# Exercise 4: More Fun

Navigate to http://www.math.union.edu/locate/MFA-09-2008/voting-rules.html to complete this assignment.

Click the bar at the bottom of the simulation window (immediately above the mean and mediancentre buttons) to open the control panel.

1. Trails: Clicking the "point trails" button on the control panel helps you track the path that the red point follows as it approaches its equilibrium position. Observe some point trails for a variety of configurations of blue points and initial positions of the red point, toggling back and forth between "mean" and "mediancentre" mode. You should begin to see some systematic differences between the shapes of these trails for the mean, and those for the median. What is the difference?

(Caution — it's possible to "throw" the red point by releasing it with some initial velocity, but resist the temptation. For this part, only release the red point "from rest." That is, hold the red point still for a moment before releasing it.)

2. Look for configurations that produce very curvy point trails. If you find a very good one, send me an e-mail (`zwickerw` at `union` dot `edu`) and tell me about it. To do so, use the "save" button at the lower right of the control panel; this opens a small window containing data about your current configuration. Copy this text and paste it into an email to me.

3. Is the opposite situation also possible (a profile that yields a tie in the MCBorda election but no tie in the Borda election?

4. The point trails for the mean have a particularly simple shape.

• What is that simple shape?
• How can this be explained? (This is a hard question!)

5. Weight and Friction: Click the lowest of the four triangles to the right of the simulation window (marked "The Control Panel") and read about adjusting weight and friction.

• How do you expect changes in weight to affect the motion of the red point?
• What about changes in friction?
• What will happen if friction is set to zero?