Preconditions for a self-organizing system based on Lyapunov Functions

Self-organizing systems based on Lyapunov Functions

Lyapunov functions can explain how a city/system self-organizes.

Consider Disneyland with 5 rides and 5 customers. Each ride can only service one customer.

• Each customer first waits at the gate and writes down the sequence of rides they plan to take
• Once the gate opens, each customer goes to the first ride on their list
• If more than one customer goes to the same ride, then someone have to wait and a ride somewhere is empty.
• To avoid waiting, customers in queue will then choose rides with no queue.
• This behaviour collectively result in every ride being occupied and no queues.
• The system self-organizes.

Pre-conditions for self-organization

However, self-organization happens only if certain preconditions are true:

1. Perfect information - e.g. all customers know exactly which rides are empty (say within visual range)
2. No cost to switching - e.g. customer can switch to another ride without cost
3. No barriers to switching - e.g. all customer can switch to any other ride at any time
4. No externalities - e.g. customers behaviour are independent, one taking a ride does not affect another taking a ride

Consider if the the opposite conditions hold:

1. Imperfect information
• e.g. customers cannot see which rides are empty (e.g. rides are in housed the buildings) so they'd rather continue queuing rather than risk going to another ride which may have a longer queue
2. Costly switching
• e.g. rides are very far apart and to walk to the next nearest ride takes half a day - then customers may rather stay put
3. Barriers to switching
• e.g. rides are segregated to VIP zone and normal zone - normal customer cannot access VIP rides even if they are empty.
4. Externalities
• e.g. when Customer A is a sloppy person and leaves behind a trail of food and grime everywhere - other customers will then be reluctant to go on a rides that Customer A has been on

Other examples of self-organizing systems

• Checkout queues in a hypermarket encourages self-organizing customers
• Consider what happens if the checkout queues are split into 3 separate groups
• Multi-storey parking buildings with the number of free slots at each floor prominently displayed
• Drivers get better information and make better decisions
• Consider a crowded parking building with zero information