I live in Drotningsvik. My everyday routine for several years now is to take a bus to the city, and back again. Sotrasambandet, the highway to Sotra/Øygarden – Bergen, and the Sotra bridge affects my daily life, negatively, as it suffers from heavy congestion nearly every rush hour. The municipality has plans to alleviate this with a new, larger bridge and an expanded route.
The goal with this blog post is to essentially explore the old and the new in regards to network game theory, and find the drivers strategies to both, with focus on the two bridges.
Current and planned state of the Sotra Bridge
The Sotra bridge is a two-lane bridge built in 1971. It is built to accommodate up to 12 000 cars a day at average, however today’s count goes up to 27 000 (Sørensen, Fjell Municipality, 2016, p. 13).
The planned route and bridge runs close to the old one; the difference is that it is a four-lane route, likely capable to hold a higher number of vehicles. The new bridge will run right next to the old one. We can assume that it will be able to hold twice as many cars compared to the old bridge.
The Game Setup and Strategy Given the Old
Let’s establish some shared concepts in the game:
- The game takes place during rush hour, with travel times to reflect it.
- We’ll assume all players is from Bergen and wants to head to Straume.
- Everybody wants to minimise their traveling time, as their payoff.
- The X number of players is 1 400. This comes from an analysis done by Statens Vegvesen (Public Roads Administration) who measured the average number of vehicles passing through a point on the route. We are using the number of cars given rush hour, at 16:00. (Statens Vegvesen 2010, p.14 fig 9).
In reality there is no strategy when it comes to accessing Sotra/Øygarden by car today. There is only one way; driving across the Sotra bridge, meaning all drivers have only this one strategy. However this lets us measure travel times given that all drivers chooses a single path, giving us more accurate max travel times.
Finding appropriate travel times was a challenge. According to Google Maps, the route from Festplassen to Straume Terminal is est. 16 km.
- From Bergen – Storavatnet is 10 km. The analysis by Statens Vegvesen states Storavatnet is often congested (p. 15). From data provided by TomTom Traffic Index, it is shown most congested spots is relatively close to Storavatnet and throughout to Sotra bridge and Straume. We can simplify and say Bergen – Storavatnet is insensitive to congestion and takes 7,5 minutes, given that drivers follow the speed limit of 80. The rest of the route is congested.
- According to Statens Vegvesen’s analysis, during rush hour the drivers average speed is about 25-30 km/h. (Statens Vegvesen 2010, p. 15. fig.10). If the average speed here is 25 km/h, the remaining 6 km will take about 14.5 min, making a total of 22 minutes, which matches Google Maps’ own estimate. Now if we say the max number of drivers take 14,5 minutes to drive two 3 km paths, divided by Storavatnet- Sotra bridge, Sotra bridge – Straume, then 1400/14, 5 = ca. 97 and 1400/7,25 = ca. 193. These are our travel time estimates: X/193 for both paths.
Then the resulting graph should be:
Strategy given the new
With a new bridge planned there is now a choice between what brigde to take.
Unfortunately we have no statistics to figure out its travel time, but we know it has twice as many lanes (Sørensen, Fjell Municipality, 2016, p. 6). We can say it is non-congested, and give it a stable and more general travel time. The new road is supposed to have a speed limit of 80 km/h (Statens Vegvesen, 2016, p. .2 1.3.1). The new route is 9.4 km from Storavatnet to Kolltveit. (Sørensen, Fjell Municipality, 2016, p. 5). Google Maps states the old road is est. 9 – 9.3 km. Since the roads are so close to each other in length and placement, we can say the length of the old and new road is the same, including for Straume-Storavatnet, for 6 km.
So we again have two 3 km paths Storavatnet- Sotra bridge, Sotra bridge – Straume, and given the drivers pass a non-congested route at 80 km/h, it will take them 2.25 minutes for both paths.
These assumptions makes this resulting graph:
Now finally we can apply some game theory and find out possible strategies. We are just going to find potential Nash equilibriums and dominant strategies, if they exists.
- With this graph we have a path from Bergen to Straume that has no congestion whatsoever. It takes 12 minutes using the new path no matter how many drivers picked the new strategy ( 7,5 + 2,25 + 2,25 = 12).
- If all players takes the route to old-bridge – Straume, players will have an incentive to switch to the new route, for sticking to current strategy takes them 22 minutes, instead of the 12 minutes it takes to drive the new route.
- If 700 takes the old route and the other 700 drive the new route, it will take 14.7 minutes to drive the old route (700/193= 3.6 ->7.5 + 3.6 + 3.6 = 14.7). There is still incentive to switch over to the new route, for a better payoff of 12 minutes.
- When there are exactly 434 drivers using the old route, we have an equal payoff for both routes. (434/193 = 2,25 -> 7.5 + 2,25 + 2,25 = 12). This means a balance of 434 drivers using the old path, and 966 drivers taking the new path is a Nash equilibrium, drivers on either strategy has no incentive to switch. Drivers on the new route will not switch to the old route because it will become more congested and take more time, worsening their payoff. New drivers from Bergen will pick the old route if there are less than 434 drivers on it, because there is a better payoff to gain.
So we have in this game found a Nash equilibrium. One thing to note is that the introduction of the new road and a Nash equilibrium has improved travel time across the board, according to our game. The total traveling time and payoff for our 1400 drivers have gone from 22 minutes to 12 minutes, a substantial decrease . It is the opposite of a Braess Paradox in that regard (Kleinberg, Easley, 2010, p. 231).
The end result when the new bridge is done might be different that the result of the game, because of a variety of outside factors. We’ll have for now wait and see, until it is done.
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- Google Maps app. Google Inc.