Risk theory attempts to explain the decisions people make when they are faced with uncertainty about the future. Typically, a situation in which risk theory may be applied involves a number of possible states of the world, a number of possible decisions and an outcome for each combination of state and decision. The theory predicts a decision according to the distribution of outcomes it will produce. The theory is important for people who make decisions whose success hinges on the way the risks in the world turn out. For example, people involved with insurance companies, whose success depends on predicting the frequency and magnitude of claims, use risk theory to help determine their optimum exposure to risks.

Any decision people make about the future must take into account a certain amount of uncertainty. In some cases, like the decision to invest in a company that may default, the uncertainty affects the price the investor is willing to pay. In others, uncertainty can make the difference between whether or not a person should take an action at all. Those cases are the ones in which risk theory is used.

The first step in applying risk theory to a situation is to determine what the outcomes are. Each combination of a state and a decision yields an outcome according to some function. In mathematical terms, what the function does is called mapping: it takes each point in a graph illustrating possible states and decisions, and defines a corresponding point on a graph of outcomes.

Next, a value must be assigned to each outcome. As with any theory that attempts to explain individual choices, an important component of risk theory is the quantification of qualitative conditions. One must assign values to each outcome in order to compare them to each other. These values, which combine all of the benefits and drawbacks of each outcome, are called utility values. The absolute value of each utility value is not important; what matters is the relative value of each to the others, because this determines how much each affects the final decision.

Finally, the analyst must assign a probability to each state. These probabilities determine the weight each outcome has. The weighted outcomes that may arise from each decision are added together to yield an overall value for each decision. The theory recommends the decision with the highest overall value.

These abstract instructions may be illustrated best with an example. Imagine you are deciding between planting cacti or flowers in a window box outside your kitchen. The relative precipitation will influence the health of the plants. In a wet year, the flowers will flourish, and the cacti will also thrive, though not to the same level. In a dry year, neither will do as well. The cacti, however, will do considerably better than the flowers.

The next step is to assign values to these outcomes based on the utility you will get from the different boxes in their different states. You might decide that flowers in a wet year will give you a utility of 10, while cacti in a wet year will give eight units of utility. In a dry year, the cacti will give you seven units, and the flowers will give you three. Finally, you must estimate the probability of having a wet year and the probability of having a dry year.

Consider two different probability scenarios. If you believe that there is a 90 percent chance of having a wet year, then your expected utility from planting flowers is 0.9*10+0.1*3=9.3, while your expected utility from planting cacti is 0.9*8+0.1*7=7.9. You should plant the flowers. If the probability of a wet year is only 60 percent, however, then your expected utility from planting flowers is 0.6*10+0.4*3=7.2, and your expected utility from cacti is 0.6*8+0.4*7=7.6. Risk theory tells you that, even though flowers give you the most utility in the more likely state, your overall utility is best served by planting cacti.