1. August 12, 2010Online Mechanisms without Money On-line Mechanisms without Money Sujit Gujar E-Commerce Lab Dept of Computer Science and Automation Indian Institute of Science August 12, 2010 Presented at XRCE, Grenoble

2. August 12, 2010Online Mechanisms without Money Agenda Motivating Examples and Mechanism Design On-line Mechanisms Without Money –One-sided Markets (Dynamic House Allocation) –Two-sided Markets (Dynamic Matchings) Conclusion

3. August 12, 2010Online Mechanisms without Money Motivating Examples Dynamic allocations of goods –clean up task on wiki, science collaborators to perform useful task for society Dynamic re-allocation of goods –University dorm assignments, office space reassignments Dynamic Matchings –Campus recruitments

4. August 12, 2010Online Mechanisms without Money Dynamic allocation of goods A 3 : O1O1 O1O1 t t=3 t=5 O3O3 O3O3 O2O2 O2O2 › › A 3 Available A 1 : O1O1 O1O1 t t=1 t=2 O3O3 O3O3 O2O2 O2O2›› A 1 Available A 2 : O1O1 O1O1 t t=2t=4 O3O3 O3O3 O2O2 O2O2 ›› A 2 Available

5. August 12, 2010Online Mechanisms without Money M T M

6. August 12, 2010Online Mechanisms without Money M T TW

7. August 12, 2010Online Mechanisms without Money TW W

8. August 12, 2010Online Mechanisms without Money M T M TW W ?

9. August 12, 2010Online Mechanisms without Money Company 1 t=1t=2 Company 2

10. August 12, 2010Online Mechanisms without Money Company 3 t=1t=2 Company 2

11. August 12, 2010Online Mechanisms without Money Company 3Company 1 t=1t=2 Company 2 ?

12. August 12, 2010Online Mechanisms without Money Intellectual Challenges Preferences of the agents are private –(Incomplete information) Manipulative agents On-line nature of agents –any decision pertaining to the agents should be taken before he/she leaves the system No Money Repeated usage of static solutions fail All the above characteristics naturally fit into mechanism design problem solving framework

13. August 12, 2010Online Mechanisms without Money Mechanism Design Game Theory: Analysis of strategic interaction among players Mechanism Design: Reverse engineering of game theory Mechanism Design is the art of designing rules of a game to achieve a specific outcome in presence of multiple self-interested agents, each with private information about their preferences.

14. August 12, 2010Online Mechanisms without Money On-line mechanism without money One-sided markets Without initial Endowments (Dynamic allocation Mechanisms Ex 1) With initial Endowments (Dynamic house Allocation) Ex 2 Two-sided markets (Dynamic Matchings) EX 3

15. August 12, 2010Online Mechanisms without Money Dynamic House Allocation 1 1 Sujit Gujar, James Zou, David C Parkes, “Dynamic House Allocation”, In the proceedings of 5th Multidisciplinary Workshop on Advances in Preference Handling. (MPREF-2010)

16. August 12, 2010Online Mechanisms without Money M T M TW W ?

17. August 12, 2010Online Mechanisms without Money Dynamic House Allocation The above problem fits in dynamic house allocation framework Object owned by an agent – house Agents have private preferences Agents arrive-depart dynamically Feasibility: No agent can receive a house that does not arrive before his departure

18. August 12, 2010Online Mechanisms without Money System Setup  Agent A i – owns house h i (n agents)  Preferences over set of houses h 1,h 2,…,h n  Agent A i enters market in period a i and depart at period d i.  Preferences are private  Agents may mis-report preferences to receive better house  We assume agents do not lie about a i – d i  The allocation to agent i should be done within time periods a i – d i

19. August 12, 2010Online Mechanisms without Money Desirable Properties Strategyproof Mechanism is strategyproof if at all type profiles and at all arrival- departure schedules reporting preferences truthfully is dominant strategy equilibrium Individual Rationality (IR) Agent either receives better house than current one or keeps its own house Pareto Efficiency No other way of reallocation such that every agent is equally happy and at least one is strictly better off. Rank efficiency = average true rank for allocated houses. Minimize expected rank of allocated houses

20. August 12, 2010Online Mechanisms without Money State of the Art (Static Problem)

21. August 12, 2010Online Mechanisms without Money Top Trading Cycle Algorithm (TTCA) 1  Each agent points to his most preferred house among the available houses  Cycles – Trades  Remove all agents in the cycles  Continue till nobody is left 1 L. Shapley and H. Scarf, ‘On cores and indivisibility’, J. of Math. Econ., 1(1), 23–37, (1974).

22. August 12, 2010Online Mechanisms without Money A1 – h1 A2 – h2 A4 – h4 A5 – h5 A3 – h3 A4 – h4 A3 – h3A1 – h1 A2 – h2 A5 – h5 h2 > h4 > h3 > h1 > h5 h3 > h4 > h5 > h1 > h2 h2 > h3 > h1 > h4 > h5 h5 > h2 > h3 > h4 > h1 h1 > h4 > h2 > h3 > h5

23. August 12, 2010Online Mechanisms without Money A1 – h1 A2 – h3 A4 – h4 A5 – h5 A3 – h2 A4 – h4 A1 – h1 A5 – h5 h2 > h3 > h5 > h1 > h5 h3 > h4 > h5 > h1 > h2 h2 > h3 > h1 > h4 > h5 h5 > h2 > h3 > h4 > h1 h1 > h4 > h2 > h3 > h5

24. August 12, 2010Online Mechanisms without Money A1 – h5 A2 – h3 A4 – h4 A5 – h1 A3 – h2 A4 – h4 h2 > h3 > h5 > h1 > h5 h3 > h4 > h5 > h1 > h2 h2 > h3 > h1 > h4 > h5 h5 > h2 > h3 > h4 > h1 h1 > h4 > h2 > h3 > h5 Final house allocation: A1- h5, A2-h3, A3-h2, A4-h4, A5-h1

25. August 12, 2010Online Mechanisms without Money Static Case Results TTCA Strategyproof 1 Core (No subset of agents can improve allocation) ⇒ Pareto optimality, IR Ma 2 –TTCA is unique mechanism that is strategyproof, individually rational and in the core 1 A. E. Roth, ‘Incentive compatibility in a market with indivisible goods’, Economics Letters, 9(2), 127–132, (1982). 2 J. Ma, ‘Strategy-proofness and the strict core in a market with indivisibilities’, Int. J. of Game Theory, 23(1), 75–83, (1994).

26. August 12, 2010Online Mechanisms without Money Coming to On-line Settings In on-line settings our main result is: A simple strategyproof mechanism cannot use agent’s reported preference to decide at which time to let it participate in TTCA

27. August 12, 2010Online Mechanisms without Money In on-line setting no mechanism can achieve efficiency as well as individual rationality Compromise IR? Look for maximal efficiency in mechanisms that are IR? Recall: in Static Settings, TTCA achieves efficiency + Individual Rationality We show:

28. August 12, 2010Online Mechanisms without Money In on-line setting no mechanism can achieve efficiency as well as individual rationality Compromise IR? Look for maximal efficiency in mechanisms that are IR? ⅹ Recall: in Static Settings, TTCA achieves efficiency + Individual Rationality We show:

29. August 12, 2010Online Mechanisms without Money Naïve Mechanism: On-line TTCA In each period: The agents those are present, trade using TTCA Fails to be strategyproof (For more than three agents and two or more period)

30. August 12, 2010Online Mechanisms without Money M T M T ⅹ

31. August 12, 2010Online Mechanisms without Money M T T ⅹ

32. August 12, 2010Online Mechanisms without Money M T M T

33. August 12, 2010Online Mechanisms without Money M T T

34. August 12, 2010Online Mechanisms without Money Preliminaries:  Instead of houses: Classes of houses (To avoid corner cases in the proofs)  Simple mechanism: Suppose for a given mechanism, for any agent, A there exists, a perfect match set, occurrence of which guarantees the agent his most preferred house and the agents not similar to A do not trade with the perfect set agents

35. August 12, 2010Online Mechanisms without Money Characterization Result A simple strategyproof mechanism cannot use agent’s reported preference to decide at which time to let it participate in TTCA (Informal) In any strategyproof, simple on-line house allocation mechanism, if agent A participates in TTCA trade in period t by reporting some preference, then he continues to participate in TTCA trade in the period t for any other report If an online house allocation mechanism is SP and simple and agent A participates in TTCA in period t(> a ) for some report > a, then fixing scenario w(0,t) in regard to all agents except A, agent A continues to participate in period t(> a ) for all reports > a ’.

36. August 12, 2010Online Mechanisms without Money Partition Mechanisms Easy: Partition Mechanisms are strategyproof and IR Partition agents into sets such that all the agents in a set are simultaneously present in some period The partition is independent of agent’s preferences. It may depend upon a i -d i Execute TTCA among the agents in each partition

37. August 12, 2010Online Mechanisms without Money DO-TTCA DO-TTCA (Departing agents On-line TTCA) Partition agents based on departure time (All agents with d i = t trade in period t) a large number of agents that are present but may depart at distinct times T-TTCA T-TTCA (Threshold TTCA) If more than THRSHLD number of agents that have not participated in TTCA,are present, execute TTCA with these agents. Otherwise, if there are any departing agents, execute TTCA with these agents DO-TTCA, T-TTCA ⅹ

38. August 12, 2010Online Mechanisms without Money SO-TTCA SOTTCA Stochastic Optimization TTCA Adopts a sample-based stochastic optimization 1 method for partitioning the agents In each period, generate samples of possible future arrival- departure For each sample, find an offline partition using greedy heuristic (the bigger the each set in the partition, the better) For each agent, find out how often he is getting scheduled in the current period. Include the agent in current period if he is getting scheduled in current period more often than the fraction of agents yet to arrive 1 P. Van Hentenryck and R. Bent, Online Stochastic Combinatorial Optimization, MIT Press, 2006.

39. August 12, 2010Online Mechanisms without Money Partition Mechanism Simple DO-TTCA, T-TTCA and SO-TTCA induce partition of the agents which is independent of the agents preference reports and hence partition mechanisms DO-TTCA T-TTCA and SO-TTCA are simple mechanisms

40. August 12, 2010Online Mechanisms without Money Simulation Results E1: Poisson Arrival rate L = n/8 waiting period exponential distribution u = 0.01 L SP Not SP Static

41. August 12, 2010Online Mechanisms without Money Simulation Results E2: Poisson Arrival rate L = n/8 waiting period exponential distribution u = 0.1 L SP Not SP Static

42. August 12, 2010Online Mechanisms without Money Simulation Results E3: Uniform Arrival in period {1,2,…,T} (T=30) Departure period uniform between {ai, ai+T/8} truncate to T if di > T SP Not SP Static

43. August 12, 2010Online Mechanisms without Money Simulation Results E4: Preferences are not skewed (Some houses are more demanded than other) Arrival Departure as in E1 SP Not SP Static

44. August 12, 2010Online Mechanisms without Money Story so far… We have seen, Trade-off between efficiency and individual rationality in the on-line version For strategyproofness: agents cannot trade with different subsets of agents by changing their preference report Partition Mechanisms SO-TTCA, T-TTCA out perform DO-TTCA Immediate question: What if agents can mis-report arrival-departure schedules?

45. August 12, 2010Online Mechanisms without Money Dynamic Matchings 1 1 Sujit Gujar and David C Parkes, “Dynamic Matching with a Fall-back Option”, In the proceedings of 19 th European Conference on Artificial Intelligence, ECAI- 2010

46. August 12, 2010Online Mechanisms without Money Company 3Company 1 t=1t=2 Company 2 ?

47. August 12, 2010Online Mechanisms without Money Observations This can be considered as a matching problem Students are static Companies arrive-depart the system dynamically The preferences of the students are private We can assume that a company won’t lie about its preferences Typically it would be known what grades, skill-sets are required for a particular position in the company We consider students as Men (M) and Companies as Women (W)

48. August 12, 2010Online Mechanisms without Money Desirable Properties Blocking pair: (m,w) blocks the matching if they prefer to match with each other than their current match Stability: no blocking pair Strategyproof: For each agent, for any arrival- departure schedule, for any preferences it is a best response to report preferences truthfully Good Rank Efficiency. Lower the Rank(f), better

49. August 12, 2010Online Mechanisms without Money Male Proposal Deferred Acceptance Gale-Shapley 1 proposed, Each man proposes to his most preferred woman Each woman keeps her most preferred man among the proposals received and rejects all the others The men who are rejected in the above step propose to their next preferred woman If a woman receives a match better than her current match, she is matched with the new man and the previous man is not matched The process continues till all the men are matched 1 D. Gale and L. S. Shapley, “College admissions and the stability of marriage”, The American Mathematical Monthly, 69(1), 9-15, (January 1962)

50. August 12, 2010Online Mechanisms without Money Important Static Case Results Deferred acceptance is stable Deferred acceptance is strategyproof for men 2 No stable algorithm is strategyproof 2 2 A. E. Roth, “The economics of matching: Stability and incentives”, Mathematics of Operations Research, 7(4), 617-628, (1982).

51. August 12, 2010Online Mechanisms without Money On-line Deferred Acceptance In each period, run deferred acceptance. All these matches are temporary If any woman is departing in a period, the man with whom she is matched with, is now committed to her

52. August 12, 2010Online Mechanisms without Money But It Fails

53. Company 1 t=1 Company 2

54. Company 3 t=2 Company 2

55. Company 3Company 1 t=1t=2 Company 2

56. Company 1 t=1 Company 2

57. Company 1 t=1 Company 2

58. Company 3 t=2 Company 2

59. Company 3 t=2 Company 2

60. Company 3Company 1 t=1t=2 Company 2

61. August 12, 2010Online Mechanisms without Money We propose: GSODAS Why does the On-line Deferred Acceptance fail? What if a man has option to withdraw from previous match if he gets a better match? Generalized Stable, On-line Deferred Acceptance with Substitutes Algorithm: –Run Male Proposal Deferred Acceptance in each period. –If a man receives a better match, he decommits from the current match –If his previous match has left the system, she receives a substitute for him –The final period match is final match

62. August 12, 2010Online Mechanisms without Money Properties of the GSODAS Stable Strategyproof What about substitutes?

63. August 12, 2010Online Mechanisms without Money the GSODAS achieves stability optimally Worst Case Optimality of the GSODAS GSODAS: stability at the cost of substitutes Trade-off between usage of substitutes and stability Let n = α T, α ∈ {1, 2, 3,...} We use the metric, S(μ) = # Unstable men in μ + # Substitutes used in μ We show, Any on-line algorithm for matching, in worst case, S(μ) ≥ α (T − 1) and for the GSODAS S(μ) ≤ α (T − 1) with equality in the worst case. This implies,

64. August 12, 2010Online Mechanisms without Money Other Strategyproof Approaches ROMA: Randomized On-line Matching Algorithm ROMA1 In each period select |DW(t)| men randomly and run deferred acceptance with these men and departing women This match is final ROMA2 In each period, if |W(t)| > τ, select |W(t)| men randomly and run deferred acceptance with these men and the W(t) This match is final

65. August 12, 2010Online Mechanisms without Money Benchmark: Consensus This algorithm employs techniques of on-line stochastic optimization In each period, simulate the future arrival of the yet to arrive women by sampling sufficient number of future possible scenarios For each woman find out which man is getting matched most frequently. Based on this select the men to be considered for deferred acceptance in the current period Commit only those matches that involve departing women Clearly, ROMA1, ROMA2 are strategyproof and Consensus is not strategyproof

66. August 12, 2010Online Mechanisms without Money Number of Substitutes required by GSODAS

67. August 12, 2010Online Mechanisms without Money

68. August 12, 2010Online Mechanisms without Money Stability vs Rank Efficiency

69. August 12, 2010Online Mechanisms without Money

70. August 12, 2010Online Mechanisms without Money We have seen The naive usage of deferred acceptance fails in on-line settings We introduce the fall-back option Based on fall-back option we propose a scheme GSODAS for dynamic matching which is stable as well strategyproof for men at the cost of substitutes GSODAS achieves stability with minimal number of substitutes on worst case analysis Experiments show the GSODAS requires less substitutes than worst case bounds GSODAS performs quite well for rank efficiency

71. August 12, 2010Online Mechanisms without Money From here… In experiments: as T increase, as bad as 55% men use substitutes on worst case Will be unacceptable in many practical situations Relax the offline stability notion Look for better algorithms

72. August 12, 2010Online Mechanisms without Money Conclusion Naïve usage of static solutions fail to be strategyproof in on-line settings Need for weaken the efficiency and stability of static settings when problems are dynamic Any weaker solution concept for truthfulness?

73. August 12, 2010Online Mechanisms without Money Questions?

74. August 12, 2010Online Mechanisms without Money Thank You!

75. August 12, 2010Online Mechanisms without Money Few details Proof of the claim for GSODAS Technical Description of Simple mechanisms and DO-TTCA is simple

76. August 12, 2010Online Mechanisms without Money Proof of the Claim for GSODAS For GSODAS, # Unstable men = 0 and # Substitutes ≤ α(T − 1) i.e. S(μ) ≤ α(T − 1) Construct preference profile for which for any on-line matching algorithm, S(μ) is at least α(T − 1)

77. August 12, 2010Online Mechanisms without Money Preference

78. August 12, 2010Online Mechanisms without Money Preliminaries:  Instead of houses: Classes of houses (To avoid corner cases in the proofs)  Sample Path: w(>,ρ) An instance of an dynamic house allocation problem  w(t): Restriction of w(<,ρ) to the agents arrived before period t  w(t,t’): instance of reports of agents arrived in period [t,t’]  Λ: Perfect Match Set { A i,h i, > i,t > a i, d i }

79. August 12, 2010Online Mechanisms without Money Simple Mechanism A simple mechanism: Given a scenario w(0, t) and agent A available at t, there exists a perfect match set agents Λ = {A i,h i, > A,t > a, d} such that (a)If a continuation w(t+) contains Λ, then A receives his most preferred house under the scenario w(0, t)+w(t+), and (b) if B is present in w(0,t) and not similar to A then B does not trade with any agent in Λ

80. August 12, 2010Online Mechanisms without Money Partition Mechanism Simple DO-TTCA : For agent A (A,h 0, > A,a, d), the perfect match set: A set of n’ identical agents (A^, h^, > A^,d,d) h^ : A most preferred house and A^ ’ s second most h 0 : A^ ’ s most preferred house and n’ : the number of agents similar to A^ T-TTCA and SO-TTCA are simple mechanisms