# Rational Decision under Uncertainty

How would an individual make a rational decision under uncertainty? This article explains how economists characterize decision-making under uncertainty. Economic models always focus first on decisions under certainty and then introduce uncertainty later in the learning process. The focal point here is how this affects the rationality of agents’ decisions to use scarce resources.

## Introduction to Rational Decision under Uncertainty

We are all faced with daily situations where we have to make decisions that will lead to different outcomes immediately or later. Such decisions occur under uncertainty of results. Economic theory explains this problem in the models of decision-making under uncertainty. The literature is vast, from simple to more complex models. In this article, we want to focus on the general understanding of rationality under uncertainty.

We have experienced a practical example of decision-making under uncertainty during the 2020 Corona Pandemic. One of the decisions governments made was to keep the economy open during the Corona Pandemic or lock down the economy to contain the spread of the COVID19 virus. Consequently, the corona pandemic forced the governments to make decisions that would have led to different states (outcomes), which they did. All nations have faced the health and economic risks caused by the COVID19 Pandemic.

For example, in the United States of America, the government decided to ignore the certainty that the virus existed or impacted their economy. By doing that, the government made decisions that led to several outcomes in the USA. Should the government have made a rational decision under uncertainty to enable achieving other economic states? Currently, Brazil is looking into the case of atrocities against humanity due to similar circumstances to charge its current president. Why should decisions under uncertainty be understood in depth?

The world executives branches of governments engaged in a conflict between rational thinking and the club of irrational thinking. The lesson should be that irrationality or ‘alternative facts’ does not resolve decisions under uncertainty.

## Decision Parameters of Rational Decision-Making under Uncertainty

Let us first look at the decision parameters of decision-making under uncertainty. Any decision made by an individual contributes some utility to them. Economists embed such knowledge in the risk preferences of an individual via some utility function.

### Utility and Risk Preferences

Literature about risk covers risk preferences in great depth. Economists describe three types of choices in such models: risk-neutral, risk-averse, and risk-loving preferences using utility functions $U(x)$. The utility function obeys the general assumptions of decisions under certainty. Let us view each on its own.

#### Risk-neutral Preferences

Economists describe risk-neutral preferences as those that show that the decision-makers are always indifferent between risky and non-risky decisions and between significant and slight variations. These indifferences are because such preferences indicate that the certainty equivalent choice under certainty offers identical utility as the utility of the expected value of all possible outcomes of a game. Economists further assume that the markets are risk-neutral while coordinating market transactions between buyers and sellers. Finally, risk-neutral preferences are linear in the payoffs of a game and have constant marginal utility, e.g., $U(x)=\alpha x$.

#### Risk-averse Preferences

Risk-averse preferences are such that the decision-makers prefer minor variations of outcomes over significant variations under similar decision-making circumstances. In the Economic theory of risk, Risk-averse behavior is a standard assumption. The literature on basic economics introduces a simple model of risk using risk-averse preferences. Most models assume that agents in an economy behave risk-averse. Finally, risk-averse choices are non-linear in the payoffs of a game and have diminishing marginal utility, e.g., $U(x)=\alpha x^{\beta}$ with $0<\beta<1$.

#### Risk-loving Preferences

Risk-loving preferences are such that the decision-maker prefers significant variations over minor variations under similar decision-making circumstances. Although the risk-loving assumption is rare in economic theory, there are some practical examples. For example, who would sell an insurance contract to whom? Probably a risk-loving agent would sell an insurance contract to a risk-averse agent in an economy. Finally, risk-loving preferences are non-linear in the payoffs of a game and have increasing marginal utility, e.g., $U(x)=\alpha x^{\beta}$ with $\beta>1$.

### The Expected Value of a Game and its Utility Level

The expected value of a game is the weighted sum of all outcomes given the probabilities of all outcomes or the average outcome. The average result guarantees the individual a certain level of utility, which we call the utility of the expected value.

### The Expected Utility and the Certainty Equivalent

The expected utility of a game is the weighted sum of all utility levels of outcomes given the probabilities of all achievable results. The value that guarantees the expected utility is called the security equivalent, which means the individual would be indifferent to receiving the same outcome with certainty.

## Difference between the Expected Value and the Certainty Equivalent

The difference between the expected value and the expected utility is that the expected value is the average of all game outcomes. In contrast, the expected utility is the average utility the individual gains from all game outcomes.

## Fair Insurance Premium for Risk-averse Agents

Assume you are a risk-averse agent in an economy where you have the opportunity to earn 35000 € but face a health risk that would incur an income loss of 15000 € at the 50 % probability of getting sick.

How much premium would be fair for complete insurance coverage worth 15000 €? The expected income, in this case, will be the average income in both states of the world: you are sick, and the other you are not ill.

Under risk-averse preferences, you will earn more utility with the expected income as with the certainty equivalent. Therefore, you will prefer to secure the average payment/income over the certainty equivalent. Adequate insurance should at least guarantee an income level above or equivalent to the average income. Hence, the Fair insurance premium is the difference between the current income (35,000.00 €) and the expected value of the game (27500 €): (35000 – 27500=7.500 €).

What would an unfair insurance premium be worth? Consequently, an unfair insurance premium would guarantee a lower utility than the utility of the expected income. Subsequently, the unfair insurance should at least allow for a higher utility than, or equivalent to, the expected utility. Therefore, the unfair insurance premium is the difference between the current income (35000 €) and the security equivalent of the expected utility of the game (24000 €): (35000 – 24,000=11,000 €). Insurance coverage that guarantees the certainty equivalent charges the maximum premium.

Unfair insurance deals, which demand higher risk premiums, and guarantee a utility level that is lower than the expected utility of the game, will be ignored by decision-makers. An individual will not close such an unfair insurance contract if they are fully informed.

## References

Rogg, M. (2020). The Pandemic and Its Impact on Security Policy. PRISM, 8(4), 54–67. https://www.jstor.org/stable/26918234 Cite
Dixig, A. (1989). Entry and Exit Decisions under Uncertainty. Journal of Political Economy, 97(3), 620–638. http://www.jstor.org/stable/1830458 Cite
Marinacci, M. (2015). MODEL UNCERTAINTY: Model Uncertainty. Journal of the European Economic Association, 13(6), 1022–1100. https://doi.org/10.1111/jeea.12164 Cite
Nicholson, W., & Snyder, C. (2010). Intermediate microeconomics and its applications (11th ed). South-Western Cengage Learning. Cite
Pollak, R. A. (1967). Additive von Neumann-Morgenstern Utility Functions. Econometrica, 35(3/4), 485. https://doi.org/10.2307/1905650 Cite
Mankiw, N. G., & Taylor, M. P. (2021). Grundzüge der Volkswirtschaftslehre (M. Herrmann, C. Müller, & D. Püplichhuysen, Trans.; 8., überarbeitete Auflage). Schäffer-Poeschel Verlag. Cite
Gillespie, A. (2016). Foundations of economics (Fourth edition). Oxford University Press. Cite
Varian, H. R. (2016). Grundzüge der Mikroökonomik (9., aktualisierte und erweiterte Auflage). De Gruyter Oldenbourg. Cite
Pindyck, R. S., & Rubinfeld, D. L. (2018). Mikroökonomie (9., aktualisierte Auflage). Pearson. Cite
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