An Introduction to Mathematics of Risk

As we consider uncertainty, we may use rigorous quantitative studies of chance, the recognition of its empirical regularity in uncertain situations. Many of these methods are used to quantify the occurrence of uncertain events that represent intellectual milestones. As we create models based upon probability and statistics, you will likely recognize that probability and statistics touch nearly every field of study today. Our discussion of these probabilities will focus on simple calculations for illustrative purposes.

First, we need to learn the idea of expected monetary value. The expected value of a situation is the sum of the probabilities of certain events multiplied by the numbers associated with those events.

For instance, supposed we played a game in which a coin flip landing heads paid out $10, and a coin flip landing tails paid out $0. Assuming a fair coin, the expected value of the game is 0.5 * $10 + 0.5 * $0 = $5. Using this information, if I charged you less than $5 to play, you might consider it a good deal, as “in expectation” you would make money from the game. If I charged you more, say $9, to play, you might consider it a bad deal, because “in expectation” you will only receive $5 back. Of course, you might get lucky, flip heads, and make a dollar on the game (the $10 payout for heads minus the $9 cost to play), but many people would not find this to be a compelling wager.

Now, consider if the game changed in two ways. First, change the payouts to $1000 for heads and $0 for tails, the amount you’d likely pay to play the game would increase. What you might have considered an unwise investment at $9 now seems like a great deal. This is because the expected value has changed from 0.5 * $10 + 0.5 * $0 = $5 to 0.5 * $1,000 + 0.5 * $0 = $500.

Finally, if the probabilities associated with the game changed, you might also change your mind about playing. Suppose instead of using a fair coin, the coin used landed on heads 90% of the time (and since probabilities must sum to 100%, it means the chance of tails becomes 10%). Now the expected value associated with the game becomes 0.9 * $1,000 + 0.1 * $0 = $900. You might be willing to pay a substantial sum to play this kind of game.

Expected value calculations are not limited to coin flips. If we have more than two potential outcomes, we can just add up those outcomes multiplied by the probabilities, making sure the probabilities sum to 100%. For example, suppose a firm was considering using a particular trademark that might possibly infringe on another’s mark. The firm thinks there is a 50% chance that the other firm won’t care about it and the company will reap $50,000 in additional profits from using the mark, a 20% chance the other firm will win a lawsuit for $100,000, and a 30% chance the other firm will sue but lose, costing the firm only $10,000 in attorney fees. The expected value from using the trademark is 0.5 * $50,000 + 0.2 * ($50,000-$100,000) + 0.3 * ($50,000-$10,000) = $27,000. As the expected profit is greater than zero, perhaps the firm should consider proceeding. If the profit expected from the mark were less than zero, then perhaps the firm should revisit the trademark and come up with something new.

In the next section, we will offer a technique that attempts to quantify how firms view this decision.

It is important to note that the calculated expected value does not represent the actual outcome of a decision, since only one outcome will occur. It is used to inform decision-making when more than one outcome is possible.

Utility Theory

There is a simple way to adjust our expected value calculation from the last section to incorporate attitude towards risk. We do it by drawing on a bit of economics called utility theory and then weighting the probabilities in our expected value calculations.

Utility theory bases its beliefs upon individuals’ preferences. It is a theory postulated in economics to explain behaviour of individuals based on the premise people can consistently rank order their choices depending upon their preferences. Each individual will show different preferences, which appear to be hard-wired within each individual. We can thus infer that individuals’ preferences are intrinsic. Any theory, which proposes to capture preferences, is, by necessity, abstraction based on certain assumptions. Utility theory is a positive theory that seeks to explain the individuals’ observed behaviour and choices.

This contrasts with a normative theory, one that dictates that people should behave in the manner prescribed by it. Instead, it is only since the theory itself is positive, after observing the choices that individuals make, we can draw inferences about their preferences.

When we place certain restrictions on those preferences, we can represent them analytically using a utility function—a mathematical formulation that ranks the preferences of the individual in terms of satisfaction different consumption bundles provide. Thus, under the assumptions of utility theory, we can assume that people behaved as if they had a utility function and acted according to it. Therefore, the fact that a person does not know his/her utility function, or even denies its existence, does not contradict the theory. Economists have used experiments to decipher individuals’ utility functions and the behaviour that underlies individuals’ utility. If you’ve studied economics, you will recognize that we are simplifying a vast amount of theory in this section, but our aim is to provide a framework for legal analysis.

The key to utility theory is to recognize that people will value the same good in different ways. For example, dining out at an Italian restaurant may provide a tremendous amount of utility for some people but very little for others. The dining experience might convey 1,000 “utils” worth of enjoyment for me, but only 100 “utils” of enjoyment for you. Similarly, watching a ballet might convey thousands of “utils” worth of enjoyment to you, but negative “utils” of enjoyment for me. In general, the higher the utility from an activity, the greater one’s willingness to pay for it.

Expected Value Calculations with Utility

We can use the concept of utility to provide further guidance for companies facing legal decisions. The key is that we use probabilities multiplied by utilities in our expected value calculation, rather than probabilities multiplied by financial outcomes alone. For instance, consider our trademark example from above, but now assume that the firm only has $75,000 in assets, so losing a lawsuit for $100,000 will bankrupt the company.

Nobody at the company wants to risk bankruptcy, and so our simple calculation from above needs to be modified. Recall, the firm thinks there is a 50% chance that the other firm won’t care about it, a 20% chance the other firm will win a lawsuit for $100,000, and a 30% chance the other firm will sue but lose, costing the firm $10,000 in attorney fees. The expected value from using the trademark we calculated above was 0.5 * $50,000 + 0.2 * ($50,000-$100,000) + 0.3 * ($50,000-$10,000) = $27,000.

Let’s change this from dollars to “utils”. We will use functional notation, which uses u(x) to denote the utility from x. Here, let’s assume that u($50,000) = 500, that is the company derives 500 in utils from making $50,000, as they are able to use the new mark without trouble. Let’s also assume that u($50,000-$100,000) = -2,000 utils, as the company is very afraid of bankruptcy, and so they weight that risk much higher than the monetary loss alone, and that u($50,000-$10,000) = 400 utils, as the company wouldn’t like that outcome but doesn’t risk bankruptcy. Our expected value calculation is now 0.5 u($50,000) + 0.2 u($50,000 – $100,000) + 0.3 u($50,000 – $10,000) = 0.5 * 500 utils + 0.2 * (-2,000 utils) + 0.3 * 400 utils = -30 utils. Under this calculation, the huge risk of bankruptcy leads to a negative utility so the firm will likely avoid using the trademark.

A firm that puts greater weight on losses is a risk-averse firm, such as in the example above. They will shy away from risky situations. A firm that puts greater weight on gains, such as a venture capital firm, will engage in greater risk. For example, suppose a firm valued profits more than potential losses, as bankruptcy was common in the industry and would not scare future investors. Their expected utility from using the trademark might be 0.5 u($50,000) + 0.2 u($50,000 – $100,000) + 0.3 u($50,000 – $10,000) = 0.5 * 500 utils + 0.2 * 0 utils + 0.3 * 400 utils = 370 utils. This firm would be much more likely to proceed with using the trademark.

Finally, a firm might be risk neutral. In that case, they would value the utility from gains and losses at the monetary value alone. In this case, u($1) = 1, u($5) = 5, and so on. For this firm, the utility from the trademark transaction is simply the expected value of the monetary gains and losses. For them, 0.5 u($50,000) + 0.2 u($50,000 – $100,000)+ 0.3 u($50,000 – $10,000) = 0.5 * 50,000 utils + 0.2 * (-50,000 utils) + 0.3 * (-40,000 utils) = 27,000 utils. Thus, the calculations for a risk neutral firm are the simplest of the three.