Algorithmic Game Theory - hni.uni-paderborn.de fileRandomization Combinatorial Auctions IC in Expectation De nition A randomized mechanism isincentive compatible in expectationif truth-telling

Randomization Combinatorial Auctions

Mechanism Design

Algorithmic Game Theory

Alexander Skopalik Algorithmic Game Theory

Mechanism Design


Randomized Mechanisms

Single-Minded Combinatorial Auctions


Mechanism Design


Randomizing the Dictator

Reconsider the social choice problem...

Random-Ballot MechanismEach player reports a preferenceChoose at random one of these preferences

Is it incentive compatible?


Mechanism Design



Definition

I A randomized mechanism is a distribution over deterministic mechanisms(all with the same players, type spaces Vi and outcome space A).

I A randomized mechanism is incentive compatible in the universal sense ifevery deterministic mechanism in the support is incentive compatible.


Mechanism Design


IC in Expectation

DefinitionA randomized mechanism is incentive compatible in expectation if truth-tellingis a dominant strategy in the game induced by expectation.

I Outcome and payments are random variables

I Denote (a, pi ) random variable for reporting vi

I Denote (a′, p′i ) random variable for reporting v ′iI E [·] is expectation over the randomness of the mechanism

I For all i , all true valuations vi , all v−i and v ′i we have

E [vi (a)− pi ] ≥ E [vi (a′)− p′i ].


Mechanism Design


Single Parameter Domains

Recall: Single Parameter Domain, vi is value for winning alternatives in Wi ,value 0 otherwise.

Deonte by wi (vi , v−i ) = Pr [f (vi , v−i ) ∈Wi ] the winning probability for player iwith bid vi given v−i .

We use pi (vi , v−i ) directly for the expected payment.

Normalized mechanism: Lowest possible bid v 0i = t0 loses completely

wi (t0, v−i ) = 0 and pi (t

0, v−i ) = 0 for all v−i .


Mechanism Design


Characterization

TheoremA normalized randomized mechanism in a single parameter domain is incentivecompatible in expectation if and only if for every i and every fixed v−i we havethat

1. the function wi (vi , v−i ) is monotonically non-decreasing in vi and

2. pi (vi , v−i ) = vi · wi (vi , v−i )−∫ viv0i

wi (t, v−i )dt.


Mechanism Design



Single-Minded Combinatorial Auctions


Mechanism Design


Ad Auctions


Mechanism Design


Spectrum Auctions


Mechanism Design


General Setting

Combinatorial Auction:

I Set M of m indivisible items (e.g., ad slots) auctioned simultaneously

I n bidders, valuations for each subset of items

I Who should get which items and pay how much?

I General Allocation Problem of Interrelated Resources

Valuation vi for bidder i :

I vi (S) ∈ R when getting assigned set S ⊆ M

I free disposal: S ⊆ T ⇒ v(S) ≤ v(T )

I normalized: v(∅) = 0.


Mechanism Design


Allocation

I Allocation of the items:S1, . . . , Sn where

⋃i Si ⊆ M and Si ∩ Sj = ∅ for i 6= j .

I Valuation of a player independent of items received by other players(no externalities)

I Social Welfare:∑

i vi (Si ).An efficient allocation S∗1 , . . . , S

∗n maximizes social welfare.

I Quasi-linear utilities: vi (Si )− pi (vi , v−i )

I VCG is truthful with S∗, but computing S∗ is NP-hard!

I Let us restrict attention to a special case.


Mechanism Design


Single-Minded extends Single-Parameter

DefinitionA valuation vi is single-minded if there exists a threshold bundle S t and valuev t ∈ R+ such that vi (S) = v t for all S ⊇ S t , and vi (S) = 0 otherwise. Asingle-minded bid is (S t , v t).

Single-Minded extends Single-Parameter:

I Single parameter domain, vi (a) = v t for all a ∈Wi and 0 otherwise.

I For single-parameter domains Wi ⊆ A is a publicly known set.

I Single-minded bidders can lie about S t .

I A single-parameter bid is v t , a single-minded bid (S t , v t).


Mechanism Design


Allocation Problem

DefinitionThe allocation problem among single-minded bidders is given by:

INPUT: (S ti , v

ti ) for each bidder i = 1, . . . , n

OUTPUT: Set of winners W ⊆ {1, . . . , n} with maximum social welfare∑i∈W v t

i and such that S ti ∩S t

j = ∅ for each i , j ∈W with i 6= j

Even for the restricted case of single-minded bidders, computing the optimalallocation is very hard.

TheoremThe allocation problem among single-minded bidders is NP-hard.


Mechanism Design


Reduction from Independent Set

Proof:INDEPENDENT SET Problem:Has a graph an independent set of size at least k?

Vertices → Bidders, Edges → Items(S t

i , vti ) = (Set of incident edges, 1)


Mechanism Design


Approximation Algorithms

I c-approximation algorithm: Returns an allocation T such that with theefficient allocation S∗ we have∑

i

vi (Ti ) ≥∑

i vi (S∗i )

c

I Simple n-approximation algorithm:Player with the maximum valuation gets M.Trivially yields an IC mechanism, essentially single-item VCG auction.

TheoremFor any ε > 0 it is NP-hard to approximate INDEPENDENT SET to within afactor of n1−ε.

Corollary

For any ε > 0 it is NP-hard to approximate the allocation problem amongsingle-minded bidders to within a factor of n1−ε.


Mechanism Design


Approximation Algorithms

The graph in the reduction has at most m < n2 edges/items, hence

Proposition

For any ε > 0 it is NP-hard to approximate the allocation problem to within afactor of m1/2−ε.

Note:√m < n for sparse instances.

So far, our best algorithm yields an n-approximation. Can we get a truthfulmechanism that returns an allocation that is a

√m-approximation?


Mechanism Design


Greedy Mechanism for Single-Minded Bidders

INPUT: (S ti , v

ti ) for each bidder i

OUTPUT: A set of winners W , payments pj for all 1 ≤ j ≤ n.

Initialization:

1. Reorder bids:v t1√|St

1|≥ . . . ≥ v tn√

|Stn|

2. W ← ∅, pi = 0 for all i

Iteration:

3. For i = 1 . . . n do: If S ti ∩(⋃

j∈W S tj

)= ∅ then W ←W ∪ {i}

Payments:

4. For each i ∈W do

5. find smallest index j such that

S ti ∩ S t

j 6= ∅ and for all k < j , k 6= i it holds S tk ∩ S t

j = ∅

6. if j exists, set pi =v tj√|St

j |/|Sti |


Mechanism Design


Example

Phone Headset Power Mary Jack John

x 50 0 0x 0 0 0

x 0 0 0

x x 50 60 0x x 50 0 65

x x 0 0 0

x x x 50 60 65


Mechanism Design


Example

Reordering:

S ti v t

i v ti /√|S t

i |1. Mary Phone 50 50

2. Jack Phone, Power 65 45.96...

3. John Phone, Headset 60 42.42...

Algorithm determines W and pi :

I 1. Mary: W = ∅, so W = {1}I 2. Jack: S t

1 ∩ S t2 = {Phone}

I 3. John: S t1 ∩ S t

3 = {Phone}I Winner is Mary

I First player blocked by Mary, which could be in W , is Jack (2)

I Payments: p1 = v t2/√|S t

2 |/|S t1 | = 65/

√2/1 = 45.96...


Mechanism Design


Incentive Compatibility

LemmaA mechanism for single-minded bidders with pi = 0 whenever i 6∈W is IC ifand only if for every player i and fixed other bids (S t

−i , vt−i ) the following holds:

I Monotonicity: If bidder i wins with (S ti , v

ti ), then he remains a winner for

any v ′i > v ti and S ′i ⊂ S t

i .

I Critical Payment: A winning bidder pays the minimum value needed forwinning – the infimum of all values v ′i such that (S t

i , vi ) still wins.


Mechanism Design


Greedy is IC

Monotonicity: (S ti , v

ti ) wins, then it wins with any v ′i > v t

i and S ′i ⊂ S ti

Critical Payment: Winner pays infimum of all v ′i such that (S ti , v′i ) wins.

Does Greedy satisfy it?

I Increasing v ti or reducing S t

i increases v ti /√|S t

i |I i moves up in order and remains winning

I Payment is the switching point between i and j :

x√|S t

i |≤

v tj√|S t

j |⇒ x ≤ v t

j

√|S t

i |√|S t

j |=

v tj√

|S tj |/|S t

i |


Mechanism Design


Proof of Lemma (if-part)

Initial Observations

I Truthful bidder has always positive utility

I Bidder has (S , v) and bids (S ′, v ′) 6= (S , v)

I If (S ′, v ′) is a losing bid, reporting (S , v) can only help.

I If S 6⊆ S ′, reporting (S , v ′) can only help.

Assumption: (S ′, v ′) is winning bid and S ⊆ S ′.

Winner is never worse off to bid (S , v ′):

I Denote payment p′ for (S ′, v ′) and p for (S , v ′).

I If (S , x) with x < p loses, then (monotone) (S ′, x) loses.

I Thus, for the critical payments p′ ≥ p.

I (S , v ′) causes at most the payments of (S ′, v ′).It can win in cases, in which (S ′, v ′) loses.


Mechanism Design


Proof of Lemma (if-part)

If bidders reveal their true sets S , truthful bidding of v follows by the previoussingle-parameter arguments about critical value payments:

I Assume (S , v ′) wins and (S , v) also

I Critical payment p for (S , v)

I For v ′ > p same payments, for v ′ < p losing ⇒ IC

I Assume (S , v ′) wins and (S , v) loses

I v smaller than critical payments, negative utility for (S , v ′)


Mechanism Design


Approximation of Social Welfare

LemmaThe greedy mechanism computes a

√m-approximation for the corresponding

allocation problem.

Proof:

I Denote optimal winner set W ∗, output of greedy W .

I For each i ∈W consider W ∗i = {j ∈W ∗, j ≥ i | S tj ∩ S t

i 6= ∅}.

I Every j ∈W ∗ appears in at least one W ∗i , so∑

i

∑j∈W∗

iv tj ≥

∑i∈W∗ v

ti .

I Claim:∑j∈W∗

i

v tj ≤ v t

i

√m.

I Then lemma follows with intuitive accounting argument:Consider value that greedy loses compared to optimum because of addingi to W . This is at most a factor of

√m larger than the value he secures

by adding i to W .


Mechanism Design


Proving the Claim

For every j ∈W ∗i we have j ≥ i , and so by the order

v tj√|S t

j |≤ v t

i√|S t

i |⇒ v t

j ≤v ti

√|S t

j |√|S t

i |.

Summing over all j ∈W ∗i we get∑j∈W∗

i

v tj ≤

v ti√|S t

i |

∑j∈W∗

i

√|S t

j | .

The following Cauchy-Schwartz inequality∑j∈W∗

i

1 ·√|S t

j |

2

≤

∑j∈W∗

i

12

·∑

j∈W∗i

(√|S t

j |)2

yields a bound on the last term:∑j∈W∗

i

√|S t

j | ≤√|W ∗i |

√∑j∈W∗

i

|S tj | .


Mechanism Design


Proving the Claim

Combining the last two bounds we have so far:∑j∈W∗

i

v tj ≤

v ti√|S t

i |

√|W ∗i |

√∑j∈W∗

i

|S tj | .

I Every S tj intersects S t

i for j ∈W ∗i .

I W ∗ yields allocation, so S tj ∩ S t

k = ∅ for j , k ∈W ∗iI This means |W ∗i | ≤ |S t

i |.I W ∗ is allocation, so

∑j∈W∗ |S t

j | ≤ m

This gives ∑j∈W∗

i

v tj ≤ v t

i

√∑j∈W∗

|S tj | ≤ v t

i

√m

and finishes the proof.


Mechanism Design


Can it really be that bad?

Two bidders, m items M = {g1, . . . , gm}

(S∗1 , v∗1 ) = ({g1}, 1 + ε), (S∗2 , v

∗2 ) = (M,

√m)

Greedy winner: Player 1, 1 + ε Optimal winner: Player 2,√m


Mechanism Design


Special Cases and Heuristics

Special Cases:

Matching: Each bidder wants at most 2 items, |S∗i | ≤ 2Solved by algorithms for Weighted (Non-bipartite) Matching

Intervals: Items can be ordered on the real line such that S∗i includesexactly the items from an intervalSolved by a dynamic programming algorithm

Heuristics:

Various heuristics to solve the associated Integer Linear Program, mostallocation problems with tens of thousands of items are “practically solvable”.


Mechanism Design


Recommended Literature

I Chapter 9 and 11 in the AGT book.

I D. Lehmann, L.I. O’Callaghan, Y. Shoham. Truth revelation inapproximately efficient combinatorial auctions. Journal of the ACM,49(5):577–602, 2002.


Mechanism Design

Algorithmic Game Theory - hni.uni-paderborn.de fileRandomization Combinatorial Auctions IC in Expectation De nition A randomized mechanism isincentive compatible in expectationif truth-telling

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