## Abstract

This study considers a model of road congestion with average cost pricing. Subjects must choose between two routes—Road and Metro. The travel cost on the road is increasing in the number of commuters who choose this route, while the travel cost on the metro is decreasing in the number of its users. We examine how changes to the road capacity, the number of commuters, and the metro pricing scheme influence the commuters’ route-choice behavior. According to the Downs-Thomson paradox, improved road capacity increases travel times along both routes because it attracts more users to the road and away from the metro, thereby worsening both services. A change in route design generates two Nash equilibria; and the resulting coordination problem is amplified even further when the number of commuters is large. We find that, similar to other binary choice experiments with congestion effects, aggregate traffic flows are close to the equilibrium levels, but systematic individual differences persist over time.

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## Notes

- 1.
The data are from the 2011 Urban Mobility Report, Texas Transportation Institute. This cost excludes the environmental cost of pollutants produced by the extra fuel and the resulting health cost. Further, “in round numbers, the evidence suggests that each additional ten minutes in daily commuting time cuts involvement in community affairs by ten percent” (Putnam 2000, p. 213).

- 2.
To be consistent with the literature, hereafter we will refer to transportation modes as ‘route’ choices.

- 3.
This support for the paradox played a crucial role in the implementation of an urban pricing scheme in London in the subsequent years.

- 4.
In our experiment, subjects choose between a road, which is assumed to be a congestible route and the metro, which is assumed to be a non-congestible route.

- 5.
- 6.
As anyone who has stood throughout a long metro ride can attest, there is a point after which the cost of the metro increases with greater use intensity. In our analysis, we do not consider this end of the cost spectrum.

- 7.
Notice that

*x*^{∗}is not the efficient number of road users. In fact, the equilibrium number of road users is exactly double the efficient or socially optimal level. Suppose, for example, that*x*(≤*k*) commuters choose the road and the remaining (*n*−*x*) choose the metro, then minimization of the aggregate travel cost of the entire commuter population,*x*(*a*+*bx*)+(*n*−*x*)*t*_{1}, leads to \(x_{1}^{**}=\frac{t_{1}-a}{2b}=\frac{x_{1}^{*}}{2}\) commuters choosing the road. If, on the other hand,*x*(≥*k*+1) commuters choose the road, then the optimal number of road users is \(x_{2}^{**}=\frac{t_{2}-a}{2b}=\frac{x_{2}^{*}}{2}\). The result that the efficient number of road users is half the equilibrium rate is consistent with the notion that when users individually try to minimize their personal cost, they fail to internalize the social cost their decisions impose on others. However, the efficient outcome does not constitute an equilibrium because there is always an incentive for any individual commuter to switch to the road. - 8.
In addition, there are \(C^{n}_{x^{*}-1}=\frac{n!}{(x^{*}-1)!(n-x^{*}+1)!}\) weak pure strategy equilibria in which

*x*^{∗}−1 commuters choose the road and*n*−*x*^{∗}+1 commuters choose the metro. - 9.
The above cited studies examine the impact of population size in minimum-effort coordination games. Ziegelmeyer et al. (2008) is the only study on traffic congestion to consider number of commuters as an experimental parameter. However, they examine the impact of population size in a bottleneck model where commuters have to choose their departure time in order to reach a common destination.

- 10.
Although the congestion parameter,

*b*, has increased, this increase is marginal compared to the large drop in the fixed cost parameter,*a*. We selected these parameter values so that the equilibrium number of commuters on each route is an integer. For our parameter specification, this change entails improved road service for all commuters, as long as the total number of commuters does not exceeds 37. - 11.
The instructions and decision screens refer to choices as Route A and Route B. We use this transportation terminology to help subjects more readily understand the decision they face. However, to prevent personal biases from influencing the results, we do not make any reference to the ‘public transit’ option. The terminology is held constant across all sessions, so framing cannot affect the conclusions regarding comparative static hypotheses that are the focus of this research.

- 12.
Most experiments on traffic (Helbing 2004; Ziegelmeyer et al. 2008; Rapoport et al. 2009) and market entry games (Sundali et al. 1995), with the notable exception of Morgan et al. (2009), provide subjects with the precise description of the associated costs. Duality of equilibrium prediction in our setup creates strategic uncertainty and Helbing (2004) conjecture that improved information could facilitate better “adaptation performance.” Therefore, to create conditions most conducive to coordination and to be consistent with the literature, we provide subjects with the entire cost structure. It is important to bear in mind that even with complete information of the underlying cost, coordination of 10 to 16 subjects on a particular strategy is by no means trivial. Nevertheless, as a robustness check, we did run two additional sessions where subjects were not informed about the entire cost schedule and only knew the minimum and maximum possible travel cost for Route A. There is no statistical difference between the complete and incomplete information treatments (see also footnote 13). The details for the incomplete information sessions are available from the authors upon request.

- 13.
In reference to footnote 12, these results are confirmed even in the no information treatment. When subjects did not have complete information about the underlying cost structure for Route A, the observed number of road users and the total travel were significantly higher than the Pareto dominant equilibrium in AC16 (6.34 and 176.52, respectively) and significantly lower in AC′16 (10.58 and 201.91, respectively).

- 14.
Two-sided Wilcoxon sign-rank test—for AC16:

*n*=8,*z*=2.32,*p*-value = 0.02; AC′16:*n*=8,*z*=−2.52,*p*-value = 0.02; AC10:*n*=12,*z*=2.01,*p*-value = 0.04; CC10:*n*=12,*z*=−2.75,*p*-value = 0.01. - 15.
Despite concentrating on the last 15 periods of each sequence run, we find that the time trend is significant in AC′16 and CC10. In particular, the difference between the observed and the predicted number of road users declines over time in CC10, but it increases over time in AC′16. The results are fairly similar irrespective of whether ln(period) or 1/period is used to control for the time trend. As a robustness check, we also estimate the model using session dummy variables instead of random effects. There are no substantial differences between the fixed effects and the random effects regressions with regards to the equilibrium comparison. However, the fixed effects regressions indicate that treatment ordering may have a minor effect on behavior. Specifically, in these regressions, the sequence dummy is significant in AC′16, where it indicates a difference of one road user between the first and the second sequence. However, since the estimated sequence dummy is negative, this does not affect our finding that the observed number of road users is lower than predicted. All regression results are available as supplementary material on this journal’s website.

- 16.
In Table 4, switching propensity = [probability of road change from one period to the next × number of opportunities for each player × number of players in a session]/total number of possible changes. For instance, in case of AC16, switching propensity=([2×0.2×(1−0.2)]×14×16)/14×16=0.32.

- 17.
This relation is strictly negative in all four cases, but is statistically insignificant for AC′16 (

*p*-value = 0.46). Furthermore, formal non-parametric tests reject the null hypothesis that these correlation coefficients are not different across all treatments. - 18.
The estimation is performed in Maple. The code is available from the authors upon request.

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## Additional information

This research is funded by the National Science Foundation (Grant 0527534). We thank Tim Cason, Oleg Korenok, the editor Jacob Goeree, two anonymous referees, and seminar participants at the University of Siena, Italy, the International Meeting of the Economic Science Association, and the European Meeting of the Economic Science Association for helpful comments.

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Dechenaux, E., Mago, S.D. & Razzolini, L. Traffic congestion: an experimental study of the Downs-Thomson paradox.
*Exp Econ* **17, **461–487 (2014). https://doi.org/10.1007/s10683-013-9378-4

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### Keywords

- Congestion
- Laboratory experiments
- Downs-Thomson Paradox
- Coordination

### JEL Classification

- C91
- C92
- D83
- R40
- R41