CE 8214: Transportation Economics: Introduction

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CE 8214 Transportation Economics Introduction

• David Levinson

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Introductions

• Who are you?
• State your name, major/profession, degree goal,
  research interest

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Syllabus

• Handouts
• Textbook

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Paper reviews

• handouts

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The game

• 1. An indefinitely repeated round-robin
• 2. A payoff matrix
• 3. Odds Evens
• 4. The strategy (write it down, keep it secret
  for now)
• 5. Scorekeeping (record your score honor
  system)
• 6. The prize The awe of your peers

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The Payoff Matrix
Player B Odd Player B Even
Player A Odd 3, 3 0, 5
Player A Even 5, 0 1, 1
Payoff A, Payoff B
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Roundrobin Schedules

• How many students

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11 Players
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12 Players
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13 Players
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14 Players
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15 Players
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16 Players
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17 Players
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Discussion

• What does this all mean?
• System Rational vs. User Rational
• Tit for Tat vs. Myopic Selfishness

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Next Time

• Email me your reviews by Tuesday 530 pm.
• Talk with me if you have problem with your
  assigned Discussion Paper.
• Discuss Game Theory

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Game Theory

• David Levinson

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Overview

• Game theory is concerned with general analysis of
  strategic interaction of economic agents whose
  decisions affect each other.

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Problems that can be Analyzed with Game Theory

• Congestion
• Financing
• Merging
• Bus vs. Car
• who are the agents?

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Dominant Strategy

• A Dominant Strategy is one in which one choice
  clearly dominates all others while a non-dominant
  strategy is one that has superior strategies.
• DEFINITION Dominant Strategy Let an individual
  player in a game evaluate separately each of the
  strategy combinations he may face, and, for each
  combination, choose from his own strategies the
  one that gives the best payoff. If the same
  strategy is chosen for each of the different
  combinations of strategies the player might face,
  that strategy is called a "dominant strategy" for
  that player in that game.
• DEFINITION Dominant Strategy Equilibrium If, in
  a game, each player has a dominant strategy, and
  each player plays the dominant strategy, then
  that combination of (dominant) strategies and the
  corresponding payoffs are said to constitute the
  dominant strategy equilibrium for that game.

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Nash Equilibrium

• Nash Equilibrium (NE) a pair of strategies is
  defined as a NE if A's choice is optimal given
  B's and B's choice is optimal given A's choice.
• A NE can be interpreted as a pair of expectations
  about each person's choice such that once one
  person makes their choice neither individual
  wants to change their behavior. For example,
• DEFINITION Nash Equilibrium If there is a set of
  strategies with the property that no player can
  benefit by changing her strategy while the other
  players keep their strategies unchanged, then
  that set of strategies and the corresponding
  payoffs constitute the Nash Equilibrium.
• NOTE any dominant strategy equilibrium is also a
  Nash Equilibrium

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A Nash Equilibrium
B B
i j
A i 3,3 2,2
j 2,2 1,1
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Representation

• Payoffs for player A are represented is the first
  number in a cell, the payoffs for player B are
  given as the second number in that cell. Thus
  strategy pair i,i implies a payoff of 3 for
  player A and also a payoff of 3 for player B.
  The NE is asterisked in the above illustrations.
  This represents a situation in which each firm or
  person is making an optimal choice given the
  other firm or persons choice. Here both A and B
  clearly prefer choice i to choice j. Thus i,i
  is a NE.

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Prisoners Dilemma

• Last week in class, we played both a finite
  one-time game and an indefinitely repeated game.
  The game was formulated as what is referred to
  as a prisoners dilemma.
• The term prisoners dilemma comes from the
  situation where two partners in crime are both
  arrested and interviewed separately .
• If they both hang tough, they get light
  sentences for lack of evidence (say 1 year each).
  
• If they both crumble in interrogation and
  confess, they both split the time for the crime
  (say 10 years).
• But if one confesses and the other doesnt, the
  one who confesses turns states evidence (and
  gets parole) and helps convict the other (who
  does 20 years time in prison)

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P.D. Dominant Strategy

• In the one-time or finitely repeated Prisoners'
  Dilemma game, to confess (toll, defect, evens) is
  a dominant strategy, and when both prisoners
  confess (states toll, defect, evens), that is a
  dominant strategy equilibrium.

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Example Tolling at a Frontier

• Two states (Delaware and New Jersey) are
  separated by a body of water. They are connected
  by a bridge over that body. How should they
  finance that bridge and the rest of their roads?
• Should they toll or tax?
• Let rI and rJ are tolls of the two
  jurisdictions. Demand is a negative exponential
  function.
• (Objective, minimize payoff)

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Objectives
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Payoffs

• The table is read like this Each jurisdiction
  chooses one of the two strategies (Toll or Tax).
  In effect, Jurisdiction 1 (Delaware) chooses a
  row and jurisdiction 2 (New Jersey) chooses a
  column. The two numbers in each cell tell the
  outcomes for the two states when the
  corresponding pair of strategies is chosen. The
  number to the left of the comma tells the payoff
  to the jurisdiction who chooses the rows
  (Delaware) while the number to the right of the
  column tells the payoff to the state who chooses
  the columns (New Jersey). Thus (reading down the
  first column) if they both toll, each gets
  1153/hour in welfare , but if New Jersey Tolls
  and Delaware Taxes, New Jersey gets 2322 and
  Delaware only 883.

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Solution

• So how to solve this game? What strategies are
  "rational" if both states want to maximize
  welfare? New Jersey might reason as follows "Two
  things can happen Delaware can toll or Delaware
  can keep tax. Suppose Delaware tolls. Then I get
  only 883 if I don't toll, 1153 years if I do,
  so in that case it's best to toll. On the other
  hand, if Delaware taxes and I toll, I get 2322,
  and if I tax we both get 1777. Either way, it's
  best if I toll. Therefore, I'll toll."
• But Delaware reasons similarly. Thus they both
  toll, and lost 624/hour. Yet, if they had acted
  "irrationally," and taxed, they each could have
  gotten 1777/hour.

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Coordination Game

• In Britain, Japan, Australia, and some other
  island nations people drive on the left side of
  the road in the US and the European continent
  they drive on the right. But everywhere,
  everyone drives on the same side as everywhere
  else, even if that side changes from place to
  place.
• How is this arrangement achieved?
• There are two strategies drive on the left side
  and drive on the right side. There are two
  possible outcomes the two cars pass one another
  without incident or they crash. We arbitrarily
  assign a value of one each to passing without
  problems and of -10 each to a crash. Here is the
  payoff table

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Coordination Game Payoff Table
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Coordination Discussion

• (Objective Maximize payoff)
• Verify that LL and RR are both Nash equilibria.
• But, if we do not know which side to choose,
  there is some danger that we will choose LR or RL
  at random and crash. How can we know which side
  to choose? The answer is, of course, that for
  this coordination game we rely on social
  convention. Conversely, we know that in this
  game, social convention is very powerful and
  persistent, and no less so in the country where
  the solution is LL than in the country where it
  is RR

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Issues in Game Theory

• What is rationality ?
• What happens when the rational strategy depends
  on strategies of others?
• What happens if information is incomplete?
• What happens if there is uncertainty or risk?
• Under what circumstances is cooperation better
  than selfishness? Under what circumstances is
  cooperation selfish?
• How do continuing interactions differ from
  one-time events?
• Can morality be derived from rational
  selfishness?
• How does reality compare with game theory?

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Discussion

• How does an infinitely or indefinitely repeated
  Prisoners Dilemma game differ from a finitely
  repeated or one-time game?
• Why?

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Problem

• Two airlines (United, American) each offer 1
  flight from New York to Los Angeles. Price
  /pax, Payoff /flight. Each plane carries 500
  passengers, fixed cost is 50000 per flight,
  total demand at 200 is 500 passengers. At 400,
  total demand is 250 passengers. Passengers choose
  cheapest flight. Payoff Revenue - Cost
• Work in pairs (4 minutes)
• Formulate the Payoff Matrix for the Game

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Solution
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Zero-Sum

• DEFINITION Zero-Sum game If we add up the wins
  and losses in a game, treating losses as
  negatives, and we find that the sum is zero for
  each set of strategies chosen, then the game is a
  "zero-sum game."
• 2. What is equilibrium ?

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• 200,200
• SOLUTION Maximin criterion For a two-person,
  zero sum game it is rational for each player to
  choose the strategy that maximizes the minimum
  payoff, and the pair of strategies and payoffs
  such that each player maximizes her minimum
  payoff is the "solution to the game."
• 3. What happens if there is a third price 300,
  for which demand is 375 passengers.

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3 Possible Strategies

• At 300,300 Each airline gets 375/2 share
  187.5 pax 300 56,250, cost remains 50,000
• At 300, 400, 300 airline gets 375300 112,500
  - 50000

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Mixed Strategies?

• What is the equilibrium in a non-cooperative, 1
  shot game?
• 200,200.
• What is equilibrium in a repeated game?
• Note No longer zero sum.
• DEFINITION Mixed strategy If a player in a game
  chooses among two or more strategies at random
  according to specific probabilities, this choice
  is called a "mixed strategy."

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Microfoundations of Congestion and Pricing

• David Levinson

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Objective of Research

• To build simplest model that explains congestion
  phenomenon and shows implications of congestion
  pricing.
• Uses game theory to illustrate ideas, informed by
  structure of congestion problems
• simultaneous arrival
• arrival rate gt service flow
• first-in, first-out queueing,
• delay cost,
• schedule delay cost

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Game Theory Assumptions

• Actors are instrumentally rational
• (actors express preferences and act to satisfy
  them)
• Common knowledge of rationality
• (each actor knows each other actor is
  instrumentally rational, and so on)
• Consistent alignment of beliefs
• (each actor, given same information and
  circumstances, would make same choice)
• Actors have perfect knowledge

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Application of Games in Transportation

• Fare evasion and compliance (Jankowski 1990)
• Truck weight limits (Hildebrand 1990)
• Merging behavior (Kita et al. 2001)
• Highway finance choices (Levinson 1999, 2000)
• Airports and Aviation (Hansen 1988, 2001)

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Two-Player Congestion Game

• Penalty for Early Arrival (E), Late Arrival (L),
  Delayed (D)
• Each vehicle has option of departing (from home)
  early (e), departing on-time (o), or departing
  (l)
• If two vehicles depart from home at the same
  time, they will arrive at the queue at the same
  time and there will be congestion. One vehicle
  will depart the queue (arrive at work) in that
  time slot, one vehicle will depart the queue in
  the next time slot.

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Congesting Strategies

• If both individuals depart early (a strategy pair
  we denote as ee), one will arrive early and one
  will be delayed but arrive on-time. We can say
  that each individual has a 50 chance of being
  early or being delayed.
• If both individuals depart on-time (strategy oe),
  one will arrive on-time and one will be delayed
  and arrive late. Each individual has a 50
  chance of being delayed and being late.
• If both individuals depart late (strategy ll),
  one will arrive late and one will be delayed and
  arrive very late. Each individual has a 50
  change of being delayed and being very late.

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Payoff Matrix
Note Payout for Vehicle 1, Payout for Vehicle
2 Objective to Minimize Own Payout, S.t. others
doing same
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Example 1 (1,0,1)
Note Indicates Nash Equilibrium Italics
indicates social welfare maximizing solution
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Example 2 (3,1,4)
Note Indicates Nash Equilibrium Italics
indicates social welfare maximizing solution
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Payoff matrix with congestion pricing
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What are the proper prices?

• Normally use marginal cost pricing
• MC ? TC/?Q
• But Total Costs (TC) are discrete, so we use
  incremental cost pricing
• IC ?TC/?Q
• Total Costs include both delay costs as well as
  schedule delay costs.
• ?o ?l 0.5(LD)
• ?e MAX(0.5(D-E),0)

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Subtleties

• Vehicles may affect other vehicles by causing
  them to change behavior.
• Total costs do not include these pecuniary
  externalities such as displacement in time, just
  what the cost would be for that choice, given the
  other person is there, compared with the cost for
  that choice if one player were not there.
• You cant blame departing early on the other
  player.

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Example 1 (1,0,1) with congestion prices
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Example 2 (3,1,4) with congestion prices
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Two-Player Game Results
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Three-Player Congestion Pricing Game

• The model can be extended. With more players, we
  need to add one departure from home (arrival at
  the back of the queue) time period, and two
  arrival at work (departure from the front of the
  queue) time periods.

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Delay

• Expected delay
• Cost of delay
• where
• D delay penalty
• Qt standing queue at time t
• At arrivals at time t.

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Schedule Delay

• Schedule delay is the deviation from the time
  which a vehicle departs the queue and the
  desired, or on-time period.
• Where
• dt delay
• ta time of arrival at back of queue
• to desired time of departure from front of
  queue (time to be on-time)
• The cost of schedule delay is thus

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Probabilistics

• We only know the delay probabilistically, so
  schedule delay is also probabilistic
• Where
• P() probability function for traveler i,
  summarized in Table 9.
• ?t penalty function (2E, E, 0, L, 2L, 3L)
• are the periods of departure from the queue
  (very early, early, on-time, late, really late,
  super late).

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Nomenclature

• V - Very Early
• E - Early
• O - On-time
• L - Late
• R - Really Late
• S - Super Late

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Three-Player Game Arrival and DeparturePatterns
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Departure Probability Given Arrival Strategies
v,_,_
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Three-Player Game Results
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Conclusions

• Presented a simple (the simplest?) model of
  congestion and pricing.
• A new way of viewing congestion and pricing in
  the context of game theory.
• Illustrates the effectiveness of moving
  equilibria from individually to socially optimal
  solutions.
• Extensions empirical estimates of E, D, L risk
  uncertainty and stochastic behavior simulations
  with more players.

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Break
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On Whom The Toll Falls A Model of Network
Financing

• by David Levinson

Man in Bowler Hat To Boost The British
Economy, Id Tax All Foreigners Living Abroad --
Chapman et al. (1989)
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Outline

• Research Questions, Motivation, Hypotheses
• Historical Background
• Actors Actions
• Free Riders Cross Subsidies
• Analytical Model
• Empirical Values
• Model Evaluation
• Conclusions

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Research Questions

• How and why has the preferred method of highway
  financing changed over time between taxes and
  tolls?
• Who wins and who loses under various revenue
  mechanisms?
• How does the spatial distribution of winners and
  losers affect the choice?

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Motivation

• New Capacity Desired
• New Concerns Social Costs
• New Fleet EVs
• New Networks ITS
• New Toll Technology ETC
• New Owners Privatization
• New Rules ISTEA 2
• New Priorities
• Capital -gt Operating

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Hypothesis

• Hypothesis Jurisdiction Size Collection Costs
  Influence Revenue Choice.
• Cross-subsidies from non-locals to locals will be
  more politically palatable than vice versa.
• Small jurisdictions can affect cross-subsidies
  more easily with tolls than large jurisdictions.
• New technologies lower toll collection costs.

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Actors and Actions

• Jurisidiction/ Road Authority
• Operates Local Roads
• Serves Local Non-Local Travelers
• Sets Revenue Mechanism Rate
• Has Poll Tax Authority
• Objective Local Welfare Maximization (Sum of
  Profit to Road and Consumers Surplus of
  Residents)
• Travelers
• Travel on Local Non-Local Roads
• Collectively Own Jurisdiction of Residence

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Revenue Instrument
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Why No Gas Tax ?

• The Gas Tax is bounded by two cases
• Odometer Tax (where all gas purchased in the home
  jurisdiction) and
• Perfect Toll (where all gas purchased in the
  jurisdiction of travel).
• What is proper behavioral assumption about
  location of purchase?

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Long Road Trip Classes
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Free Riders
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Cross Subsidy by Instrument Class
Assumes Total CostTotal Revenue Fair is
proportional to distance traveled
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Model Parameters

• Demand
• Distance,
• Price of Trip,
• Fixed User Cost.
• Network Cost
• Fixed Network Costs,
• Variable Network Costs,
• Fixed Collection Costs,
• Variable Collection Costs.
• Network Revenue
• Rate of Toll, Tax,
• Basis.

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Equilibrium Cooperative vs. Non-Cooperative

• Non-Cooperative (Nash) Assume other
  jurisdictions policies are fixed when setting
  toll.
• Cooperative Assume other jurisdictions behave by
  setting same toll rate as J0. Results in higher
  welfare. Not equilibrium in one-shot game.

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Empirical Values
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Cases Considered
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Application

• Welfare vs. Tolls
• Tolls vs. Tolls
• General Tax vs. Cordon
• Equilibrium Cooperative vs. Non-Cooperative
• Game Policy Choice
• Perfect Tolls
• Odometer Tax

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Representative Game

• Two Choices
• revenue mechanism,
• rate given revenue mechanism
• Form of Prisoners Dilemma
• Payoff Toll, Toll Lower Than Payoff Tax,
  Tax.

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Welfare in J0 as a function of J0 Toll
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Welfare in J0 at Welfare Maximizing Tolls vs.
Jurisdiction Size in an All-Tax Environment
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Welfare in J0 at Welfare Maximizing Tolls vs.
Jurisdiction Size in an All-Toll Environment
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Tolls by Location of Origin and Destination.
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Policy Choice as a Function of Fixed Collection
Costs and Jurisdiction Size
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Policy Choice as a Function of Variable
Collection Costs and Jurisdiction Size
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Reaction Curves Best J0 Toll as Tolls Vary in
Toll Environment
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Uniqueness, Non-Cooperative Welfare Maximizing J0
Toll as Initial Toll for Other Jurisdiction
Varies in Toll Environment
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Elasticity About Mean
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Comparison of Tolls and Welfare for Different
Jurisdiction Sizes
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Rate of Toll Under Various Policies
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General Trip Classification
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Conclusions

• Necessary Conditions
• For Tolls to Become Widespread, Need
• Relatively Low Transaction Costs,
• Sufficiently Decentralized (Local) Decisions
  About Placement of Tolls.

• Actual Conditions
• Policy Environment Becoming More Favorable to
  Road Pricing
• Localized Decisions (MPO),
• Federal encouragement (ISTEA 2 pilot projects),
• Longer trips,
• Lower transaction costs (ETC).

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Demand (1)

• f(z) flow past point z F flow between
  sections
• ?(PT(x,yPI))dxdy demand function representing
  the number of trips that enter facility between x
  and x dx and leave between y and y dy
• PT(x,yPI) generalized cost of travel to users
  defined below)
• x,y where trip enters,exits road
• PI price of infrastructure

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Demand (2)

• PTtotal user cost
• PIvector of price of infrastructure
• ??coefficient (relates price to demand), ? lt 0
• ? coefficient (trips per km (_at_ PT 0)), ? gt 0
• ? fixed private vehicle cost
• ?? variable private vehicle cost per unit
  distance
• x,y location trip enters, exits road
• VT value of time
• SF freeflow speed
• indicates absolute value

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Consumers Surplus

• U - denotes consumers surplus
• a,b - jurisidction borders
• n - counter for tollbooths crossed
• d - spacing between tollbooths

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Model Outcomes

• As the size of jurisdiction J0 increases, that is
  as b-a gets large
• 1. F-0 / F- increases.
• 2. F 0 / F- increases.
• 3. The total number of trips originating in or
  destined for jurisdiction J0 (F00, F 0, and F-0)
  increase.

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Transportation Revenue
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Total Network Cost

• where
• CT Total Cost
• CCV Variable Collection Cost
• CCF Fixed Collection Cost
• C? Variable Network Cost
• CS Fixed Network Cost
• ??????????? model coefficients

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Tolls in All-Cordon Environment
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Price of Infrastructure
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Rate of Toll Under Various Policies
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Odometer Tax