## Complexity of Cooperation Web Site

### Exercise 5

Exercise 5. Human Waves. People are seated in a football stadium. Someone gets up and shouts, "Wave." A human wave can be formed if everyone follows rule A: stand up for one second if either (or both) immediate neighbors are standing.

To keep things simple, suppose there are only 20 people, and they are seated in a circle around the field. These people can be represented as being on a line, with the understanding that person 20 is adjacent to person 1 and as well as being adjacent to person 19. The result would look like this:

time 1: . . . x. . . . . . . . . . . . . . . .
time 2: . . x. x. . . . . . . . . . . . . . .
time 3: . x. x. x. . . . . . . . . . . . . .
time 4: x. x. x. x. . . . . . . . . . . . .
time 5: . x. x. x. x. . . . . . . . . . . x
time 6: x. x. x. x. x. . . . . . . . . x.

Note that at time 6 six people are standing including the person in seat 2. Also note that the person who started the process keeps standing up and sitting down.

Now suppose that people also follow rule B which says that regardless of rule A, once a person sits down, he or she stays seated for at least two seconds. Now the result looks like this:

time 1: . . . x. . . . . . . . . . . . . . . .
time 2: . . x. x. . . . . . . . . . . . . . .
time 3: . x. . . x. . . . . . . . . . . . . .
time 4: x. . . . . x. . . . . . . . . . . . .
time 5: . . . . . . . x. . . . . . . . . . . x
time 6: . . . . . . . . x. . . . . . . . . x.

Note that at time 6 only two people are standing.

The first part of the exercise is to compare what will happen under rule A alone, and under the combination of rules A and B. The second part of the exercise is to modify the A-B model to take into account some or all of the following observations:

1. A human wave actually moves in one rather than two directions.
2. A human wave can continue even if there is one person who never stands.
3. A human wave eventually dies out.
4. Some people need to stay seated for more than 2 seconds.

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