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CE 8214: Transportation Economics: Introduction


CE 8214: Transportation Economics: Introduction David Levinson Introductions Who are you? State your name, major/profession, degree goal, research interest Syllabus ... – PowerPoint PPT presentation

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Title: CE 8214: Transportation Economics: Introduction

CE 8214 Transportation Economics Introduction
  • David Levinson

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

  • Handouts
  • Textbook

Paper reviews
  • handouts

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
  • 6. The prize The awe of your peers

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
Roundrobin Schedules
  • How many students

11 Players
12 Players
13 Players
14 Players
15 Players
16 Players
17 Players
  • What does this all mean?
  • System Rational vs. User Rational
  • Tit for Tat vs. Myopic Selfishness

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

Game Theory
  • David Levinson

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

Problems that can be Analyzed with Game Theory
  • Congestion
  • Financing
  • Merging
  • Bus vs. Car
  • who are the agents?

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.

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

A Nash Equilibrium
i j
A i 3,3 2,2
j 2,2 1,1
  • 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.

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)

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.

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
  • (Objective, minimize payoff)

  • 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.

  • 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.

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
  • 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

Coordination Game Payoff Table
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

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
  • How does reality compare with game theory?

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

  • 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

  • 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 ?

  • 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.

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

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."

Microfoundations of Congestion and Pricing
  • David Levinson

Objective of Research
  • To build simplest model that explains congestion
    phenomenon and shows implications of congestion
  • 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

Game Theory Assumptions
  • Actors are instrumentally rational
  • (actors express preferences and act to satisfy
  • 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

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)

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
  • 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.

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.

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

  • 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

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

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

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

  • 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).

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

Three-Player Game Arrival and DeparturePatterns
Departure Probability Given Arrival Strategies
Three-Player Game Results
  • 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
  • Extensions empirical estimates of E, D, L risk
    uncertainty and stochastic behavior simulations
    with more players.

On Whom The Toll Falls A Model of Network
  • by David Levinson

Man in Bowler Hat To Boost The British
Economy, Id Tax All Foreigners Living Abroad --
Chapman et al. (1989)
  • Research Questions, Motivation, Hypotheses
  • Historical Background
  • Actors Actions
  • Free Riders Cross Subsidies
  • Analytical Model
  • Empirical Values
  • Model Evaluation
  • Conclusions

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

  • 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

  • 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.

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
  • Travelers
  • Travel on Local Non-Local Roads
  • Collectively Own Jurisdiction of Residence

Revenue Instrument
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?

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

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

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

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

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

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
  • x,y location trip enters, exits road
  • VT value of time
  • SF freeflow speed
  • indicates absolute value

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

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)

Transportation Revenue
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

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