COU 3: Modeling Simple Preferences with Utility Functions

The severity of the penalities the judicial system applies to any given category of crime can be readily depicted with a numerical spatial model. For instance, imagine a set of policies that each set a minimum term of imprisonment for persons convicted of selling drugs. A numerical spatial model of such policies would locate policies that assign longer prison terms at higher numbers, like so:

{
  const margin = {top: 10, right: 10, bottom: 20, left: 10}
  const chartHeight = 150;
  let svg = d3.create('svg')
    .attr('viewBox', `0 0 ${width} ${chartHeight}`)
  const pScale = d3.scaleLinear()
    .domain([-12,12])
    .range([margin.left, width - margin.right])
  const axis = d3.axisBottom(pScale)
    .ticks(25)
  svg.append('g')
    .attr('transform', `translate(0,${chartHeight-margin.bottom})`)
    .call(axis)
  const policies = [
    {
      label: 'A',
      desc: '1 year minimum',
      pos: -10
    },
    {
      label: 'B',
      desc: '8 year minimum',
      pos: 0
    },
    {
      label: 'C',
      desc: '10 year minimum',
      pos: 4
    }
  ]
  svg
    .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)}, ${chartHeight-margin.bottom})`)
    .append('path')
      .attr('d', 'M0 0 l-5 -10 l10 0 l-5 10 m0 -10 l0 -20')
      .attr('stroke', 'black')
      .attr('stroke-width', '1px')
      .attr('fill', 'none')
  svg.selectAll('g.policy')
    .data(policies)
    .append('text')
      .attr('x', 0)
      .attr('y', -32)
      .text(d => d.label)
  svg.selectAll('g.name')
    .data(policies)
    .enter()
    .append('g')
      .attr('class', 'name')
      .attr('font-size', '14')
      .attr('font-family', 'sans-serif')
      .attr('text-anchor', 'start')
      .attr('transform', (d,idx) => `translate(${0.4*width}, ${margin.top + idx*20})`)
    .append('text')
      .attr('x', 0)
      .attr('y', 0)
      .text(d => `${d.label}: ${d.desc}`)
    
  
  return svg.node();
}

Write four different utility functions that represent a person’s preferences over these policies. Each utility function you write must represent the person as preferring shorter prison terms to longer prison terms.

For each utility function you write, illustrate the fact that it represents the person as preferring shorter to longer terms by picking two arbitrary locations, computing the value of the utility function at each of those locations, and pointing out that the value at the lower location is higher than the value at the higher location.

Rubric

You can earn up to 8 points on this COU, 2 points for each of the four different utility functions you write. For each of those utility functions, you get…

• 2 points if what you write is a utility function that is not the same function as any of the other functions you write and that represents the person as preferring shorter to longer terms and you illustrate (as described in the instructions above) that it represents the person as preferring shorter to longer terms.
• 1 point if what you write is a utility function that is not the same function as any of the other functions you write and that represents the person as preferring shorter to longer terms but you don’t illustrate that in a way that follows all the requirements of the instructions above.
• 0 points otherwise.