I came home from work and began leafing through new technologies. I came to the column for this week's issue, mini-problem, and started immediately with a solution. Weekly issue is about traffic congestion and reads:

A graph of the problem

A very large number of cars, after an event in the A drive to D. The paths AC, BC and BD has a high capacity. Here, the running time is always the same, 2 h, 0.25 h and 2 h road AB has limited capacity, and the running time is (1 + p) h, where 0 <p <1, the proportion of cars that AB. Analogously, the route CD, with 0 <q <1.

One must first learn that the way BC is off. When planning about half of the drivers to run ABD and ACD rest. In both cases, it takes 3.5 hours, but just before the event is over you get to know that BC has been opened, and you also get continuous reports of traffic flow on roads.

This new capability means that all journeys are taking fourth longer. How can this be?

Since all cars constantly have information about how the other cars act, all the cars to take as much time on the road. This means that all possible routes will go just as fast to go, so:

1 + p + 2 = 2 + 1 + q = 1 + p + 0.25 + q + 1 (1)

<=> 3 + p = 3 + q <=> q ​​= p

=> 3 + p = 2.25 + 2p => p = q = 0.75

Stopped this into (1) the total åktiden 3.75 hours, ie 15 minutes longer than if no extra path had been opened! An additional way that intuitive should mean more space and less time, giving instead the opposite result. This is called a Nash equilibrium that occurs when all individuals try to optimize for themselves.

Example: Suppose there are two players who choose between two options: to take the shortest path for themselves or to jointly optimize with respect to both players simultaneously. In Box (1) optimizes the individuals with respect to the collective, in box (4) with respect to themselves. In Box (2) and (3) optimize any individual with regard to himself and the other with respect to the collective. Since the outcome in these boxes is worse than when both optimize with respect to themselves, then an equilibrium occur in (4).

Another example of Nash equilibrium is when two players picking mushrooms before it is ripe (because someone else could take it before then). This is an example of a so-called prisoners dilemma that occurs when you can take advantage of that act immediately rather than waiting.