august 12, 2010online mechanisms without money on-line mechanisms without money sujit gujar...

August 12, 2010 Online Mechanisms without Money

On-line Mechanisms without Money

Sujit GujarE-Commerce Lab

Dept of Computer Science and AutomationIndian Institute of Science

August 12, 2010

Presented at XRCE, Grenoble


Agenda

• Motivating Examples and Mechanism Design

• On-line Mechanisms Without Money– One-sided Markets (Dynamic House Allocation)– Two-sided Markets (Dynamic Matchings)

• Conclusion


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


Dynamic allocation of goods

A3 :O1O1 t

t=3 t=5O3O3 O2O2

›› ››A3 Available

A1 : O1O1 tt=1 t=2

O3O3O2O2›› ››

A1 Available

A2 : O1O1 tt=2 t=4

O3O3 O2O2

›› ››A2 Available


M T

M


M T

T W


T W

W


M T

M

T W

W

??


Company 1

t=1 t=2

Company 2


Company 3

t=1 t=2

Company 2


Company 3Company 1

t=1 t=2

Company 2

??


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


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.


On-line mechanismwithout money

One-sided marketsTwo-sided marketsTwo-sided markets

(Dynamic Matchings)EX 3

Without initial Endowments

(Dynamic allocation Mechanisms Ex 1)

With initial With initial EndowmentsEndowments

(Dynamic house Allocation) Ex 2


Dynamic House Allocation1

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)


M T

M

T W

W

??


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


System Setup

Agent Ai – owns house hi (n agents)Preferences over set of houses h1,h2,…,hn

Agent Ai enters market in period ai and depart at period di.

Preferences are privateAgents may mis-report preferences to receive

better houseWe assume agents do not lie about ai – di

The allocation to agent i should be done within time periods ai – di


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 EfficiencyNo 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


State of the Art(Static Problem)


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 left1L. Shapley and H. Scarf, ‘On cores and indivisibility’, J. of Math. Econ., 1(1), 23–37, (1974).


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


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


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-h1A1- h5, A2-h3, A3-h2, A4-h4, A5-h1


Static Case ResultsTTCA

• Strategyproof1

• Core (No subset of agents can improve allocation) • ⇒ Pareto optimality, IR

Ma2

– TTCA is unique mechanism that is strategyproof, individually rational and in the core

1A. E. Roth, ‘Incentive compatibility in a market with indivisible goods’, Economics Letters, 9(2), 127–132, (1982).

2J. Ma, ‘Strategy-proofness and the strict core in a market with indivisibilities’, Int. J. of Game Theory, 23(1), 75–83, (1994).


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


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:


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:


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)


M T

M

T

ⅹ


M T

T

ⅹ


M T

M

T


M T

T


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


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’.

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’.


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 ai-di

Execute TTCA among the agents in each partition


DO-TTCADO-TTCA (Departing agents On-line TTCA)

Partition agents based on departure time (All agents with di = t trade in period t)

a large number of agents that are present but may depart at distinct times

T-TTCAT-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


SO-TTCA

SStochastic OOptimization TTCATTCAAdopts a sample-based stochastic optimization1 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

1P. Van Hentenryck and R. Bent, Online Stochastic Combinatorial Optimization, MIT Press, 2006.


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


Simulation ResultsE1: Poisson Arrival rate L = n/8 waiting period exponential distribution u = 0.01 L

SP

Not SP

Static


Simulation ResultsE2: Poisson Arrival rate L = n/8 waiting period exponential distribution u = 0.1 L

SP

Not SP

Static


Simulation ResultsE3: 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


Simulation ResultsE4: Preferences are not skewed (Some houses are more demanded than other)Arrival Departure as in E1

SP

Not SP

Static


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 MechanismsSO-TTCA, T-TTCA out perform DO-TTCA

• Immediate question: What if agents can mis-report arrival-departure schedules?


Dynamic Matchings1

1 Sujit Gujar and David C Parkes, “Dynamic Matching with a Fall-back Option”, In the proceedings of 19th European Conference on Artificial Intelligence, ECAI-2010


Company 3Company 1

t=1 t=2

Company 2

??


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)


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


Male Proposal Deferred Acceptance

Gale-Shapley1 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

1D. Gale and L. S. Shapley, “College admissions and the stability of marriage”, The American Mathematical Monthly, 69(1), 9-15, (January 1962)


Important Static Case Results

• Deferred acceptance is stable

• Deferred acceptance is strategyproof for men2

• No stable algorithm is strategyproof2

2A. E. Roth, “The economics of matching: Stability and incentives”, Mathematics of Operations Research, 7(4), 617-628, (1982).


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


But It Fails

Company 1

t=1

Company 2

Company 3

t=2

Company 2

Company 3Company 1

t=1 t=2

Company 2

Company 1

t=1

Company 2

Company 3

t=2

Company 2

Company 3Company 1

t=1 t=2

Company 2


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


Properties of the GSODAS

• Stable

• Strategyproof

• What about substitutes?


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,


Other Strategyproof Approaches

ROMA: Randomized On-line Matching AlgorithmROMA1• In each period select |DW(t)| men randomly and run

deferred acceptance with these men and departing women

• This match is finalROMA2• 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


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


Number of Substitutes required by GSODAS


Stability vs Rank Efficiency


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


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


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?


Questions?


Thank You!


Few details

• Proof of the claim for GSODAS

• Technical Description of Simple mechanisms and DO-TTCA is simple


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)


Preference


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 {Ai,hi,>i,t > ai, di}


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 = {Ai,hi, >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


Partition Mechanism Simple

• DO-TTCA: For agent A (A,h0, >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

h0 : A^ ’ s most preferred house and

n’ : the number of agents similar to A^

• T-TTCA and SO-TTCA are simple mechanisms

august 12, 2010online mechanisms without money on-line mechanisms without money sujit gujar...

Documents

set of houses h1

dynamic house allocationthe

preferred house

ai dithe allocation

period ai

hnagent ai

agent points

agent i