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An impossible board decision

 

JEREMIAH JOVEN JOAQUIN

IMAGINE that you, along with other company board members, are at a meeting. It is one of the most important meetings of the board as you will all decide the fate of your company. The board’s chair presents three options — A, B, C — and she tells everyone that it is imperative to have a majority decision as to what option the company should take.
After much discussion and deliberation, one board member moved that instead of the usual one voting per motion, a ranked-choice voting system must be used in casting one’s vote.

He reasoned that by considering each member’s preferences, rather than just his or her first choice, the board will have more information on what each member truly prefers. And besides, this is another voting procedure prescribed by Robert’s Rules of Order. You seconded the motion and was unanimously agreed upon by the body. The chair then proceeded to discuss how to voting will proceed:


Each board member will rank the options according to his or her personal preference. For example, if one prefers A over B and B over C, then her preference ranking is A, B, C. If one prefers B over A and A over C, then her preference ranking is B, A, C. And so on. After casting the votes, the chair will aggregate them. And whatever the best-ranked option might turn out to be, it would be the board’s decision and would be the course of action that the company will take.

Each of you is given some time to think about your decision. An hour has passed and the chair called for a vote. One by one, you all gave your rankings. After tallying the results, the chair announced that option A will be the board’s decision. Everyone is happy that such a democratic process led to the best choice. But is this board decision really a product of a democratic, non-dictatorial process?

According to the Nobel Prize Laureate for Economics, Kenneth Arrow, this is sadly not so. In his 1950 paper, “A Difficulty in the Concept of Social Welfare,” Arrow presents his now-famous impossibility theorem, that “the only methods of passing from individual tastes (preferences) to social preferences (collective choices) which will be satisfactory and which will be defined for a wide range of sets of individual orderings are either imposed or dictatorial.” A simple version of Arrow’s proof runs as follows.

Suppose that we have a group where each member ranks a given set of alternatives according to their personal preference. If we base a decision from the aggregate of these personal rankings, we will see that there is a subgroup that is decisive. This subgroup is decisive in the sense that its preference is pivotal. This subgroup might be a supermajority, a simple majority, a decisive minority, or whatnot. By iterating this process, we arrive at another decisive sub-sub group of the decisive subgroup, and a decisive sub-sub-sub group of the decisive sub-sub group, and so on, until we arrive at the decisive nth subgroup that contains only one individual member. If that individual is the sole member of that decisive nth subgroup, then that individual is a dictator. Thus, for any choice that any group may make, there will be this decisive dictator whose preference will always be the choice of the group.

Now you might wonder whether this “dictator” is really a dictator in our ordinary sense or whether it is just an unfortunate use of the term. Whoever this person might be, it seems that he or she has no real intention to curtail other people’s freedoms. He or she might just be a product of a mathematical happenstance, an innocent bystander while Arrow’s theorem comes gliding through.

We could, of course, think of Arrow’s dictator in such benign terms. The pivotal voter may have imposed his or her will on others unintentionally. He or she might not even know that his or hers is the decisive vote. But here’s the thing: we won’t ever know whether or not this is the case. We won’t have any conclusive proof either way because we simply can’t know the pivotal voter’s innermost thoughts.

Arrow’s theorem implies that one can have a collective decision but it is imposed or dictatorial. The math is impeccable! And this is a gloomy prospect for believers of the democratic process.

So next time that you go to a board meeting, ask yourself, who will be today’s decisive dictator?

Dr. Jeremiah Joven Joaquin is an associate professor of Philosophy at De La Salle University. Email: jeremiah.joaquin@dlsu.edu.ph

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Today’s Front Page February 08, 2020

Today’s Front Page February 08, 2020