Decision Theory

Table of Contents

Core Idea

The core idea of decision theory is expected value:
- The expected value of a state-of-affairs is the product of its probability and value.
- The expected value of an action is the sum of the expected values of the possible outcomes.
For example, the expected value of betting $100 on red in roulette in Las Vegas is:
- Expected value of winning = value of winning x probability of winning = $100 x (18/38) = $47.37
- Expected value of losing = value of losing x probability of losing = –$100 * (20/38) = –$52.63
- So, the expected value of betting $100 on red = expected value of winning + expected value of losing = $47.37 – $52.63 = –$5.26
- [A Las Vegas roulette wheel has 38 grooves: 18 red, 18 black, 2 zero.]

Decision Making using Expected Value

Having bought a refrigerator, you’re offered an extended warranty: pay $200 and any repairs within five years are at no charge.
Suppose the chance of a breakdown over five years is 1 of 5 and the average cost of a repair is $500.
If you forego the warranty,
- the expected cost of repairs is $100 (= ⅕ x $500)
- the expected cost of no repairs is $0 (= ⅘ x $0),
- so the total expected cost of foregoing the warranty is $100 (= $100 + $0).
Since the expected cost of declining the warranty, $100, is less than the cost of the warranty, $200, you should skip the warranty, other things being equal.
The reasoning is represented by a Decision Tree:

Is a bird in the hand worth two in the bush?

Suppose a bird is worth $10 and the probability of catching a bird in the bush is ½. Let’s assume you have two options:
- keep the bird in your hand
- let your bird go and try to catch the two birds in the bush.
The expected value of the first option is $10 multiplied by probability 1.0.
The second option has four equally likely outcomes, with a total expected value of $10.

A decision tree:

Thus, indeed, a bird in the hand is worth two in the bush.

Decision Trees for Complex Decisions

Decision Trees are useful in making complex decisions, e.g. the decision to stay or leave JCPOA, the Iran Nuclear Agreement.

Decisions under Certainty, Risk, and Ignorance

A decision under certainty is a decision where the possible consequences of the options are reasonably certain, so only their value needs to be considered.
A decision under risk is a decision where the (im)probabilities of possible outcomes are known, as well as their value.
- For example, the decision on the extended warranty.
A decision under ignorance is a decision where the (im)probabilities of the possible outcomes are unknown.

Decisions under Ignorance

An example from Martin Peterson’s An Introduction to Decision Theory

“In the 1960s, Dr. Christiaan Barnard in Cape Town experimented on animals to develop a method for transplanting hearts. In 1967 he offered 55-year-old Louis Washkansky the chance to become the first human to undergo a heart transplant. Mr. Washkansky was dying of severe heart disease and was in desperate need of a new heart…. The decision made by Mr. Washkansky was a decision under ignorance. This is because it was virtually impossible for him (and Dr. Barnard) to assign meaningful probabilities to the possible outcomes. No one knew anything about the probability that the surgical method would work. However, it was nevertheless easy for Mr. Washkansky to decide what to do…. He had nothing to lose.”

Expected Value of PowerBall

The expected value of buying a $2.00 Powerball ticket where the jackpot is $302 million is:
- Expected value of winning the jackpot = the value of winning x the probability of winning = $302,000,000 x 1/292,201,338 = $1.04
- Expected value of losing = the value of losing x the probability of losing = –$2.00 x (292,201,337 / 292,201,338) = –$2.00
- So, the expected value of buying a Powerball ticket = expected value of winning + expected value of losing = $1.04 – $2.00 = –$0.96
This calculation ignores the possibility of multiple winners and alternative ways of winning.