False-Name-Proof Mechanisms for hiring a team

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

Problem • 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 • 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! • Mechanism Design

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

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

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

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

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

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

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

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

Previous Work • [Archer, Tardos 02, ESS04] For two node-disjoint 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 • Idea : Penalize paths with many edges

Finding frugal false-name-proof mechanisms for hiring a team

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

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

Self-Division • Pretending multiple distinct agents involved in task of an element • Auctioneer uncertain about true set system (E, F) 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. 1,b 0,a 1,b 0,a 0,a’’ 0,a’

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

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

Algorithm Polynomial for path Auctions • Given ‘s • Choose set minimizing among all feasible sets • Each agent e in the winning set is paid : “Best” solution among feasible sets not containing e. Steep disincentive to self-divide

Results Theorem 1 • MP is false-name-proof. • 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

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

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

Proof idea of theorem 2 Theorem2 • M : any strategy-proof mechanism on path auction • Frugality ratio of M on C is : Threshold bid of agent in Claim: there exists an edge in such that:

Proof (simplified) • 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 :

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

Proof idea of theorem 2 1 2 d

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

Summary

Open Questions • 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

False-Name-Proof Mechanisms for hiring a team