1 / 107

CE 8214 Transportation Economics Introduction

- David Levinson

Introductions

- Who are you?
- State your name, major/profession, degree goal,

research interest

Syllabus

- 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

system) - 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

Discussion

- 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

Overview

- 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

B B

i j

A i 3,3 2,2

j 2,2 1,1

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.

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

function. - (Objective, minimize payoff)

Objectives

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.

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.

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

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

selfishness? - How does reality compare with game theory?

Discussion

- How does an infinitely or indefinitely repeated

Prisoners Dilemma game differ from a finitely

repeated or one-time game? - Why?

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

Solution

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 ?

- 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

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

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

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

(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.

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)

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.

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.

Delay

- 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

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).

Nomenclature

- 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

v,_,_

Three-Player Game Results

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.

Break

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)

Outline

- 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

tolls? - Who wins and who loses under various revenue

mechanisms? - How does the spatial distribution of winners and

losers affect the choice?

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

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.

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

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

toll. - 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

Application

- 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,

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

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).

(No Transcript)

(No Transcript)

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

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

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)

increase.

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