Traffic Games Econ / CS166b Feb 28, 2012 John Musacchio Associate Professor Technology and Information Management University of California, Santa Cruz johnm@soe.ucsc.edu
Traffic Games l Basics l Braess paradox l Pricing to improve social welfare l Selfish pricing
Traffic Routing Games -- Intro l Which route would you choose? The one with the lower delay of course!
Traffic Routing Games -- Intro delay traffic delay traffic l Which route would you choose? Depends on what routes the other cars choose! This is a game.
Traffic Routing Games -- Intro delay traffic l In this game There are many small players l assume that each player has negligible effect on total traffic Payoff functions are the same l - Delay delay traffic
Traffic Routing Games -- Intro delay traffic delay traffic l What is the strategy space of each player? The path taken, i.e. (up, down) l What is a Strategy Profile? An assignment of route to each car But since, utility functions identical l just specify fraction that goes on each route. l Call this a traffic assignment
Traffic Routing Games -- Intro delay traffic delay traffic l What is a Nash Equilibrium of this type of game? A traffic assignment such that no car has an incentive to switch paths. This is known as a Wardrop Equilbirum l For this game: the rates (½, ½) specify the Wardrop Equilbrium
Traffic Routing Games -- Intro l For a traffic assignment to be a Wardrop Equilbirum Traffic on used paths must encounter the same delay l Otherwise cars would switch paths Any unused path must have a delay greater than the used paths delay traffic delay traffic delay traffic
Example 1/2 Delay proportional to traffic Delay fixed x 1 1 1 x 2 1/2 l What is the Wardrop equilibrium? (½, ½) l What is the delay the cars face? 1.5
Example x 1 1 0 1 x 2 A new road l What is the Wardrop equilibrium?
Example 1/2 x 1 1 0 1 x 2 1/2 l Is this the Wardrop equilbirium? No Cars can do better
Example 1/3 x 1 1 0 1/3 1 x 2 l Is this a Wardrop equilibrium? 1/3 No Cars can do better Red paths have a delay 5/3, green only 4/3 Cars want to switch to green
Example x 1 1 0 1 1 x 2 l Is this a Wardrop equilbirium? Yes The delay is 2. One car switching to red path still finds a delay of 2.
Braess s Paradox 1/2 x 1 1 1 x 2 1/2 l But this network had less delay in Wardrop Equilibriumum! l Delay: 1.5 x 1 l This network has an additional road l Delay: 2 l 1 0 1 1 If a social planner could re-route traffic we could achieve a delay of 1.5 x 2
Generalizations of Example l We looked at one particular network With affine latency functions l And found that the social cost with Selfish routing is 2 Social optimal routing is 3/2 l The ratio: Is sometimes referred to as the price of anarchy
Generalizations of Example l It turns out With affine latency functions And arbitrary network topology The maximum price of anarchy is 4/3 l The Braess paradox example is the worst case. l With arbitrary latency functions The price of anarchy can be arbitrarily bad!
Selfishness vs. Social Planner l Why is it that selfish traffic has higher delay than optimally routed traffic? l Notation Let l i (x i ) be the latency (delay) on link I l i (x i ) = a i x i + b i l In our example For the asphalt roads: a i = 1, b i = 0 For the dirt roads: a i = 0, b i = 1
Selfishness vs. Social Planner l Social Planner Problem: Delay on a link Amount of traffic that Suffers that delay
Selfishness vs. Social Planner l When a social planner considers adding a small amount, ε, of flow to link i, The additional (marginal) cost is proportional to the derivative (2 a i x i + b i ) ε l In contrast, a driver sees the added cost as l i (x i ) ε = ( a i x i + b i ) ε
Pricing to achieve optimality l Suppose we can charge a price on each link l What price should we charge to get users to route optimally? l Drivers already see cost as ( a i x i + b i ) ε l Social planner sees cost as (2 a i x i + b i ) ε l Add some cost to drivers so their costs match social cost Charge a i x i per unit traffic
Pricing to achieve optimality l What does charging a i x i mean? l It s in units of time l But of course time is money l Figure out how many dollars a unit of time is worth, and multiply by that l For simplicity, suppose people s time is worth a dollar per time unit
Braess Paradox Network with Pricing Delay: X 1 Price: X 1 1 0 1 Delay: X 2 Price: X 2 l Is it a Wardrop Equilibrium for all traffic to take the green path? Cost to driver of Green path: 4 Cost to driver of upper or lower path: 3 No traffic switches.
Braess Paradox Network with Pricing Delay: X 1 Price: X 1 1 1/2 0 l Is this the Wardrop Equilbirum? Yes Cost to driver of red path(s): 2 Cannot improve by switching 1 1/2 Delay: X 2 Price: X 2
Braess Paradox Network with Pricing Delay: X 1 Price: X 1 1 1/2 0 1 1/2 l The Delay along each path is now 1.5 Reduced from 2 before we installed pricing. Delay: X 2 Price: X 2
What about selfish pricing? l What if link owners are selfish, and try to maximize profit? Path 1 p 1 Path 2 p 2
Overview l Model Single source-destination pair Competing providers Non-atomic users Traffic dependent latency Elastic user demand l Model due to Acemoglu and Ozdaglar (2007) l Elastic user demand extension: Hayrapetyan, Tardos, Wexler (2007) John Musacchio
Overview l Parallel-Serial topologies without elastic demand studied by Acemoglu and Ozdaglar (06) John Musacchio
Wardrop Equilibrium for given prices Non-Atomic Users Path 1 p 1 + l 1 (f 1 ) Path 2 p 2 + l 2 (f 2 ) Choose lowest disutility : p i + l(f i ) Path 1 Path 2 Delay+Price Delay+Price Delay p 2 Delay p 1 Traffic 80% 100% Traffic 100% 20% John Musacchio
Elastic Demand? Demand or Disutility Curve Disutility User Surplus Key assumption: Concave Decreasing Total Flow (#of non-atomic users that connect) John Musacchio
Overview l Acemoglu and Ozdaglar show that for competing providers in parallel l Musacchio and Wu (07) Show same result using circuit Analogy l Present work Same technique, more complex topologies Presented at ITA 09 and OptimA09 John Musacchio
Compare Social Optimum Path 1 p 1 Path 2 Nash Path 1 p 2 p 1 Path 2 p 2 John Musacchio
Arbitrary Parallel-Serial Topology Source Disutility Flow Price: p 111 Latency: a 111 f 111 Price: p 31 Latency: a 31 f 31 Latency: b 1 Latency: b 3 Destination
Simple Parallel-Serial Topology Source Disutility Flow Price: p 11 Latency: a 11 f 11 Latency: b 1 Latency: b 3 Destination
Very Inefficient equilibria U max Flow P 1 = U max P 2 = U max l Each price is a best response to the other l Social Welfare 0 l PoA infinite John Musacchio
Class of Equilibria l Strict Equilibria Every player plays strict best response May not be possible some branches have too high a fixed latency l Zero-Flow Zero-Price Equilibria Providers that carry zero flow reduce their prices to zero in attempt to get some flow. John Musacchio
Circuit Analogy for Wardrop Equilibrium Source Voltage Current p 31 p 111 a 31 f 31 a 111 f 111 b 1 b 3 Destination
Thévenin Equivalent Source Voltage Slope: s Current V s Thévenin Equivalent p 111 V th a 111 ± 1111 a 111 b 1 b 3 Destination
Nash Equilibrium Condition: l Consider profit from ² price change John Musacchio
Nash Circuit Source Voltage Current δ 1111 + a 1111 a 1111 δ 31 + a 31 a 31 b 1 b 3 Destination
Almost on branches V=1 s=1 But then, 2 nd branch undercut p 1 d=3/4 1+1 d< 5/7 1+1/2 p 2 For provider 1, a small increase in price may see a different Thévenin equivalent than a small decrease a 1 f 1 1 1 Can show : 0 + - + - 0.74 Thévenin resistance with branches that would carry pos. flow with small price increase. Thévenin resistance with branches carrying positive flow
Existence l Theorem: A zero-flow, zero-price equilibrium exists. Proof sketch l δ i (S) = Thévenin equivalent with braches in set S on. l For each fixed set S, Nash circuit has unique solution for resistors {δ i (S) } l As S increases, voltage across branches decreases witch reduces set ON = branches actually on either fixed point S = On find almost on branches reduce δ s in other branches so that flow in almost on branches is 0. John Musacchio
Nash Circuit Source Voltage Slope: s Current d 1111 δ 1111 + a 1111 a 1111 d 31 δ 31 + a 31 a 31 b 1 b 3 Destination
Social optimum pricing l Price so that users see the cost they impose on society. l Latency cost on link i: (a i f i * + b i )f i * l Marginal cost: 2a i f i * + b i l Latency seen by user: a i f i* + b i l Difference: a i f i * l Conclusion: p i* = a i f i * achieves social optimum
Social Optimum Circuit Source a 31 a 1111 a 31 a 1111 b 1 b 3 Destination
Nash vs. Social Opt. Original Game Nash Equilibrium: Social Optimum: -s d d* John Musacchio
Nash vs. Social Opt. Modification 1 Nash Equilibrium: Social Optimum: -s d d* Flow & Social Welfare Unchanged Flow & Social Welfare Not Reduced Price of Anarchy Not Reduced John Musacchio
V Nash vs. Social Opt. Modification 2 V Nash Equilibrium: Social Optimum: -s d d* Flow Unchanged Social Welfare reduced by light green area Flow Unchanged Social Welfare reduced by light green area Price of Anarchy Not Reduced
Notation l Primary Branches i = = a 11 a 1 a δ 12 1 a 2 a 3 a 4 s a 13 l Group 11 in branch 1 sees Thévenin resistance δ 11 δ 1 a 12 a 13 l If group 12 consists of providers in parallel: δ 121 = a 122 a 123 δ 21
Notation l For simple Parallel-Serial, let m i = #providers on each branch i
Nash Welfare simple-parallel serial V s l Welfare truncated disutility d 11 δ 11 + a 11 a 11 l Becomes l Kirchoff V. L.: b 1 John Musacchio
Soc. Opt. Welfare Simple-Parallel Serial V s a 11 a 11 (Relax positivity constraint on social optimum flow. Gives upper bound on true social optimum.) b 1 John Musacchio
Finding Bounds l z times Nash welfare minus social opt welfare l Algebra, Matrix Inversion Lemma l Off diagonal elements positive, focus on diagonal John Musacchio
Finding Bounds l Diagonal elements of form Where N i are polynomials in z and ta i - monotone increasing in z l Find sufficiently high z to make them all positive. John Musacchio
PoA Bound Simple Parallel-Serial l For zero-flow zero-price equilibria l Where m is max number of players connected serially John Musacchio
m=1, Parallel competition 1.5 m=1 player connected serialy 1.45 1.4 1.35 1.3 PoA 1.25 1.2 1.15 1.1 1.05 1 0 0.2 0.4 0.6 0.8 1 Conductance Ratio
m=3,5 serially, N in parallel 2.5 m=3 player connected serialy 2 PoA 1.5 1 0 0.2 0.4 0.6 0.8 1 3.5 3 Conductance Ratio m=5 player connected serialy PoA 2.5 2 1.5 1 0 0.2 0.4 0.6 0.8 1 Conductance Ratio
General Parallel-Serial l Consider a parallel grouping within network: a 11 +δ 1 1 a 12 +δ 1 2 a 11 a 12 a 13 +δ 13 a 13 < = a 1 +δ 1 a 1 +d 1 δ 11 a 1 f 1 d 1 <1 l Profit at least l But for simple parallel-serial it was l Consequence: - General parallel-serial requires X 2
Conclusion l Circuit analogy to prove existence of Nash Eq. and find POA bounds l Future work: Tighten bound on general parallel-serial. John Musacchio
References D. Acemoglu and A. Ozdaglar, Competition and Efficiency in Congested Markets, Math. of OR, Feb. 2007. D, Acemoglu and A. Ozdaglar, Competition in Parallel-Serial Networks, JSAC, 2006 A. Hayrapetyan, E. Tardos and T. Wexler, A Network Pricing Game for Selfish Traffic, Distributed Computing, March 2007. A. Ozdaglar, ``Price Competition with Elastic Traffic,' Networks, 2008. J. Musacchio, S. Wu, The price of Anarchy in a Network Pricing Game, Allerton 2007. John Musacchio