16  Continuity

Warning

This section contains more advanced material and can be skipped on a first reading.

Suppose that \(X=\{a, b, c\}\) and that a rational decision maker strictly prefers \(a\) to \(b\) and \(b\) to \(c\) (so item \(a\) is the favorite, item \(b\) is the second-favorite, and item \(c\) is the least-favorite). The decision maker is offered series of choices between taking \(b\) for sure, or a gamble between \(a\) and \(c\). That is, the decision maker is asked to compare the following two lotteries for different values of \(r\in [0, 1]\): \[[b:1]\qquad\mbox{vs.}\qquad [a:r, c:1-r].\] When \(r=1\), the gamble is strictly preferred to the sure-thing (i.e., \([a:r, c:1-r]\mathrel{P} [b:1]\)) and when \(r=0\), the sure-thing is strictly preferred to the gamble (i.e., \([b:1] \mathrel{P} [a:r, c:1-r]\)). This means that as \(r\) ranges from \(0\) to \(1\), at some point, the preference between the sure-thing and the gamble flips. Assuming that this change of opinion is “continuous” means that there must be some value of \(r\) such that the decision maker is indifferent between the sure-thing \([b:1]\) and the gamble \([a:r, c:1-r]\).

For example, suppose that the decision maker ranks lotteries by comparing their expected utilities for a fixed utility function \(u:\{a, b, c\}\rightarrow\mathbb{R}\). In the following graph, you can choose different utilities for \(a\), \(b\), and \(c\) defining a utility function \(u:\{a, b, c\}\rightarrow\mathbb{R}\). The graph displays the utility of \(a\) (the orange line), the utility of \(b\) (the red line), the utility of \(c\) (the green line), and \(EU(L, u)\) where \(L=[a:r, c:1-r]\) as \(r\) ranges between \(0\) and \(1\) (the blue line). We also display the value of \(r\) such that \(EU([a:r, c:1-r], u)=EU([b:1], u)\), when it exists (the dashed red line).

viewof a = Inputs.range([0,10], {value: 7, step:0.01,  label:  tex.block`u(a)=`}
)
viewof b = Inputs.range([0,10], {value: 4,  step:0.01, label: tex.block`u(b)=`}
)

viewof c = Inputs.range([0,10], {value: 0, step:0.01,  label: tex.block`u(c)=`}
)

p = Math.round(((b-c) / (a-c)) * 1000) / 1000;

display_p = p >= 0 && p <= 1 ? p : -1;

(c <= b && b <= a) || (a <= b && b <= c) ? 
tex.block`
  EU([a:${p}, c:${Math.round((1-p)*1000) / 1000}], u) = ${Math.round((p * a + (1-p) * c) * 100) / 100} = EU([b:1], u)
` : tex.block` \text{There is no } r \text{ such that } EU([a:r, c:1-r], u) = EU([b:1], u)`

data_a = [
    {"x": 0, "y": a, "type":  "utility of a"},
    {"x": 1, "y": a, "type":  "utility of a"},
]
data_b = [
    {"x": 0, "y": b, "type":  "utility of b"},
    {"x": 1, "y": b, "type":  "utility of b"},
]

data_c = [
    {"x": 0, "y": c,  "type":  "utility of c"},
    {"x": 1, "y": c, "type":  "utility of c"},
]

data_eu={
  let d_eu = [];
  let x = 0;
  for (let i = 0; i <= 10000; i++) {
    d_eu[i] = {"x": x, "y": x * a + (1-x) * c, "type":  "expected utility of L"};
    x += 0.0001;
  }
  return d_eu;
}

Plot.plot({
 style: "overflow: visible;",
  inset: 10,
  x: {
    domain:[0, 1]
  },
  y: {
    grid: true,
    domain: [0, 10]
  },
    color: {
    legend: true
  },

  marks: [
    Plot.line([{x:0, y:a, "type":  "utility of a"}, {x:1, y:a,"type":  "utility of a"}], Plot.windowY({k: 14, x: "x", y: "y", stroke: "type", strokeWidth: 2})),
    Plot.ruleX([{"x": display_p, "init_y": -0.1, "end_y": b}], {x:"x", y1: "init_y", y2: "end_y", stroke: "red", strokeDasharray:"1"}),
    Plot.dot([{"x": display_p, "y": 0.3}], {x:"x", y:"y", fill: "white", r: 7}),
    Plot.text([{"x": display_p, "y": 0.3, "text": p}], {x: "x", y: "y", text: "text"}),

    Plot.line(data_b, Plot.windowY({k: 14, x: "x", y: "y", stroke: "type", strokeWidth: 2})),
    Plot.line(data_c, Plot.windowY({k: 14, x: "x", y: "y", stroke: "type",strokeWidth: 2})),
    Plot.line(data_eu, Plot.windowY({k: 30, x: "x", y: "y", stroke: "type", strokeWidth: 3})),

  ]
})

The Continuity Axiom ensures that a decision maker’s preference are continuous in the sense described above.

Continuity Axiom

Suppose that \(\mathcal{L}\) is a set of lotteries and \((P, I)\) is a rational preference over \(\mathcal{L}\). For all \(L, L', L''\in \mathcal{L}\), if \(L\mathrel{P} L'\mathrel{P} L''\), then there exists an \(r\in (0, 1)\) such that \[L'\mathrel{I} [L:r,\ L'':(1-r)].\]

16.1 Exercises

1. Suppose that \(u:\{a, b, c\}\rightarrow\mathbb{R}\) is a utility function with \(u(a)=2\), \(u(b)=1\) and \(u(c)=0\). Let \(U\) be a utility function on \(\mathcal{L}(X)\) where for all \(L\in\mathcal{L}(X)\), \(U(L) = EU(L, u) + 0.5\) if \(L\) is not a sure-thing, \(U(L) = EU(L, u)\) if \(L\) is a sure-thing. Show that the preference generated from this utility function violates the Continuity Axiom.

2. Suppose that \(u:\{a, b, c\}\rightarrow\mathbb{R}\) is a utility function with \(u(a)=2\), \(u(b)=1\) and \(u(c)=0\). Let \(U\) be a utility function on \(\mathcal{L}(X)\) where for all \(L\in\mathcal{L}(X)\), \(U(L) = EU(L, u)\) if \(L\) is not a sure-thing, \(U(L) = 2*EU(L, u)\) if \(L\) is a sure-thing. Show that the preference generated from this utility function violates the Continuity Axiom.