The decision-making process: an overview – Part 3: Decision-making under risk

Expected value theory

This theory states that human decision-makers are perfectly rational individuals and excellent mathematicians, comparing the average outcome of their options using this formula:

Here,  each of the option’s outcome is weighted by the probability of this outcome’s occurrence, where:

- EV(X): expected value of an option X

- Xi: ith outcome of the option

- P(Xi): probability of the ith outcome of the option to occur

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Consider the following options:

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According to this theory, we should choose the option on the left as it gives, on average, more reward (24€) than the one on the right (6€). The math behind is this:

EV left = P(30€)30€+P(0€)0€ = 0.830€+0.20€= 24€

EV right = P(30€)30€+P(0€)0€ = 0.230€+0.80€= 6€

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One major limitation of this approach is that the expected value is objective, while our decisions can be subjective. More specifically, the same outcome can have different effects on different individuals. The same gain (loss) can make a person happier (sadder) than it makes another one. For instance, the option on the left could be less gratifying for someone accustomed to higher rewards (i.e., an 80€ win instead of 30€), whereas the option on the right could still interest a person more used to earning 3€ at best.

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Expected utility theory

This theory means to overcome the limits of the above theory and better explain human behavior. It states that while we are rational individuals with great mathematical skills who pick an option based on its average outcome, our choices are still subjective rather than objective. This average outcome is not calculated based on actual outcomes but on perceived ones. In other words, the value of an outcome is based on context rather than reality. Therefore, the expected utility is calculated according to this formula:

Where:

EU(X) = Expected utility of option X

Xi: ith outcome of option X

P(Xi): Probability of the occurrence of outcome Xi

u(Xi): utility of Xi, meaning subjective perception of value Xi

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Bernoulli defined the utility function as logarithmic or u(x) = log(x). The utility function proposes to explain how people perceive the value of the same option differently. The expected utility of an option increases as the objective value of an option does (earning 10,000€ is better perceived than earning 100€). Yet, the estimated value of a choice depends on its starting point. For instance, when receiving 20€, a billionaire would be far less happy than someone in need. In other words, the psychological impact of the same amount of money decreases as the original value is higher. This behavior is known as the diminishing marginal utility of wealth.

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The latter helps explain why a consumer is more likely to buy further items if the price for additional units is lower. For instance, a thirsty person can be willing to spend 2€ to buy a bottle of water, but knowing that a bottle is likely sufficient, he is less willing to pay another 2€ for an extra one. Businesses apply the expected utility theory in their marketing campaigns. For instance, they propose a second item for half the price of the first. As they know that consumers are less willing to buy the second item at full price, selling it at a reduced price increases its chance of being purchased.

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One significant limit of this theory is that it supposes that we are subjective decision-makers regarding outcome evaluation U(Xi) but objective mathematicians when it comes to outcome probability P(Xi). Nonetheless, our perception of an event probability is usually relative, impacted by different factors: whether we experienced a specific outcome in the past, were told about it by a friend, remember having witnessed this outcome before etc. Suppose you are buying plane tickets for a vacation, and the website suggests purchasing extra travel insurance in case of sickness. As you think back on when you or one of your friends got sick abroad, you might overestimate the probability of such an event and therefore be more susceptible to buying insurance.

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Prospect theory

This theory proposes to overcome the Expected utility theory’s limits and better explain human behavior. It states that:

1- People are subjective decision-makers regarding outcome estimation and probability evaluation.

2- Rare events tend to have a stronger impact on our decisions than predicted by classical rational theory. By contrast, common, predictable events will have less impact on our decision-making than classical theories would posit. For instance, when booking a flight, even though the probability of canceling their trip is very low, people can be interested in paying extra fees for cancellation insurance.

Thus, the choice of an option is based on its utility, defined by:

Where:

Xi: ith outcome of the option X

v(Xi): Subjective evaluation of Xi

pi: probability of the Xi outcome

π(pi):  the probability weighting function, reflecting the assumption that people overestimate rare events and underestimate common events

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A key aspect also introduced by this theory is that our subjective evaluation of outcomes is inconsistent across contexts. Instead, it depends on a reference point in our minds: are we losing or gaining something? In a series of experiments, Daniel Kahneman and Amos Tversky, the authors of this theory, showed that people are more loss-averse than gain-seeking, meaning that our choices are more impacted by the risk of losing something than the possibility of winning something. We can even make the “irrational” choice of a suboptimal option to avoid losses. Consequently, we tend to prefer a guaranteed gain, even if suboptimal, to a risky but superior one. Such behavior can be inverted depending on how likely the loss/gain is to happen. So next time you claim to be a risk-seeking person, think twice and consider the context!

Loss aversion is widely used in marketing. For instance, in discount offers, companies tend to present an offer as "subscribe and save 100€,” implying a loss of 100€ averted rather than "subscribe and win 100€," which frames the subscription as a gain-related decision.

Table explaining different choice situations based on context (gain/loss) and probability (high/low). Insights taken from the book “Thinking, Fast and Slow” by Daniel Kahneman.

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