17 Decision theory
Learning objectives:
Students will be able to:
- Distinguish different theories of decision making.
- Calculate the optimal decision under certainty, uncertainty, and under risk.
- Describe and use various criteria of decision making.
- Simplify complex decision making situations and use formal approaches of decision making to guide the decision making behavior of managers.
Required readings: Finne (1998)
Recommended readings: Bonanno (2017, sec. 3)
17.1 Payoff table
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
The payoff or outcome depends on both the chosen alternative and the future state of nature that occurs. For instance, if alternative
The payoff is a numerical value that represents either profit, cost, or more generally, utility (benefit) or disutility (loss).
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.
17.2 Certainty
When a decision must be made under certainty, the state of nature is fully known, and the optimal choice is to select the alternative with the highest payoff. However, determining this payoff can be complex, as it may be the result of a sophisticated function involving multiple variables.
For example, imagine you need to choose between four different restaurants (
In this scenario,
Please note that the characteristics
Domination
To arrive at an overall outcome for each alternative and make an informed decision, the first step is to determine whether any alternatives are dominated by others. An alternative is considered dominated if it is not superior in any characteristic compared to at least one other alternative.
Dominated alternatives can be excluded from consideration. For example, in Table 17.3, we can see that alternative 2 outperforms alternative 3. This makes it unnecessary to consider alternative 3 in the decision-making process.
Weighting
No preferences
Still, we have three alternative left. How to decide? Well, we need to become clear what characteristics matter (most). Suppose you don’t have any preferences than you would go for restaurant
Clear preferences
Suppose you have a preference for the first three characteristics, that are quality of the food (
This means, for example, that you value music (
To determine the best decision, you can calculate the aggregated expected utility for each alternative as follows:
where
Thus, alternative
Maximax (go for cup)
If you like to go for cup, that is, you search for a great experience in at least one characteristic, then, you can choose the alternative that gives the maximum possible output in any characteristic. The choice would in our example be (see Table 17.6):
Minimax (best of the worst)
The Minimax (or maximin) criterion is a conservative criterion because it is based on making the best out of the worst possible conditions. The choice would in our example be (see Table 17.7):
Körth’s Maximin-Rule
According to this rule, we compare alternatives by the worst possible outcome under each alternative, and we should choose the one which maximizes the utility of the worst outcome. More concrete, the procedure consists of 4 steps:
- Calculate the utility maximum for each column
of the payoff matrix (see Table 17.8):
- Calculate for each cell the relative utility (see Table 17.9),
3/5 | 0/1 | 7/7 | 1/3 | 4/4 | |
4/5 | 1/1 | 4/7 | 2/3 | 1/4 | |
4/5 | 0/1 | 3/7 | 2/3 | 1/4 | |
5/5 | 1/1 | 2/7 | 3/3 | 1/4 |
- Calculate for each row
the minimum (see Table 17.10):
3/5 | 0/1 | 7/7 | 1/3 | 4/4 | 0 | |
4/5 | 1/1 | 4/7 | 2/3 | 1/4 | 1/4 | |
4/5 | 0/1 | 3/7 | 2/3 | 1/4 | 0 | |
5/5 | 1/1 | 2/7 | 3/3 | 1/4 | 1/4 |
- Set preferences by maximizing
:
17.3 Uncertainty
When a decision must be made under uncertainty, the state of nature is fully unknown. That is, different possible states of nature exist but no information on their probability of occurrences are given. The optimal rational choice can’t be made without a criterion that reflect preferences such as risk aversion. In the following, I discuss some popular criteria.
Laplace criterion
The Laplace criterion assigns equal probabilities to all possible payoffs for each alternative, then selects the alternative with the highest expected payoff. An example can be found in Finne (1998). In Table 17.12 are the data for another example. According to the expected average utility, the decision should be
Maximax criterion (go for cup)
If you’re aiming for the best possible outcome without regard for the potential worst-case scenario, you would choose the alternative with the highest possible payoff. This “go for cup” approach focuses on maximizing the best-case outcome.
In the example of Table 17.12, the decision applying the Maximax strategy is
Minimax criterion (best of the worst)
The Minimax (or Maximin) criterion is a conservative approach, aimed at securing the best outcome under the worst possible conditions. This approach is often used by risk-averse decision-makers. For examples on how to apply this criterion, see Finne (1998).
In the example of Table 17.12, the decision applying the Maximax strategy is
Savage Minimax criterion
The Savage Minimax criterion minimizes the worst-case regret by selecting the option that performs as closely as possible to the optimal decision. Unlike the traditional minimax, this approach applies the minimax principle to the regret (that is, the difference or ratio of payoffs), making it less pessimistic. For more details, see Finne (1998).
Using the example data of Table 17.12, the regret table is constructed by subtracting the maximum payoff in each state from the payoffs in that state, see Table 17.13.
Alternatives | Regret for |
Regret for |
Regret for |
Maximum Regret |
---|---|---|---|---|
30 | ||||
50 | ||||
50 |
Based on the Savage Minimax criterion, the alternative with the smallest maximum regret should be chosen and the decision is
Hurwicz criterion
The Hurwicz criterion allows the decision-maker to calculate a weighted average between the best and worst possible payoff for each alternative. The alternative with the highest weighted average is then chosen.
For each decision alternative, the weight
This formula allows for flexibility in decision-making by adjusting the value of
The Hurwicz criterion calculates a weighted average between the best and worst payoffs for each alternative. Using the data of Table 17.12 once again, we need to assume an optimism index. Let’s say we are slightly optimistic and willing to take some risks by setting
In
In
In
Thus, the decision is
The example that is shown in Figure 7 of Finne (1998, p. 401) contains some errors. Here is the correct table including the Hurwicz-values (we assume a
36 | 110 | 73 | |
40 | 100 | 70 | |
58 | 74 | 66 | |
61 | 66 | 63.5 |
Thus, the order of preference is
17.4 Risk
When some information is given about the probability of occurrence of states of nature, we speak of decision-making under risk. The most straight forward technique to make a decision here is to maximize the expected outcome for each alternative given the probability of occurrence,
However, the expected utility hypothesis states that the subjective value associated with an individual’s gamble is the statistical expectation of that individual’s valuations of the outcomes of that gamble, where these valuations may differ from the Euro value of those outcomes. Thus, you should better look on the utility of a respective outcome rather than on the outcome itself because the utility and outcome do not have to be linked in a linear way. The St. Petersburg Paradox by Daniel Bernoulli in 1738 is considered the beginnings of the hypothesis.
Infinite St. Petersburg lotteries
Suppose a casino offers a game of chance for a single player, where a fair coin is tossed at each stage. The first time head appears the player gets $1. From then onwards, every time a head appears, the stake is doubled. The game continues until the first tails appears, at which point the player receives
Given the rules of the game, what would be a fair price for the player to pay the casino in order to enter the game?
To answer this question, one needs to consider the expected payout: The player has a 1/2 probability of winning $1, a 1/4 probability of winning $2, a 1/8 probability of winning $4, and so on. Thus, the overall expected value can be calculated as follows:
In the context of the St. Petersburg Paradox, it becomes evident that relying solely on expected values is inadequate for certain games and for making well-informed decisions. Expected utility, on the other hand, has been the prevailing concept used to reconcile actual behavior with the notion of rationality thus far.
Finite St. Petersburg lotteries
Let us assume that at the beginning, the casino and the player agrees upon how many times the coin will be tossed. So we have a finite number I of lotteries with
To calculate the expected value of the game, the probability
Thus, the expected payoff grows proportionally with the maximum number of rolls. This is because at any point in the game, the option to keep playing has a positive value no matter how many times head has appeared before. Thus, the expected value of the game is infinitely high for an unlimited number of tosses but not so for a limited number of tosses. Even with a very limited maximum number of tosses of, for example,
In the real world, we understand that money is limited and the casino offering this game also operates within a limited budget. Let’s assume, for example, that the casino’s maximum budget is $20,000,000. As a result, the game must conclude after 25 coin tosses because
The impact of output on utility matters
Daniel Bernoulli (1700 - 1782) worked on the paradox while being a professor in St. Petersburg. His solution builds on the conceptual separation of the expected payoff and its utility. He describes the basis of the paradox as follows:
``Until now scientists have usually rested their hypothesis on the assumption that all gains must be evaluated exclusively in terms of themselves, i.e., on the basis of their intrinsic qualities, and that these gains will always produce a utility directly proportionate to the gain.’’ (Bernoulli, 1954, p. 27)
The relationship between gain and utility, however, is not simply directly proportional but rather more complex. Therefore, it is important to evaluate the game based on expected utility rather than just the expected payoff.
In experiments in which people were offered this game, their willingness to pay was roughly between 2 and 3 Euro. Thus, the suggests logarithmic utility function seems to be a pretty realistic specification. The main reason is mathematically that the increasing expected payoff has decreasing marginal utility and hence the utility function reflects the risk aversion of many people.