1. Truthful Mechanisms for One-parameter Agents Aaron Archer, Eva Tardos Presented by: Ittai Abraham

2. 2 Truthful Mechanisms for One-parameter Agents Introduction Introduction Terminology and notation Related work Characterization of truthful mechanisms Examples: –Scheduling to minimize makespan –LP –Uncapacitated facility location Lower bounds

3. 3 Introduction: Q||C max Scheduling jobs on related parallel machines to minimize makespan Each job j has processing requirement p j Each machine i runs in speed s i If job j is scheduled on machine i it takes p j /s i Goal: allocate jobs so that last job finishes as early as possible (makespan) Its NP-complete, and there is a known PTAS

4. 4 Introduction: the Q||C max game Each machine i is a distinct economic agent which incurs a cost proportional to the total time it spends processing Only machine i knows its true speed s i Our mechanism : –Asks each machine to report its speed –Allocates jobs using some output function o –Hands payments p i to each machine i using some payment function p

5. 5 Introduction: mechanism design of the Q||C max game We would like our mechanism to: –Cause truth-telling to be a (weakly) dominant strategy –Reach a (near) optimal allocation –Use polynomial resources –Never give truth tellers negative profits –Pay as little as possible For the PTAS allocation there is no payment scheme that causes profit-interested agents to be truthful A 3-approximation allocation combined with a payment scheme (both polynomial and shown later) cause truth-telling to be a dominant strategy

6. 6 Truthful Mechanisms for One-parameter Agents Introduction Introduction Terminology and notation Related work Characterization of truthful mechanisms Examples: –Scheduling to minimize makespan –LP –Uncapacitated facility location Lower bounds

7. 7 Terminology and notation M agents, represented by index set I Each agent i has a private value t i  Each agent reports a bid b i  t1t1 t2t2 titi tmtm Mechanism 12i m b1b1 b2b2 bibi bmbm

8. 8 Terminology and notation O is the set of allowable outcomes Output is a function o:  m  O Payment is a function p:  m   m Mechanism is a pair Mechanism t1t1 t2t2 titi tmtm 12i m b1b1 b2b2 bibi bmbm o p1p1 p2p2 pipi pmpm

9. 9 Terminology and notation Each outcome assigns work w:O  m Each Agent i wants to maximize her profit A mechanism is truthful if truth-telling is a dominant strategy. Formally fix any b -i then for all b i profit i (b -i,t i )  profit i (b -i,b i ) An output function o admits a truthful payment scheme if there exists a payment scheme p such that: Mechanism is truthful p i (b)- t i w i (o(b))

10. 10 Main questions Characterization: What output functions admit truthful payment schemes ? Mechanism design: For an output function that admits a truthful payment scheme, what is the payment scheme ?

11. 11 Truthful Mechanisms for One-parameter Agents Introduction Terminology and notation Related work Characterization of truthful mechanisms Examples: –Scheduling to minimize makespan –LP –Uncapacitated facility location Lower bounds

12. 12 Related Work Vickery-Clarke-Groves Mechanism maximizes the sum of the agent valuations (social welfare) Algorithmic mechanism design. (Ronen, Nisan) focus on scheduling unrelated machines through a Vickery auction for each job (reach m-approximation, 2 is best known) Algorithms for rational agents. (Ronen) characterize all truthful 0-1 load functions

13. 13 Truthful Mechanisms for One-parameter Agents Introduction Introduction Terminology and notation Related work Characterization of truthful mechanisms Examples: –Scheduling to minimize makespan –LP –Uncapacitated facility location Lower bounds

14. 14 Characterization of truthful mechanisms Definition: for a given b -i the load on agent i is: l i (x)= w i (o(b -i,x)) Definition:The output function o(b) is decreasing if for all b -i and for all i: l i is decreasing Theorem 1: The output function o(b) admits a truthful payment scheme only if it is decreasing

15. 15 Pictorial proof of Theorem 1 profit i =pay i -cost i : pay i =p i (b) cost i =t i w i (o(b)) If t i =y then cost i (x)+A+B=cost i (y) If t i =x then cost i (x)+A=cost i (y) So p i (y)- p i (x) is at least A+B and at most A But B is Positive ! AB xy l i (x) l i (y) Cost(x) when t=y

16. 16 Characterization of truthful mechanisms Theorem 2: A decreasing output function o(b) admits a truthful payment scheme if and only if it is of the form: For example if b=y then p=c-A y l i (y) A

17. 17 Proof of Theorem 2 (only if) Suppose l i (x) is differentiable, so for all b -i and for all t i the point t i is a maximum of p i (b -i,x)- t i w i (o(b -i,x ))

18. 18 Pictorial proof of Theorem 2 (if) Profit is p i (b)- t i w i (o(b)) Bidding truthfully gives –T Bidding lower gives –L titi l i (t) l l i (l) G

19. 19 Pictorial proof of Theorem 2 (if) Bidding truthfully gives –T Bidding higher gives –H. titi l i (t) h l i (h) G

20. 20 Characterization so far The output function o(b) admits a truthful payment scheme if only if it is decreasing. In this case the mechanism is truthful if and only if the payments p i (b -i,b i ) are of the form Where the h i are arbitrary functions

21. 21 Characterization: Voluntary participation A mechanism satisfies the voluntary participation condition if agents who bid truthfully never incur a loss Need to set h i (b -i ) to be at least as large as the integral 0 to t i for all t i Theorem 3: A decreasing output function o(b) admits a truthful payment scheme that satisfies the voluntary participation condition if and only if and we can choose it to be

22. 22 Generalization of Vickery auction The Vickery auction is a special case were agents bid their costs load is 0 or 1 and the lowest bidder pays the amount of the second lowest bid Critical value

23. 23 Truthful Mechanisms for One-parameter Agents Introduction Introduction Terminology and notation Related work Characterization of truthful mechanisms Examples: –Scheduling to minimize makespan –LP –Uncapacitated facility location Lower bounds

24. 24 Scheduling jobs on related parallel machines to minimize makespan Each job j has processing requirement p j Each machine i runs in speed s i, so t i =1/ s i If job j is scheduled on machine i it takes p j /s i Output function: allocate jobs to minimize makespan w i (o(b)) is the sum of the p j assigned to machine i So we need a decreasing allocation function

25. 25 Known allocations are not decreasing The PTAS of Hochbaum and Shmoys uses rounding and dynamic programming, announcing a slightly slower speed may cause receiving a different set of jobs and the load could increase because of rounding The greedy is not decreasing: two machines of almost equal speeds and jobs 2,1+ ,1+ . First, fast machine gets job 2 then, slow machine gets both 1+  jobs – so slower gets more work !

26. 26 From Scheduling to Bin Packing and fractional relaxations Equivalent to bin packing with uneven bins Let Cmax be the optimal makespan Given a guess T at the value of Cmax, create bins of size T/b i for each machine i T  Cmax iff exists an assignment of jobs s.t. each bin is at least as large as the total size of jobs assigned to it Get lower bound by relaxing the requirement and allowing fractional assignments of jobs

27. 27 Valid fractional assignments A fractional assignment is valid if –Each bin is at least as large as the total size of all fractional jobs assigned to it –Every bin receiving a piece of a job is large enough to contain the entire job The smallest T for which there exists a valid assignment is a lower bound for C max Given such a T the greedy algorithm finds the allocation: Assign the largest unassigned job to the largest bin that is not full yet Number bin and jobs from largest to smallest b 1  …  b m and p 1  …  p n

28. 28 Finding the lower bound of valid fractional allocation with greedy When is greedy valid ? For every job j, let i(j) denote the last bin that is as large as job j Greedy is valid iff for all j, the total capacity of the first i(j) bins is at least as large as the total size of the first j jobs

29. 29 Remember we need decreasing allocations Lemma: Sizing bins as T LB, greedy yields a valid fractional assignment s.t. each bin contains some full jobs and at most two partial jobs So round each split job to the faster machine and we get a 2-approximation But suppose p j b i is a bottleneck and job j exactly finishes bin i For b i +  T LB gets bigger so job j+1 gets partially in bin i increasing the load on bin i Seems difficult to overcome deterministically

30. 30 Randomized allocations for truthful mechanisms What does it mean for a randomized allocation to be truthful ? Agents aim to maximize their expected profit Truth telling a dominant strategy for agent i if bidding t i maximizes her expected profit regardless of what other agents bid A mechanism is truthful if for all agents truth telling is a dominant strategy So now interpret w i (o(b)) as the expected load on agent i

31. 31 Randomized Rounding Start with the greedy valid fractional assignment given by T LB Randomly assign the partial jobs in the following way: Job j is assigned to machine i with probability equal to the proportion of j that is fractionally assigned to bin i

32. 32 Theorem and proof of 3 approximation Theorem: allocation admits a truthful payment scheme satisfying voluntary participation and deterministically yields a polytime 3-approximation mechanism for Q||C max 3-approximation follows from valid allocation and at most two partial jobs

33. 33 Proof of decreasing expected work load and voluntary participation The expected load on bin i is the load on the greedy fractional assignment: T LB /b i Suppose some machine claims she is slower replacing b i with  b i where  >1 Clearly T LB  T ’ LB but T ’ LB  T LB (check) so T LB /b i  T ’ LB /  b i thus allocation is decreasing If a machine is very slow it will not receive any job (The T without it divided by b i is smaller than any job)

34. 34 Computing payments in polynomial time For a given b -i the load on agent i is: l i (x)= w i (o(b -i,x))= T LB (x)/x T LB (x)is either (1) constant, (2) or of the form cx, (3) or of the form c/(d+1/x) Breakpoint occur only when x coincides with another agent ’ s bid or when the two terms in T LB (x) cross ((3) to (2)) The number of intervals is polynomial

35. 35 More examples LP Uncapacitated facility location Scheduling to minimize sum of completion times Max flow

36. 36 Truthful Mechanisms for One-parameter Agents Introduction Terminology and notation Related work Characterization of truthful mechanisms Examples: –Scheduling to minimize makespan –LP –Uncapacitated facility location Lower bounds

37. 37 Lower bounds Scheduling on machines with speeds to minimize weighted sum of completion times Theorem: No truthful mechanism for Q||  w j C j can achieve an approximation ratio better than 2/  3, even with just two jobs and two machines

38. 38 Lower bounds Proof sketch: –Job 1 has weight and processing requirement 1 –Job 2 has weight w and processing req. p>1 –Machine 1 runs at speed 1, machine 2 at speed s Set pw<1 so that opt will be non-monotone OPT: –For small s both jobs will be on 1, then split, then swap, then both on 2. Not monotone A decreasing allocation: –For s >1 both on 2

39. 39 Summery For the model where profit i =p i (x)-t i *l i (x) gave a full characterization of allocation functions that admit truthful payment schemes For scheduling related machines to minimize makespan, shown a 3-approx truthful mechanism Can be used for other combinatorial allocations (max flow, facility location, … ) Can be used to prove lower bounds