So-called paradoxes fire the imagination of mathematicians and non-experts alike. Actually, these are not really paradoxes in the sense of an illogical conclusion, like 1+1=3, because such contradictory statements do not happen in mathematics (but see Russell’s Paradox in this page). They are called paradoxes because they lead to conclusions that are counter-intuitive.
Braess’ Paradox
The road network in the figure is used by motorists to travel from A to D. The roads marked in green are long, but travelling along them takes six minutes flat. The roads marked in yellow are shorter but they suffer from congestion. One car goes through it in one minute, two cars in two minutes, and so on. The red road from B to C is short and fast. It takes one minute flat to go from B to C.
Four motorists enter the network at A at the same time. The first motorist has three options: A-C-D, taking seven minutes; A-B-D, also taking seven minutes; or A-B-C-D, taking three minutes. The obvious solution for the first motorist is to take route A-B-C-D. The second motorist follows hot on the first’s (w)heels. The possible routes now are: A-C-D, taking eight minutes (because she might encounter the first motorist on the C-D stretch); A-B-D, also eight minutes; and A-B-C-D, taking five minutes. So the second motorist takes route A-B-C-D. With a similar argument, so do the next two. The natural choice for all four motorists is therefore A-B-C-D, for a total commute time of nine minutes each.
The following day the four motorists arrive at the same spot at the same time, but they see a road-sign saying that the route B-C is closed for the day for repairs. The first motorist calculates that it makes no difference what route he takes, so he chooses route A-B-D. The second motorist calculates that now her best route is A-C-D, the third motorist can take any of the two routes, and the fourth motorist takes a route different from the third’s. So two motorists will take route A-B-D while the two others will take route A-C-D for a total commute time of 6 + 2 = eight minutes each. The closure of the fast route has made their time on the road shorter.
The reason that this happens is that the motorists had to act ‘selfishly’ because they could not communicate with each other. Perhaps the most well-known example of this conundrum is the Prisoners’ Dilemma.
The Prisoners’ Dilemma
The most famous situation in which agents are unable to communicate, so have to decide ‘selfishly’, thereby leading to their choosing a stable solution that is not optimal is perhaps the Prisoners’ Dilemma. Two prisoners are in police cells being investigated for a burglary. They are kept in separate cells and each one is told they can either betray the other prisoner or remain silent. If both remain silent, they will each spend one year in prison for a lesser charge. If they both betray each other, they will each serve two years in prison. However, if one betrays the other and the other remains silent, then the silent prisoner will get five years in jail and the other runs free. With no means of communication (that is the stronghold the police have over them), they might be driven to act selfishly and choose the stable solution: betray each other, netting themselves two years each in prison, whereas the optimal solution for both is to remain silent and spend one year each. This problem can be the starting point to motivate a course in Game Theory, a highly developed branch of modern applied mathematics, which also has much to say about situations like Braess’ Paradox. The Prisoners’ Dilemma, first formulated in 1950 by two researchers at RAND Corporation, has a long history and a number of variants which can model real world solutions involving cooperative versus non-cooperative behaviour.
Josef Lauri is a professor at the Department of Mathematics at the Faculty of Science of the University of Malta.
Sound bites
• What do these have in common: the planet Neptune, the element Scandium and the Higgs boson? They were all predicted theoretically before their existence was actually confirmed.
• Some errors are good errors. Einstein introduced the cosmological constant so that his theory of relativity would be consistent with the theory of a static universe which was the accepted theory at the time. Although this theory of the Universe has now been abandoned and the cosmological constant was deemed to be zero for some time, it is now considered to be the best explanation for current theories of the universe.
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Did you know?
Did you know that elementary mathematics is not necessarily easy? Igor Vladimirovich Arnold (1900-1948) devised 20 questions all of which have the answer “3 - 1 = 2”. We present a selection below. Should be easy now that you know the answer, yes?
• Maria said: “I have three more brothers than sisters.” How many more boys than girls are there in Maria’s family?
• A train was due to arrive one hour ago. We are told it is three hours late. When can we expect it to arrive?
• A brick and a spade weigh the same as three bricks. What is the weight of the spade?
• My brother is three times as old as me. How many times my age now was his age when I was born?
• It takes a minute for a train 1 km long to completely pass a telegraph pole by the track side. At the same speed the train passes right through a tunnel in three minutes. What is the length of the tunnel?
For more trivia, see: www.um.edu.mt/think