Blackbox Reductions from
Mechanisms to Algorithms
Algorithm Design
v1
v2
v3
Input v
Feasibility constraints
on outcome space
Output x
v4
v5
GOAL: maximize (or minimize) some function f(x,v)
Mechanism Design
v1 b1
v2 b2
v3 b3
Input b
v
bi chosen to maximize
utility = vixi(b)-pi(b)
Feasibility constraints
on outcome space
Allocation x
Payment p
v4 b4
v5 b5
GOAL: maximize (or minimize) some function f(x,v)
behind every greatincentive
mechanism
computation
is a great algorithm
Black-Box Transformations
Algorithm
Input vb
Transformation
Allocation x
Payment p
GOAL: for every algorithm, transformation
preserves quality of solution in equilibrium.
and is incentive compatible.
Black-Box Transformations
Algorithm
Input v
Transformation
Allocation x
… and is incentive compatible (IC), i.e., monotone:
ex-post IC (truthful in expectation): allocation to agent i
is increasing in i’s bid for all bid profiles of others
Bayesian IC: allocation to agent i is increasing in i’s bid
in expectation w.r.t. prior of over bid profiles of others
Optimal
Algorithm
Input v
VCG Transformation
Allocation x
EXAMPLE: Vickrey-Clark-Groves auction transforms
any optimal algorithm into an optimal ex-post IC
mechanism for any monotone objective function.
Single Item Auction
Selection
Algorithm
Input v
VCG Transformation
Find agent
w/max value
Allocation x
one item, agent i has value vi for item
(single-parameter)
Combinatorial Auction
i1
i2
i3
many items, agent i has value vij for subset Sj
(multi-parameter)
Combinatorial Auction
Input v
???
Find max value
non-overlapping
collection of sets
VCG Transformation
Allocation x
many items, agent i has value vij for subset Sj
(multi-parameter)
^
approximation
BIC Transformation
Positive Result: Transform approximation algorithms
into Bayesian IC mechs with small loss in social welfare.
Single-parameter: (single private value for allocation)
1. Monotonization.
For dist. F and algorithm A, there is a Bayesian IC
transformation TA,F satisfying E[TA,F(v)] ≥ E[A(v)].
2. Blackbox computation.
TA,F can be computed in polytime with queries to A.
3. Payment computation.
Payments can be computed with two queries to A.
Monotonization
xi(vi) = E[alloc. to i | vi]
Not BIC
vi
BIC
Fact. There’re payments that make an alg. Bayesian IC if and only if
for all i, expected allocation is monotone non-decreasing in value vi.
Monotonization
Goal: construct yi from xi s.t.
1. Monotonicity. yi(.) non-decreasing monotone
2. Surplus-preservation. Evi[viyi(vi)] ≥ Evi[vixi(vi)]
3. Distribution-preservation.
(can apply construction independently to each j)
Monotonization
Idea 1: remap values.
Monotonization
Idea 2: resample values.
Monotonization
allocation
cumulative
curve
Idea 3: resample values in region where
cumulative allocation is not monotone.
Monotonization
xi(vi)
yi(vi)
Construction of yi(vi) from xi(vi) preserves:
1. Distribution-preservation.
2. Monotonicity. yi non-decreasing monotone
Monotonization
xi(vi)
yi(vi)
Construction of yi(vi) from xi(vi) preserves:
3. Surplus-preservation. Evi[vi(yi - xi)] ≥ 0
E[v(y-x)] = ∫ ba v(y-x) d f(v)
= v(Y-X)|ba – ∫ba v’(Y-X) d f(v)
= 0 – (non-neg.) x (non-pos.)
≥0
(integration by parts)
(v , X dominates Y)
(2nd term non-pos.)
BIC Transformation for Welfare
Positive Result: Transform approximation algorithms
into Bayesian IC mechs with small loss in social welfare.
Single-parameter: (single private value for allocation)
1. Monotonization.
For dist. F and algorithm A, there is a Bayesian IC
transformation TA,F satisfying E[A(v)] ≥ E[TA,F(v)].
2. Blackbox computation.
TA,F can be computed in polytime with queries to A.
3. Payment computation.
Payments can be computed with two queries to A.
Blackbox Computation
BIC Transformation for Welfare
Positive Result: Transform approximation algorithms
into Bayesian IC mechs with small loss in social welfare.
Single-parameter: (single private value for allocation)
1. Monotonization.
For dist. F and algorithm A, there is a Bayesian IC
transformation TA,F satisfying E[A(v)] ≥ E[TA,F(v)].
2. Blackbox computation.
TA,F can be computed in polytime with queries to A.
3. Payment computation.
Payments can be computed with two queries to A.
Payment Computation
p(v) = v y(v) –
v
∫0
y(z) dz
payment identity
Idea: compute random variable P with E[P] = p(v)
Payment Computation
payment identity p(v) = v y(v) –
1st call to A
2nd call to A
v
∫0
y(z) dz
1. Y indicator random variable for
whether agent wins in A (with y(v))
2. z drawn uniformly from [0,v]
3. Yz indicator random variable for
whether agent wins in A (with y(z))
4. P = v (Y – Yz) const. # calls per agent
Idea: compute random variable P with E[P] = p(v)
Payment Computation
goal: given A, find an alg. A’ that computes
allocation and payments with just 1 call to A
only call to A
1. Pick agent k uniformly at random
and draw wk from Fk
2. Calculate outcome y’ for A(wk, v-k)
3. For each agent i ≠ k, set p’i = viy’i
4. For agent k, set p’k = 0 if wk > vk
and p’k = -(n – 1)y’k/fk(wk) otherwise
5. Output (y’, p’)
Payment Computation
Thm. Algorithm A’ is Bayesian IC.
Proof.
1. Monotone. y’ linear transformation of y.
y’(v) = (1 - 1/n) y(v) + 1/n E[y(w)]
2. Payment Identity.
v
∫ 0 y’(z)
p’(v) = v y’(v) –
dz
v
p’(v) = (1 - 1/n) vy(v) – (1/n)(n - 1)∫ 0 y(z) dz
payment for i ≠ k
payment for i = k
(see ugly formula)
Payment Computation
Thm. Welfare is E[A’(v)] ≥ E[A(v) – max(v)]
Proof.
1. Each buyer has welfare ≥ (1 - 1/n) vy(v)
2. Since y(v) is a probability, vy(v) ≤ max(v)
3. Lose at most max(v) in total buyer welfare
4. Expected payments are the same, so lose
nothing in seller welfare
Finds (alloc, payments) with 1 call to monotone alg.
[Babaioff, Kleinberg, Slivkins’10]
Approx.
Algorithm
Input v
(drawn from
known dist.)
Dist. of
values
Transformation
Allocation x
Payment p
POSSIBILITY: can transform any approximation algorithm
into a Bayesian IC mech. with small loss for f(x,v) = Σixivi.
[Hartline-Lucier’10]
Multi-parameter Transformation
Goal: construct allocation from algorithm s.t.
1. “Monotonicity”.
2. Surplus-preservation.
3. Distribution-preservation.
By mapping types of an agent to surrogates in a
way that preserves above properties.
Replicas and Surrogates
replicas
surrogates
surrogate
(drawn from F)
(drawn from F) allocations
max-weight
original
type t
matching
v(t,x(t’))
surrogate
type t’
x(t’)
Set payment equal to VCG payment for type t.
Set allocation equal to output on surrogate type profile.
Replicas and Surrogates
Thm. Transformation is distribution-preserving.
Thm. Transformation is Bayesian IC.
Thm. Transformation doesn’t lose much welfare.
Prf. Because replicas are “close” to matched
surrogates in values for outcomes.
[Hartline, Kleinberg, Malekian’11]
[Bei, Huang’11]
Strengthening the Result
Solution concept: black-box transformations for
social welfare that preserve approximation and
are truthful in expectation?
Social objective: black-box transformations that
preserve approximation, are Bayesian IC, and
work for other social objectives?
Multi-parameter Transformations
Thm. There’s no truthful in expectation mech.
for combinatorial auctions with submodular
valuations that guarantees a sub-linear approx.
Note: there is a (1-1/e)-approximation alg.
[Dughmi, Vondrak’11]
Single-parameter Transformations
Truthful in Expectation.
For all algorithms A, TA is truthful in expectation,
i.e., expected allocation is monotone for all i.
Worst-case approximation preserving.
For all values vectors v and algorithms A,
expected welfare of transformation is close to
expected welfare of algorithm.
Proof Outline
1. Define welfare instance (feasible allocations,
values of agents).
2. Find algorithm with high welfare.
3. Use monotonicity to show any ex-post
transformation has low worst-case welfare.
row ave. of x1 increasing
Intuition
v1
Bayesian IC
(.5,.5)
(x1,x2)
v2
column ave. of x2 increasing
Intuition
(.7,.9)
(.6,.2)
v1
(.6,.3) (.3,.4) (.5,.5) (.1,.6) (.7,.7)
(.4,.1)
(.3,.9)
(.2,.7)
Ex-post IC
v2
Intuition
Input vector
v1
(.2,.2) (.1,.3) (.5,.5) (.3,.4) (.8,.7)
Query Query
Query Query
v2
Transformation must fix non-monotonicities
in every row and column.
Intuition
Make all allocations
constant on these agents.
(.6,.2) (.3,.3)
𝑣1
(.5,.5) (.2,.6)
𝑣2
Idea: hide non-monotonicity on high-dim. diagonal.
Truthful in Expectation
Thm. Any truthful-in-expectation transformation
loses a polynomial factor in welfare
approximation.
[Chawla, Immorlica, Lucier’12]
Thank You