Dynamic Housing Allocation Malvika Rao and Alice Gao Dynamic Housing Allocation by Morimitsu Kurino Presented by Malvika Rao and Alice Gao House Allocation with Overlapping Agents: A Dynamic Mechnism Design Approach House Allocation with Overlapping Agents: A Dynamic Mechanism 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! House Allocation with Overlapping Agents: A Dynamic Mechnism Design Approach

Introduction Spot mechanism Mechanism properties Impact of orderings 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 properties Impact of orderings Futures mechanism House Allocation with Overlapping Agents: A Dynamic Mechnism Design Approach

Assumptions Preferences Same # of agents arriving. Each agent stays for same amount of time. Same set of houses every period. Preferences Period preferences: (h1, h2) < (h2, h1) But (h1, h1) ? (h2, h2) Time-separable preferences. Time-invariant preferences. House Allocation with Overlapping Agents: A Dynamic Mechnism Design Approach

Model Time starts at t = 1, agents live in houses for T periods (A, H, R, e) A: set of agents; A = E + N H: set of houses R: set of preference profiles e: set of endowment profiles E: existing tenants; N: new tenants D: endowed agents; U: unendowed agents House Allocation with Overlapping Agents: A Dynamic Mechnism Design Approach

Model Continued Period t matching µ(t) Matching plan µ: collection of period t matchings Set of all matching plans M Period t static mechanism: (D(t), U(t), H, R(t), e(t)) Dynamic mechanism π: R  M House Allocation with Overlapping Agents: A Dynamic Mechnism Design Approach

Desirable Properties Acceptability Strategyproofness Pareto efficiency Each agent is weakly better off as time goes on. Strategyproofness History-independent strategy of revealing true period preferences is weakly better than any other HI strategy. Pareto efficiency A 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. House Allocation with Overlapping Agents: A Dynamic Mechnism Design Approach

Impossibility Result Theorem 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. House Allocation with Overlapping Agents: A Dynamic Mechnism Design Approach

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)] House Allocation with Overlapping Agents: A Dynamic Mechnism Design Approach

SD Spot Mechanism Spot mechanism without property rights transfer Dynamic problem without endowments Proposition 1: SD Spot Mech. is strategy-proof Proof: Each SD period mechanism is independent of past assignments. House Allocation with Overlapping Agents: A Dynamic Mechnism Design Approach

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

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

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 No (Thm 3) TTC Spot SD Futures House Allocation with Overlapping Agents: A Dynamic Mechnism Design Approach

AS-TTC Static Mechanism Static serial dictatorship mechanism with squatting rights is not Pareto efficient. AS-TTC static mechanism (YRMH-IGYT) – Pareto efficient, individually rational, and strategyproof. House Allocation with Overlapping Agents: A Dynamic Mechnism Design Approach

TTC Spot Mechanism Acceptable? Pareto efficient? House Allocation with Overlapping Agents: A Dynamic Mechnism Design Approach

TTC Spot Mechanism Strategy-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. House Allocation with Overlapping Agents: A Dynamic Mechnism Design Approach

TTC Spot Mechanism House Allocation with Overlapping Agents: A Dynamic Mechnism Design Approach

TTC Spot Mechanisms Theorem 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 House Allocation with Overlapping Agents: A Dynamic Mechnism Design Approach

Malvika Rao and Alice Gao TTC Spot Mechanisms Theorem 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. House Allocation with Overlapping Agents: A Dynamic Mechnism Design Approach House Allocation with Overlapping Agents: A Dynamic Mechanism Design Approach

Dynamic Mechanisms under Time-Invariant Preferences Acceptable Strategy-proof Pareto efficient General SD Spot Yes Constant SD Spot Favoring E Yes* SD Spot Favoring N General TTC Spot Yes (Thm 4) TTC Spot Favoring E Yes** (Thm 5) Yes** (Thm 7) TTC Spot Favoring N No (Thm 6) No (Thm 8) SD Futures Yes (Thm 9) Yes* - the spot mechanism is acceptable for ND Yes** - Strategyproof (Pareto efficient) for ND and Strategyproof (Pareto efficient) among all agents except initial existing tenants for WD. House Allocation with Overlapping Agents: A Dynamic Mechnism Design Approach

SD Futures Mechanisms Dynamic 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. House Allocation with Overlapping Agents: A Dynamic Mechnism Design Approach

Dynamic Mechanisms under Time-Invariant Preferences Acceptable Strategy-proof Pareto efficient General SD Spot Yes Constant SD Spot Favoring E Yes* SD Spot Favoring N General TTC Spot Yes (Thm 4) TTC Spot Favoring E Yes** (Thm 5) Yes** (Thm 7) TTC Spot Favoring N No (Thm 6) No (Thm 8) SD Futures Yes (Thm 9) Yes* - the spot mechanism is acceptable for ND Yes** - Strategyproof (Pareto efficient) for ND and Strategyproof (Pareto efficient) among all agents except initial existing tenants for WD. House Allocation with Overlapping Agents: A Dynamic Mechnism Design Approach

Thank you! Discussion Questions… House Allocation with Overlapping Agents: A Dynamic Mechnism Design Approach