The original “Allais questions” consist of two pairwise lottery choices. A subject is first asked to choose between lotteries $A$ and $B$ where

$A=\text{Certainty}\text{of}\$1\text{million}$ and $B=\{\begin{array}{l}1\%\text{chance}\text{of}\$0\\ 89\%\text{chance}\text{of}\$1\text{million}\\ 10\%\text{chance}\text{of}\$5\text{million}\end{array}$

Then, a subject is asked to choose between lotteries $A^{\prime }$ and $B^{\prime }$ where

$A^{\prime }=\{\begin{array}{l}89\%\text{chance}\text{of}\$0\\ 11\%\text{chance}\text{of}\$1\text{million}\end{array}$ and $B^{\prime }=\{\begin{array}{l}90\%\text{chance}\text{of}\$0\\ 10\%\text{chance}\text{of}\$5\text{million}\end{array}$

The four possible pairs of choices are $A{A}^{\prime }$, $B{B}^{\prime }$, $A{B}^{\prime }$, and $B{A}^{\prime }$. Out of these four possibilities, only the first two are apparently consistent with expected utility theory and the last two are not.¹¹ In practice, most subjects choose $A{B}^{\prime }$.¹²

In this article, we argue that the brain also considers anticipated stress-response costs in these choices. Consider option $A$. As this is a risk-free option, there are no associated stress-response costs; hence, utility from option $A$ is simply $u\left(A\right)$. In option $B$, there is a chance of getting nothing, which has anticipated stress-response costs associated with it as it may activate the stress-response system. These costs include BRC1, BRC2, and BOC, and are influenced by the properties CS and IS (see the discussion in Section 2). Denoting the probability of the stress-response system getting activated in response to the bad outcome of $0 by $\gamma$, the anticipated/expected stress-response costs can be written as:

$\bar{C}\left(CS,IS\right)=\gamma \ast C\left(CS,IS\right)+\left(1-\gamma \right)\ast 0$

Such that:

$\frac{\partial C\left(CS,IS\right)}{\partial IS}>0\left(IS\text{Property}\right)$ (3.1)

$\frac{\partial C\left(CS,IS\right)}{\partial CS}>0\left(CS\text{Property}\right)$ (3.2)

(3.1) captures the finding that if a bad outcome generates further stressors such as thoughts that become internal stressors in their own right (keep the stress-response system switched on long after the initial arrival of bad news), then the stress-response costs are higher. (3.2) captures the finding that higher the cumulative stress on the DM, higher are the stress-response costs of an additional stressor.

With the inclusion of the stress-response costs, the expected utility from lottery $B$ is:

$u\left(B\right)=0.89\ast u\left(\$1M\right)+0.10\ast u\left(\$5M\right)+0.01\ast u\left(\$0\right)-0.01\ast {\bar{C}}_B\left(CS,IS\right)$ (3.3)

where ${\bar{C}}_B\left(CS,IS\right)$ captures the anticipated stress-response costs, which are associated with the bad outcome in lottery $B$.

With the original “Allais questions”, ${\bar{C}}_B\left(CS,IS\right)$ is expected to be quite high. The reason is that the bad outcome of nothing is expected to generate the emotion of regret for not choosing the certain $1M earlier.¹³ The associated thoughts of self-blame for giving up such a large sum of money (for a chance at winning $5M) would keep the stress-response system switched on long after learning the bad outcome. By becoming an internal stressor, such thoughts would keep the stress-response system switched on for a long time substantially adding to the stress-response costs (BRC1, BRC2, and BOC).¹⁴ With ${\bar{C}}_B\left(CS,IS\right)$ sufficiently large, $A$ is preferred to $B$ even if one has low risk-aversion or even if one is risk neutral.

With the inclusion of anticipated stress-response costs, the utility from lotteries $A^{\prime }$ and $B^{\prime }$ are:

$u\left(A^{\prime }\right)=0.11\ast u\left(\$1M\right)+0.89\ast u\left(\$0\right)-0.89\ast {\bar{C}}_{A^{\prime }}\left(CS,IS\right)$ (3.4)

$u\left(B^{\prime }\right)=0.10\ast u\left(\$5M\right)+0.90\ast u\left(\$0\right)-0.90\ast {\bar{C}}_{B^{\prime }}\left(CS,IS\right)$ (3.5)

where:

${\bar{C}}_{A^{\prime }}\left(CS,IS\right)=\gamma \ast C_{A^{\prime }}\left(CS,IS\right)+\left(1-\gamma \right)\ast 0$ (3.5a)

${\bar{C}}_{B^{\prime }}\left(CS,IS\right)=\gamma \ast C_{B^{\prime }}\left(CS,IS\right)+\left(1-\gamma \right)\ast 0$ (3.5b)

The stress-response costs in lotteries $A^{\prime }$ and $B^{\prime }$, which are $C_{A^{\prime }}\left(CS,IS\right)$ and $C_{B^{\prime }}\left(CS,IS\right)$ respectively, are likely to be much smaller than ${\bar{C}}_B\left(CS,IS\right)$ due to the absence of the internal stressor of self-blame in these lotteries. In both $A^{\prime }$ and $B^{\prime }$, the odds are that one is not going to win anything, so there is little reason for self-blame to arise from not picking the other option. Indicating this by writing ' $-$' in the place-holder for $IS$ in $C_{A^{\prime }}\left(CS,IS\right)$ and $C_{B^{\prime }}\left(CS,IS\right)$:

$C_B\left(CS,IS\right)\gg C_{A^{\prime }}\left(CS,-\right)$ (3.5c)

$C_B\left(CS,IS\right)\gg C_{B^{\prime }}\left(CS,-\right)$ (3.5d)

The typical response of preferring $B^{\prime }$ to $A^{\prime }$ implies:

$\begin{array}{l}0.10\ast u\left(\$5M\right)+0.90\ast u\left(\$0\right)-0.90\ast {\bar{C}}_{B^{\prime }}\left(CS,-\right)\\ >0.11\ast u\left(\$1M\right)+0.89\ast u\left(\$0\right)-0.89\ast {\bar{C}}_{A^{\prime }}\left(CS,-\right)\end{array}$ (3.6)

With ${\bar{C}}_{B^{\prime }}\left(CS,-\right)~{\bar{C}}_{A^{\prime }}\left(CS,-\right)=\bar{C}\left(CS,-\right)$ and setting $u\left(\$0\right)=0$, the condition for preferring $B^{\prime }$ to $A^{\prime }$ is:

$0.10\ast u\left(\$5M\right)>0.11\ast u\left(\$1M\right)+0.01\ast \bar{C}\left(CS,-\right)$ (3.7)

It follows that, unless $\bar{C}\left(CS,-\right)$ is very large, $B^{\prime }$ would continue to be preferred over $A^{\prime }$.

The above discussion shows that even though the observed outcome of $A{B}^{\prime }$ is considered inconsistent with the expected utility theory (without the stress-response costs), including the stress-response makes it consistent.

Previous analysis gives rise to testable predictions regarding the observed behavior in Allais type questions. The predictions follow from the properties of IS and CS, which are factors in determining the magnitude of the stress-response costs.

The anticipated stress-response costs associated with the bad outcome in lottery $B$, ${\bar{C}}_B\left(CS,IS\right)$, play a key role in giving rise to observed violations from expected utility theory without stress-response costs (EUT). In particular, it is the self-blame thoughts for passing up on a life-changing amount of $1M for a small chance of winning an additional $4M that become internal stressors. Such internal stressors keep the stress-response system switched on long after realization of the bad outcome. It immediately follows that reducing the large payoffs to smaller ones would reduce ${\bar{C}}_B\left(CS,IS\right)$. For example, if the original payoffs in Allais questions are proportionately reduced such that the sure outcome in $A$ is $5 and the payoffs in $B$ are 89% chance of $5, 10% chance of $25, and 1% chance of $0, then the thoughts of self-blame from choosing $B$ and ending up with $0 are expected to be much milder or none at all. This is because, unlike $1M, $5 is not a life-changing amount of money. It immediately follows that the anticipated stress-response costs without the $IS$ factor, ${\bar{C}}_B\left(CS,-\right)$ are much smaller: ${\bar{C}}_B\left(CS,-\right)\ll {\bar{C}}_B\left(CS,IS\right)$. Proposition 1 immediately follows.

Proposition 1 (Payoff Size Matters): The observed violations of EUT in Allais type questions fall as payoff size falls.

Another prediction follows from manipulating the Allais questions in accordance with the CS property. Recall that the CS property refers to the finding that higher the level of cumulative stress, higher is the stress-response cost (see Section 2). That is, the stress-response costs without any cumulative stress, ${\bar{C}}_B\left(-,IS\right)$, are lower: ${\bar{C}}_B\left(-,IS\right)<{\bar{C}}_B\left(CS,IS\right)$. It follows that for subjects which are facing a lower level of cumulative stress, the observed violations should be lower. People with steady jobs, financial assets, and high level of education are likely to have fewer pre-existing stressors when compared with people without jobs, financial assets, and education.

Proposition 2 (Cumulative-Stress Correlates Matter): The observed violations of EUT in Allais type questions are lower if subject characteristics (such as education, steady job, financial assets) correspond to a lower level of cumulative stress when compared with subject characteristics (such as lack of education, unemployment, no financial assets) that correspond to a higher level of cumulative stress.

Both the predictions above regarding payoff size and cumulative-stress correlates hold up well in both lab and field experiments. Huck and Muller (2012) conduct field experiments with Allais type questions with a large representative sample of general population as well as with students in the lab. Consistent with the predictions here, they report fewer violations with lower payoffs and with subject characteristics such as incomes, financial assets, and education.

The original demonstration of “Ellsberg paradox” involves an urn containing 90 balls, 30 are red while the remaining 60 are either black or yellow in unknown proportions. The balls are well mixed so each ball is as likely to be drawn as any other. A subject is first asked to choose between the following scenarios:

$A=\$100$ if a red ball is drawn and $B=\$100$ if a black ball is drawn

Then, the subject makes a choice between the following scenarios with the same situational parameters given earlier:

$A^{\prime }=\$100$ if a red or a yellow ball is drawn and $B^{\prime }=\$100$ if a black or a yellow ball is drawn

Out of the 4 possible outcomes, $A{A}^{\prime }$, $B{B}^{\prime }$, $A{B}^{\prime }$, and $B{A}^{\prime }$, only the first two are consistent with expected utility theory (without stress-response costs) (EUT). However, typically subjects choose $A{B}^{\prime }$, apparently in violation of EUT.¹⁵ However, as shown in this section, the typical outcome $A{B}^{\prime }$ is not inconsistent with EUT inclusive of the stress-response costs.

In scenario $A$, if the anticipated stress-response costs are also considered, then the expected utility is:

$u\left(A\right)=\frac{1}{3}\ast u\left(\$100\right)+\frac{2}{3}\ast u\left(\$0\right)-\frac{2}{3}\ast {\bar{C}}_0\left(CS,IS\right)$ (3.8)

where ${\bar{C}}_0\left(CS,IS\right)$ is the anticipated stress-response cost associated with the $0 outcome:

${\bar{C}}_0\left(CS,IS\right)=\gamma \ast C_0\left(CS,IS\right)+\left(1-\gamma \right)\ast 0$ (3.8a)

Compared to scenario $A$, in scenario $B$, in the absence of objectively known odds of drawing a black ball, a subject assigns a subjective probability; however, she is aware that there is missing information which could be adverse. In this case, the missing information may reveal that the objective probability is less than the assumed subjective probability. Such adverse information may trigger a stress-response. If the subject assigns a (subjective) probability of $\pi$ to the missing information being adverse (the objective probability of drawing a black ball is less than assumed subjective probability), then the expected utility is:

$\begin{array}{c}u\left(B\right)=P_B\ast u\left(\$100\right)+\left(1-\frac{1}{3}-P_B\right)\ast u\left(\$0\right)\\ -\left(1-\frac{1}{3}-P_B\right)\ast {\bar{C}}_0\left(CS,IS\right)-\pi \ast {\bar{C}}_B\left(CS,IS\right)\end{array}$ (3.9)

where ${\bar{C}}_B\left(CS,IS\right)$ is the anticipated stress-response cost associated with the missing information being adverse. If the missing information turns out to be adverse, which has a probability, $\pi$, then the stress-response system may be triggered with a probability, $\gamma$, creating the anticipated stress-response costs of ${\bar{C}}_B\left(CS,IS\right)=\gamma \ast C\left(CS,IS\right)+\left(1-\gamma \right)\ast 0$.

A comparison of (3.9) and (3.8) indicates that even if the subject assigns a subjective probability which is higher than 1/3 to the drawn ball being black, that is $P_B>1/3$, a subject may still choose $A$ over $B$. This is because the subject knows that the true probability can be different and such missing information could be adverse, which may trigger the stress-response system with the anticipated stress-response costs of ${\bar{C}}_B\left(CS,IS\right)$.

It is illustrative to consider the case when the assumed subjective probability of drawing a black ball, $P_B$, is 1/3:

$u\left(B\right)=\frac{1}{3}\ast u\left(\$100\right)+\frac{2}{3}\ast u\left(\$0\right)-\frac{1}{3}\ast {\bar{C}}_0\left(CS,IS\right)-\pi \ast {\bar{C}}_B\left(CS,IS\right)$ (3.10)

Note that $u\left(B\right)$ in (3.10) is less than $u\left(A\right)$ in (3.8) due to an additional term, $\pi \ast {\bar{C}}_B\left(CS,IS\right)$. This term is a product of the subjective probability of missing information being adverse, $\pi$, and the associated expected stress-response costs, ${\bar{C}}_B\left(CS,IS\right)$.

Extending the above discussion to scenarios $A^{\prime }$ and $B^{\prime }$:

$\begin{array}{c}u\left(A^{\prime }\right)=\left(\frac{1}{3}+P_Y\right)\ast u\left(\$100\right)+\left(1-\frac{1}{3}-P_Y\right)\ast u\left(\$0\right)\\ -\left(1-\frac{1}{3}-P_Y\right)\ast {\bar{C}}_0\left(CS,IS\right)-\pi \ast {\bar{C}}_{A^{\prime }}\left(CS,IS\right)\end{array}$ (3.11)

where the subjective probability of drawing a yellow ball is $P_Y$.

$u\left(B^{\prime }\right)=\frac{2}{3}\ast u\left(\$100\right)+\frac{1}{3}\ast u\left(\$0\right)-\frac{1}{3}\ast {\bar{C}}_0\left(CS,IS\right)$ (3.12)

Even if the subjective probability of drawing a yellow ball, $P_Y$, is more than 1/3, a subject may still choose $B^{\prime }$ over $A^{\prime }$, because of the awareness that missing information about probabilities could be adverse (objective probability of drawing a yellow ball is less than the assumed subjective probability). This introduces an additional stress-response cost term in (3.11) of ${\bar{C}}_{A^{\prime }}\left(CS,IS\right)$. There is no corresponding term in (3.12) as there is no missing information about probabilities in lottery $B^{\prime }$.

Considering the illustrative case, $P_Y=1/3$, it follows that:

$\begin{array}{c}u\left(A^{\prime }\right)=\frac{2}{3}\ast u\left(\$100\right)+\frac{1}{3}\ast u\left(\$0\right)-\frac{1}{3}\ast {\bar{C}}_0\left(CS,IS\right)-\pi \ast {\bar{C}}_{A^{\prime }}\left(CS,IS\right)\\ <\frac{2}{3}\ast u\left(\$100\right)+\frac{1}{3}\ast u\left(\$0\right)-\frac{1}{3}\ast {\bar{C}}_0\left(CS,IS\right)=u\left(B^{\prime }\right)\end{array}$ (3.13)

Hence, the typical outcome in Ellsberg type scenarios, $A{B}^{\prime }$, which is not consistent with EUT without stress-response costs, is consistent with EUT inclusive of stress-response costs. It is intriguing that both the Allais paradox and the Ellsberg paradox, which generally serve as exhibit 1 and exhibit 2 against EUT in the literature, are reconcilable with EUT inclusive of the stress-response costs.

Previous analysis is based on the idea that the anticipated stress-response costs cannot be ignored in a comparison of a scenario where objective probabilities are known with a scenarios where objective probabilities are unknown. When objective probabilities are unknown, subjective probabilities are assigned; however, a subject is aware that they could be wrong. That is, missing information about probabilities may be adverse with the objective probability of good outcomes turning out to be less than the assumed subjective probability. Such adverse information may trigger the stress-response system. The subjective probability of missing information being adverse, $\pi$, and the anticipated stress-response costs arising from such adverse information, $\bar{C}\left(CS,IS\right)$, are jointly considered in the expected utility calculation: $\pi \ast \bar{C}\left(CS,IS\right)$. Ignoring this term is the source of Ellsberg type violations from EUT.

From the above discussion, it immediately follows that the subjective probability of missing information being adverse is a critical factor in determining whether a subject displays Ellsberg type violations or not. Where does the subjective probability of the missing information being adverse, $\pi$, comes from? We assert that they come from a subject’s past experiences with missing information. Every subject carries with her an inventory of prior experiences with missing information. In some cases, the missing information might have turned out to be positive whereas in some others it might have been negative. It makes sense to think of $\pi$ as some function of the frequency of prior negative experiences with missing information. Proposition 3 follows.

Proposition 3 (Prior Experiences Matter): The observed violations of EUT in Ellsberg type questions is higher if the frequency of prior negative experiences with missing information is higher.

Malmendier and Nagel (2011) present evidence that people who have lived through the depression years exhibit much lower stock market and bond market participation rates when compared with the general population. In the stress-response decision framework developed here, buying a financial asset such as a stock requires assigning subjective probabilities to various outcomes while being aware that further information may reveal these probabilities to be different than what has been assumed. Exposure to pre-dominantly adverse missing information during the depression years implies assigning a higher probability to the missing information being negative in the new gambles (financial assets) that one faces. Akin to higher violations of EUT in Ellsberg type scenarios, this implies not accepting the gambles offered by financial assets resulting in the observed low participation rates.

Ellsberg paradox has spawned a large literature on ambiguity aversion (see Machina and Siniscalchi (2014) and references therein). However, as discussed here, what appears as ambiguity aversion may be arising from ignoring the anticipated stress-response costs associated with missing information.