Issues on the border of economics and computation

נושאים בגבול כלכלה וחישוב

Speaker: Dr. Michael SchapiraTopic: VCG and Combinatorial Auctions

II

QuickRecap

Mechanism Design Scheme

t1

t2

t3

t4

r1

r2

r3

r4

types reports

outcome

paymentsp1,p2,p3,p4

Social planner

VCG Basic Idea

Welfare of the other players from the chosen outcome

Optimal welfare (for the other players) if

player i was not participating.

• You can maximize efficiency by:– Choosing the efficient outcome (given the

bids)– Each player pays his “social cost” (how

much his existence hurts the others).

pi =

VCG: Formal Definition

• The VCG mechanism:– Outcome w* is chosen.– Each bidder pays:

The total value for the others when

player i is not participating

ij

jjij

ijj wtvwtv ),(),( **

The total value for the others when i

participates

• Bidders are asked to report their private values ti

• Terminology: (given the reported ti’s)– w* outcome that maximizes the efficiency.– Let w*-i be the efficient outcome when i is not playing.

Truthfulness

Conclusion: welfare maximization can always be achieved in dominant strategies.• No Bayesian distributional assumptions.• No real multiple-equilibria problem as in Nash.• Very simple strategy for the bidders.

Theorem (Vickrey-Clarke-Groves):In the VCG mechanism, truth-telling is a dominant strategy for all players.

Combinatorial Auctions• Set M of m indivisible items• Set N of n bidders• Preferences are on subsets S – bundles – of

items • Valuation function vi: 2M R

– vi(S) – bidder i’s value for bundle S

– monotone: vi(S) not decreasing in S

– normalized: vi() = 0

Allocation: mutually-disjoint subsets S1, S2, … Sn

Social welfare of allocation: i vi(Si)

Single Minded Auctions

• A valuation v is single minded if there is a bundle of items S* and value a such that – v(S) = a if S contains S*– v(S) = 0 for all other S

• Very simple to represent: (S*, a)

• Allocation problem for single minded bidders:– Given bids {(Si*, ai)}i for bidders i=1..n – Find a feasible subset W of winning

bids with maximum social welfare j in

W aj*

What Do We Want?

1. “Good” (w.r.t. efficiency) outcomes (preferably optimal)

2. Incentive compatibility (preferably in dominant strategies)

3. Low running time (in the “natural parameters”: n and m)

Cannot Simply Use VCG!

• Finding optimal allocation is computationally (=NP) hard!

• Cannot compute “approximate” VCG payments.

• The “clash” between Econ and CS. What can we do?

Approximating the Best Allocation

€

∀T1,..,TnVi(Ti)∑Vi(Si)∑

≤ γ

2/1m

• Allocation S1,..,Sn is a g-approximation if:

• Even approximating optimal allocation of items in single-minded auctions within factor of is NP-hard!

Incentive-Compatible

Mechanism forSingle-Minded

Auctions

Mechanism for Single-Minded Auctions

• Approximation factor of (m is #items)

• Incentive compatible in dominant strategies

• Efficiently computable (obvious)€

m

Proof of Incentive Compatibility

• Lemma: A mechanism for single minded bidders in which losers pay 0 is incentive compatible iff it satisfies:

– Monotonicity: if a bid (S,a) is a winning bid, the bid (S*,a*), where S* is contained in S, or a*>a, is also winning.

– Critical payment: A bidder who wins with bid (S,a) pays the minimum needed for winning: the infimum of all values b such that (S,b) wins

• The two conditions are met by the greedy algorithm. Why?

•Monotonicity

• Critical payment

Proof of Incentive Compatibility

• We prove that the two conditions imply incentive compatibility (in dominant strategies).

• Exercise: Prove the reverse direction.

• Let B=(S,a) be the true input of a bidder, and let B*=(S*, a*) be a possible bid

• If B* loses or S* does not contain S, it makes no sense to bid B*

• Let p be the bidder’s critical payment for bid B, and p* be the critical payment for bid B*

• Critical payment: for every x < p, the bid (S,x) loses• Monotonicity: so, for every x < p, the bid (S*,x) also loses• Hence: p ≤ p*• Bidding (S, a*) instead of B*=(S*, a*) is no worse• But, B=(S, a) is no worse than (S, a*)

– If B wins payment is always p– If B loses, a < p and therefore it is not worth to win

Proof of Incentive Compatibility

Proof of Approximation RatioTheorem: Let OPT be allocation maximizingiOPTvi* and let W be the output of the greedy algorithm. Then iOPTvi* < √m(jWvj*)

Proof:• For each i in W let

OPTi={j OPT, i≤j| Si*Sj* ≠ }– the set of elements in OPT that did not

enter W “because” of i (also including i)

• Observe that OPT iW OPTi

• Will show: jOPTivj* ≤ (√m)vi* for all i in W

Proof of Approximation Ratio• For all jOPTi we know that vj*≤vi*√(|Sj*|/|Si*|)

• Hence, jOPTivj* ≤ (vi*/√|Si*|)(jOPTi

√|Sj*|)

• Using the Cauchy-Schwartz inequality we get that:jOPTi

√|Sj*| ≤ (√|OPTi |)(√jOPTi|Sj*|)

• For jOPTi, Si*Sj*≠• Since OPT is an allocation:

– these intersections are disjoint and so |OPTi | ≤ |Si*|

– jOPTi |Sj*| ≤ m

– jOPTi √|Sj*| ≤ √|Si*|√m

– Plugging into first inequality: jOPTivj* ≤ (√m)vi*

Other Interesting Combinatorial

Auctions

Natural Restrictions on Bidders

• Defn: A valuation v is subadditive (complement-free) if for all S,TM, v(ST) ≤ v(S) + v(T).

• Defn: A valuation v is submodular if for all S,TM, v(ST) ≤ v(S) + v(T).

• Equivalent definition of submodularity: for all STM, and j not in T,

v(T{j})-v(T) ≤ v(S{j})-v(S)

(decreasing marginal utilites)

• Fact: Submodularity implies subadditivity.

Computational Hardness

• Thm: Finding an optimal allocation in combinatorial auctions with submodular bidders is NP-hard.

• We now prove the theorem.

Proof

• We show a reduction of the PARTITION problem: We are given k real numbers {a1,…,ak} and the goal is to determined whether they can be partitioned into two disjoint subsets, W1 and W2, so that iW1

ai = jW2 ai

• Given an instance of PARTITION, we construct an auction with two identical bidders with valuation function:

v(S) = min{jS aj, ½iai}

• Observe that this valuation is submodular.• Observe that a social welfare of iai is achievable iff it

is possible to partition {a1,…,ak} as desired.

Approximating the Optimum?

• Thm: A 2-approximation to the optimal allocation in combinatorial auctions with submodular bidders can be computed in a computationally-efficient manner.

• How?

Greedy Algorithm for Submodular Auctions

• Set S1=S2=…=Sn=

• Go over the items in some order, WLOG, j=1,…,m

– Let k be the bidder for which the marginal value for item j, i.e., vi(Si{j})-vi(Si), is maximized.

– Allocated item j to bidder k, i.e., set Sk=Sk {j}

Approximability for Submodular Bidders

• Thm: The greedy algorithm outputs a2-approximation to the optimal allocation in combinatorial auctions with submodular bidders.

• Remark: There exists a (different!)2-approximation algorithm for the more general case of subadditive bidders.

• We now prove the theorem.

Proof• We prove by induction on the number of items.

Suppose that the statement is true for m-1 items.

• Let ALG(I) be the allocation the algorithm outputs for a given instance I of a combinatorial auction with submodular bidders. Let OPT(I) be the optimal allocation for instance I.– We will abuse notation and use ALG(I) and OPT(I) to denote

both allocations and social-welfare of allocations.

• Let k be the bidder to which item 1 is allocated in ALG(I). Let I* denote the instance derived from instance I by removing item 1 and setting v’k(S)=vk(S{1})-vk({1}) for all S– Observe that the bidders remain submodular!

• Let ALG(I*) and OPT(I*) denote the algorithm’s output and optimal allocation for instance I*, respectively

Proof• Clearly ALG(I)=ALG(I*)+vk({1})

• We will now show that OPT(I) ≤ OPT(I*)+2vk({1})

• We will then use the fact that OPT(I*) ≤ 2ALG(I*)– the induction hypothesis

• To conclude that:OPT(I) ≤ OPT(I*)+2vk({1}) ≤ 2ALG(I*)+2vk({1}) =2ALG(I)

• So, let’s prove that OPT(I) ≤ OPT(I*)+2vk({1})

Proof• We wish to show that OPT(I) ≤ OPT(I*)+2vk({1})

• Let OPT(I)={O1,…,On}. Suppose that item 1 is in Or. Let T1,…,Tn be the allocation of items {2,…,m} as in OPT(I).

• T1,…,Tn is a possible solution to I*. We now compare its value to OPT(I).

• All bidders but r get the exact same bundle in T1,…,Tn and in OPT(I). All bidders but k have the exact same valuation function in I and in I*.

• How much does bidder r lose? vr(Or)-vr(Tr) = vr(Tr{1})-vr(Tr) ≤ vr({1}) ≤ vk({1})

• How much does bidder k lose?vk(Ok)-(vk(Tk{1})-vk({1}) = vk(Ok)-vk(Ok{1}+vk({1}) ≤ vk({1}

• So, OPT(I) ≤ OPT(I*)+2vk({1})

So…

• We have a 2-approximation algorithm for combinatorial auctions with submodular bidders.

• The analysis for this algorithm is tight– better approximation ratios are

achievable.

• Is this algorithm incentive compatible?

Simple Example

• 2 items, 2 bidders:– v1(1)=1+ , v1(2)=2- , v1({1,2})=2-– v2(1)=1, v2(2)=1, v1({1,2})=1

• What will the algorithm do?

• Is this incentive compatible?

• Thm: The greedy algorithm cannot be rendered incentive compatible (via any payment rule).

• Lemma: If an algorithm A is incentive compatible in dominant strategies then:

pi(v, v-i) = pi (a, v-i ), where A(v) = a.

• Proposition: (incentive compatibility weak monotonicity):

Suppose A(vi ,v-i) = a and A(ui,v-i ) = b. Then pi(a,v-i) - vi(a) > pi(b,v-i ) - vi(b),(otherwise bidder i would declare ui instead of vi).

And, pi(b,v-i) - ui (b) > pi(a,v-i) - ui (a),(otherwise bidder i would declare ui instead of vi).

vi (a) + ui (b) ≤ ui (a) + vi (b).• •

Proof

Proof

• Now, let us revisiting the 2-item 2-bidder example:– v1(1)=1+ , v1(2)=2- , v1({1,2})=2-– v2(1)=1, v2(2)=1, v1({1,2})=1

• Now, consider v1 above and the following u1: u1(1)=0, u1(2)=2-e, u1({1,2})=2-e

• Observe that weak monotonicity does not hold!