Existence, Convergence and Efficiency Analysis of Nash Equilibrium and Its Application to Traffic Networks Lihua Xie School of Electrical and Electronic Engineering Nanyang Technological University, Singapore (Joint work with Xuehe Wang) 1
Outline Motivation Related Work Road Pricing Strategies: A Nash Equilibrium Perspective - Distributed consensus in non-cooperative games - Routing problem - Price of anarchy Conclusion and Opportunities 2
Outline Motivation Related Work Road Pricing Strategies: A Nash Equilibrium Perspective - Distributed consensus in non-cooperative games - Routing problem - Price of anarchy Conclusion and Opportunities 3
Motivation Rapid Vehicle & Population Growth vs Limited Road Development (Singapore 2002~2012) 40% Vehicle 30% Population 20% Road 10% 0% 2002 2003 2004 2005 2006 2007 2008 2009 2010 2011 2012 Source: Land Transport Authority of Singapore Traffic control makes full Vehicle Population Road utilization of existing infrastructure without 2012 0.97 million 5.3 million 3,425 km road expansion 2030 1.2 million 6.9 million 3,700 km Increase ~24% ~30% ~8% 4
Motivation Traffic Congestion Problem • Traffic congestion causes significant efficiency losses, wasteful energy consumption, and excessive air pollution • It is difficult to enlarge the roadway capacity in major urban areas Congestion Losses • In Europe, the external costs of road traffic congestion amount to 0.5% of Paper — Community GDP (White European Transport Policy for 2010) • Sri Lanka loses 1.5% of the GDP (Rs 32 billion) due to traffic congestion (Business Times 2011) • In UK, traffic congestion is costing the economy more than GBP 4.3 billion a Relationship between economic losses and traffic congestions in key South- year (Centre for Economics and East Asian cities. World Resources Institute, World Resources 1996-97 , 1997 Business Research 2012) 5
Traffic Control Systems Component Illustration Plant Dynamics of traffic flows Basic Control system (1) Loop detectors Noise Sensor (2) Pneumatic tube counter (3) Cameras Plant (1) Road price Actuator Sensor Controller (2) Traffic signals (3) Variable-message sign Controller Driver’s action and/or decision Actuator Objective (1) Demand variations Noise (2) Accidents, raining, fog (1) Minimization of travel delay and/or Objective number of stops (optimized performance) (2) Boundedness of vehicle queues (stability) A control system to establish traffic regulations and their communications to the driver 6
Traffic Sensors Loop Detector Camera Pneumatic Tube Counter Traffic Controllers Road Pricing Variable-message Sign Traffic Signals Other controllers Signs and markings Car pooling programs Number plate auction 7
ERP in Singapore Electronic Road Pricing (ERP System) Alleviate traffic congestion by • affecting road users’ routing choices • refraining road users from travelling during peak hours 3500 ERP gantry in Singapore Total 3000 Cost (x1000) Effect of Road Pricing (ERP system) 2500 • 2000 Traffic flow shoots up quickly just 1500 before the ERP begins 1000 • Sharp decrease just at the time ERP 500 begins 0 Time 00:00 Time 01:10 Time 02:20 Time 03:30 Time 04:40 Time 05:50 Time 07:00 Time 08:10 Time 09:20 Time 10:30 Time 11:40 Time 12:50 Time 14:00 Time 15:10 Time 16:20 Time 17:30 Time 18:40 Time 19:50 Time 21:00 Time 22:10 Time 23:20 • Sharp rise just after ERP ends Fig. 1 The effect of ERP to number of vehicles on Anson Road in Singapore 8
ERP2 in Singapore • Expected to be implemented progressively from 2020 by the consortium of NCS Pte Ltd and Mitsubishi Heavy Industries Engine System Asia Pte Ltd, at a cost of S$556 million • Based on Global Navigation Satellite System (GNSS) and Dedicated Short Range Communication (DSRC) • Allow for more flexibility in managing traffic congestion through distance- based road pricing • Provide services for motorists’ convenience, such as disseminating information on traffic advisories and facilitating e-payments 9
Objective Traffic Network • Routing Problem: Multiple origin- destination pairs and each origin- destination pair has several routes • Trip Timing Problem: Different departure time to avoid peak hour Objective : To develop road pricing strategies based on game theory and consensus control to manage traffic flows in an optimal manner and minimize traffic congestion • To estimate the mass behavior of all players via consensus control • To analyze the efficiency of Nash equilibrium and the performance in evolution of repeated games • To design dynamic pricing control to improve the efficiency of Nash equilibrium and the overall efficiency in repeated games for trip timing and routing problem 10
Outline Motivation Related Work Road Pricing Strategies: A Nash Equilibrium Perspective - Distributed consensus in non-cooperative games - Routing problem - Price of anarchy Conclusion and Opportunities 11
Related Work Game theory: deals with strategic interactions among multiple players, where each player tries to maximize his/her own utility (R. Gibbons, 1992). Equilibrium concept (Nash) Elements of a game 𝑡 𝑜𝑓 = (𝑡 𝑗 𝑜𝑓 , 𝑡 −𝑗 𝑜𝑓 ) is a Nash equilibrium • • Players: road users if for any player 𝑗 , • Strategies: route choices or trip timing 𝑜𝑓 = max 𝑜𝑓 , 𝑡 −𝑗 𝑜𝑓 𝑉 𝑗 𝑡 𝑗 𝑡 𝑗 ∈𝑆 𝑉 𝑗 𝑡 𝑗 , 𝑡 −𝑗 available to each player 𝑗, 𝑡 𝑗 ∈ 𝑆 • Utility function: 𝑉 𝑗 (𝑡) , where 𝑡 = (𝑡 𝑗 , 𝑡 −𝑗 ) None of players can improve his/her • utility by a unilateral move • Not always exist and may be inefficient 12
Related Work Congestion game: The utility/cost of each player depends on the strategy/resource it chooses and the number of players choosing the same strategy/resource (R. Rosenthal, 1973). Potential game: 1 , 𝑡 −𝑗 − 𝑉 𝑗 𝑡 𝑗 2 , 𝑡 −𝑗 = 𝛸 𝑡 𝑗 1 , 𝑡 −𝑗 − 𝛸 𝑡 𝑗 2 , 𝑡 −𝑗 𝑉 𝑗 𝑡 𝑗 • Guarantee the existence of Nash equilibrium (D. Monderer et al, 1996). • All players tend to jointly optimize the potential function. (R. Rosenthal, 1973) Congestion game Potential Game Existence of Nash equilibrium 13
Related Work Learning in games: Allows players to adapt their strategies in response to the available information gathered over prior stages (D. Fudenberg et al, 1998). • Update perceptions of traffic conditions based on information broadcasted by government and/or obtained from other drivers through V2X • Inertia (intuitively): Some reluctance to change previous travel pattern Driver’s decision process (from J.R. Marden etc. 2009) 14
Learning of Games: Fictitious Play Complete information of all utility functions is generally not available for individual player ( Monderer 1996) Fictitious play: each player assumes that other players make decision independently according to observed empirical frequencies. The empirical frequencies generated by fictitious play of a potential game converge to a mixed strategy NE Shortcoming: when number of players is large, actual action for player i at every stage is computationally infeasible since it depends on a mapping over a joint space (Marden et al. 2009) Joint strategy fictitious play: each player assumes that other players make decisions randomly and jointly according to joint empirical frequencies. It still ensures the convergence for potential games and reduces the computational burden of standard fictitious play 15 Shortcoming: utility updating process is required for each player at every stage
Related Work Discrete-time consensus protocol : To estimate the number of players choosing each strategy for binary strategies case in inventory games (D. Bauso et al, 2009). • Exchange information with a set of neighbors. • Initial state of each agent – initial strategy of each player. • Global objective: Average-consensus – the percentage of players choosing each strategy. Pricing schemes: To improve the efficiency of Nash equilibrium. • The first-best pricing (marginal-cost pricing): the difference between the marginal social cost and the marginal private cost (M. J. Beckmann, 1967; M. Smith, 1979; H. Yang et al, 2004). • The second-best pricing: the location of the toll-gate, how much to charge, and the different impacts of the pricing schemes on different users (M. Marchand, 1968; T. Tsekeris et al, 2009). • Dynamic road pricing: vary according to real time road condition (R. B. Dial, 1999; T. Wongpiromsarn et al, 2012). 16
Outline Motivation Road Pricing Strategies: A Nash Equilibrium Perspective - Distributed consensus in non-cooperative games - Routing problem - Price of anarchy Conclusion and Opportunities 17
Distributed Consensus in Non-cooperative Congestion Games Model Setup Public transportation or private car and departure time? Notations: • : public transportations. • : trip timing choices . : player 𝑗 ’s cho ices, • where with . • : the action profile of all players. 18
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