Read the information given below and answer the question that follows.

Four cars need to travel from Akala (A) to Bakala (B). Two routes are available, one via Mamur (M) and the other via Nanur (N).
The roads from A to M, and from N to B, are both short and narrow. In each case, one car takes 6 minutes to cover the distance, and each additional car increases the travel time per car by 3 minutes because of congestion. (For example, if only two cars drive from A to M, each car takes 9 minutes.) On the road from A to N, one car takes 20 minutes, and each additional car increases the travel time per car by 1 minute. On the road from M to B, one car takes 20 minutes, and each additional car increases the travel time per car by 0.9 minute.

The police department orders each car to take a particular route in such a manner that it is not possible for any car to reduce its travel time by not following the order, while the other cars are following the order.

If all the cars follow the police order, what is the difference in travel time (in minutes) between a car which takes the route A-N-B and a car that takes the route A-M-B?

The correct option is B (0.1).

Let the number of cars taking route A-M-B be x and route A-N-B be y. Since there are 4 cars in total, we have:

$x+y=4$

Now let's write the travel time equations for each road segment as a function of the number of cars using them:

For road A-M: Since x cars use it, the travel time is:
$T_{A-M}=6+3(x-1)=3+3x$

For road M-B: Since x cars use it, the travel time is:
$T_{M-B}=20+0.9(x-1)=19.1+0.9x$

For road A-N: Since y cars use it, the travel time is:
$T_{A-N}=20+1(y-1)=19+y$

For road N-B: Since y cars use it, the travel time is:
$T_{N-B}=6+3(y-1)=3+3y$

The total travel times for the two routes are:

$T_{A-M-B}=(3+3x)+(19.1+0.9x)=22.1+3.9x$

$T_{A-N-B}=(19+y)+(3+3y)=22+4y$

The police department's order represents a Nash Equilibrium, where no car can benefit by unilaterally changing its route. Let's analyze the distributions:

If x = 2 and y = 2:
Travel time on A-M-B: $22.1+3.9\left(2\right)=29.9$ minutes.
Travel time on A-N-B: $22+4\left(2\right)=30$ minutes.

If a car on A-M-B switches to A-N-B (making y = 3), its new travel time would be:
$22+4\left(3\right)=34$ minutes, which is longer than 29.9. Thus, it will not switch.

If a car on A-N-B switches to A-M-B (making x = 3), its new travel time would be:
$22.1+3.9\left(3\right)=33.8$ minutes, which is longer than 30. Thus, it will not switch.

Therefore, x = 2, y = 2 is the stable distribution under the police order. The difference in travel time is:
$30-29.9=0.1$ minutes.