Thinking, Fast and Slow

With the foundation of prospect theory in place, we’ll explore a few implications of the model.

Probabilities are Overweighted at the Edges

Consider which is more meaningful to you:

Going from 0% chance of winning $1 million to 5% chance
Going from 5% chance of winning $1 million to 10% chance

Most likely you felt better about the first than the second. The mere possibility of winning something (that may still be highly unlikely) is overweighted in its importance . (Shortform note: as Jim Carrey’s character said in the film Dumb and Dumber , in response to a woman who gave him a 1 in million shot at being with her: “ so you’re telling me there’s a chance! ”)

More examples of this effect:

We fantasize about small chances of big gains.

Lottery tickets and gambling in general play on this hope.
A small sliver of chance to rescue a failing company is given outsized weight.

We obsess about tiny chances of very bad outcomes.

The risk of nuclear disasters and natural disasters is overweighted.
We worry about our child coming home late at night, though rationally we know there’s little chance anything bad happened.

People are willing to pay disproportionately more to reduce risk entirely.

Parents were willing to pay 24% more per bottle of insect spray to reduce a child’s risk of poisoning by 2/3, but they were willing to pay 80% as much to reduce it to 0.
People are not willing to pay half price for insurance coverage that covers you only on odd days. Technically, this is a good deal, because there are more odd days in a year than even, so you’re getting more value for your money. But the reduction from 100% risk to 50% risk is far less valuable than the reduction from 50% risk to 0% risk.

We Feel Better About Absolute Certainty

We’ve covered how people feel about small chances. Now consider how you feel about these options on the opposite end of probability:

In a surgical procedure, going from 90% success rate to 95% success rate.
In a surgical procedure, going from 95% success rate to 100% success rate.

Most likely, you felt better about the second than the first. Outcomes that are almost certain are given less weight than their probability justifies. 95% success rate is actually fantastic! But it doesn’t feel this way, because it’s not 100%.

As a practical example, people fighting lawsuits tend to take settlements even if they have a strong case. They overweight the small chance of a loss.

(Shortform note: how we feel about 0% and 100% are similar and are inversions of each other. A 100% gain can be converted into 0% loss—we feel strongly about both. For example, say a company has a 100% chance of failure, but a new project reduces that to 99%. It feels as though the chance of failure is reduced much more than 1%. Inversely, a project that increases the rate of success from 0% to 1% seems much more likely to work than 1% suggests.)

How We Feel About Probabilities

The above points give a feel for how people feel about probabilities, but let’s be specific. Here’s a chart showing the weight that people give each probability:

Probability (%)

100

Decision weight

5.5

8.1

13.2

18.6

26.1

42.1

60.1

71.2

79.3

87.1

91.2

100

At the two extremes, people feel how you would expect—at 0% probability, people weigh it at 0; at 100% probability, people weigh at 100.

But in between, there are interesting results:

The weight for 1% is 5.5 (when it should rationally be 1.0) and the weight for 99% is 91.2 (far from 99). People overestimate low probabilities, and they discount high probabilities that aren’t certain.
- Imagine that you’re told there’s a 99% chance your operation will be successful, with a 1% rate of permanent disability. You will certainly be more anxious than the miniscule 1% suggests!
Note that 50% is weighted at 42.1.
- Imagine that you have a 50-50 coin toss where heads wins you a dollar, tails loses you a dollar. You really would not value this as a 50-50 coin toss—you feel the possibility of a loss more strongly than the gain.
The range between 5% and 95%, which covers 90% in probability, ranges in weight from 13 to 79, covering just 66 points. This means we are not adequately sensitive to changes in probability.
- Similarly, we can’t easily tell the difference between a 0.001% cancer risk and 0.00001% risk, even though this translates to 3000 cancers for the US population vs 30.

The Fourfold Pattern of Prospect Theory

Putting prospect theory into another summary form, here’s a 2x2 grid showing how people feel about risk in different situations. The upper left quadrant shows how people feel about a high probability of a gain, the upper right shows how people feel about a high probability of a loss, and so on.

GAINS

LOSSES

HIGH PROBABILITY

Certainty Effect

95% chance to win $10,000 vs 100% chance to win $9,500

Fear of disappointment

RISK AVERSE

Accept unfavorable settlement

Example: Lawsuit settlement

95% chance to lose $10,000 vs 100% chance to lose $9,500

Hope to avoid loss

RISK SEEKING

Reject favorable settlement

Example: Hail mary to save failing company

LOW PROBABILITY

Possibility Effect

5% chance to win $10,000 vs 100% chance to win $500

Hope of large gain

RISK SEEKING

Reject favorable settlement

Example: Lottery

5% chance to lose $10,000 vs 100% chance to lose $500

Fear of large loss

RISK AVERSE

Accept unfavorable settlement

Example: Insurance

Putting it all together - two factors are at work in evaluating gambles:

Diminishing sensitivity, so that more of the same causes less of a change in psychological value (in the graph, the slope decreases as you move further from the y-axis)
Inaccurate weighting of probabilities at the edges

In the first row of this table, the two factors work in the same direction:

In the upper right quadrant, diminishing sensitivity causes loss aversion: a sure loss is painful. On the prospect theory graph, 100% of -900 is more negative than 90% of -1,000.
Making matters worse is the inaccurate weighting of probabilities. While 100% is weighted at 100, 90% is weighted only at 71. This certainty effect causes the 100% loss to feel much more painful than a very high chance of loss.
Similarly, in the positive situation on the upper right, the diminishing sensitivity makes a certain lower gain more attractive, and the certainty effect reduces the attractiveness of the gamble.

In the bottom row, the two factors work in opposite directions :

In the lower left corner, diminishing sensitivity still makes the sure gain more attractive than the chance of a gain. But the overweighting of low probabilities overcomes this effect, so people in this quadrant tend to choose the 5% gamble.

Kahneman notes that many human tragedies happen in the upper right quadrant. People who are between two very bad options take desperate gambles, accepting a high chance of making things worse to avoid a certain loss. The certain large loss is too painful, and the small chance of salvation too tempting, to decide to cut one’s losses.

Miscellaneous Implications

Opposing Incentives in Litigation

Litigation is a nice example of where all of the above can cause tumult:

Plaintiffs file frivolous lawsuits. They have a low chance of winning, and they overweight the probability of winning (lower left corner in the fourfold pattern).
Defendants prefer settling frivolous lawsuits to lower the risk of a more expensive loss (lower right corner). Yet if the defendant does this habitually for each lawsuit, it invites more frivolous lawsuits, and can thus be costly in the long run.

We can also reverse the situations:

A plaintiff has a strong case and has an almost certain chance to win, but wants to avoid the small chance of a loss. She is prone to risk-aversively take a settlement (upper left corner).
The defendant knows she’s likely to lose, but has a small chance of winning. She’s willing to drag the case on, because the certain loss from a settlement with the plaintiff is painful and the small chance of winning is gratifying (upper right corner).
In this case, the defendant holds the stronger hand, and the plaintiff will settle for less than the case strength suggests.

Theory-Accepted Blindness

Why did it take so long for someone to notice the problems with Bernoulli’s conception of utility? Kahneman notes that once you have accepted a theory and use it as a tool in your thinking, it is very difficult to notice its flaws . Even if you see inconsistencies, you reason them away, with the impression that the model somehow takes care of it, and that so many smart people who agree with your theory can’t all be wrong.

In Bernoulli’s theory, even when people noticed inconsistencies, they tried to bend utility theory to fit the problem. Kahneman and Tversky instead made the radical choice to abandon the idea that people are rational decision-makers, and instead took a psychological bent that assumed foibles in decision-making.

Blind Spots in Prospect Theory

Prospect theory has holes in its reasoning as well. Kahneman argues that it can’t handle disappointment - that not all zeroes are the same. Consider two scenarios:

1% chance to win $1 million and 99% chance to win nothing
99% chance to win $1 million and 1% chance to win nothing.

In both these cases, prospect theory would assign the same value to “winning nothing.” But losing in case 2 clearly feels worse. The high probability of winning has set up a new reference point—possibly at say $800k.

Prospect theory also can’t handle regret, in which failing to win a line of gambles causes losses to become increasingly more painful.

People have developed more complicated models that do factor in regret and disappointment, but they haven’t yielded enough novel findings to justify the extra complexity.

Part 4-2: Implications of Prospect Theory