: The VCG mechanism w/ virtual functions f1,…, fm
interpretation: virtual value derived by bidder i
when given item j, if his type is A
Characterizing the Vertices
is a polytope whose corners are implementable by virtual
VCG allocation rules.
[CDW ’12]: The interim allocation rule of any feasible mechanism
can be implemented as a distribution over virtual VCG allocation rules.
Feasibility of Interim Rules
set of feasible
interim rules
Claim 1:
We can optimize over the set of
feasible reduced forms, using the
ellipsoid algorithm.
The optimal reduced form can be
converted to a convex combination
of max-welfare computations.
Claim 2: Given max-welfare algorithm for allocation constraints
can find a separation oracle for
(and vice versa).
MD version of Grötschel-Lovász-Schrijver
equivalence of optimization and separation
Claim 3: Every vertex of the polytope is the reduced form of a virtual
welfare maximizing allocation rule.
Reality Check
►
Connection to Grötschel-Lovász-Schrijver
 Wishful thinking: given algorithm for welfare optimization under
allocation constraints
, can get linear optimization algorithm for
.
 Reality: Can’t do this efficiently; but can get additive-approx FPTAS.
 Trouble: GLS requires exact linear optimization, and its extensions
multiplicative approximations.
 Solution: Specialized for MD.
 Our result provides an extension of GLS to additive approximation
algorithms, exploiting the structure of
.
►
End product: FPRAS
 OPT-ε revenue, in time poly(1/ε).
 [This still happens at BNE and not ε-BNE]
De-mystifying Myerson
►
Unexpected properties of Myerson’s optimal single-item auction:
 the auction is deterministic (no internal coin-flips) ;
 while it is optimal among all auctions, it is itself a Dominant Strategy
Truthful auction.
►
Through our combinatorial optimization perspective we can
easily show that
 for all single-dimensional settings; and
 for all objective functions that are linear in the interim rule (e.g. welfare,
revenue, linear combinations thereof)
the optimal auction is deterministic, DST, and a virtual welfare maximizer.
►
Why? The optimal solution of the LP must be a corner. (No
regularity assumption or ironing is needed for the argument.)
Summary
►
Welfare optimization is a well-understood problem:
 the VCG auction provides a reduction from mechanism to algorithm design
►
The same is not true for revenue:
 Myerson’s auction optimizes revenue in single-dimensional settings;
 but multi-item settings are not well understood.
►
I showed a general result providing the natural generalization of
Myerson’s theorem to all multi-item settings.
 “The revenue optimal auction is a virtual-welfare maximizer; which can be
computed with polynomially many queries to a welfare-maximizing algorithm.”
►
Techniques: geometry, ellipsoid algorithm;
 can optimize over reduced forms using welfare algorithm as a separation oracle.
Further Results/Open Problems
►
Approximation Preserving reduction?
 Given a-approximation for welfare, can obtain a-approximation for revenue?
 Yes [Cai-D-Weinberg SODA’13];
 Complication: Can’t get separation oracle for polytope of reduced forms from
approximation algorithm for welfare.
size of support of bidder
i’s type-distribution
►
Implicit type-distributions?
 our running-time is polynomial in
;
 but suppose type-distribution is given implicitly
► e.g.
consider additive bidder whose value for item j is uniform in {aj, bj}
independently of other items
 Q: can running time be improved for such implicit disn’s?
 [D-Deckelbaum-Tzamos]: Problem is #P-hard even when there is a single
additive bidder whose values are independent, rational, of support two.
 But FPTAS still not precluded.
Further Results/Open Problems
►
Nature of Virtual Functions




►
both Myerson’s auction and its generalization are virtual welfare maximizers;
Myerson provided a closed form formula for each bidder’s virtual function;
in the generalized auction, the virtual functions are computed by an algorithm;
any structure for simple multi-item settings?
Value Oracle Model?
 in our work valuations in support are described explicitly;
 what if valuations are implicit?
► i.e.
given S, a circuit outputs value of bidder for subset S of items
 Problem is NP-hard in general [Dobzinski-Fu-Kleinberg’11]
 but what if valuations have structure?
Thanks for listening