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CE 8214: Transportation Economics: Introduction David Levinson

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

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Syllabus Handouts Textbook

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Paper reviews handouts

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

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

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Roundrobin Schedules How many students …

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11 Players

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12 Players

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13 Players

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14 Players

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15 Players

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16 Players

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17 Players

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

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Next Time Email me your reviews by Tuesday 5:30 pm. Talk with me if you have problem with your assigned Discussion Paper. Discuss Game Theory

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Game Theory David Levinson

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

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

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

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

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A Nash Equilibrium B ij Ai[3,3]*[2,2] j [1,1]

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

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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)

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

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

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Objectives

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

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

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

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Coordination Game Payoff Table

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

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

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Discussion How does an infinitely or indefinitely repeated Prisoner’s Dilemma game differ from a finitely repeated or one-time game? Why?

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

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Solution

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

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

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3 Possible Strategies At [300,300] Each airline gets 375/2 share = 187.5 pax * $300 = $56,250, cost remains $50,000 At [300, 400], 300 airline gets 375*300 = 112,500 - 50000

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

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Microfoundations of Congestion and Pricing David Levinson

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

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

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

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

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

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

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Example 1: (1,0,1) Note: * Indicates Nash Equilibrium Italics indicates social welfare maximizing solution

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Example 2: (3,1,4) Note: * Indicates Nash Equilibrium Italics indicates social welfare maximizing solution

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Payoff matrix with congestion pricing

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What are the proper prices? Normally use marginal cost pricing – MC = ∂ TC/∂Q But Total Costs (TC) are discrete, so we use incremental cost pricing –IC = TC/ Q Total Costs include both delay costs as well as schedule delay costs. – o = l =0.5*(L+D) – e = MAX(0.5*(D-E),0)

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

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

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

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Two-Player Game Results

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

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Delay Expected delay Cost of delay where: D = delay penalty Q t = standing queue at time t A t = arrivals at time t.

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Schedule Delay Schedule delay is the deviation from the time which a vehicle departs the queue and the desired, or on-time period. Where: 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

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

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

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Three- Player Game Arrival and Departure Patterns

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

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Three-Player Game Results

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

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Break

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On Whom The Toll Falls: A Model of Network Financing by David Levinson Man in Bowler Hat: To Boost The British Economy, I’d Tax All Foreigners Living Abroad -- Chapman et al. (1989)

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

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

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Motivation New Capacity Desired New Concerns: Social Costs New Fleet: EVs New Networks: ITS New Toll Technology: ETC New Owners: Privatization New Rules: ISTEA 2 New Priorities: –Capital -> Operating

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

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

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Revenue Instrument

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Why No Gas Tax ? The Gas Tax is bounded by two cases: Odometer Tax (where all gas purchased in the home jurisdiction) and Perfect Toll (where all gas purchased in the jurisdiction of travel). What is proper behavioral assumption about location of purchase?

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Long Road & Trip Classes

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Free Riders

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Cross Subsidy by Instrument & Class Assumes Total Cost=Total Revenue; “Fair” is proportional to distance traveled

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Model Parameters Demand: –Distance, –Price of Trip, –Fixed User Cost. Network Cost: –Fixed Network Costs, –Variable Network Costs, –Fixed Collection Costs, –Variable Collection Costs. Network Revenue: –Rate of Toll, Tax, –Basis.

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Equilibrium: Cooperative vs. Non-Cooperative Non-Cooperative (Nash): Assume other jurisdictions’ policies are fixed when setting toll. Cooperative: Assume other jurisdictions behave by setting same toll rate as J 0. Results in higher welfare. Not equilibrium in one- shot game.

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Empirical Values

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Cases Considered

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

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Representative Game Two Choices: –revenue mechanism, –rate given revenue mechanism Form of Prisoner’s Dilemma: –Payoff [Toll, Toll]* Lower Than Payoff [Tax, Tax].

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Welfare in J 0 as a function of J 0 Toll

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Welfare in J 0 at Welfare Maximizing Tolls vs. Jurisdiction Size in an All- Tax Environment

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Welfare in J 0 at Welfare Maximizing Tolls vs. Jurisdiction Size in an All- Toll Environment

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Tolls by Location of Origin and Destination.

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

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

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Reaction Curves: Best J 0 Toll as Tolls Vary in Toll Environment

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Uniqueness, Non-Cooperative Welfare Maximizing J 0 Toll as Initial Toll for Other Jurisdiction Varies in Toll Environment

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Elasticity About Mean

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Comparison of Tolls and Welfare for Different Jurisdiction Sizes

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Rate of Toll Under Various Policies

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General Trip Classification

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Conclusions Necessary Conditions For Tolls to Become Widespread, Need: »Relatively Low Transaction Costs, »Sufficiently Decentralized (Local) Decisions About Placement of Tolls. Actual Conditions Policy Environment Becoming More Favorable to Road Pricing: »Localized Decisions (MPO), »Federal encouragement (ISTEA 2 pilot projects), »Longer trips, »Lower transaction costs (ETC).

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Demand (1) f(z) = flow past point z; F = flow between sections (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

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Demand (2) P T =total user cost P I =vector of price of infrastructure =coefficient (relates price to demand), < 0 = coefficient (trips per km (@ P T =0)), > 0 = fixed private vehicle cost = variable private vehicle cost per unit distance x,y = location trip enters, exits road V T = value of time S F = freeflow speed | | indicates absolute value

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

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Model Outcomes As the size of jurisdiction J 0 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 J 0 (F 00, F 0+, and F -0 ) increase.

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Transportation Revenue

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Total Network Cost where: C T = Total Cost C CV = Variable Collection Cost C CF = Fixed Collection Cost C = Variable Network Cost C S = Fixed Network Cost = model coefficients

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Tolls in All-Cordon Environment

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Price of Infrastructure

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Rate of Toll Under Various Policies

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Odometer Tax

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