issues on the border of economics and computation נושאים בגבול כלכלה וחישוב
DESCRIPTION
Issues on the border of economics and computation נושאים בגבול כלכלה וחישוב. Speaker: Dr. Michael Schapira Topic: VCG and Combinatorial Auctions II. Quick Recap. Mechanism Design Scheme. types. reports. t 1. r 1. t 2. r 2. outcome. payments. t 3. r 3. Social planner. - PowerPoint PPT PresentationTRANSCRIPT
![Page 1: Issues on the border of economics and computation נושאים בגבול כלכלה וחישוב](https://reader030.vdocument.in/reader030/viewer/2022032612/5681330b550346895d99c877/html5/thumbnails/1.jpg)
Issues on the border of economics and computation
נושאים בגבול כלכלה וחישוב
Speaker: Dr. Michael SchapiraTopic: VCG and Combinatorial Auctions
II
![Page 2: Issues on the border of economics and computation נושאים בגבול כלכלה וחישוב](https://reader030.vdocument.in/reader030/viewer/2022032612/5681330b550346895d99c877/html5/thumbnails/2.jpg)
QuickRecap
![Page 3: Issues on the border of economics and computation נושאים בגבול כלכלה וחישוב](https://reader030.vdocument.in/reader030/viewer/2022032612/5681330b550346895d99c877/html5/thumbnails/3.jpg)
Mechanism Design Scheme
t1
t2
t3
t4
r1
r2
r3
r4
types reports
outcome
paymentsp1,p2,p3,p4
Social planner
![Page 4: Issues on the border of economics and computation נושאים בגבול כלכלה וחישוב](https://reader030.vdocument.in/reader030/viewer/2022032612/5681330b550346895d99c877/html5/thumbnails/4.jpg)
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 =
![Page 5: Issues on the border of economics and computation נושאים בגבול כלכלה וחישוב](https://reader030.vdocument.in/reader030/viewer/2022032612/5681330b550346895d99c877/html5/thumbnails/5.jpg)
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.
![Page 6: Issues on the border of economics and computation נושאים בגבול כלכלה וחישוב](https://reader030.vdocument.in/reader030/viewer/2022032612/5681330b550346895d99c877/html5/thumbnails/6.jpg)
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.
![Page 7: Issues on the border of economics and computation נושאים בגבול כלכלה וחישוב](https://reader030.vdocument.in/reader030/viewer/2022032612/5681330b550346895d99c877/html5/thumbnails/7.jpg)
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)
![Page 8: Issues on the border of economics and computation נושאים בגבול כלכלה וחישוב](https://reader030.vdocument.in/reader030/viewer/2022032612/5681330b550346895d99c877/html5/thumbnails/8.jpg)
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*
![Page 9: Issues on the border of economics and computation נושאים בגבול כלכלה וחישוב](https://reader030.vdocument.in/reader030/viewer/2022032612/5681330b550346895d99c877/html5/thumbnails/9.jpg)
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)
![Page 10: Issues on the border of economics and computation נושאים בגבול כלכלה וחישוב](https://reader030.vdocument.in/reader030/viewer/2022032612/5681330b550346895d99c877/html5/thumbnails/10.jpg)
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?
![Page 11: Issues on the border of economics and computation נושאים בגבול כלכלה וחישוב](https://reader030.vdocument.in/reader030/viewer/2022032612/5681330b550346895d99c877/html5/thumbnails/11.jpg)
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!
![Page 12: Issues on the border of economics and computation נושאים בגבול כלכלה וחישוב](https://reader030.vdocument.in/reader030/viewer/2022032612/5681330b550346895d99c877/html5/thumbnails/12.jpg)
Incentive-Compatible
Mechanism forSingle-Minded
Auctions
![Page 13: Issues on the border of economics and computation נושאים בגבול כלכלה וחישוב](https://reader030.vdocument.in/reader030/viewer/2022032612/5681330b550346895d99c877/html5/thumbnails/13.jpg)
Mechanism for Single-Minded Auctions
• Approximation factor of (m is #items)
• Incentive compatible in dominant strategies
• Efficiently computable (obvious)€
m
![Page 14: Issues on the border of economics and computation נושאים בגבול כלכלה וחישוב](https://reader030.vdocument.in/reader030/viewer/2022032612/5681330b550346895d99c877/html5/thumbnails/14.jpg)
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?
![Page 15: Issues on the border of economics and computation נושאים בגבול כלכלה וחישוב](https://reader030.vdocument.in/reader030/viewer/2022032612/5681330b550346895d99c877/html5/thumbnails/15.jpg)
•Monotonicity
• Critical payment
Proof of Incentive Compatibility
![Page 16: Issues on the border of economics and computation נושאים בגבול כלכלה וחישוב](https://reader030.vdocument.in/reader030/viewer/2022032612/5681330b550346895d99c877/html5/thumbnails/16.jpg)
• 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
![Page 17: Issues on the border of economics and computation נושאים בגבול כלכלה וחישוב](https://reader030.vdocument.in/reader030/viewer/2022032612/5681330b550346895d99c877/html5/thumbnails/17.jpg)
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
![Page 18: Issues on the border of economics and computation נושאים בגבול כלכלה וחישוב](https://reader030.vdocument.in/reader030/viewer/2022032612/5681330b550346895d99c877/html5/thumbnails/18.jpg)
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*
![Page 19: Issues on the border of economics and computation נושאים בגבול כלכלה וחישוב](https://reader030.vdocument.in/reader030/viewer/2022032612/5681330b550346895d99c877/html5/thumbnails/19.jpg)
Other Interesting Combinatorial
Auctions
![Page 20: Issues on the border of economics and computation נושאים בגבול כלכלה וחישוב](https://reader030.vdocument.in/reader030/viewer/2022032612/5681330b550346895d99c877/html5/thumbnails/20.jpg)
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.
![Page 21: Issues on the border of economics and computation נושאים בגבול כלכלה וחישוב](https://reader030.vdocument.in/reader030/viewer/2022032612/5681330b550346895d99c877/html5/thumbnails/21.jpg)
Computational Hardness
• Thm: Finding an optimal allocation in combinatorial auctions with submodular bidders is NP-hard.
• We now prove the theorem.
![Page 22: Issues on the border of economics and computation נושאים בגבול כלכלה וחישוב](https://reader030.vdocument.in/reader030/viewer/2022032612/5681330b550346895d99c877/html5/thumbnails/22.jpg)
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.
![Page 23: Issues on the border of economics and computation נושאים בגבול כלכלה וחישוב](https://reader030.vdocument.in/reader030/viewer/2022032612/5681330b550346895d99c877/html5/thumbnails/23.jpg)
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?
![Page 24: Issues on the border of economics and computation נושאים בגבול כלכלה וחישוב](https://reader030.vdocument.in/reader030/viewer/2022032612/5681330b550346895d99c877/html5/thumbnails/24.jpg)
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}
![Page 25: Issues on the border of economics and computation נושאים בגבול כלכלה וחישוב](https://reader030.vdocument.in/reader030/viewer/2022032612/5681330b550346895d99c877/html5/thumbnails/25.jpg)
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.
![Page 26: Issues on the border of economics and computation נושאים בגבול כלכלה וחישוב](https://reader030.vdocument.in/reader030/viewer/2022032612/5681330b550346895d99c877/html5/thumbnails/26.jpg)
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
![Page 27: Issues on the border of economics and computation נושאים בגבול כלכלה וחישוב](https://reader030.vdocument.in/reader030/viewer/2022032612/5681330b550346895d99c877/html5/thumbnails/27.jpg)
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})
![Page 28: Issues on the border of economics and computation נושאים בגבול כלכלה וחישוב](https://reader030.vdocument.in/reader030/viewer/2022032612/5681330b550346895d99c877/html5/thumbnails/28.jpg)
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})
![Page 29: Issues on the border of economics and computation נושאים בגבול כלכלה וחישוב](https://reader030.vdocument.in/reader030/viewer/2022032612/5681330b550346895d99c877/html5/thumbnails/29.jpg)
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?
![Page 30: Issues on the border of economics and computation נושאים בגבול כלכלה וחישוב](https://reader030.vdocument.in/reader030/viewer/2022032612/5681330b550346895d99c877/html5/thumbnails/30.jpg)
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).
![Page 31: Issues on the border of economics and computation נושאים בגבול כלכלה וחישוב](https://reader030.vdocument.in/reader030/viewer/2022032612/5681330b550346895d99c877/html5/thumbnails/31.jpg)
• 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
![Page 32: Issues on the border of economics and computation נושאים בגבול כלכלה וחישוב](https://reader030.vdocument.in/reader030/viewer/2022032612/5681330b550346895d99c877/html5/thumbnails/32.jpg)
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!