Networks, Crowds, and Markets:Reasoning about a Highly Connected World

David EasleyDept. of Economics

Cornell University

Jon KleinbergDept. of Computer Science

Cornell University

Cambridge University Press, 2010Draft version: June 10, 2010.

8.2. BRAESS’S PARADOX 231

This all fits very naturally into the framework we’ve been using. One thing to notice,

of course, is that up to now we have focused primarily on games with two players, whereas

the current game will generally have an enormous number of players (4000 in our example).

But this poses no direct problem for applying any of the ideas we’ve developed. A game can

have any number of players, each of whom can have any number of available strategies, and

the payoff to each player depends on the strategies chosen by all. A Nash equilibrium is still

a list of strategies, one for each player, so that each player’s strategy is a best response to

all the others. The notions of dominant strategies, mixed strategies and Nash equilibrium

with mixed strategies all have direct parallels with their definitions for two-player games.

In this traffic game, there is generally not a dominant strategy; for example, in Figure 8.1

either route has the potential to be the best choice for a player if all the other players are

using the other route. The game does have Nash equilibria, however: as we will see next,

any list of strategies in which the drivers balance themselves evenly between the two routes

(2000 on each) is a Nash equilibrium, and these are the only Nash equilibria.

Why does equal balance yield a Nash equilibrium, and why do all Nash equilibria have

equal balance? To answer the first question, we just observe that with an even balance

between the two routes, no driver has an incentive to switch over to the other route. For the

second question, consider a list of strategies in which x drivers use the upper route and the

remaining 4000−x drivers use the lower route. Then if x is not equal to 2000, the two routes

will have unequal travel times, and any driver on the slower route would have an incentive to

switch to the faster one. Hence any list of strategies in which x is not equal to 2000 cannot

be a Nash equilibrium; and any list of strategies in which x = 2000 is a Nash equilibrium.

8.2 Braess’s Paradox

In Figure 8.1, everything works out very cleanly: self-interested behavior by all drivers causes

them — at equilibrium — to balance perfectly between the available routes. But with only a

small change to the network, we can quickly find ourselves in truly counterintuitive territory.

The change is as follows: suppose that the city government decides to build a new, very

fast highway from C to D, as indicated in Figure 8.2. To keep things simple, we’ll model its

travel time as 0, regardless of the number of cars on it, although the resulting effect would

happen even with more realistic (but small) travel times. It would stand to reason that

people’s travel time from A to B ought to get better after this edge from C to D is added.

Does it?

Here’s the surprise: there is a unique Nash equilibrium in this new highway network,

but it leads to a worse travel time for everyone. At equilibrium, every driver uses the

route through both C and D; and as a result, the travel time for every driver is 80 (since

4000/100 + 0 + 4000/100 = 80). To see why this is an equilibrium, note that no driver can

232 CHAPTER 8. MODELING NETWORK TRAFFIC USING GAME THEORY

BA

C

D

x/100

45

45

x/100

0

Figure 8.2: The highway network from the previous figure, after a very fast edge has beenadded from C to D. Although the highway system has been “upgraded,” the travel time atequilibrium is now 80 minutes, since all cars use the route through C and D.

benefit by changing their route: with traffic snaking through C and D the way it is, any

other route would now take 85 minutes. And to see why it’s the only equilibrium, you can

check that the creation of the edge from C to D has in fact made the route through C and

D a dominant strategy for all drivers: regardless of the current traffic pattern, you gain by

switching your route to go through C and D.

In other words, once the fast highway from C to D is built, the route through C and

D acts like a “vortex” that draws all drivers into it — to the detriment of all. In the new

network there is no way, given individually self-interested behavior by the drivers, to get

back to the even-balance solution that was better for everyone.

This phenomenon — that adding resources to a transportation network can sometimes

hurt performance at equilibrium — was first articulated by Dietrich Braess in 1968 [76],

and it has become known as Braess’s Paradox. Like many counterintuitive anomalies, it

needs the right combination of conditions to actually pop up in real life; but it has been

observed empirically in real transportation networks — including in Seoul, Korea, where the

destruction of a six-lane highway to build a public park actually improved travel time into

and out of the city (even though traffic volume stayed roughly the same before and after the

change) [37].

Some reflections on Braess’s paradox. Having now seen how Braess’s paradox works,

we can also appreciate that there is actually nothing really “paradoxical” about it. There are

many settings in which adding a new strategy to a game makes things worse for everyone.

For example, the Prisoner’s Dilemma from Chapter 6 can be used to illustrate this point: if

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Generated Traffic and Induced Travel Implications for Transport Planning

2 January 2017

Todd Litman

Victoria Transport Policy Institute

Abstract

Traffic congestion tends to maintain equilibrium; traffic volumes increase to the point that congestion delays discourage additional peak-period trips. If road capacity increases, peak-period trips also increase until congestion again limits further traffic growth. The additional travel is called “generated traffic.” Generated traffic consists of diverted traffic (trips shifted in time, route and destination), and induced vehicle travel (shifts from other modes, longer trips and new vehicle trips). Research indicates that generated traffic often fills a significant portion of capacity added to congested urban road. Generated traffic has three implications for transport planning. First, it reduces the congestion reduction benefits of road capacity expansion. Second, it increases many external costs. Third, it provides relatively small user benefits because it consists of vehicle travel that consumers are most willing to forego when their costs increase. It is important to account for these factors in analysis. This paper defines types of generated traffic, discusses generated traffic impacts, recommends ways to incorporate generated traffic into evaluation, and describes alternatives to roadway capacity expansion.

A version of this paper was published in the ITE Journal, Vol. 71, No. 4, Institute of Transportation

Engineers (www.ite.org), April 2001, pp. 38-47.

Generated Traffic: Implications for Transport Planning Victoria Transport Policy Institute

5

travel (induced travel) which affects overall benefits and costs. Table 1 describes various types of generated traffic. In the short term, most generated traffic consists of trips diverted from other routes, times and modes, called Triple Convergence (Downs 1992). Over the long term an increasing portion is induced travel. In some situations, adding roadway capacity can reduce the network’s overall efficiency, a phenomena called Braess’s Paradox (Youn, Jeong and Gastner 2008). Highway capacity expansion can induce additional vehicle travel on adjacent roads (Hansen, et al. 1993) by stimulating more dispersed, automobile-dependent development. Although these indirect impacts are difficult to quantify they are potentially large and should be considered in transport planning (Louis Berger & Assoc. 1998). Table 1 Types of Generated Traffic

Type of Generated Traffic

Category

Time Frame

Travel Impacts

Cost Impacts

Shorter Route Improved road allows drivers to use more direct route.

Diverted trip

Short term

Small reduction

Reduction

Longer Route Improved road attracts traffic from more direct routes.

Diverted trip

Short term Small increase Slight increase

Time Change Reduced peak period congestion reduces the need to defer trips to off-peak periods.

Diverted trip.

Short term

None

Slight increase

Mode Shift; Existing Travel Choices Improved traffic flow makes driving relatively more attractive than other modes.

Induced vehicle trip

Short term

Increased driving

Moderate to large increase

Mode Shift; Changes in Travel Choice Less demand leads to reduced rail and bus service, less suitable conditions for walking and cycling, and more automobile ownership.

Induced vehicle trip

Long term

Increased driving, reduced alternatives

Large increase, reduced equity

Destination Change; Existing Land Use Reduced travel costs allow drivers to choose farther destinations. No change in land use patterns.

Longer trip

Short term

Increase

Moderate to large increase

Destination Change; Land Use Changes Improved access allows land use changes, especially urban fringe development.

Longer trip

Long term

More driving and auto dependency

Moderate to large increase, equity costs

New Trip; No Land Use Changes Improved travel time allows driving to substitute for non-travel activities.

Induced trip

Short term

Increase

Large increase

Automobile Dependency Synergetic effects of increased automobile oriented land use and transportation system.

Induced trip

Long term

Increased driving, fewer alternatives

Large increase, reduced equity

Some types of generated traffic represent diverted trips (trips shifted from other times or routes) while others increase total vehicle travel, reduce travel choices, and affect land use patterns.