Game Theory, Braess’s Paradox and Hotelling’s Law

“Whether one street is preferable to another depends not only on the quality of the road, but also on the density of the flow. If every driver takes the path that looks most favorable to him, the resultant running times need not be minimal.”

–  Dietrich Braess, German mathematician

Two mathematicans Horald Hotelling and Dietrich Braess have made significant contribution to Game theory.

Dietrich Braess came with concept of Braess’s Paradox, which is useful in traffic management. Sometimes building a flyover or bypass road may not result in less travel time, infact if traveller uses flyover or bypass road it may take more time than travelling by normal road.

Following example will explain Braess’s paradox.

braess paradox

Suppose T is the number of drivers and 5000 drivers want to travel from start point to end point. One route is Start-> A-> End, other route is Start-> B-> End. Let us assume that drivers give equal preference to both the routes so 2500 divers will take first route and equal number will take second route.

In first case travel time is equal to 2500/100+45 = 70 minutes, in second case travel time is equal to 45+ 2500/100 = 70 minutes. Now suppose due to political pressure, municipality constructs a flyover from A to B and travel time between A to B is not even a minute, let us take it as 0 minute.

bandra flyover

Now all drivers ( in game theory players think rationally) decide to travel from start to A, A to B and B to End, to take advantage of flyover. But now travel time is 5000/100+ 5000/100 = 100 minutes. In other words if all drivers think rationally they will end up taking more time. This is Braess’s paradox, without flyover, drivers were taking less time; with flyover (constructed to reduce travel time) it takes more time.

So constructing a flyover or bypass road may not always be answer to traffic congestion.

42nd_Street_in_New_York

On Earth Day, New York City’s Transportation Commissioner decided to close 42d Street, which as every New Yorker knows is always congested. “Many predicted it would be doomsday,” said the Commissioner, Lucius J. Riccio. “You didn’t need to be a rocket scientist or have a sophisticated computer queuing model to see that this could have been a major problem.” …But to everyone’s surprise, Earth Day generated no historic traffic jam. Traffic flow actually improved when 42d Street was closed

–   New York Times, 25th December 1990

Have you ever wondered why, if there are two medicine shops on street, they are always in the middle of street and next to each other and not at the ends of street?

This is explained by  Horald Hotelling’s Law.Take example of two ice cream sellers on beach ( assuming both sell same quality  of ice cream). Assume one starts at the south end of the beach and one starts at the north. Again assuming a rational consumer and equal distribution along the beach, each cart will get 50% of the customers, divided along an invisible line equidistant from the carts. But, each cart owner will be tempted to push his cart slightly towards the other, in order to move the invisible line so that it encompasses more than 50% of the beach. Eventually, the ice cream sellers end up next to each other in the centre of the beach.

0198606737_hotelling-model_1

 

What if they put their ice cream carts at two ends of beach? Theoretically each would still draw half of the customers (the northern or southern half) and the customers would enjoy a shorter travel distance. However, neither would be willing to do this independently, as it would then allow the other seller to relocate and capture more than half the market i.e. 50% of his side, plus say 25% of other’s market i.e. one in middle will have 75% market and one at end will have only 25% of market.

 

 

 

 

 

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