MARKETS, MECHANISMS, MACHINES University of Virginia, Spring 2019 Class 18: Hardness of Auctions 21 March 2019 cs4501/econ4559 Spring 2019 David Evans and Denis Nekipelov https://uvammm.github.io Trial Auction starts Now!

Trial Auction Starts Now! 1

Recap: Multi-Unit Auction 2 Input: Valuations for the players: !" , … , !% . Output: Allocation, '" , … , '% where ∑ ) ') ≤ ' that maximizes ∑ ) !) ') . Query model: !) 0 = 0, ∀. ∈ 0, … , ' − 1 : !) . ≤ !) (. + 1) Single-Minded Step Bids: !) . = 0 for . < .) ∗; !) . = <) ∗ for . ≥ .) ∗. Downward Sloping: ∀. ∈ 0, … , ' − 1 : !) . + 1 − !) . ≤ !) . − !) (. − 1) No polynomial (in log ') time solution Know polynomial (in log ') time solution ???

Second most Interesting Complexity Class? 3 NP Non-deterministic Polynomial Time

q 1 1 1 1 0 1 0 … q 0 1 1 1 0 0 0 … q 6 0 1 1 0 1 0 … Deterministic Turing Machine Nondeterministic Turing Machine q 0 1 1 1 0 0 0 … Tries all possible transitions; accepts if any path leads to accepting state. q 1 1 1 1 0 1 0 … q 7 1 1 1 0 0 0 …

Computability: Is NDTM more powerful than DTM?

Computability: Is NDTM more powerful than DTM? No! We can simulate a NDTM with a DTM. Use a tape to keep track of which paths to try (breadth-first search, not depth first!)

Complexity: Are there problems that are “hard” to solve with a DTM that are “easy” to solve with a NDTM?

Complexity Classes NP P Problems that a DTM can solve in polynomial time. Problems that a NDTM can solve in polynomial time.

P NP P NP We know P ⊆ NP: no need to take advantage of non-determinism.

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P NP P NP Option 1: There are problems in Class NP that are not tractable Option 2: All problems in Class NP are tractable P = NP? P ⊊ NP P = NP

Assuming P ⊊ NP, how can we prove a problem is not in P? 12

Class NP-Complete 13

NP-Complete A problem B is in NP-Complete iff: 2. There is a polynomial-time reduction from every problem A Î NP to B. 1. B Î NP B NP B NP What does NP-Complete look like?

NP-Hard A problem B is in NP-Hard iff: There is a polynomial-time reduction from every problem A Î NP to B. B NP What does NP-Hard look like?

NP-Hard (if P Ì NP) P NP-C Option 1a: P Ì NP, NP-C È P Ì NP Option 1b: P Ì NP, NP-C È P = NP NP-Hard P NP-C

NP-Hard (if P = NP) P NP Option 2: P = NP NP-C ≈ NP-Complete

What’s a Reduction? 18 There is a polynomial-time reduction from every problem A Î NP to B. B NP

Valid Reductions Transform ! into " Transformation must not take too long: finish in polynomial time Transformation must not expand input size too much: polynomial in original input size There is a polynomial-time reduction from every problem A Î NP to B. B NP Invalid transformation: “do an exponential search to find the answer, and output that”

Reduction Proofs Goal: Prove A does not have some property Y. We already know B does not have property Y. Proof by Contradiction: Assume ! has some property ". Since ! has ", there is a solution $ to ! that satisfies ". Show how $ could be used to solve %.

Reduction Proofs Goal: Prove A does not have some property Y. We already know B does not have property Y. Proof by Contradiction: Assume ! has some property ". Since ! has ", there is a solution $ to ! that satisfies ". Show how $ could be used to solve %. Since we know B does not satisfy Y, but having S would imply B satisfies Y, S cannot exist. Thus, S cannot exist, and A does not have Y.

NP-Hardness Reduction Proofs Goal: Prove A does not have some property Y. We already know B does not have property Y. Proof by Contradiction: Assume ! has some property ". Since ! has ", there is a solution $ to ! that satisfies ". Show how $ could be used to solve %. NP-Hardness: " = “is NP-Hard”, % = a known NP-Hard problem, $ = “a TM that solves ! in polynomial-time”

Proving NP-Hardness Show there is a polynomial-time reduction from every problem A Î NP to B. B NP Show there is a polynomial- time reduction from one problem X Î NP-Hard to B. X NP B This assumes we already know some problem X that is in NP-Hard. To get the first one, we need to prove it the hard way!

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KNAPSACK Problem Input: !" , $" , !% , $% , … , !' , $' !( , $( ∈ ℝ+, maximum weight , Output: - ⊆ {1, … , 1} that maximizes ∑ ( ∈4 $( subject to ∑ ( ∈4 !( ≤ , 25 Known (but won’t prove): KNAPSACK problem is in NP-Hard.

Multi-Unit Auction: Input Representation 26 Input: Valuations for the players: !" , … , !% . Output: Allocation, '" , … , '% where ∑ ) ') ≤ ' that maximizes ∑ ) !) ') . Query model: !) 0 = 0, ∀. ∈ 0, … , ' − 1 : !) . ≤ !) (. + 1) Single-Minded Step Bids: !) . = 0 for . < .) ∗; !) . = <) ∗ for . ≥ .) ∗. Downward Sloping: ∀. ∈ 0, … , ' − 1 : !) . + 1 − !) . ≤ !) . − !) (. − 1) No polynomial (in log ') time solution Know polynomial (in log ') time solution ???

Proof: Multi-Unit Auction with Single-Minded Bids is NP-Hard 27 Goal: Prove A does not have some property Y. We already know B does not have property Y. Proof by Contradiction: Assume ! has some property ". Since ! has ", there is a solution $ to ! that satisfies ". Show how $ could be used to solve %. NP-Hardness: " = “is NP-Hard”, % = a known NP-Hard problem, $ = “a TM that solves ! in polynomial-time”

Proof: Multi-Unit Auction with Single-Minded Bids is NP-Hard 28 Goal: Prove Multi-Unit Auction with Single-Minded Bids is NP-Hard. We already know KNAPSACK is NP-Hard. Proof by Contradiction: Assume !"# is not NP-Hard. Since !"# is not NP-Hard, there is a polynomial-time solution % to !"#. Show how % could be used to solve &'#(%#)&. NP-Hardness: * = “is NP-Hard”, + = KNAPSACK, % = “a TM that solves !"# in polynomial-time”

Reduction 29 MUA-SMB Input: Valuations for the players: !" , … , !% . !' ( = 0 for ( < (' ∗; !' ( = 1' ∗ for ( ≥ (' ∗. Output: Allocation, 3" , … , 3% where ∑ ' 3' ≤ 3 that maximizes ∑ ' !' 3' . KNAPSACK Input: 1" , !" , 16 , !6 , … , 1% , !% 1' , !' ∈ ℝ9, maximum weight : Output: ; ⊆ {1, … , ?} that maximizes ∑ ' ∈A !' subject to ∑ ' ∈A 1' ≤ :

Multi-Unit Auction: Input Representation 30 Input: Valuations for the players: !" , … , !% . Output: Allocation, '" , … , '% where ∑ ) ') ≤ ' that maximizes ∑ ) !) ') . Query model: !) 0 = 0, ∀. ∈ 0, … , ' − 1 : !) . ≤ !) (. + 1) Single-Minded Step Bids: !) . = 0 for . < .) ∗; !) . = <) ∗ for . ≥ .) ∗. Downward Sloping: ∀. ∈ 0, … , ' − 1 : !) . + 1 − !) . ≤ !) . − !) (. − 1) No polynomial (in log ') time solution Know polynomial (in log ') time solution NP-Hard: no polynomial time solution unless P=NP

31 Trial Auction Results

“If P = NP, then the world would be a profoundly different place than we usually assume it to be. There would be no special value in ‘creative leaps’, no fundamental gap between solving a problem and recognizing the solution once it’s found. Everyone who could appreciate a symphony would be Mozart; everyone who could follow a step-by-step argument would be Gauss...” Scott Aaronson Charge Project 4: Final auction next Thursday Project 4 reports Due next Friday