Approximation Mechanisms for Preference Aggregation

1. Approximation Mechanisms for Preference Aggregation Moshe Tennenholtz Technion –Israel Institute of Technology

2. Approximate Mechanism design without money [Procaccia&Tennenholtz’09] • Approximation has been considered before in mechanism design (with money) to circumvent computational complexity [Nisan&Ronen’99] • In this work: approximation is used to obtain strategyproofness in settings without money • Money cannot be used due to ethical, legal or technical (e.g., security) considerations • Strategyproofness may come at the expense of optimality • Can consider computationally tractable optimization problem

3. Where to locate a library on a street? • Want to locate a public facility (library, train station, supermarket) on a street • n agents A, B, C,... report their ideal locations • A mechanism receives the reported locations as input, and returns the location of the facility • Given facility location, cost of an agent = its distance from the facility

4. Take 1: average • Suppose we have two agents, A and B • Mechanism: take the average • A mechanism is strategyproof if agents can never benefit from lying = the distance from their location cannot decrease by misreporting it • Problem: average is not strategyproof

5. Take 2: leftmost location • Mechanism: select the leftmost reported location • Mechanism is strategyproof • A mechanism is group strategyproofif a coalition of agents cannot all gain by lying = the distance from at least one member does not decrease • Mechanism is group strategyproof A B B B C D E

6. Social cost and approximation • Social cost (SC) of facility location = sum of distances to the agents • Leftmost location mechanism can be bad in terms of social cost • One agent at 0, n-1 agents at 1 • Mechanism selects 0, social cost MECH = n1 • Optimal solution selects 1, social cost OPT = 1 • Mechanism gives -approximationif for every instance, MECH/OPT   • Leftmost location mechanism has ratio  n1

7. Take 3: the median • Mechanism: select the median agent • The median is group strategyproof • The median minimizes the social cost B A C D D D E

8. Single-peaked preferences • Suppose candidates are ordered on a line • Every voter prefers candidates that are closer to her most preferred candidate • Let every voter report only her most preferred candidate (“peak”) • Choose the median voter’s peak as the winner • This will also be the Condorcet winner • Nonmanipulable! v5 v4 v2 v1 v3 a1 a2 a3 a4 a5

9. Facility location on a network • Agents located on a network, represented by a graph • Examples: • Network of roads in a city • Telecommunications network: • Line, Hierarchical (tree), Ring (circle) • Scheduling a daily task: circle A B C

10. Model • N={1,…,n} set of agents • Network represented by a graph G • Agents are located at x={x1,…,xn} • A mechanism is a function f:Gn G • If facility located at y, costi(y,xi)=d(xi,y) • A mechanism f is strategy-proof (SP) if:for every x  Gn, i N, and deviation xi’  G costi(f(xi’,x-i),xi)  costi(f(x),xi)

11. Median on trees • Suppose network is a tree • Mechanism: start from root, move towards majority of agents as long as possible • Mechanism minimizes social cost • Mechanism is (group) strategyproof A A B B D F F C C G E

12. Strategyproof mechanisms in general networks • Schummer and Vohra [JET 2004] characterized the strategyproof mechanisms on several networks • Corollary: if network contains a cycle, there is no strategyproof mechanism with approx ratio < n1 for SC

13. A randomized mechanism • A randomized mechanism is a function f:Gn(G) • Cost of agent = expected distance from the facility • If f(x)=P, costi(P,xi)=Ey~P[ d(xi,y) ] • Social cost = sum of costs = sum of expected distances • Random dictator mechanism: select an agent uniformly and return its location • Theorem: random dictator is a strategyproof (22/n)-approx mechanism for SC on any network

14. Random dictator is not always group strategyproof • Consider a star with three arms of length one, with three agents at leaves • Cost of each agent = 4/3 • After moving to center, cost of each agent = 1 1/3 A A 1 N 1 1 1/3 1/3 B B C C

15. Random dictator is sometimes group strategyproof • If the network is a line, random dictator is group strategyproof • Theorem: if the network is a circle, random dictator is group strategyproof

16. More generally • Theorem: Let x1,…,xn,y1,…,yn circle. There exists i  {1,…,n} such that: D X and are “nearly-antipodal” if there is no agent such that:  ( , ) or  ( , ) For example, and are nearly-antipodal X Y Z C B X Z Y X Y Z A A C A The proof establishes that there exists a pair of nearly-antipodal agents that do not both benefit from deviation B C D

17. Summary of social cost ? ?

18. Minimizing the maximum cost • Maximum cost (MC) of facility location = max distance to the agents • Example: facility is a fire station • Optimal solution on a line = average of leftmost and rightmost locations, its max cost = d(A,E)/2 • Mechanism: select A • Mechanism is group strategyproof and gives a 2-approximation to MC • Theorem [Procaccia&Tennenholtz09]:There is no deterministic strategyproof mechanism with approx ratio for MC smaller than 2 on a line A C D E B

19. The Left-Right-Middle Mechanism[Procaccia&Tennenholtz’09] • Left-Right-Middle (LRM) Mechanism: select leftmost location with prob. ¼, rightmost with prob. ¼, and average with prob. ½ • Approx ratio for MC is[½  (2  OPT) + ½  OPT] / OPT = 3/2 • LRM mechanism is strategyproof • Theorem:There is no randomized strategyproof mechanism with approximation ratio better than 3/2 for MC on a line 2d d B B C A D E 1/2 1/4 1/2 1/4 1/4

20. Minmax on general networks • Mechanism: choose A • Gives a 2-approximation to the maximum cost • O = optimal location, X = some agent • d(A,X)  d(A,O) + d(O,X)  2  OPT • Lower bound of 2 (for deterministic mechanism on a line) still holds

21. Minmax on a circle • Semicircle like an interval on a line • If all agents are on one semicircle, can apply LRM • Otherwise: LRM is meaningless 1/4 A B C F D 1/2 E 1/4

22. How about: Random Point • Random Point (RP) Mechanism: choose a point on the circle uniformly at random • RP is strategyproof • Approx ratio 7/4 if agents are not on one semicircle [worst case is where the nodes are “a bit more” than semi-circle, and then we have with probability ½ (out of the “semi-circle”) worst case ½, and with probability half (in the semi-circle) around 3/8 (average of ½ and ¼). And all together ½ x ½ +3/8 x ½ =7/16. While the optimal is 1/4)] • Expected cost of each agent in RP is ¼

23. Hybrid Mechanism 1 (Best of all Worlds) Are all agents on one semicircle? yes no LRM (approx 3/2 and SP) (RP gives bad approx) RP (approx 7/4 and SP) (LRM is meaningless) So, is Hybrid Mechanism 1 an SP mechanism with approx 7/4?

24. Hybrid Mechanism 1 is SP • Theorem: Mechanism 1 is SP • Proof:simple cases (all on one semicircle before and after the deviation, or not all on one semicircle before and after the deviation): (almost) trivially OKnot-so-simple cases: next

25. Hybrid Mechanism 1 is SP • Case 1: one semicircle  not one semicircle Cost = 1/41/2 + 1/21/4 = 1/4 A 1/4 1/4 1/2 LRM

26. Hybrid Mechanism 1 is SP • Case 1: one semicircle  not one semicircle  Cost = 1/4 Cost = 1/41/2 + 1/21/4 = 1/4 A A LRM RP

27. Hybrid Mechanism 1 is SP • Case 2: not one semicircle  one semicircle • Suffices to prove: cost after deviation (c’)  1/4 R’ A A C’ = ¼ d(A,L) + ¼ d(A,R) + ½ d(A,Y) = ¼ ( d(A,L) + d(A,R) ) + ½ d(A,Y) L’ L  ¼  ½ (“long” arc) 1/4 R d(L’,y)  ¼ d(R’,y)  ¼ A  [L’,R’] 1/2 1/4 A’ y=cen(L,R)  ¼  ½ + ½  ¼ = ¼

28. Intermediate conclusions • We found an SP mechanism with upper bound 7/4 • Can one do better? • Recall: lower bound of 3/2 still holds (even on a line)

29. Random Midpoint • Look at points antipodal to agents’ locations • Random Midpoint Mechanism: choose midpoint of arc between two antipodal points with prob. proportional to length • Theorem: mechanism is strategyproof • Approx ratio 3/2 if agents are not on one semicircle, but  2 if they are B 1/4 A 3/8 C C A B 3/8

30. Hybrid Mechanism 1 Are all agents on one semicircle? yes no LRM (approx 3/2 and SP) Random Point (approx 7/4 and SP)

31. 2 Hybrid Mechanism 1 Are all agents on one semicircle? yes no LRM (approx 3/2 and SP) Random Point (approx 7/4 + SP) Random Midpoint 3/2 Theorem: Mechanism is SP and gives 3/2-approximation for MC when network is a circle

32. PROOF… q.e.d.

33. Proof of Approximation 3/2 • If all agents are on one semicircle: LRM gives approximation 3/2 • If not all agents are on one semicircle: a a ≤ ½ B A A B OPT = (1-a)/2

34. A randomized lower bound on trees • Theorem:there is no randomized strategyproof mechanism with approximation ratio better than 2o(1) on trees • Randomization cannot help us much if the network is a tree

35. Summary of maximum cost ?

36. (Asymptotic) Approximate Mechanism design without money [NST] [Procaccia&Tennenholtz’09]: • Approximation has been considered before in mechanism design (with money) to circumvent computational complexity [Nisan&Ronen’99] • In this work: approximation is used to obtain strategyproofness in settings without money • Money cannot be used due to ethical, legal or technical (e.g., security) considerations • Strategyproofness may come at the expense of optimality • Can consider computationally tractable optimization problem [Nissim, Smorodinsky, Tennenholtz]: Asymptotic approximate mechanism design without money can go even further! (Nissim, Smorodinsky, T).

37. 2 facilities • n agents • 2 facilities • Agent cost = distance from its assigned facility. • Agents’ are assigned to facilities based on their reports using a randomized mechanism. • Optimal solution – choose locations for the facilities, and assign each agent to his closest facility, in a way that minimizes the sum of distances (social welfare).

38. A (general) idea [Nissim, Smorodinsky, and T] • The exponential mechanism from different privacy makes an agent’s declaration to effect its costs by no more than exp(ε ). (notice that this does not necessarily encourage truthfulness!) • With small probability the above is augmented with a well-designed “punishment” mechanism where an agent is assigned based on his reports to the closest among randomly selected locations. • By carefully adjusting parameters any non-negligible deviation becomes dominated by truthfulness, and non-dominated strategy profiles become asymptotically close to optimal!

39. An algorithm • Let ε > 0. Given declarations b: • With probability 1-p: choose facility location r={c,d} with probability exp(-ε cost(b,r)) where cost(b,r) is the social cost when agents’ location are b, facility location is r, and each agent is assigned to his closest location. • With probability p: choose i between 1 and log(n); choose uniformly among the pairs of locations with distance and assign each agent to the closest location among those.

40. Asymptotic Optimality • Let ε = , δ = , p = Then, deviation from truth telling by more than δ is dominated by truth telling. and when agents all use non-dominated strategies the social cost is OPT+O

41. Thank Y u!

42. Open problems • Randomized SP mechanisms on a circle, when the target is the social cost. • Gap between 1+e and 2 • GSP mechanisms on a circle when the target is the maximum cost • Gap between 3/2 and 2

43. Selection systems (k=2)

44. The In-degree k-Selection System: • A natural and popular objective: select a set of k nodes that maximize the sum of in-degrees. • Variants are applicable to tweeter popularity on one side and to survivor-like competitions on the other side, but also to internet-based reputation systems.

45. Incentive Compatibility • LetG=(V,E) and G'=(V,E') be graphs that differ only in the outgoing edges from vertex v. • A selection system is F incentive compatible, if for any such graphs G and G’ and vertex v, where if v is selected, and 0 otherwise.

46. Approximation Ratios • A selection has an approximation ratio of l, if the sum of in-degrees in it is at most l times lower than in an optimal selection. • We are interested in finding incentive compatible selection systems which are approximately optimal for some small l.

47. A general impossibility result • For any k < n, there is no incentive compatible selection systems that give any finite approximation ratio. • That is, you can not guarantee to always select in non-empty graphs an agent that someone votes for it.

48. Reality Games

49. In a variant of the TV show ``Survivor’’ each tribe member can recommend at most one other trusted member The mechanism selects a member to be eliminated in the tribal council, based on these recommendations Axiom: If there is a unique tribe member that received positive recommendations, then this member cannot be the eliminated one.

50. Alon, Fischer, Procaccia, Tennenholtz (2009): No such scheme can be strategy-proof, that is, there must be a scenario in which a member, knowing the scheme and the recommendations of all others, can gain (=avoid being eliminated) by mis-reporting his recommendation.