What it means
Decision theory divides into two halves that are frequently confused. Normative theory describes how a perfectly rational actor ought to choose, while descriptive theory studies how real people actually choose, which turns out to be systematically different.
The central idea is expected utility. Rather than ranking options purely by their average cash outcome, you first convert each outcome into a measure of its value to you, which captures the fact that losing $500,000 usually hurts a great deal more than gaining $500,000 pleases.
That distinction explains behaviour that looks irrational on a spreadsheet. A company insuring a factory pays more in premiums over time than it expects to claim, and it is entirely right to do so, because the ruin caused by an uninsured fire is worth paying a price to avoid.
Decision theory also supplies rules for updating beliefs as evidence arrives, and criteria for choosing when probabilities are simply unknown. Maximin picks the option with the least bad worst case, while minimax regret picks the option you are least likely to kick yourself over afterwards.
Its limits deserve stating plainly. Real decision makers have inconsistent preferences, shaky probability estimates and limited attention, so the theory works best as a check on reasoning rather than as a machine that hands over the answer.
In practice
Real-world examples.
Example
A shipping company insures its fleet against total loss even though the premiums exceed the expected value of claims. The board accepts the negative expected value because a single uninsured sinking would end the business, which is a textbook expected utility judgement.
Example
A treasurer choosing between two currency hedges faces one that is cheaper on average and one that caps the worst case. She chooses the cap, applying a maximin criterion because the company cannot survive the tail outcome the cheaper hedge leaves open.
Example
A venture fund happily backs 30 startups it expects to fail individually, because at portfolio level the maths works. The fund can take positive expected value bets that any single founder would be irrational to take, since the fund gets to repeat the gamble and the founder does not.
Formula
Calculation
Expected value = Sum of (probability x payoff)
Expected utility = Sum of (probability x utility of payoff)
Certainty equivalent = the guaranteed amount that carries the same utility as the gamble
A founder is offered a certain $1,000,000 to sell her stake today, or she can hold on for an outcome that carries a 50% chance of $2,500,000 and a 50% chance of nothing.
Expected value of holding = 0.50 x $2,500,000 + 0.50 x $0 = $1,250,000
On expected value alone she should hold, since $1,250,000 beats $1,000,000. Now apply a utility function that reflects diminishing value, where the utility of an amount is its square root.
Utility of selling = square root of $1,000,000 = 1,000
Utility of holding = 0.50 x square root of $2,500,000 = 0.50 x 1,581.1 = 790.6
Selling scores higher, so under this utility function she should take the certain money. The certainty equivalent of the gamble is 790.6 squared, which is $625,000 to the nearest thousand, meaning she is willing to give up $1,250,000 - $625,000 = $625,000 of expected value in order to remove the risk of ending up with nothing at all.Case study
Seen in the real world.
The company and people here are fictional, and the case is illustrative only. Larkfield Robotics had built a warehouse-picking system and faced a genuine fork in the road. A large integrator offered $3,000,000 for an exclusive licence, while going to market alone offered, on the founders' own estimates, a 40% chance of being worth $9,000,000 and a 60% chance of being worth nothing much at all.
On expected value the independent route looked better: 0.40 x $9,000,000 = $3,600,000, comfortably ahead of the certain $3,000,000. When the founders applied a square-root utility function to reflect what the money actually meant to them, the picture reversed. The utility of the licence was the square root of $3,000,000, about 1,732, while the utility of going alone was 0.40 x the square root of $9,000,000, which is 0.40 x 3,000 = 1,200, giving a certainty equivalent of 1,200 squared, or $1,440,000.
The founders took the licence, and the useful part was not the arithmetic but the conversation it produced. Writing down the utility function forced them to say out loud that a 60% chance of walking away with nothing after seven years was not a risk they were prepared to run, which was a statement about their own circumstances rather than about the technology. A venture fund holding thirty such positions would have reached the opposite conclusion, and both answers would have been correct.
Watch out
Common mistakes.
- Assuming the highest expected value is always the right choice. Expected value ignores how badly the worst case would hurt, which matters enormously when one bad outcome is fatal.
- Confusing risk aversion with irrationality. Preferring a smaller certain sum to a larger risky one is perfectly rational once you account for what each outcome is worth to that particular decision maker.
- Applying the theory to decisions where the probabilities are pure invention. Dressing guesses in formal notation makes them look authoritative without making them any more accurate.
Questions
People also ask.
What is utility in plain terms?
A measure of how much an outcome is worth to a specific person or organisation, which is not the same as the cash amount, since the second million matters less than the first.
What is the difference between risk and uncertainty?
Risk means you can put credible probabilities on outcomes, whereas uncertainty means you cannot, and different decision criteria apply in each case.
Is decision theory the same as decision analysis?
Decision theory is the underlying framework, while decision analysis is the applied practice of using it on a specific business problem.
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