Selfish Routing in Public Service

電機四 王鴻鈞 [ 指導老師 ] 陳和麟 教授2015/1/16

Vincent A. Knight , Paul R. HarperSchool of Mathematics, Cardiff University, Cardiff, UK

2

Why Find this Paper

● Previous Working on the Paper Survey of Electronic Health Care Data System● Want to Find Some Relation Between Them

ROUTING GAME ?

3

Motivation of Paper & Previous Works

Equilibrium Behavior of a queueing system

● Social optimization versus selfish optimization in waiting lines I. Adler, P. Naor 1969

How to Evaluate the Efficiency● Selfish Routing and the Price of Anarchy Tim Roughgarden, 2005

When Player Have Choice[Removing Central Control for a Given System]

A Survey OnNetworking Games in Telecommunications

E. Altman 2006

Price of Anarchy in the Markovian Single Server

Queue

G. Gilboa-Freedman, R. Hassin2008

The Price of Anarchy in Basketball

B. Skinner 2010

Public Servicee.g.

Health Care System?

4

Main Contributions of the Paper

Placing Choice between Public Serviceswithin the Formulation of Routing Games

Theoretical results of the effect of demand and worth of service

Numerical approach based on heuristics to calculate

the price of anarchy in a real world setting

The above ideas are demonstrated with a large scale real worldcase study

With an adequate capacity to provide the perceived worth of service, a high price of anarchy is to be expected

5

Introduction of Routing Game

Here we focus on non-atomic routing game:

𝐺=(𝑉 ,𝐸 )

(𝐶𝑜𝑚𝑚𝑜𝑑𝑖𝑡𝑖𝑒𝑠) A 𝑠𝑒𝑡𝑜𝑓 𝑠𝑜𝑢𝑟𝑐𝑒−𝑠𝑖𝑛𝑘𝑝𝑎𝑖𝑟𝑠 :{(𝑠1 ,𝑡 1) ,(𝑠2 , 𝑡 2) , . . ., (𝑠𝑘 , 𝑡𝑘)}

𝑃𝑖 : h𝑇 𝑒𝑠𝑒𝑡 𝑜𝑓 𝑎𝑙𝑙𝑝𝑜𝑠𝑠𝑖𝑏𝑙𝑒 (𝑠𝑖 ,𝑡𝑖) h𝑝𝑎𝑡 𝑠( 𝑓𝑜𝑟 𝑝𝑙𝑎𝑦𝑒𝑟 𝑖∈[𝑘])

h𝐸𝑎𝑐 𝑒𝑑𝑔𝑒 h𝑎𝑠𝑎𝑐𝑜𝑠𝑡 𝑓𝑢𝑛𝑐𝑡𝑖𝑜𝑛𝑚𝑎𝑝 h𝑡 𝑒𝑡𝑟𝑎𝑓𝑓𝑖𝑐 𝑡𝑜 h𝑡 𝑒𝑐𝑜𝑠𝑡 :𝐶𝑒(𝑛𝑜𝑛−𝑛𝑒𝑔𝑎𝑡𝑖𝑣𝑒 ,𝑐𝑜𝑛𝑣𝑒𝑥 ,𝑛𝑜𝑛−𝑑𝑒𝑐𝑟𝑒𝑎𝑠𝑖𝑛𝑔)

𝐶 ( 𝑓 )≔ ∑𝑃∈𝑃

𝐶𝑃 ( 𝑓 ) 𝑓 𝑃≔∑𝑒∈𝐸

𝑐𝑒 ( 𝑓 ) 𝑓 𝑒

h𝑇 𝑒𝑅𝑜𝑢𝑡𝑖𝑛𝑔𝐺𝑎𝑚𝑒𝑖𝑠𝑑𝑒𝑛𝑜𝑡𝑒𝑑𝑏𝑦 (𝐺 ,𝑟 ,𝑐)

6

Some Definition & Theorem of Routing Game

𝑃𝑜𝐴 (𝐺 ,𝑟 ,𝑐 )=𝐶 ( 𝑓 )𝐶 ( 𝑓 ∗)

7

Link to A Case – POA in the Public Service System

◎ h𝑇 𝑒𝑎𝑏𝑜𝑣𝑒 h𝑔𝑟𝑎𝑝 𝑖𝑠 𝑛𝑜𝑡 h𝑡 𝑒 h𝑔𝑟𝑎𝑝 𝑤𝑒𝑑𝑒𝑓𝑖𝑛𝑒𝑖𝑛 h𝑡 𝑒𝐺𝑎𝑚𝑒 .

m demand nodesBriefly two choice :1) Balking (not seek service)2) Seeking serviceEach demand node has 1) Demand rate: 2) Cost of balking: 3) Weight of caring about the distance to service node :

n service nodes1) Cost function of the facility : (Assume that the service node doesn’t have the choice of rejecting the query of demand node )

From demand nodes to service nodes1) Arrival rate : 2) Distance :

8

Link to A Case – POA in the Public Service System

𝑖𝑠 h𝑡 𝑒𝑟𝑎𝑡𝑒𝑜𝑓 𝑏𝑎𝑙𝑘𝑖𝑛𝑔 𝑜𝑓 𝑑𝑒𝑚𝑎𝑛𝑑𝑛𝑜𝑑𝑒𝑖

9

Link to A Case – POA in the Public Service System

10

Special Case – Single Source, Single Facility

11

Special Case – Single Source, Single Facility

12

13

A Real Case Study – Price of Anarchy in the Welsh Health Service

The choice of facilityfor elective knee replacement surgery across Wales

14

A Real Case Study – Price of Anarchy in the Welsh Health Service

15

A Real Case Study – Price of Anarchy in the Welsh Health Service

16

A Real Case Study – Price of Anarchy in the Welsh Health Service

17

A Real Case Study – Price of Anarchy in the Welsh Health Service

Simple single hospital single demand node example with two patient groups

18

A Real Case Study – Price of Anarchy in the Welsh Health Service

Simple single hospital single demand node example with two patient groups Effect on the price of anarchy of in the example with

19

Conclusion

●The price of anarchy increases with worth of service, up to a point.● In a system with insufficient capacity the price of anarchy is low● When considering the public service system as a whole, the highest level of inefficiency occurs when the capacity of the system matches the perceived worth of service.

20

Further Discussion

h𝑇 𝑒𝑟𝑒 𝑎𝑟𝑒𝑚𝑜𝑟𝑒𝑎𝑛𝑑𝑚𝑜𝑟𝑒𝑎𝑝𝑝𝑙𝑖𝑐𝑎𝑡𝑖𝑜𝑛 h𝑡 𝑎𝑡 𝑛𝑒𝑒𝑑 h𝑡 𝑒𝑎𝑛𝑎𝑙𝑦𝑠𝑖𝑠𝑜𝑓 h𝑡 𝑒𝑒𝑓𝑓𝑖𝑐𝑖𝑒𝑛𝑐𝑦 𝑜𝑓 h𝑡 𝑒𝑠𝑦𝑠𝑡𝑒𝑚 .

𝐵𝑢𝑡 h𝑡 𝑒𝑟𝑒𝑎𝑟𝑒 𝑡𝑜𝑜𝑚𝑎𝑛𝑦 𝑑𝑖𝑓𝑓𝑒𝑟𝑒𝑛𝑐𝑒𝑖𝑛𝑑𝑒𝑡𝑎𝑖𝑙 ,𝑠𝑜 h𝑡 𝑒𝑑𝑜𝑚𝑎𝑖𝑛𝑘𝑛𝑜𝑤𝑙𝑒𝑑𝑔𝑒𝑖𝑠 𝑠𝑡𝑟𝑜𝑛𝑔𝑙𝑦𝑛𝑒𝑒𝑑𝑒𝑑 .