COU 4: Analyzing a Parameterized Probability Distribution

Imagine a person named Lysander who is considering running for as a candidate in an upcoming election. Lysander knows that if he runs, winning will require getting more votes than another person named Ruhi who has already declared her candidacy. Lysander also knows that there is yet another person named Marit who is considering entering the race. So if Lysander enters the race, he will either be running head-to-head against Ruhi, or will be in a three-way contest with Ruhi and Marit.

These three persons’ ideal points are positioned like this:

Suppose that Lysander knows that each person who votes in the election will vote for the candidate whose ideal point is closest to their own. Further the election will be decided by plurality rule – i.e. the candidate with the most votes wins. This means that the winner of the election will be determined by the relative numbers of voters who have ideal points in each of four regions. Specifically, let “L-M” be the midpoint between Lysander and Marit, “L-R” be the midpoint between Lysander and Ruhi, and “M-R” be the midpoint between Marit and Ruhi, like this:

These midpoints define four regions of the policy space, ‘Far Left’, ‘Mid-Left’, ‘Mid-Right’ and ‘Far-Right’, like so:

Notice that if Lysander runs but Marit does not, then all the voters in the Far-Left and Mid-Left regions vote for Lysander, while all the voters in the Mid-Right and Far-Right regions vote for Ruhi, like this:

On the other hand if both Lysander and Marit run, then Lysander gets voters only from the Far-Left, Marit gets voters from the Mid-Left and Mid-Right, and Ruhi gets voters only from the Far-Right, like this:

This of course means that Lysander’s chance of winning the election (if he runs) depends on whether Marit runs. Assume that exactly one of the following possibilities about the proportion of voter ideal points that lie in each region is true:

And assume the relative likelihood of these possibilities is given by the following parameterized probability distribution:

Event Probability
Far-Left Majority \frac{1}{2}\lambda
Mid-Left Majority \frac{1}{2}(1-\lambda)
Mid-Right Majority \frac{1}{4}
Far-Right Majority \frac{1}{4}

where \lambda (the Greek letter pronounced “lamb-dah”) is a number between 0 and 1.

Prompt 1

Suppose Lysander runs but Marit does not. Then Lysander wins if there is a Far-Left Majority or a Mid-Left Majority, and Ruhi wins if there is a Mid-Right majority or a Mid-Left majority. In other words, uncertainty about the election outcome is described by the probability distribution

Outcome Probability
Lysander Wins P(\text{Far-Left Majority}) + P(\text{Mid-Left Majority})
Ruhi Wins P(\text{Mid-Right Majority}) + P(\text{Far-Right Majority})

Where P(\cdots) just means “probability of”. For instance, using the table above listing the probabilities of the four possibilities, we have P(\text{Far-Left Majority}) = \frac{1}{2}\lambda and P(\text{Mid-Left Majority}) = \frac{1}{2}(1-\lambda) so P(\text{Far-Left Majority}) + P(\text{Mid-Left Majority}) = \frac{1}{2}\lambda + \frac{1}{2}(1-\lambda) = \frac{1}{2}

By reading off the table listing the probabilities of the four possibilities, fill in the expressions or values for the probabilities in the table below:

Probability Distribution over Election Outcomes if Lysander Runs and Marit Does Not Run
Outcome Probability
Lysander Wins
Ruhi Wins

Prompt 2

On the other hand, if both Lysander and Marit run as candidates, then the probability distribution over the outcome of the election is:

Outcome Probability
Lysander Wins P(\text{Far-Left Majority})
Marit Wins P(\text{Mid-Left Majority}) + P(\text{Mid-Right Majority})
Ruhi Wins P(\text{Far-Right Majority})

By reading off the table listing the probabilities of the four possibilities, fill in the expressions or values for the probabilities in the table below:

Probability Distribution over Election Outcomes if Lysander and Marit both Run
Outcome Probability
Lysander Wins
Marit Wins
Ruhi Wins

Prompt 3

So, Lysander’s probability of winning in a head-to-head race against Ruhi is P\left(\text{Far-Left Majority}\right) + P\left(\text{Mid-Left Majority}\right) And Lysander’s probability of winning in a three-way race is P\left(\text{Far-Left Majority}\right)

So, suppose Lysander has entered the race as a candidate. How much will Marit reduce the probability that Lysander will win if she enters the race? P\left(\text{Far-Left Majority}\right) + P\left(\text{Mid-Left Majority}\right) - P\left(\text{Far-Left Majority}\right) = P\left(\text{Mid-Left Majority}\right)

Write the expression for the amount Marit reduce’s Lycander’s chance of winning by entering the race. (I.e. just write the expression for P\left(\text{Mid-Left Majority}\right)!)

Prompt 4

Draw a graph showing how much Marit reduces Lysander’s probability of winning the race as a function of the parameter \lambda. In other words, draw a graph in which \lambda (which ranges from 0 to 1 is on the horizontal axis) and P\left(\text{Mid-Left Majority}\right) is on the vertical axis.

Rubric

You get up to 1 point on Prompt 1 – one-half of a point for each of the two required expressions that you write correctly.

You get up to 1 point on Prompt 2 – one-third of a point for each of the three required expressions that you write correctly.

You get up to 1 point on Prompt 3 – one point for the one required expression that you write correctly.

You get 1 point on Prompt 4 if you draw the graph correctly and 0 points otherwise.