The Cost ofAnarchy

John Hubbard

April 30, 2011

Sunday, May 1, 2011

References• I benefitted enormously from the web page

http://images.math.cnrs.fr/Le-prix-de-l-anarchie.html

by Etienne Ghys• The original research is largely due to

=

T. Roughgarden and E. Tardos

Sunday, May 1, 2011

http://images.math.cnrs.fr/Le-prix-de-l-anarchie.htmlhttp://images.math.cnrs.fr/Le-prix-de-l-anarchie.html

An example from traffic flow

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In winding river county, there is a river...

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and a town called sleepyville.

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There is also a factory, and to get from the town to the factory

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There are two roads, and

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two bridges.

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Thus there are two ways from sleepyville to the factory:

this one

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Thus there are two ways from sleepyville to the factory:

this one

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and this one.

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and this one.

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The roads are wide:however many cars take them,

it takes 1:05 to drive on either road.

On the other hand, the bridges are narrow and easily congested:

if x vehicles/hour take a bridge, they each takex/1000 hours to cross the bridge.

During morning rush hour, 1000 vehicles/hourdrive from sleepyville to the factory.

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1:05

1:05

x/1000

x/1000

Everyone takes the path that willget him or her to the factory fastest.

Everyone has perfect information about the traffic.

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1:05

1:05

x/1000

x/1000

Everyone takes the path that willget him or her to the factory fastest.

Everyone has perfect information about the traffic.

How long does it take to drive from sleepyville to the factory in the morning?

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Yes, the answer is 1:35.If at some stage there are more people taking the green path than the red path, the next driver will

gain by taking the red path,and vice versa.

Thus there will always be about 500 vehicles/hour

taking each path.

Each will take 1:05 on the road and 0:30 on the bridgesince there will be 500 vehicles/hour on each bridge.

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Now at the urging of the mayor of Sleepytownand the urging of the factory manager,Senator Ear Mark secures funding for

a third bridge.

It is so big that it never gets congested, andit takes 2 minutes to cross it.

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1:05

1:05

x/1000

x/1000

Everyone still takes the path that willget him or her to the factory fastest.

Everyone still has perfect information about the traffic.

0:02

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1:05

1:05

x/1000

x/1000

Everyone still takes the path that willget him or her to the factory fastest.

Everyone still has perfect information about the traffic.

Now how long does it take to drive from sleepyville

to the factory in the morning?

0:02

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Yes, the answer is 2:02Everyone will take all three bridges.

No one can gain by changing his path.

In the worst case, taking a small bridge and the big bridge takes 1:02

whereas the road is always 1:05,so you can never gain by taking the road.

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You might also wonder how “psychological”this all is.

I will show that even straighforwardphysical systems, with springs and strings,

exhibit the same sort of behavior.

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The setup consists of a spring,then a yellow string,then another spring,

then a weight.

In addition, there are two (slightly) loose strings one pinkgoing from the top to the top of

the bottom spring,and one green, going from the

bottom of the top string to the weight.

The yellow string holds all the weight.

Sunday, May 1, 2011

The setup consists of a spring,then a yellow string,then another spring,

then a weight.

In addition, there are two (slightly) loose strings one pinkgoing from the top to the top of

the bottom spring,and one green, going from the

bottom of the top string to the weight.

The yellow string holds all the weight.

What will happen if I cut it?Sunday, May 1, 2011

As the experiment shows, the weight goes up!

You should think of the two loose strings as the roads,the springs as the small bridges, and the yellow string

as the big bridge.

If you cut the yellow string,i.e., if you blow up the big bridge,

The weight goes up, i.e., the traffic goes faster.

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To analyze this sort of difficulty, we need to define the problem more precisely

A traffic network is a graph with oriented arrows:think of them as roads joining cities.

For each pair of cities A and B, we will assumethat we know how many vehicles wish to go

from A to B.A routing is an assignment of a number of vehicles to

each oriented edge of the graph,compatible with the given travel between cities.

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A possible graph, consisting of 22 cities and roads joining

them

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A routing is a selfish routing if no one canimprove his time by changing his route.

A routing is a social routing if it minimizes total travel time.

A selfish routing will always take more total time

than a social routing, of course.

The question is:

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A routing is a selfish routing if no one canimprove his time by changing his route.

A routing is a social routing if it minimizes total travel time.

A selfish routing will always take more total time

than a social routing, of course.

The question is:

How bad can it get?Sunday, May 1, 2011

The price of anarchyis by definition the smallest ratio of the total timeof a selfish routing to the time of a social routing.

The main result (Roughgarden and Tardos)

says that

The price of anarchy is at most 4/3.

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Actually, for this to be true you have to makean assumption about how congestion affects

travel time:

You must assume that the travel time for each road r

is of the form Tr(x)= arx+br

if x vehicles/hour travel on that edge.

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Roughgarden and Tardos also found the formula if the

travel time is a polynomial of degree d≥1:1

1− 1d(d+1)(d+1)/d

Note that if d=1, then

11− 1

d(d+1)(d+1)/d= 1

1− 11·22

= 43

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You might wonder if the price of anarchy applies to real situations.

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Three physicists, Young, Jeong and Gastner,studied the price of anarchy for drivers going from

Harvard square to the Boston common.As you can see, at about 10000 vehicles/hour,

the price of anarchy peaks at about 1.3Sunday, May 1, 2011

How do we prove any of this?

The key is to introduce an appropriate energyfor a routing.

Suppose that for a particular routing R road r is taken by xr vehicles.

Define the energy E(r) of road r to be

Er(R)=Tr(1)+Tr(2)+...+Tr(xr)

and the energy of the routing to be the sum

E(R) = ∑ Er(R) all roads r

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The main thing to notice is that a routing R

is a selfish routing if and only if it is a minimum of E .

Indeed, suppose that Rold is a minimum of Ecalled Eold

and that one driver changes his path to form Rnewwith energy Enew

The new energy is obtained from the old energy bysubtracting the time the driver took to drive his

previous path, and adding in the time it takes to drive the new course.

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We see that Enew> Eold means that the driver took more time to drive to his destination,

in other words, a minimum of E is a selfish routing.

It is not hard to see that conversely, a selfish routing is a minimum of E .

Note that in particular this shows that selfish routingsexist: there are only finitely many routings, so some

routing must be a minimum of the energy.

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We will now show that the price of anarchy is at most 2;this is not sharp (4/3 is sharp) but the precise result is

too technical for a public lecture.

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First inequality:

Er(R)=Tr(1)+Tr(2)+...+Tr(xr)≥ 1/2 xr Tr(xr)

(a+ b) + (2a+ b) + · · ·+ (xa+ b)= a(1 + 2 + · · ·+ x) + b(1 + 1 + · · ·+ 1)

= ax(x+1)2 + xb ≥12x(ax+ b)

Second inequality:

Er(R)=Tr(1)+Tr(2)+...+Tr(xr)≤ xr Tr(xr)

Tr(1) + Tr(2) + · · ·+ Tr(xr)≤ Tr(xr) + Tr(xr) + · · ·+ Tr(xr) = xrTr(xr)

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Thus if I have a selfish routing with energy Eselfand total time Tself

and a social routing with energy Esocand total time Tsoc then

inequality 1 inequality 2A selfish routing is a minimum of the energy

Thus a selfish time is at most twice the social time.

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That’s all folks!!

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