When we evaluate complex objects (mother-in-law, gadgets), we assign weights to their characteristics: some have a bigger influence than others, which we might not be aware of. When we evaluate an uncertain situation, we assign weights to the possible outcomes. These weights are correlated with the probabilities of the outcomes: a 40% chance of winning the jackpot is more appealing than a 2% chance. Assigning weights sometimes happens deliberately, but often it is an automatic process of System 1.

The decision making in gambling provides a natural rule for the assignment of weights to outcomes: the more probable an outcome, the more weight it gets. The expected value of a gamble is the average of the outcomes, all weighted by their probability. This is called the ‘expectation principle’. Bernoulli applied this principle to the psychological value of the outcomes: the utility of a gamble is the average of the utilities of the outcomes, all weighted by their probability.

The expectation principle is flawed, because it does not describe how we think about the probabilities associated with risky prospects. Consider the following example. The chance of winning the jackpot improves by 5%. Is every option equally good?

1. From 0 to 5 %

2. From 5% to 10%

3. From 50% to 55%

4. From 95% to 100%

Expectation principle: utility increases by 5% in each option, but this does not describe how you experience it. From 0-5 and from 95-100 appears more impressive than the other two options. The first option creates a (previously non-existing) possibility, which gives hope and therefore is a qualitative change. This impact is known as the ‘possibility effect’: highly unlikely outcomes are weighted disproportionally more than they should. The option 5-10 is merely a quantitative change: it doubles your chance of winning, unlike the psychological value of the prospect. The change from 95-100 is also a qualitative change that induces the ‘certainty effect’: almost certain outcomes are assigned less weight than it should.

Certainty and possibility both have powerful effects when it comes to losses. When your mother needs to have surgery, with a 5% risk of dying, that risk feels worse than half as bad as a 10% risk. The possibility effect causes us to overweight small risks and being more willing to pay a lot more than expected value to avoid those risks. The psychological difference between 95% risk of a bad event happening or 100% (certainty) seems even bigger: a tiny bit of hope looms large. The overweighting of small probabilities increases the appeal of insurance policies and gambling.

What is the ‘Paradox of Allais’?

Maurice Allais introduced the theory that people are susceptible to a certainty effect and thus violate expected utility theory and the axioms of rational choice. Consider the following puzzles.

X. 62% chance of winning € 410.000 or 64% chance of winning € 390.0000

Y. 98% chance of winning ,€ 410.000 or 100% chance of winning € 390.000

Most people go for the first option in puzzle X and the second option in puzzle Y, but that is considered a logical sin and violates the rules of rational choice. It is explained by the certainty effect. The 2% difference between 98-100 is more impressive than the 2% difference between 62-64. There have been several attempts to provide a plausible justification for the certainty effect, but so far all failed.

The prospect theory describes the choice making of people, whether they are rational or not. In this theory, decision weight do not equal probabilities. At the extremes of 0 and 100, the decision weights match the corresponding probabilities. Unlikely events are generally overweighted (possibility effect). Example: the corresponding decision weight of a 5% chance of a gain is 13.2. The decision weight would be 5, if the axioms of rational choice were met. The other end of the probability scale demonstrates the certainty effect: a 5% risk of not winning (95% of winning) reduces the utility of the gamble by 21% (from 100 to 79). People are inadequate sensitive to intermediate probabilities: the range of probabilities between 5% and 95% correspond with a much smaller range of decision weights.

Amos and Kahneman found that decision weights assigned to outcomes differ from probabilities and that people attach values to losses and gains (not to wealth). Both conclusions explain the ‘fourfold pattern’, a pattern of preferences which is the main achievement of the prospect theory.

1. Illustrative prospect (gains: 95% chance of winning money / 5% chance of winning money), losses: 95% chance of losing money / 5% chance of losing money).

2. Focal emotion evoked by the prospect (gains: 95% chance: fear of disappointment / 5% chance: hope of large gain), losses: 95% chance: hope to avoid loss / 5% chance: fear of large loss).

3. Behavior of people when offered a choice between a gamble and sure gain/loss that corresponds to the expected value. Gains: 95% chance: risk averse / 5% chance: risk seeking, losses: 95% chance: risk seeking / 5% chance: risk averse.

4. Expected attitudes of a plaintiff and a defendant when discussing a settlement (gains: 95% chance: accept unfavorable settlement / 5% chance: reject favorable settlement, losses: 95% chance: reject favorable settlement / 5% chance: accept unfavorable settlement.

People are averse to risk when they consider prospects with a substantial chance of a large win. They are willing to accept less than the expect value of a gamble if it means a certain win. The possibility effect explains the popularity of lotteries. When the jackpot is huge, people appear indifferent to a minuscule winning chance. Lottery tickets are the best example of the possibility effect. Buying one gives a chance to win and dream about a nice life. Insurance is bought ‘in the fourth row’. People are willing to pay a lot more for insurance than expected value. People do not only buy protection against an unlikely disaster, they purchase a comfortable feeling and eliminate worrying.