Tuesday, December 6, 2011

Decision Making According to Economists, Part 1: The Expected Value Criterion

In a previous post, I introduced two approaches in studying how people make decisions. In this post, I will discuss the first of these approaches in greater detail: normative decision making.

The normative or prescriptive approach deals with how people should make decisions: it looks at decision making from the point of view of an ideal decision maker--one who is fully informed, able to compute with perfect accuracy, and fully rational--in a environment where information about all available alternatives are known. The normative approach is generally quantitative and involves formulas that range from the basic to the insanely incomprehensible, and thus usually falls within the exclusive purview of economists and mathematicians. The first important normative decision making criteria--expected value--shall be the focus of this post.

Making decisions using expected value

The scenario below illustrates a typical decision problem

ABC Realty has recently purchased land for the development of a new luxury condominium complex. ABC is in the process of selecting the size of the project that will lead to the largest profit given the uncertainty of the demand for condominiums.

Using the four steps in making good decisions that we discussed previously, let's take a closer look at the details of this problem.

1. Identify all available alternatives. ABC needs to choose the size of the condominium project. Let's say the following alternatives are available:

d1 = a small condominium complex
d2 = a medium condominium complex
d3 = a large condominium complex

3. Identify uncontrollable or unpredictable circumstances that may affect the payoffs of alternatives. The project's profits will be affected by the demand for condominiums. Since these possible occurrences, referred to as states of nature, are beyond the control of the decision maker, they are just assigned likelihoods or probabilities. For example, let's say that according to ABC's analysts, there is an 80% chance that demand for condominiums in the foreseeable future will be strong and a 20% chance that it will be weak, or

probability of strong demand = P(s1) = 80%
probability of weak demand = P(s2) = 20%

It should be pointed out that since s1 and s2 cover all possibilities for the demand for condominiums, P(s1) and P(s2) should sum to 1 or 100%.

2. Determine the costs and benefits of alternatives. Both ABC's choice for the size of the project and the demand for condominiums would affect the project's profits or payoffs, which are presented in the following table

We see from this payoff table that the decision is not straightforward since the best alternative--the one with the highest payoff--changes with the demand for condominiums: whereas a large complex would take advantage of strong demand, it would lead to losses if demand turns out to be weak because of (presumably) the higher cost of construction.

4. Evaluate alternatives using some criteria or rule and make a decision. One commonly used criterion in making a decision given the information presented above is the expected value criterion.

The expected value of an alternative is the weighted average of its payoffs under different states of nature, using the probabilities as weights

This concept was first formalized in Jakob Bernoulli's groundbreaking work, Ars Conjectandi, published in 1713.

Jakob Bernoulli
We compute for the expected value of each alternative as follows

According to this rule, since alternative d3 or building a large condominium complex results in the highest expected value, ABC should choose this alternative.

While the expected value rule does make a lot of practical sense, we should be careful in interpreting the numbers that result from our analysis. An expected value of 14.2 million does not mean ABC will earn that much if it decides to build a large complex: ABC will either earn 20 million or lose 9 million depending on what demand for condominiums will be. 14.2 million is ABC's average profit if it faces this scenario several times and makes the same decision to build a large complex each time; in other words, it's what ABC stands to earn in the long run.

Expected value in practice

The most common practical application of the expected value concept that I can think of is in gambling, particularly in poker. If you play the Texas Hold'em variety or any similar variant, you may have heard of this strategy rule: join the game if the probability of making one of your outs (i.e., your number of outs divided by the number of cards remaining) times the pot is greater than the required bet. The first part of this rule--the probability of making one of your outs times the pot--is the expected value or payoff of joining the game, so if this is higher than the cost of joining, then it makes sense to call the bet.

Something to think about

We'll end this post with something that has puzzled the greatest minds for the longest time until it led to another groundbreaking concept in the study of decision making, which will be the topic of Part 2 of this post.

Think about this scenario for a while.

In a game of chance, you pay a fixed fee to enter, and then a fair coin will be tossed repeatedly until a tail first appears, ending the game. The pot starts at 1 peso and is doubled every time a head appears. You win whatever is in the pot after the game ends. Thus you win 1 peso if a tail appears on the first toss, 2 pesos if on the second, 4 pesos if on the third, 8 pesos if on the fourth, etc. In short, you win 2^(k−1) pesos if the coin is tossed k times until the first tail appears.

What would be a fair price to pay for entering the game? Read Part 2 to find out.

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