Braesss Paradox: The Price of Free Choice

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Braesss ParadoxThe Price of Free Choice
College of the Redwoods Monday, October 3, 2005
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Introduction

• Steve McKelvey
• Assoc. Professor of Mathematics, Saint Olaf
  College, Northfield, MN

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Congestion in Traffic Flow
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Why Do Math Modeling?

• Some experiments cost too much or involve too
  much risk.
• Some experiments take too much time to be
  helpful.
• The rigorous thinking imposed by mathematics can
  lead to unexpected yet true results.

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Todays Example Network Flow Equilibrium

• A mathematical model of human behavior.
• Computations are easy. (Arithmetic Only)
• Yields counterintuitive results.
• Upon looking, one sees these results exist in the
  real world.
• Ultimate goal update our notion of common
  sense.

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Network Vocabulary

• Link
• Node
• Graph/Network
• Path

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Vocabulary of Network Equilibrium

• Origin/Destination Pairs
• Feasible Flow Pattern
• User Link Cost
• User Path Cost
• Total Social Cost

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Notation
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Example
b
a
c
d
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Three Paths

• Path 1

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Three Paths

• Path 2

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Three Paths

• Path 3

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Example of a Feasible Flow Pattern
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Definition of Equilibrium Flow

• An Equilibrium Flow Pattern is a flow pattern
  in which no user of the system has an incentive
  to make a unilateral change in his/her choice of
  path from the origin to the destination.
• Mathematically, this is equivalent to saying that
  for any O/D pair d, there is a cost u so that for
  every path p connecting d we have

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Example of a Feasible Flow Pattern
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An Equilibrium Flow Pattern
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General Networks Can Be Solved
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Questions

• Are equilibrium flow patterns unique?
• Do equilibrium flow patterns minimize total
  social cost?
• Can the addition of new links to the network,
  leaving all else unchanged, lead to worse user
  and total social costs?

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Are Equilibrium Flow Patterns Unique?

• The short answer is yes, under certain reasonable
  conditions. In the model we have been
  considering, equilibrium is unique as long as the
  cost functions on each link are strictly
  increasing as link flow increases.

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Do equilibrium flow patterns minimize total
social cost?

• Well see.

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Can the addition of new links to the network,
leaving all else unchanged, lead to worse user
and total social costs?

• Who knows? (We will, soon enough.)

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A Simple Yet Instructive Example
a
c

• Demand 6
• Four Links

b
d
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Two Paths, Symmetric Cost Functions
c
a
a
c
b
b
d
d
Path 1
Path 2
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Equilibrium Flow On This Network
a
c
b
d
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Equilibrium Flow On This Network Also Minimizes
Total Social Cost
a
c
b
d
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Adding a New Link To This Network

• Only change is the new link. All cost functions
  on the original four links remain the same.

a
c
e
d
b
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New Network Has Three Paths
Path 3
Path 1
Path 2
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Three Path Cost Functions
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Path Costs Under Old Equilibrium
Note Path 3 is attractive to users.
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Transition to New Equilibrium

• The new path, Path 3, is cheaper than paths 1 and
  2, so some users will shift from their old
  equilibrium choice to the new path.
• This will increase flows on the links in Path3,
  specifically links a, e and d.
• A new equilibrium flow pattern will develop.

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New Equilibrium Flow
c
a
e
d
b
Path 2 Flow2
Path 3 Flow2
Path 1 Flow2
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Path Costs Under New Equilibrium
Note No user has incentive to change paths
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Consequences of New Link

• New link will be used by two users.
• After shift, no user has incentive to change
  their choice of path.
• The user cost to each driver has increased from
  28 to 34. (Everyone is worse off.)
• The total social cost of the new pattern has
  increased from 168 to

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Bottom LineFreedom Isnt Free

• The construction of link e represents a new
  choice to the network users.
• All prior choices are still available to the
  users. No options have been taken away. No
  coercion is being used.
• Everyone would be better off if link e did not
  exist or its use were curtailed.
• The increased costs to users and society results
  from freedom of choice.

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Does This Really Happen?

• Braesss Paradox has been observed frequently on
  urban road networks. The most famous and
  thoroughly documented case occurred in Amsterdam,
  The Netherlands. A short new street was opened
  in an attempt to improve traffic flow in downtown
  Amsterdam, and horrible traffic snarls developed.
  Authorities then closed the street and traffic
  flow improved to the previous conditions.

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Common Sense

• Common Sense indicates that increasing the number
  of choices (paths) should improve the situation
  or, at worst, not hurt anything. After all,
  everyones set of choices is expanding.
• This is true for optimization problems where
  expanding choices cannot hurt anything.
• Equilibrium problems are fundamentally different
  and intuition from optimization can be very
  misleading when applied to equilibrium problems.

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Possible Solutions (All Are Coercive)

• Limit alternatives by failing to build links like
  e.
• Remove freedom of choice for drivers and ration
  access to paths.
• Apply tolls to links in a way that drives the
  choices of individual drivers into preferred
  traffic patterns. (Preferred being defined
  either by minimizing user costs or minimizing
  total social costs.)

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References

• "Braess's Paradox A Puzzler from Applied Network
  Analysis," by Steve McKelvey, The UMAP Journal 13
  (4) (1992) 303-312.
• D. Braess, Über ein Paradoxon aus der
  Verkehrsplanung. Unternehmensforschung 12, 258 -
  268 (1968)
• S. Dafermos and A. Nagurney, On some traffic
  equilibrium theory paradoxes. Transpn. Res. B 18,
  101 - 110 (1984)

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References

• New York Times. What if they closed 42nd Street
  and nobody noticed? NYT 25 December 1990, p.38.
• P.A. Samuelson, Tragedy of the open road
  Avoiding paradox by use of regulated public
  utilities that charge corrected Knightian tolls.
  J. of Int. and Comparative Econ. 1, 3 - 12 (1992)