Choice — the basics

Off we go on our long, long journey through economic theory. We should start at the beginning, so we start with a discussion of how a person makes choices, in the view of the economist.

This seems simple, but it merits attention as it is the foundation of economic theory. We will start with some assumptions that seem to turn our choice-maker (let’s use decision maker from now on) into a robot-like person, the infamous economic man (homo economicus, for those who want to sound pretentious). I will tell you more later in our journey about how we can make our decision-maker act more realistically. But first, we need to start with the simplest version of the story.

For a choice/decision to be made, we must have: (1) a decision maker (let’s call her/him D for short), (2) some alternatives that D can choose from, and (3) a method that D follows to make a decision.

1. For now, think of D as a person. Later we will see that D can be a computer algorithm, an animal, or a group of people.
2. Assume there is a set of alternatives that are available to be chosen by D.
3. We have a choice for the method; either D ranks the available alternatives from best to worst and chooses the best, or D applies some other method that results in D’s decisions from all the possible sets of alternatives being a stable collection of choices.

Notation

Using a bit of shorthand will make our life easier; nobody likes mouthfuls, whether giving a speech or reading a text. Our shorthand will make the text look more like math, but do not be afraid! I will use the shorthand notation only in the service of the noble aim of letting us be lazy in a productive manner. This is exactly why all sciences use notation. Nobody needs to clutter their precious, short temporary memory with long phrases when a small collection of symbols will do. And with an electronic text like this, you can always search for a notation you may have forgotten. We’ve already started using notation, by naming our decision maker D.

All right, so let’s make up some notation so we can refer to the set of alternatives more readily. Let us call the set of alternatives X.

An example would be a menu you receive when you visit your favorite restaurant for dinner. Then X might look like this (of course, with more items and fancier names):

X = { salad, soup, pasta, chicken, steak }.

Sometimes we have a limitation to a subset of X, such as when one of the items on the dinner menu is unavailable (no more chicken tonight, sorry!). A subset of X is another set which contains some of the members of X and nothing that does not belong to X.

There is nothing to restrict our discussion to dinner, though; it was just a convenient example. Other examples of sets of alternatives of interest include the following, and there are many, many others that economists study.

• All items for sale in a store we are in that D can afford.
• All candidates who are running in an election in which D is voting.
• All the possible drafts of a law that D wants to propose to a legislative body.
• All the possible prices a firm may charge for its products.
• All the measures a government may take to reduce unemployment.

Incidentally, are you surprised that voting and legislation showed up? Aren’t we talking about economics, you say? Ah, but the economic way of thinking applies to many choices and that’s why economics has been accused by sociologists, political scientists, and others to be imperialistic. By the end of our journey, you’ll be able to make your own mind on the merits of this accusation.

Preferences

Take the first of the two possibilities from point 3 above (how D makes a decision). For this, we assume that D has a stable preference ordering. While this sounds fancy, it simply means that D has thought about all the alternatives in X and has ranked them all from best to worst.

We may decide to allow ties (a tie is being indifferent between two distinct alternatives) but I will spare you this for now, as it complicates notation and theory construction a bit and you don’t need complications this early in the journey.

So a preference ordering on a set of alternatives X looks like this: name the members of X a, b, c, and so on, and write aPb , bPc and so on to mean that a ranks higher (is preferable) to b, b ranks higher than c, and so on.

With this tool in our toolbox, we can now describe the method that D follows to make a decision: (A) bring to mind D’s own preference ordering, and then (B) decide to choose the best-ranked available alternative. That’s the one, call it x, that when compared to any other available alternative y, stands to it in the relation xPy — in words, x is preferred, according to D, to y.

Simple! Isn’t it? Rank all the things you may have to choose from. Look at which ones are actually available in any given decision situation. Choose the top-ranked one among these. Done.

Choice functions

The preference ordering method assumes a lot about D. It assumes that D has thought long and hard about all potential alternatives that D might face, has ranked them all, the ranking does not depend on the presence of particular alternatives in the subset actually available for choice in a given situation. That’s why it feels like D is acting like a robot, or at least like Mr. Spock of Start Trek.

This is why economic theorists have developed another method to describe D’s choice procedure, a method that can allow a wider range of behavior than the preference ordering method does.

Start with the set of all potential alternatives again, X. Consider all its possible subsets, except the empty one (of course). Call S any particular subset of X. The method that D uses to decide on a choice from any such S is to have a choice function, C, which is a little choice-factory, if you like. Give C a subset S of available alternative, and C produces a choice. Simple, like the preference ordering method, right?

Well, if the overall set X has many members, then it has a huge number of subsets. This means that the single letter C conceals a huge number of choices waiting to be made, depending on which subset S of X is available in any given situation. D still has to have a big, Mr. Spockian brain to hold it inside.

Next steps

This has been a tiny little first step on our long journey. The next step will be to consider some examples of the operation of the two decision-making methods, the one via a preference ordering and the one via a choice function, so you can get a better idea how the methods work. Until next time!

[This draft 2014–11–04]