Equilibrium Behavior in Competing Dynamic Matching Markets Zhuoshu - - PowerPoint PPT Presentation

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equilibrium behavior in competing dynamic matching markets
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Equilibrium Behavior in Competing Dynamic Matching Markets Zhuoshu - - PowerPoint PPT Presentation

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Equilibrium Behavior in Competing Dynamic Matching Markets Zhuoshu Li , Neal Gupta, Sanmay Das, John P . Dickerson 1 Motivation: Kidney Exchange 2 Motivation: Kidney Exchange Wife Recipients Brother 2 Motivation: Kidney

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Equilibrium Behavior in Competing Dynamic Matching Markets

  • 1

Zhuoshu Li, Neal Gupta, Sanmay Das, John P . Dickerson

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Motivation: Kidney Exchange

2

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SLIDE 3

Motivation: Kidney Exchange

2

Wife Brother

Recipients

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SLIDE 4

Motivation: Kidney Exchange

2

Wife Husband Brother Brother

Donors

Recipients

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Motivation: Kidney Exchange

2

Wife Husband Brother Brother

Donors

Recipients

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Motivation: Kidney Exchange

2

Wife Husband Brother Brother

Donors

Recipients

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SLIDE 7

Motivation: Kidney Exchange

3

Wife Husband Brother Brother

Donors

Recipients

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SLIDE 8

Motivation: Kidney Exchange

3

Wife Husband Brother Brother

Donors

Recipients

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Motivation: Kidney Exchange

3

Wife Husband Brother Brother

Donors

Recipients

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Kidney Exchange is Dynamic

4

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Kidney Exchange is Dynamic

4

  • Patient-donor pairs (agents) arrive gradually over time
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Kidney Exchange is Dynamic

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  • Patient-donor pairs (agents) arrive gradually over time
  • stay in the market to find a compatible pair
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Kidney Exchange is Dynamic

4

  • Patient-donor pairs (agents) arrive gradually over time
  • stay in the market to find a compatible pair
  • may leave if the patient’s condition deteriorates to

the point where kidney transplants become infeasible

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Planner / Clearinghouse Platform

5

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Planner / Clearinghouse Platform

5

  • Minimizes the number of agents who perish (leave the

exchange without finding a match)

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Planner / Clearinghouse Platform

5

[1] M. Akbarpour, S. Li, and S. O. Gharan. Dynamic matching market design. 2017

  • Minimizes the number of agents who perish (leave the

exchange without finding a match)

  • Knows agent’s expiration time[1]
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Planner / Clearinghouse Platform

5

[1] M. Akbarpour, S. Li, and S. O. Gharan. Dynamic matching market design. 2017

  • Minimizes the number of agents who perish (leave the

exchange without finding a match)

  • Knows agent’s expiration time[1]
  • Has only probabilistic knowledge about future

incoming agents

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Planner / Clearinghouse Platform

5

[1] M. Akbarpour, S. Li, and S. O. Gharan. Dynamic matching market design. 2017

  • Minimizes the number of agents who perish (leave the

exchange without finding a match)

  • Knows agent’s expiration time[1]
  • Has only probabilistic knowledge about future

incoming agents

  • Selects a subset of acceptable transactions at any

point in time

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Greedy and Patient Exchanges

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  • M. Akbarpour, S. Li, and S. O. Gharan. Dynamic matching market design. 2017

Greedy algorithm Patient algorithm

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Greedy Exchange

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Greedy algorithm

  • M. Akbarpour, S. Li, and S. O. Gharan. Dynamic matching market design. 2017
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Greedy Exchange

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Greedy algorithm

  • M. Akbarpour, S. Li, and S. O. Gharan. Dynamic matching market design. 2017
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Greedy Exchange

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Greedy algorithm

  • M. Akbarpour, S. Li, and S. O. Gharan. Dynamic matching market design. 2017
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Greedy Exchange

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Greedy algorithm

  • M. Akbarpour, S. Li, and S. O. Gharan. Dynamic matching market design. 2017
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Greedy Exchange

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Greedy algorithm

  • M. Akbarpour, S. Li, and S. O. Gharan. Dynamic matching market design. 2017
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Greedy Exchange

7

Greedy algorithm

  • M. Akbarpour, S. Li, and S. O. Gharan. Dynamic matching market design. 2017
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Greedy Exchange

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Greedy algorithm

  • M. Akbarpour, S. Li, and S. O. Gharan. Dynamic matching market design. 2017
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Patient Exchange

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Patient algorithm

  • M. Akbarpour, S. Li, and S. O. Gharan. Dynamic matching market design. 2017
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Patient Exchange

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Patient algorithm

  • M. Akbarpour, S. Li, and S. O. Gharan. Dynamic matching market design. 2017
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Patient Exchange

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Patient algorithm

  • M. Akbarpour, S. Li, and S. O. Gharan. Dynamic matching market design. 2017
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Patient Exchange

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Patient algorithm

  • M. Akbarpour, S. Li, and S. O. Gharan. Dynamic matching market design. 2017
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Patient Exchange

8

Patient algorithm

  • M. Akbarpour, S. Li, and S. O. Gharan. Dynamic matching market design. 2017
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Patient Exchange

8

Critical

Patient algorithm

  • M. Akbarpour, S. Li, and S. O. Gharan. Dynamic matching market design. 2017
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Patient Exchange

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Patient algorithm

  • M. Akbarpour, S. Li, and S. O. Gharan. Dynamic matching market design. 2017
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Two Questions: Strategic Agents

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Greedy algorithm Patient algorithm

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Two Questions: Strategic Agents

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Greedy algorithm Patient algorithm Short-lived: Ts

𝜄

Long-lived: Tl

1 - 𝜄

Ts < Tl

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Two Questions: Strategic Agents

9

Greedy algorithm Patient algorithm

Which market to enter? How is the social welfare affected?

Short-lived: Ts

𝜄

Long-lived: Tl

1 - 𝜄

Ts < Tl

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Patient(𝛽1) Patient(𝛽2)

Two Questions: Strategic Markets

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10

Patient(𝛽1) Patient(𝛽2)

Two Questions: Strategic Markets

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10

Patient(𝛽1)

(1-𝛿1)𝛿2

Patient(𝛽2)

Two Questions: Strategic Markets

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Patient(𝛽1)

(1-𝛿1) (1-𝛿2) (1-𝛿1)𝛿2

Patient(𝛽2)

Two Questions: Strategic Markets

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10

Patient(𝛽1)

𝛿1 (1-𝛿1) (1-𝛿2) (1-𝛿1)𝛿2

Patient(𝛽2)

Two Questions: Strategic Markets

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10

Patient(𝛽1)

How do interactions between overlapping pools, different matching rate affect social welfare?

𝛿1 (1-𝛿1) (1-𝛿2) (1-𝛿1)𝛿2

Patient(𝛽2)

Two Questions: Strategic Markets

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Utility Functions

11

Agents Markets Number of matches

Short-lived Long-lived

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Model I: Strategic Agents

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Model I: Strategic Agents

  • Fixed matching policy: one is Greedy, the other is Patient

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Model I: Strategic Agents

  • Fixed matching policy: one is Greedy, the other is Patient
  • Random Agents: allow a 𝜚 fraction of random-choice agents
  • choose either market with 0.5 probability

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Model I: Strategic Agents

  • Fixed matching policy: one is Greedy, the other is Patient
  • Random Agents: allow a 𝜚 fraction of random-choice agents
  • choose either market with 0.5 probability
  • Strategic Agents: 1 - 𝜚, decide which market to enter upon

arrival based on her expected utility

  • 𝜄: short-lived, 1 - 𝜄: long-lived

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Model I: Strategic Agents

  • Fixed matching policy: one is Greedy, the other is Patient
  • Random Agents: allow a 𝜚 fraction of random-choice agents
  • choose either market with 0.5 probability
  • Strategic Agents: 1 - 𝜚, decide which market to enter upon

arrival based on her expected utility

  • 𝜄: short-lived, 1 - 𝜄: long-lived
  • Analyze equilibrium strategies of strategic agents given 𝜄 and 𝜚

12

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Model I: Agent’s Tradeoff

  • Matching probability vs utility
  • Patient market: higher matching probability due to

market thickness, lower utility due to waiting

  • Greedy market: lower matching probability due to

market thinness, higher utility due to immediate matching

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Model I: Experimental Results

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Pooling Equilibria: Short-lived: Greedy Long-lived: Greedy Separating Equilibria: Short-lived Patient, Long-lived: Greedy Pooling Equilibria: Short-lived: Patient Long-lived: Patient

𝜚 = 0.4

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Model I: Experimental Results

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Pooling Equilibria: Short-lived: Greedy Long-lived: Greedy Separating Equilibria: Short-lived Patient, Long-lived: Greedy Pooling Equilibria: Short-lived: Patient Long-lived: Patient

Increasing proportion of short-lived agents

𝜚 = 0.4

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0.2 0.4 0.6 0.8 1 0.4 0.5 0.6 0.7 0.8 0.9 1

Expected utility = 0.4

Competing Greedy Patient

Model I: Experimental Results

Pooling: Greedy Pooling: Patient

Separating

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Model II: Strategic Markets

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Model II: Strategic Markets

  • Still a two-market system: Patient(𝛽1) , Patient (𝛽2)

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Model II: Strategic Markets

  • Still a two-market system: Patient(𝛽1) , Patient (𝛽2)
  • 𝛽: parameter for the degree of patience (Akbarpour et al. 2017),

higher 𝛽 means more patient

16

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Model II: Strategic Markets

  • Still a two-market system: Patient(𝛽1) , Patient (𝛽2)
  • 𝛽: parameter for the degree of patience (Akbarpour et al. 2017),

higher 𝛽 means more patient

  • Agents stochastically enter either Market 1, Market 2 or both markets

(Das et al. 2015)

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Model II: Strategic Markets

  • Still a two-market system: Patient(𝛽1) , Patient (𝛽2)
  • 𝛽: parameter for the degree of patience (Akbarpour et al. 2017),

higher 𝛽 means more patient

  • Agents stochastically enter either Market 1, Market 2 or both markets

(Das et al. 2015)

  • Markets respond to each other under best response dynamics. At

any time period

  • one observes the matching rate of its competitor
  • chooses maximum payoff strategy for perpetuity for the next time

period

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Model II: Market’s Tradeoff

  • Faster matching rate
  • Increased share of agents that enter both

markets

  • Match fewer agents that only enter this Market

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Model II: Experimental Results

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Model II: Experimental Results

  • Convergence to the Patient strategy under

appropriate initial conditions (𝛽1,𝛽2)

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Model II: Experimental Results

  • Convergence to the Patient strategy under

appropriate initial conditions (𝛽1,𝛽2)

  • For markets with sufficient overlap, and low initial

values of (𝛽1,𝛽2), convergence to a (Greedy, Greedy) equilibrium

18

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Model II: Experimental Results

  • Convergence to the Patient strategy under

appropriate initial conditions (𝛽1,𝛽2)

  • For markets with sufficient overlap, and low initial

values of (𝛽1,𝛽2), convergence to a (Greedy, Greedy) equilibrium

  • No other phenomena occur in more than 5% of

bootstrap samples

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Policy Implications

  • Significant social welfare losses through

fragmentation

  • Race to the bottom: Suboptimal matching policies
  • The United Network for Organ Sharing (UNOS)

matches per month to now 2+ times per week due to competition with fast-matching the National Kidney Registry (NKR)

  • Our model informs the debate

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