In the previous article, we started discussing multiple criteria decision analysis (MCDA). We went over the whole process of solving a multiple-criteria decision problem. Well, almost. We skipped two crucial parts: a detailed look at the concept of criteria and, more importantly, I haven't introduced any specific MCDA methods that would produce something useful.

Today we are going to address the second issue and talk about Even Swaps which is a very simple decision-making method that does not require any complex mathematical formulas. I want to introduce you to a tool you can start using quickly without much hassle. Unlike other methods, Even Swaps doesn't require advanced knowledge about criteria, so for the time being we can postpone a discussion on this topic. We will get back to it pretty soon though. But for now, back to Even Swaps ...

Which apartment should I rent?

The best way to start is by introducing an example problem. Let's say I am looking for a new apartment to rent. My goal is to find an apartment that fulfills my requirements the best. I am interested only in apartments in my current city and since I don't yet have a family, I am fine with 2 rooms plus a kitchen corner or a separate kitchen. There are hundreds of such apartments currently available but for illustrative purposes, I've limited my set of alternatives to six.

I consider the following criteria as important to my decision: (1) monthly rent including basic services such as water and heating, (2) apartment size, (3) condition of the apartment, (4) neighborhood rating and (5) commute time to work.

We can summarize the evaluation of my 6 alternatives according to these 5 criteria in a so-called decision matrix. A decision matrix is a simple table with rows representing alternatives and columns representing criteria (or vice-versa if you prefer it that way):

Dominated alternatives

We start the Even Swaps procedure by looking for so-called dominated alternatives. An alternative is dominated by some other alternative if this other alternative is better at least according to one criterion and equally good according to all other criteria.

This definition might sound a bit confusing, so let's get specific. Consider alternatives F and C. We can see that apartment C is cheaper, larger, located in a better neighborhood, and closer to my company's offices. The condition of both apartments is the same. Given these circumstances, we can say that F is dominated by C. If we explore the whole decision matrix we discover that F is dominated by every other alternative. We are sure there are alternatives better than F. If we were to rationally select the best alternative, it wouldn't be F. We can remove it from our decision matrix:

Alternative	Price [EUR]	Size [m2]	Condition	Neighborhood	Commute time [min]
A	580	55	good	average	26
B	575	53	good	average	41
C	615	53	average	good	27
D	480	47	good	average	39
E	755	51	very good	good	11

Let's also consider alternatives A and B. We can see that B is smaller and more distant from my workplace. Apartment quality and neighborhood rating are the same. B is a bit cheaper, but the difference is negligible - not enough to compensate for its weaknesses. We can say that B is pseudo-dominated by A and get rid of B as well:

Alternative	Price [EUR]	Size [m2]	Condition	Neighborhood	Commute time [min]
A	580	55	good	average	26
C	615	53	average	good	27
D	480	47	good	average	39
E	755	51	very good	good	11

Even swaps

Now that we got rid of dominated alternatives, we can start performing even swaps. The idea is to select one criterion that we want to equalize for all alternatives. We can start with neighborhood rating. Let's say we would like all alternatives to have a value of "good". Alternatives C and E already are in a good neighborhood. No issue there. But what about A and D? We can't just replace "average" with "good". This would essentially create a more valuable alternative. We have to compensate for the virtual neighborhood upgrade by a downgrade in some other criterion so its overall value stays the same. Let's pick the apartment size.

We now ask ourselves: "How much apartment area would I be willing to give up if the neighborhood ranking went from average to good?" Answering such trade-off questions is the most difficult part of the process. In my case, I would be willing to give up 3 square meters in case of A. As for D, I am willing to give up only 1 square meter, because I am approaching the minimal area I would consider comfortable. After trading some apartment area for neighborhood rating, we get a modified decision matrix. We have just performed our first even swap.

	Price [EUR]	Size [m2]	Condition	Neighborhood	Commute time [min]
A	580	52	good	good	26
C	615	53	average	good	27
D	480	46	good	good	39
E	755	51	very good	good	11

Since now all alternatives have the same neighborhood rating, this criterion is no longer useful and we can remove it from the problem description:

	Price [EUR]	Size [m2]	Condition	Commute time [min]
A	580	52	good	26
C	615	53	average	27
D	480	46	good	39
E	755	51	very good	11

Getting to the best alternative

We now have all the tools we need to solve our decision problem. First, we remove dominated alternatives. Second, we perform even swaps and remove a criterion. All we have to do is repeat these two steps until only a single alternative remains.

Now that we got rid of the neighborhood rating criterion, we should again look for dominated alternatives. We can't find any cases of true domination in our current decision matrix. What about pseudo-domination? We can see that compared to C, apartment A is cheaper, in better condition and slightly closer to work. The only advantage of C is its slightly bigger size. We can again consider C as being pseudo-dominated by A and remove it from the decision matrix.

	Price [EUR]	Size [m2]	Condition	Commute time [min]
A	580	52	good	26
D	480	46	good	39
E	755	51	very good	11

We have resolved domination, so it's again time for even swaps. Let's now focus on equalizing commute time by trading it for price. We would like all alternatives to have a commute time of 26 minutes. Apartment D has a commute time of 39 minutes, that's 13 minutes more. So we ask: "What price increase would we accept in exchange for a 13-minute commute time decrease?" My answer is 30 EUR.

As for alternative E, we have to ask the opposite question: "What price decrease would compensate for a 15-minute commute time increase? For me, 26 minutes is still quite an acceptable commute time so I would be fine with a relatively small price decrease of 20 EUR.

After considering both changes and removing the now equalized commute time criterion, we get a new decision matrix:

	Price [EUR]	Size [m2]	Condition
A	580	52	good
D	510	46	good
E	735	51	very good

Let's have a look at dominance. There are again no cases of true dominance. At the same time, I don't see any alternative being pseudo-dominated by another. We start the next round of even swaps without removing any alternative.

We can equalize the apartment condition criterion on the value of "very good" and again use price as our trade-off criterion. Since a "good" condition is enough for me and doing small repairs from time to time sounds fun, I don't value the improvement that much. If the apartment condition went up from "good" to "very good" I would be willing to pay only 15 EUR more. After considering this tradeoff and removing the condition criterion, we get a new decision matrix:

	Price [EUR]	Size [m2]
A	595	52
D	525	46
E	735	51

We see that E is dominated by A. It is both smaller and more expensive. We can get rid of it:

	Price [EUR]	Size [m2]
A	595	52
D	525	46

Now it's only between price and size. Let's equalize the apartment size to 52 square meters. How much would I be willing to pay to go from 46 to 52 square meters? Given that I now live in an apartment that is smaller than 46 square meters and am doing quite okay, I would be willing to pay 50 EUR at max. This gives us:

	Price [EUR]
A	595
D	575

According to our last remaining criterion, A is dominated by D. We can remove this alternative from the decision matrix. Now only a single alternative remains. We have found a solution to our problem. Even Swaps recommends renting apartment D.

Pros and Cons of Even Swaps

Now that we have seen how Even Swaps works, let's talk a bit about its strengths and weaknesses. Starting with the positive, the biggest advantage of Even Swaps is its relative simplicity - no mathematical formulas, we just remove dominated alternatives and equalize criteria by doing trade-offs.

Simple does not mean easy though. Evaluating individual trade-offs between criteria is hard. The HBR article that originally introduced the Even Swaps method offers a few recommendations:

• Make the easier swaps first.

• It might be tempting to start thinking about the importance of different criteria. This might be misleading. As we will discuss in future articles, there are many MCDA methods where criteria importance plays a crucial role. But when it comes to Even Swaps, we need to stay focused on doing trade-offs.

• Consider your starting point. A salary increase of 500 EUR can have very a different value if your starting point is 1500 EUR compared to starting at 4000 EUR.

• Seek out solid information.

When Denis Bouyssou and Marc Pirlot discuss Even Swaps in Multiple Criteria Decision Analysis: State of the Art Surveys, they mention more disadvantages:

• Even Swaps is only usable when solving smaller problems. Imagine if you had to perform Even Swaps with 50 alternatives and 10 criteria. You would get crazy.

• The result is only a single alternative. As such Even Swaps has limited use. It can only address choice problematic problems. There is no way to rank alternatives, sort them into different categories, or get a better understanding of the problem at hand. You might have two objections here: maybe we could rank alternatives by the order in which they were eliminated. This might be misleading though. The order of elimination depends on the order in which you eliminate criteria. The second objection might be that we are creating a decision matrix, which seems like a good tool to describe and analyze our problem. This might be true, but the concept of a decision matrix is not something unique to Even Swaps. As we will discuss soon, it is a very general tool deployed by many MCDA methods.

• If you add a new alternative, you need to start the whole procedure all over again.

• It's hard to refer to Even Swaps when explaining your decision to someone else.

There is also an interesting article by Lahtinen and Hämälainen, describing two additional issues with Even Swaps:

• Path dependence: This is a generalization of a problem I already described. Order of criteria elimination can have an impact not only on the "ranking" of alternatives but also on our final result.

• Cognitive biases can influence the Even Swaps procedure. The article mentions two such biases. First, there is a measuring stick bias which states that we put too much importance on a criterion that we use as our "currency" when buying or selling a criterion value. The second bias is loss aversion: we fear losses more than we are happy about gains. In our example, we might put more value on a decrease in size from 52 to 51 square meters than on an increase from 51 to 52 square meters. The authors offer two recommendations to mitigate these biases. First, when it comes to the choice of your "currency" or "measuring stick", you should choose the criterion in which your alternatives vary the least. Second, you shouldn't stick to a single alternative as your basis for equalization. It's better to use a different base alternative in each iteration.

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To sum up, Even Swaps is a very simple decision making method. But this simplicity comes at a cost of limited usability. My recommendation is to use Even Swaps only when solving small closed problems with only a few criteria and alternatives. A second requirement is a presence of a good measuring stick - a criterion that can be used as a currency for making trade-offs.

Now that you've read to the end, there is one last thing you should do: Find a decision problem you care about and try to solve it with Even Swaps. Practice is the best teacher. If you encounter difficulties, don't hesitate to write a comment.

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Cover image by starline on Freepik