CE 8214: Transportation Economics: Introduction

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Transcript 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

• Handouts • Textbook

Syllabus

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

Player A Odd Player A Even

The Payoff Matrix

Player B Odd [3, 3] Player B Even [0, 5] [5, 0] [1, 1] [Payoff A, Payoff B]

Roundrobin Schedules

• How many students …

11 Players

11. Round robin schedule for 11 teams.

A B C D E Bye 6- 7 5- 8 4- 9 3-10 2-11 1 7- 8 6- 9 5-10 4-11 3- 1 2 8- 9 7-10 6-11 5- 1 4- 2 3 9-10 8-11 7- 1 6- 2 5- 3 4 10-11 9- 1 8- 2 7- 3 6- 4 5 11- 1 10- 2 9- 3 8- 4 7- 5 6 1- 2 11- 3 10- 4 9- 5 8- 6 7 2- 3 1- 4 11- 5 10- 6 9- 7 8 3- 4 2- 5 1- 6 11- 7 10- 8 9 4- 5 3- 6 2- 7 1- 8 11- 9 10 5- 6 4- 7 3- 8 2- 9 1-10 11

12 Players

12. Round robin schedule for 12 teams.

A B C D E Bye 7- 8 6- 9 5-10 4-11 3-12 1 2 8- 9 7-10 6-11 5-12 3- 1 2 4 9-10 8-11 7-12 5- 1 4- 2 3 6 10-11 9-12 7- 1 6- 2 5- 3 4 8 11-12 9- 1 8- 2 7- 3 6- 4 5 10 11- 1 10- 2 9- 3 8- 4 7- 5 6 12 12- 2 11- 3 10- 4 9- 5 8- 6 1 7 1- 2 12- 4 11- 5 10- 6 9- 7 3 8 2- 3 1- 4 12- 6 11- 7 10- 8 5 9 3- 4 2- 5 1- 6 12- 8 11- 9 7 10 4- 5 3- 6 2- 7 1- 8 12-10 9 11 5- 6 4- 7 3- 8 2- 9 1-10 11 12 3-10 2-11 1-12 4 5 6 7 8 9 6- 7 5- 8 4- 9 1 2 3 10 11 12

13 Players

13. Round robin schedule for 13 teams.

A B C D E F Bye 7- 8 6- 9 5-10 4-11 3-12 2-13 1 8- 9 7-10 6-11 5-12 4-13 3- 1 2 9-10 8-11 7-12 6-13 5- 1 4- 2 3 10-11 9-12 8-13 7- 1 6- 2 5- 3 4 11-12 10-13 9- 1 8- 2 7- 3 6- 4 5 12-13 11- 1 10- 2 9- 3 8- 4 7- 5 6 13- 1 12- 2 11- 3 10- 4 9- 5 8- 6 7 1- 2 13- 3 12- 4 11- 5 10- 6 9- 7 8 2- 3 1- 4 13- 5 12- 6 11- 7 10- 8 9 3- 4 2- 5 1- 6 13- 7 12- 8 11- 9 10 4- 5 3- 6 2- 7 1- 8 13- 9 12-10 11 5- 6 4- 7 3- 8 2- 9 1-10 13-11 12 6- 7 5- 8 4- 9 3-10 2-11 1-12 13

14 Players

14. Round robin schedule for 14 teams.

A B C D E F Bye 8- 9 7-10 6-11 5-12 4-13 3-14 1 2 9-10 8-11 7-12 6-13 5-14 3- 1 2 4 10-11 9-12 8-13 7-14 5- 1 4- 2 3 6 11-12 10-13 9-14 7- 1 6- 2 5- 3 4 8 12-13 11-14 9- 1 8- 2 7- 3 6- 4 5 10 13-14 11- 1 10- 2 9- 3 8- 4 7- 5 6 12 13- 1 12- 2 11- 3 10- 4 9- 5 8- 6 7 14 14- 2 13- 3 12- 4 11- 5 10- 6 9- 7 1 8 1- 2 14- 4 13- 5 12- 6 11- 7 10- 8 3 9 2- 3 1- 4 14- 6 13- 7 12- 8 11- 9 5 10 3- 4 2- 5 1- 6 14- 8 13- 9 12-10 7 11 4- 5 3- 6 2- 7 1- 8 14-10 13-11 9 12 5- 6 4- 7 3- 8 2- 9 1-10 14-12 11 13 6- 7 5- 8 4- 9 3-10 2-11 1-12 13 14 4-11 3-12 2-13 1-14 5 6 7 8 9 10 7- 8 6- 9 5-10 1 2 3 4 11 12 13 14 You may wish to swap rounds 2 and 13, in order to distribute the byes more evenly.

15 Players

15. Round robin schedule for 15 teams.

A B C D E F G Bye 8- 9 7-10 6-11 5-12 4-13 3-14 2-15 1 9-10 8-11 7-12 6-13 5-14 4-15 3- 1 2 10-11 9-12 8-13 7-14 6-15 5- 1 4- 2 3 11-12 10-13 9-14 8-15 7- 1 6- 2 5- 3 4 12-13 11-14 10-15 9- 1 8- 2 7- 3 6- 4 5 13-14 12-15 11- 1 10- 2 9- 3 8- 4 7- 5 6 14-15 13- 1 12- 2 11- 3 10- 4 9- 5 8- 6 7 15- 1 14- 2 13- 3 12- 4 11- 5 10- 6 9- 7 8 1- 2 15- 3 14- 4 13- 5 12- 6 11- 7 10- 8 9 2- 3 1- 4 15- 5 14- 6 13- 7 12- 8 11- 9 10 3- 4 2- 5 1- 6 15- 7 14- 8 13- 9 12-10 11 4- 5 3- 6 2- 7 1- 8 15- 9 14-10 13-11 12 5- 6 4- 7 3- 8 2- 9 1-10 15-11 14-12 13 6- 7 5- 8 4- 9 3-10 2-11 1-12 15-13 14 7- 8 6- 9 5-10 4-11 3-12 2-13 1-14 15

16 Players

16. Round robin schedule for 16 teams.

A B C D E F G Bye 9-10 8-11 7-12 6-13 5-14 4-15 3-16 1 2 10-11 9-12 8-13 7-14 6-15 5-16 3- 1 2 4 11-12 10-13 9-14 8-15 7-16 5- 1 4- 2 3 6 12-13 11-14 10-15 9-16 7- 1 6- 2 5- 3 4 8 13-14 12-15 11-16 9- 1 8- 2 7- 3 6- 4 5 10 14-15 13-16 11- 1 10- 2 9- 3 8- 4 7- 5 6 12 15-16 13- 1 12- 2 11- 3 10- 4 9- 5 8- 6 7 14 15- 1 14- 2 13- 3 12- 4 11- 5 10- 6 9- 7 8 16 16- 2 15- 3 14- 4 13- 5 12- 6 11- 7 10- 8 1 9 1- 2 16- 4 15- 5 14- 6 13- 7 12- 8 11- 9 3 10 2- 3 1- 4 16- 6 15- 7 14- 8 13- 9 12-10 5 11 3- 4 2- 5 1- 6 16- 8 15- 9 14-10 13-11 7 12 4- 5 3- 6 2- 7 1- 8 16-10 15-11 14-12 9 13 5- 6 4- 7 3- 8 2- 9 1-10 16-12 15-13 11 14 6- 7 5- 8 4- 9 3-10 2-11 1-12 16-14 13 15 7- 8 6- 9 5-10 4-11 3-12 2-13 1-14 15 16 4-13 3-14 2-15 1-16 5 6 7 8 9 10 11 12 8- 9 7-10 6-11 5-12 1 2 3 413 14 15 16

17 Players

17. Round robin schedule for 17 teams.

A B C D E F G H Bye 9-10 8-11 7-12 6-13 5-14 4-15 3-16 2-17 1 10-11 9-12 8-13 7-14 6-15 5-16 4-17 3- 1 2 11-12 10-13 9-14 8-15 7-16 6-17 5- 1 4- 2 3 12-13 11-14 10-15 9-16 8-17 7- 1 6- 2 5- 3 4 13-14 12-15 11-16 10-17 9- 1 8- 2 7- 3 6- 4 5 14-15 13-16 12-17 11- 1 10- 2 9- 3 8- 4 7- 5 6 15-16 14-17 13- 1 12- 2 11- 3 10- 4 9- 5 8- 6 7 16-17 15- 1 14- 2 13- 3 12- 4 11- 5 10- 6 9- 7 8 17- 1 16- 2 15- 3 14- 4 13- 5 12- 6 11- 7 10- 8 9 1- 2 17- 3 16- 4 15- 5 14- 6 13- 7 12- 8 11- 9 10 2- 3 1- 4 17- 5 16- 6 15- 7 14- 8 13- 9 12-10 11 3- 4 2- 5 1- 6 17- 7 16- 8 15- 9 14-10 13-11 12 4- 5 3- 6 2- 7 1- 8 17- 9 16-10 15-11 14-12 13 5- 6 4- 7 3- 8 2- 9 1-10 17-11 16-12 15-13 14 6- 7 5- 8 4- 9 3-10 2-11 1-12 17-13 16-14 15 7- 8 6- 9 5-10 4-11 3-12 2-13 1-14 17-15 16 8- 9 7-10 6-11 5-12 4-13 3-14 2-15 1-16 17

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 5:30 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

i i [3,3]* B j [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.

Prisoner’s 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 ‘prisoner’s dilemma’. • The term prisoner’s 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 doesn’t, the one who confesses turns state’s 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 r I and r J are tolls of the two jurisdictions. Demand is a negative exponential function. • (Objective, minimize payoff)

Objectives

Objective Local welfare Func tion max

r I W L



U ij

 2 *

R ij

 2*

C Nij

 2 *

C Vij

Componen t Flow Consumer's surplus Network use cost Coll ection cost Revenue Equa tions

f b U ij

 

C Nij

    



 

 

r J



r I



r J

 

    

r I



 

r J

 

 

r I



r I



r J C Vij

 

 

r J



R ij



r I



 

r I



r J

  (1) (2) (3) (4) (5)

Payoffs

J I Delaware Toll Tax Toll [1153, 1153]* [883, 2322] J J New Jersey Tax [2322, 883] [1777, 1777] • 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

Buick

L R

Mercedes

L [1,1] [-10,-10] R [-10,-10] [1,1]

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 Prisoner’s 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

United

Pric e=$200 Pric e=$400

American

Pric e=$200 [0,0 ] [-50000, 50000 ] Pric e=$400 [50000, -50000] [0,0]

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

United

Pric e=$200 Pric e=$300 Pric e=$400

American

Pric e=$200 [0,0] [-50000, 50000 ] [-50000, 50000 ] Pric e=$300 [50000, -50000] [6250, 6250] [-50000, 62500 ] Pric e=$400 [50000, -50000 ] [62500, - 50000] [0,0] • 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 375*300 = 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 > 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

Early Early [0.5*(

E+D

), 0.5*(

E+D

)] Vehicle 1 On-time [0,

] Late [

L, E

] Vehicle 2 On-time [

, 0] [0.5*(

L+D

), 0.5*(

L+D

)] [

, 0] Late [

E ,L

] [0,

] [

+0.5*(

L+D

+0.5*(

L+D

)] Note: [Payout for Vehicle 1, Payout for Vehicle 2] Objective to Minimize Own Payout, S.t. others doing same

Example 1: (1,0,1)

E, D, L

1, 0, 1 Early Vehicle 1 On-time Late Early

[0.5,0.5] [0,1]

[1,1] Vehicle 2 On-time

[1,0]

Late [1,1]

[0.5, 0.5] * [0,1] [1,0]

[1.5, 1.5] Note: * Indicates Nash Equilibrium Italics indicates social welfare maximizing solution

Example 2: (3,1,4)

E, D, L

3, 1, 4 Early Vehicle 1 On-time Late Early [2,2]

[0,3]

[4,3] Vehicle 2 On-time

[3,0]

Late [3,4] [2.5, 2.5] * [0,4] [4,0] [6.5, 6.5] Note: * Indicates Nash Equilibrium Italics indicates social welfare maximizing solution

Payoff matrix with congestion pricing

Vehicle 2 Early Early [0.5*(E+D)+  e , 0.5*(E+D) +  e ] Vehicle 1 On-tim e [0, E] On-tim e [E, 0] [0.5*(L+D)+  o , 0.5*(L+D) +  o ] Late [L, E] [L, 0] Late [E, L] [0, L] [L+0.5*(L+D)+  l , L+0.5* (L+D) +  l ]

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*( L+D )  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 can’t blame departing early on the other player.

Example 1 (1,0,1) with congestion prices

E, D, L

1, 0, 1 Early Vehicle 1 On-time Late Early

[0.5,0.5] [0,1]*

[1,1] Vehicle 2 On-time [1,0]* [1,1]*

[1,0]*

Late [1,1]

[0,1]*

[2,2]

Example 2 (3,1,4) with congestion prices

E, D, L

3,1,4 Ve hicle 1 Early On-time Late Early [2.5,2.5] [0,3]* [4,3] Ve hicle 2 On-time [3,0]* [5,5] [4,0] Late [3,4] [0,4] [9,9]

Two-Player Game Results

E,D,L

0,0,0

Number of Nash Equilibria (Unpriced)

9 6 0,1,0 0,0,1 0,1,1 1,0,0 1,1,0 3 2 4 2 1 1,0,1 5 1,1,1 3,1,4 4,0,3 4,1,3 3,0,4 1 1 1 1

Solutions (Unpriced)

all

eo, el, oe, ol, le, lo ee, eo, oe oe, eo oo, ol, lo, ll ol, l o oo eo, oe, oo, ol, lo oo oo oo oo

Total Cost

0 0 0 0 0 0 1 1 or 2 5 3 4 4

Minimum Total Cost

0 0 0 0 0 0 1 1 3 3 3 3

Number of Nash Equilibria (Priced)

9 6 3 2 4 2 5 4 2 3 2 2

Solutions (Priced)

all

eo, el, oe, ol, le, lo ee, eo, oe oe, eo oo, ol, lo, ll ol, l o eo, oe, oo, ol, l o eo, oe, ol, lo oe, eo oo, ol, lo ol, l o eo, oe

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 • • Q t A t = arrivals at time t .

 t  

Q t

 0.5

A t

  

Q t

 0.5

A 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:

S i



t a



d t



t o

• • • d t = delay t a = time of arrival at back of queue t o = desired time of departure from front of queue (time to be on-time)  • The cost of schedule delay is thus  

S i

if S i

 0

S i

if S i

 0 



Probabilistics

  • We only know the delay probabilistically, so schedule delay is also probabilistic 1    • Where:   

P t t



P t



      * 2

   *      

   *   

   * 2

   * 3

   • • • P() Table 9.

 t = i , summarized in 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

Arrival Frequency Departure Frequency Pattern (/64) Patterns (/64)

vvv vve vvo veo vee

1 3 3 6 3

veo

vel

vvl vel voo vol

3 6 3 6

vol

vll eee eeo eel eoo eol

3 1 3 3 3 6

vlr eol

3 16

ell ooo ool oll lll

3 1 3 3 1

elr olr lrs

3 7 1

Three Player Game Arrival and Departure Patterns

Departure Probability Given Arrival Strategies [v,_,_]

Player A

      Player B

Player C

v v e

Departure Probability   0.33

    0.33

0.33

      0 0 0 0.5

0.5

0 0 0 0

v o

0.5

0 0 0 0

l v

0.5

0 0 0 0

e e

1 0 0 0 0 0

e o

1 0 0 0 0 0

l e

1 0 0 0 0 0

o o

1 0 0 0 0 0

l o

1 0 0 0 0 0

l l

1 0 0 0 0 0

Three-Player Game Results

E,D,L 0,0,0 0,1,0 0,0,1 0,1,1 1,0,0 1,1,0 1,0,1 1,1,1 3,1,4 4,0,3 4,1,3 3,0,4 Numb er of Nash Equilib ria (Unpriced) 64 24 13 6 8 7 5 9 1 2 4 1 Solutions (Unpriced) all

veo …, vel …, vol …, eol … , vvv, vvo …, vee … , veo, … veo, … ooo,lll,oll, …, ool … ooo, ool …, oll, … ooo, eee, … eeo, … eol, … , eoo, … eee ooo, eee ooo, eoo, ...

eee

Numb er of Nash Equilib ria (Priced) 64 24 13 6 8 3 4 6 3 4 9 1 Solutions (Priced) all

veo …, vel …, vol …, eol … , vvv, vvo …, vee … , veo, … veo, … ooo,lll,llo, …, ool … llo, … eee, eeo, … eol, … eoo, … eee, eoo eol, …, eoo eee

Lowest Total Cost 0 0 0 0 0 1 2 2 7 7 7 7

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, I’d 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 -> 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

Instrument System Access Tax Use-Based Tax Cordon Toll Perfect Toll Example Poll Tax Odometer Tax Toll to Cross Cordon Toll on Every Segment Where Collected home jurisdiction home jurisdiction, jurisdiction of use jurisdiction of use

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 -• S Class G 00 Trips a J 0 b J +1 - • a J 0 x y s.t. y > x for all trips b Class G 0+ Trips - • a J 0 x b y Class G -0 Trips - • - • x Class G -+ Trips - • - • x a a J 0 J 0 y b b y • • • • • • J +2 • S +

Long Road & Trip Classes

Free Riders

Class of Rider Perfect Free Riders Share (S) of Full Cost Paid 0% Imperfect Free Riders or Easy Riders Fair Riders 0% < S < 100% 100% Overburdened or Hard Riders 100% < S

Cross Subsidy by Instrument & Class

User Group Origin Destination Local Local Instrument System Access Tax (G 00 ) Hard Use-Based (Odometer) Tax Hard Cordon Toll Free Perfect Toll Fair Local Non-Local (G 0+ ) Hard residents Free - non residents Hard residents Free - non residents Hard/Fair/ Easy Fair Non-Local Local (G -0 ) Hard residents Free - non residents Hard residents Free - non residents Hard/Fair/ Easy Fair Non-Local Non-Local (Through) (G -+ ) Free Free Hard Fair Assumes Total Cost=Total 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 J 0.

Results in higher welfare. Not equilibrium in one shot game.

Empirical Values

Coefficient Description density of trips (  ) (Trips Loading/km) Value 180 user total price (  ) user fixed cost (  ) ($) user distance cost (  ) ($/vkt) network fixed cost (  ) ($/km) network variable cost (  ) ($/vkt) toll collection fixed cost (  ) ($/toll-booth) toll collection variable cost (  ) ($/crossing) -1 1.23

0.15

0 0. 018 90 0.08

Cases Considered

J 0 : Environ ment: General Tax (  ) General Tax (  ) N N Cordon Toll (  ) Odometer Tax (  ) Perfect Toll (  ) Cordon Toll (  ) N N, C Odometer Tax (  ) Perfect Toll (  ) C C

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 Prisoner’s Dilemma: – Payoff [Toll, Toll]* Lower Than Payoff [Tax, Tax]. Tax Representative Jurisdiction All-Tax [3087, 3087  All-Cordon Toll [2309,

3555

 J 0 Cordon Toll [3555 ,

2309

 [2418, 2418 

Welfare (W, U,



) ($)

4 0 0

Welfare in J

function of J

as a Toll

Collect io 0 5 0 0 0 -5 0 0 0 1 0 00 1 5 00 Welf are-Tax Enviornment 0 .5

W* 1  * 1 .5

2 2 .5

J 0 Toll ( $ / Crossing)

Cons umers' Surplus-Tax Environment 3 3 .5

Prof it -Tax Environment 4 4 .5

Welfare in J

at Welfare Maximizing Tolls vs. Jurisdiction Size in an All Tax Environment

Welfare (W*, U*,



*) ($)

500 .00

400 .00

300 .00

200 .00

100 .00

0 .00

1 -100 .00

10 100 1000

Size (k m)

 /k m W-Toll/k m U/k m W-Tax/ km

Welfare in J

at Welfare Maximizing Tolls vs. Jurisdiction Size in an All Toll Environment

Welfare (W*, U*,



*) ($)

500 .00

400 .00

300 .00

200 .00

100 .00

0 .00

1 -100 .00

10 100 1000

Size

 /k m W-Toll/k m W-Tax/ km

Tolls by Location of Origin and Destination

G -+ -oo

Toll

0 r  +r  r  +r  2r  +2r  2r  +4r  2r  +6r  a J 0 K -1 exit K 0 ent b J +1 J +2 oo K 0 exit K +1 ent K +1 exi K +2 ent K +2 exi K +3 ent

Policy Choice as a Function of Fixed Collection Costs and Jurisdiction Size

Fixe d Collection Costs ($)

8 0 0 7 0 0 6 0 0 5 0 0 4 0 0 3 0 0 2 0 0 Toll if En viro nment All-Tax, Tax if Enviro nme nt All-Cordo n Toll 1 0 0 0 1 1 0 CCF' - T ax Env ironmen t Always Toll

Size (k m)

CCF' - T oll Environm ent 1 0 0 CCF - Empirical Always Tax 1 0 00

Policy Choice as a Function of Variable Collection Costs and Jurisdiction Size

Variable Collection Cost, Toll ($)

1 .4

1 .2

1 0 .8

0 .6

0 .4

0 .2

0 1 Thet a - Empir ical Thet a' - Tax Env iro nment 1 0 Always Tax Always Toll

Size (k m)

Thet a' - Toll Env ironment 1 0 0 Toll - T ax Env iro nment 1 0 00 Toll - T oll Env ironmen t

Reaction Curves: Best J

Toll as Tolls Vary in Toll Environment

Toll J 0 ($ )

1 .20

1 .00

0 .80

0 .60

0 .40

0 .20

0 .00

0 .20

0 .40

0 .60

Toll Environment



( $ )

0 .80

W-Juris dict ion Size=1 0 1 .00

km km 1 .20

Uniqueness, Non-Cooperative Welfare Maximizing J

Toll as Initial Toll for Other Jurisdiction Varies in Toll Environment

Toll ($)

12 10 8 6 4 2 0 0 -2 1 2 S=-1

Ite rat ion #

S=0 S=1 3 S=10 4 5

Elasticity About Mean

Variable |Alpha| Delt a Zeta Psi sensiti vit y o f demand to cost density of trip origin s private fixed costs sensiti vit y o f demand to trip leng th Welf are -3.070

Revenue -3.437

Cost -1.758

Cons.

Surp lus -2.785

Profit -5.320

1.038

1.000

0.699

1.000

1.337

-0.424

-0.094

-1.150

-0.143

-0.498

-0.119

-0.239

-0.065

-1.879

-0.156

Gamm a Phi Theta fixed co sts associated wit h network leng th variable cost associated wit h network use variable coll ection costs -0.044

-0.049

-0.016

0.000

-0.028

-0.013

0.250

0.277

0.053

0.000

-0.017

-0.009

-0.596

-0.251

-0.065

Comparison of Tolls and Welfare for Different Jurisdiction Sizes

Jurisdic tion Size (km) 10 2x10 20 Tolls $0.65

$1.30

$0.68

Welf are 2367 4734 5451

Rate of Toll Under Various Policies

tax J 0 Poli cy toll Envi ronment Poli cy tax toll r    = 0 r    = 0 r    = 0 r   = r   r    =  r   r    = 0 r    =  r   r    =  r  

General Trip Classification

Section o f Origin (x) S J 0 S + Section o f S G - G 0 G + Destination J 0 G -0 G 00 G +0 (y) S + G -+ G 0+ G ++

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).

(

) 

 

Demand (1)

 

  (

;

P I

) 

dydx F x

2  

 2

2 

1   

;

P I

 

dydx

• f(z) = flow past point z; F = flow between sections   (P T (x,y;P I ))dxdy = demand function representing the number of trips that enter facility between x and x + dx and leave between y and y + dy • • P T (x,y;P I ) = generalized cost of travel to users defined below) x,y = where trip enters,exits road • P I = price of infrastructure

 

P T



;

P I

Demand (2)

   



P T



;

P I



P T



;

P I

 

P I

    



 

V T





S F x

 • P T =total user cost • P I =vector of price of infrastructure   =coefficient (relates price to demand),  < 0     = coefficient (trips per km (@ P = fixed private vehicle cost T =0)),  > 0   = variable private vehicle cost per unit distance • • • • x,y = location trip enters, exits road V T S F = value of time = freeflow speed | | indicates absolute value

Consumers’ Surplus





    

P T



;

 

dpdydx a x P I

  

 1





 1 



n d

 

P I

 

P T



;

 

dpdydx

U - denotes consumer’s surplus a,b - jurisidction borders n - counter for tollbooths crossed d - spacing between tollbooths

Model Outcomes

• As the size of jurisdiction J 0 as |b-a| gets large: increases, that is 1. F -0 / F -+ increases. 2. F 0+ / F -+ increases. 3. The total number of trips originating in or destined for jurisdiction J 0 increase.

(F 00 , F 0+ , and F -0 )

Transportation Revenue

Policy in J 0 Total Revenue General Tax (  ) ($) Cordon Tolls (  ) ($/Crossing) 0 Odometer Tax (  ) ($/km) Perfect Toll (  ) ($/km)





 1

(

z k

) 

b a

 







  

P T



P I

  





b a



b x



 

b a

 



b x



  

P T



P I

  



+ 



  

P T



P I

  



+ 



a b

 





  

P T



P I

  



+ 

  







  

P T



P I

  



Total Network Cost

C T



K i

   



a C S



  

z z

  

b a f

(

)



C CV



C CF

 

K i



 1

(

z k

)

 

K i

where: C T C CV = Total Cost = Variable Collection Cost C CF C  = Fixed Collection Cost = Variable Network Cost C S = Fixed Network Cost  = model coefficients

Tolls in All-Cordon Environment

G -+ -•

Toll

2r  + 2r  2r  + 4r  2r  + 6 r  a J 0 K -1 exit K 0 ent b J +1 J +2 • K 0 exit K +1 ent K +1 exi K +2 ent K +2 exi K +3 ent

Policy General Tax (  )

Price of Infrastructure

User Group Amount paid to J 0 0 Amount paid outside J 0 0 All Cordon Tolls (  ) G 0 0 0 0

  2

 1  G 0 + , G -0



  2

 2

 2  G -+ 2



 



 Odometer Tax (  ) G 0 0 , G 0 + 0 G -0 , G -+ 0 0 Perfect Toll (  ) G 0 0 G 0 + G -0 G -+

 





 





 





 



 0

 





 





 



  





J 0 Policy

Rate of Toll Under Various Policies

tax toll Enviro nment tax r    = 0 r    = 0 r    =  r   r    = 0 toll r    = 0 r   = r   r    =  r   r    =  r  

Odometer Tax



C T U

  

a x



 

 



 

 

 



    

  

  

 

   2   



   



 

dydx

  

  2 

 

  

  