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False-Name-Proof Mechanisms for hiring a team Mahyar Salek Joint work with Atsushi Iwasaki, David False-Name-Proof Mechanisms for hiring a team Mahyar Salek Joint work with Atsushi Iwasaki, David Kempe, Yasumasa Saito and Makoto Yokoo CS/SS 241 a presentation California Institute of Technology 1

Problem n n n Hire a team to perform a task Each agent incurs Problem n n n Hire a team to perform a task Each agent incurs a cost by performing her sub-task Know which teams are capable of performing the task n n n Feasible Sets Don’t know how much cost a member of a team (an Agent) incurs to get her sub-task done Agents are selfish and opportunistic. Might lie about the required cost! n Mechanism Design 2

Definition n Set system (E, F) n n E: set of n elements (agent): Definition n Set system (E, F) n n E: set of n elements (agent): F of feasible sets n S: n S’: : cost of agent e n $10 $1 3

First Price Auction n n Cheapest path is bought. Agents are paid their own First Price Auction n n Cheapest path is bought. Agents are paid their own bids. n $10 Incentive to lie about cost! Searching for truthful Mechanisms… $10 $20 $1 4

VCG : truthful mechanism n Selection Rule n n Pick the cheapest feasible set VCG : truthful mechanism n Selection Rule n n Pick the cheapest feasible set Payment Rule n D Pay an agent the highest amount she could have bid to still be part of the winning set: Threshold bid owns: AD c : $1 owns: AB CD c : $0 C A owns: BC owns: c : $0 B bid e-name Fals 5

False-name manipulations [Yokoo, Sakurai, Matsubara-00] Self Division Identifier Splitting 1, b 2, b 0, False-name manipulations [Yokoo, Sakurai, Matsubara-00] Self Division Identifier Splitting 1, b 2, b 0, a 1, a Auctioneer uncertain about the graph structure 1, b 0, a 2, b 0, a’ 2, a’’ False-name-proof mechanisms : Agents’ best interest to reveal true ownership and cost 6

Impossibility Result n There is no false-name proof mechanism for hiring a team that Impossibility Result n There is no false-name proof mechanism for hiring a team that is individually rational and Pareto efficient. [Du, Sami, Shi 06, YSM 00] n A winning path selection is Pareto-Efficient if the mechanism selects a path with minimum cost. 7

VCG and Overpayment n Selection Rule n n Pick the cheapest feasible set Payment VCG and Overpayment n Selection Rule n n Pick the cheapest feasible set Payment Rule n Pay an agent the highest amount she could have bid to still be part of the winning set. n Overpayment compared to what? n n Cheapest solution? “Second” cheapest solution? n Cost of the solution : 0 Cost of the most expensive solution : 1 Payment of VCG : n Cost 1 Cost 0 VCG overpays a lot! 8

The second cheapest… n Cheapest solution disjoint from our solution n Might not exist The second cheapest… n Cheapest solution disjoint from our solution n Might not exist even in monopoly-free graphs! 1 0 0 0 1 Need more robust definition… 9

Frugality Ratio [Karlin et. Al. 05]: Intuition : cheapest total payment in a first Frugality Ratio [Karlin et. Al. 05]: Intuition : cheapest total payment in a first price auction Let S be cheapest feasible set with respect to cost is value minimizing : Subject to : for all e for all For every there is a such that: and Frugality Ratio : : Total payment of M when the true cost is c 10

Previous Work n n [Archer, Tardos 02, ESS 04] For two nodedisjoint s-t paths Previous Work n n [Archer, Tardos 02, ESS 04] For two nodedisjoint s-t paths of length n/2 each, no truthful mechanism with [Karlin, Kempe, Tamir 05] introduce mechanism, within constant factor of best frugality ratio n Idea : Penalize paths with many edges 11

Finding frugal false-nameproof mechanisms for hiring a team 12 Finding frugal false-nameproof mechanisms for hiring a team 12

Preliminaries n Owned Set system ((E, F), A) n n n E : set Preliminaries n Owned Set system ((E, F), A) n n n E : set of n elements F of feasible sets : cost of agent e n n n Auction: n n Private to agent : set of elements owned by i. Agents submit their bid consisting of cost and ownership. Auctioneer runs an algorithm to determine winning set and payments. n n n Winning set: Payments for each (pseudo) agent (could own multiple elements) Profit of agent i: 13

Identifier Splitting n Defined on a set system 0, a 1, b 0, a’ Identifier Splitting n Defined on a set system 0, a 1, b 0, a’ 14

Self-Division Pretending multiple distinct agents involved in task of an element • Single-element ownership: Self-Division Pretending multiple distinct agents involved in task of an element • Single-element ownership: F’ : keep every set that didn’t contain e, and replace e by its new set in every set that contained e. • Auctioneer uncertain about true set system (E, F) 1, b 0, a’’ 15

“Reachability” and Closure n set system (E’, F’) is reachable from (E, F). n “Reachability” and Closure n set system (E’, F’) is reachable from (E, F). n class C of set systems closed under subdivision n (E, F) if for any (E, F) in C, all set systems reachable from (E, F) also in C. (E’, F’) 16

The Multiplicative Penalty (MP) Mechanism n Assumption: n Each agent only owns one element The Multiplicative Penalty (MP) Mechanism n Assumption: n Each agent only owns one element n n Identify elements with agents Idea: n Penalize long paths n n Agents lose interest to subdivide Lose efficiency (honest economic long paths might not be winning anymore) 17

Algorithm n n n Pol yno mia Given ‘s Choose set minimizing among all Algorithm n n n Pol yno mia Given ‘s Choose set minimizing among all feasible sets Each agent e in the winning set is paid l fo rp ath Au ctio ns : “Best” solution among feasible sets not containing e. Steep disincentive to self-divide 18

Results Theorem 1 n MP is false-name-proof. n n so long as each agent Results Theorem 1 n MP is false-name-proof. n n so long as each agent only owns one element. It has frugality ratio of : payment of mechanism when the cost is c : “second” cheapest solution 19

Results… Theorem 2 n C : any class of monopoly free set systems closed Results… Theorem 2 n C : any class of monopoly free set systems closed under self-division n M : any false-name-proof mechanism n Frugality Ratio of M on C is Nearly matches MP’s overpayment What if agents own multiple elements… 20

The Additive Penalty (AP) Mechanism similar to MP but with additive penalty and reserve The Additive Penalty (AP) Mechanism similar to MP but with additive penalty and reserve cost n n n Agents can own multiple edges Only purchase a solution when total penalized cost does not exceed the reserve cost Reserve cost n n Buyer has own feasible set with a cost r Requires choice of r by the auctioneer Theorem 3: AP is false-name-proof, even if agents can own multiple elements and split identifiers. : does not always buy a path 21

Proof idea of theorem 2 Theorem 2 n M : any strategy-proof mechanism on Proof idea of theorem 2 Theorem 2 n M : any strategy-proof mechanism on path auction n Frugality ratio of M on C is : Threshold bid of agent Claim: there exists an edge in in such that: 22

Proof (simplified) n Claim: for all d, there exists an h no bigger than Proof (simplified) n Claim: for all d, there exists an h no bigger than d such that: Proof by induction on d: • Base case is trivial • Incentive compatibility for each agent requires that : 23

Summing up over all agents i=h … h+k: Taking l to be the minimum Summing up over all agents i=h … h+k: Taking l to be the minimum Using IH and h’ = h + l 24

Proof idea of theorem 2 1 2 d 25 Proof idea of theorem 2 1 2 d 25

Proof idea … bids 0 Wins and gets $1 Overpayment = 26 Proof idea … bids 0 Wins and gets $1 Overpayment = 26

Summary VCG n MP n Truthful n. Always buys a path n. Pareto Efficient Summary VCG n MP n Truthful n. Always buys a path n. Pareto Efficient Truthful and false-nameproof. nalways buys a path n“Reasonable” n False-Name susceptible Assumption: Each agent only owns a single element n overpayment compared to lower-bound AP Truthful and false-nameproof n. No Assumption on ownership n Might not buy a path n. Overpayment depends on reserve cost n 27

Open Questions n n Mechanism that always buys a path and is false-name-proof even Open Questions n n Mechanism that always buys a path and is false-name-proof even when each agent has multiple elements Matching upper-bound and lower-bound in overpayment for MP 28