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Rational Choice

The theory of rational choice can be summarized in one sentence: When faced with several courses of action, people usually do what they believe is likely to have the best overall outcome.

To perform a rational choice an individual must first gather information. Next, the individual forges beliefs, based on the gathered information, by using reasoning, assessment and judgment. Examples for what we call "beliefs" include how the decision maker assesses the relative value of different outcomes, the probabilities of the occurrence of various events, and which actions appear to provide efficient means for pursuing one's own objectives. Finally the decision maker, based on his beliefs, has to choose the action which provides the best outcome.

This process is complex, as problems can occur in each of the three stages. The first stage requires collecting facts and information. The main question here is: how much to collect? We would like the amount of collected evidence to be, in some sense, optimal. Collecting too little evidence makes it more likely that the end decision will be in error. Collecting too much evidence costs time and energy. The costs of deliberating may exceed the benefits. For example, if a doctor's examination is too long, the patient may die.

This is particularly relevant for deciding whether to commit suicide. Pro-life proponents encourage suicidal people to seek for other options. In other words, to search for more facts and evidence in order to be able to assess the option of staying alive. However, these proponents do not specify whether and when one should stop looking for more evidence. Consider that undergoing various forms of treatment takes much time and energy. In the meanwhile, a person in distress might suffer for many years, without relief. Indeed a suicidal person should seek for other solutions, but after some time, the suffer caused by prolonging the decision process is more costly than the benefit which might reasonably be gained from further exploration.

The second step of decision making requires that beliefs should be "rational" with respect to available information. For example, If there are dark grey clouds in the sky (information) then most people would believe that there is an increased possibility of rain (belief). Ignoring this information (and thus not adapting your beliefs) while trying to decide whether or not to take an umbrella, is not rational.

The last step, of choosing the best outcome based upon the beliefs, is the focus of the rest of this article.

Good Decisions and Bad Outcomes

Suppose you heard the weather report on the radio. The forecast is rain. You decide to take your umbrella, but there is no rain that day, and you had to take your cumbersome umbrella with you all day long. Did you make a good decision?

Some people would say that you made a mistake. That you should have not taken the umbrella, because the outcome (no rain and you being umbrella-free) would be the best overall outcome. However, it is important to distinguish between good decisions and good outcomes.

At the time you had to make the decision, of whether or not to take an umbrella, you only had so much information on which to base a decision. On that particular day, the outcome was not as good as it could have been, but assuming that weather reports are 80% accurate, taking the umbrella, when the forecast recommends, would be the best action 4 times out of 5. The decision was good, but on any particular day, the outcome may not be optimal.

A similar argument, made by some pro-life advocates, claims that suicide is never a good action, since there is always a chance that if the person continues to live, their situation will improve. This observation does not distinguish between the quality of the decision and the quality of the outcome. Indeed if the person continues to live and the situation improves then this is the best outcome. However, if the probability of improvement is small enough, deciding to live may not be a rational decision.

Maximizing Expected Value

The simplest model uses the expected value [3]. Intuitively, the expected value is what we expect the average to be. For example, lets say we have a dice. When thrown, it can fall on one of 6 faces ranging from 1 to 6. Let f(i) be the value of the dice when it is thrown for the i-th time. If we throw it N times we can calculate the average as :

A = ( f(1) + f(2) ... + f(N) ) / N

For increasing N, A will approach the number 3.5 . This is the expected value. It is computed by calculating the weighted average of the possible values ( in our case, the dice values ) using the probabilities as weights. In our case, the probability of each face is 1/6 . So the expected value is :

E = (1/6) * 1 + (1/6) * 2 + (1/6) * 3 + (1/6) * 4 + (1/6) * 5 + (1/6) * 6 =

(1/6) * (1 + 2 + 3 + 4 + 5 + 6) = 21/6 = 3.5

The difference between the average and the expected value, is that the expected value is something we can calculate in advance, before we make any experiment in real life ( under the assumption that we know the probabilities for each outcome ). The average, on the other hand, is computed according to the results of experiments or real data.

Using the expected value is useful since it is easy to use it to perform various calculations which involve uncertainty. For example, consider the following lottery. There are two choices: A) you do not participate at all ( the monetary value of this option is 0 with probability 1), or B) you pay 10$ for a lottery ticket to get 1,000,010$ with probability 1/2,000,000 . We can calculate the expected values:

E(A) = 1.0 * 0$ = 0

E(B) = 1/2,000,000 * (1,000,010$ - 10$) + ( 1- 1/2,000,000 ) * (-10$) =

1/2 - 9.999995 = -9.499995

So E(A) = 0 and E(B) is negative. According to this, it is not a good decision to buy a ticket. In fact, this is true for most real life lotteries, otherwise, the organizers of the lottery would not make money.

However, there are several problems using expected values. Sometimes the values are not in terms of money or in terms of any other numerical form. For example, suppose you have a choice of eating either:

1) an apple, or an orange, each with probability 1/2, or

2) a banana with probability 1

Which is better? In such cases, in order to use expected values, the outcomes must be assigned some numerical value.

Such problems occur even when using money. For example, is the value of two billion dollars really twice the value of one billion dollars? Obviously, it is worth twice numerically, but an individual cannot really enjoy 2 billion dollars twice as much as he can enjoy one billion. Being able to buy twice as much stuff does not imply twice as much happiness, for example, one can buy to expensive cars instead of one, but at any given time, only one of them would be used and enjoyed. Even for money, we might need convert to different values, in order to compare to other options.

Another difficulty of using expected value is that the model does not account for the fact that different people value different things differently. For a millioner, 50,000$ is worth much less than for a poor person. For example, take the following lottery with two choices: either take 50,000$ with probability 1 ( i.e. guaranteed), or take one million dollars with probability 1/10 . The millioner would be more inclined to take the second option, since 50000$ earned would make little difference to his life. For a poor person 50000$ is worth much, therefore, he would rather take the first option. According to expected value, the first option is better ( 1,000,000 * 0.1 = 100,000 ) however, it is difficult to argue with the poor man's decision to take the option with no risk. In fact, this may provide good reasoning why buying lottery tickets might be rational after all.

See [4] for examples of applying the expected value model for the decision of suicide.

For these reasons we sketch a more elaborate model called expected utility, which is commonly used in the social sciences, especially in economics.

Preference Ordering

To determine what is the best overall outcome, the decision maker must be able to compare different outcomes. Suppose that you are hungry. Is it more rational to choose to eat cake or ice cream? As another example, suppose you have a free afternoon. Is it better to work late, or go meet a friend? Note that in the first example, there is a common goal, to relieve the feeling of hunger. The second example does not even seem to have a common goal.

However, it is always possible to present actions as means of attaining a given end, if that end is to act according to the preferences of the decision maker. By asking a person, or by observing his behavior, we can find out how he ranks different options. One might prefer a piece of cake to one portion of ice cream, but prefer two portions of ice cream to one piece of cake. A list of such pairwise comparisons is called the person's preference ordering.

Note that each person has a different preference ordering, upon which rational decision making will be based. The use of a preference ordering formalizes the intuition that different people may be in the same situation yet prefer different things. This observation rules out any possibility of making recommendations of whether or not to exit, based solely on one's objective situation. The preference ordering reflects the subjective preferences of an individual, it may even change as time passes.

Utility

One is said to be behaving rational if one is acting in such a way as to maximize a utility function. That is, it is assumed that possible outcomes (of a person's decision) can be assigned numeric values and that the person decides in such a way that the outcome with the highest value is chosen.

Using some mathematics, a preference ordering can be converted into a utility function. However, there are cases for which no utility function exists. The possibility of such a conversion is guaranteed if the preference ordering has the following characteristic:

The person must be able to judge, for every two options, which one is preferable, or whether they are equally preferred.
The preference ordering must be consistent ( or in mathematical terms transitive ). This means that if A is more preferred than B and B is more preferred than C, then necessarily A is more preferred than C. For example the relation A "is higher than" B is transitive, but the relation A "loves" B is not transitive. If John "is higher than" Jill, and Jill "is higher than" Gary, then it must be the case that John "is higher than" Gary. However, If John "loves" Jill, and Jill "loves" Gary, then it is not necessary that John "loves" Gary (in fact, it is quite likely that John hates Gary due to jealousy). In short, our requirement is that the relation "is more prefered than" should be transitive.
The person must be able to trade off values against each other.

Stating the third assumption mathematically is beyond the scope of this essay, see [6] for futher details and for more specifics on how to calculate u.

Given these three conditions it is possible to define a utility function, u, which has the following two properties:

x is more preferred than y if and only if u(x) > u(y)
The utility of obtaining outcome x with probability p and outcome y with probability 1 - p is given by the formula: p * u(x) + (1-p) * u(y)

Property 1 means that the utility function reflects preference by numbers. Higher numbers are more preferable.

The second property states that using utility, we can use the same mathematical rules as we have done for calculating expected values. Instead of using numeric values, we use the values obtained by the utility function.

Before we continue, lets summarize what has been said so far. For making decisions under uncertainty, expected value (which we know from high-school) is easy to use. However, expected value is problematic because it requires numerical outcomes, and it does not account for personal preference. These problems can be solved by using expected utility instead of expected value. The utility function, u, gives for each outcome a number. More preferred outcomes have higher numbers. Using u solves the problem of comparing between outcomes which do not have a numeric value. The utility function can be obtained from a preference ordering which is simply a list of comparisons (e.g., ice cream is more preferred than pizza) between outcomes or events. The comparisons are made by a particular individual, so the utility function u is different for each person. This solves the problem of personal preference.

Maximizing Expected Utility

Using the model of expected utility, a rational person computes the sum of the multiplications of the utilities with their corresponding probabilities and then takes the action that maximizes expected utility. If we have actions A1, A2, ... An, and for each action Ai there are m possible outcomes Oi1, Oi2, ... ,Oim , each with a corresponding probability, P(Oij) then the expected utility of Ai is calculated as:

U(Ai) = P(Oi1) * u(Oi1) + ... + P(Oim) * u(Oim)

where U is the expected utility and u(Oij) is the individual utility of an outcome. To choose the best action we calculate U(Ai) for all i from 1 to n, and choose the i for which U(Ai) is maximal.

See [5] for how this model can be applied for the decision of suicide.

EverDawn

REFERENCES

[1] Elster Jon, Nuts and bolts for the social sciences, 1989.

[2] Peter C. Fishburn, Decision and Value Theory, 1964.

[3] Kyle Siegrist and Jason York, "Expected Value" http://www.fmi.uni-sofia.bg/vesta/Virtual_Labs/expect/expect0.html

[4] Answering Pro-Life Arguments, There is Hope, 2b_badlife.html

[5] Normative Models for Rational Suicide, 2b_normative.html

[6] From Preference to Utility, 2b_utility.html