# Evaluating alternatives

On a daily basis we have to choose between many different alternatives that the world imposes on us. Some of them are good, some are misleading, some are unaffordable, others are available only after signing up a contract so that you can’t switch for the duration it is active. Some are “free” only after registration, some are based on recommendations from friends and family, some are the most starred alternative that other people have previously chosen, some are based on a recommendation from the seller (usually by a form of discount or to increase profits). There is no guarantee that in any of these situations we will choose the “best” (or at least approximately so) option that is presented to us. And if we make suboptimal or impulsive decisions repeatedly and over a period of time, this accumulates and starts to make our life more miserable.

Any time we enter a store, we have to evaluate alternatives. Bread? 6 offers. Choco biscuits? 10 offers. Ice cream? 11 offers. Just choose. This can quickly become mentally exhausting. But it doesn’t happen only in supermarkets. Need a car? Many car companies, many different types of cars (sedan, van, SUV) with many models each. Choose. Need a furniture for your home? Many furniture providers, many models, many geometrical, visual and comfort constraints. Choose. Getting ill? Many doctors, many recommendations, many diagnoses, many possible treatments. Choose. Want to hire a programmer? 20 candidates for a single position. Choose.

When faced with so much choice, it is easy to try to shut down the brain and just approach the first alternative that comes to your way. In fact, many people seem to do so. Or they rely on any verbal recommendation they come across. Usually this leads to bad choices and some form of regret later. What could help in a similar situation is to ask ourselves, especially when the stakes are high (like planning a bigger financial investment or looking at possible paths of a career as a long-term consequence), “What are the most important attributes I’d like to see in this alternative and what is the relative value of each of them, to me, personally?”. This might seem a strange question at first, but may give you a surprising amount of clarity if you are ready to describe this on paper. What it also highlights, is that no two choices will be alike, given the same information, due to the individual preferences of the decision makers. One might prefer a more comfortable furniture piece, while another will look that it can fit in a small area dedicated for it. Subjectively, the attribute comfort will have a higher value for the first person, while geometric fit will be more important for the second person. You could try to combine all possible attributes you could think of, especially if you want to truly select the “best” alternative, but this could make the decision that much harder. The question above mentioned something about relative importance of each attribute. And it is one thing to try to determine it when you have 5 attributes and very different when you have 20. It is unrealistic to assume that each attribute will be equally important (thus having a weight of 0.2 in the first and 0.05 in the second case). We would need to think of each one separately and assign a weight that makes sense to us, under the constraint that the sum of all weights is equal to one. But this too is unfortunately not enough. We would need to normalize between the alternatives before doing any calculations.

Assume that one of your important attributes is the number of times that a given product or service has been starred (having access to such information). Product A has been starred 101 times, product B - 79 times and product C - 93 times. We could then take the maximum of the three numbers and assume that it corresponds to 1. So the row of the “starred” attribute given these three products would contain the numbers 1.0000 (101/101), 0.7821 (79/101) and 0.9207 (93/101). Lets assume that the prices of the products are 60$, 44$ and 80$. Then we get numbers like 0.75 (60/80), 0.55 (44/80) and 1.0000 (80/80).

This gives us the following table (intentionally kept simple):

As you can see, the values of the weights and the values of the attributes "comfort", "geometric fit" and "visual fit" are entirely subjective, based on our own preferences. As a side note, I’d like to add that it has been shown that we are already relatively good at determining relative values, because the rules we apply to them tend to be the same. But this also means that noone else should see or fill even part of our own table.

Now we want to evaluate these alternatives and find the "best" choice (at least statistically) given our preference. But there is a small problem. We will seek a maximal number, where the price has to be kept minimal. In other words, we want to see 1, 1 and 1 for comfort, geometric fit and visual fit, but we don’t want to see 1 for the highest price (as it is currently set). Instead, 1 has to be reserved for the lowest price and anything above that price should decrease the score we put into the table. Fortunately, only a small change is needed. Instead of calculating current/maximal, we calculate 1/(current/minimal). Now the scores become 0.7333 (1/(60/44)), 1.0000 (1/(44/44)) and 0.55 (1/(80/44)).

Does this remind us of something? Well, the weights can be represented by the vector w and the products—by the vectors p1, p2 and p3. What we would like to compute are the dot products: dot(w, p1), dot(w, p2) and dot(w, p3). The dot product is the sum of all component-wise multiplications of the two vectors, so we are looking for the maximal sum across all cases that we have. Specifically, this means:

Product A: 0.35 * 0.36 + 0.15 * 1 + 0.12 * 0.9 + 0.08 * 1 + 0.3 * 0.7333 = 0.126 + 0.15 + 0.108 + 0.08 + 0.2199 = 0.6039

Product B: 0.35 * 0.72 + 0.15 * 0.52 + 0.12 * 1 + 0.08 * 0.7821 + 0.3 * 1 = 0.252 + 0.078 + 0.12 + 0.0625 + 0.3 = 0.8125

Product C: 0.35 * 1 + 0.15 * 0.58 + 0.12 * 0.74 + 0.08 * 0.9207 + 0.3 * 0.55 = 0.35 + 0.087 + 0.0888 + 0.0736 + 0.165 = 0.7644

We can see that based on our criteria, product B would be a better choice than product C, which would be again a better choice than product A.

What if we had missing values in our table? Could we then still evaluate the alternatives? The answer is no, but we could try to fill an approximate number as long as we feel that it is accurate or a good representation of something we have observed. Then we could proceed as before. Is this approach always applicable? No, since it carries the cost of requiring a sufficiently detailed description. This means that we might choose to apply it only where it will truly matter, only for a relatively small number of products or attributes and where the data can be filled in a reasonable amount of time. In addition, since we decide to omit data this way, we could only approximately evaluate the alternatives, potentially neglecting bad data about some attributes that we haven’t considered.

If we had to implement this, we could highlight the best option after the user clicked on the “Evaluate alternatives” button. This would provide them with immediate feedback and reduce the psychological burden of having to think about all of the alternatives separately. The calculations can be made automatically, as long as the attributes, weights and values are described precisely. And here is the problem, since we don’t know the names of these attributes or the individual’s preferences about the weights or any other attribute-specific data that is known in advance. Every situation will be different, so the only thing that we could provide is a dynamically growing table where this varying data can be filled and where most fields accept only numerical data. But this wouldn’t feel as an application that is specific for the purpose of evaluating alternatives; it could be literally an application about anything.

In the real world there aren’t just three alternatives, but frequently around a dozen or more. This makes sense, because it allows companies to more fully exploit the potentials within the markets. But for the end consumer, this means dramatically more choices. For instance, if you are looking to buy a laptop from Lenovo, there will be ThinkPad, IdeaPad and maybe other series. Each series will have different models. If you are an older person that has always used Windows, for instance, you might still want to stay with it, not necessarily looking for the fastest performance or a particular model. If you are a gamer, the GPU becomes a quite important attribute for you, since it may determine whether you’ll be able to play the latest games. If you are frequently dealing with lots of data, you might need a big and fast storage device, big memory (for online analysis) and fast CPU so you can make transformations on this data, compute occurrences, scores or something else. If you are teaching, having the Office package may be important, even though it may not come pre-installed or would be available separately at a higher total cost. The weight of the laptop will also be an important attribute, since you probably don’t want to carry 5kg (including the projector) across floors and rooms for an entire day. If you are a frequent traveler, in addition to the laptop weight, the battery will also likely have an increased numeric weight in your calculations. All this illustrates how important it is to know in advance what you are looking for, in many situations of the everyday life.

With enough data, all alternatives can be evaluated. While this simple model isn’t perfect, it shows one way in which we can rely less on impulsive decisions and common wisdom and more on the data we collect and examine. If you know of other interesting ways to evaluate alternatives, feel free to share them here.