1. CS/SS 241a presentation California Institute of Technology1 False-Name-Proof Mechanisms for hiring a team Mahyar Salek Joint work with Atsushi Iwasaki, David Kempe, Yasumasa Saito and Makoto Yokoo

2. 2 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

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

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

5. 5 VCG : truthful mechanism 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 A C B D owns: AD owns: AB owns: BC CD c : $1 owns: c : $0 False-name bid

6. 6 False-name manipulations [Yokoo, Sakurai, Matsubara-00] 2,b 0,a 2,b 0,a0,a’ Identifier Splitting 1,b 0,a 0,a’ 1,b Self Division 1,a 2,a’’ Auctioneer uncertain about the graph structure False-name-proof mechanisms : Agents’ best interest to reveal true ownership and cost

7. 7 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.

8. 8 VCG and Overpayment 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. n Cost 1 Cost 0 Cost of the solution : 0 Cost of the most expensive solution : 1 Payment of VCG : n Overpayment compared to what? Cheapest solution? “Second” cheapest solution? VCG overpays a lot!

9. 9 The second cheapest… Cheapest solution disjoint from our solution Might not exist even in monopoly-free graphs! 1 1 000 Need more robust definition…

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

11. 11 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

12. 12 Finding frugal false-name- proof mechanisms for hiring a team

13. 13 Preliminaries 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. 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:

14. 14 Identifier Splitting Defined on a set system 0,a 1,b 0,a0,a’

15. 15 Self-Division 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. 0,a 0,a’ 1,b Pretending multiple distinct agents involved in task of an element Auctioneer uncertain about true set system (E, F) 0,a’’

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

17. 17 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)

18. 18 Algorithm 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. Polynomial for path Auctions Steep disincentive to self-divide

19. 19 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

20. 20 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 What if agents own multiple elements… Nearly matches MP’s overpayment

21. 21 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 Theorem 3: AP is false-name-proof, even if agents can own multiple elements and split identifiers. Mechanism similar to MP but with additive penalty and reserve cost : does not always buy a path

22. 22 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:

23. 23 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 :

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

25. 25 Proof idea of theorem 2 1 2 d

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

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

28. 28 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