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