On Optimal Multi-dimensional
Mechanism Design
Constantinos Daskalakis
EECS, MIT
costis@mit.edu
Joint work with Yang Cai, Matt Weinberg
Multi-Item Multi-Bidder Auction
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Multi-Item Multi-Bidder Auction
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if the vij’s known exactly, can
compute optimal allocation
and extract full surplus.
Revenue Maximization?
Additional Constraints: Demands, Budgets
can go around computational hardness with randomization.
Multi-Item Multi-Bidder Auction
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if the vij’s are unknown,
then all bets are off...
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Revenue Maximization?
Natural approach: online optimization
[wait till 10.30]
Additional Constraints: Demands, Budgets
can go around computational hardness with randomization
Multi-Item Multi-Bidder Auction
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Suppose Bayesian
information is known
for bidders’ values
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Revenue Maximization?
Optimal Mechanism Design
Additional Constraints: Demands, Budgets
can go around computational hardness with randomization
Single-Parameter Optimal Mechanisms
►
[Myerson ’81]: If setting is single-parameter and bidders have
independent values, there exist closed-form revenue-optimal
mechanisms.
 single-parameter: each bidder has a fixed value for winning an
item, no matter what item she gets; that value is drawn from
known distribution;
 in our running example: if bidder wins painting, her value for
the painting is uniform in [$10,$100], no matter what painting
she gets…
Multiple Copies of Same Item
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i.e. one (unknown) parameter
determines each bidder’s values
Single-Parameter Optimal Mechanisms
►
►
[Myerson ’81]: If setting is single-parameter and bidders have
independent values, there are closed-form revenue-optimal
mechanisms.
 single-parameter: each bidder has a fixed value for winning an
item, no matter what item she gets; that value is drawn from
known distn’
 In our running example: if bidder wins painting, her value for
the painting is uniform in [$10,$100], no matter what painting
she gets
 closed-form: revenue optimization reduces to VCG
Closed-form ≠ Computationally Efficient
Multi-dimensional Optimal MD
►
Central Open Problem: Are there closed-form, efficient
revenue-optimal mechanisms, when the bidders are multidimensional, i.e. have different values for different allocations?
 large body of work in Economics for restricted settings;
see, e.g., survey paper by [Vincent-Manelli ’07]
 Constant-factor approximations are known:
►
multi-item auctions, and certain matroidal settings;
poly-time for regular distn’ [Chawla, Hartline, Malec, Sivan ’10]
► multi-item auctions + budget constraints
[Bhattacharya, Goel, Gollapudi, Munagala ’10]
Multi-dimensional Optimal MD
►
[D-Weinberg ’11]: Closed-form, efficiently computable, nearly
optimal revenue mechanisms when the number of items or the
number of bidders is a constant.
 In the first case, we allow the values of each bidder for the items
to be arbitrarily correlated, but assume that bidders are i.i.d.
 In the second case, we require each bidder to have i.i.d. values for
the items, but allow different bidders to have different
distributions.
Multi-dimensional Optimal MD
constant #items
constant #bidders
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(i.i.d)
Arbitrary joint
distribution
possibly different
per bidder
Multi-dimensional Optimal MD
►
[D-Weinberg ’11]: Closed-form, efficiently computable, nearly
optimal revenue mechanisms when the number of items or the number
of bidders is a constant.
 In the first case, we allow the values of each bidder for the items to
be arbitrarily correlated, but assume bidders are i.i.d.
 In the second case, we require each bidder to have i.i.d. values for
all items, but allow different bidders to have different distributions.
NOTES:
a. Can do arbitrary budget, demand constraints. Also explicitly price
bundles of items.
b. Solution concept: Bayesian Incentive Compatibility (BIC), or
ε-Incentive Compatibility (IC).
c. Nearly optimal: OPT- ε, for any desired ε >0, when support of
distributions is normalized to [0,1]; or (1- ε)OPT if distn’s MHR.
a glimpse of the techniques
giant
Optimal MD it’s just an LP…
Let
be the joint distribution of all bidders’ values for the items
(supported on a subset of
): specifies all vij’s.
For every point in the support of
, mechanism needs to determine a
(randomized) allocation of items to bidders and the charged prices.
Write down an LP that looks for such (allocation distribution, price list)
pair for every point in
, and enforces incentive compatibility,
individual rationality, budget, demand constraints, etc.
both are
exponential
LP lives in
dimensions
[Birkhoff–von Neumann theorem]: Sufficient to compute the marginals
of the allocation distribution for each valuation vector.
Improved LP lives in
dimensions
Ingredient 1: The Role of Symmetries
Theorem: Let
be the distribution of bidders’ values for the
items (supported on a subset of
).
Let also
be a set of permutations such that:
Then there exists an optimal randomized mechanism M, respecting
all the symmetries in .
i.e.
c.f. Nash’s symmetry theorem: “In every game there exists a Nash equilibrium
that simultaneously satisfies all symmetries satisfied by the game.”
NOTES:
a. Above symmetry does not hold for deterministic mechanisms.
b. Certifies existence of a mechanism of small description complexity,
if setting is sufficiently symmetric. However, not clear how to
modify LP to locate the succinct mechanism…
Ingredient 2: Monotonicity
Theorem: If mechanism M is truthful and item-symmetric, then it
satisfies a natural monotonicity property.
Theorem:
Monotonicity + Symmetry => succinct LP, if enough symmetry.
Ingredient 3: Continuous to Discrete
Continuous distributions -> discretize to get finite support
Previous Approach results in ε-BIC
Use mechanism computed by LP as a back-end, and design a
VCG front-end whereby continuous bidders “buy” discretized
representatives.
Transfer approximation from truthfulness to revenue.
(cf recent surrogate-replica construction of [Hartline-KleinbergMalekian ’11])
Ingredient 4: Extreme Value Theorems
• So far, OPT – ε, when distributions normalized to [0,1].
• From additive ε-approx. to multiplicative (1-ε)-approx?
• Theorem [Cai-D ’11]: Let X1,…,Xn be independent (but not
necessarily identically distributed) MHR random variables.
Then there exists anchoring point β such that:
X1
X2 (1-ε)OPT is extracted from
Corollary:
Xn
values in (ε β, 1/ε log1/ε β).
X3
Pr[max Xi ≥ β] = Ω(1)
contribution to E[max Xi]
from values here is ≤ ε β
Beyond symmetries?
Single Unit-Demand Bidder
If the vi’s are i.i.d.,
already know how to
find optimal
mechanism.
Multidimensional Pricing
If the vi’s are
independent (but not
necessarily i.i.d.), can
we at least find
optimal prices?
[CHK’07]
Optimal Multi-dimensional Pricing
►
[Cai-D ’11]: Nearly-optimal, efficient pricing algorithms for a
single unit-demand bidder whose values are independent (but not
necessarily identically distributed).
NOTES:
a. Nearly optimal: OPT- ε, for any desired ε >0, when support of
distributions is normalized in [0,1]; or (1- ε)OPT if distn’s are
MHR or regular.
b. Efficient = PTAS, and quasi-PTAS for regular.
Techniques
►
►
►
Symmetry lemma does not apply to prices.
Not enough symmetry, to find optimal randomized mechanism with
previous approach.
Our approach:
 Search space: price vectors.
 Instead focus attention on revenue distributions induced by price
vectors.
 Identify appropriate distance measure in this
space, so that closeness reflects closeness
in revenue.
  implicit polynomial-size cover of this space.
Structural Results (e.g. MHR)
• A Constant Number of Prices Suffices
Theorem[Cai-D ’10]
For all
,
distinct prices suffice to get
revenue, where
is an increasing
function that does not depend on Fi or n .
• A Single Price Suffices for i.i.d.
Theorem[Cai-D ’10]
For all
, if
, then a single price suffices
to get
revenue, where
is an increasing
function that does not depend on Fi or n .
Summary
constant #bidders
constant #items
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(i.i.d)
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possibly different
per bidder
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single unit-demand bidder, pricing
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Arbitrary joint
distribution
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Open Problems
• Complexity of the exact problem.
• Conjecture: #P-hard to find optimal mechanism even for single
bidder with independent values for the items.
NOTE: If values are correlated, it is already known that the
problem is highly-inapproximatble. [Briest ’08]
• Beyond item/bidder symmetric settings
• Combine LP approach with probabilistic covering theorems.
Thank you for your attention
Multidimensional Pricing
Our approach:
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Revenue Distribution
Multidimensional Pricing
Smoothness properties that would be useful:
- what happens to the objective if I replace
where
?
with
- what happens to the objective if I replace
where
?
with
- what happens to the objective if I restrict the prices to the
support of the value distributions?
Multidimensional Pricing
Our approach:
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Revenue Distribution
Geometric Approach
 implicit polynomial-size
- cover of this space
“Implicit”: it is output by an
algorithm, given {Fi}i
distance function:
TV
Geometric Approach
distance function:
Multidimensional Pricing
►
►
[Chawla,Hartline,Kleinberg ’07]: poly-time constant factor
approxi-mations for regular distributions; even the i.i.d. case is
not easier.
[Cai-D ’10]: PTAS, for independent MHR distributions, or when
support is balanced
Multi-dimensional Optimal MD
►
[Cai-D ’10]: Closed-form, efficiently computable, (1-ε)optimal revenue mechanism for the SINGLE bidder case.
Symmetries in Nash’s paper
Symmetric Games: Suppose each player p has
- the same strategy set: S = {1,…, s}
- the same payoff function: u = u (σ ; n1, n2,…,ns)
Description size: s ns-1
(instead of: n sn )
strategy of p
E.g. : - prisoner’s dilemma
number of the other
players choosing each
strategy in S
- 1000 drivers from Pasadena to Westwood
Nash ’50: Always exists an equilibrium in which every player
uses the same randomization.
Does this make computation easier?
[Gale-KuhnTucker 1950]
Symmetrization
y
x
x
y
x
0, 0
C, R
y
RT, CT
0, 0
R,C
Equilibrium
In fact […]
Equilibrium
Symmetric Equilibrium
Any Equilibrium
Symmetrization
x
y
y
x
0,0
C, R
y
RT,CT
0,0
x R,C
Equilibrium
In fact […]
Equilibrium
Hence, PPAD to solve
symmetric 2-player games
Symmetric Equilibrium
Any Equilibrium
Open: - Reduction from 3-player games to symmetric 3-player games
Multi-player symmetric games
If n is large, s is small, a symmetric equilibrium
x = (x1, x2, …, xs)
can be found as follows:
- guess the support of x, 2s possibilities
- write down a set of polynomial equations an
inequalities corresponding to the equilibrium
conditions, for the guessed support
- polynomial equations and inequalities of degree n
in s variables
can be solved
approximately
in time
ns log(1/ε)
(recall description
complexity is s ns-1)
Multidimensional Pricing
If vi’s unknown can
we set prices to
optimize revenue ?
revenue?
(if vi’s known trivial to optimize)
Multidimensional Pricing
If vi’s unknown can
we set prices to
optimize revenue ?
expected
revenue: