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

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 Exchange Husband Wife Recipients Donors Brother Brother � 2

Motivation: Kidney Exchange Husband Wife Recipients Donors Brother Brother � 2

Motivation: Kidney Exchange Husband Wife Recipients Donors Brother Brother � 2

Motivation: Kidney Exchange Husband Wife Recipients Donors Brother Brother � 3

Motivation: Kidney Exchange Husband Wife Recipients Donors Brother Brother � 3

Motivation: Kidney Exchange Husband Wife Recipients Donors Brother Brother � 3

Kidney Exchange is Dynamic � 4

Kidney Exchange is Dynamic • Patient-donor pairs (agents) arrive gradually over time � 4

Kidney Exchange is Dynamic • Patient-donor pairs (agents) arrive gradually over time ‣ stay in the market to find a compatible pair � 4

Kidney Exchange is Dynamic • 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 � 4

Planner / Clearinghouse Platform 5

Planner / Clearinghouse Platform • Minimizes the number of agents who perish (leave the exchange without finding a match) 5

Planner / Clearinghouse Platform • Minimizes the number of agents who perish (leave the exchange without finding a match) • Knows agent’s expiration time [1] [1] M. Akbarpour, S. Li, and S. O. Gharan. Dynamic matching market design. 2017 5

Planner / Clearinghouse Platform • 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 [1] M. Akbarpour, S. Li, and S. O. Gharan. Dynamic matching market design. 2017 5

Planner / Clearinghouse Platform • 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 [1] M. Akbarpour, S. Li, and S. O. Gharan. Dynamic matching market design. 2017 5

Greedy and Patient Exchanges Greedy Patient algorithm algorithm M. Akbarpour, S. Li, and S. O. Gharan. Dynamic matching market design. 2017 6

Greedy Exchange Greedy algorithm M. Akbarpour, S. Li, and S. O. Gharan. Dynamic matching market design. 2017 7

Greedy Exchange Greedy algorithm M. Akbarpour, S. Li, and S. O. Gharan. Dynamic matching market design. 2017 7

Greedy Exchange Greedy algorithm M. Akbarpour, S. Li, and S. O. Gharan. Dynamic matching market design. 2017 7

Greedy Exchange Greedy algorithm M. Akbarpour, S. Li, and S. O. Gharan. Dynamic matching market design. 2017 7

Greedy Exchange Greedy algorithm M. Akbarpour, S. Li, and S. O. Gharan. Dynamic matching market design. 2017 7

Greedy Exchange Greedy algorithm M. Akbarpour, S. Li, and S. O. Gharan. Dynamic matching market design. 2017 7

Greedy Exchange Greedy algorithm M. Akbarpour, S. Li, and S. O. Gharan. Dynamic matching market design. 2017 7

Patient Exchange Patient algorithm M. Akbarpour, S. Li, and S. O. Gharan. Dynamic matching market design. 2017 8

Patient Exchange Patient algorithm M. Akbarpour, S. Li, and S. O. Gharan. Dynamic matching market design. 2017 8

Patient Exchange Patient algorithm M. Akbarpour, S. Li, and S. O. Gharan. Dynamic matching market design. 2017 8

Patient Exchange Patient algorithm M. Akbarpour, S. Li, and S. O. Gharan. Dynamic matching market design. 2017 8

Patient Exchange Patient algorithm M. Akbarpour, S. Li, and S. O. Gharan. Dynamic matching market design. 2017 8

Patient Exchange Critical Patient algorithm M. Akbarpour, S. Li, and S. O. Gharan. Dynamic matching market design. 2017 8

Patient Exchange Patient algorithm M. Akbarpour, S. Li, and S. O. Gharan. Dynamic matching market design. 2017 8

Two Questions: Strategic Agents Patient algorithm Greedy algorithm 9

Two Questions: Strategic Agents 𝜄 Patient algorithm Short-lived: T s T s < T l Greedy 1 - 𝜄 algorithm Long-lived: T l 9

Two Questions: Strategic Agents 𝜄 Patient algorithm Short-lived: T s Which market to enter? T s < T l How is the social welfare affected? Greedy 1 - 𝜄 algorithm Long-lived: T l 9

Two Questions: Strategic Markets Patient( 𝛽 1 ) Patient( 𝛽 2 ) 10

Two Questions: Strategic Markets Patient( 𝛽 1 ) Patient( 𝛽 2 ) 10

Two Questions: Strategic Markets Patient( 𝛽 1 ) (1- 𝛿 1 ) 𝛿 2 Patient( 𝛽 2 ) 10

Two Questions: Strategic Markets Patient( 𝛽 1 ) (1- 𝛿 1 ) 𝛿 2 (1- 𝛿 1 ) (1- 𝛿 2 ) Patient( 𝛽 2 ) 10

Two Questions: Strategic Markets Patient( 𝛽 1 ) (1- 𝛿 1 ) 𝛿 2 𝛿 1 (1- 𝛿 1 ) (1- 𝛿 2 ) Patient( 𝛽 2 ) 10

Two Questions: Strategic Markets Patient( 𝛽 1 ) (1- 𝛿 1 ) 𝛿 2 How do interactions 𝛿 1 between overlapping pools, different matching rate affect social welfare? (1- 𝛿 1 ) (1- 𝛿 2 ) Patient( 𝛽 2 ) 10

Utility Functions Agents Long-lived Short-lived Markets Number of matches 11

Model I: Strategic Agents 12

Model I: Strategic Agents • Fixed matching policy: one is Greedy, the other is Patient 12

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 12

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 12

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

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 13

Model I: Experimental Results 𝜚 = 0.4 Separating Equilibria: Pooling Equilibria: Pooling Equilibria: Short-lived Short-lived: Greedy Short-lived: Patient Patient, Long-lived: Greedy Long-lived: Patient Long-lived: Greedy 14

Model I: Experimental Results 𝜚 = 0.4 Separating Equilibria: Pooling Equilibria: Pooling Equilibria: Short-lived Short-lived: Greedy Short-lived: Patient Patient, Long-lived: Greedy Long-lived: Patient Long-lived: Greedy Increasing proportion of short-lived agents 14

Model I: Experimental Results = 0.4 1 Competing Greedy 0.9 Patient Expected utility 0.8 0.7 0.6 0.5 Pooling: Pooling: Separating Greedy Patient 0.4 0 0.2 0.4 0.6 0.8 1 15

Model II: Strategic Markets 16

Model II: Strategic Markets • Still a two-market system: Patient( 𝛽 1 ) , Patient ( 𝛽 2 ) 16

Model II: Strategic Markets • Still a two-market system: Patient( 𝛽 1 ) , Patient ( 𝛽 2 ) - 𝛽 : parameter for the degree of patience (Akbarpour et al. 2017), higher 𝛽 means more patient 16

Model II: Strategic Markets • Still a two-market system: Patient( 𝛽 1 ) , Patient ( 𝛽 2 ) - 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) 16

Model II: Strategic Markets • Still a two-market system: Patient( 𝛽 1 ) , Patient ( 𝛽 2 ) - 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 16

Model II: Market’s Tradeoff • Faster matching rate • Increased share of agents that enter both markets • Match fewer agents that only enter this Market 17

Model II: Experimental Results 18

Model II: Experimental Results • Convergence to the Patient strategy under appropriate initial conditions ( 𝛽 1 , 𝛽 2 ) 18