Price of anarchy is independent of network topology

The Price of Anarchy is Independent of the Network Topology Tim Roughgarden (presented by Aleksandr Yampolskiy)

Outline ,[object Object],[object Object],[object Object],[object Object]

Q: Which route would you take? suburb train station ? wide, circuitous road: 1 hour delay narrow, straight road: 20 minute delay

A: Most drivers would take the narrow road suburb train station resulting in traffic congestion

What if… ,[object Object],[object Object],suburb train station red sedan must take the longer route

The Model ,[object Object],[object Object],[object Object],[object Object],[object Object],} ( G , r , l ) s 1 t 1 v x x 1 1 0 r 1 = 1 w s 1 ->w->t 1 s 1 -> v -> w ->t 1 s 1 ->v->t 1 Simple Paths

Flows ,[object Object],[object Object],[object Object],s 1 t 1 v x x 1 1 0 r 1 = 1 w f p 2 = ½ f p 1 = ½ edge e = ( w , t 1 ): f e = ½ + ½ = 1 l e (f e ) = 1

The Cost of a Flow ,[object Object],[object Object],s 1 t 1 v x x 1 1 0 r 1 = 1 w f p 2 = ½ f p 1 = ½ l p 1 ( f ) = ½ + 1 = 1.5 l p 2 ( f ) = ½ + 0 + 1 = 1.5 C ( f ) = ½ * 1.5 + ½ * 1.5 = 1.5

Some Assumptions ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]

Nash Flows ,[object Object],[object Object],[object Object],[object Object],[object Object]

More on Nash Flows ,[object Object],s 1 t 1 1 x r 1 = 1 flow = ½ flow = ½ s 1 t 1 1 x r 1 = 1 flow = 0 flow = 1 C ( f* ) = ½ * 1 + ½ * ½ = ¾ C ( f ) = 0 * 1 + 1 * 1 = 1 optimal flow Nash flow [Pigou 1920]

Optimal Flows ,[object Object],[object Object],s 1 t 1 1 x r 1 = 1 s 1 t 1 1 2x r 1 = 1 latency functions marginal cost functions flow = ½ flow = ½ flow = ½ flow = ½ optimal flow Nash flow

The Price of Anarchy ,[object Object],[object Object]

Linear Latency Functions ,[object Object],s 1 t 1 1 2x r = 1 flow = ½ flow = ½ s 1 t 1 1 x r = 1 flow = 0 flow = 1 C ( f* ) = ½ * 1 + ½ * ½ = ¾ C ( f ) = 0 * 1 + 1 * 1 = 1 latency functions marginal cost functions Nash flow: Optimal flow:

Linear Latency Functions (cont.) ,[object Object],[object Object],[object Object]

Proof Idea ,[object Object],[object Object],[object Object],[object Object],s 1 t 1 1 x r = 1 flow = ½ flow = 0 flow = ½ flow =0 r = ½

Proof Idea ,[object Object],[object Object],[object Object],[object Object],cost of f* at rate r = cost of f/2 at rate r/2 + cost of augmenting flow to rate r ≥ ¼ C ( f ) (easy) ≥ ½ C( f ) (hard) ≥ ¾ C( f )

General Latency Functions ,[object Object],[object Object],s 1 t 1 1 x p r = 1 s 1 t 1 1 r = 1 latency functions marginal cost functions ( p +1) x p flow = 1 flow = 0 flow = ( p + 1) -1/p flow = 1 – ( p + 1) -1/p Nash flow: Optimal flow: C ( f ) = 1 C ( f* ) = 1 – p ¢ ( p +1) -(p+1)/p =  ( )

Main Theorem ,[object Object],[object Object]

Upper Bound: ρ ( G , r , l ) ≤  ( L ) ,[object Object],[object Object]

Upper Bound: ρ ( G , r , l ) ≤  ( L ) (cont.) ,[object Object],[object Object]

Upper Bound: ρ ( G , r , l ) ≤  ( L ) (cont.) ,[object Object],[object Object],[object Object],[object Object],s 1 t 1 1 x 2 r = 1 flow = 1/√3 flow = 1 - 1/√3 l(x) = x 2 , l * (x) = 3x 2 . Want 3(  x ) 2 = x 2 )  = √ 1/3

Upper Bound: ρ ( G , r , l ) ≤  ( L ) (cont.) ,[object Object]

Lower Bound: ρ ( G , r , l ) ≥  ( L )

Computing the Price of Anarchy ,[object Object],[object Object],s 1 t 1 constant  · l r

Computing the Price of Anarchy (cont.)

Price of anarchy is independent of network topology

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