Arrow's theorem

Main article: Voting Methods

MCDM can be used for voting but runs into Arrow’s impossibility theorem. Kenneth Arrow, an economist, won a Nobel prize in 1972 for his contention that it is impossible to extract a social order of preferences from individual preferences while also adhering to fair voting principles. To get some understanding of this, it is useful to imagine a scenario where we are trying to choose between Policy A, B, or C. Suppose 3 people come up with their order of preferences:

1. ABC
2. BCA
3. CAB

Now our task is to devise a single order of preferences which best reflects the views of all three. Voter 1 prefers A to B and this preference is shared by Voter 3. We could therefore say that A > B. We can further see that B > C because both Voters 1 and 2 prefer it. We would therefore expect that A > C by a presumed transitive property but in fact quite the opposite is true. Two voters prefer C to A. If we continue like this we get, rather than a social ordering, a cycle. It turns out to be impossible to construct a social ordering that makes any sense.

We obtain the same result if we do this a little differently. Suppose we look at the ordering BC which means that B is preferred to C. We can see that this is the case for 1 and 2 and since this preference is satisfied by two of the options on the list, we will give 2 points to each option that has it. Thus option 1 receives 2 points and option 2 receives 2 points for its BC preference.

Let’s do the same for ordering AB, and again we can give 2 points to Option 1 and 2 points to Option 3 for having it.

If we continue along these lines, we will come up with the following points for each option:

1. ABC – +2 pt for BC, +2 pt for AB, +1 pt for AC
2. BCA – +2 pt for BC, +2 pt for CA, +1 pt for BA
3. CAB – +2 pt for CA, +2 pt for AB, +1 pt for CB

Each option wins 5 points, meaning there is no clearly preferred social choice. It doesn’t help to invoke preferences which no voter listed in the hope that it will somehow result in a better compromise:

4. CBA – +1 pt for CB, +1 pt for BA, +2 pt for CA
5. ACB – +1 pt for CB, +1 pt for AC, +2 pt for AB
6. BAC – +1 pt for BA, +2 pt for BC, +1 pt for AC

Each of these options are equal as well, although they are all worse than the first three.

Let’s make this a little more realistic by assuming the 100 people voted for preferences 1,2,3. Let’s say:

1. 30 votes for ABC
2. 30 votes for BCA
3. 40 votes for CAB

We obtain the following vote tallies:

• 60 votes for B > C
• 70 votes for C > A ==> here we would expect that B > A, but…
• 70 votes for A > B

The last vote tally contradicts the first two and leads to the same cycle we mentioned above.

Arrow’s Theorem (aka Arrow’s Impossibility Paradox) generalizes this result by saying that in any ranked choice voting system (except trivial ones) it is impossible to determine a social choice preference given the following fair voting principles:

1. Social ordering – The results must be an ordering of alternatives, not a cycle.
2. Unrestricted domain – The aggregation procedure must handle any individual preferences.
3. Pareto efficiency – Unanimous individual preferences must be respected. If every voter prefers A to B then A should always win, regardless of any change in the preferences of other alternatives.
4. Non-dictatorship – The wishes of multiple voters needs to be considered. The decision cannot mimic the choice of a single voter.
5. Independence of irrelevant alternatives – If a choice is removed, the ordering of the rest of the choices should be preserved.

With this in mind, let’s suppose instead that we tallied the votes by awarding every ordering (1,2,3) with the total for every pairwise ordering:

1. ABC would get 70 for AB, 30 for AC, and 60 for BC ==> Total = 160
2. BCA would get 60 for BC, 70 for CA, and 30 for BA ==> Total = 160
3. CAB would get 70 for CA, 40 for CB, and 70 for AB ==> Total = 180

It would appear that CAB is the winner, even though we suspect that it won just because it has a plurality of the votes. In fact, though, it violates Condition 5 of Arrow’s theorem. If we remove, for instance, choice A we would get the following:

1. BC would get 30 votes
2. BC would get 30 votes
3. CB would get 40 votes

Now, clearly BC is the winner even though in our previous analysis we scored CB the winner because CAB won.

Arrow’s paradox does not always lead to failure. For example, in cases where a clear majority favors one ordering of choices, the winner is obvious. It is also not a problem when a simple two-choice decision is made (as in many elections) because then, indeed, a clear majority will favor one or the other. This case can only fail when the electorate is evenly split, a generally unlikely event where a large number of people vote. In smaller voting scenarios (eg panels of judges) a majority will usually be guaranteed by restricting the membership to an odd number.

Nevertheless, Arrow’s insight about voting methods presents real difficulties for consensus-based systems. We should note at this point that it applies to ordinal voting systems only. We can get around its difficulties, perhaps, by looking at cardinal voting systems.