Braess paradox working itself out in Target and Walmart?
A study from 2010 conducted by researchers from University of Massachusetts Amherst proposed a hypothesis that suggests the possible disappearance of the Braess paradox under high demands of traffic networks. The main idea of the hypothesis is that when the travel demand is too high and the traffic congestion has become too problematic that the additional path will no longer be used and the cost of the travelers will not increase as expected in the Braess paradox either. The formula derived by the researchers provides insight into the design of congested and noncooperative networks. However, can we confidently say that, for any existing network infrastructure, there is no need to design any new route or pathway to optimize the equilibrium of traffic? Or more specifically, can we interpret all the pathways to have the same significance in real-life applications?
The theorem in the research article claims that there exists a positive mathematical relationship between the travel demand (d) and the path capacity (r) that allow the Braess paradox to occur within only a fixed level and when the demand exceeds the particular value dw1, the paradox stops to happen and the new alternative path will not be used. The cost of travel thus returns to the original equilibrium when there is no interchangeable path C built between A and B as demonstrated in the lecture. The interpretation and implication of this theorem is that the Braess paradox will ultimately be negated by a high traffic demand. The new route expected to be used and cost an increase in the cost of travel will only be used within a specific level of traffic demand, once the demand has exceeded the upper limit of the demand level, the phenomenon of “wisdom of crowds” will take over.
The proposed theorem of the negation of the Braess paradox is intriguing and intuitively straightforward if we think of its application in real-life situations such as the checkout lines at grocery stores. Instead of picturing each line as a parallel traffic pathway, we can think of them as a collection of paths to a single destination of B, which is the exit of Target or Walmart, and A would then be the place right after the last section of aisles. Shoppers would like to get to the entrance by picking a line from the collection of checkout counters. The status and length of line, which equate the efficiency and cost of the “route,” are visible to all customers waiting in the lines. The equilibrium in this network and the dominant strategy can be achieved by measuring the average checkout time in each line multiplied by the total number of customers waiting in the line. And the customers should balance themselves evenly between all lines in a Nash equilibrium, in which everyone is performing their best response to other customers’ decisions. If any line shows an unequal “travel time” due to the items in the carts before them or the inefficiency of the cashiers, there is an incentive for people to switch to another line that is faster.
As explained in the class, such self-interested behavior by the customers in an equilibrium is in fact counterintuitive and that opening a new line at the checkout counters when there is a high demand of customers will only lead to an even worse waiting time. It seems reasonable that when a new counter is opened, the customers waiting at the back of other lines will inevitably consider the new line to be faster than the current one and move to the new line, which makes it to be even slower and customers in the new line may have to spend more time waiting. However, given the theorem in the research paper, the Braess phenomenon will only occur at a particular level of traffic demand and will disappear if the demand is high enough. If we apply the theorem to our everyday observation, we can examine the pattern described in the theorem. Imagine there is a huge group of customers walking close to the lines deciding which line they should choose and a cashier comes to open a new line, we can expect that a fair amount of customers will switch from their current line to the new line. Once the new line has approached the average number of total customers at the scene, the other customers who just arrived at the line or not fast enough to move will no longer have the incentive to switch anymore. The new line now becomes exactly like the original lines and the network will recover to equilibrium again.
Source: https://iopscience.iop.org/article/10.1209/0295-5075/91/48002/pdf