Inner-city traffic congestion –
Prisoners’ dilemma and social preferences
Torsten Marner
Institute of Transport Economics
University of Munster
Am Stadtgraben 9
48143 Munster
Germany
phone: +49 (0)251 8322818
email: torsten.marner@iv-muenster.de
Inner-city traffic congestion –
Prisoners’ dilemma and social preferences
Torsten Marner
Abstract
This paper presents an analysis of inner-city traffic congestion in order to evaluate different
instruments and their ability to cope with congestion. It is assumed that users do have
heterogeneous preferences that are differentiated in materialistic, reciprocally motivated and
altruistic preferences. Dependent on the interaction of users with different preferences,
various instruments and their impacts on the model are analyzed. It is shown that the
consideration of social preferences may change the user equilibrium, so that instruments that
are proposed under the assumption of strictly selfish user behaviour need to be scrutinized.
However, if materialistic preferences are predominant, measures such as road pricing are
appropriate instruments. If social preferences are predominant, it may be better not to
implement road pricing.
1. Introduction
Traffic congestion is to be regarded as one of the central problems of inner cities. Traffic
volume systematically exceeds road capacity because of an inefficient use of the restricted
road capacity. Thus, congestion and its well known consequences such as time losses,
environmental pollution and noise arise. As a consequence, significant welfare losses are
originated. For instance for Germany, congestion costs are estimated in a range between 16,35
and 193,46 bill. € (Frank and Sumpf, 1997; Link, 2002; INFRAS and IWW, 2004), dependent
on the method of evaluation. An overview of international congestion costs is given by
Krause (2003), Kossak (2004) and INFRAS and IWW (2004). Especially in inner-cities
congestion costs are significant. Based on INFRAS and IWW (2004), the proportion of innercity congestion costs amounts between 26% and 32% of the whole congestion costs in
Germany. Compared with federal highways and motorways, inner-city streets suffer from
significantly higher marginal congestion costs (Newbery, 1990; Sansom et al., 2001; INFRAS
and IWW, 2004). An overview of investigations of marginal congestion costs is presented by
Mayeres (1996) or Schrage (2005).
Assuming individual rationality and selfish user behaviour inner-city traffic congestion is
presentable as stable Nash-equilibrium. In spite of congestion, users do not have incentive to
change their strategy “use the car” to drive into the inner-city. This causes a classical
prisoners’ dilemma, because all users were better off, if they would use a cooperative strategy
such as using public transport instead of using the car (Rapoport, 1967; Tucker, 1983). In
chapter 2 we model the binary decision problem of the users to choose either car or public
transport to arrive at the inner-city at a particular time. Referring to Schelling (1978) and
McCain (2003) we use a game theoretic n-person framework, but in contrast to Schelling and
McCain we additionally consider capacity limits and bus tracks and trams, as it is usual in
larger inner cities.
Based on Schelling (1978) and Dawes (1980) we show in chapter 3 that our model represents
a prisoners’ dilemma. Under the neoclassical assumption of individual rationality and thus
selfish user behaviour the users do not have any chances to break this dilemma, so that
external interventions such as road pricing are essential. In economic literature road pricing is
regarded as efficient instrument to mitigate congestion (Pigou, 1920; Knight, 1924; Walters,
1961; Eisenkopf, 2002).
The aim of the paper is to evaluate, if an external intervention is essential to battle inner-city
congestion. Fur this purpose it is reasonable to take the heterogeneity of users in account.
Following this basic approach, we introduce heterogeneity of users in chapter 4. Thus, we
reject the assumption of unconditional selfish user behaviour by considering social
preferences such as altruism and reciprocity (Hackett, 1993; 1994; Fehr and Schmidt, 1999;
Bolton and Ockenfels, 2000). In contrast to common economic literature we try to show the
impacts of the existence of social preferences on the user equilibrium. Numerous
experimental investigations document the existence of social preferences in real life (Güth et
al., 1982; Roth, 1995; Güth, 1995; Fehr and Fischbacher, 2002), especially by analyzing
games such as the ultimatum game. Hence, it seems to be recommendable to use these
thoughts for transport economy too.
However, if materialistic preferences are predominant, an external intervention will be
essential. Chapter 5 consequently demonstrates the impacts of different measures to battle
congestion: capacity extensions, road pricing and improvements of public transport (Marner,
2004). If social preferences are predominant, it may be not reasonable to battle congestion by
coercion, because in the case of existing social preferences users may be able to mitigate
congestion without coercion. Chapter 6 analyzes the interrelation between social preferences
and cooperative behaviour, before chapter 7 draws a conclusion.
Numerous game theoretic investigations of traffic behaviour are available so far. An
interesting and actually overview of these analyses is presented by Hollander and Prashkar
(2006).
2. Modelling inner-city traffic congestion
Inner-city congestion can be seen as a logical consequence of users’ individual rationality.
Individual rationality means that users will choose those alternatives with highest expected
utility. Our model shows that at restricted capacity, congestion can be understood as stable
user equilibrium with actually realized traffic volume exceeding capacity (Marner, 2004).
With the aid of figure 1 we analyze the users’ decision to use either the car or public transport
at a particular time starting from a point A outside the inner-city to a point B inside the innercity.i Figure 1 shows the utility of a particular user on his way towards the inner-city either as
car user or as public transport user given a particular allocation of traffic.ii
- Insert figure 1 here The ordinate of the coordinate system documents the users’ respective utility, whereas the
abscissa illustrates the proportion α of the users who choose the car to cover the distance AB
at a particular period to arrive at the inner-city. The continuous line represents the utility of
the public transport users. If α = b (or as well α = 0), his utility will be y. The dashed line
however documents the car users’ utility. Using the car at α = b (or as well α = 0) implies a
utility x > y.
There is a nearly perfect correlation between α and traffic density (Bauriedl and Winkler,
2004). The lower the proportion of car users at AB, the more players uses public transport to
enter the inner-city. Thus, fewer vehicles are needed to transport the fixed number of road
users. Because of the higher capacities of busses or trams a particular number of cars will be
substituted by a bus or a tram. An articulated bus for example is able to transport as much
passengers as some 74 passenger cars (Verkehrsbetriebe STI, 2006), while the passenger car
unit (PCU) of a bus is – dependent on the length – 2,5-3 (Transport Research Laboratory,
1996). According to this the net effect of the substitution of a car by a bus or a tram is a 25 to
30 fold relieve of the streets. Independent from a concrete dimension, traffic density decreases
significantly, so that α can be interpreted as traffic density. Consequently, the strategy of each
player “to use public transport” can be defined as cooperative strategy C, whereas the strategy
“to use the car”, that tends to result in a stimulation of congestion, can be defined as defecting
strategy D.
b can be interpreted as the capacity limit of the considered road. For all α ≤ b road capacity is
not yet exhausted and therefore road users are not confronted with time losses while driving
from A to B. If α > b, the demand for utilization of the road exceeds capacity limit resulting in
congestion. Thus, for all α > b, the users are faced with opportunity costs of time. The graphs
of the utility functions of car users and public transport users run parallel for all α ≤ b.
Nevertheless, the utility of car users exceeds the utility of public transport users in this range.
This assumption is based on two groups of arguments:

cognitive-rationale arguments, that make allowance for trading off perceived costs
and utilities,

emotional arguments, that make allowance for aspects as reputation and feelings
(Steg et al., 2001).
Cognitive-rationale reasons for a higher utility of choosing the alternative “car” in the case of
“no congestion” can be differentiated in flexibility, independence, reliability, accessibility,
speed, convenience and easier use of the car as means of transportation of goods (Hensher and
Stopher, 1979; Recker and Golob, 1976; Jakubowski et al., 1998; Steg, 2003). Examples for
emotional reasons can be craving for sensation, feeling of power, feeling of superiority,
enjoying motoring, feeling of freedom and privacy protection (Näätänen and Summala, 1976;
Lawton et al., 1997).
For all α > b congestion implies different impacts on the utility functions of car users and
public transport users. Although in the case of congestion the utility of the car user decreases
as well as the utility of the public transport users, however we identify three reasons for a
lower mitigation of utility in the case of public transport use than in the case of car use: First,
car users do not have the same options to escape congestion as public transport users often do
have by using non-congested bus tracks. Second, for rail-bound traffic as one part of public
transport there is no congestion problem anyway.iii Third, stress-related mitigation of utility in
case of congestion is assumed to be higher for car users because of their active driving role.
Thus, an increasing α determines a sharper decline of the graphs of the utility functions in the
car users’ case (Marner, 2004).
The different slopes of the two curves imply the existence of an intersection point N. N’s
position is dependent on the capacity limit b and the devolution of the utility curves that are
determined by users’ preferences. Assuming users to be individual rational, N with α = c
emerges as stable Nash-equilibrium (in pure strategies). A Nash-equilibrium exists if and only
if “no player has incentive to deviate from his strategy given that the other players do not
deviate” (Rasmusen, 1994).iv If N is therefore identified as stable user equilibrium, there will
be no incentive for any user to change, neither from public transport to car nor vice versa.
Thus, the proportion of car users is defined α = c, the proportion of public transport users is
defined  = 1-c and the resulting utility is equal for every user z.
Given the above mentioned and absolutely realistic assumptions, the distribution (α,) of
users on the regarded route AB converges independent from the actual proportion α towards
intersection point (c, 1-c). This can be explained as follows: Assume α to be c with the actual
equilibrium point (c, 1-c). Is there incentive for any player to change her strategy? If e.g. a car
driver wishes to use public transport from now on, the proportion of car users α changes
marginally to α’ = c -  with  symbolizing the proportion of exactly one new car driver of all
users on the regarded route. Consequently, at the point α’ = c - , for the new public transport
user the continuous line of the utility function is valid, which however at α’ = c -  proceeds
below the dashed line so that at α’ = c -  each player has incentive to use the strategy “car”.
Thus, the regarded player would do better returning to the starting point N with at α = c.
Hence, there is no incentive to change the strategy. In analogy, at user equilibrium N there is
no incentive for any public transport user to change her strategy. N is not only a stable Nashequilibrium but also a unique equilibrium, because all thinkable distributions converge
towards N. Each initial distribution below α = c, e.g. at α’ = c - 2, α’ = c - 3 etc. a fortiori
implies incentive for a change from public transport use to car use and each initial distribution
above α = c, e.g. at α’ = c + 2, α’ = c + 3 etc. a fortiori implies incentive to change from car
use to public transport use (Marner, 2004). Thus, in the case of n users, equilibrium point N
with (α,) = (c, 1-c) is realized.
3. The prisoners’ dilemma
For each user the existence of equilibrium point N implies a utility of z < y, so that in the case
of mutual choice of the cooperative strategy “public transport use” the utility for each of the
users would be higher. Assuming individual rationality, at least a proportion α = c of users
choose the defective strategy “car use”, resulting in an undesirable outcome for each user and
therefore in an inefficient equilibrium N with (α,) = (c, 1-c). This inefficiency can be
characterized as an alleviated multi-person prisoners’ dilemma. The prevalent definition of a
multi-person prisoners’ dilemma consists of two conditions:
a) Each user benefits more from a socially defecting choice, i.e. using the car to arrive at
the inner-city, than from a socially cooperating choice, i.e. using public transport,
independent from other users’ strategies and
b) all users are better off if all choose the cooperative strategy than if all choose the
defective strategy (Dawes, 1980).
The mathematical structure is as follows: Each player chooses one of two available strategies:
Defecting (D) or Cooperating (C). Each players’ utility depends on the own strategy and on
the number of players choosing strategy D or C. Let D (β) be the utility of the users who
defect in the considered multi-person game with α being the proportion of players who defect
and β being the proportion of players who cooperate. Let C (β) be the utility of the users who
cooperate when β is the proportion of players who cooperate (including themselves). A
prisoners’ dilemma is characterized by the following inequalities:
c) D (β) > C (β+ε) und
d) D (β=0) < C (β=1).
Inequality c) means that in the case of parallel utility curves defecting is a dominating strategy
for all users. Inequality d) indicates that collective defection confronts all players with less
preferred results as in the case of collective cooperation (Dawes, 1980).v
However, in our case of non-parallel curves with an interception point N, condition a) as well
as inequality c) is not fulfilled, because a proportion of users is better off by choosing strategy
“cooperate”. Hence, by definition, the players are not confronted with a prisoners’ dilemma,
but in fact, they are. The only difference between the case of parallel lines of utility, that
satisfies the conditions of a prisoners’ dilemma, and the case of intersecting curves, that does
not satisfy the conditions of a prisoners’ dilemma, is that in the first case, at the equilibrium
no player uses the cooperative strategy, but in the second case, at the intersection point N,
some players do use the cooperative strategy. Both equilibriums are inefficient and in both
cases the players’ utility would be greater than in the case of collective cooperation. The only
difference is the extent of the dilemma. With intersecting curves, the player’s utility is greater
than in the case of parallel curves, but by all means less than the collective cooperative utility
(Schelling, 1978). So in fact, our binary decision problem can be interpreted as an alleviated
multi-person prisoners’ dilemma.
Based on Schelling’s definition of a prisoners’ dilemma with an inefficient potential
equilibrium, our binary decision problem must be called a prisoners’ dilemma, because it
exactly satisfies the following conditions: “There is a unique equilibrium with some choosing
…. [D], some …. [C]. …All would be better off if all chose…[C]. The collective total would
then be a maximum. The collective total would be even higher if some still choose … [D],
everybody still being better off than at the equilibrium, but not equally so.” (Schelling, 1978).
The existence of a prisoners’ dilemma implies an inefficiency that calls for measures to
change the incentive structures of the players. Social dilemmas in general are defined in terms
of social payoff structures. Thus, the simplest method for stimulating cooperative behaviour is
to change those payoff structures. This is possible by providing incentive to use the
cooperative strategy, in our case “use public transport”. Incentives can be differentiated in
internal and external incentives to be cooperative.
Internal incentives are based on social preferences of the users. Those can be differentiated in
aspects such as moral, normative concerns, fairness, reciprocity, inequity aversion and
altruistic concerns (Hackett, 1993; 1994; Fehr and Schmidt, 1999; Bolton and Ockenfels,
2000). By providing internal incentives to the players to be cooperative equilibrium may
convert from an inefficient to an efficient equilibrium and the prisoners’ dilemma breaks up.
To ensure cooperative behaviour in dilemma situations the often used assumption of
completely rational and selfish behaviour must be rejected. This is true for a) the players have
to contemplate the situation and understand the complexion of the dilemma, so that possibly
existing social preferences as well as external interventions can influence the players’
decisions and b) the players must have beliefs that the other players will not defect, because
they are able to anticipate a greater utility in the case of collective cooperation than in the case
of non-cooperation, albeit individual rationality suggests defective behaviour (Dawes, 1980).
If social preferences exist in an adequate proportion, no coercion by means of political
interventions will be necessary to procure an efficient equilibrium.
External incentives are based on the idea, that individuals are not able to break a dilemma
situation without external intervention. Thus, a dilemma could legitimate a political measure.
External authorities could implement a political measure to correct existing but inefficient
incentive structures by coercion.vi There are three categories of reasonable measures to break
a dilemma situation: a) providing incentives not to defect, i.e. implementing measures that
lower the attractiveness of using the car, b) providing incentives to cooperate, i.e.
implementing measures to strengthen alternatives to using the car, e.g. improving public
transport, c) capacity extensions (Marner, 2004).
Both, internal and external incentives tend to result in a change of social payoff structures and
therefore are appropriate measures to break up the dilemma. The efficacy and adequacy of the
nature of the incentives are dependent on the characteristics of the users. So, in contrast to
many theoretical models, we absolutely realistic have to assume that users are heterogeneous.
4. Heterogeneity of users
The conventional assumption referring to economic behaviour is that there is no heterogeneity
of users concerning their preferences. In fact, users do have different preferences, so that the
assumption of collective selfish behaviour, that is caused by unconfined materialistic
preferences, must be rejected (Hackett 1993;1994; Fehr and Schmidt, 1999; Bolton and
Ockenfels, 2000; Poulsen and Poulsen, 2002). Not only real life suggests evidence for the
existence of social preferences. Charity or tipping the waiter in face of the knowledge, that
you will never see him again, so that you cannot expect a return flow, can be interpreted as
consequences of social preferences. But also scientific evidence is given by numerous
experimental investigations (Güth et al., 1982; Roth, 1995; Güth, 1995; Fehr and Fischbacher,
2002), so that it is reasonable to consider heterogeneity of users concerning their preferences.
We distinguish two categories of preferences (Fershtman and Weiss, 1998; Guttmann, 2000):

rational, selfish preferences and

social preferences.
Rational preferences are to be defined as follows: a rational and selfish user chooses the
alternative with highest expected utility for him, completely independent from other players’
utilities. For users with rational, selfish preferences, exclusively materialistic aspects are
relevant for their utility functions, so that those users seek to maximize their monetary return.
Thus, we restrict rational and selfish preferences to materialistic preferences (Poulsen and
Poulsen, 2002).
In the following we restrict social preferences to contain the following types of preferences:
altruism and reciprocity. Altruism may be defined as unconditional niceness. Independent
from the behaviour of other users, altruistic users always tend to be kind (Hirshleifer, 1977;
Dawes, 1980; Becker, 1982). Altruism is to understand as the phenomenon that the welfare of
other users positively influences the utility of the regarded altruistic user (Axelrod, 2000).
Reciprocity may be defined as “conditional niceness” (Poulsen and Poulsen, 2002). Basically
these users are kind, until they face unkind behaviour of others, so that their initially kind
behaviour turns to unkind behaviour. Thus, reciprocity means to be kind towards players, who
are expected to be kind and to be unkind towards players, who are expected to be unkind
(Axelrod, 2000, Poulsen and Poulsen, 2002). Consequently, the possible outcomes of the
existence of reciprocally motivated users are ambiguous and depend on the composition of the
user group (Ehrmann et al., 2006).
As shown above, we restrict the analysis to three relevant types of preferences: the
materialistic preference type (m), the reciprocator preference type (r) and the altruist
preference type (a) (Poulsen and Poulsen, 2002). Let a simplified prisoners’ dilemma have the
following monetary payoffs that are illustrated by figure 2.
-
Insert figure 2 here –
Let A > 1 > 0 > b be true and C and D stand for “cooperate” and “defect”. Individual
rationality leads players exclusively to care about their own monetary utility, so that outcome
0 is realized, that is worse than 1.
Different preference orderings generate different user behaviour. From there preference
orderings, we can derive following best replies. Best replies document the decision players
choose, given that opponent players choose either C or D. For players with materialistic
preferences (m), D is a strictly dominant strategy. For players with altruistic preferences (a), C
is strictly dominant. Reciprocally motivated players choose C, if they expect opponent players
to be cooperative and D, if they expect them to be defective. Here, the outcome (C,C) is
strictly preferred to other outcomes with (C,C) > (D,C) > (C,D) > (D,D). You have to keep in
mind, that these simplified accomplishments with dominant strategies refer to the simplified
illustration of figure 2 to simplify the understanding of the impact of social preferences.
Materialistic preferences (m) and altruistic preferences (a) are non-critical for the analysis,
because m-players generally use strategy “defect” and a-players generally use strategy
“cooperate”, but the choice of reciprocally motivated players remains ambiguous, because it
depends on the decisions of the unknown opponents. Thus, in the case of reciprocally
motivated players, the interaction of players is of vital importance. There are different
determinants that explain the impacts of interaction of heterogeneous players. First, the
proportions of players with m-, r- and a-preferences play an important role and second, the
beliefs of the reciprocally motivated users are important. If they expect to meet rather
altruistic or reciprocally motivated than materialistic motivated players, they will tend to use
cooperative strategies and therefore, the proportion of cooperative strategies increases and the
user equilibrium tends to move towards a non-congested situation. Figure 3 illustrates the
impacts of the consideration of social preferences by showing the interaction between users
with heterogeneous preferences in an extended game.
-
Insert figure 3 here –
The tree of figure 3 documents, that for selfish players D is a dominant strategy, so that they
always use the defective strategy, independent on their belief of the composition of the user
group. In contrast to selfish players, altruists do have a dominant strategy C. Reciprocally
motivated players do not have dominant strategies. If they are confronted with a defective
strategy, they will react by using a defective strategy in the next move, if they are confronted
with a kind player choosing a cooperative strategy, they will be kind too and choose a
cooperative strategy in the next move, too.
Starting with a materialistic player, we see, that in the first move of our game, exclusively
defective strategies are used (3 out of 3). The second move consists of three subgames with
completely nine best replies. In the left subgame the m-player makes the choice, in the middle
subgame it is the r-player and in the left subgame it is the a-player who makes the choice. In
the second move, in 66, 67% (six out of 9) of the cases the respective player’s best reply is D.
In the third move the proportion of D as best reply decreases to 55, 56% and in the fourth
move to 51, 85%, in the fifth move to 50, 62% and so on. The limiting value, still for starting
with a materialistic player, is 50%. If you start with an altruistic player at root of the tree, the
left and right subgames remain unmodified and only the middle subgame changes. But even
the middle subgame tends to a limiting value of 50% for n → ∞. Thus, assuming a rectangular
distribution of preferences, independent from the starting point, 50% of the strategies are
cooperative. This can be interpreted as an evidence for the functioning of cooperation, even
without coercion.
The following chapters 5 and 6 take up these thoughts and assume that a dominance of selfish
or materialistic preferences causes the necessity of coercion, because in the case of a higher
rate of defecting the user equilibrium tends to be rather at a congested state, while with a
dominance of social preferences coercion rather should be rejected, because natural
cooperation tends to be higher, so that the user equilibrium is rather near the capacity limit.
5. Rationality, selfish behaviour and instruments to minimize congestion
Assuming a dominance of rational and selfish user behaviour, the efficacy of social
preferences concerning users’ behaviour is negligible. Thus, internal incentives do not have
any relevance for the user equilibrium. Consequently, there is a need to revert to external
incentives by means of political intervention. In this context, road pricing can be seen as the
most efficient instrument to mitigate welfare losses caused by congestion (Eisenkopf, 2002).
Two central concepts concerning road pricing are user equilibrium and system optimum of
traffic flow distribution in road networks. User equilibrium means that all users try to
minimize their individual travel costs, whereas system optimum means, that users cooperate
to minimize total network cost. The difference between user equilibrium and system optimum
causes a welfare loss. Road pricing and other instruments to battle congestion have the aim to
minimize this difference.
The effectiveness of an implementation of road user charges corresponds to the fundamental
principal of marginal-cost pricing, which suggests, that at congested roads users have to pay
an optimal toll equal to the difference between marginal-social cost and marginal-private cost
with the objective of maximizing social surplus (Dupuit, 1844; Pigou, 1920; Knight, 1924;
Walters, 1961). By internalizing the external effects of congestion by means of optimal
pricing, incentives for an efficient use of the scarce good road infrastructure are provided.
Thus, density decreases, until capacity utilisation is welfare optimal and the welfare loss
caused by the external effect “congestion” is avoided.
In economic literature it is common sense, that first best solutions are not feasible, so that
numerous second best solutions are proposed (May and Milne, 2000; Sumalee, 2004; Yang
and Huang, 2005). Albeit those solutions do not generate efficiency like first best solutions
do, they at least are more feasible than first bests because of lower transaction costs and major
practicability. Furthermore, do not generate considerably higher efficiency than conveniently
implemented second bests appear to do (Proost and van Dender, 1999).
Empirical evidences from the implementation of the London City Congestion Charge
(Marner, 2004; Eichinger and Knorr, 2004; Prud’homme and Bocajero, 2005) and
Stockholm’s recently successfully completed trial (Armelius and Hultkrantz, 2005; Abboud
and Clevstrom, 2006) demonstrate, that second best solutions my be successful for traffic
control purposes. Already the experiences of the pioneer of road pricing, Singapur with its
electronic road pricing ERP can be evaluated predominantly positive (Olszewski and Xie,
2005). Especially after these positive experiences and after the recently implemented tolling
of the inner-city of Bologna it is not amazing, that numerous large cities such as Paris,
Brussels, Oslo, Milan or Helsinki are concerned with concrete plans for an implementation of
a pricing scheme for the congested inner-cities (de Palma et al. 2006a; de Palma et al. 2006b;
Vold, 2006; de Palma/ Lindsey, 2006; Proost/ Sen, 2006; o.V., 2006a; o.V., 2006b). Adequate
overviews of the concrete theory of urban road pricing are presented by for instance May and
Nash (1996), Arnott (2001) or Yang and Huang (2005).
In the regarded model framework of figure 1, road pricing causes the following, whether by
optimal pricing or second best pricing: Users are assumed to be heterogeneous and therefore
have different willingness’s to pay. Thus, a differentiated analysis of user groups is required.
Those users with a willingness to pay exceeding the toll continue using the defective strategy
by using the car and are confronted with two effects: first, their utility is mitigated by the
increasing price of using the road infrastructure. Second, users’ utility will increase by saving
opportunity costs of time, if congestion is mitigated.
If net effect is positive, so that the average of the regarded category of users is better off or
after implementing pricing, predominantly depends on the amount of the toll and the
preferences of the users. As the aim of pricing is the mitigation of congestion, it has to be
guaranteed, that the amount of the toll is high enough to burden the average of the user group.
In this case, the average incentive to use the car decreases. Figure 4 illustrates the impact of
pricing by shifting the utility curves of the car users downward.
- Insert figure 4 here –
Users with a willingness to pay that is lower than required to cover the user price – that could
be interpreted as the sum of individual conventional costs and the toll – will be crowded out
by either making them change their strategy to be cooperative or putting them out of the
market. Exclusively considering the problem of congestion, this development is enjoyable,
because hence  decreases and therefore the equilibrium point N of figure 1 is shifted towards
a non-congested state. A sufficient toll tends to activate a crowding out of car users and
therefore enforces cooperative behaviour by thenceforward at least a few users will have
incentive to use public transport. Thus, similar to the idea of internalisation of external effects
by Pigou, car users are crowded out, traffic density decreases and the congestion-caused
welfare loss decreases too.
In Figure 4, the thick drawn, dashed line illustrates the utility of a car user after
implementation of pricing, while the thin drawn, dashed line illustrates his utility before
pricing. It is shown, that road pricing shifts the proportion of car users from c0 to c1 with a
significant decrease of congestion, equivalent to the minimization of the distance c-b. c0-b is
significantly greater than c1-b, so that the Nash-equilibrium changes from N0, that is true for
the non-pricing case, to N1, that is true for the pricing case. The prisoners’ dilemma is
minimized, because the difference between z1 and y is significantly minor to the difference
between z0 and y. Thus, basically the implementation of a pricing system can be evaluated as
an appropriate measure to solve the inner-city congestion problem. However, choosing an
appropriate amount of the toll that is dependent from several determinants such as politicoeconomic aspects (Evans, 1992; Gomez-Ibanez, 1992; Lehmann, 1996; Jones, 1998) and
individual urban frameworks (Marner, 2004) is of great importance. In general, for an
adequate amount of toll is true, that the toll has to be exactly so, that the car users exactly
from this amount on have incentive to change their strategy. Figure 5 shows the impacts of an
optimal charge on the users’ strategies simplified for two players.
-
Insert figure 5 here –
First matrix illustrates the utilities of the players in the case without charging. Assuming a) an
ordinal ranking T > R and P > S and b) 2R > T+S > 2P, the Nash equilibrium is the strategy
combination (car use, car use) with an utility of 2P for both users, that is lower than 2R, that
could be achieved by a collective rational choice of the strategy (public transport use, public
transport use). This implies a stable prisoners’ dilemma. Second matrix illustrates the case
with optimal charging. A price topt causes a more expensive use of the regarded route AB for
the car users with an initially mitigated utility. Consequently, for a part of users T-topt < R and
P-topt < S is true. So, this is true for those users with their willingness to pay not covering the
increase in prices. Those users do not only have a modified payoff structure as every user
does have, for them it is furthermore individual rational to change their strategy and use
public transport after pricing. They will be crowded out, so that  decreases and converges to
capacity limit b. For all car users, topt causes an average mitigation of utility, so that the
considered utility curve at figure 4 is shifted downward. It must be pointed out, that the
positive welfare effect that is caused by optimal pricing will not be overcompensated by
prohibitive high transaction costs or system costs that could generate a governmental failure
(Prud’homme and Bocajero, 2005; Baum et al., 2005).
Further actual discussed alternatives to reduce inner-city traffic congestion are a) capacity
extensions and b) improvements of public transport.
a) capacity extensions
Capacity extension in general causes an increase of supply of street space and tends to reduce
congestion by shifting the capacity limit b of figure 1 rightwards (Small et al., 1989).
However, there are several significant restrictions to capacity extensions:

There are substantial urbanistic, political and spatial barriers, so that capacity
extension would be either impossible or at least prohibitive expensive.

You have to consider – perhaps in the context of cost-benefit-analyses –
substantial investment costs.

Capacity extensions imply induced traffic (DeCorla-Souza and Cohen, 1998;
Marte, 2005), so that, following Downs (1962) and his fundamental law of
congestion extension of infrastructure coercively generates an increase in
volume of traffic. Induced demand will cause a new short-term congestion, if
capacity of the road network is extended (Downs, 1962). Figure 6 illustrates
the consistence of this idea with the implications of our model.
-
Insert figure 6 here –
Capacity extension causes a shift of the capacity limit b0 to b1. Consequently, the thin drawn
utility curves are shifted rightwards, so that in the case of capacity extension the thick drawn
curves are true. At (c1,z1) a new stable Nash-equilibrium (in pure strategies) N1 is originated,
but however, c1 and b1 are equidistant to c1 and b0, so that congestion remains constant. The
new equilibrium N1 implies z1 = z0, so that capacity extension does not influence the utility of
the concerned users. In our model, induced traffic (c1-c0) follows Downs’ prediction and
compensates capacity extension that remains inefficient. Indeed, you have to take into
consideration, that infrastructure investments generally induce significant positive welfare
effects (Aschauer, 1989; Gramlich, 1994; Baum and Behnke, 1997; Baum and Kurte, 1999;
Vickerman, 2000, Hartwig and Armbrecht, 2005).
b) improvements of public transport
We investigate two categories of an improvement of public transport that have different
impacts on the utility curves of our model. First, we analyze substantial improvements of the
capacities of busses by implementing new routes, higher frequencies or larger busses and by
supplying an improved service, for instance by implementation of a 24 hour-service. In our
model, these measures cause a parallel translation of the utility curves of the public transport
users. Second, we analyze an improvement of bus lanes or a consideration of tram or subway
systems. These measures cause a more even decline of the regarded utility curve. Figure 7
illustrates the impacts of both cases.
-
Insert figure 7 here –
After consideration of an improvement of public transport, at (c2, z2) a new stable Nashequilibrium N2 is originated. Compared with N1, N2 converges to the capacity limit b, so that
the dimension of the social dilemma significantly decreases. Empirical evidences, i.e. of the
London case, show that improvements of public transport can be seen as appropriate
accompanying measures, if the revenues of road pricing are used for public transport (Marner,
2004; Santos and Bhakar, 2006). In this context, Armelius and Hultkrantz (2005) show that in
larger cities with a greater significance of public transport, such as Stockholm or London, it is
more unproblematic to compensate users by using the revenues for public transport. The
aggregate impact of the measure package “road pricing plus improvement of public transport”
is documented by figure 8.
-
Insert figure 8 here –
The thin drawn curves illustrate the status quo that is shown in figure 1, while the thick drawn
curves show the impact of the package “road pricing plus improvement of public transport”.
Equilibrium converges from c0 to c3 = b and implies – dependent on the intensity of
intervention – a non-congested situation. In the case shown in figure 8 an intensive
intervention breaks the dilemma. However, at least, we identify a relocation of the
equilibrium towards a non-congested situation. The utility of the users increase from z0 to z3,
that is however lower than x, but higher than y. The dilemma is solved, congestion decreases
or eventually disappears and the congestion caused welfare loss is eliminated or at least
lowered.
6. Social preferences and cooperative behaviour
Assuming a dominance of social preferences, road pricing or other instruments that could be
implemented by coercion, will be redundant. Given the assumption, that a significant
proportion of preferences are social, the users will make the user equilibrium shift from N0 to
N3, as figure 9 shows.
-
Insert figure 9 here –
The realization of N3 implies a utility of z4 for the users and therefore a decreasing welfare
loss. However, dependent on the degree of social preferences, congestion may not be
completely eliminated, but following arguments may speak against an intervention: First,
congestion is a temporary phenomena. The route AB is not congested permanently and
furthermore you have to consider temporal and spatial relocation, so that second, a cost-based
reduction of traffic demand would generate welfare losses. In economy it is common sense
that traffic generates substantial welfare effects (Aschauer, 1989; Gramlich, 1994; Baum and
Behnke, 1997; Baum and Kurte, 1999; Vickerman, 2000; Hartwig and Armbrecht, 2005).
Thus, an optimal congestion is existent at b + γ. γ albeit tends to be small and depends on the
structure of traffic flow and tends. The existence of an optimality of congestion means, that
the occurrence of social preferences could be adequate for a minimization of congestion.
Third, the system costs and transaction costs of an intervention have to be taken into account.
Prohibitive high costs, as identified for the London case (Prud’homme and Bocajero, 2005),
involve the danger of a state failure.
There are two further effects that influence the impact of social preferences. First, the
analyzed daily problem of intermodal choice may be seen as a typical repeated game. The
regarded decision problem recurs day by day. Repeated games tend to favour cooperative
behaviour, because players learn to be trustful and eventually learn that the game not only
consists of materialistic players, but of a mix of materialistic players and players with social
preferences. Thus, in repeated games the probability of learning that collective behaviour
tends to increase the utilities of all players is much higher than in one-shot games (Axelrod,
2000). Compared with this, anonymity is one important determinant of defective strategies.
The probability of defecting is much larger in an anonymously environment. On congested
routes, n → ∞ players are involved in the game, so that the degree of anonymity is very high.
At this you have to differentiate between nearly permanent congestion at large cities such as
London or Paris and probably temporary congestion at small cities. In small cities anonymity
is significantly lower, because the probability that persons know each other is substantially
higher. However, anonymity tends to favour defective behaviour, because especially in small
cities with n being significantly lower than in larger cities each player rather has
reinforcement control over other players (Dawes, 1980). Users in smaller cities with a lower
degree of anonymity rather can punish other users for defective behaviour than in large cities
with a degree of anonymity near 1. If users have the chance to punish defective users, they
can influence others’ choices. For this case, Rapoport (1967) shows that prisoners’ dilemma is
not a dilemma anymore.
7. Conclusions
The model documented, that users are captured in a prisoners’ dilemma, if all users are
assumed to be completely selfish. Allowing for heterogeneity, it is shown, that you have to
differentiate users in players with materialistic, altruistic and reciprocally motivated
preferences. If materialistic preferences dominate the regarded user group, instruments to
provide users with incentive for an efficient use of the inner-city roads are essential. It is
shown that road pricing in combination with improvement of public transport then is an
appropriate measure to battle congestion. Capacity extension is not able to mitigate
congestion because of the expected induced traffic. If social preferences dominate, laissezfaire may be the best policy, because social preferences cause users to behave cooperative, so
that the prisoners’ dilemma tends to break. If social preferences are predominant, it will be
furthermore easier to strengthen the consciousness of the users for cooperative behaviour. It is
shown that the interaction of users plays an important role and that it is dangerous to ignore
the heterogeneity of users and commend policies that are solely based on unconditional selfish
users.
What remains nebulous is the actual proportion of social preferences. However, numerous
studies suggest a significant proportion of social preferences in real life, especially by
analyzing ultimatum games (Güth et al., 1982; Rabin, 1993). Further experimental research
could investigate social preferences especially in the transport sector with its specific
characteristics such as anonymity of the users and repeated games. The analyzed daily
problem of intermodal choice may be seen as a typical repeated game. While repeated games
tend to favour cooperative behaviour, the fact, that especially in larger cities the users do not
know each other and therefore the degree of anonymity is high, tends to favour defective
behaviour. Thus, it would be interesting to learn more about the actual proportions of social
preferences.
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Captions to illustrations
Fig.1: The model – Inner-city traffic and the prisoners’ dilemma
Fig.2: Payoffs in a simplified prisoners’ dilemma game
Fig.3: Interaction between users with heterogeneous preferences
Fig.4: The model – Impacts of road pricing
Fig.5: Impacts of an optimal charge in a simplified 2x2-game
Fig.6: The model – Impacts of capacity extensions
Fig.7: The model – Impacts of an improvement of public transport
Fig.8: The model – Impacts of the measure package “road pricing plus improvement of public
transport”
Fig.9: The model – Impacts of the consideration of social preferences
Figure 1: The model – Inner-city traffic and the prisoners‘ dilemma
utility of the user either as car user or as public transport user
x
utility of the car user
y
utility of the public
transport user
N
z
0%
b
Source: Own illustration
c
proportion of
100% car users
Figure 2: Payoffs in a simplified prisoners‘ dilemma game
C
D
C
1
b
D
a
0
Source: Poulsen/Poulsen (2002), p. 6.
Figure 3: Interaction between users with heterogeneous preferences
m
D
D
m
D D
D
Move 3: D
vs. C 15:12
m
r
a
D D D DDD C CC
m r a
r
D D
m r a m r a
m
D DD
m r a
D
a
C C
D
r
a
DDD C CC
m r a
Move 4: D vs. C 42:39
Source: Own Illustration
Move1: D vs. C 3:0
m r a
m
r
Move 2: D
vs. C 6:3
C
a
C
DDD CC C CC
m r a
m r a
m r a
Figure 4: The model – Impacts of road pricing
utility of the user either as car user or as public transport user
x
utility of the car user
y
utility of the public
transport user
N1
z1
N0
z0
0%
b
c1
Source: Own illustration
c0
100%
proportion of
car users
Figure 5: Impacts of an optimal charge in a simplified 2x2-game
without
road
pricing
public
transport
use
car use
with
road
pricing
public
transport
use
car use
public
transport
use
R, R
S, T
public
transport
use
R, R
S, T-topt
car use
T, S
P, P
car use
T-topt, S
P-topt, P-topt
Source: Own illustration
Figure 6: Impacts of capacity extensions
utility of the user either as car user or as public transport user
x
utility of the car user
y
utility of the public
transport user
N1
z0 = z1
N0
0%
b0
b1
Source: Own illustration
c0
c1
proportion of
100% car users
Figure 7: Impacts of an improvement of public transport
utility of the user either as car user or as public transport user
x
N2
utility of the car user
z2
utility of the public
transport user
y
N0
z0
0%
b
c2
Source: Own illustration
c0
100%
proportion of
car users
Figure 8: Impacts of the measure package „road pricing plus improvement of public transport“
utility of the user either as car user or as public transport user
x
N3
utility of the car user
z3
y
utility of the public
transport user
N0
z0
0%
b = c3
Source: Own illustration
c0
proportion of
100% car users
Figure 9: The model – Impacts of the consideration of social preferences
utility of the user either as car user or as public transport user
x
utility of the car user
N1
z4
y
utility of the public
transport user
z0
N0
0%
b
c4
Source: Own illustration
c0
proportion of
100% car users
To simplify the analysis we create a binary decision problem – car versus public transport - by ignoring less
relevant alternatives as cycling, taxi or walking.
ii
The model assumes a particular allocation of traffic of a sufficient large number of vehicles because otherwise
a high proportion of car users could generate equilibrium beneath capacity limit, particularly in the case of just
one car on the network. This assumption is insofar non-critical as the violation of the assumption will not result
in an economical problem.
iii
Even if the proportion of rail-bound traffic is low, it at least will tend to result in a reduction of the mitigation
of utility.
iv
See Binmore (1992), who identifies a Nash-equilibrium by the players mutually playing best responses to the
strategies of the other players.
v
See Tucker (1983) for the first technical formulation of the 2-person prisoners’ dilemma. See Hamburger
(1973) for a technical analysis of the multi-person prisoners’ dilemma.
vi
There are many thinkable possibilities to provide incentives to cooperate, by coercion as well as without
coercion, that are not analysed in this framework, ranging from the constitution of a “Leviathan” (Hobbes, 1651,
1947) to public appeals.
i