COU 7: Candidate Entry

This COU is only for extra credit

Any points you earn will count towards your total COU points for the semester, but the points available on this COU do not add to the total COU points available used to calculate your grade at the end of the semester.

In this COU you’ll explore a model in which a two persons each decide whether or not to run in an election in which a single “current candidate” is already running. The two potential candidates each face a situation similar to the one depicted in the model of the decision to run studied in Lesson 3. Specifically, they can each benefit from running because if they run and win, they gain the power to set policy on an issue, but they each must pay a personal cost to run.

Suppose that whoever wins the election sets the location of policy as in the spatial model. If the current candidate wins, she will set policy at location R. Suppose that one of the potential challengers, whom we’ll call the “middle challenger”, has ideal point M < R and if he wins he will set policy at location M. Suppose that the other potential challenger, whom we’ll call the “left challenger”, has ideal point L < M and if she wins she will set policy at location L. So, the locations of the ideal points of the left challenger, the middle challenger and the current candidate look something like this:

The left challenger and middle challenger must each choose between two alternatives, to run or to not run. In choosing between these alternatives, each considers the personal costs that must be paid to run and the location of policy that will ultimately result from the election. Specifically, the left challenger’s utility level, given the action she takes and the outcome of the election is:

action election outcome utility level
run left challenger wins -C
run current candidate wins L-R-C
run middle challenger wins L-M-C
not run current candidate wins L-R
not run middle challenger wins L-M

The middle challenger’s utility level, given the action he takes and the outcome of the election is:

action election outcome utility level
run left challenger wins L-M-C
run current candidate wins M-R-C
run right challenger wins -C
not run left challenger wins L-M
not run current candidate wins M-R

If neither the left challenger nor the right challenger run, then the current candidate wins the election with certainty. If either runs, on the other hand, then each candidate in the race has some chance of winning. Specifically:

Who Runs Probability Left Challenger Wins Probability Middle Challenger Wins Probability Current Candidate Wins
left and middle challengers \lambda \mu 1-\lambda-\mu
left challenger only \lambda + \delta\mu 0 1 - \lambda - \delta \mu
middle challenger only 0 \lambda + \mu 1-\lambda - \mu
neither challenger 0 0 1

where \lambda, \mu and \delta are each numbers between 0 and 1.

These probabilities might look totally arbitrary greek-alphabet soup. But there is a method to the madness! Imagine that once the challengers each decide whether to run, voters choose which candidate to vote for in part according to their views of the policy each candidate will implement if she wins. Thus, a candidate’s likelihood of winning depends in part on the portion of voters who prefer that candidate’s policy position to the policy positions of her opponents. Those portions in turn are determined by the midpoints between the ideal points of the candidates. For instance, suppose that both the left challenger and middle challenger run. Then the relevant midpoints between three candidates’ ideal points look like this:

with “L\leftrightarrow M” denoting the midpoint between L and M and “M \leftrightarrow R” denoting the midpoint between M and R.

Suppose that a portion \lambda of the electorate have ideal points to the left of the midpoint L \leftrightarrow M, another portion \mu have ideal points between L \leftrightarrow M and M \leftrightarrow R, and the remaining portion 1-\lambda-\mu have ideal points to the right of M\leftrightarrow R. Like this:

Then we might expect that the likelihood that the left challenger wins is proportional to \lambda, the likelihood that the middle challenger wins is proportional to \mu and the likelihood that the current candidate wins is proportional to 1-\lambda-\mu. Moreover, if only the middle challenger runs against the current candidate, then the proportion \lambda of the electorate that preferred policy L to policy M, will prefer policy M to R, and so the left challenger will win with likelihood proportional to \lambda + \mu, while the right-challenger wins with likelihood proportional to 1-\lambda - \mu, like this:

What if only the left challenger runs? In that case, an additional midpoint is relevant – i.e. the midpoint L \leftrightarrow R between L and R:

The midpoint L \leftrightarrow R is to the right of the midpoint L \leftrightarrow M and to the left of the midpoint M \leftrightarrow R. Thus the portion of the electorate with ideal points to to the left of L \leftrightarrow R – i.e. the proportion that prefers policy L to policy R – is equal to the proportion \lambda to the left of L \leftrightarrow M plus a proportion \delta \mu, where \delta is a number between 0 and 1 and \mu is the proportion of voters between L \leftrightarrow M and M \leftrightarrow R, like this:

Anyway, to complete the model, we need to represent each challenger’s preferences over her available actions, given her uncertainty over the outcomes that will result from each action. Since the outcome of the election is uncertain, we’ll do this by applying the expected utility formula to the above assumptions as follows:

(If you’re on a tablet, the following is easier to read in landscape mode!)

Left Challenger Middle Challenger Left Challenger’s Expected Utility Middle Challenger’s Expected Utility
run run \mu(L-M) + (1-\lambda-\mu)(L-R) - C \lambda(L-M) + (1-\lambda-\mu)(M-R) -C
run not run (1-\lambda - \delta \mu)(L-R) - C (\lambda + \delta \mu)(L-M) + (1-\lambda - \delta \mu)(M-R)
not run run (\lambda + \mu)(L-M) + (1-\lambda-\mu)(L-R) (1-\lambda - \mu)(M-R) - C
not run not run L-R M-R

Prompt 1

Does this game depict the two potential challengers as strategically interdependent? That depends on the values of the parameters L, M, R, C, \lambda, \mu and \delta. Specifically, consider the left challenger. Is it the case that her preference ordering over her available actions (to run or to not run) depends on whether she expects the middle challenger to run?

  1. Write the inequality that describes the conditions under which the left challenger’s expected utility from running is greater than or equal to her expected utility from not running when she expects the middle challenger to run.
  2. Rearrange the inequality you wrote in response to (a) so that C is on one side and \lambda, L and M are on the other. (Note: (1-\lambda-\mu)(L-R) appears on both sides of the inequality in (a) and so cancels out!)
  3. Write the inequality that describes the conditions under which the left challenger’s expected utility from running is greater than or equal to her expected utility from not running when she expects the middle challenger to not run.
  4. Rearrange the inequality you wrote in response to (c) so that C is on one side and R, L, \lambda, \delta and \mu are on the other.

Prompt 2

If you correctly answered Prompt 1, you’ll know that if C is neither too large nor too small, the left challenger prefers to run instead of not run when she expects the middle challenger to not run, and prefers to not run instead of run when she expects the middle challenger to run. Without using any mathematical expressions or computations, explain why this is so.

(Hint: Notice that if the middle challenger does not run, then by running instead of not running, the left challenger reduces the probability that the current candidate wins. On the other other hand, if the middle challenger runs, then by running instead of not running, the left challenger reduces the probability that the middle challenger wins but has no effect on the probability with which the current candidate wins. Thus, when the middle challenger does not run, by running instead of not running the left challenger reduces the probability that the policy adopted is R. On the other hand, when the middle challenger runs, by running instead of not running the left challenger reduces the probability that the policy adopted is M. Recall that L is closer to M than to R. So by reducing the probability that the policy adopted is M, the left challenger is reducing the likelihood of an outcome that for her is “not too bad”, while by reducing the probability that the policy adopted is L, the left challenger is reducing the probability of an outcome that for her is “very bad”.)

Prompt 3

When M and L are sufficiently close to one another, the middle challenger is always better off when the left challenger runs than when the left challenger does not run. (Why? Because when M and L are close, the left challenger’s policy L is not too far from the middle challenger’s policy M. Thus the middle challenger benefits when the left challenger bears the burden of running and pulling policy away from R and closer to L and M.) From Prompt 1 you know that if C is neither too large nor too small, the left challenger prefers to run when she expects the middle challenger to not run. Thus, when M and L are sufficiently close to one another and C is neither too small nor too large, the middle challenger wants the left challenger to expect that the middle challenger will not run.

Imagine you are the middle challenger. In no more than one page of double-spaced text, describe a tactic you would use to attempt to cause the left challenger to expect that you will not run for office and explain why you think that tactic would be effective.

Rubric

Prompt 1

The answer to this prompt gets one point for each of parts (a) through (d) for which it gives a correct answer. No partial credit available.

Prompt 2

A successful answer does all of the following:

  1. Points out that if the left challenger expects the middle challenger to not run, then she expects that by running instead of not running she will cause policy to be equal to her ideal point L with some likelihood, instead of equal to R with certainty.
  2. Points out that if the left challenger expects the middle challenger to run, then by running instead of not running she causes policy to be equal to her ideal point L with some likelihood, instead of being either M or R.
  3. Points out that because of (a) and (b) the change in policy the left challenger causes by running is larger when she expects the middle challenger to not run than when she expects the middle challenger to not run.
  4. Is sufficiently free of errors of spelling, grammar and usage as to be readily and easily comprehensible.
  5. If hand written, is immediately and easily legible.

An answer gets…

Four points…
if it meets all of the above criteria.
Two points…
if it meets criteria (d) and (e) and either (a) and (b) but not (c).
Zero points…
if it misses any of criteria (a), (b), (d) or (e).

Prompt 3

A successful answer does all of the following:

  1. Describes a tactic a person could employ that could cause another person to expect that she will not run for office.
  2. Explains why that tactic would be effective.
  3. Is sufficiently free of errors of spelling, grammar and usage as to be readily and easily comprehensible.
  4. If hand written, is immediately and easily legible.

An answer gets…

Four points…
if it meets all of the above criteria.
Zero points…
if it misses any of the above criteria.