COU 3: Voting

Choose One

For the Lesson 7 COUs due on November 24, you are only required to do ONE of either this COU or COU 4. If you choose to do both COU 3 and COU 4, any points beyond the 12 available on either one of them will count for extra credit.

Imagine an election in which seven voters are each choosing which of three candidates to vote for. Imagine that whichever candidate wins the election will determine the location of policy towards an issue as in the spatial model. Specifically, label the three candidates “Candidate A”, “Candidate B” and “Candidate C”. Let “A” be the policy Candidate A will implement if he wins, “B” be the policy Candidate B will implement if he wins and “C” be the policy Candidate C will implement if he wins, and suppose that those policies are located like so:

policySpace({
  policies: [
    {
      x: 10,
      label: 'C'
    },
    {
      x: 60,
      label: 'B'
    },
    {
      x: 90,
      label: 'A'
    },
  ],
})

Suppose that each of the seven voters have single-peaked preferences over the policy issue that will be decided by the election. Label the voters “Voter 1”, “Voter 2”, and so on through “Voter 7”, label their ideal points “1”, “2”, and so on through “7”, and assume that their ideal points are located as follows:

policySpace({
  policies: [
    {
      x: 10,
      label: 'C'
    },
    {
      x: 60,
      label: 'B'
    },
    {
      x: 90,
      label: 'A'
    },
  ],
  persons: [
    {
      x: 1,
      label: "1"
    },
    {
      x: 3,
      label: "2"
    },
    {
      x: 7,
      label: "3"
    },
    {
      x: 62,
      label: "4"
    },
    {
      x: 65,
      label: "5"
    },
    {
      x: 85,
      label: "6"
    },
    {
      x: 88,
      label: "7"
    },
  ]
})

Assume that whichever candidate gets the most votes wins the election. And for the moment, don’t worry about what happens if two candidates tie with three of the seven votes each – we will come back to that later! Assume further that Voters 1, 2 and 3 will vote for Candidate C. Thus going into the election, Candidate C already has three of the seven votes in hand.

Notice that Candidate C is the worst candidate from the points of view of Voters 4 through 7. And because Voters 4 through 7 constitute a majority of the seven voters, if all four of them vote for Candidate B or all four of them vote for Candidate A, Candidate C will lose. However, Voters 4 through 7 face a dilemma: they disagree with one another about Candidates B and A. Examine the diagram above and notice that Voters 4 and 5 each prefer Candidate B to A and Voters 6 and 7 each prefer Candidate A to B. So, since Candidate C already has three votes locked down, if each of Voters 4 through 7 votes for her favorite candidate, Candidates B and A will each only get two votes and thus Candidate C will win the election. In short, if each of Voters 4 through 7 votes for her favorite candidate, the winner of the election will be each of those voters’ least favorite candidate.

All this means that Voters 4 through 7 are strategically interdependent. For instance, examine the diagram again and think about the election from the point of view of Voter 4:

policySpace({
  policies: [
    {
      x: 10,
      label: 'C'
    },
    {
      x: 60,
      label: 'B'
    },
    {
      x: 90,
      label: 'A'
    },
  ],
  persons: [
    {
      x: 1,
      label: "1"
    },
    {
      x: 3,
      label: "2"
    },
    {
      x: 7,
      label: "3"
    },
    {
      x: 62,
      label: "4"
    },
    {
      x: 65,
      label: "5"
    },
    {
      x: 85,
      label: "6"
    },
    {
      x: 88,
      label: "7"
    },
  ]
})

The best possible outcome for Voter 4 would be for Candidate B to win the election. But her preferences over how to cast her vote depend on what she expects Voters 5 through 7 to do. For instance:

Voter 4’s Expectations About the Actions of Voters 5, 6 and 7	Voter 4’s Preferences Over Her Own Action
Voters 5, 6 and 7 will vote for Candidate B	Voter 4 prefers to vote for B instead of A
Voters 5, 6 and 7 will vote for Candidate A	Voter 4 prefers to vote for A instead of B

(Again, for the moment, don’t worry about what happens if three votes go to each of two candidates. We’ll come back to that!)

This kind of strategic interdependence is to be expected whenever voting is used to choose between more than two alternatives. Elections in which there are more than two candidates, like the one in the model above, are just one example.

In this COU, you’ll construct a simultaneous move game that depicts the strategic interdependence faced by Voters 4 through 7 in the model above.

Persons, Actions and Profiles of Actions

The game you construct will have four persons: Voter 4, Voter 5, Voter 6 and Voter 7. It will depict each of these persons as choosing between two actions: Vote for Candidate B or Vote for Candidate A. Therefore, a profile of actions in this game will be a list naming the candidates for which each of Voters 4 through 7 vote, like so… (B,A,B,A) …where the first element in the list is the candidate for whom Voter 4 votes, the second is the candidate for whom Voter 5 votes, the third is the candidate for whom Voter 6 votes and the fourth is the candidate for whom Voter 7 votes. Thus, in the above profile, Voter 4 votes for Candidate B, Voter 5 votes for Candidate A, Voter 6 votes for Candidate B, and Voter 7 votes for Candidate A.

Thus the payoff function for this game can be written as a table like this:

Profile of Actions	Voter 4’s Utility	Voter 5’s Utility	Voter 6’s Utility	Voter 7’s Utility
(B,B,B,B)
(B,B,B,A)
etc.
etc.

Prompt 1

Write a table like the one above. Use the table to write out the entire set of all profiles of actions in the game by filling out all the rows in the Profile of Actions column. If you correctly identify all the profiles of actions, your table will have 16 rows, not including the header. Leave the cells indicating each voter’s utility level at each profile blank.

Ties

Consider the two following two profiles of actions: \begin{array}{c} (B,B,A,B) \\ (B,B,A,A) \end{array} In each of these these two profiles, Voters 4 and 5 vote for Candidate B and Voter 6 votes for Candidate A. Voter 7’s vote varies between the two profiles: She votes for Candidate A in the first profile and votes for Candidate C in the second.

Continue to assume that Voters 1, 2 and 3 will vote for Candidate C. Thus if Voters 4 through 7 vote as in the profile (B,B,A,B), the election will result in a tie, with Candidates C and B each getting three votes. On the other hand, if Voters 4 through 7 vote as in the profile (B,B,A,A), Candidate C gets three votes while the other two candidates gets two votes each, and thus Candidate C wins the election outright.

With all this in mind, examine the diagram again that shows the locations of the voters’ ideal points and the candidates’ policy positions:

policySpace({
  policies: [
    {
      x: 10,
      label: 'C'
    },
    {
      x: 60,
      label: 'B'
    },
    {
      x: 90,
      label: 'A'
    },
  ],
  persons: [
    {
      x: 1,
      label: "1"
    },
    {
      x: 3,
      label: "2"
    },
    {
      x: 7,
      label: "3"
    },
    {
      x: 62,
      label: "4"
    },
    {
      x: 65,
      label: "5"
    },
    {
      x: 85,
      label: "6"
    },
    {
      x: 88,
      label: "7"
    },
  ]
})

Suppose Voter 7 expects Voters 4, 5 and 6 to vote as in the profiles (B,B,A,B) and (B,B,A,A) – i.e. she expects Voters 4 and 5 to vote for Candidate B and Voter 6 to Vote for Candidate A. If Voter 7 has this expectation, then she expects her vote will determine whether (i) there is a tie between Candidate C and Candidate B or instead (ii) Candidate C wins the election outright.

Notice from the above diagram that even though Voter 7’s favorite candidate is A, she prefers Candidate B to Candidate C. So there is presumably a significant difference from her point of view between Candidate C winning the election outright and the election resulting in a tie between Candidates C and B. So to fully depict the strategic interdependence between Voters 4, 5, 6, and 7, we have to make an assumption about what happens when the election results in a tie.

Since there are seven voters in this model, there is only one way a tie can occur: One candidate gets three votes, another candidate gets three votes, and the third candidate gets one vote. Moreover, we’ve assumed that Voters 1, 2 and 3 will all vote for Candidate C. So the only possible ties that can occur are between Candidate C and B and between Candidate C and A. So with that in mind, we’ll make the following assumption about how ties are resolved:

Assumption About Ties

If there is a tie between Candidates C and B, a second “runoff” election is held in which Candidates C and B are the only candidates. In the runoff election, Voters 4 through 7 will all vote for Candidate B, and thus Candidate B will be the ultimate winner.
If there is a tie between Candidates C and A, a second “runoff” election is held in which Candidates C and A are the only candidates. In the runoff election, Voters 4 through 7 will all vote for Candidate A, and thus Candidate A will be the winner.

Under this assumption, the difference between the profiles (B,B,A,B) and (B,B,A,A) from Voter 7’s point of view is unambiguous: The former profile results in a tie between Candidates C and B, and thus (under the assumption above), a runoff election that will be won by Candidate B. The latter results in victory for Candidate C. Thus if Voter 7 expects Voters 4 and 5 to vote for Candidate B and Voter 6 to vote for Candidate A as in these two profiles, she clearly prefers to vote for Candidate B instead of voting for Candidate A.

The assumption about ties above, then, suggests utility levels we might assign to Voter 7 at the profiles (B,B,A,B) and (B,B,A,A). Since Voter 7 prefers the outcome that results at the profile (B,B,A,B) to the outcome that results at (B,B,A,A), we should assign her a higher utility level at the former profile than at the latter. So, we can add utility levels for Voter 7 to the rows in the table corresponding to those two profiles as follows:

Profile of Actions	Voter 4’s Utility	Voter 5’s Utility	Voter 6’s Utility	Voter 7’s Utility
\cdots
\cdots
(B,B,A,B)				1
(B,B,A,A)				0
\cdots
\cdots

Prompt 2

To make sure you understand the implications of the assumption we just made about how ties in the election are resolved, re-write the table you wrote in response to Prompt 1. Then fill in Voter 7’s utility levels at the profiles (B,B,A,B) and (B,B,A,A) as in the example above. Then fill in utility levels for Voter 7 at the profiles (A,A,B,A) (A,A,B,B) in a way that is consistent with Voter 7’s preferences about the outcome and with the assumption above about how ties are resolved.

Prompt 3

In up to one page of double-spaced text, explain how the utility levels you assigned to Voter 7 at the profiles (A,A,B,A) and (A,A,B,B) in combination with the utility levels already assigned to Voter 7 at (B,B,A,B) and (B,B,A,A) depict the strategic interdependence between Voters 4 through 7 in this election.

In formulating your explanation, keep in mind that strategic interdependence only occurs when a person’s preferences over her available actions vary according to her expectations about the actions other persons will take! Thus the key thing your explanation should focus on is how the utility levels you’ve assigned depict Voter 7’s preferences over her available action changing when her expectations change about what Voters 4, 5 and 6 will do!

Prompt 4

To solidify your understanding of how the utility levels assigned by a payoff function depict strategic interdependence, consider the following two profiles: \begin{array}{c} (B,B,A,A)\\ (B,A,A,A)\\ \end{array}

Assign utility levels to Voter 5 at each of the these two profiles. Assign utility levels that are consistent with Voter 5’s preferences over the election outcome and with the assumption above about how ties are resolved.

Prompt 5

Assign utility levels to Voter 5 at each of the following profiles: \begin{array}{c} (B,A,B,A)\\ (B,B,B,A)\\ \end{array}

Assign utility levels that are consistent with Voter 5’s preferences over the election outcome and with the assumption above about how ties are resolved.

Prompt 6

In up to one page of double-spaced text, explain how the utility levels you assigned to Voter 5 in your answers to Prompts 4 and 5 depict the strategic interdependence between Voters 4 through 7 in this election.

In formulating your explanation, keep in mind that strategic interdependence only occurs when a person’s preferences over her available actions vary according to her expectations about the actions other persons will take! Thus the key thing your explanation should focus on is how the utility levels you’ve assigned depict Voter 5’s preferences over her available action changing when her expectations change about what Voters 4, 6 and 7 will do!

Rubric

Prompt 1

The answer gets one point if it consists of the correct table, with the column containing the profiles of actions fully and correctly filled in. The answer gets zero points otherwise.

Prompt 2

The answer gets one point if it consists of the correct table, with the column containing the profiles of actions fully and correctly filled in, and with the columns for Person 7’s utility level filled in as 1 and 0 at the profiles (B,B,A,B), (B,B,A,A), and with the utility levels for Voter 7 at the profiles (A,A,B,A), (A,A,B,B) filled in, with the former higher than the latter.

The answer gets one point if it consists of the correct table filled in as required. The answer gets zero points otherwise.

Prompt 3

A successful answer does all of the following

It points out that the utility levels assigned in the answer to Prompt 2 imply that Voter 7 prefers to vote for A instead of B when she expects Voters 4 and 5 to vote for A and Voter 6 to vote for B, but prefers to vote for B instead of A when she expects Voters 4 and 5 to vote for B and Voter 6 to vote for A. Therefore, Voter 7’s preference ordering over her actions switches if her expectations about the actions the other voters will take changes.
It points out that a group of persons are strategically interdependent when each one’s preferences over her actions depend on her expectations about what the other persons in the group will do.
It must be sufficiently free of errors of spelling, grammar and usage as to be readily and easily comprehensible.
If hand written, it must be immediately and easily legible.

An answer gets…

Four points…: if it meets all of the above criteria.
Two points…: if it meets criteria (c) and (d) but misses either criterion (a) or (b)
Zero points…: if it misses either (c) or (d).

Prompt 4

The answer gets one point if it consists of the correct table, with the column containing the profiles of actions fully and correctly filled in, and with the columns for Person 5’s utility level filled at the profiles (B,B,A,A), (B,A,A,A) with the latter higher than the former.

The answer gets one point if it consists of the correct table filled in as required. The answer gets zero points otherwise.

Prompt 5

The answer gets one point if it consists of the correct table, with the column containing the profiles of actions fully and correctly filled in, and with the columns for Person 5’s utility level filled at the profiles (B,A,B,A), (B,B,B,A) with the latter higher than the former.

The answer gets one point if it consists of the correct table filled in as required. The answer gets zero points otherwise.

Prompt 6

A successful answer does all of the following

It points out that the utility levels assigned in the answer to Prompts 4 and 5 imply that Voter 5 prefers to vote for A instead of B when she expects Voter 4 to vote for B and Voters 6 and 7 to vote for A, but prefers to vote for B instead of A when she expects Voters 4 and 6 to vote for B and Voter 7 to Vote for A. Therefore, Voter 5’s preference ordering over her actions switches if her expectations about the actions the other voters will take changes.
It points out that a group of persons are strategically interdependent when each one’s preferences over her actions depend on her expectations about what the other persons in the group will do.
It must be sufficiently free of errors of spelling, grammar and usage as to be readily and easily comprehensible.
If hand written, it must be immediately and easily legible.

An answer gets…

Four points…: if it meets all of the above criteria.
Two points…: if it meets criteria (c) and (d) but misses either criterion (a) or (b)
Zero points…: if it misses either (c) or (d).

function policySpace(params){
  function vArrowDown(x,y){
    return `M${x} ${y} l4 -20 l-8 0 Z m0 -20 l0-20`
  }
  function vArrowUp(x,y){
  return `M${x} ${y} l4 20 l-8 0 Z m0 20 l0 20`
  }
  const chartHeight = 150;
  const margin = ({top: 5, right: 10, bottom: 5, left: 10})
  const pScale = d3.scaleLinear()
        .domain([0,100])
        .range([margin.left, width - margin.right])
  const pSpacePoints = [{x: 0}, {x: 100}]
  const pPath = d3.line()
        .x(d => pScale(d.x))
        .y(d => chartHeight-margin.bottom-60)
  const axisLabel = htl.html`
    <div class="axis-label-container">
      <div>
        ${params.axisLabel}
        <span class="svg-icon svg-baseline">
        <svg viewBox="0 0 11 4">
          <path d="M0 1 l10 0 l-2 -2 m2 2 l-2 2" stroke="black" fill="none" stroke-width="1px" />
        </svg>
        </span>
      </div>
    </div>
  `
  const view = htl.html`
    <div class="spatial-display">
      <svg viewBox="0 0 ${width} ${chartHeight}" style="font: 16px sans-serif;">
          ${params.policies ? params.policies.map(policy => htl.svg`
            <g transform="translate(${pScale(policy.x)},${chartHeight-margin.bottom-60})">
              <text x=0 y=-45 text-anchor="middle" >${policy.label}</text>
              <path d=${vArrowDown(0,0)} stroke="black" fill="none" stroke-width="1px" />
            </g>
          `) : null}
          <path d=${pPath(pSpacePoints)} stroke="black" fill="none" stroke-width="6px" stroke-linecap="round"/>
          ${params.persons ? params.persons.map(person => htl.svg`
            <g transform="translate(${pScale(person.x)},${chartHeight-margin.bottom-60})">
              <text x=0 y=58 text-anchor="middle" >${person.label}</text>
              <path d=${vArrowUp(0,0)} stroke="black" fill="none" stroke-width="1px" />
            </g>
          `) : null}
      </svg>
      ${params.axisLabel ? axisLabel : null}
      ${params.persons ? params.persons[0].desc ? htl.html`
      <div class="policies-list-container">
        <ul class="policies-list">
          ${params.persons.map(person => htl.html`
            <li><b>${person.label}</b>: ${person.desc}</li>
          `)}
        </ul>
      </div>
      `
      : null : null}
      
    </div>
  `
  return view;
}