3 Payoff table
Required: Finne (1998)
Recommended: Bonanno (2017, sec. 3)
Every decision has consequences. While these consequences can be complex, economists often simplify decision analysis using the concept of utility. In this framework, anything positive is regarded as utility, and anything negative is viewed as disutility. Ultimately, these outcomes can be represented by a single value, which we will refer to as the “payoff.”
In this sense, decision-making becomes straightforward: we simply choose the alternative that provides the highest payoff. For example, if you know the weather will be sunny, you might choose to wear a T-shirt and shorts. On a cold day, however, you would opt for warmer clothing. But since the state of nature (the weather) is not completely certain, your decision involves some degree of risk, assuming you have some idea of the likelihood of sunshine or cold weather. If you have no information about the weather at all, your decision is made under uncertainty.
In the following, I introduce the payoff table as a tool to stylize a situation in which a decision must be made. Specifically, I will describe three modes of decision-making: under certainty, under uncertainty, and under risk.
A payoff table, also known as a decision matrix, can be a helpful tool for decision making, as shown in the table below. It presents the available alternatives denoted by \(A_i\), along with the possible future states of nature denoted by \(N_j\). A state of nature (or simply “state”) refers to the set of external factors (he has no direct power in it) that are relevant to the decision maker.
The payoff or outcome depends on both the chosen alternative and the future state of nature that occurs. For instance, if alternative \(A_i\) is chosen and state of nature \(N_j\) occurs, the resulting payoff is \(O_{ij}\). Our goal is to choose the alternative \(A_i\) that yields the most favorable outcome \(O_{ij}\).
The payoff is a numerical value that represents either profit, cost, or more generally, utility (benefit) or disutility (loss).
| State of nature (\(N_j\)) | \(N_1\) | \(N_2\) | \(\cdots\) | \(N_j\) | \(\cdots\) | \(N_n\) |
| Probability (p) | \(p_1\) | \(p_2\) | \(\cdots\) | \(p_j\) | \(\cdots\) | \(p_n\) |
| Alternative (\(A_i\)) | ||||||
| \(A_1\) | \(O_{11}\) | \(O_{12}\) | \(\cdots\) | \(O_{1j}\) | \(\cdots\) | \(O_{1n}\) |
| \(A_2\) | \(O_{21}\) | \(O_{22}\) | \(\cdots\) | \(O_{2j}\) | \(\cdots\) | \(O_{2n}\) |
| \(\cdots\) | \(\cdots\) | \(\cdots\) | \(\cdots\) | \(\cdots\) | \(\cdots\) | \(\cdots\) |
| \(A_i\) | \(O_{i1}\) | \(O_{i2}\) | \(\cdots\) | \(O_{ij}\) | \(\cdots\) | \(O_{in}\) |
| \(\cdots\) | \(\cdots\) | \(\cdots\) | \(\cdots\) | \(\cdots\) | \(\cdots\) | \(\cdots\) |
| \(A_m\) | \(O_{m1}\) | \(O_{m2}\) | \(\cdots\) | \(O_{mj}\) | \(\cdots\) | \(O_{mn}\) |
If we assume that all states are independent from each other and that we are certain about the state of nature, the decision is straightforward: just go for the alternative with the best outcome for each state of nature. However, most real-world scenarios are not that simple because most states of nature are more complex and needs further to be considered.
Decision making under uncertainty assumes that we are fully unaware of the future state of nature.
If a decision should be made under risks, then we have some information about the probability that certain states appear. A decision under uncertainty simple means we have no information, that is, no probabilities.
Unless stated otherwise, the outputs in a payoff table represent utility (or profits), where a higher number indicates a better outcome. However, the outputs could also represent something negative, such as disutility (or deficits). In such cases, the interpretation—and the decision-making process—changes significantly. Please keep this in mind.