Saverio Bolognani
Senior Scientist at ETH Zurich
The mathematics of cities of the future
Saverio Bolognani
Nicolas Lanzetti
(Gioele Zardini)
(GIUSEPPE BELGIOIOSO)
(Christian Hartnik)
https://nccr-automation.ch
https://control.ee.ethz.ch
https://idsc.ethz.ch
https://www.ethz.ch
Game theory
What is a game in mathematics?
Spieler / players +
Aktionen / actions +
Gewinn / payoff
The rules
Mathematical model
- What is the equilibrium of the game? What does a rational player do?
- How can we change the rules of the game to obtain a different (and better) behavior?
Analysis
Complex games
Micro-economics
Macro-economics
Politics
Biology
Engineering
Social science
Does an equilibrium always exist?
John Nash
Nobel Prize in Economics in 1994
Traffic and games
A game in which 200 agents travel from A to B and there are two paths
What is the equilibrium? Who takes which path?
How much time does it take to reach the destination?
The Braess Paradox
A game in which 200 agents travel from A to B and there are two paths
... and now also a bridge!
What is the equilibrium? Who takes which path?
How much time does it take to reach the destination?
The Braess Paradox
- The previous options are still available (road + ferry).
- The bridge is "perfect": no delay.
- Everyone takes the bridge:
but... travel time is worse for everyone!
What if They Closed 42d Street and Nobody Noticed?
25 December 1990
“On Earth Day this year, New York City’s Transportation Commissioner decided to close 42d Street, which as every New Yorker knows is always congested. [...] But to everyone's surprise, Earth Day generated no historic traffic jam. Traffic flow actually improved when 42d Street was closed.”
Optimal solution (approx. 64 minutes on average):
- 50 agents road-bridge-road (55 minutes)
- 75 agents road-ferry (67.5 minutes)
- 75 agents ferry-road (67.5 minutes)
... but this is not an equilibrium!
Let’s better utilize the bridge
The price of anarchy
VS
\mathrm{Social~ cost} = \mathrm{cost~1} +\ldots+\mathrm{cost~200}
\mathrm{PoA} =
\frac{
\text{social cost at equilibrium}
}{
\text{best possible social cost}
}
Let’s introduce a toll for the bridge:
Can we obtain 65 minutes of travel time?
With what toll?
Optimal solution (approx. 64 minutes on average):
- 50 agents road-bridge-road (55 minutes)
- 75 agents road-ferry (67.5 minutes)
- 75 agents ferry-road (67.5 minutes)
1/4 of the agents prefers to pay the toll.
Which toll yields this optimal equilibrium?
toll
12.5 minutes
What if the system is more complex than our small example?
Lugano 2030
Shared mobility
BUS
AMOD
BIKE SHARING
Autonomous Mobility
On Demand
From mathematics to technology
1
2
3
At each stage:
- peer review
- scientific papers
- conferences
https://nccr-automation.ch
https://control.ee.ethz.ch
https://idsc.ethz.ch
https://www.ethz.ch
The mathematics of cities of the future
By Saverio Bolognani
