Title: CE 8214: Transportation Economics: Introduction
1CE 8214 Transportation Economics Introduction
2Introductions
- Who are you?
- State your name, major/profession, degree goal,
research interest
3Syllabus
4Paper reviews
5The 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
6The 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
7Roundrobin Schedules
811 Players
912 Players
1013 Players
1114 Players
1215 Players
1316 Players
1417 Players
15Discussion
- What does this all mean?
- System Rational vs. User Rational
- Tit for Tat vs. Myopic Selfishness
16Next Time
- Email me your reviews by Tuesday 530 pm.
- Talk with me if you have problem with your
assigned Discussion Paper. - Discuss Game Theory
17Game Theory
18Overview
- Game theory is concerned with general analysis of
strategic interaction of economic agents whose
decisions affect each other.
19Problems that can be Analyzed with Game Theory
- Congestion
- Financing
- Merging
- Bus vs. Car
- who are the agents?
20Dominant 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.
21Nash 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
22A Nash Equilibrium
B B
i j
A i 3,3 2,2
j 2,2 1,1
23Representation
- 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.
24Prisoners 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)
25P.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.
26Example 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)
27Objectives
28Payoffs
- 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.
29Solution
- 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.
30Coordination 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
31Coordination Game Payoff Table
32Coordination 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
33Issues 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?
34Discussion
- How does an infinitely or indefinitely repeated
Prisoners Dilemma game differ from a finitely
repeated or one-time game? - Why?
35Problem
- 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
36Solution
37Zero-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 ?
38- 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.
393 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
40Mixed 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."
41Microfoundations of Congestion and Pricing
42Objective 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
43Game 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
44Application 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)
45Two-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.
46Congesting 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.
47Payoff Matrix
Note Payout for Vehicle 1, Payout for Vehicle
2 Objective to Minimize Own Payout, S.t. others
doing same
48Example 1 (1,0,1)
Note Indicates Nash Equilibrium Italics
indicates social welfare maximizing solution
49Example 2 (3,1,4)
Note Indicates Nash Equilibrium Italics
indicates social welfare maximizing solution
50Payoff matrix with congestion pricing
51What 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)
52Subtleties
- 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.
53Example 1 (1,0,1) with congestion prices
54Example 2 (3,1,4) with congestion prices
55Two-Player Game Results
56Three-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.
57Delay
- Expected delay
- Cost of delay
- where
- D delay penalty
- Qt standing queue at time t
- At arrivals at time t.
58Schedule 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
59Probabilistics
- 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).
60Nomenclature
- V - Very Early
- E - Early
- O - On-time
- L - Late
- R - Really Late
- S - Super Late
61Three-Player Game Arrival and DeparturePatterns
62Departure Probability Given Arrival Strategies
v,_,_
63Three-Player Game Results
64Conclusions
- 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.
65Break
66On Whom The Toll Falls A Model of Network
Financing
Man in Bowler Hat To Boost The British
Economy, Id Tax All Foreigners Living Abroad --
Chapman et al. (1989)
67Outline
- Research Questions, Motivation, Hypotheses
- Historical Background
- Actors Actions
- Free Riders Cross Subsidies
- Analytical Model
- Empirical Values
- Model Evaluation
- Conclusions
68Research 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?
69Motivation
- 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
70Hypothesis
- 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.
71Actors 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
72Revenue Instrument
73Why 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?
74Long Road Trip Classes
75Free Riders
76Cross Subsidy by Instrument Class
Assumes Total CostTotal Revenue Fair is
proportional to distance traveled
77Model 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.
78Equilibrium 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.
79Empirical Values
80Cases Considered
81Application
- Welfare vs. Tolls
- Tolls vs. Tolls
- General Tax vs. Cordon
- Equilibrium Cooperative vs. Non-Cooperative
- Game Policy Choice
- Perfect Tolls
- Odometer Tax
82Representative Game
- Two Choices
- revenue mechanism,
- rate given revenue mechanism
- Form of Prisoners Dilemma
- Payoff Toll, Toll Lower Than Payoff Tax,
Tax.
83Welfare in J0 as a function of J0 Toll
84Welfare in J0 at Welfare Maximizing Tolls vs.
Jurisdiction Size in an All-Tax Environment
85Welfare in J0 at Welfare Maximizing Tolls vs.
Jurisdiction Size in an All-Toll Environment
86Tolls by Location of Origin and Destination.
87Policy Choice as a Function of Fixed Collection
Costs and Jurisdiction Size
88Policy Choice as a Function of Variable
Collection Costs and Jurisdiction Size
89Reaction Curves Best J0 Toll as Tolls Vary in
Toll Environment
90Uniqueness, Non-Cooperative Welfare Maximizing J0
Toll as Initial Toll for Other Jurisdiction
Varies in Toll Environment
91Elasticity About Mean
92Comparison of Tolls and Welfare for Different
Jurisdiction Sizes
93Rate of Toll Under Various Policies
94General Trip Classification
95Conclusions
- 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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98Demand (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
99Demand (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
100Consumers Surplus
- U - denotes consumers surplus
- a,b - jurisidction borders
- n - counter for tollbooths crossed
- d - spacing between tollbooths
101Model 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.
102Transportation Revenue
103Total Network Cost
- where
- CT Total Cost
- CCV Variable Collection Cost
- CCF Fixed Collection Cost
- C? Variable Network Cost
- CS Fixed Network Cost
- ??????????? model coefficients
104Tolls in All-Cordon Environment
105Price of Infrastructure
106Rate of Toll Under Various Policies
107Odometer Tax