Dynamic Housing Allocation

by Morimitsu Kurino

Presented by Malvika Rao and Alice Gao

Dynamic Housing Allocation

1House Allocation with Overlapping Agents: A Dynamic Mechnism Design Approach

Introduction – an Example


•Two available houses h1 and h2.•Each agent prefers h1 to h2 in each period.•Each agent prefers (h2,h1) to (h1,h2).

Static allocation is not dynamically Pareto efficient!

IntroductionSpot mechanism

W/o property rights transfer – for problems w/o endowments

With property rights transfer – for problems with and w/o endowments

SD versus TTC

Mechanism propertiesImpact of orderingsFutures mechanism


Preferences

Same # of agents arriving. Each agent stays for same amount of time.Same set of houses every period.


Assumptions

Period preferences: (h1, h2) < (h2, h1)But (h1, h1) ? (h2, h2)

Time-separable preferences.Time-invariant preferences.

ModelTime starts at t = 1, agents live in houses

for T periods

(A, H, R, e)A: set of agents; A = E + NH: set of housesR: set of preference profilese: set of endowment profiles

E: existing tenants; N: new tenantsD: endowed agents; U: unendowed agents


Model ContinuedPeriod t matching µ(t)Matching plan µ: collection of period t

matchingsSet of all matching plans M

Period t static mechanism: (D(t), U(t), H, R(t), e(t))

Dynamic mechanism π: R M


Desirable PropertiesAcceptability

Each agent is weakly better off as time goes on.

StrategyproofnessHistory-independent strategy of revealing true

period preferences is weakly better than any other HI strategy.

Pareto efficiencyA matching plan is PE if there exists no other

matching plan that makes all agents weakly better off and at least one agent strictly better off.


Impossibility ResultTheorem 1: For a dynamic problem with or

without endowments, there is no dynamic mechanism that is Pareto efficient and acceptable, if there are at least 2 newcomers in each period who live for at least 3 periods.


A different notion of acceptability?Acceptability (their version):

τ = t+1, …, t+T-1: µ(τ) Ra(τ) µ(τ-1)

Acceptability (different version):

τ = t+1, …, t+T-1: [µ(τ), …, µ(t+T-1)] Ra [µ(τ-1), …, µ(τ-1)]


SD Spot MechanismSpot mechanism without property rights

transferDynamic problem without endowments

Proposition 1: SD Spot Mech. is strategy-proofProof: Each SD period mechanism is

independent of past assignments.


SD Spot Mech. – Pareto efficient ?What period orderings can induce Pareto

efficient SD Spot mechanisms?

Theorem 2: Without endowments, constant SD Spot Mech. favoring existing tenants is Pareto efficient.


When is SD Spot Mech. undesirable?Pareto efficiency depends on the ordering

structure

Theorem 3: SD spot mech. favoring newcomers under time-invariant preferences is NOT Pareto efficient.


Dynamic Mechanisms under General Preferences

Acceptable Strategy-proof

Pareto efficient

General SD Spot Yes (Prop 1)

Constant SD Spot Favoring E

Yes Yes (Thm 2)

SD Spot Favoring N Yes No (Thm 3)

TTC Spot Yes

SD Futures Yes Yes


AS-TTC Static MechanismStatic serial dictatorship mechanism with

squatting rights is not Pareto efficient.

AS-TTC static mechanism (YRMH-IGYT) – Pareto efficient, individually rational, and

strategyproof.


TTC Spot MechanismAcceptable?

Pareto efficient?


TTC Spot MechanismStrategy-proof?

Theorem 5: For WD and time-invariant preferences, a constant TTC spot mechanism favoring existing tenants is strategy-proof among all agents except initial existing tenants.


TTC Spot Mechanism


TTC Spot MechanismsTheorem 6: For WD or ND and time-

invariant preferences, TTC spot mechanism favoring newcomers is NOT strategy-proof among all agents except initial existing agents


TTC Spot MechanismsTheorem 7: For WD and time-invariant

preferences, a constant TTC spot mechanism favoring existing tenants is Pareto efficient among all agents except initial existing tenants, but not Pareto efficient for all agents.

Theorem 8: For WD or ND and time-invariant preferences, a TTC spot mechanism favoring newcomers is NOT Pareto efficient among all agents except initial existing tenants, if there are at least 2 newcomers in each period.


Dynamic Mechanisms under Time-Invariant Preferences


Pareto efficient

General SD Spot Yes


Yes* Yes Yes

SD Spot Favoring N Yes

General TTC Spot Yes (Thm 4)

TTC Spot Favoring E Yes Yes** (Thm 5) Yes** (Thm 7)

TTC Spot Favoring N Yes No (Thm 6) No (Thm 8)

SD Futures Yes (Thm 9) Yes (Thm 9)

Yes* - the spot mechanism is acceptable for NDYes** - Strategyproof (Pareto efficient) for ND and Strategyproof (Pareto efficient) among all agents except initial existing tenants for WD.


SD Futures MechanismsDynamic problem without endowments

Agents report preferences over “assignments” during the period when he is in the market, and are given “assignments” of houses

Theorem 9: For ND, a SD futures mechanism is strategy-proof and Pareto efficient but not acceptable under same assumptions as the Impossibility Theorem.


Dynamic Mechanisms under Time-Invariant Preferences


Pareto efficient

General SD Spot Yes


Yes* Yes Yes

SD Spot Favoring N Yes

General TTC Spot Yes (Thm 4)

TTC Spot Favoring E Yes Yes** (Thm 5) Yes** (Thm 7)

TTC Spot Favoring N Yes No (Thm 6) No (Thm 8)

SD Futures Yes (Thm 9) Yes (Thm 9)

Yes* - the spot mechanism is acceptable for NDYes** - Strategyproof (Pareto efficient) for ND and Strategyproof (Pareto efficient) among all agents except initial existing tenants for WD.


Thank you!

House Allocation with Overlapping Agents: A Dynamic Mechnism Design Approach

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Dynamic Housing Allocation

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