14  Compound Lotteries

Consider the following two lotteries:

Both lotteries \(L_1\) and \(L_2\) assign the same probabilities to the outcomes. That is, we can simplify lottery \(L_2\) as follows: \[[a:(0.5*0.4 + 0.5 *0.6), b: 0.5*0.6, c: 0.5 *0.4].\]

More generally, given a compound lottery \(L\), we can define a simplified version of \(L\):

Definition 14.1 (Simplified Lottery) Suppose that \(L=[L_1:p_1,\ldots, L_n,p_n]\) is a compound lottery, where for each \(i=1,\ldots, n\), we have \(L_i=[x_1:p_{1,i},\ldots, x_n:p_{n,i}]\). Then the simplification of \(L\), denoted \(s(L)\), is: \[[x_1:(p_1p_{1,1}+ p_2p_{1,2} + \cdots p_np_{1,n}),\ldots, x_n:(p_1p_{n,1}+ p_2p_{n,2} + \cdots p_np_{n,n})].\]

For example, suppose that \(L=[[a:0.2, b:0.8]:0.4, [b:0.3, c:0.7]:0.6]\). Then, \[s(L)=[a:0.4 * 0.2, b: (0.4 * 0.8 + 0.6 * 0.3), c:0.6*0.7] = [a:0.08, b: 0.5, c:0.42].\]

If a decision maker is always indifferent between a lottery \(L\) and its simplified version, then the decision maker does not get any utility from the “thrill of gambling”. That is, all that matters to the decision maker when comparing lotteries is how likely she is to receive prizes that she prefers.

Compound Lottery Axiom
For any lottery \(L\), the decision maker is indifferent between \(L\) and the simplification of \(L\). Formally, if \(I\) represents the decision maker’s indifference relation, then for all lotteries \(L\), \(L\mathrel{I}s(L)\).

14.1 Exercises

  1. Find the simplifications of the following compound lotteries:

    1. \(L=[[a: 0.5, b:0.5]: 0.1, b:0.9]\)
    2. \(L=[[a: 0.5, c:0.5]: 0.1, b:0.9]\)
    3. \(L=[[a: 0.5, b:0.5]: 0.75, [a:0.2, b:0.8]:0.25]\)
    4. \(L=[[a: 1]: 0.75, [a:0.2, b:0.8]:0.25]\)
    5. \(L=[[a:0.75, b:0.25]: 0.75, [a:0.25, b:0.75]:0.25]\)

  1. Find the simplifications of the following compound lotteries:

    1. \(L=[[a: 0.5, b:0.5]: 0.1, b:0.9]\)
      \[\begin{align*} s(L)&=[a:(0.5 * 0.1), b: (0.5 * 0.1 + 0.9)] \\ &=[a:0.05, b: 0.95] \end{align*}\]
    2. \(L=[[a: 0.5, c:0.5]: 0.1, b:0.9]\)
      \[\begin{align*} s(L) &=[a:(0.5 * 0.1), c: (0.5 * 0.1), b:0.9] \\ &= [a:0.05, b: 0.9, c:0.05] \end{align*}\]
    3. \(L=[[a: 0.5, b:0.5]: 0.75, [a:0.2, b:0.8]:0.25]\)
      \[\begin{align*} s(L) &=[a:(0.5 * 0.75 + 0.2 * 0.25), b: (0.5 * 0.75 + 0.8*0.25)] \\ &= [a:0.425, b: 0.575] \end{align*}\]
    4. \(L=[[a: 1]: 0.75, [a:0.2, b:0.8]:0.25]\) \[\begin{align*} s(L) &=[a:(1 * 0.75 + 0.2 * 0.25), b: 0.8 * 0.25] \\ &= [a:0.8, b: 0.2] \end{align*}\]
    5. \(L=[[a:0.75, b:0.25]: 0.75, [a:0.25, b:0.75]:0.25]\) \[\begin{align*} s(L) &=[a:(0.75*0.75 + 0.25 * 0.25), b: (0.25 * 0.75 + 0.75 * 0.25)] \\ &= [a:0.625, b: 0.375] \end{align*}\]