4  Lotteries

Suppose that \(X\) is a finite set. Elements of \(X\) are called outcomes or prizes. A lottery on \(X\) is a probability function on \(X\).

Definition 4.1 (Lottery) Suppose that \(X\) is a finite set. A lottery, or probability, on \(X\) is a function \(p:X \rightarrow [0, 1]\) such that \(\sum_{x\in X} p(x) = 1\).

Note

There are a number of mathematical details about probability that we are glossing over here. Our discussion in this course is greatly simplified since we assume that the set of objects \(X\) is finite.

To simplify notation, we represent a lottery \(p:X\rightarrow [0, 1]\) on a set \(X=\{x_1,\ldots, x_n\}\) as a list associating each outcome with its probability: \([x_1:p(x_1), x_2:p(x_2), \ldots, x_n:p(x_n)]\).

For instance, if \(X=\{a, b, c\}\), then the following are examples of three lotteries on \(X\):

1. \([a:0, b:1, c:0]\): There is a 100% chance of getting \(b\).
2. \([a:0.25, b:.35, c:0.4]\): There is a 25% chance of getting \(a\), 35% chance of getting \(b\), and a 40% chance of getting \(c\).
3. \([a:0.25, b:.75, c:0]\): There is a 25% chance of getting \(a\) and a 75% chance of getting \(b\).

We will make use of the following notation about lotteries in these notes:

• Lotteries in which one outcome is assigned probability 1 are called sure-things. We associate each element of \(x\in X\) with the sure-thing lottery \([x:1]\).
• We may not include outcomes that are assigned probability 0 by a lottery. For instance, suppose that \(X=\{a, b, c, d\}\). If we say that \([b:0.5, c:0.5]\) is a lottery on \(X\), then this is the lottery \([a:0, b:0.5, c:0.5, d:0]\).
  
• If lotteries contain the same prize with different probabilities, we can simplify by summing the probabilities for that prize. For instance, the lottery \([a:0.2, b:0.1, a:0.3, c:0.1, b:0.3]\) is the same lottery as \([a:(0.3 + 0.2), b:(0.1 + 0.3), c:0.1] = [a:0.5, b:0.4, c:0.1]\).

4.1 Compound Lotteries

Suppose that \(L_1=[a: 0.5, b: 0.5]\) and \(L_2 = [b:0.25, c:0.75]\) are two lotteries on \(X=\{a, b, c\}\). Now, consider the lottery in which a fair coin is flipped and if it lands heads, then the lottery \(L_1\) is played, otherwise the lottery \(L_2\) is played. This compound lottery can be represented as \([L_1:0.5, L_2:0.5]\). In fact, any set of lotteries can be combined to form a compound lottery.

Definition 4.2 (Compound Lottery) Suppose that \(L_1, \ldots, L_n\) are lotteries on a set \(X\). Then, \([L_1:p_1,\ldots, L_n:p_n]\) is compound lottery, where \(\sum_{i=1}^n p_i=1\).

We can display compound lotteries as a tree in which \([L_1:p_1, \ldots, L_n:p_n]\) is the tree in which there is an edge from the root node to the tree representing \(L_i\) labeled by \(p_i\).

• The lottery \([a:0.5, b:0.3, c:0.2]\) can be pictured as follows:

• The lottery \([[a:0.4, b:0.6]:0.5, [a:0.6, c:0.4]:0.5]\) can be pictured as follows:

• The lottery \([[[a:0.5, b:0.5]: 0.25, [a:0.3, b:0.7]: 0.75]: 0.5, c: 0.5]\) can be pictured as follows:

4.2 Exercises

1. Consider the lottery in which a fair coin is flipped. If it lands heads, then you win \(\$100\) and if it lands tails, you lose $100. Write this lottery down using the notation described above.

2. Consider the lottery in which a biased coin is flipped. If it lands heads, then you win \(\$100\) and if it lands tails, you lose $100. Suppose that bias of the coin is that the chance for heads is 3-times the chance for tails. Write this lottery down using the notation described above.

3. Consider the lottery in which a fair coin is flipped. If it lands heads, then the you lose \(\$5\) and if it lands tails, then you roll a fair die (with 6-sides) and you win the amount in dollars shown on the die. Write this lottery down using the notation described above.

4. Consider \([\$10: 0.5, [\$10: 0.3, \$5:0.7]: 0.5]\). What is the probability that you will \(\$10\)? What is the probability that you win \(\$5\)?

5. Draw the tree that depicts the following lotteries:

   1. \([a:0.2, b:0.4, a:0.1, c:0.3]\)
      
   2. \([[a:0.5, b:0.5]:0.2, [a:1]:0.8]\)
   3. \([[[a:0.5, b:0.5]:0.5, [a:0.5, b:0.5]:0.5]:0.5, b:0.5]\)

Solutions

1. Consider the lottery in which a fair coin is flipped. If it lands heads, then you win \(\$100\) and if it lands tails, you lose $100. Write this lottery down using the notation described above.

   \[[\$100: 0.5, -\$100:0.5]\]

2. Consider the lottery in which a biased coin is flipped. If it lands heads, then you win \(\$100\) and if it lands tails, you lose $100. Suppose that bias of the coin is that the chance for heads is 3-times the chance for tails. Write this lottery down using the notation described above.

   \[[\$100: 0.75, -\$100:0.25]\]

3. Consider the lottery in which a fair coin is flipped. If it lands heads, then the you lose \(\$5\) and if it lands tails, then you roll a fair die (with 6-sides) and you win the amount in dollars shown on the die. Write this lottery down using the notation described above.

   \[[-\$5: \frac{1}{2}, [1:\frac{1}{6}, 2:\frac{1}{6}, 3:\frac{1}{6}, 4:\frac{1}{6}, 5:\frac{1}{6}, 6:\frac{1}{6}]:\frac{1}{2}]\]

4. Consider \([\$10: 0.5, [\$10: 0.3, \$5:0.7]: 0.5]\). What is the probability that you will \(\$10\)? What is the probability that you win \(\$5\)?

   • What is the probability that you will \(\$10\)? \(0.5 + 0.5 * 0.3 = 0.65\)
   • What is the probability that you win \(\$5\)? \(0.5 * 0.7 = 0.35\)

5. Draw the tree that depicts the following lotteries:

1. \([a:0.2, b:0.4, a:0.1, c:0.3]\)
   
2. \([[a:0.5, b:0.5]:0.2, [a:1]:0.8]\)
3. \([[[a:0.5, b:0.5]:0.5, [a:0.5, b:0.5]:0.5]:0.5, b:0.5]\)