Trump's Triumph: The Paradox of Voting (an Old Theory, a New Perspective)

“You are out!”

The U.S. recently held its 45th presidential election, and the plot twist was strong in this one. Donald Trump as the president-elect is an outcome many did not foresee. His rising to the throne has been causing non-stop chaotic political, social and economic responses ever since the announcement. While his actual performance as the U.S. president has yet been realized (and we still do not expect good things), there is admittedly still a possibility that the projected disaster is an over-estimation. Nonetheless, this election result per se has resulted in many questioning the power of the people, the democracy itself, whether it is as good as we once thought. While I do not believe in democracy as a one-size-fit-all system of governance, I do think it is the best practice out of other worse alternatives.

But, if you want to understand the U.S. election and democracy as a ruling mechanism in practice, it is essential that you grasp the mismatch between individual and social preferences – the fruit of, not democracy, but the human-designed electoral system that we so believe to be imbued with democratic quality.

Let’s get four things out of the way, which are the four factors mentioned countless times by the media as the root causes of the recent U.S. electoral result:

1. Many people resent the U.S. government, accusing it of stealing their jobs (due to immigration and relocation of manufacturing facilities to foreign countries, etc)

2. Almost half of those eligible for voting did not vote

3. The convoluted indirect electoral system (electoral college) in which voters only get to vote for the electors who have pledged to vote for one of the two presidential candidates. This means that the popular vote (direct vote from the qualified voters) does not decide the victor.

4. You can only vote between two choices. To some people, it is like having to pick between rotten apple and spoiled egg which might have been a part of the reason of the high abstention rate.

These four factors will not be discussed here as mainstream media has extensively covered them. The objective of this article, however, is to explain about individual and social preferences and then relate them to the key idea of Arrow’s Impossibility Theorem and the paradox of voting.

First things first, to distinguish between the two, it is imperative that we understand what preference really is in economics. As a common terms, preference is simply how you order/rank among different alternatives. In economics, the definition is quite similar, except the fact that economics prefer a certain type of preference over the others. For the so-called “Preference” to be valid, for it to produce consistent and scientifically sound outcome, economics require preference to possess the following features (non-exhaustive list):

1. Completeness of preference: All outcomes can be ranked. That is to say if you have two objects A and B, you must be able to decide the order (which comes first) between the two, whether A is preferred to B (A is at least as good as B) or B is preferred to A (B is at least as good as A) or you are indifferent between A and B (i.e., A is just as good as B).

2. Transitivity of preference: Preference has to be transitive. That is to say your ordering has to conform to logical flow. If you have to rank among three objects: A, B, and C, and you prefer A to B and B to C, then without having to ask you any question, I have to be able to correctly infer that you prefer A to C (A > B > C). If you like Apple better than Burger and Burger better than Cat (not to say that you eat cat), then given the three choices in front of you, you have to pick Apple for your preference to be transitive. If you pick Cat over Apple, then this means that your preference is not consistent and it will be very expensive to design economic policies to suit people like you (because revealing your preference does not say much about what improves your welfare).

3. Independent of irrelevant alternatives: Simply put, this means that introducing a new option to you will not change your final ordering of things. If you prefer A to B, then the introduction of C into your list of available choices should not cause your preference to reverse. In other words, say, if you prefer small-size drink to large-size drink, then introducing a medium-size drink should not cause you to switch to large-size drink. This is a rule quite often violated at place like movie theatre where the medium size drink’s price is so close to the large-size one that people just choose the large one instead (it was an experiment conducted long ago). Such preference can be easily exploited and is not considered economically sound.

Unfortunately for economists out there, people often unconsciously violate these rules. This means individual ordering is not so nice and neat as we usually think, and the implication is that even at individual level alone, the mechanism behind the U.S. voting does not produce the socially best outcome. What’s worse is that the soundness of social preference is resting on an even more delicate balance than individual preference. An optimal social preference has two additional requirements:

4. Pareto optimality: If every individual prefers A to B, then the society should ultimately choose A to B. That is to say if the society prefers A to B, it is enough information for us to know that every individual in that society prefers A to B. What it translates into is the concept of pareto improvement, that is to say if social preference reflects individual preference, then given proper compensation, an increase in social welfare will make everyone at least as better off as before and no one worse off. Of course, this is a very strong assumption, and in most cases, social preference can only reflect the majority’s preference and an improvement in one part almost always come at the cost of the others.

5. Non-dictatorship: a rather trivial requirement. It just says that one person cannot be the sole decision maker of what the social preference should be without accounting for the ranking of other members.

Only when these two qualities are further satisfied can we say that the social preference, the voting result, is truly a reflection of democracy at its best. Only then can we confidently say that the outcome is determined by a force equivalent to the market force. Only then can we judge democracy itself and not the people who design the self-proclaimed democratic system. The sad reality is that the election is subject to manipulation either unintentionally or by design, not because of democracy’s core principles but because how we interpret it and how we use that interpretation to construct and shape our ideal democratic society. The former can be considered a systematic error, but the latter is a clear sign of an inefficiently designed ruling system that needs change. So, what do I mean when I say ‘the mechanical design of the election can be the manipulative force that alters the outcome according to certain individuals’ will?’

This has to do with what is known as “the Arrow’s Impossibility Theorem”. Arrow conducts a thought experiment that explains the voting conundrum. He then extends the result to conclude that it is impossible to find a social preference that represents every one of its members’. Let me elucidate the idea here.

Think about the following scenario where the population is divided equally into three groups: The democrats, the republicans, and the monarchists. These three groups hold different ideology and each has its own complete and transitive preference (as shown in the table). That is to say, preference within individual group is logically sound. Let’s suppose that there are three presidential candidates available: Jane (a democrat), Bill (a republican), and Sam (a monarchist). Can the electoral system that requires you to pick between two choices at a time yield the best outcome for the society as a whole? Is it free of manipulation? The answer is No.

Let’s look at the table below that showcases each group’s ranking of the presidential candidates.

Carefully consider the followings:

– If we held an election between Jane and Bill, then the Democrats pick Jane, Republicans pick Bill, and Monarchists pick Jane (since Sam is excluded). So, Jane would win by two-third of the vote (Jane > Bill).

– If we held an election between Jane and Sam, then the Democrats pick Jane, the Republicans pick Sam (since Bill is excluded), and the Monarchists pick Sam. So, Sam would win by two-third of the vote (Sam > Jane).

– If we held an election between Bill and Sam, then the Democrats pick Bill (since Jane is excluded), the Republicans pick Bill, and the Monarchists pick Sam. So, Bill would win by two-third of the vote (Bill > Sam).

How does this ultimately affect the election result? Well, because the preference in this case is not an economically preferred one. The preference is not transitive. Jane is preferred to Bill (Jane > Bill). Bill is preferred to Sam (Bill > Sam). BUT, Sam is preferred to Jane (Sam > Jane) when it should be the other way around. This social preference does not make sense. This allows it to be easily manipulated by any third-party wishing to benefit from their preferred candidate.

Think about it for a bit. I can alter the result however I want. In other words, I can pick the winner even before the election.

If I want Jane to win, then I can first ask people to vote between Bill and Sam. Bill would win. Then I can arrange another vote between Jane and Bill, and Jane would win.

If I want Bill to win, then I can first ask people to vote between Jane and Sam. Sam would win. Then, just like before, I hold the grand election between Bill and Sam. Bill would easily beat Sam by 33% margin.

Same logic if I want Sam to win. I first ask voters to pick between Jane and Bill. Jane would win. Then I simply hold the finale between Jane and Sam, and I know for sure that Sam would win.

What the above implies basically is that the voting system we have now does not represent a well-built and coherent collective preference that captures individual preferences in a consistent manner. It is thus subject to manipulation at the very basic level without even having to go through technologically advanced methods like vote-hacking. This is the underlying economics. Well, probably not purely economics, but the idea is well embedded within micro-economics on the study of individual preference in order to understand and maximize people’s welfare. I consider this to be economics enough. Of course, I understand the reality is way more complicated, but by extracting only the most necessary elements for simplicity, it allows us to understand the more complex problem at hand.

Was this really why Trump won? It is a possible explanation though I doubt it explains the whole story. Every other field has their own way of unraveling the mystery behind that historical election, and it is unwise to only consider each separately. So, what I put forth in this article is a powerful idea worth considering, not solely, but in conjunction with other economic and non-economic factors that we did not bring up for discussion.

Arrow’s Impossibility Theorem (and the paradox presented) is not without criticism. But, it remains to be very influential in economics, in public choice theory, the theory related to the political market (market of legislation) that we will discuss in our next article. For now, I hope you have learned something from this post and can use it to help assess and revise whatever voting system relevant to you within your close and expanded circle.

Look forward to the next article, a continuation of the previous one on economic modelling of the legislation market.