COU 5: Bribery

To make sure you understand how the model of the potential candidate’s trade-offs works, you’ll work in this COU with a model of a completely different phenomenon: Bribery. As a depiction of bribery, this model is simplistic and uninteresting. But it has a structure very similar to that of the potential candidate’s choice to run, and so it’s useful for checking your understanding of that model.

Imagine a government officeholder who has the power to determine the location of policy in a policy space. Suppose that this government officeholder is approached by a wealthy and unscrupulous person who offers to pay the officeholder a bribe in exchange for moving policy from its current location, which -5, to a new location 0. It looks like this:

{
  const margin = {top: 10, right: 10, bottom: 20, left: 10}
  const chartHeight = 150;
  let chart = d3.create('svg')
    .attr('viewBox', `0 0 ${width} ${chartHeight}`)
  const pScale = d3.scaleLinear()
    .domain([-6,2])
    .range([margin.left, width - margin.right])
  const axis = d3.axisBottom(pScale)
    .ticks(9)
  chart.append('g')
    .attr('transform', `translate(0,${chartHeight-margin.bottom})`)
    .call(axis)
  const policies = [
    {
      label: '',
      desc1: "policy is currently",
      desc2: "located here",
      pos: -5
    },
    {
      label: '',
      desc1: "officeholder gets a bribe",
      desc2: "if she moves policy here",
      pos: 0
    }
  ]
  chart
    .selectAll('g.policy')
    .data(policies)
    .enter()
    .append('g')
      .attr('class', 'policy')
      .attr('font-size', '10')
      .attr('font-family', 'sans-serif')
      .attr('text-anchor', 'middle')
      .attr('transform', d => `translate(${pScale(d.pos)}, ${chartHeight-margin.bottom})`)
    .append('path')
      .attr('d', 'M0 0 l-5 -10 l10 0 l-5 10 m0 -10 l0 -20')
      .attr('stroke', 'black')
      .attr('stroke-width', '1px')
      .attr('fill', 'none')
  chart.selectAll('g.policy')
    .data(policies)
    .append('text')
      .attr('x', 0)
      .attr('y', -60)
      .text(d => `${d.desc1}`)
  chart.selectAll('g.policy')
    .data(policies)
    .append('text')
      .attr('x', 0)
      .attr('y', -45)
      .text(d => `${d.desc2}`)
  return chart.node();
}

Suppose the officeholder cares about the location of policy. Specifically, she has single-peaked preferences with an ideal point of -3, like so:

{
  const margin = {top: 10, right: 10, bottom: 20, left: 10}
  const chartHeight = 150;
  let chart = d3.create('svg')
    .attr('viewBox', `0 0 ${width} ${chartHeight}`)
  const pScale = d3.scaleLinear()
    .domain([-6,2])
    .range([margin.left, width - margin.right])
  const axis = d3.axisBottom(pScale)
    .ticks(9)
  chart.append('g')
    .attr('transform', `translate(0,${chartHeight-margin.bottom})`)
    .call(axis)
  const policies = [
    {
      label: '',
      desc1: "policy is currently",
      desc2: "located here",
      pos: -5
    },
    {
      label: '',
      desc1: "officeholder's",
      desc2: "ideal point",
      pos: -3
    },
    {
      label: '',
      desc1: "officeholder gets a bribe",
      desc2: "if she moves policy here",
      pos: 0
    }
  ]
  chart
    .selectAll('g.policy')
    .data(policies)
    .enter()
    .append('g')
      .attr('class', 'policy')
      .attr('font-size', '10')
      .attr('font-family', 'sans-serif')
      .attr('text-anchor', 'middle')
      .attr('transform', d => `translate(${pScale(d.pos)}, ${chartHeight-margin.bottom})`)
    .append('path')
      .attr('d', 'M0 0 l-5 -10 l10 0 l-5 10 m0 -10 l0 -20')
      .attr('stroke', 'black')
      .attr('stroke-width', '1px')
      .attr('fill', 'none')
  chart.selectAll('g.policy')
    .data(policies)
    .append('text')
      .attr('x', 0)
      .attr('y', -60)
      .text(d => `${d.desc1}`)
  chart.selectAll('g.policy')
    .data(policies)
    .append('text')
      .attr('x', 0)
      .attr('y', -45)
      .text(d => `${d.desc2}`)
  return chart.node();
}

In addition to caring about the location of policy, the officeholder also likes money. Thus, when she chooses whether or not to accept the deal offered by the wealthy and unscrupulous person, she faces a tradeoff. On the one hand (and as you can see from the diagram above) accepting the deal requires the officeholder to move policy to a location further from her ideal point than the location of current policy. On the other, accepting the deal entails getting money!

Given the location x of the policy that results from the officeholder’s choice and the amount of money m that the officeholder collects, suppose the officeholder’s utility level is given by the function: - | x - (-3) | + m

Notice that (like the utility function used in the model of the potential candidate’s choice in the lesson), this utility function consists of two terms – a “Policy Term” and a “Money Term” – like so: \begin{array}{ccc} \underbrace{ - | x - (-3) | } & + & \underbrace{m} \\ \text{Policy Term} & & \text{Money Term} \end{array}

Assume that events in the model occur as follows:

The wealthy and unscrupulous person offers the officeholder a specific amount of money M.
The officeholder then chooses between one of two available options: accept the offer or reject the offer.
If the officeholder accepts the offer, the location of policy x is set equal to 0 and the officeholder receives the amount M of money offered. If the officeholder rejects the offer, x is set equal to -5 and the officeholder collects 0 units of money.

Prompt 1

Suppose the amount of money M that the wealthy and unscrupulous person offers is 0.5. (a) Write the value of the Policy Term, the value of the Money Term, and the overall utility level that results if the officeholder accepts the offer, (b) write the value of the Policy Term, the value of the Money Term, and the overall utility level that results if the officeholder rejects the offer, and (c) say whether or not the officeholder prefers to accept the offer or reject the offer.

Prompt 2

Suppose the amount of money M that the wealthy and unscrupulous person offers is 2. (a) Write the value of the Policy Term, the value of the Money Term, and the overall utility level that results if the officeholder accepts the offer, (b) write the value of the Policy Term, the value of the Money Term, and the overall utility level that results if the officeholder rejects the offer, and (c) say whether or not the officeholder prefers to accept the offer or reject the offer.

Prompt 3

Suppose the amount of money M that the wealthy and unscrupulous person offers is 4. (a) Write the value of the Policy Term, the value of the Money Term, and the overall utility level that results if the officeholder accepts the offer, (b) write the value of the Policy Term, the value of the Money Term, and the overall utility level that results if the officeholder rejects the offer, and (c) say whether or not the officeholder prefers to accept the offer or reject the offer.

Prompt 4

What is the exact amount of money that the wealthy and unscrupulous person would have to offer to make the officeholder just indifferent between accepting and rejecting the offer – meaning that the officeholder gets the same utility level from each option? State your answer, explain how you know it’s the correct answer, and show how you used the officeholders utility function to derive your answer.

Prompt 5

For the purpose of this prompt, we’ll modify the model to depict the idea that officeholders vary in the extent to which they care about policy relative to money. Specifically, some officeholders care a lot about the location of policy relative to money, and so have to be offered very large bribes to get them to make changes to policy they disagree with. On the other hand, some officeholders do not care much about policy relative to money, and so can be bought relatively cheaply.

More specifically, suppose that just as in the model used for Prompts 1 through 4, the wealthy and unscrupulous individual offers the officeholder an amount of money M. If the officeholder accepts the offer, the location of policy is set to 0 and the officeholder is paid M units of money. And if the officeholder rejects the offer, the location of policy is set to -5 and the officeholder collects 0 money. Unlike the model used for the previous prompts, however, the officeholder’s utility as a function of the final location of policy x and the amount of money m she collects is: -P | x - (-3) | + m where P is some positive number.

In this utility function, the parameter P depicts the extent to which the officeholder’s utility level decreases as the final policy x moves away from her ideal point -3, relative to the extent to which the officeholder’s utility increases as she collects more money.

Answer the following prompts:

Write the expression (which will be a function of the parameter P) for the amount of money that makes the officeholder just indifferent between accepting the offer and rejecting the offer. Show and explain in complete and coherent sentences the steps you went through to derive this expression.
Is the amount of money that makes the officeholder just indifferent between accepting the offer and rejecting the offer increasing or decreasing as a function of P. Use your answer to (a) to explain how you know the answer.

Rubric

For each of Prompts 1 through 3, you get 1 point if you correctly answer all three parts (a, b and c) and 0 points otherwise.

For Prompt 4, you get 2 points if you give a correct explanation of the answer you give and correctly show how to derive that answer and 0 points otherwise.

For Prompt 5 part (a) you get 1 point if you write the correct expression, plus an additional 2 points for giving a correct and complete explanation in complete and coherent sentences of the steps used to derive that expression.

For Prompt 5 part (b) you get 1 point only if you got full credit (3 points) part (a) and if you give the correct answer, plus an additional 2 points for giving a correct and complete explanation of how that correct answer follows from the correct answer to part (a).