Prospect Theory — Daniel Kahneman and Amos Tversky on Decision-Making Under Risk

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Prospect Theory — Daniel Kahneman and Amos Tversky on Decision-Making Under Risk

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Gamblers and helmets

Humans are odd creatures. We pay insurers large sums of money to cover potential damage to our health, homes and cars. Yet, at the same time, we are happy to hand over hard-earned cash to casinos, lotteries, and stock markets in hopes of striking it big.

A colleague of mine, for example, pays for extra coverage for his health insurance plan, but rides his bike to work without a helmet. He doesn’t want to upset his gelled hairdo. Perhaps, that’s reasonable? People are heterogeneous, after all. They assess their prospects, probabilities, and payoffs differently.

Prospect Theory

As psychologists Daniel Kahneman and Amon Tversky found in their development of Prospect Theory, people assess potential gains and losses in a nuanced but predictable way. As we’ll see, the psychologists showed how certainty, reference points, and framing can affect decision-making under risk—seminal ideas that led to Kahneman’s Nobel prize in 2002.

The certainty effect

For starters, Kahneman and Tversky observed a tendency for people to overweight outcomes that feel certain. They called this a “certainty effect”. Consider, for example, the gamble below that Kahneman and Tversky devised (Gamble I). They presented participants with two options, and asked them to identify which of the two they prefer (results in brackets):

Gamble I (N = 95)
Options:A. 80% chance to win $4,000B. $3,000 with certainty
Results:(20%)(80%)
Results refer to the percentage of total participants (N) that chose each option. Source: Adapted from Problem 3 in Daniel Kahneman and Amon Tversky (1979). Prospect Theory: An Analysis of Decision under Risk.

Would you go for Option A or B? Why?

Now consider the following gamble:

Gamble II (N = 95)
Options:C. 20% chance to win $4,000 D. 25% chance to win $3,000
Results:(65%)(35%)
Results refer to the percentage of total participants (N) that chose each option. Source: Adapted from Problem 4 in Daniel Kahneman and Amon Tversky (1979). Prospect Theory: An Analysis of Decision under Risk.

Would you go for Option C or D? Why?

If you’re like most people, then you would prefer the certainty of Option B relative to A under Gamble I, and the chance of a bigger reward in Option C relative to D under Gamble II.

The result appears reasonable and intuitive, of course. But if you look carefully, you will see that Gamble II is actually a “probability mixture” of Gamble I with a 25 percent chance of occurring. The flip in preferences from Gamble I to Gamble II suggest that people do not always abide by the “substitution axiom of utility theory”, which “asserts that if B is preferred to A, then any probability mixture of B must be preferred to the [same probability] mixture [of A].” That’s not to say, however, that the results imply that people are irrational. It does say that the axioms of standard theory may not be reflective of human behaviour. In this case, a certainty effect appears to weigh on our preferences and decision-making. 

A trip to London

The authors find a similar result in non-monetary gambles. Consider the holiday packages below (Holiday I and II). Under each package, which option would you choose and why?

Holiday I (N = 72)
Options:A. 50% chance to win a three-week tour of England, France, and ItalyB. A one-week tour of England with certainty
Results:(22%)(78%)
Results refer to the percentage of total participants (N) that chose each option. Source: Adapted from Problem 5 in Daniel Kahneman and Amon Tversky (1979). Prospect Theory: An Analysis of Decision under Risk.
Holiday II (N = 72)
Options:C. 5% chance to win a three-week tour of England, France, and ItalyD. 10% chance to win a one-week tour of England
Results:(67%)(33%)
Results refer to the percentage of total participants (N) that chose each option. Source: Adapted from Problem 6 in Daniel Kahneman and Amon Tversky (1979). Prospect Theory: An Analysis of Decision under Risk.

Again, most respondents preferred B over A under Holiday I, but C over D under Holiday II. Most of them liked the sound of a guaranteed holiday in the first package, but the chance of something grander in the second. Notice, however, that Holiday II is simply a probability mixture of Holiday I with a 10 percent chance of occurring. The reversal in preferences speaks again to the certainty effect at work.

The reflection effect

The gambles and holidays above, however, refer only to the prospect of potential gains. If humans aren’t complicated enough, things seem to “reflect” when we talk about potential losses. Below are the results of positive-prospect gambles and their negative-prospect (mirrored) gambles, and the preferences that respondents exhibited:

Gamble I: Original and Mirrored
Options:80% chance to win $4,000Win $3,000 with certainty
Results:(20%)(80%)
Options (mirrored):80% chance to lose $4,000Lose $3,000 with certainty
Results:(92%)(8%)
Results refer to the percentage of total participants (N) that chose each option. Bolded cells reflect the preferred option under each prospect. Source: Adapted from Problem 3, 4, 3’ and 4’ in Daniel Kahneman and Amon Tversky (1979). Prospect Theory: An Analysis of Decision under Risk.
Gamble II: Original and Mirrored
Options:20% chance to win $4,00025% chance to win $3,000
Results:(65%)(35%)
Options (mirrored):20% chance to lose $4,00025% chance to lose $3,000
Results:(42%)(58%)
Results refer to the percentage of total participants (N) that chose each option. Bolded cells reflect the preferred option under each prospect. Source: Adapted from Problem 3, 4, 3’ and 4’ in Daniel Kahneman and Amon Tversky (1979). Prospect Theory: An Analysis of Decision under Risk.

Notice that when questions about potential gains turn into potential losses, preferences flipped. In the positive setting of Gamble I, respondents preferred a sure-thing over probable windfalls. But in the negative, they preferred to test their luck than to accept a guaranteed loss. Kahneman and Tversky refer to this phenomenon as the “reflection effect”. 

Similarly, in the positive setting of Gamble II, respondents preferred to gamble on the chance of a larger reward. But in the negative, they preferred the higher likelihood of a smaller loss. Differences in Gamble I and II suggest that the certainty effect and reflection effect can interact together.

As Kahneman and Tversky explain:

“The reflection effect implies that risk aversion in the positive domain is accompanied by risk seeking in the negative domain… [But] our data are incompatible with the notion that certainty is generally desirable. [So] it appears that certainty increases the aversiveness of losses as well as the desirability of gains.”

Daniel Kahneman and Amon Tversky (1979). Prospect Theory: An Analysis of Decision under Risk.

The fourfold pattern

So we can begin to see why people might buy lottery tickets and insurance at the same time. A cleaner manifestation of this is summarized in Kahneman’s fourfold pattern in Thinking, Fast and Slow (Table 1). It describes the “distinctive pattern of preferences” when we distinguish between certainty and possibility, and gains and losses, in a given problem.

Table 1. The Fourfold Pattern
SituationExampleThought processTypical behavior
1. High probability
of gains		95% chance to win $10,000Fear of loss of big gainsRisk averse
2. High probability
of losses		95% chance to lose $10,000Hope to avoid big lossesRisk seeking
3. Low probability
of gains		5% chance to win $10,000Hope to a win big gainsRisk seeking (e.g., buy lottery tickets)
4. Low probability
of losses		5% chance to lose $10,000Fear of big lossesRisk averse (e.g., buy insurance)
Source: Daniel Kahneman. (2011). Thinking Fast and Slow. Part IV. Choices. 29. The Fourfold Pattern.

As Kahneman and Tversky write:

“This analysis suggests that a person who has not made peace with his losses is likely to accept gambles that would be unacceptable to him otherwise. The well known observation that the tendency to bet on long shots increases in the course of the betting day provides some support for the hypothesis that a failure to adapt to losses or to attain an expected gain induces risk seeking.”

Daniel Kahneman and Amon Tversky (1979). Prospect Theory: An Analysis of Decision under Risk.

The isolation effect

Moreover, dealing with probabilities is hard work. We often seek ways to separate and discard components of the problem to simplify what we are dealing with. While this can be helpful, it can also transform our preferences and assessment of the problem itself. Kahneman and Tversky call this the “isolation effect”.

Consider, for example, the differences between these two games:

  • Game A is a two stage game. In stage-one, there is a 75% chance the game ends immediately (zero payoff) and a 25% chance to proceed to stage-two. In stage-two, you get to choose between a 20% chance to win $4,000 OR $3,000 with certainty.
  • Game B is a single-stage game. You get to choose between a 20% chance to win $4,000 OR a 25% chance to win $3,000. 

If you had to play Game A, which choice would you go for and why? What about Game B?

Curiously, in Kahneman and Tversky’s study, 78% of respondents in Game A say they would choose $3,000 with certainty (if they made it to the second stage), while 65% of respondents in Game B say they would choose a 20% chance to win $4,000.  

Note, however, that the final outcomes and probabilities in Game A and B are identical. Indeed, in the two-stage game, you also have a 20% chance to win $4,000, OR a 25% chance to win $3,000. While both problems are identical, separating the stages in Game A led to a reversal in preferences. The new structure gave the certainty-effect an opportunity to predominate. 

Reference points and aspirations

Kahneman and Tversky provide one more example for illustration. This time, it’s between two gambles with guaranteed prizes.

Game I (N = 70)

$1,000 plus your choice of Option A or B:

Options:(A) 50% chance to win $1,000(B) 100% chance to win $500
Results:16%84%
Results refer to the percentage of total participants (N) that chose each option. Bolded cells reflect the preferred option under each prospect. Source: Adapted from Problem 11 in Daniel Kahneman and Amon Tversky (1979). Prospect Theory: An Analysis of Decision under Risk.
Game II (N = 68)

$2,000 plus your choice of Option C or D:

Options:(C) 50% chance to lose $1,000(D) 100% chance to lose $500
Results:69%31%
Results refer to the percentage of total participants (N) that chose each option. Bolded cells reflect the preferred option under each prospect. Source: Adapted from Problem 12 in Daniel Kahneman and Amon Tversky (1979). Prospect Theory: An Analysis of Decision under Risk.

For each game, what choice did you make and why?

Again, both problems are the same if you focus only on final values. Options A and C give you a 50% chance to win $2,000 or $1,000; while Options B and D give you $1,500 with certainty. The reversal in preferences “implies that the carriers of value or utility are changes of wealth, rather than final asset positions that include current wealth”, Kahneman and Tversky write. 

“The prevalent failure to integrate riskless and risky prospects, dramatized in the isolation effect, suggests that people are unlikely to perform the operation of subtracting the cost from the outcomes in deciding whether to buy a gamble… The location of the reference point, and the manner in which choice problems are coded and edited emerge as critical factors in the analysis of decisions.”

Daniel Kahneman and Amon Tversky (1979). Prospect Theory: An Analysis of Decision under Risk.

Herbert Simon made a similar note in his theory of bounded rationality—that we evaluate alternatives relative to “aspiration levels” that are sometimes arbitrary and in flux. Indeed, reference points are all the rage. Many CEOs, I suspect, would cry foul if their compensation packages were lowered. But what if they were paid too high in the first place?

Lions and wildebeests

To belabor the point, Kahneman and Tversky say that the “essential feature of [Prospect] theory is that the carriers of value are changes in wealth or welfare, rather than final states.” Indeed, whether we deem something as loud or soft, or hot or cold, depends on our context and reference point.

When we watch a documentary about lions, we hope that the pride will succeed in their hunt. We do not want to see their cubs go hungry, after all. Yet when a similar documentary is made about zebras, antelopes, or wildebeests, we are relieved when the herd survives the chase to live another day.

A tale of two investors

Former poker world champion Annie Duke observes a similar tale-of-two-investors in her book Thinking in Bets. Consider, for instance, how people feel when the price of their stock rises by 50 percent before falling back to its original purchase price. Many investors, you can imagine, may regret not selling-high when they had the chance.

Now, consider a second scenario in which the stock price falls instead by 50 percent before recovering to its original purchase price. This time, many investors will feel relieved to break even. While the final position in both scenarios are the same, the emotional experience during that journey differs greatly. The moral of Duke’s tale is this: reference points matter.

Prospect Theory may explain broader behaviors in financial markets too. Nicholas Barbaeris notes, for example, that people continue to invest in high-profile initial public offerings even when they know the empirical track record for IPOs are lackluster. But they do so anyhow in hopes of finding the next Microsoft, Google, or Amazon.

As the Fourfold Pattern suggests, the prospect of improbable outsized gains, much like a lottery ticket, makes these investors risk-seeking. But if these investors focus instead on how much money they might lose, the certainty effect and loss aversiveness might dominate—making them risk averse instead.

Indeed, Prospect Theory encourages us to go beyond the rational axioms of standard theory to understand how people edit and evaluate their problems under risk. As we’ve seen, certainty, reflection, and isolation effects can manifest themselves in peculiar but predictable ways.

Of course, there’s a lot more to Prospect Theory than we have room for here—concepts like diminishing sensitivity and decision-weights play a role in our decision-making processes too. But I’ll leave that to you for further reading. 

Sources and further reading