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 0, to a new location -4. 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: 0
    },
    {
      label: '',
      desc1: "officeholder gets a bribe",
      desc2: "if she moves policy here",
      pos: -4
    }
  ]
  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 -1, 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: 0
    },
    {
      label: '',
      desc1: "officeholder's",
      desc2: "ideal point",
      pos: -1
    },
    {
      label: '',
      desc1: "officeholder gets a bribe",
      desc2: "if she moves policy here",
      pos: -4
    }
  ]
  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: \begin{Bmatrix} x-(-1) & \text{if $x < -1$} \\ 0 & \text{if $x = -1$} \\ -1-x & \text{if $-1 < x$} \end{Bmatrix} + 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{ \begin{Bmatrix} x-(-1) & \text{if $x < -1$} \\ 0 & \text{if $x = -1$} \\ -1-x & \text{if $-1 < x$} \end{Bmatrix} } & + & \underbrace{m} \\ \text{Policy Term} & & \text{Money Term} \end{array}

Assume that the wealthy and unscrupulous person offers the officeholder a specific amount of money if the officeholder accepts the deal. The officeholder then chooses between one of two available options: accept the deal or reject the deal. If the officeholder accepts the deal, x is set equal to -4 and m is set equal to the amount of money offered. If the officeholder rejects the deal, x is set equal to 0 and m is set equal to 0.

Prompt 1

Suppose the amount of money that the wealthy and unscrupulous person offers is 1. (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 deal, (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 deal, and (c) say whether or not the officeholder prefers to accept the deal or reject the deal.

Prompt 2

Suppose the amount of money 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 deal, (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 deal, and (c) say whether or not the officeholder prefers to accept the deal or reject the deal.

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 deal, (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 deal, and (c) say whether or not the officeholder prefers to accept the deal or reject the deal.

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 deal – 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.

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.